UNIVERSITY OF CAMBRIDGE
INSTITUTE OF ASTRONOMY
A DISSERTATION SUBMITTED TO THE
UNIVERSITY OF CAMBRIDGE FOR THE DEGREE
OF DOCTOR OF PHILOSOPHY
PROTOPLANETARY DISC
EVOLUTION AND DISPERSAL
James Edward Owen
CHURCHILL COLLEGE
Submitted to the Board of Graduate Studies
June 30, 2010
UNDER THE SUPERVISION OF
PROFESSOR CATHIE J. CLARKE

For my friends and family

CONTENTS
Contents v
Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of Tables xiv
List of Figures xiv
Nomenclature xvii
1 Introduction 1
1.1 The origin of protoplanetary discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Observation of protoplanetary discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 The Spectral Energy Distribution of Discs . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 Observational tracers of gas and dust . . . . . . . . . . . . . . . . . . . . . . . 6
1.2.3 Observations of Disc Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.2.4 Properties of ‘Transition’ Discs . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Structure and Evolution of Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3.1 Disc Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.3.2 Disc Evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.3.3 Origin of the viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.4 Disc Clearing Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.1 Planet Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4.2 Photoevaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.4.3 Observations of Photoevaporation at Work . . . . . . . . . . . . . . . . . . . . 30
1.5 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Photoevaporation Theory 33
2.1 Heating of protoplanetary discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.1.1 Optical & IR heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.1.2 EUV heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.3 FUV heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.1.4 X-ray heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.1.5 Photoevaporation as a result of disc heating. . . . . . . . . . . . . . . . . . . . . 37
2.1.6 Basic results of thermally driven winds . . . . . . . . . . . . . . . . . . . . . . 38
2.2 The theory of X-ray Photoevaporation: scaling relations . . . . . . . . . . . . . . . . . . 39
2.2.1 Primordial discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
v
vi Contents
2.2.2 Discs with inner holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 The theory of X-ray photoevaporation: absolute flow rates . . . . . . . . . . . . . . . . 43
2.3.1 Role of the sonic surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.3.2 Importance of Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.4 Theory of UV & X-ray Photoevaporation . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.1 EUV and X-ray heated flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.2 FUV and X-ray heated flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3 Numerical Models 51
3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.1.2 Radiation-Hydrodynamic Methods . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.2 Primordial Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Inner Hole Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1 Primordial Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.2 Inner Hole Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5 Numerical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.5.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.5.2 Inclusion of ‘FUV’ heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.6.1 Final Clearing of the Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.6.2 Comparison with previous work . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4 Discs evolution and dispersal 81
4.1 X-ray Luminosity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.2 Photoevaporating Viscous Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
4.2.1 Viscosity Law and Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 A population of viscously evolving, photoevaporating discs . . . . . . . . . . . . . . . . 86
4.4 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.4.1 Photoevaporation starved accretion . . . . . . . . . . . . . . . . . . . . . . . . 89
4.4.2 The nature of transition discs: accreting and non-accreting . . . . . . . . . . . . 92
4.4.3 Final clearing of the disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5 Detecting the wind I 101
5.1 The status of current observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.2 Radiative transfer post-processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
Contents vii
5.2.1 Calculation of the line profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.3.1 Line profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
5.3.2 NeII 12.8µm and OI 6300A˚ radial profiles . . . . . . . . . . . . . . . . . . . . 107
5.4 Discussion: The emission line spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.1 Oxygen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.4.2 Neon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.4.3 Sulphur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4.4 Magnesium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5 A direct comparison: TW-Hya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.5.1 The structure of TW Hya’s inner gas disc . . . . . . . . . . . . . . . . . . . . . 116
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
6 Detecting the wind II 121
6.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.1.1 Hydrodynamic photoevaporative wind . . . . . . . . . . . . . . . . . . . . . . . 123
6.1.2 Model Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
6.1.3 Dust Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1.4 Radiative Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
6.1.5 Calculations of Crystallinity Structure . . . . . . . . . . . . . . . . . . . . . . . 127
6.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
6.3 Crystallinity Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.4 Comparisons to Observed Edge-on Objects . . . . . . . . . . . . . . . . . . . . . . . . 130
6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7 Summary & Outlook 135
7.1 Outstanding Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
7.2 Implications and future direction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
A Numerical Fits to the Mass-loss Rates 139
A.1 Primordial Discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
A.2 Discs with Inner holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Bibliography 143

DECLARATION
I hereby declare that my thesis entitled Protoplanetary disc evolution and dispersal is not substantially
the same as any that I have submitted for a degree or diploma or other qualification at any other Univer-
sity. I further state that no part of my thesis has already been or is being concurrently submitted for any
such degree, diploma or other qualification. This dissertation is the result of my own work and includes
nothing which is the outcome of work done in collaboration except where specifically indicated in the
text. Those parts of this thesis which have been published or accepted for publication are as follows:
• Chapter 2 is based on work that has been completed in collaboration with C. J. Clarke and B.
Ercolano and has been submitted to Monthly Notices of the Royal Astronomical Society as:
– Owen, J.E.; Clarke, C.J. & Ercolano, B. ‘On the theory of photoevaporation’
• Chapters 3 & to 4 has been completed in collaboration with C.J. Clarke, B. Ercolano & R.D.
Alexander and is based on three articles:
– Owen, J.E.; Ercolano, B.; Clarke, C.J. & Alexander, R.D. (2010) ‘Radiation-hydrodynamic
models of X-ray and EUV photoevaporating protoplanetary discs’, Monthly Notices of the
Royal Astronomical Society, 401, 1415-1428.
– Owen, J.E.; Ercolano, B. & Clarke, C.J. (2011) ‘Protoplanetary disc evolution and dispersal:
the implications of X-ray photoevaporation’, Monthly Notices of the Royal Astronomical
Society, 412, 13-25.
– Owen, J.E.; Clarke, C.J. & Ercolano, B. ‘On the theory of photoevaporation’, subm.
• The work presented in Chapter 5 is based on work completed in collaboration with B. Ercolano
and is published as:
– Ercolano, B. & Owen, J.E. (2010) ‘Theoretical spectra of photoevaporating protoplanetary
discs: an atlas of atomic and low-ionization emission lines’, Monthly Notices of the Royal
Astronomical Society, 406, 1553-1569.
• The work presented in Chapter 6 has been completed in collaboration with C.J. Clarke & B. Er-
colano and is published as:
– Owen, J.E.; Ercolano, B. & Clarke, C.J. (2011) ‘The imprint of photoevaporation on edge-on
discs’, Monthly Notices of the Royal Astronomical Society, 411, 1104-110.
This thesis contains fewer than 60,000 words.
James Edward Owen
Cambridge, June 30, 2011
ix

ACKNOWLEDGEMENTS
There are many people to whom I am deeply indebted for their help and support, both during years at the
IoA, Cambridge and long before that; I have enjoyed (almost) every minute of the last three years. Firstly,
I would like to thank my supervisor, Cathie Clarke for all her hard work and support, I couldn’t have
asked for a better supervisor. Cathie has given me the freedom to pursue fruitful areas of research, and
for indulging in many long conversation on different topics. Secondly, I must of course thank Barbara
Ercolano, much of the foundations for this thesis lie in Barbara’s work. Barbara has certainly gone above
and beyond the call of duty; the rapid edits of her MOCASSIN code, for allowing myself to bouncing
ideas of her and of course for tireless correcting my ‘Yoda style’ sentences.
Other people at the IoA who deserve a special mention are Mark Wyatt my second supervisor who
providing his data on debris discs used in this thesis, Jim Pringle for many insightful discussion and
valuable advise on getting a job, and of course Paul Hewitt for his advice on how astronomy works.
Many astronomers outside the IoA have aided me either with providing help or giving me their data:
Richard Alexander for introducing me to the finer points of the ZEUS code and for providing me with
his OPEN-MP version, along with his expert knowledge on photoevaporation theory which has proved
invaluable. Manuel Gu¨edel, Thomas Preibisch and Jermey Drake have been invaluable in explaining the
X-ray observations to myself. Thanks must also go to Ilaria Pascucci (particuarly for providing the NeII
and OI data used in this work, and for involving myself in their interpretation), Al Glassgold, Eugene
Chiang, Greg Herczeg, Uma Gorti, David Hollenbach, Nate Bastian, Phil Armitage and Tim Harris for
the interesting discussions we have had either in person or over e-mail.
Of course I would have never made it to Cambridge or the IoA, without the support and encourage-
ment people have provided over the years. My parents and grandparents have always encouraged me to
be inquisitive and supported my career choice. Thanks must go to Tim Joliff, who encouraged me to
apply to Cambridge although I was eventually lured away from Chemistry! Of course I must thank all
for my year: Warrick, Amy, Jon, Alex, Becky, Stephanie, Yin-Zhe, James, Ryan, Adrian, and Dom, for
entertaining coffee and tea time conversation. I am grateful to my past and present office mates: Luke,
Richard, Quincy, Jayne, Stephanie, Susie, Phil, Alan and of course Hernande´z, many of whom proof
read parts of this Thesis.
Finally, to Claire for your support and everything else, I would not have made it without you.
James Edward Owen
Cambridge, June 30, 2011
xi

SUMMARY
In this thesis I have studied how discs around young stars evolve and disperse. In particular, I build
models which combine viscous evolution with photoevaporation, as previous work suggests that pho-
toevaporation can reproduce the observed disc evolution and dispersal time-scales. The main question
this thesis attempts to address is: Can photoevaporation provide a dominant dispersal mechanism for the
observed population of young stars?
Photoevaporation arises from the heating that high energy (UV and X-ray) photons provide to the
surface layers of a disc. Before I started this work, only photoevaporation from a pure EUV radiation
field was described within a hydrodynamic framework. Therefore, I start by building a hydrodynamic
solution to the pure X-ray photoevaporation problem, and then extend this solution to the entire high
energy spectrum. This hydrodynamic model leads me to conclude that it is the X-ray radiation field that
sets the mass-loss rates. These mass-loss rates scale linearly with X-ray luminosity, are independent of
the underlying disc structure and explicitly independent of stellar mass. I build a radiation-hydrodynamic
algorithm, based on previous work, to describe the process of X-ray heating in discs. I then use this
algorithm to span the full range of observed parameter space, to fully solve the X-ray photoevaporation
problem. I further extend the algorithm to roughly approximate the heating an FUV radiation field
would have on the photoevaporative flow, as well as separately testing the effect an EUV radiation field
will have. These numerical tests are in agreement with the hydrodynamic model derived. Specifically, it
is the X-rays that are driving the photoevaporative flow from the inner disc.
Armed with an accurate description of the photoevaporative mass-loss rates from young stars, I con-
sider the evolution of a population of disc-bearing, young (0.7M) stars, in order to asses photoevapo-
ration’s role as a dispersal mechanism. This study shows that the observed spread in X-ray luminosity
of young stars is sufficient to drive the dispersal of the entire population of discs, reproducing both the
required time-scales and the required spread in observables (disc lifetime, accretion rate). I also show
that a large fraction of the observed population of ‘transition’ discs are consistent with being created
through photoevaporation.
Having shown photoevaporation can provide a dominant dispersal mechanism for a population of
discs, I attempt to describe some direct observable consequences of photoevaporation, both through gas
tracers and dust emission. During this work, the first direct evidence of a photoevaporative flow emerging
from a young star was obtained from TW Hya, in the NeII 12.8µm line. Therefore, I discuss this result
within the framework of the X-ray photoevaporation model. Furthermore, I suggest that emission from
the photoevaporative flow is the origin of the unexplained, blue-shifted, OI 6300A˚ line observed around
all young stars that possess discs. I then described the properties of the dust particles that may be
entrained within the photoevaporative flow. The total dust mass in the flow is found to be small compared
to the disc, although such a region becomes observable once the disc presents as edge-on, obscuring the
central star and hot inner disc. I discuss the emission from these regions and compare them to the sample
of currently imaged edge-on discs.
The presented photoevaporation model reproduces all of the current observations, and I discuss some
predictions it makes with regard to future observations. Finally, photoevporation may have some intrigu-
ing consequences on planet formation and dust evolution that warrant further investigation.
xiii
xiv List of Figures
LIST OF TABLES
3.1 Simulation setup for photoevaporating primordial discs . . . . . . . . . . . . . . . . . . 56
3.2 Simulation setup for photoevaporating discs with inner holes . . . . . . . . . . . . . . . 59
5.1 Luminosity of emission lines from primordial disc models . . . . . . . . . . . . . . . . 105
5.2 Luminosity of emission lines from discs with inner holes . . . . . . . . . . . . . . . . . 105
5.3 Properties of the line profiles from primordial disc models . . . . . . . . . . . . . . . . 111
5.4 Properties of the line profiles from discs with inner holes . . . . . . . . . . . . . . . . . 112
6.1 Scattered light luminosities from a photoevaporative wind . . . . . . . . . . . . . . . . . 129
LIST OF FIGURES
1.1 Kinematic structure of Serpens molecular cloud . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Structure of collapsing cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Median SED of young stars in Taurus . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Disc fraction against cluster age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5 Accreting fraction against cluster age . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Colour-Colour plot of young stars in Taurus . . . . . . . . . . . . . . . . . . . . . . . . 11
1.7 Observed accretion rate variation with stellar age . . . . . . . . . . . . . . . . . . . . . 12
1.8 SED of ‘transition’ disc TW Hya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Observed inner hole size and accretion rate of ‘transition’ discs . . . . . . . . . . . . . . 14
1.10 Calculation of disc clearing by planet formation . . . . . . . . . . . . . . . . . . . . . . 22
1.11 Cartoon of disc photoevaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.12 Evolution of disc undergoing photoevaporation . . . . . . . . . . . . . . . . . . . . . . 26
1.13 Evolution of disc undergoing indirect and direct photoevaporation . . . . . . . . . . . . 27
1.14 Hydrostatic disc structure with X-ray heating . . . . . . . . . . . . . . . . . . . . . . . 28
1.15 Observed OI 6300A˚ line profile of CW Tau . . . . . . . . . . . . . . . . . . . . . . . . 31
2.1 Velocity of Parker wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.2 Surface mass-loss rate through the sonic surface . . . . . . . . . . . . . . . . . . . . . . 45
List of Figures xv
2.3 Schematic of possible photoevaporation topologies . . . . . . . . . . . . . . . . . . . . 47
3.1 Input X-ray spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2 Temperature ionization parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Flow structure of photoevporating primordial disc . . . . . . . . . . . . . . . . . . . . . 60
3.4 Variation of mass-loss rate with mass and X-ray luminosity . . . . . . . . . . . . . . . . 60
3.5 Comparison of the flow structure around different masses . . . . . . . . . . . . . . . . . 61
3.6 Cumulative mass-loss profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.7 Variation of mass-loss rate with inner hole size and X-ray luminosity for M∗ = 0.7M . . 62
3.8 Variation of mass-loss rate with inner hole size and X-ray luminosity for M∗ = 0.1M . . 63
3.9 Photoevaporation from a disc with an inner hole . . . . . . . . . . . . . . . . . . . . . . 63
3.10 Disc being cleared by ‘thermal sweeping’ . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.11 Streamline properites for discs with inner holes . . . . . . . . . . . . . . . . . . . . . . 65
3.12 Simple FUV model compared with real calculation . . . . . . . . . . . . . . . . . . . . 67
3.13 Bernoulli potential along streamlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.14 Histogram of temperature errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.15 Comparison of temperatures between ZEUS and MOCASSIN . . . . . . . . . . . . . . . . 70
3.16 Specific energy density of the flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.17 Temperature at the sonic surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.18 Flow structure for various disc structures . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.19 Ionization structure of photoevaporative wind . . . . . . . . . . . . . . . . . . . . . . . 73
3.20 Photoevaporation calculations including ‘FUV’ like heating . . . . . . . . . . . . . . . . 75
3.21 Mid-plane column as a function of inner hole radii for clearing discs . . . . . . . . . . . 76
3.22 Comparison between the surface mass-loss profiles for X-ray and EUV photoevaporation 78
4.1 Cumulative X-ray luminosity function for Taurus and Orion . . . . . . . . . . . . . . . 83
4.2 Accretion rate evolution of the ‘null’ model . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Disc fraction with time for a synthetic disc population . . . . . . . . . . . . . . . . . . . 86
4.4 Surface density evolution for a median X-ray luminosity . . . . . . . . . . . . . . . . . 88
4.5 Accretion rate evolution of the synthetic population . . . . . . . . . . . . . . . . . . . . 89
4.6 Comparison of photoevaporating and non-photoevaporating discs . . . . . . . . . . . . . 90
4.7 Correlation between X-ray luminosity and accretion rate . . . . . . . . . . . . . . . . . 91
4.8 X-ray luminosities of CTTs compared to WTTs . . . . . . . . . . . . . . . . . . . . . . 92
4.9 Transition disc probability variation with inner hole radius . . . . . . . . . . . . . . . . 93
4.10 Transition discs in the accretion rate and inner hole size plane for stars with M∗ = 0.7M 94
4.11 Transition discs in the accretion rate and inner hole size plane for low mass stars . . . . . 95
4.12 Accretion rate and inner hole size compared to X-ray luminosity for transition discs . . . 97
4.13 Photoevaporation relics compared to observations of debris discs. . . . . . . . . . . . . . 99
5.1 Line profiles for the NeII and OI line from primordial discs . . . . . . . . . . . . . . . . 106
5.2 Line profiles for the NeII and OI line from discs with cleared inner holes . . . . . . . . . 107
5.3 Wind structure with regions of high emissivity for the NeII and OI line shown . . . . . . 108
5.4 Representative shapes of the theoretical line profiles . . . . . . . . . . . . . . . . . . . . 109
xvi List of Figures
5.5 Radial intensity profiles for the NeII and OI line . . . . . . . . . . . . . . . . . . . . . . 110
5.6 Histogram of equivalent widths for the OI line . . . . . . . . . . . . . . . . . . . . . . . 113
5.7 Comparison of theoretical NeII profile compared to the observed line from TW Hya . . . 115
5.8 Cartoons of the structure of TW Hya’s inner gas disc . . . . . . . . . . . . . . . . . . . 117
5.9 Comparisons of theoretical OI line profiles compared to the observations of TW Hya . . 118
6.1 Gas structure of an EUV wind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
6.2 Maximum dust grain size entrained in a photoevaporative wind . . . . . . . . . . . . . . 125
6.3 Dust density of a photoevaporative wind . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.4 Scattered light images of a photoevaporative wind . . . . . . . . . . . . . . . . . . . . . 128
6.5 Cystallinity fraction in a photoevaporative wind . . . . . . . . . . . . . . . . . . . . . . 130
6.6 Three colour plot of PDS 144 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
6.7 Colour variation through photoevaporative wind . . . . . . . . . . . . . . . . . . . . . . 132
7.1 Variation of the global dust to gas ratio of discs with time . . . . . . . . . . . . . . . . . 138
NOMENCLATURE
Unless otherwise stated, it can be assumed the following symbols have this definition
Co-ordinate Systems
Cartesian r = [x xˆ,y yˆ,z zˆ]
Cylindrical Polar r = [R Rˆ,φ φˆ ,z zˆ]
Spherical Polar r = [r rˆ,θ θˆ ,ϕ ϕˆ]
Physical Constants
Boltzmann’s Constant kb
Gravitational Constant G
Plank’s Constant h
Fluid Mechanical Variables
Fluid Velocity u
Mass Density ρ
Number Density n
Gas Temperature T
Stream-bundle Area A
Kinematic Viscosity ν
Stress Tensor T
Length Along a Streamline l
Net Cooling Rate Λ
Energy Density e
Pressure P
Sound Speed cs
Dynamical Viscosity η
Components of Stress Tensor Ti j
Accretion Disc Variables
Surface Density Σ
Accretion Rate M˙∗
Surface Mass-loss Rate Σ˙w
Viscous ‘alpha’ Parameter α
Initial Disc Scale Radius R1
Radial Pressure Scale Height HR
Integrated Photoevaporation Rate M˙w
Disc Mass Md
Initial Viscous Time tν
Global Viscous Time τν
Vertical Pressure Scale Height H
Inner Hole Radius Rhole
Radiative Transfer Variables
Wavelength λ
Frequency ν
Bolometric Luminosity Lbol
Energy Density Mean Opacity κE
Dust Temperature Td
X-ray Luminosity LX
Flux f
Column N
Plank Mean Opacity κp
EUV Luminosity Φ∗
xvii
xviii Nomenclature
Miscellaneous Variables
Free fall time-scale t f f
Sound crossing time-scale tsc
Dust Grain Radius a
Angular Velocity Ω
Time t
Ionization Parameter ξ
Ratio of Specific Heats γ
Stellar Mass M∗
Gravitational Potential Φ
Effective Potential ψ
Stellar Radius R∗
Big O Notation (of order) O
Telescope Names
ALMA (Atacama Large Millimeter Array) - Future Millimetre/sub-millimetre interferometer
Herschel Just launched space based MIR/FIR observatory
Spitzer In use, space based NIR/MIR observatory
Logarithms
In this thesis I use log to refer to logarithms in any base, the implied base should be clear from the
context. Although for clarity: numerical values are always given in base 10; while algebraic expressions
use the natural base (e).
Never lose the child-like wonder.
It’s just too important.
It’s what drives us.
The Last Lecture
RANDY PAUSCH (1960-2008)

1
Introduction
ONE of the ultimate goals of modern Science is understanding how our solar system, newlydiscovered solar systems and currently undetected solar systems are assembled. It is certainlytrue that the final stages of star formation and planet formation are coupled processes, with star
formation setting conditions in which planet formation will begin. Currently, one can argue that how,
where and when planets form is a question that is far from being answered. In fact one of the only results
that the current range of planet formation models agree on, is that the initial conditions are crucial to
the final result. Therefore, a detailed understanding of the physical processes, and the time-scales on
which they operate will provide stringent conditions with which to begin constraining models of planet
formation.
Therefore, I shall begin our discussion by describing the processes which lead to the formation of the
reservoir of gas and dust from which planets will form: the protoplanetary disc. Protoplanetary discs are
an inevitable outcome of star formation and their existence was postulated during the 18th century within
the framework of the nebular hypothesis.
1.1 The origin of protoplanetary discs
In order to initiate the star formation process, the gas from which the star is going to form must sat-
isfy a basic requirement: namely that the gravitational collapse can proceed unimpeded by the thermal
pressure, which can stabilise a region against collapse. This simply means that the free-fall time must
be considerably less than the sound crossing time, where t f f /tsc ∝ ρ−1/6T 1/2, meaning we need a cold,
dense region. These stellar nurseries have been observationally detected and studied for many years. At
the densities (n> 10 cm−3) and temperatures (T < 50K) expected for these regions the gas is molecular,
and these regions have been aptly named ‘molecular clouds’. One property that is shared by all star
forming regions is that they are slowly rotating (e.g. Figure 1.1; Myers & Benson 1983; Goldsmith &
1
2 Chapter 1. Introduction
Figure 1.1: Three colour plot of the gas (traced by C18O) in the Serpens molecular cloud, blue indicates
gas moving towards the observer, green indicates gas stationary with respect to the observer and red
indicates gas moving away from the observer. At a distance of 230pc, 10’ corresponds to 0.6pc. The
East-West velocity gradient clearly indicates that the Serpens molecular cloud is globally rotating about
an axis running approximately North-South with an angular frequency of ∼ 10−15 s−1. (Figure from
Graves et al. 2010)
Arquilla 1985; Goodman et al. 1993; Burkert & Bodenheimer 2000), and therefore given their size (of
order parsecs) the gas possesses significant amounts of angular momentum. The formation of an individ-
ual stellar system begins when an individual region within the molecular cloud becomes gravitationally
unstable, and begins to collapse. When the collapse begins the gas is optically thin and it proceeds
isothermally. An isothermal collapse is a runaway process, thus the collapse proceeds until the cloud
becomes dense enough that it is optically thick to its own cooling radiation. The size of a region that be-
comes gravitationally unstable (O 0.1pc - the Jeans radius - Jeans 1902) can be compared to the size of a
formed solar system (O 100 AU), i.e. the collapse proceeds through many orders in magnitude in radius.
Therefore, conservation of angular momentum will result in a large increase in the angular velocity of
the collapsing gas. This large increase in rotation will necessarily cause a flattening of the rotating cloud
and the beginning of a disc like structure. As the collapse proceeds a centrally condensed object forms
in the centre, with most of the material falling onto this object. However, gas particles in the cloud that
posses significant angular velocities (resulting from the conservation of angular momentum), will be on
1.1 The origin of protoplanetary discs 3
Figure 1.2: The density and velocity structure of collapsing, rotating cloud shown after 8× 104 years.
Density contours range from log10ρ = −9.8 to -16 at constant logarithmic steps (Figure from Yorke et
al. 1999).
an orbital path that has a periapsis that will miss the central condensed structure. In an approximately
symmetric collapse, any gas parcel that approaches the mid-plane1 of the system will have an oppos-
ing gas parcel that arrives at the mid-plane at the same time, but with equal and opposite momentum.
Therefore, the gas parcels will terminate their orbits at the mid-plane. This populates the mid-plane with
large amounts of gaseous material, and is the origin of the protoplanetary disc. This process is shown
in Figure 1.2 taken from a simulated collapse of a rotating 1M axis-symmetric gas cloud (Yorke et al.
1999), after 8×104 years a 0.45M protostar is surrounded by a stable disc structure.
Once the disc has first formed it begins to grow by accumulating material, as it grows it intercepts
material at a height above the mid-plane, that may have otherwise fallen onto the protostar, or the disc
at small radii. This results in a growing and expanding disc until the majority of in-falling material falls
onto the disc rather than the protostar. The most basic approximation we can make is a collapsing and
slowly rotating core conserves angular momentum, in this case we can write the initial disc radius as
(Cassen & Mossman 1981; Lin & Pringle 1990):
Rd = 40AU
(
Ωc
1×10−16s−1
)2( Rc
0.1pc
)4( Mc
1M
)−1
(1.1)
where Ωc, Rc and Mc are the initial angular velocity, radius and mass of the collapsing core respectfully,
although the redistribution of angular momentum through tidal torques can reduce this radius (Lin &
Pringle 1990). During this disc formation phase, the central protostar is continuing to collapse, heating
up from the energy realised due to the collapse. Eventually this protostar will reach the point at which it
is hot enough for nuclear fusion to begin. Once this begins the protostar has formally become a star, and
it will begin a process of attaining equilibrium and becoming a fully fledged main sequence star.
Of course, the description above has swept under the carpet some of the poorly understood and hotly
debated issues of star formation. Perhaps the most divisive issue that remains in understanding star
formation is what sets which regions of the molecular cloud collapse to form stars and which don’t (the
1The plane that includes the centre of mass of the system and intersects the axis of rotation at a right angle
4 Chapter 1. Introduction
process of fragmentation), along with the final mass of the stars these collapsing regions form, and the
multiplicity of the final stellar systems. A problem that is better understood is the time-scale dichotomy
that must exist between the formation of low mass stars and high mass stars. In the scenario described
above, there was no discussion of the effects of radiative feedback from the forming star itself on the
collapse and disc formation stage itself. While this is a reasonable assumption for the formation of low
mass stars, at higher masses the luminosity of the central protostar/star2 becomes large enough that the
effects from heating of the gas, and radiative feedback due to momentum transfer from photons onto
dust grains and hence the gas (which becomes a dominant effect at ∼ 10M, e.g. Wolfire & Cassinelli
1986,1987; Sonnhalter & Yorke 2000; Edgar & Clarke 2003; Krumholz et al. 2007,2009; Kuiper et al.
2010,2011) cannot be neglected during the star/disc formation stage (e.g. Zinnecker & Yorke, 2007). No
detailed, and accurate calculations of the formation of massive stellar systems have been undertaken, and
the differences between the formation of low mass stars and high mass stars is unknown.
Therefore, in this thesis I will restrict my discussion to low mass stars (M∗ > 2M), where the forma-
tion of an individual disc systems is fairly well understood and in good agreement with the observations.
1.2 Observation of protoplanetary discs
We have now seen that discs are a natural outcome of the star formation process, and it is these discs
which must provide the material from which planets form. Therefore, it is important to summarise the
basic characteristics before we consider the more detailed aspects of their evolution. The gas cloud from
which a star/disc system is born, is composed of (in terms of mass fraction) 99% gas and 1% small dust
particles (with radii in the range 5× 10−3µm to 1µm, Mathis et al. 1977). Thus, observations from
both the dust component and gas component will be vital to a detailed understanding of protoplanetary
discs. The first observational indication of a disc around a forming star was through the detection of
an excess of emission above what would be expected from a pure stellar atmosphere at Infra-Red (IR)
wavelengths (Mendoza 1966, 1968; Cohen 1975). This excess was interpreted as the thermal emission
arising from dust in a disc which surrounds the forming star, where the disc is colder than the forming
star itself, hence the excess appearing at IR wavelengths (Adams & Shu 1986). These early indirect
detections of protoplanetary discs have now been supplemented with imaging in both gas and dust.
Modern observations now include a large range of broadband photometery, to study the dust emission
(thermal & scattered) and emission arising from accretion, and spectroscopy to study the emission from
gas phase diagnostics.
1.2.1 The Spectral Energy Distribution of Discs
One of the most widespread observational tools for studying the forming star/disc system is through
analysis of its spectral energy distribution (SED), with broadband photometry ranging from the Ultra-
Violet (UV), through the optical, IR to sub-millimetre and millimetre wavelengths available for many
objects. Figure 1.3 shows the median SED from the Taurus star forming region (D’alessio et al. 2001).
Compared to a pure stellar photosphere, the IR excess above the photosphere is clearly visible in this
2Due to the time-scale for stellar evolution compared to formation, the difference between a protostar and star becomes
blurred as the time-scale for stellar evolution becomes comparable to the formation time-scale and a forming massive star can
be considered to be an actively evolving star.
1.2 Observation of protoplanetary discs 5
Figure 1.3: Points show the median SED observations of objects in the Taurus star forming region. The
dot-dashed line shows emission arising due to a forming star, the solid line and dotted lined show the fits
of various protoplanetary disc models (Figure from D’alessio et al. 2001).
plot, and its slope can be used to determine the temperature structure of the disc’s photosphere. At
longer wavelengths (sub-millimetre/millimetre) the disc’s emission falls off rapidly, indicative of the disc
becoming optically thin to its own thermal emission. If the emission is optically thin the total flux can be
related directly to the total amount of opacity at these wavelengths, and hence allows for model dependant
determination of (dust) disc masses. At wavelengths shorter than the peak of the stellar photosphere,
the SED again shows an excess above the photosphere. This hot continuum excess is thought to arise
from an accretion shock on the stellar surface, caused by gas free-falling onto the stellar surface from
magnetically mediated accretion funnels (e.g Muzerolle et al. 2000). Therefore, this UV excess tell us
that these protoplanetary discs are actively accreting onto the forming star, which is unsurprising, since
from the discussion above a natural question to ask is: If a large fraction of the in-falling material falls
onto the disc rather than the forming star, how does it accrete onto the central star? This is a question
I will discuss further in Section 1.3 and relates to angular momentum transport. Furthermore, this UV
excess provides a direct way to measure the rate at which material is accreted onto the forming star and is
therefore crucial to discussions of protoplanetary disc evolution. Finally, the optical photometry allows
us to characterise the forming star, allowing constraints to be placed on its mass and age. Therefore,
the SED of a forming star is an extremely powerful tool for characterising star/disc systems allowing
measurements of the accretion rates, temperature structure and mass measurements of protoplanetary
discs, which have been extremely successful in constraining the basic models of disc evolution.
6 Chapter 1. Introduction
The SED as an indicator of Evolution
The ubiquity of SED observations has lead to their use as an observational proxy for disc evolution. The
use of the SED in classifying forming stars was first used by Lada (1987) and has been updated several
times since (Andre´ et al. 1993; Andre´ & Montmerle 1994). Objects are typically classified into four
classes (Class 0, I, II & III) based on the slope of the SED at IR wavelengths. The physical basis for this
classification is that since emission at IR wavelengths is indicative of thermal dust emission, a ‘redder’
SED is indicative of more dust surrounding the forming star. A Class 0 is an object is that is heavily
embedded and shows very little emission in the IR but is detectable in the sub-millimetre, and is thought
to represent the early stages of collapse and star/disc formation. A class I object is thought to indicate the
later stages of star/disc assemblage, with in-fall onto the disc still occurring, often being associated with
a molecular outflow/jet. Class II indicates an evolving star/disc system with very little in-fall occurring
and Class III represents a naked photosphere/debris disk (see Wyatt 2008 for an introduction into debris
disks) and is thought to indicate the by-products of the star and planet formation phase.
1.2.2 Observational tracers of gas and dust
As discussed above, most of the emission produced by the disc is thermal continuum emission, while
this is important for SED modelling, and gives us an idea of roughly the temperature of the dust, it gives
us relatively few specific details. However, there is a growing range of tracers that can give clues as to
the properties of the gas and dust at different temperature and radial ranges within the discs. The most
commonly used is strong optical emission lines from the gas (mostly commonly Hα), being optical lines
they typically have very high Boltzmann temperatures and therefore trace hot gas. Furthermore, these
emission lines are broad, with velocities indicative of free-fall close to the star, indicating these emission
lines are tracing the hot gas falling onto the star, i.e. accretion. Methods of fitting the observed emission
with multiple lines and the UV continuum have become quite sophisticated and successful, and perhaps
provide the most accurate measurements of accretion rates in young stars (e.g. Muzerolle et al. 1998,
2001,2005; Natta et al 2004; Herczeg & Hillenbrand 2008).
Specifically the detectability of Hα has lead to another observational classification scheme (Herbig
& Bell 1988). Young stars with strong Hα emission indicative of active accretion and are known as Clas-
sical T Tauri stars (CTTs), whereas young stars with negligible Hα emission indicative of no accretion
are know as ‘Weak-line’ T Tauri stars (WTTs). Typically, CTTs are also Class II objects (under the SED
classification scheme mentioned above) and indicative of a primordial disc, whereas WTTs are Class III
objects and associated with young stars that have lost their discs. Although the mapping between the
accretion classification and SED classification is not one to one, it is true in general. In recent years,
these classifications have wandered away from their original meaning and some authors use CTTs/Class
II/Primordial disc or WTTs/Class III/disc-less interchangeably, although throughout this thesis I will try
to remain true to the original definitions.
As with the dust emission, the direct tracers of gas also occur in distinct temperature ranges, hence
they probe separate radial ranges. At cold temperatures and thus large radii, the most useful probes are
the rotational transitions of molecules in the gas, with CO (1→0, 2→1 & 3→2) rotational transitions
being the strongest (Dutrey et al. 1998; Guilloteau & Dutrey 1998; Qi et al. 2004; Hughes et al. 2008),
due to the cold temperatures and low spatial resolution the information from this type of observations is
1.2 Observation of protoplanetary discs 7
limited, other than rough bounds on the total CO mass in the disc, but should prove much more fruitful
once high spatial resolution imaging with ALMA is possible. At warmer temperatures, CO ro-vibrational
emission from within the inner disc has provided estimates of the inner disc masses (e.g. Brittain et al.
2007), as well as demonstrating that the inner gas disc extends inside the dust destruction radius (where
the dust is so hot it evaporates) until the disc is truncated by the stellar magnetic field (e.g Edwards et al.
1994; Muzerolle et al. 2001; Calvet, 2004).
Furthermore, forbidden and low ionization lines e.g. OI 6300A˚ (Hartigan et al. 1995) and NeII
(Pascucci & Sterzik 2009), are providing details of the upper layers of the gas discs, with many of
these lines showing blue-shifts indicative of gas escaping from the surface of the disc. These higher
temperature lines, along with the expected lower temperature lines expected from Herschel are beginning
to provide a wealth of information on the upper layers of the gas disc at a range of radii.
Dust properties in the discs are even more difficult to asses than gas properties. This is because most
of the disc’s emission is thermal black-body emission from an optically thick dust disc, and is therefore,
to first order insensitive to the dust properties of grain sizes. However, it is possible to use higher order
effects to constrain certain properties from this dust emission. Of course, any dust that exists above the
photosphere will by definition be producing optically thin emission, and for irradiated discs this optically
thin emission is always hotter than the continuum. While small in total, this emission has been used to
constrain properties about grain size and composition (Natta et al. 2000; Furlan et al. 2006), with detailed
modelling by D’alessio et al. (1998,1999,2001) showing that dust has evolved from the standard ISM
dust population, arguing that the dust grains must have grown in size and be settled (i.e. not well mixed
throughout the entire disc). Therefore, dust is likely to be being actively processed in the disc formation
and evolution stage (Dullemond & Dominik 2005; Dullemond et al. 2006). This dust processing con-
clusion is backed up by complementary observations of the silicate dust grains. Silicate dust grains are
useful as their optical properties possess a feature at 10µm, which is sensitive to the underlying crystal
structure of the grain. In the ISM silicate dust grains are almost entirely in the amorphous state; however,
analysis of the 10µm silicate feature indicates that most discs possess a significant fraction of crystalline
silicate dust grains (Natta et al. 2000; Furlan et al. 2006). In order to transform from the amorphous
state to the crystalline state, the dust particle must undergo thermal annealing at temperatures greater
than 800K. Spatially resolved observations of the 10µm feature, indicate that the crystalline fraction is
significant, even at radii too large for the grains to be actively undergoing the process there (Apai et al.
2005), suggesting there must be some form of radial outward movement of crystalline silicate grains.
At sub-millimetre and millimetre wavelengths, the dust emission becomes optically thin allowing
estimates of the dust mass to be made along with surface density profiles, and these are often turned
into gas masses by assuming the same dust to gas mass ratio as the ISM from which the disc formed
(0.01). Typical discs masses are around ∼ 10−2M, but posses an order of magnitude or larger, scatter
around that value (e.g. Andrews & Williams 2005, 2007). The measurement of disc masses posses
significant sources of systematic errors, the emissivity of the dust varies as the fourth power of the dust
temperature (typically taken to be 10K), and small errors in the dust temperature will lead to large errors
in derived dust masses. Furthermore, the observed emission is not sensitive to dust grains larger than a
few mm, if the grains have been significantly processed (as suggested by IR observations) then a large
mass reservoir of dust grains could be missed with this technique. Therefore, disc masses are perhaps
the least well constrained of the disc’s properties and may carry systematic errors as large as an order of
8 Chapter 1. Introduction
Figure 1.4: Fraction of objects in a cluster which show an NIR excess, plotted against cluster age (Figure
from Wyatt et al. 2008).
magnitude. The most likely case is that the current measurements are under predictions of the actual disc
mass (Andrews & Williams, 2005). While the actual values of the derived disc masses are sensitive to the
assumptions made, relative comparisons are less sensitive to these assumptions. Andrews et al. (2010)
used resolved sub-millimetre maps of protoplanetary discs to fit for the surface density distribution of
the discs, finding the discs show a surface density profile of Σ ∝ R−0.9±0.2 at small radius, consistent with
the theoretically preferred value of Σ ∝ R−1 which I will discuss later.
1.2.3 Observations of Disc Evolution
In this thesis I attempt to build a model for the evolution and dispersal of discs. Therefore, the ob-
servations of disc evolution should form the basis for any model I develop later and it is important to
summarise the current status of the observations here. Unfortunately due to the lack of observations at
Far-IR wavelengths, the current observations of disc evolution probe two different scales. Near to Mid-
IR observations probe the radial scales around ∼ 1AU, while the sub-millimetre/millimetre observations
probe radial scales around ∼ 100 AU, and therefore we must consider the evidence from both before
making any definitive statements of global disc evolution. Perhaps one of the most important indicators
of disc evolution is how long discs live around young stars. Figure 1.4 shows the fraction of young stars
with masses ? 0.2M (Haisch et al. 2001b) in a cluster with an IR excess (JHKL bands) as function of
age.
This clearly shows that at early ages > 1Myr most (∼ 80%) young stars are surrounded by (dust)
1.2 Observation of protoplanetary discs 9
Figure 1.5: Fraction of young stars in a cluster with detectable signs of accretion as a function of cluster
age. Crosses represent accreting fraction (determined from the equivalent width of Hα), where as pluses
represent NIR disc fraction which are slightly offset (to the right) from the accreting fraction in age. The
following clusters are shown: 1) Taurus; 2) Cha I; 3) IC348; 4) Tr 37; 5) NGC 2362; 6) OB1bc; 7) Upper
Sco; 8) B1a/250ri; 9) NGC 7160 (Figure from Kennedy & Kenyon, 2009)
discs at small radii, consistent will all stars being born with discs (but some losing them due to binary
truncation; e.g. Alexander & Armitage, 2009). However, over the next 10Myr of evolution the majority
of stars lose their discs, with a median inner dust disc lifetime of∼ 3Myr (Haisch et al. 2001b). As men-
tioned above we must be careful not to generalise results to all radii and to both gas and dust components.
While the lifetimes of inner dust discs have been known for a long time, a lot of attention has been re-
cently paid to whether this is indicative of the lifetime of the entire protoplanetary disc at all radii in both
gas and dust. In order to study the outer disc a comparison between IR and sub-millimetre emission was
employed (Duvert et al. 2000, Andrews & Williams 2005,2007). The results of these studies were quite
conclusive, there is almost a one to one mapping between objects indicated as having a disc in the IR
and a sub-millimetre detection and objects indicated as disc-less and no sub-millimetre detection. This
indicates the destruction of a disc is spatially coordinated, the disc appears to disappear at all radii at the
same time (compared to the disc’s lifetime; I will discuss the interesting and special case of ‘transition’
discs a bit later). Furthermore, Kennedy & Kenyon (2009) performed a similar analysis, but instead of
using IR observations of dust discs to indicate disc fraction, they used the fraction of discs that show
signs of accretion. The results are shown in Figure 1.5, which shows an almost identical form to the
decline in NIR dust discs. This indicates that the clearance of a dust disc is indicative of the clearance
10 Chapter 1. Introduction
of the gas and dust disc. Therefore, the observations appear to indicate that gas and dust discs around
stars live on average∼ 3Myr, but have a spread of around an order of magnitude, but after 10Myr almost
all discs appear to be devoid of detectable dust emission in the IR and sub-millimetre, along with no
detectable gas emission (Zuckerman et al. 1995). The mass dependence of this evolution is currently
unknown although Ercolano et al. (2011) present a preliminary, but inconclusive attempt to understand
the affect of stellar mass of disc lifetimes.
IR photometery can tell us much more than estimates of disc lifetimes, one of the most fruitful
analysis is the comparison of NIR disc colours. This analysis compares the emission from different
radii of the disc at the same time. Longer wavelengths probe thermal emission at lower temperature
and hence larger radii. Larger values of the colours indicate ‘redder’ emission, i.e. more emission from
colder regions of the disc and hence larger radii. In Figure 1.6 I show colour-colour plots for young stars
binned by spectral type, where the colours are scaled such that a colour of 0 indicates the value expected
from a pure Rayleigh-Jeans photosphere. This plot clearly shows that most discs appear in two separate
regions of the parameter space, either those consistent with pure stellar photosphere (bottom left) and
those consistent with optically thick primordial discs (top right). The crosses indicate the limiting case of
a flat disc which is on the point of transitioning from optically thick to optically thin. This indicates that
there are very few young stars that posses discs, which lie in between the point of being classified as a
primordial disc and being classified as disc-less. This indicates that time-scale for the transition between
a star surrounded by a primordial disc and a disc-less star is rapid, considerably shorter than the discs
lifetime.
The points circled in Figure 1.6 are discs that are inconsistent with both a optically thick primordial
disc and a stellar photosphere. Based on the fact these objects lie in between primordial and disc-less,
these objects have been labelled as ‘transition’ discs, under the assertion that these discs are caught in the
act of transitioning from primordial to disc-less. These ‘transition’ discs could fall into two categorises.
They could in fact be discs that are transitioning from primordial to cleared, and by comparing the
number of these objects to the number of primordial discs, an estimate of the transitional time-scale as
∼ 10% of the disc’s total lifetime can be found (e.g. Strom et al. 1989; Skrutskie et al. 1990; Kenyon
& Hartmann 1995; Muzerolle et al. 2010; Luhman et al. 2010). Or of course, these discs could be rare
and long lived and uncorrelated with disc dispersal, but the evolution from primordial to disc-less must
still be rapid in order to explain the gap. Furthermore, if we assume that these ‘transition’ discs are discs
caught in the act of clearing, then the colour-colour plot indicates that this clearing must proceed from
inside out, as the discs loose their excess at shorter wavelengths (smaller radii) before they loose their
excess at longer wavelengths. This opacity deficit at small radii has been interpreted as evidence as an
inner hole in their dust (and possibly gas) discs (e.g. Calvet et al. 2002), and that disc clearing proceeds
from inside out.
One important evolutionary indicator is the accretion rate onto the star. Accretion rate measurements
can be made using the UV excess above the photosphere discussed earlier, or through line emission
that is thought to correlate with the UV excess (e.g. Herczeg & Hillenbrand 2007). In Figure 1.7 is an
example of analysis of accretion rates. There is a clear and possibly power law decline in accretion rates
over many orders of magnitude in accretion rate, with a best fitting value of M˙∗ ∝ t−3/2. However, for a
given age there is clearly a large spread in the observed accretion rates, much larger than expected from
pure random errors (Hartmann et al. 1998). At this point it is also worth pointing out that there is a
1.2 Observation of protoplanetary discs 11
Figure 1.6: Colour-Colour plot of young stars in Taurus, shown for different spectral types. Circled points
are those which are not consistent with at pure stellar photosphere or optically thick IR disc (Figure from
Luhman et al. 2010).
current debate about the accurary of stellar ages, that may apply some form of transform to the x-axis
of Figure 1.7 changing the exact results, but what is not in doubt is the decline with time and spread in
accretion rates.
1.2.4 Properties of ‘Transition’ Discs
‘Transition’ discs are a poorly defined (both observationally and theoretically) class of objects which
represent a small but important subgroup of disc-bearing systems. As mentioned above the hypothesis
that these objects are discs which are transitioning from an optically thick primordial disc, to a disc-
less system has placed an upper bound on the time-scale for this transition. Therefore, if these objects
truly represent a set of objects caught in act of clearing, then studying their properties can provide clues
as to the physical mechanism which disperses the discs. The main method for classifying objects as
12 Chapter 1. Introduction
Figure 1.7: Accretion rate measurements plotted against the age of the young stars, for the Taurus and
Chameleon I star forming regions. Dashed lines, least-square best-fit assuming that the error in log t
is 0.3, while the error in log dM∗/dt is 0.6. Dotted lines, least-square best-fit assuming that the error
in both log t and log dM∗/dt are 0.4. The dashed line fit is preferred because the errors in estimating
accretion rates, along with the intrinsic dispersion of accretion at a given age, are likely to be larger than
the differential errors in stellar ages (Figure from Hartmann et al. 1998).
‘transition’ discs is through the shape of their SED. An SED which shows a significant drop (compared
to a primordial disc) in emission at NIR wavelengths, but then returns to the emission levels expected for
a primordial disc at MIR-FIR wavelengths is the essential defining feature (e.g. Figure 1.8). However,
there is no consensus of what drop in emission relative to the flux at longer wavelength constitutes
a ‘transition’ disc and observational samples do not span the same regions of parameter space (e.g.
Merin et al. 2010). This drop in emission at short wavelengths is indicative of a drop in opacity at
those temperatures, and has lead to the suggestion that it represents a cleared inner hole in the dust
disc. This hypothesis has been tested with resolved imaging of the largest holes, where the hole sizes
estimated from the SEDs are roughly consistent with the holes seen in the resolved images (Hughes
et al. 2008). Furthermore, interferometric measurements of the dust structure near the inner holes is
also providing a fruitful avenue of research (e.g. Ratzka et al. 2007), although the number of resolved
images or interferometric measurements of ‘transition’ discs remains limited, and SED fitting is still the
most common and widespread method for measuring the radii of the regions of low opacity. One of
the most interesting distinctions that can split the population of ‘transition’ discs, is whether they are
accreting or not. Evidence of accretion implies that there must be a gas reservoir close to the central star
and ro-vibrational CO observations (Salyk et al. 2007) have confirmed that the well know ‘transition’
disc GM Aur has a gas mass in the inner disc (R < 5AU) of 0.3M⊕, while still heavily accreting at
∼ 10−8M yr−1, with a dust inner hole radii of ∼ 20AU (Calvet et al. 2005). Furthermore, Espaillat et
1.2 Observation of protoplanetary discs 13
Figure 1.8: The observed SED of the ‘transition’ disc TW Hya shown as points, the lines represent best
fit models of a pure stellar photosphere or a primordial disc (Figure from Calvet et al. 2002).
al. (2008,2010) has found evidence for optically thin dust emission from within the hole itself and the
possibility of an optically thick inner dust disc at very small radii ∼ 0.1 AU. All of the Espaillat et al.
discs are heavily accreting ? 10−8M yr−1, although the significance of this inner dust disc correlated
with heavily accretion has yet to be identified, as the sample only contains 4 objects.
Perhaps one of the most interesting ways to examine the properties of ‘transition’ discs is in the M˙∗
- Rhole plane. In Figure 1.9 is a large sample of objects from the literature classified as ‘transition’ discs
by their authors. Two things are immediately clear from this plot, the selection criterion used by one
author does not in general probe the entire range of parameter space, as different selection criterion have
been applied on similar regions by different authors, underlying the lack of a cast iron definition for
‘transition’ discs. Furthermore, the possible dichotomy in the sample between those with small inner
holes R < 20 AU that have a wide range of accretion rate, to those with a large inner hole R > 20AU
which either have a significant accretion rate M˙∗ ∼ 10−8M yr−1or appear to be not accreting, indicating
the full population ‘transition’ discs may not be a homogeneous one, with several possible mechanisms
giving rise to their creation. While the lack of objects with large inner holes and low acccretion rates
is certainly a real effect, the number of non-accreting objects with large holes is unknown and certainly
something the FIR observations provided by Herschel should help with.
While the total statistics of the number of ‘transition’ discs, compared to primordial discs has placed
an upper bound for the clearing time-scale of the inner disc. The fact that the ‘transition’ disc sample does
not appear to be homogeneous, and the possibility of incompleteness at large hole sizes does not allow
us to place any constraints on the speed of the inner disc clearing compared to the outer disc clearing.
14 Chapter 1. Introduction
Inner Hole Radius [AU]
lo
g 1
0 
(A
cc
ret
ion
 R
ate
) [M
  y
r−1
]
1 10 100
−11
−10.5
−10
−9.5
−9
−8.5
−8
−7.5
−7
⊙
Figure 1.9: The inner hole radius and accretion rate of low-mass objects classified as ‘transistion’ discs
by varies authors: cyan triangles (Brown et al. 2009); blue filled triangles & squares (Cieza et al. 2010);
black open squares (Ercolano et al. 2009a); black filled squares (Espaillat et al. 2008,2010); black open
circles (Hughes et al. 2009, 2010); red filled circles (Kim et al. 2009); magenta filled diamonds (Merin et
al. 2010); green filled star (Najita et al. 2010). Where error-bars or ranges are listed they are shown. As
Cieza et al. (2010) list the wavelength of the inner hole rather than the radius, I conservatively estimate
an inner hole radius of < 5 AU for λhole < 5.8µm, and < 10 AU for 8 < λhole < 24µm. Objects with
no sign of accretion are placed on the plot with an accretion rate of 10−11M yr−1, indicative of the
observational detection limit.
1.3 Structure and Evolution of Discs
Now we have discussed discs from an observational point of view it is useful to consider the models
of disc structure and evolution. The starting point for this discussion must be from the point of view
that these protoplanetary discs are actively accreting onto their central star, i.e. they are accretion discs.
Accretion discs are prevalent on all scales in astrophysics and the main problem remains the same. If
the discs are actively accreting, then they must be transporting angular momentum. While very heavy
protoplanetary discs (compared to the central star) may transport angular momentum in spiral density
waves, through self gravitating interactions (e.g. Lodato & Rice 2005; Clarke, 2009), this is thought
to be only important at the very earliest times, during the disc’s assemblage and early evolution phase,
thought to be closely correlated with the Class I phase. However, discs spend most of their time in the
Class II phase, where the disc mass is not large enough to drive self-gravitating interactions. A viable
alternative angular momentum transport mechanism is a viscosity in the disc. A viscosity will allow
gas particles to diffuse radially, carrying specific angular momentum with it, allowing the disc to accrete
material by diffusing its angular momentum to large radius.
The time-scale associated with this process can be roughly estimated by dividing the discs mass by
1.3 Structure and Evolution of Discs 15
the accretion rate, which gives a time-scale OMyrs. As we shall see this time-scale is much longer than
the time-scales associated with obtaining dynamical equilibrium in the disc, therefore we can consider
the structure and evolution of a protoplanetary disc as two separate problems. As the disc is primarily
composed of gas then the governing equations are those of fluid mechanics. Furthermore, since the
disc’s mass is considerably less than the stellar mass then we can assume the gravitational potential is
simply given byΦ=−GM∗/r. Therefore, the governing equations of the system are continuity and mass
conservation:
∂ρ
∂ t
+∇ · (ρu) = 0 (1.2)
∂u
∂ t
+u ·∇u = −∇P
ρ
−∇Φ+ 1
ρ
∇ ·T (1.3)
where ρ is the gas density, u is the gas velocity vector, P is the thermal pressure and T is the viscous stress
tensor. The viscosity in the disc can also be a source of heating, through viscous dissipation. However,
in order to match the SED observations a significant source of heating from the viscosity is inconsistent
with the observations, and protoplanetary discs are closer to ‘passive’ discs, where the thermal structure
is set by the radiation field of the star (e.g. Kenyon & Hartmann 1987; Hartmann, 2009). As the gas
and dust will be in thermal equilibrium with the stellar radiation, the discs local sound speed will be
considerably less than the escape velocity. In order to simplify the problem further I assume that the disc
is axis-symmetric and is locally isothermal, or cs(R,φ ,z) ≈ cs(R), which is fairly robust if most of the
disc’s mass is either very optically thick or very optically thin to its own heating/cooling radiation.
1.3.1 Disc Structure
Radial Structure
I will first consider the radial, mid-plane structure of an axis-symmetric disc: the momentum equation in
the radial direction becomes:
ρ
(
∂uR
∂ t
+uR
∂uR
∂R
+uz
∂uR
∂ z
− u
2
φ
R
)
=
∂P
∂R
− GM∗ρ
R2
+
1
R
∂ (RTRR)
∂R
+
∂TRz
∂ z
(1.4)
where {R,φ ,z} describe the cylindrical co-ordinate system and ui & Ti j are the components of the
velocity and viscous stress tensor respectfully. So clearly, any variations are going to arise for two
physical reasons: velocity adjustments due to a change in the disc’s radial and hence pressure structure
and velocity adjustments to viscous processes. Above I argued that the observations indicate that the
time-scales associated with the viscous processes are long (comparable to the discs age). The time-
scale associated with pressure variations is simply HR/cs, where HR is the radial pressure scale length.
For a passively heated protoplanetary disc, with a surface density profile that falls off something like
Σ ∝ R−1, HR ∼ 0.1R. Therefore, the radial dynamical time-scale is considerably shorter than the time-
scale associated with the viscous processes and we can separate Equation 1.3.1, into it’s viscous and
dynamical components, and on the time-scales associated with disc evolution can consider the disc to be
in dynamical equilibrium in the radial direction. Furthermore, since uφ  uR at all radii then the equation
16 Chapter 1. Introduction
for dynamical equilibrium simply becomes:
1
ρ
∂P
∂R
=
u2φ
R
− GM∗
R2
(1.5)
Thus, the solution to the above equation allows us to find the azimuthal velocity in the disc. The magni-
tude of the pressure term on the LHS is simply (R/H)c2s/R (where H is the vertical scale height of the
disc), whereas the magnitude of the gravitational force is v2esc/R (where vesc is the escape velocity), as
observations tell us in order to match the SEDs of protoplanetary discs the discs sound speed is many
orders of magnitude less than the escape velocity, to first order we can simply neglect the radial pres-
sure gradient in the force balance and take the disc to be Keplerian with uφ =
√
GM∗/R. It is worth
mentioning, that although this radial pressure gradient is small is has important implications for dust
in discs. A slightly more detailed calculation indicates that the gases’ azimuthal velocity in the disc is
uφ ≈ [1−O(HR/R)2]
√
GM∗/R (e.g. Armitage, 2010), as dust will not be effected by the pressure gra-
dient it will be orbiting at a Keplerian velocity and will be travelling faster than gas. Therefore, the dust
particles will feel a headwind drag and begin to slowly loose angular momentum and spiral in to the star
(Weidenschilling, 1977) and in some cases this drift time-scale can be as short as a couple of orders of
magnitude larger than the orbital time-scale (e.g. Takeuchi & Lin 2002; Takeuchi et al. 2005), certainly
shorter than the discs lifetime. While this certainly has some important issues that need to be resolved,
namely the retention of mm grains in the outer disc (e.g. Alexander & Armitage, 2007), it will certainly
become important when the formation of ‘transition’ discs are discussed.
Vertical Structure
The momentum equation in the vertical direction for the disc is:
ρ
(
∂uz
∂ t
+uR
∂uz
∂R
+uz
∂uz
∂ z
)
=
∂P
∂ z
+
∂
∂ z
(
GM∗
r
)
+
1
R
∂RTRz
∂R
+
∂Tzz
∂ z
(1.6)
As before, this equation can be separated into the viscous components and dynamical component and
assuming that uz is small (compared to the sound speed), the disc can be considered to be in vertical
hydrostatic equilibrium. If we restrict ourselves to regions close to the mid-plane, then the equation for
hydrostatic equilibrium is directly integrable and we find:
ρ(z) ∝ exp
(
− z
2
2H(R)2
)
(1.7)
so the density profile is Gaussian about the mid-plane for an isothermal disc, where the vertical scale
height is given by:
H(R) =
cs
Ω
(1.8)
where Ω is the angular velocity. This results tell us two important results, since the sound speed is
considerably less than the escape velocity, this implies that H(R) R, nominally observations suggest
H(R) ≈ 0.1R. Thus the majority of the disc’s mass is contained close to the mid-plane and the density
profile derived above is to a high degree of accuracy the vertical density structure for the entire isothermal
1.3 Structure and Evolution of Discs 17
disc. Finally, the time-scale for the disc to attain dynamical equilibrium vertically is Ω−1. This time-
scale is comparable to the time-scale to attain dynamical equilibrium radially. Therefore, the disc can be
considered to be globally in dynamical equilibrium on time-scales longer than the disc’s orbital period,
thus when we are interested in the disc’s evolution due to viscous processes, dynamical equilibrium
arising from variations in disc structure can be assumed, and vica-versa.
1.3.2 Disc Evolution
As we have seen above the disc is thin and not radially pressure supported, allowing us to make two
further simplifications when we discuss the discs evolution. The fact that the disc is thin and not radially
pressure supported allows us to assume the disc is Keplerian, where the angular velocity at each radius
is just given by:
uφ =
√
GM∗
R
(1.9)
Therefore, the continuity equation and eφ component of the momentum equation, for an axis-symmetric
disc become:
2piR
∂ρ
∂ t
+
∂
∂R
(2piRρuR) = 0 (1.10)
2piρuR
dh
dR
=
∂
∂R
(
2piR2TRφ
)
+
∂
∂ z
(
2piR2Tφz
)
(1.11)
These equations can be simplified even further by integrating them over the z direction (as the disc is
thin physical variables can be averaged over - weighted by mass - in the z direction), to deal with the
evolution of the discs surface density rather than density. Also adopting zero shear boundary conditions
at the disc’s surface and the fact that TRφ = ηRdΩdR for a Navier-Stokes type viscosity, where I will discuss
this approximation and the origin of the viscosity in Section 1.3.3. Therefore, we can re-arrange the
equations into:
2piR
∂Σ
∂ t
+
∂
∂R
(2piRΣuR) = 0 (1.12)
2piRΣuR
dh
dR
=
∂
∂R
(
2piνΣR3
dΩ
dR
)
(1.13)
The physical significance of these two equations is now clear, where 1.12 is just still just the continuity
equation, or more physically conservation of the discs mass. Equation 1.13 is a statement of angular
momentum conservation, since its of the form m˙h′ =−τ ′ where τ can be thought of as a viscous torque.
Eliminating uR from above we arrive at the 1D equation for the evolution of the disc’s surface density:
∂Σ
∂ t
=
3
R
∂
∂R
[
R1/2
∂
∂R
(
R1/2νΣ
)]
(1.14)
This clearly has the form of a diffusion equation, consistent with our earlier assertion that viscous trans-
port leads to the diffusion of angular momentum. Therefore, we can envisage the evolution of the an
accretion disc as follows: The viscosity within the disc leads to diffusion of gas parcels, and its as-
sociated angular momentum, a disc with an initially finite outer radius will begin to spread (‘viscous
spreading’), taking with it the disc’s angular momentum, allowing material to accrete onto the central
18 Chapter 1. Introduction
star. This process will continue, until eventually all the disc’s mass has accreted onto the central star, and
all the disc’s angular momentum is at R= ∞ on none of the mass.
Of course, with exact knowledge of the viscosity law the solution of Equation 1.14 is trivial to de-
termine. However, as I shall discuss in Section 1.3.3, the viscosity law is far from being determined. In
the special case that ν contains no special scales then power-law similarity solutions are obtainable for
Equation 1.14 and these are discussed in detail by Ruden (1993). If one assumes ν is a function of radius
only, then Equation 1.14 becomes linear, and if ν is a power-law with radius, then the problem possesses
the similarity solutions found by Lynden-Bell & Pringle (1974). These were discussed in detail when
compared to protoplanetary disc evolution by Hartmann et al. (1998). Therefore, adopting ν ∝ Rγ , the
similarity solution for Σ takes the form:
Σ(R, t) ∝
Md(0)
Rγ
T−
5/2−γ
2−γ exp
(
−R
2−γ
T
)
(1.15)
where T = t/tν+1 and tν is the disc’s initial viscous time. Therefore, at small radii and/or late times the
surface density profile has an almost pure power law form of Σ ∝ Rγ . Also even though this similarity
solution implies an initial disc structure, that is unlikely to be indicative of the actual disc’s initial struc-
ture, which will depend more on gravity than viscous processes. Any initial surface density profile with
the same viscosity law will tend to this solution on a time-scale comparable to tν . Furthermore, we can
integrate this solution to find the accretion rate onto the star and time-evolution of the disc’s mass is
M˙∗ ∝ T−
5/2−γ
2−γ (1.16)
Md(t) ∝ T
− 1[2(2−γ)] (1.17)
respectfully. The observations suggest that Σ ∝ R−1 and M˙∗ ∝ t−3/2. Both these results are consistent
with the solutions discussed above if ν ∝ R as argued for by Hartmann et al. (1998). Therefore, a
reasonable approach is to adopt a viscosity law of ν ∝ R and use the similarity solutions above. I will
use the results of these similarity solutions throughout this work and therefore it is useful to summarise
them here. In this case then Equation 1.14 becomes:
Σ(R, t) =
Md(0)
2piRR1
1
T 3/2
exp
(
− R
R1T
)
(1.18)
where R1 is the discs scale factor, or more physically the point where the discs initial (global) viscous
time (tν ) equals the local viscous time, which is given by:
tν =
R21
3ν(R1)
=
Md(0)
2M˙∗(0)
(1.19)
Furthermore, we find that the evolution of the accretion rate and disc mass are given by:
M˙∗(t) = M˙∗(0)
(
1+
t
tν
)−3/2
(1.20)
Md(t) = Md(0)
(
1+
t
tν
)−1/2
(1.21)
1.3 Structure and Evolution of Discs 19
At late times t tν these become power-law functions of time, and combinations of these equations lead
to useful quantities, the current disc mass divided by the current accretion rate yields:
Md(t)
M˙∗(t)
≈ 2t ≈ 2τν (1.22)
Namely, the disc’s mass divided by the current age yields twice the disc’s age, which is also the disc’s
global viscous time τν , so we find that the global viscous time of the disc, i.e. the time with which
the disc’s evolutionary properties are changing is approximately the disc’s current age. However, one
important fact becomes apparent, this solution is entirely inconsistent with a rapid clearing of the disc, as
observations indicate then even though the discs appear to evolve as above for their entire lifetimes, when
they begin to dissipate they do so on a time-scale much less than their lifetime. In fact discs that evolve
under pure viscous evolution should be observable in NIR dust observations for Gyrs (e.g. Ercolano et
al. 2011a), rather than the Myrs observed. Therefore, a pure viscous model agrees very well with the
evolution indicators of CTTs, provided ν ∝ R; however, there must be some other process that operates
on a time-scale much less that the disc’s global viscous time-scale in order to clear the disc rapidly out
to large radius.
1.3.3 Origin of the viscosity
In the previous discussion I managed to completely ignore the origin and magnitude of the viscosity,
only supposing its existence based on the requirement that the discs must be transporting angular mo-
mentum. The first most obvious source of viscosity is molecular viscosity (arising from the diffusion
of molecules due to random motions of the gas particles), a simple calculation yields that the viscous
time-scale molecular viscosity will produce are ∼ 1012 times too long (e.g. Armitage, 2010). Therefore,
the viscosity in the disc must arise through some other mechanism. A viscosity is simply a resistance to
shear (and possibly tensile stress), and a protoplanetary disc is an obvious example of a shearing flow,
as each annulus of the disc is rotating at Keplerian velocity, there exists large radial velocity gradient
between each of the annuli. Therefore, if there was any diffusion of gas particles between annuli, there
would be a shear stress and hence a viscosity. One proposed mechanism for the transport of angular mo-
mentum is self-gravitating interactions (e.g Lin & Pringle 1987), where spiral density waves re-distribute
angular momentum. While not formally a local viscous process, under the conditions found in a pro-
toplanetary discs it can behave as an ‘effective viscosity’ where the torques dissipate locally (Lodato &
Rice, 2004). However, for self-gravitating interactions to become important the disc need to be massive
(compared to the central star) with the nominal value being Md/M∗ > 0.1 (e.g. Lodato & Rice 2005;
Clarke 2009). Given this requirement, self-gravity cannot be the source of angular momentum transport
for the majority of the disc’s lifetime and certainly not the source of the angular momentum transport
during the disc clearing phase we are interested in here.
One other solution would be if the disc was turbulent, then this could transport angular momentum
and act as a type of viscosity. This form of the viscosity was discussed by Shakura & Sunyaev (1973) by
assuming the turbulent viscosity can be parametrised in terms of a representative velocity and scale for
the turbulence. Shakura & Sunyaev glossed over our ignorance of the origin of the turbulence and how it
acts as a viscosity by introducing the famous α parameter, where the viscosity in this formalism is given
20 Chapter 1. Introduction
by:
ν = αcsH(R) (1.23)
where the scale size of the turbulence is taken to scale with the disc’s scale height and the velocity of
turbulence is taken to scale with the local sound speed, and the actual magnitude is described within
the dimensionless α parameter. Therefore, a disc in which the viscosity scales linearly with radius and
constant α parameter requires that the disc is slightly ‘flared’ with H(R) ∝ R5/4 in agreement with the
current models of vertical disc structure (e.g. Chiang & Goldreich 1997; D’alessio et al. 1998).
While turbulence is the leading candidate for angular momentum transport in T Tauri discs, the ori-
gin of the turbulence is still a matter of debate. It is well known that a Keplerian disc is stable to pure
hydrodynamic interchange instabilities (e.g. Ruden, 1993), thus we still require an instability to drive
the turbulence. Lin & Papaloizou (1980) suggested thermal convective instabilities as a possible driv-
ing mechanism, with estimates of the magnitdue of the viscosity in the range required by observations
(Ruden & Lin 1986; Cabot et al. 1987; Ruden et al. 1988). Although it has been pointed out that a
convectively unstable disc may be unable to sustain turbulence during the disc’s lifetime (Korycansky,
1992) or even transport angular momentum the correct way (e.g. Ryu & Goodman, 1992)! Currently the
leading candidate is the Magneto-Rotational Instability (MRI) discovered and rediscovered by several
authors, but it was Balbus & Hawley (1991) who applied it to accretion discs. In the limit of ideal MHD
(where magnetic field lines are coupled to gas parcels), a rotating shear flow is unstable to linear MRI
modes if:
dΩ2
dR
< 0 (1.24)
This inequality is clearly satisfied in a Keplerain protoplanetary disc, and leads to a globally turbulent
accretion disc (e.g. Flock et al. 2011; Beckwith et al. 2011) and α parameters in the range 10−3−10−1,
sufficient to explain the accretion rates and viscous time-scales observed. However, all these results rely
on the ‘Ideal’ MHD approximation, where the effects from ambipolar diffusion, ohmic dissipation and
the Hall effect are neglected. The most studied of these ‘non-ideal’ effects to date is ambipolar diffusion;
the ‘ideal’ approximation assumes that the magnetic field lines are coupled to the gas parcels, this is true
in the case of fully ionized gas. However, a protoplanetary disc is composed mainly of cold molecular
gas. Low ionization levels from FUV or hard X-ray photons are possible, enough to make the regions
of the disc close to the ‘ideal’ approximation. Though, at radii around ∼ 1AU (for solar mass stars) the
gas is too cold for thermal ionization, and the disc’s column too large to allow high ionization rates from
FUV/X-ray photons. Thus, it is possible there are large regions of the disc where the gas parcels are not
coupled to the magnetic field lines, and unable to drive turbulence. The idea of these ‘dead zones’ were
suggested by Gammie (1996), and detailed calculations seem to suggest that these dead zones may exist,
not allowing high accretion rates at small radii. Latests calculations by (Perez-Barker & Chiang, 2011),
show that in their model the MRI is unable to sustain the full range of observed accretion rates all the
way to the star.
Therefore, it appears at this stage a full understanding of the turbulence processes in the disc are far
from understood and the process of converting this into a predictive viscosity law in accretion discs is
a long way off, although it remains an extremely active and fruitful area of research. Therefore, in this
thesis I will follow the example of many others and parametrise the viscosity in terms of ν = ν1R/R1 as
1.4 Disc Clearing Mechanisms 21
indicated by observations.
1.4 Disc Clearing Mechanisms
We have seen that while the standard theory of accretion disc evolution, matches the observations of
discs during the CTTs phase, it is completely unable to explain the rapid removal of disc material at
late times. The two-time-scale nature of disc evolution is something that cannot be reproduced by pure
viscous processes, thus in order to explain disc dispersal another mechanism must be found. When a disc
disperses, there are only three things that can happen to the disc material, it can either accrete onto the
central star, be removed from the disc/star system or end up in planets. While I have shown above that
accreting material onto the star cannot explain the final disc dispersal stage, we are left with the final two
options, either place the material in planets or remove it entirely from the system.
1.4.1 Planet Formation
Planet formation as the origin of disc destruction has an appealing closed nature to it, as planet formation
which requires a disc, can also destroy the disc. Armitage & Hansen (1999) discussed planet formation
as a route to disc dispersal. Placing a Jupiter mass planet in a very massive disc (Md = 0.1M∗), they found
that this planet was able to drive self gravitating perturbations, emphasised at the Lindblad resonances.
Once the planet reaches 4-5 Jupiter masses the disc begins to fragment rapidly, forming many more
planetary bodies near both the inner and outer Lindblad resonances. Figure 1.10 shows results from
a numerical calculation of this process occurring, showing that these new planets could form rapidly
(on dynamical rather than viscous time-scales). Planet formation as a mechanism for disc destruction,
requires that one of these new planets grow large enough to begin this process again and again, until all
the disc material is exhausted. The model presented by Armitage & Hansen can successfully reproduce
the rapid clearing of the disc. It predicts that the resultant planetary system would consists initially of
numerous massive planets on co-planar, near circular orbits, but mutual interactions would eventually
leave a system with a small number of massive planets on eccentric orbits. It remains to be seen whether
the observations of planetary systems are in agreement with this model for planet formation. However,
it is unlikely this method could provide a dominant dispersal mechanism for planet formation since it
requires the disc to be massive enough to fragment, to allow the formation of either the initial planet or
secondary planets.
Origin of transition discs
The planet formation model can also account for observations of transition discs. A massive planet
embedded in a gas disc will clear a gap in the gas and dust disc (Lin & Papaloizou 1979a,b), this is
shown clearly in the top right panel of Figure 1.10. While it is clear that the planet provides a ‘leaky’
barrier to the gas, Rice et al. (2003,2006) suggested that the large radial pressure gradient present at
the edge of the gap in the gas disc could trap large dust grains, preventing them form accreting across
the planetary barrier and entering the inner disc, resulting in an opacity drop in the inner disc. Detailed
calculations of this process have yet to be performed and it is still yet to be seen whether this opacity
22 Chapter 1. Introduction
Figure 1.10: Disc surface density shown from an SPH calculation of a protoplanetary disc. The top
left panel shows an isolated disc after 512 orbital period at the inner edge, whereas the top right planet
shows a disc at the same time but with a initially Jupiter mass planet embedded in it. The bottom left
panel indicates the time (608 orbital periods) at which the planet has reached ∼ 4 Jupiter masses and is
beginning to drive resonances. At 736 orbital periods (shown in the bottom right) the disc is fragmenting
forming large numbers of new planetary bodies (Figure from Armitage & Hansen 1999).
1.4 Disc Clearing Mechanisms 23
Figure 1.11: Cartoon of disc photoevaporation. Inside the radius rg a bound atmosphere is formed,
outside rg a thermal disc wind is formed (Figure from Hollenbach et al. 2000).
drop is sufficient to explain the NIR emission in transition discs. However, it is widely believed that discs
embedded with planets are the origin for some ‘transition’ discs.
1.4.2 Photoevaporation
In order to remove disc material from the star/disc system it must be accelerated to the escape veloc-
ity. Since, the the majority of the disc is at temperatures well below the escape temperature in order
for this process to occur there must be regions close to the surface of the disc where the temperature
increases and approaches the escape temperature. If the gas reaches sufficient temperatures it may drive
a thermal driven wind instead of forming a bound atmosphere (similar to a stellar wind emerging from
a hot corona). Such a processes has been known to occur from discs that are close enough to massive
stars, where the UV luminosity can heat the surface regions of the protoplanetary discs to temperatures
close to the escape temperature. This process of external disc ‘photoevaporation’ was invoked to explain
the tailed shapes of protoplanetary discs close to the OB stars in the Orion Nebular Cluster, where gas is
being stripped off the disc at such a rate that they can only survive for a short time (Johnstone et al. 1998;
Clarke, 2007). Detailed radiation hydrodynamic calculations have now been performed that can match
the flow morphology and low ionization lines seen from these objects (e.g. Richling & Yorke, 2000).
However, external photoevaporation cannot be the origin of disc dispersal around most young stars as
only < 25% of young stars will experience even mild external photoevaporation (Adams et al. 2004).
It has been known for a long time that young stars are much more active than their evolved counter-
parts, producing large amounts of high energy photons in the UV and X-rays. It was Hollenbach et al.
(1994) who built the first self-consistent model of disc photoevaporation, driven by the central star. The
Hollenbach et al. (1994) model was a pure EUV (photons capable of directly ionizing hydrogen) only
model and these photons produce an approximately isothermal (T ∼ 104K) layer at the disc’s surface.
Figure 1.11 shows a cartoon of photoevapoation driven by the central star, inside some radius labelled
rg the gas forms a bound hydrostatic atmosphere, outside this radius the gas drives a thermally driven
flow. Hollebach argued that this radius would be related to the escape temperature, and when the internal
energy of the gas was equivalent to the gravitational energy it would be able to escape the stars potential
24 Chapter 1. Introduction
in a flow. This radius can simply be calculated and is given by:
rg =
GM∗
c2s
= 8.9
(
M∗
1 M
)(
T
104 K
)−1
AU (1.25)
Since, most of the earlier literature was only concerned with isothermal 104K winds, this scale ra-
dius is commonly described only in terms of mass, and for completeness I separately define Rg =
8.9(M∗/M)AU. This simple picture of EUV disc photoevaporation has been studied in detailed by
several authors, the picture was orginally described in Hollenbach et al. (1993) and Shu et al. (1993)
with relation to our solar system. Hollenbach et al. (1994) then performed radiative transfer calculations
based on an analytic model in order to determine the density of the base of the ionized layer. This model
was then used to estimate the total mass-loss rate by assuming that the gas outside rg had a velocity equal
to the sound speed of 10km s−1, this model gave a total mass-loss rate of:
M˙w ≈ 4.4×10−10
(
Φ∗
1041 s−1
)1/2( M∗
1 M
)1/2
M yr−1 (1.26)
where Φ∗ is the total EUV ionizing luminosity in photons per second. This initial calculation was later
improved by Font et al. (2004), who used the radiative transfer calculations of Hollenbach et al. (1994)
as the boundary conditions for a hydrodynamic calculation. This calculation verified the basic model
put forth by Hollenbach et al., while making several modifications. Firstly, it moved the inner radius
from which the flow may be launched towards the star by an order of magnitude, allowing the sub-sonic
launching of a wind outside 0.1rg, a result confirmed by Adams et al. (2004) and Alexander et al. (2008).
Secondly, it reduced the total mass-loss rate by a factor of 3 and changed the profile of the mass-loss rate
to one that was dominated at slightly smaller radii. It was originally thought that photoevaporation would
not pay an important role in disc evolution around low mass stars as Md/M˙w yields a time-scale several
orders of magnitude longer than the observed disc lifetimes.
Photoevaporation coupled with viscous evolution
It was Clarke et al. (2001), who realised that photoevaporation doesn’t have to remove all the disc ma-
terial to provide a disc dispersal mechanism. Instead photoevaporation can provide a cut-off in accretion
rate, with most of the disc’s original mass being accreted onto the star. This is suggested by the obser-
vation that there are very few discs accreting at a level below around 10−10M yr−1(e.g. Muzerolle et
al. 2000), similar to the limit implied by photoevaporation. Including photoevaporation into the viscous
evolution described above simply involves including a sink term in Equation 1.2 and the evolution of the
discs surface density is described by:
∂Σ
∂ t
=
3
R
∂
∂R
[
R1/2
∂
∂R
(
R1/2νΣ
)]
− Σ˙w(R, t) (1.27)
1.4 Disc Clearing Mechanisms 25
where Σ˙w(R, t) describes the mass-loss rate per unit area due to photoevaporation. The implications of
this extra term can be seen if we integrate Equation 1.26 over radius to find:
dMd
dt
=−M˙∗− M˙w (1.28)
Equation 1.28 tells us immediately that there are two phases of disc evolution: when M˙∗ > M˙w the disc
evolution is dominated by the viscous transport of angular momentum and hence associated accretion
rather than the removal of gas through photoevaporation. During this stage the disc will behave in a
similar manner to a standard viscously evolving disc, without photoevaportion especially when M˙∗ 
M˙w. However, when M˙∗ < M˙w it is now photoevaporation that dominates the evolution of the disc,
opening a gap in the disc and then finally removing the remaining disc material until the disc is dispersed.
As shown earlier the accretion rate of a disc evolves on the disc’s global viscous time τν , which is the
local viscous time at the outer radius, (which is of order the disc’s lifetime; e.g. Lynden-Bell & Pringle,
1974; Pringle, 1981; Ruden, 1993). The transition from a viscously evolving disc to a clearing disc
occurs when a gap opens in the inner disc and the material inside the gap drains onto the central star.
Since, the time-scale for the inner disc to drain is the viscous time at the point the gap opens (which is
typically much less than the global viscous time of the disc), the transition from a viscously accreting
disc (i.e. one that would observationally be classified as an optically thick accreting primordial disc) to
one that is being dispersed, occurs on a time-scale much shorter than the disc’s current lifetime (Clarke
et al. 2001; Ruden 2004).
Hence, the evolution of a viscously evolving photoevaporating disc can naturally explain the obser-
vations of discs which appear to evolve under the effect of viscous transport alone, but are then rapidly
dispersed. The photoevaporation process is shown occurring in the models calculated by Clarke et al.
(2001) - shown in Figure 1.12 - where the inner disc rapidly drains onto the central star between the third
and fourth snapshot, once the accretion rate through the disc falls below the photoevaporation rate. In the
model shown above, the time-scale for the outer disc to be completely cleared is similar to the total disc
lifetime. This outer disc clearing behaviour is inconsistent with the outer disc observations that show
that most WTTs show no evidence of an outer disc at large radius (e.g. Duvert et al. 2000). This was
corrected by Alexander et al. (2006a) who realised that the topology of the photoevapotive flow changes
once the inner disc has drained onto the star and there is a large gap between the central star and the inner
edge of the outer disc. Alexander et al. (2006a) performed detailed radiation hydrodynamic simulations
of photoevaporation from discs with large cleared inner holes, which showed a mass-loss profile peaked
at the inner edge of the disc and mass-loss rates that increased with inner hole radii. This was then used
by Alexander et al. (2006b) to show that once the gap has opened and the inner disc has drained onto the
central star the outer disc can be cleared very rapidly in a time-scale much less than the discs lifetime,
and this evolution is shown in Figure 1.13.
Origin of transition discs
Photoevaporation provides two phases in which ‘transition’ discs will be created. First just after the
gap opens, Alexander & Armitage (2007) showed that any dust that was present in the inner disc is
rapidly accreted onto the star under the action of dust drag. This occurs on a time-scale several orders of
26 Chapter 1. Introduction
Figure 1.12: Evolution of the disc’s surface density of a viscously evolving photoevaporating disc with
Φ∗ = 1041 s−1. Snapshots are shown at t =0, 13.0, 14.1, 14.3, 16.0, 18.0, 20.0, 24.0 & 28.0 Myr. (Figure
from Clarke et al. (2001)
magnitude faster than the time-scale for the gas disc to drain onto the star, leaving an accreting dust free
(and therefore optically thin) inner gas disc. While Alexander & Armitage (2007) made some simplifying
assumptions in order to perform the calculation, a rapid (almost instantaneous compared to the viscous
time) drop in the NIR opacity due to dust drag is highly likely, although detailed calculations of whether
this opacity drop is sufficient to explain the observed NIR emission in ‘transition’ discs are still required.
In this work I will assume the limiting case presented by Alexander & Armitage (2007), where when
dust-dust interactions are neglected the inner gas disc becomes optically thin immediately after the gap
opens (when working on a time-scale comparable with viscous evolution).
The second phase is much more certain. Once the inner disc has fully drained onto the central star
and the outer disc is being eroded this will lead to observations of non-accreting transition discs, where
the gas and dust hole line up in radius. With the inner cavity being devoid of large amount of gas and
dust emission.
Other sources of high energy irradiation
At first thoughts the EUV photoevaporation model seems to explain the observed disc clearing success-
fully. However, the low mass-loss rates ∼ 10−10 M yr−1mean it is unable to explain the majority of
1.4 Disc Clearing Mechanisms 27
Figure 1.13: Evolution of the disc’s surface density including direct EUV photoevaporation described by
Alexander et al. (2006a). Snapshots are shown at t =0, 2.0, 4.0,5.9, 6.0, 6.01, 6.02, . . . , 6.18Myr. Direct
photoevaporation takes over when the inner disc has drained onto the star (Figure from Alexander et al.
2006b).
observed ‘transition’ discs. Furthermore, it has been pointed out by several authors that due to the low
neutral column required to full absorb the EUV radiation N ∼ 1018 cm−2, it is highly unlikely that EUV
photons actually reach the surface of the disc, and therefore are unable to drive a flow (Glassgold et al.
2004, Ercolano et al. 2009b, Gu¨del et al. 2010).
Therefore, these problems have lead to research into other high energy photons as a possible mech-
anism to heat the surface layers of the disc. Young stars produce high energy photons at UV to X-ray
wavelengths, where the UV is split into two regions: Far UV (FUV) are photons with energy just below
13.6eV, the photons above 13.6eV are the EUV photons we have already discussed. FUV photons are
unable fully ionize the gas and heat it up to 104K; however, they are able provide significant heating and
low levels of ionization. X-ray heating is also able provide significant heating, with gas able to reach
∼ 104K and low levels of ionization. Currently, no hydrodynamic calculations of internal FUV or X-ray
photoevaporation have been performed, although significant progress has been made by considering the
hydrostatic structure of irradiated discs.
X-ray Heating of Discs
Young stars are extremely luminous in the X-rays with energies in the range 10−5−10−3Lbol (e.g. Guedel
et al. 2007), thus they provide a significant source of energy that impinges on the disc. Alexander et al.
(2004) was the first to consider the irradiation of a protoplanetary disc by X-rays. However, it was
not until Ercolano et al. (2008b) that the possibility that X-rays could drive vigorous photoevaporative
flows was realised, with estimates two orders of magnitude higher than the mass-loss rate from a pure
28 Chapter 1. Introduction
Figure 1.14: Density distribution of a hydrostatic disc heated by X-rays. The black line shows the region
where the gas temperature is equal to the escape temperature (Figure from Ercolano et al. 2009b).
EUV flow obtained. In order to obtain improved estimates for the photoevaporation rates due to X-ray
heating Ercolano et al. (2009b) calculated hydrostatic models of discs that included separate dust and
gas radiative transfer.
In order to estimate the mass-loss rates from hydrostatic discs Ercolano et al., uses the ρcs method,
where the mass-loss rate at a given cylindrical radii is calculated by assuming at that radius the density
structure is flowing at the local sound speed when the gas temperature equals the escape temperature
(shown in Figure 1.14). At first thought this may seem odd, however in a pure thermally driven radial
outflow the point at which the gas reaches the sonic velocity always occur at the point where the gas
temperature is approximately the escape temperature (one half to be precise, Parker 1958). In reality the
physical equation describing the flow is the momentum equation
u
du
dl
+
1
ρ
d(c2sρ)
dl
+
dψ
dl
= 0 (1.29)
whereas the equation of hydrostatic equilibrium is:
1
ρ
d(c2sρ)
dl
+
dψ
dl
= 0 (1.30)
Below the point where the temperature equals the escape temperature the udu/dl term in the momentum
equation is smaller than or of similar magnitude to the pressure and body force terms. Therefore, the
hydrostatic structure presented by Ercolano et al. (2009b) will provide a reasonable approximation to
the true density and temperature structure in the flow, and the ρcs method of calculating mass-loss rates
1.4 Disc Clearing Mechanisms 29
is certainly physically motivated, and should give a sensible order of magnitude estimate of the total
mass-loss rates. Along with providing estimates of the total mass-loss rates Ercolano et al. (2009b) also
investigated the implications of neutral columns close to the central star. No change in the mass-loss rates
until the neutral column reached N ∼ 1021cm−2, or until photons with energies less than hν ∼ 1keV are
fully screened out. This means unlike the EUV which can only penetrate very small columns, the X-rays
are able penetrate more realistic accretion columns that may form close to the star and it is ‘soft’ X-ray
photons that provide the source of heating in the surface layers of protoplanetary discs.
FUV Heating of Discs
During the accreting phase, the accretion shock on the stellar surface produces large amount of FUV
photons, with luminosities reaching as high as 1032 erg s−1 (e.g Ingleby et al. 2011). Gorti & Hollenbach
(2008,2009) and Gorti et al. (2009) have investigated the affects of high energy heating on protoplanetary
discs included FUV, EUV and X-ray irradiation. Again in order to simplify their calculations they solve
for a hydrostatic disc structure. FUV heating is driven by photoelectric heating from dust grains and
cooling from atomic and molecular lines, and in the protoplanetary disc environment the maximum gas
temperatures reached by FUV heating is ∼ 1000K an order of magnitude less than the EUV and X-rays.
Furthermore, in the atmospheres of realistic discs, the Gorti & Hollenbach calculations showed that the
FUV could penetrate to higher columns than the X-rays and produced a disc structure, whose temperature
increased with height as near the mid-plane standard dust heating is dominant, then the FUV, then X-rays
and finally EUV at the surface of the disc.
Instead of using the ρcs method to estimate the mass-loss rates from these disc structure, Gorti &
Hollenbach use a different method. The method consists of fitting an isothermal and spherical wind
solution based on the model described by Adams et al. (2004) for external photoevaporation from the
outer edge of discs. Since the temperature of the hydrostatic disc calculations determined by Gorti
& Hollenbach vary with height, the mass-flux this solution gives also varies with height. Therefore,
in order to estimate the photoevaporative mass-flux at a given cylindrical radius, Gorti & Hollenbach
selects the height in the disc which the method gives the largest mass-flux. This height is then defined
as the flow ‘base’ and the photoevaporative wind is assumed to be isothermal from this point onwards,
neglecting the thermal structure above. Using this method Gorti & Hollenbach (2009) conclude that the
flow begins sub-sonically in the deep within the FUV heated region of the disc, with mass-loss rates
∼ 10−9− 10−8M yr−1comparable to those found by Ercolano et al. (2008b,2009b) for X-ray heated
discs.
Gorti & Hollenbach (2009) point out the use of an isothermal wind solution will lead to errors and
the wind solution has come under some criticism (e.g. Ercolano & Clarke, 2010). However, it is the
definition of a flow ‘base’ and the neglected the hydrostatic structure above this region that may provide
the biggest source of error. In the photoevaporation model defining a flow base is fairly meaningless,
since streamlines must ultimately originate at the discs mid-plane, where the bound disc structure/flow
structure must be in dynamical equilibrium so the flow can be feed at the correct rate. Therefore, many
different wind solutions (with vastly different mass-loss rates) will asymptote to this correct hydrostatic
structure at low heights, and no physical reason is presented why the method of selecting the height at
which this method gives the highest mass-flux, should pick out the correct solution. Furthermore, in
30 Chapter 1. Introduction
hydrostatic disc calculations the regions of the disc with temperatures below the escape temperature will
provide a reasonably accurate (to order unity) representation of the true disc structure. By assuming an
isothermal wind solution from deep within the disc the Gorti & Hollenbach method neglects the roughly
correct structure of the region above. Since the temperature increases with height the true sonic surface
will be lower than estimated, and may lead to errors in both the mass-loss rate and the heating mechanism
(if there is one) which sets the mass-loss rate.
‘Real’ Photoevaporation from Protoplanetary discs
We have seen that while the pure EUV photoevaporation model is well developed and produces the
require phenomenology of disc dispersal, it suffers from several issues. The main one being the fact it
is unlikely that EUV photons can actually reach a protoplanetary disc’s surface. There has been much
work on the inclusion of X-ray and FUV heating on disc models that could drive much more vigorous
flows. However, the use of simplified radiative transfer and hydrostatic disc structures is limited and has
perhaps even lead to some confusion. Therefore, in order to progress a discussion of photoevaporation
undergoing heating from all three heating mechanisms within a hydrodynamic framework is the only
way forward.
1.4.3 Observations of Photoevaporation at Work
Before, I present the outline for the work in this thesis it is briefly worth discussing the small but growing
sample of observations that attempt to probe, directly or indirectly the implications of a vigorous photoe-
vaportive flow emerging from a protoplanetary disc, apart from the standard indicators of disc evolution.
The earliest observational evidence of a photoevaporative flow was the low velocity component of the
OI 6300A˚ line observed around many class II sources by Hartigan et al. (1995). An example of the
observed line profile is shown for CW Tau, the line profile clearly shows a double peak with a peak
at ∼ 100km s−1 dubbed the high velocity component and a second one at ∼ 5km s−1 dubbed the low
velocity component. Hartigan et al. (1995) identified the high velocity component with a jet. How-
ever, they were unable to ascertain the origin of the low velocity component, concluding it was possibly
due to a disc wind escaping from a radii of around ∼ 10 AU. This feature was identified around almost
all class II objects observed, and could possibly be due to a photoevaporative wind, although an EUV
photoevaporative wind and magnetic disc wind was ruled out by Font et al. (2004).
Another blue-shifted emission line, this time in the 12.8µm NeII was observed around several ob-
jects (Passcucci & Sterzik 2009), with the closest object TW-Hya (with a disc which is almost face on)
providing direct evidence for a 5−10 km s−1 disc wind, although this time the emission was consistent
with the predictions from a pure EUV model (Alexander et al. 2008).
Less direct, but still interesting observations are the correlations between X-ray luminosity and ac-
cretion rate discussed by Drake et al. (2009). The observations of X-ray luminosity and accretion rates
in the Orion Nebular Cluster seemed to indicate a negative correlation between these two observables,
although with a large scatter. Drake et al. argued that this could arise through a process of ‘photo-
evaporation starved accretion’. Since, photoevaporation starves the inner disc of material that would
have otherwise accreted, the observed accretion rate will appear suppressed from a pure viscous model.
1.5 Thesis Outline 31
Figure 1.15: Observed OI 6300A˚ line profile of CW Tau (Figure from Hartigan et al. 1995)
Therefore, higher mass-loss rates driven by higher X-ray luminosities could starve the inner disc to a
greater extend and provide the origin for the negative correlation.
Finally, perhaps the most tentative of indicators is the disc lifetime versus metalicity correlation re-
ported by Yasui et al. (2009). Observations of disc populations in low metalicity environments in the
extreme outer galaxy, showed lower disc fractions compared with local disc populations. While a model
which relied on planet formation to clear the disc would predict a much longer disc lifetime at lower met-
alicities, X-ray photoevaporation rates are expected to be higher (since the opacity is lower and larger
columns can be heated by the X-rays) leading to shorter disc lifetimes in agreement with the observations
(Ercolano & Clarke, 2010). However, this is far from conclusive and many more observations of this type
are needed before disc lifetime verus metalicity correlations are a useful test of photoevaporation models.
1.5 Thesis Outline
In this thesis I attempt to discuss whether photoevaporation can provide a dominant dispersal mechanism
for discs, and what direct and indirect observational signatures can be used to test the scenario. Since, the
driving mechanism for photoevaporation is still debated, in Chapter 2 I will present a hydrodynamic so-
lution to the X-ray photoevaporation problem and then discuss hydrodynamic flows that include UV and
X-ray heating. In Chapter 3 I obtained accurate mass-loss profiles by performing radiation hydrodynamic
simulations of photoevaporating discs, while testing some of the predictions of the model discussed in
Chapter 2: The importance of disc structure, the effects of UV luminosity and variations in stellar mass
32 Chapter 1. Introduction
and X-ray luminosity. In order to test whether photoevaporation can provide a dominant dispersal mech-
anism I build a population synthesis model of viscously evolving photoevaporating discs, in order to
make statistical comparisons with current data along with making predictions for future observations.
Chapter 4 describes a synthetic population of evolving discs that include both viscous evolution and pho-
toevaportion, where the results from population is statistically compared to observations. In Chapters
5 & 6 I turn my attention to direct observational tests of photoevaporative winds, discussing gas line
emission in Chapter 5 and dust emission in Chapter 6. Finally, I draw my conclusions in Chapter 7 and
discuss areas where further research is required.
2
The theory of photoevaporation
PHOTOEVAPORATION is likely to play an important, if not dominant role in the evolution anddispersal of protoplanetary discs. As discussed in Chapter 1, protoplanetary discs are subject tointense irradiation by high energy photons covering a large range of energies; from FUV (hν <
13.6eV), to the EUV (13.6 eV < hν <0.1 keV) and finally to the X-rays (hν > 0.1keV). While basic
calculations of disc heating from all three radiation fields has been carried out for protoplanetary discs
(Gorti & Hollenbach 2008,2009), no one has considered the problem within the framework of radiation-
hydrodynamics, which after all is important as photoevaporation is clearly a radiation-hydrodynamic
problem.
As discussed previously it is the EUV-only photoevaporation problem that has been solved hydrody-
namically. Therefore, if we wish to discuss the photoevaporation of real discs, in which all three high
energy radiation fields will be present we must develop a hydrodynamic solution to the problem. I begin
in this Chapter by presenting a hydrodynamic description of pure X-ray photoevaporation, then discuss
the hydrodynamic problem of an EUV/FUV/X-ray heated protoplanetary disc. Therefore, in this work I
discuss the heating of protoplanetary discs, then present the theoretical basis on which we can understand
photoevaporation from real discs before we move onto detailed radiation-hydrodynamic calculations in
Chapter 3.
2.1 Heating of protoplanetary discs
On time-scales much shorter than the viscous time of the disc, the equations that describe the structure
of a protoplanetary disc/wind system are the classic Euler equations:
Dρ
Dt
+ρ∇ ·u = 0 (2.1)
33
34 Chapter 2. Photoevaporation Theory
Du
Dt
=−∇P
ρ
+∇ψeff (2.2)
D
Dt
(
e
ρ
)
=−P
ρ
∇ ·u+Λ (2.3)
where this set of equations is closed with an ideal equation of state; D/Dt represents the Lagrangian
derivative, ψeff is the effective potential, e is the gas’ internal energy and Λ is the net cooling rate.
These equations neatly hide all of the complicated physics inside the net cooling rate function, where gas
heating and cooling processes are important if not dominant in setting the gas temperature. Therefore, it
is important to separately discuss the main heating mechanisms in a protoplanetary discs and their role
in driving photoevaporation.
2.1.1 Optical & IR heating
In terms of the overall energy budget the optical/NIR emission from the star is the dominant contribution
(only for the highest accretion rates is the accretion luminosity important, e.g. Calvet & D’alessio 2011).
Furthermore, as mentioned in the introduction, the SED observations indicated that the dominant heating
mechanism is stellar irradiation and not viscous dissipation (Kenyon & Hartmann 1987); therefore, the
dominant heating mechanism must be the optical/NIR emission from the star. At these wavelengths the
opacity of the dust is many orders of magnitude larger than that of the gas, and it is the dust that is
directly heated by the star. At some height above the mid-plane the dust must be optically thin to its own
and the stellar radiation, and we can simply evaluate the dust’s temperature by balancing absorption and
emission, to find:
κE(Td)T 4d = κp(T∗)T
4
∗
(
R∗
2r
)2
(2.4)
which for perfect black-body dust (κE = κP - note in general κp(T∗)> κE(Td) - so this is a lower bound)
yields:
Td = 289 K
(
T∗
6000 K
)(
R∗
1 R
)1/2( r
1 AU
)−1/2
(2.5)
The optical depth to the central star decreases with height at a given cylindrical radius, thus the stellar
flux decreases with height and the dust temperature must also decrease with height. Therefore, the mid-
plane and photospheric temperature at a given R will always be less than the dust temperature in the
optically thin regions shown in Equation 2.5.
While the dust absorbs most of the stellar radiation it only represents a small mass fraction of the disc,
therefore the gas particles are heated indirectly by the optical/IR radiation through gas-dust collisions,
where the time-scale for this process to occur is given by the dust-gas collision time-scale:
tcol =
1
ndσdgv¯g
(2.6)
where v¯g is the average velocity of the gas particles (since md  mg) and the collision cross-section
is simply given by σdg = pia2 as the dust particles are much larger than the gas particles. At the high
densities found close to the mid-plane this time-scale is short, thus the dust and gas temperatures are
tightly coupled. However, given the approximately Gaussian fall off in density, at several scale heights
2.1 Heating of protoplanetary discs 35
this time-scale can becomes longer allowing the dust and gas temperatures to become decoupled, pro-
vided sufficiently quick gas heating and cooling mechanisms exist. This decoupling has been studied by
many authors, which show that the surface layers of the disc can be heated to temperatures above the
local escape temperature. This ‘extra’ heating arises from FUV, EUV and X-rays, where the inclusion
of these heating mechanisms has been studied either separately, or in some combination by Glassgold
(1997,2004,2007); Kamp & Dullemond (2004); Gorti & Hollenbach (2004,2008,2009); Ercolano et al.
(2008b,2009b); Woitke et al. (2009); Kamp et al. (2010).
2.1.2 EUV heating
EUV heating is perhaps the easiest ‘extra’ heating mechanism to understand. EUV photons are photons
capable of directly ionizing a Hydrogen atom and as such must have an energy ≥ 13.6eV. Given that
the interaction cross-section falls rapidly with photon energy (∝ E−3), ionizing photons are by definition
photons with energies at or just above 13.6eV. Therefore, the luminosity is measured in ionizing photons
per second, with values in the range Φ∗ = 1041−1043 s−1 possible from T Tauri stars (Alexander et al.
2005), in keeping with this I follow the convention of Ercolano et al. (2008b, 2009b) and take EUV
photons to be in the energy range 13.6 eV < hν < 0.1 keV. Since EUV photons ionize hydrogen, the
main coolant for an EUV region is to emit a recombination photon, which by definition will be at 13.6eV
and this recombination photon can then ionize another Hydrogen atom and so on. Therefore, an EUV
heated region will result in an approximately fully ionized isothermal region with a temperature∼ 104K,
where the extent of the region can be found by considering recombination balance (a Stro¨mgren volume).
Hollenbach et al. (1994) discuss two limits for EUV heating of protoplanetary discs: one where the disc’s
surface is directly illuminated by stellar ionizing photons at radii ROR∗; whereas recombination photons
dominate at radii RR∗ often called the ‘diffuse’ limit, since it is the diffuse ionizing field that is heating
and ionizing the surface layers of the disc. Due to the ease with which neutral material absorbs EUV
photons, (with columns densities of N = 1018cm−2 sufficient) meaning no direct measurements are yet
possible (Alexander et al. 2005), several authors have questioned the ability of EUV photons to reach
the outer disc (e.g. Glassgold et al. 2004; Ercolano et al. 2009b).
2.1.3 FUV heating
Young stars also emit radiation in the FUV range, i.e. UV radiation longward of 13.6eV and is thus
unable to directly ionize hydrogen; however, it is an extremely important when one considers the heating
of the upper gas layers of protoplanetary discs. FUV heating is an extremely complicated process, that
is particularly sensitive to chemistry in the disc and has spawned an extensive wealth of literature that I
cannot do justice here and I point the interested reader to Tielens & Hollenbach (1985a,b) and the review
by Hollenbach & Tielens (1997) as an excellent starting point. However, it is important to summarise
the basic principles of FUV heating here. Typical FUV luminosities of stars are in the range 1029−1032
erg s−1 (e.g. Ingleby et al. 2011) and for the type of environment in protoplanetary discs by far the most
dominant heating mechanism is the ejection of photo-electrons from dust grains (Gorti & Hollenbach
2004), whereas the dominant cooling mechanism at higher temperatures (when the gas is predominantly
atomic) are atomic forbidden line emission from SI, SiII, FeI, FeII & OI in the MIR, and molecular
line emission (when the gas is predominately molecular) at cooler temperatures. Many of these cooling
36 Chapter 2. Photoevaporation Theory
lines are not optically thin and can be optically thick, furthermore when molecular lines become impor-
tant, then so does the chemistry and the abundance of these molecules is sensitive to the local density,
temperature and radiation field. As such very few calculations exist that have aimed to include all of
the necessary physics and chemistry with the calculations by Gorti & Hollenbach (2008,2009) currently
state of the art. While this complexity will prevent me from improving on these calculations, as we shall
see I can use some of the main findings to investigate the photoevaporation of discs. Unsurprisingly
FUV heating is unable to reach the temperatures that the EUV can reach, with a typical maximum gas
temperature of around T > 1000− 2000K in the T Tauri environment (e.g. Gorti & Hollenbach 2004;
Bruderer et al. 2009). However, unlike the EUV, the FUV is able to penetrate much larger columns
densities. Since at these UV wavelengths the opacity is dominated by dust extinction, the gas column the
FUV can penetrate depends on the dust properties in the gas. For ISM type dust this penetration depth is
around N ∼ 1021 cm−2, although we expect protoplanetary discs to be significantly evolved away from
this dust population. In particular for the dust depleted disc atmospheres argued for by D’alessio et al.
(2001) based on SED modelling, the penetration column of FUV photons maybe as large as 1023cm−2
(Gorti & Hollenbach 2009).
2.1.4 X-ray heating
Fortunately X-ray heating in protoplanetary discs is a much simpler process when compared to FUV
heating. While X-rays span an extremely broad range of photon energies it is only the ‘soft’ X-ray
photons that are important for X-ray heating (Ercolano et al. 2009b), these photons typically have an
energy range of∼ 1 keV and are absorbed by large neutral columns of gas N ∼ 1022 cm−2. The dominant
mechanism for X-ray absorption is photoelectron generation by the K-shells of heavy metals (in order of
decreasing importance: Oxygen, Carbon and Iron). Unsurprisingly this electron has a very large energy,
sufficient to release a large number of ‘secondary’ electrons from nearby atoms, through collisional
ionization of neutral gas. It is these secondary electrons which dominate the heating by thermalising
within the surrounding gas. The Auger effect can also occur, however, the emitted auger electrons are
significantly outnumbered by the secondary electrons produced through the photoelectric process at the
expected temperature and density ranges.
Cooling of the X-ray heated gas can occur through several processes, although at the gas temperatures
expected (Tgas> 2×104 K - e.g. Ercolano et al. 2008a,b) cooling due to X-ray recombination and thermal
bremsstrahlung is negligible. Thus the dominant cooling mechanism is line cooling, for the densities and
temperatures that will be present in an X-ray heated region of the disc the gas will always be optically thin
to its own cooling radiation. Since these photons dominate the cooling and are free to escape to infinity
they should provide an observational indication of X-ray heating of discs. This will be discussed further
in Chapter 5, particularly in relation to the dominant cooling lines: optical forbidden line emission from
OI.
All the heating and cooling mechanisms described above are purely local processes. By this I mean
that the secondary electrons are thermalised on a scale considerably smaller than any other scale on
which properties of interest vary (e.g. radiation field or flow scales) and the cooling radiation will not
interact with the gas on distances of order (or greater than) scales of interest, giving rise to non-local heat-
ing. Furthermore, an estimation of the heating and cooling time-scales will determine whether adiabatic
2.1 Heating of protoplanetary discs 37
heating/cooling will be an important process. The heating time-scale is simply given by the collisional
time-scale of the secondary electrons with gas particles which is:
tcol =
1
ngσe−gve−
≈ 1s
( n
107cm−3
)−1(σe−g
1A˚2
)−1( Te−
107 K
)−1/2
(2.7)
whereas one can calculate the cooling time-scales using the fact these are spontaneous, collisionally
excited lines. In the case the Einstein Coefficients (Anm) are large, then the cooling time-scale is just set
by the collisional time-scale with either neutral hydrogen or electrons, both of which are fast, as seen
above. However, in the case that the Einstien Coefficients are small then the cooling rate per unit volume
will be ∝ n2 (since the dominant cooling lines are forbidden). A detailed calculation (e.g. Figure 5
Ercolano et al. 2008b) yields cooling rates per unit volume of 10−14−10−12 erg cm−3 s−1, which gives
a cooling time of > 1 yr for X-ray heated gas at ∼ 5000K and n ∼ 105 cm−3. Flow time-scales for
unbound gas will be ∼ rg/cs, which is ∼ 10 yr for similar values, implying that theat tcool tflow .
Therefore, since the heating/cooling is purely local and the thermal time-scale is much shorter than
the expected dynamical time-scales (something I will verify in Chapter 3), it is a natural question to ask
if the local radiation field is known can the temperature be parametrised in terms of local variables only.
Fortunately it has been known for a while that X-ray ionization (and hence heating) is well described in
terms of the ionization parameter (ξ ) where it takes the form:
ξ =
4pi fX
n
(2.8)
where fX is the X-ray flux and n is the number density of the gas. It essentially scales as the available
amount of X-ray energy to each atom (e.g. Tarter et al. 1969) and can characterise the ionization structure
and therefore the gas temperature. Therefore, in thermal equilibrium the temperature of the X-ray heated
gas can be specified in terms of the ionization parameter alone. In Chapter 3 this prescription of the X-
ray heated gas is described in more detail, where its exact form is taken from the previous X-ray heating
calculation of Ercolano et al. (2009). However, for the following discussion I shall assume its existence
and the temperature of the X-ray heated gas is given by T = f (ξ ) (where the form of f is shown later),
where the temperature ranges between 200 > T < 2× 104K with ξ ranging between 10−7 and 102 erg
s−1 cm.
2.1.5 Photoevaporation as a result of disc heating.
The much higher gas temperatures that the UV and X-ray radiation field can reach compared to the
disc’s mid-plane, implies that at some radii the gas temperature of the disc’s atmosphere may attain
a temperature that is above the escape temperature. Gas above the escape temperature is free to flow
away, initiating a flow from the underlying disc (this is the process of photoevaporation). In fact thermal
pressure gradients will drive a wind from sub-sonic regions, through a sonic transition and into a freely
expanding supersonic wind (similar to the famous Parker wind model for a solar wind, Parker 1958).
However, the process of disc photoevaporation is more complicated as the centrifugal force is non-
negligible, along with no requirement that the photoevaporative wind is isothermal (unless it is a pure
EUV heated wind).
38 Chapter 2. Photoevaporation Theory
As mentioned in the introduction, the pure EUV photoevaporation model is effectively a solved hy-
drodynamic problem (Font et al. 2004; Alexander et al. 2006a). Therefore, in this chapter I will deter-
mine the basic physics of pure X-ray photoevaporation, before discussing the photoevaporation of discs
exposed to the full range of UV and X-ray radiation within a hydrodynamic framework.
2.1.6 Basic results of thermally driven winds
At this point it is useful to summarise some of the basic elements of thermally driven winds before going
on to discuss the more detailed aspects of photoevaporation. Therefore, in this section I shall restrict
the discussion to an isothermal driven wind, where the fundamental starting point is the steady state
momentum equation along a streamline, given by:
u
du
dl
=−c
2
s
ρ
dρ
dl
− dψ
dl
(2.9)
where l is the length along a streamline. Using continuity, we can write Equation 2.9 as:
(
u2− c2s
) dlogu
dl
= c2s
(
dlogA
dl
− 1
c2s
dψ
dl
)
(2.10)
where A is the stream-bundle area. Thus we see that in order to undergo a sonic transition, the flow
requires that:
c2s =
∂ψ
∂ logA
(2.11)
at the sonic surface. While a sonic transition is not formally required, such ‘weak’ wind solutions, which
require that the flow be in causal contact with a boundary a large radius are known to be extremely
unlikely in astrophysical situations (e.g. Parker, 1958). Therefore, this requirement at the sonic surface
can provide some information on the topology and structure of the flow. Two well known transonic wind
solutions are the de Laval nozzle and the Parker wind.
The de Laval nozzle solution describes the topology of a transonic flow in the absence of body forces,
in this case Equation 2.10 becomes:
(
u2− c2s
) dlogu
dl
= c2s
dlogA
dl
(2.12)
This tells us that when the flow is sub-sonic the flow is convergent, and when the flow is supersonic the
flow is divergent, where the sonic transition occurs when the velocity divergence is zero. Therefore, in
this problem the conditions at the sonic surface allow us to determine information on the topology of the
flow.
Parker (1958) proposed a spherical isothermal wind as a model for the solar wind (the Parker wind).
Under the assumption of a spherical outflow, Equation 2.10 becomes:
(
u2− c2s
) dlogu
dl
= c2s
(
2
r
− 1
c2s
GM∗
r2
)
(2.13)
2.2 The theory of X-ray Photoevaporation: scaling relations 39
0 0.5 1 1.5 2 2.5
0
0.5
1
1.5
2
Radius [r/r
s
]
Ve
lo
ci
ty
 [u
/c s
]
Figure 2.1: Velocity of Parker wind as a function of radius, plotted in dimensionless variables.
where in this case we find that the sonic surface must occur at a radius of:
rs =
GM∗
2c2s
(2.14)
thus in this case, fixing the flow topology also fixes the point of the sonic surface, where the physical
significance of this radius is clear, rs is the radius at which an ideal monotonic gas has a total energy
of zero and has become unbound from the star (and is a factor of two smaller than the rg defined by
Hollenbach et al. 1994). It is also interesting to look at the form of the solution for the Parker wind,
shown in Figure 2.1, where the velocity has a transcendental form of:(
u
cs
)2
−2log
(
u
cs
)
= 4log
(
r
rs
)
+4
(
r
rs
)−1
−3 (2.15)
This shows us that it is in the sub-sonic regime that the gas gains most of its velocity, which increases
rapidly as u = cs
√
exp(−4(r/rs)−1) in this regime, whereas in the supersonic regime the increase in
velocity is small increasing as u = 2cs
√
log(r/rs) and thus the density almost falls off as ρ ∝ r−2.
This form of the velocity is easy to understand, since most of the acceleration must have occurred sub-
sonically as the wind is accelerated by the pressure gradient, which is fairly ineffective at accelerating
super-sonic gas.
Furthermore, we can now see that in general flow geometries (non-spherical), if your flow becomes
super-sonic at radii smaller than rs, then Equation 2.11 tells us it must do so at a temperature higher than
the escape temperature. Therefore, we can use this argument to reverse this principle and use the gas
temperatures obtainable by the different high energy heating mechanisms to estimate the approximate
minimum radius at which the flow may transition from sub-sonic to super-sonic. So for a solar mass star,
the EUV has a minimum radius at ∼ 10 AU, ∼ 10−100AU for the X-rays and ? 100AU for the FUV.
2.2 The theory of X-ray Photoevaporation: scaling relations
In this section I develop the theory of X-ray photoevaporation from discs, with a view to understanding
how photoevaporation rates vary. I assume that the gas is heated by optically thin X-ray radiation and is in
40 Chapter 2. Photoevaporation Theory
thermal equilibrium, where the gas temperature can be described in terms of the ionization parameter. In
qualitative terms, the flow starts highly sub-sonically at the base of each streamline and is then accelerated
by a mixture of pressure gradients and the effective gravity, geff (i.e. the combined effects of the gravity
of the central star and a centrifugal force). It is assumed that viscous effects are negligible over the flow,
thus the specific angular momentum of each fluid element is conserved, being the Keplerian value at
launch (cylindrical radius Rb) with value h=
√
GM∗Rb. Now, writing the specific energy of a gas parcel
as:
ε =
1
2
v2+Φ+
e
ρ
=
1
2
(u2R+u
2
z )+(Φ+
1
2
u2φ )+
e
ρ
(2.16)
where the term in the second brackets on the far right hand side can be identified as the effective potential
(Ψeff), now using uφ = h/R and geff =−∇Ψeff we find:
geff =−GM∗r2 rˆ+
GM∗Rb
R3
Rˆ (2.17)
In a steady state (∂ρ/∂ t = 0), the continuity equation (2.1) becomes:
∇ · (ρu) = 0 = d
dl
(ρuA) (2.18)
where the second result can be recast as:
dlogρ
dl
=−dlogu
dl
− dlogA
dl
(2.19)
Now starting from the steady state (∂u/∂ t = 0) momentum equation (2.2) we can expand the convective
derivative to find:
1
2
∇u2−u×∇×u =−∇(ρc
2
s )
ρ
+geff (2.20)
where I have introduced the sound speed through P= ρc2s . Now taking the dot product of Equation 2.20
with u the term including the Lamb vector disappears by orthogonality. Furthermore, using u ·∇ = u ddl
and Equation 2.19 we can re-write Equation 2.20 as:[(
u
cs
)2
−1
]
dlogu
dl
=
dlogA
dl
− dlogc
2
s
dl
+
geff · lˆ
c2s
(2.21)
Therefore, as described above a flow will undergo a sonic transition at the point where the right hand
side of Equation 2.21 is equal to zero. However, the present case is much more complicated, since we
do not know the flow geometry a priori and because ge f f is non-negligible at all radii and, to further
compound the problem, cs is not constant. Although it is possible to solve for u(l) (and hence ρ(l) and
cs(l)) for a fixed streamline topology, there is no guarantee that such a solution would satisfy the steady
state momentum equation perpendicular to the streamlines. This is the reason that there is presently no
analytic solution for the structure of a 2D disc wind, even in the simplest isothermal case. Instead the
steady flow structures are computed through time-dependent hydrodynamical simulations (e.g. Proga et
al. 2000; Richling & Yorke, 2000; Font et al. 2004; Alexander et al. 2006a).
At first sight these observations appear to preclude further analytic arguments, but in fact we can
make useful progress through the use of scaling relations. Take the case where we have obtained a
2.2 The theory of X-ray Photoevaporation: scaling relations 41
self-consistent steady state flow structure (via a hydrodynamical simulation) for a given set of input
parameters, and we then change some parameter of the flow and obtain a new structure. We can then ask
whether there are circumstances where one case is simply a scaled version of the original case.
Suppose that every streamline in the first case can be identified with a topologically identical stream-
line in the second case, which differs only in an over all geometrical scaling. We can also assume that
the variation of flow variables along a streamline in the second case can be scaled to the variation along
the original streamline, i.e. we write all variables in the form u= u0u˜(l˜) where l˜ is the distance along the
streamline scaled to the radius at the base of the streamline (Rb0) and u0 allows for a simple re-scaling of
the velocity between the two streamlines. In this case we can write Equation 2.21 as:[(
u0
cs0
)2( u˜
c˜s
)2
−1
]
dlogu˜
dl˜
=
dlogA˜
dl˜
− dlogc˜
2
s
dl˜
+
GM∗
Rb0c2s0
g˜eff · lˆ
c˜2s
(2.22)
where
g˜eff =− 1r˜2
˜ˆr+
1
R˜3
˜ˆR (2.23)
Now in order to obtain a consistent, self-similar solution we require that the above equation should
contain only scaled variables, i.e. should not depend on the parameters Rb0, u0 and cs0 which vary from
case to case. Examining the left hand side of Equation 2.22 requires that u0 = cs0 (i.e. we require
that the Mach number of the flow is the same for the two scaled streamlines at given l˜). Furthermore the
individual values of u0 and cs0 are the same for both streamlines: if cs0 were different then the temperature
at given l˜ would be different between the two streamlines (by a constant factor) and the density would
also be different, according to the parametrisation of temperature against ionisation parameter. Since this
relation is not a simple power law scaling (there is no reason to expect it to be as the heating and cooling
rates depend on specific ionization fractions and temperatures, this is confirmed in Chapter 3) then the
density variation along the streamlines would not be self-similar, in contradiction with our assumption.
Therefore, for streamlines to map onto one another in a self-similar way, we must have the same velocity
and temperature structure as a function of l˜.
Turning now to the right hand side of Equation 2.22 we see that in order that this does not contain
variables that differ from case to case, we must have Rb0 ∝ M∗. Or more physically, the problem is
scalable in terms of escape temperature, if there is no scale to the problem, since Tesc ∝M∗/r. Namely,
streamlines can be topologically identified in terms of the escape temperature at some (scaled) point
along the streamline.
2.2.1 Primordial discs
I apply the above description to the case of ‘primordial’ discs; those that, having mid-plane variables
which have no intrinsic scale (i.e. discs which are well described by the Lynden-Bell & Pringle similarity
solutions - as suggested by observations - e.g. Andrews et al. 2010). Therefore, primordial discs appear
to be good candidates for yielding self-similar solutions when the input parameters are varied:
42 Chapter 2. Photoevaporation Theory
Variation of LX at constant M∗
We have seen above that strict self-similarity requires that Rb0 ∝M∗. Thus for fixed M∗, Rb0 is the same
and each streamline simply maps onto itself, with the same temperature and velocity variation along the
streamline, when LX varies. Since the temperature is a function of ionisation parameter, it follows that as
one varies LX , the density scaling along each streamline must simply vary uniformly with LX . The mass
flux along each streamline is the product of streambundle area, velocity and density. As the velocity
and area are unchanged from case to case and the density scales with LX , then the mass flux is simply
∝ LX . Therefore, I conclude that primordial discs which differ only in their values of incident LX should
be topologically identical, with identical velocity and temperature structure. But with a mass weighting
on each streamline that scales with LX . This agrees with the result of Ercolano et al. (2008b), whose
hydrostatic calculations indicated that the mass-loss rate may scale linearly with X-ray luminosity.
Variation of M∗ at constant LX
In this case, Rb0 ∝M∗ implies the streamlines from discs around one mass should map onto those with a
different value of M∗ by a simple geometrical scaling in proportion to M∗ (i.e. a streamline rooted at 10
AU with M = 1M should simply map on to a streamline rooted at 1 AU when M = 0.1M). As before,
we require that the two scaled streamlines should have the same variation in velocity and temperature
as a function of distance along the streamline normalised to the radius of the streamline base, i.e. the
same variation of ionisation parameter. Since ξ = LX/nr2, it follows that a radial scaling ∝M∗ results
in a scaling with density of ∝M−2∗ . The mass-loss from the streamline is proportional to the product of
the density (∝M−2∗ ), the local velocity (constant) and the area of the streambundle (∝M2∗ ): i.e. the mass
loss rate along each streamline is the same as the mass loss rate over its scaled equivalent. Therefore
(provided the disc is sufficiently radially extended that it encompasses the whole region from which
significant mass loss occurs in all cases), I conclude that, at constant LX , the mass flux in the wind is
independent of M∗.
It is of course important to note, that even though there is no explicit mass dependence on photo-
evaporation rates from discs, there will of course be a strong implicit variation with mass through the
variation of X-ray luminosity and mass. Observations suggest this approximately M∗ ∝ L
3/2−5/3
X (e.g.
Preibisch et al. 2005, Guedel 2007).
2.2.2 Discs with inner holes
Here the truncated density profile of the disc introduces a particular radial scale (Rhole) into each under-
lying disc . However, if one fixes Rhole, then the above arguments for LX and M∗ are still applicable.
For fixed M∗ and Rhole then as one varies LX the density must also vary as n ∝ LX , resulting in a total
mass-loss rate that varies linearly with X-ray luminosity. Furthermore, if one varies M∗, then one can
simply map one solution with an inner hole radius to the equivalent inner hole around another mass using
Rhole ∝M∗. This means that the mass-loss profile should be scalable in mass in terms of Rg. However, as
described above the temperature-ionization parameter relation doesn’t permit scalable solutions in tem-
perature. The only choice to obtain an exact similarity solution - cs0 ∝M∗/Rhole - is not commensurable
with this requirement. Therefore, while photoevaporation from discs with inner holes provides scalable
2.3 The theory of X-ray photoevaporation: absolute flow rates 43
solutions in mass and X-ray luminosity a complete similarity solution that encompasses mass, X-ray
luminosity and inner hole radius does not exist. Therefore, in Chapter 3 I will use the numerical methods
to perform a full parameter study in mass, X-ray luminosity and inner hole radius, in order to describe
the variation of mass-loss rate with inner hole size.
Summary
I have argued that the flow from primordial discs is expected to be self-similar (when LX and M∗ are
varied) and predict that the mass loss rate should scale linearly with LX and be independent of M∗,
and the mass-loss profile should be scale-free when considered in terms of R/Rg. Furthermore, I have
demonstrated that the photoevaporation rate from discs with inner holes should scale linearly with LX ,
and the profiles can be scaled in terms of stellar mass. However, no general similarity solution exists for
discs with inner holes and will require simulations to describe the mass-loss rate as a function of inner
hole radius and stellar mass.
2.3 The theory of X-ray photoevaporation: absolute flow rates
In order to make further progress in predicting the mass-loss from X-ray photoevaporation, I consider
the case of a primordial disc and obtain an estimate of the total mass loss rate by integrating over the
sonic surface:
M˙w = 2
∫ ∞
0
2piRns(R)cs(R)dR (2.24)
where ns and cs are the values of the number density and sound speed at the sonic surface. Note that the
true mass loss flux will differ from this by a factor of order unity due to the fact that the flow is not purely
vertical at the sonic surface.
We can use Equation 2.21 to write the condition for the sonic surface as:
c2s =
GM
2R
× R
r
× fg
fA
(2.25)
where
fg =
−geff · lˆ
GM/r2
(2.26)
and
fA =
dlogA
dl − dlogc
2
s
dl
2/r
(2.27)
The first two terms equate the temperature at the sonic surface to the usual expression for a spherical
Parker wind; fg takes into account the different component of the effective force along the streamline
compared with the Parker case (both due to the centrifugal term and the non-spherical geometry), while
fA takes into account the different divergence of non-spherical flows and also the fact the flow is not
isothermal. If fA and fg are both of order unity and if r is/ R (i.e. if the height of the sonic surface at
radius R is/ R) then we would expect that the sound speed at the sonic surface is given approximately
by the Parker wind expression, i.e. cs ≈ (GM/2R)1/2. Certainly, the numerical calculations in the next
Chapter will verify if this condition holds in the case of primordial disc photoevaporation. However,
44 Chapter 2. Photoevaporation Theory
there are several factors that indicate that this may be a reasonable assumption, the non-spherical nature
of a photoevaporative wind primary arises from the centrifugal force. Given the centrifugal force falls
away quickly (∝ R−3), if the sonic surface occurs at a cylindrical radius significantly larger than Rb0 then
the flow is likely to be approximately spherical. Furthermore, to attain the high gas temperatures required
at the sonic surface, it must occur in a region that is optically thin to X-ray radiation. Thus there is likely
to be a sufficient height of X-ray heated gas below the sonic surface, meaning gas parcels experience a
region that is hot enough to thermally drive it to larger radii, while not hot enough to undergo a sonic
transition, and the inclusion of extra heating mechanisms below the X-ray heated region (e.g. FUV
heating - Gorti & Hollenbach 2009) mean the size of the transonic region will be larger, making the
likelihood of a spherical flow at the sonic surface even more likely.
All these factors create the expectation that the wind temperature at the sonic surface should be within
a factor of order unity of the corresponding Parker value at each cylindrical radius, i.e. we may write:
c2s = β
GM
2R
(2.28)
where β is a factor of order unity. Given that we can estimate the sound speed as a function of R at
the sonic surface via Equation 2.28 and that the temperature-ionisation parameter relationship fixes the
density at given r and T we can then write the mass loss integral in the form:
M˙w =
∫ ∞
0
4piRn{ξs(R)}
√
GM∗
2R
dR (2.29)
Now taking the temperature ionization relation to be given by T = f (ξ ), and using the definition of
ξ = LX/nr2, at the sonic surface we may approximately write1 ns(Ts) = LX/R2 f−1(Tesc/2). Therefore,
if we change the variable of integration in Equation 2.29 from cylindrical radius R to escape temperature
we find:
M˙w = LX
√
8kbpi2
3µmh
∫ ∞
0
T−1/2esc
f−1(Tesc/2)
dTesc (2.30)
≈ 8×10−9
(
LX
1×1030 erg s−1
)
M yr−1 (2.31)
where kb is Boltzman’s constant, µ is the mean molecular weight, mh is the mass of a hydrogen atom
and LX is the X-ray luminosity of the central star. The second result comes from evaluating the integral
out to an escape temperature of 1000K, or a radius of O100AU for a solar mass star (indicative of
the outer radius of the disc). As I will discuss in Section 2.4 this is also the region out to which we
expect the photoevaporation to be important for disc evolution. This again demonstrates the mass and
X-ray luminosity scaling determined above, along with obtaining reasonable estimates of the mass-loss
rate which we can compare to numerical models. Furthermore, Figure 2.2 shows the surface-mass loss
rate predicted by the analytic calculation at the sonic surface for a 0.7Mstar and X-ray luminosity of
LX = 1× 1030erg s−1. This shows that the mass-loss profile of X-ray photoevaporation is likely to be
broad, with the total mass-loss dominated at large radius, rather than small radius as a pure EUV model
yields. These values are in agreement with the surface mass-loss rates obtained from the previous static
1where f−1 ◦ f = 1
2.3 The theory of X-ray photoevaporation: absolute flow rates 45
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
4
5
x 10−14
Cylindrical Radius of Sonic Surface [AU]
Su
rfa
ce
 M
as
s−
lo
ss
 R
at
e 
[g 
cm
−
2  
s−
1 ]
Figure 2.2: Analytic predictions of the mass-flux through the sonic surface plotted against cylindrical
radius, for a 0.7M star with an X-ray luminosity of LX = 1×1030erg s−1.
calculations of Alexander et al. (2004) and Ercolano et al. (2008b,2009b) indicating that the model
described above provides a good physical basis of the X-ray photoevaporation model.
2.3.1 Role of the sonic surface
It is important not to underestimate the implications of Equation 2.28, where in the case of X-ray photo-
evaporation we are free to write down the temperature at the sonic surface a priori as a pure function of
mass and cylindrical radius. As shown, this means that the mass-flux at the sonic surface is fixed by the
shape of the temperature-ionization parameter relation and X-ray luminosity alone. Equation 2.21 tell us
that the conditions at the sonic surface must be satisfied in order to attain a steady state, thermally driven
wind. Therefore, since mass flux is conserved, the mass flux in the wind must be identical to that at the
sonic surface. This means that whatever the underlying disc structure/temperature it must feed the sonic
surface at the desired rate.
This is far more restrictive than the similarity solution described in Section 2.2, as we don’t need
to make any assumptions about the properties of the underlying disc structure. Therefore, provided the
flow satisfies: (i) the sonic surface occurs at z/ R; (ii) the sonic surface occurs in a region where the
centrifugal force is small, or Rs Rb0, the photoevaporation rate is exclusively set by the X-ray physics
alone. Therefore, we can be confident that the mass-loss rates are independent of our original choice
of underlying disc structure and any heating mechanism that may or may not operate below the X-ray
heated region, since the disc must adjust itself to feed the flow with the required mass-flux rather than
the mass-flux adjusting itself to changes in disc structure. Obviously, such a physical description of
X-ray photoevaporation, along with the degree to which these conditions are satisfied is something the
numerical models presented in Chapter 3 will test.
2.3.2 Importance of Numerical Results
Given we can obtain the mass-loss rates without reference to running hydrodynamic simulations, it is
worth briefly addressing their crucial importance to disc evolution models. The above result ultimately
only tells us about the total mass-loss rates, not the mass loss rates from the surface of the disc (Σ˙w(R)).
Of course we could extract an estimate of the surface mass-loss rates by differentiating Equation 2.30
46 Chapter 2. Photoevaporation Theory
with respect to Tesc. However, this will only give an estimate of Σ˙w at the sonic surface (which is shown in
Figure 2.2), as described above, we expect the sonic surface to be at a distance far from the bound disc.
Therefore, without running hydrodynamical models it is impossible to quantitatively predict how this
sonic surface links with the bound disc and obtain an accurate Σ˙w(R), which is crucial to the behaviour
of the disc during the final clearing stages, i.e. the point at which a gap may open in the disc, and the
speed at which the clearing may proceed through a depleted outer disc.
2.4 Theory of UV & X-ray Photoevaporation
Now I have developed the physics behind X-ray heated flows from protoplanetary discs we may turn
our attention to photoevaporation from discs around stars emitting the entire spectrum of high energy
radiation from the FUV, through the EUV to the X-rays.
2.4.1 EUV and X-ray heated flows
The calculations of Ercolano et al. (2009b) also included an EUV component, of equal luminosity to the
X-rays in their input spectrum. Based on their hydrostatic disc calculations they argued that the EUV
had negligible impact on photoevaporation, since it was the X-rays that were dominating the heating in
the region where the gas was likely to undergo a sonic transition. The order of magnitude estimate in the
chapter can be used to estimate the radius the EUV may penetrate to using the fact M˙w ∝ n×R2× cs.
Therefore an X-ray driven wind with a mass-loss rate of ∼ 10−8M yr−1 out to R∼ 100 AU at a sound
speed of ∼ 5km s−1 for a solar mass star, yields a penetration depth of approximately 1AU for the
EUV based on absorbing column of 1018cm −2 (Ercolano et al. 2009b). This penetration radius is
approximately the radius at which the heated disc’s surface transitions from a bound atmosphere to a flow
(Font et al. 2004; Alexander, 2008). Thus it is likely that an X-ray driven photoevaporation flow is too
optically thick to allow EUV photons to reach the disc’s surface, where it could drive a photoevaporative
flow. Certainly such a hypothesis is easy to test using numerical methods, once the X-ray flow solution
is obtained and I perform this analysis in Chapter 3.
2.4.2 FUV and X-ray heated flows
Gorti & Hollenbach (2008,2009) considered the combined effect of UV and X-ray heating in setting the
hydrostatic disc structure and then used their static structures to predict resulting flow rates. However,
it is a most precarious step to predict flow solutions from a hydrostatic density structure since a variety
of solutions (whether static or flowing) asymptote to the same (nearly hydrostatic) structure at small z.
Although the results of Gorti & Hollenbach still represent the state of the art in terms of the thermal and
chemical structure of X-ray and FUV irradiated discs, the implications of these studies for the resulting
mass loss rate are currently unclear. In Chapter 3 I explore this further by using a simple parametrisation
of FUV heating in radiation hydrodynamic simulations. Firstly, however I discuss the general theoretical
framework for combined FUV and X-ray heated flows. As noted above, X-ray heating is effective up to
a total hydrogen column of around 1022 cm−2 (independent of the dust properties of the gas). For models
using depleted dust in the disc atmosphere (e.g. Gorti & Hollenbach 2004,2008,2009), FUV heating
extends to significantly higher columns. This means that travelling along a line from the star to a point
2.4 Theory of UV & X-ray Photoevaporation 47
Sonic Surf
ace
X-ray pene
rtration de
pth
FUV penertration d
epth
Dust Heated Disc
Sonic Surface
X
-r
a
y
p
e
n
e
rt
ra
ti
o
n
d
e
p
th
FUV penertration d
epth
Dust Heated Disc
Figure 2.3: Figure showing a cartoon sketch of photoevaporation from a disc exposed to UV and X-ray
irradiation. The flow begins in the FUV heated region, but at some height the flow transitions to the
X-ray heated region and then passes through the sonic surface in the X-ray heated region. Left shows
X-ray dominated, and right shows FUV dominated.
in the disc, one first traverses gas that is exposed to both X-rays2 and FUV, then to FUV only and then to
neither. Within ∼ 100 AU (for a solar mass star) the X-rays dominate the heating in the regions that are
exposed to both X-rays and FUV. Therefore, in this region, the sequence (as one proceeds away from the
star) is X-ray heated, FUV heated and then heated by neither - in reality heated by coupling to the dust -
as mentioned earlier.
Though we expect this sequence of X-ray heating followed by FUV heating as one proceeds away
from the star, there are still two topologically distinct possibilities which are illustrated in Figure 2.3.
The left hand figure is a schematic illustration of the case where the flow in the FUV heated region never
exceeds its local sound speed for any of the streamlines shown and the sonic transition occurs in the
X-ray heated region. In this case we can use the arguments contained in Section 2.3.1 to argue that the
sound speed at the X-ray heated sonic surface is simply given by Equation 2.28 with β of order unity
as before: in other words the temperature at the sonic surface is simply a function of cylindrical radius.
Since this also fixes the density at the sonic surface (via the ionisation parameter) this implies that the
mass flux is just a function of R and X-ray luminosity. In other words, whatever the detailed physics
of the heating and cooling processes in the X-ray dark region, they must combine to drive a flow that
is able to deliver the required mass flux to the X-ray sonic surface. Therefore, one would expect that if
the geometry is as shown in the left hand panel, the photoevaporation rate is entirely controlled by X-ray
heating.
On the other hand, the right hand panel shows three streamlines where the two at larger radius undergo
a sonic transition within the FUV heated region. Where the FUV is driving a flow that is optically thick
to X-rays photons (c.f. the EUV and X-ray case described above). In this case the flow has to self-
adjust to deliver the required conditions at the FUV sonic surface; in other words it is the structure of the
FUV heated region that controls the mass flow rate. Although in principle gas flowing out in this FUV
heated wind could become X-ray heated at some point, this region would not be causally connected to
the flow base, the two regions being separated by a shock. As is the case in the offset ionisation fronts
created when EUV radiation interacts with an FUV wind in the case of externally illuminated proplyds
2In this case the X-rays I am explicitly referring to the X-ray photons that provide significant thermal heating, with a
maximum energy of 1-2KeV and not those at higher energies that can penetrate larger columns but provide insignificant heating
(see Ercolano et al. 2009b).
48 Chapter 2. Photoevaporation Theory
(see Johnstone et al 1998). However, it is unlikely that an FUV driven wind would ever be significantly
heated by X-rays from the central star. This is because we require the condition at the sonic surface
in the FUV heated region to satisfy Equation 2.28. Given that FUV heating does not generally attain
temperatures in excess of ∼ 1000−2000K, this implies that the schematic FUV sonic surface shown in
the right hand panel of Figure 2.3 must lie at radii of ? 100 AU where X-rays are incapable of heating
the gas to greater temperatures than the FUV.
I therefore conclude, although FUV driven winds may be significant at radii > 100 AU, if the flow
topology at smaller radii is as shown in left hand panel of Figure 2.3 then the photoevaporation rate is
set entirely by the X-rays. The only situation in which FUV heating could control mass loss from much
smaller radii is if the streamlines crossing the FUV sonic surface at> 100 AU actually originated at small
radii Naturally, I need to perform a radiation hydrodynamical simulation with FUV heating to assess this
possibility (Chapter 3) though I note in advance that it would require the gas to move sub-sonically
in the FUV heated region over a much larger radial range than seen in our previous calculations, with
exceedingly large mass fluxes.
2.5 Discussion
The hydrodynamic framework of photoevaporation discussed in this Chapter starkly disagrees with the
arguments Gorti & Hollenbach (2009) present for UV and X-ray heated discs. While the suggested
thermal structure of the disc does not differ from their state of the art calculations, it is how this disc
structure drives a photoevaporative flow that is in disagreement with the model presented in this Chapter.
Here I suggest that the density-temperature coupling at a fixed radius (arising from the temperature-
ionization parameter coupling for X-ray heated gas) fixes the photoevaporation rate at the sonic surface
in the X-ray heated region. Whereas Gorti & Hollenbach assume that the mass-flux is fixed by the
density structure deep within the sub-sonic FUV heated region and ignore the thermal structure above
in estimating the mass-loss rates. Although both models suggest that the EUV radiation field will have
negligible impact on photoevaporation arising from its lack of penetration.
Therefore, the question that needs to be discussed to understand this discrepancy is: Does the sonic
surface adjust to the structure below, or does the structure below adjust to the sonic surface? In the case
where X-ray heating is dominant at the sonic surface, I have shown the answer is quite clear: the disc
structure below must adjust itself to feed the sonic surface at the required rate. This is because, to first
order one can write down the temperature at the sonic surface as approximately the Parker value. Since
to attain the required temperatures it must occur in a region optically thin to the stellar X-rays, and thus
is likely to be a significant distance from the regions of the disc where the gas parcels originate, meaning
the centrifugal force is small and the flow is likely to be approximately spherical. If at a given cylindrical
radius we know the temperature, then we know the density through the temperature - ionization parameter
coupling and thus we know the mass-flux at the sonic surface. Thus everything is determined at the sonic
surface and the sub-sonic FUV heated region below must adjust itself in order to maintain this mass-flux.
However, this does not preclude a flow where the gas reaches the FUV sonic surface before it becomes
heated by the X-rays, I argue that such a scenario is unlikely to be important for disc evolution since
this sonic surface must occur at a radius ? 100AU. As the FUV is only capable of heating the gas
to temperatures much lower than those possible by the X-rays, at these large radii the available mass
2.6 Conclusions 49
reservoir in the disc means it is unlikely to be important for the evolution of the inner disc, where the
observational indicators of disc evolution come from. Obviously, this does not preclude a situation where
the FUV drives an extremely powerful photoevaporation at small radii up to a sonic surface at a height
of ∼ 100AU (for a solar mass star) meaning the X-rays are unable to reach the discs surface. However,
this seems unlikely given the only order of magnitude difference in the penetration depths of the FUV
and X-rays and their similar luminosities.
Therefore, the hydrodynamic model of disc photoevaporation presented in this Chapter indicates that
it is the X-ray physics that is likely to be controlling the photoevaporation from the majority of the disc,
and that the disc structure adjusts itself to feed the flow at the rate required by the X-ray sonic surface.
In this case I have shown that the mass-loss rates are explicitly independent of stellar mass (although
not implicitly since the X-ray luminosity is a strong function of stellar mass), the mass-loss rates for
all discs scale linearly with X-ray luminosity and that the mass-loss profiles should scale linearly with
escape temperature (or stellar mass). Obviously, while this discussion provides a firm physical basis
for understanding photoevaporation, this will be useless unless it is backed up with detailed numerical
calculations both to thoroughly test the basic principles, underlying scalings and simplifying assumptions
made, as well as determining the correct mass-loss profiles. Where the following assumptions have been
used in this discussion and need to be verified explicitly by the numerical models:
• The sonic surface occurs predominately in the X-ray heated region.
• The sonic surface occurs in a region where the centrifugal force is small compared to gravity.
• The sonic surface occurs at a height z/ R.
At this stage, the simple analytic calculations do not provide knowledge of the mass-loss profile from
the surface of the disc which I will need to follow the evolution of a disc’s surface density. Furthermore,
the mass-loss scaling from discs with larger inner holes is yet to be characterised. To this end, the
next chapter is dedicated to performing detailed numerical simulations that aim to capture X-ray heating
accurately of physical disc structures, along with physically motivated tests of the impact of EUV and
FUV irradiation.
2.6 Conclusions
In this Chapter I have described photoevaporation from protoplanetary discs, that are irradiated by the
full spectrum of high energy radiation present in isolated young star/disc systems. In particular I have
presented a simple model for X-ray driven photoevaporation, and discuss the implications of FUV and
EUV irradiation. The main findings are summarised below:
1. X-ray photoevaporation from primordial discs can be characterised by an as yet undetermined
similarity solution that is a function of the X-ray luminosity and r/Rg only, provided suitable scale
free boundary conditions are present (namely a scale free structure to the underlying bound disc).
2. X-ray photoevaporation rates scale linearly with X-ray luminosity and are explicitly independent
of stellar mass.
50 Chapter 2. Photoevaporation Theory
3. X-ray photoevaporation is extremely vigorous, with possible total mass-loss rates in the range
10−11−10−8M yr−1 for the observed X-ray luminosities of young stars.
4. A similarity solution does not exists for discs with large inner holes, and simulations will be needed
to describe the variation of mass-loss rate with inner hole radii.
5. In the case that the X-rays dominate the heating at the sonic surface, then the mass-loss rate is
determined by the X-ray physics while the FUV heated region must adjust itself to the required
rate.
6. The EUV may be unable to provide heating at the radii important for photoevaporation as it is
unable reach the disc’s surface at these radii.
7. The FUV maybe responsible for photoevaporation at large radii from the disc R ? 100 AU, but
this cannot be important for the disc clearing indicated by the observations.
3
Numerical Radiation-Hydrodynamic Models of
Photoevaporating Protoplanetary Discs
THE simple theoretical arguments presented in the previous Chapter have given us estimates of thetotal mass-loss rates, along with demonstrating that these mass-loss rates should be independentof all parameters except the X-ray luminosity (e.g. disc structure, stellar mass, incident UV flux).
However, these simple calculations can only get us so far, they do not provide an accurate representation
of the mass-loss profile, or demonstrate the physically motivated assumptions hold true for the full range
of observed parameter space. Therefore, in this section I will use a numerical radiation-hydrodynamic
approach, to accurately solve the problem, which will also allow testing of the predictions made in the
previous chapter, indicating whether the approach described encompasses the correct physics.
3.1 Methods
I undertake 2D hydrodynamic calculations, in which - under the assumption that the gas is in radiative
equilibrium - I parametrize the gas temperature as a function of local variables. This parametrization
is based on the results of 3D Monte Carlo radiative transfer calculations using MOCASSIN (Ercolano et
al. 2003,2005,2008a). In Chapter 2 I argued that the gas will be in radiative equilibrium, based on an
estimation of the heating and cooling time-scales compared to estimates of the dynamical time-scale.
However, these calculations will allow us to verify a posteriori that the thermal time-scale is less than
the flow time-scale, justifying our assumption of radiative equilibrium.
51
52 Chapter 3. Numerical Models
3.1.1 Radiative Transfer
Ercolano et al. (2008b,2009b) employed the 3D Monte Carlo photoionization and dust radiative transfer
code MOCASSIN, to calculate the temperature and ionization structure of a typical T Tauri disc, which
was irradiated by chromospheric X-ray and EUV flux. MOCASSIN uses a stochastic approach to the
transfer of radiation and allows for arbitrary geometries and density distributions. The version of the
code used to calculate the Ercolano et al. (2009b) models was modified by Ercolano et al. (2008a), and I
refer to this article for a detailed description of the code and model setup. For clarity I summarise some
of the basic elements of MOCASSIN next.
The MOCASSIN code
The MOCASSIN code is a publicly available radiative transfer code, originally created by Barbara Er-
colano to study the photoionization of planetary nebulae as part of her PhD thesis (Ercolano et al. 2003).
Since then it has gone through two major modifications: MOCASSIN V2 included dust radiative transfer
along side the gas radiative transfer, allowing for the coupled solution of both the gas and dust temper-
ature (Ercolano et al. 2005); MOCASSIN V3 was updated to include the radiative affects arising from a
high energy spectrum, which stretches all the way from the optical to the gamma rays (Ercolano et al.
2008a). Although it was primarily designed for calculations with an X-ray radiation field. It is a fully 3D
dimensional radiative transfer code and can use an arbitrary spaced Cartesian grid with multiple radiation
sources, which are free to be placed at any point in the domain.
The fundamentals of the Monte-Carlo radiative transfer code is based on the algorithm described by
Lucy (1999). The code stochastically follows the evolution of ‘energy-packets’ through multiple pro-
cesses of absorption and re-emission (assumed to be instantaneous processes) until they leave the grid.
Furthermore, whenever an energy packet traverses a grid cell (whether it is absorbed/scattered or not) its
contribution to the local energy density is calculated and stored. This then enables the calculation of the
gas and dust temperature without requiring a large number of energy packets, particularly in optically thin
regions. Since the (gas) opacities are dependant on temperature then an iterative procedure is employed:
at each iteration, a large number of energy packets (? 107) are allowed traverse the region of interest.
Then the gas and dust temperatures as well as ionization structure (under the assumption of local ioniza-
tional balance) are calculated. The opacities are then updated to account for the current thermal structure
and the iteration procedure is repeated again and again until convergence. Furthermore, MOCASSIN V3
included an extra step that allows the current structure to be iterated into hydrostatic equilibrium, using
the procedure set out in Alexander et al. (2004). Typically the Ercolano et al. (2008b,2009a) calcula-
tions took 10-30 iteration steps to reach an equilibrium state, and approximately 1000-10000 CPU core
hours. Under the assumption that the thermal and hydrodynamical time-scales are decoupled, a coupling
of the MOCASSIN code with a hydrodynamical code is trivial in principle. It simply consists of feeding
the density fields produced by the hydrodynamical code into MOCASSIN to calculate gas temperatures
that are then in turn fed back to the hydrodynamical code. While conceptually easy, this type of cou-
pling is obviously computationally prohibitive (one would require a radiative transfer step time-scale of
 1 CPU core hour) and we must construct a different method for including radiative transfer within a
hydrodynamic calculation.
3.1 Methods 53
Figure 3.1: Input spectrum used in MOCASSIN calculation, for the hydrostatic calculations by Ercolano
et al. (2009b) (Figure adapted from Ercolano et al. 2009b).
The ionization parameter - temperature relation
As described previously, the temperature of X-ray heated gas in radiative equilibrium can be parametrised
in terms of the ionization parameter. Therefore, in order to determine this relation I make use of the
previous calculations of X-ray irradiation of hydrostatic discs preformed in Ercolano et al. (2009b). The
model used is the FS0H2Lx1 calculation of Ercolano et al. (2009b) and I summarise here the input
parameters here. The initial density structure is of a protoplanetary disc surrounding a 0.7 M star (Te f f
= 4000 K, R∗ = 2.5 R) taken from the D’alessio et al. (2001) set and was developed as a model that
best fit the median SED in Taurus. This model is in vertical hydrostatic equilibrium in the case that the
dust temperatures are set by irradiation from the central star, and under the assumption of full thermal
coupling between the gas and dust. The total mass in the disc is 0.026 M with an outer radius of
500 AU.
This disc structure, which provides the starting density distribution, employs a bimodal dust distri-
bution, where atmospheric dust follows the standard MRN model (Mathis et al. 1977) but with a low
dust to gas mass ratio (2.5×10−5 for graphite and 4.0×10−5 for astronomical silicates) and interior dust
consists of larger grains with a size distribution still described by a power law of index -3.5, but with
amin = 0.005 µm and amax = 1 mm and a dust-to-gas mass ratio of 3.1× 10−2 for graphite and 5.0× 10−2
for astronomical silicates. The transition between atmospheric and interior dust occurs at a height of 0.1
times the mid-plane gas scale height.
This initial disc was then irradiated by a synthetic coronal spectrum of X-ray luminosity LX = 2×1030erg
s−1 (see Figure 3.1 ) and total ionizing luminosity of LTOT = 4×1030erg s−1 by Ercolano et al. (2009b).
This synthetic spectrum was a thermal spectrum generated by Ercolano et al. (2008b,2009b) using the
plasma code of Kashyap et al. (2000); the emission measure distribution is based on that derived for
54 Chapter 3. Numerical Models
Figure 3.2: Plot of Temperature as a function of ionization parameter obtained using MOCASSIN with a
numerical parametrization shown in red. The black asterisks show the mean temperature in the ionization
parameter bins obtained from MOCASSIN, with a scatter about the mean of∼ 0.3 dex. The left hand panel
is for a primordial disc and the right hand panel is for a disc with an inner hole at ∼ 16AU
RS CVn type binaries by Sanz et al. (2002) and fits to Chandra spectra of T Tauri stars by Maggio et
al. (2007) (which peaks at around logT = 7.5K). The use of the input spectrum shown in Figure 3.1
implies that there is no column of neutral material close to the source. As described in Ercolano et al.
(2009b), the disc temperatures are updated in response to irradiation by the high energy spectrum and
the density structure is updated so as to attain a situation of hydrostatic equilibrium for this updated
temperature profile. These temperature and density distributions are then updated iteratively until a sit-
uation of both thermal and hydrostatic equilibrium is attained. It is these converged hydrostatic models
that form the basis of the parametrization described below, even though the hydrodynamical calculation
will evolve away from this initial static structure towards a steady flow solution. However, I will check
retrospectively that the parametrization also applies to the evolved structure.
In order to render the calculation tractable, I exploit the fact that the wind will be optically thin to
high energy photons. Figure 3.2 shows the mean relations between ionization parameter and temperature
in the X-ray heated region of the hydrostatic models of Ercolano et al. (2009b), where I define the X-
ray heated region as comprising gas for which the absorbing column to the central star is less than 1022
cm−2 (roughly the penetration depth of 1 keV photons). The scatter of the MOCASSIN grid cells about
the relation shown is small (∼ 0.3 dex), thus confirming that the ionization parameter is indeed a good
determinant of the gas temperature. This confirms the central importance of X-ray heating in determining
the thermodynamics of the heated atmosphere (note that a tight relationship between ionization parameter
and temperature would not be expected in general in the case of heating dominated by EUV radiation).
I will also consider the case of photoevaporation from discs containing inner holes, for which I need to
make sure our parametrization in the lower temperature regime is still applicable. The right hand panel of
Figure 3.2 depicts the result of a similar calculation of a disc which is truncated at an inner radius of∼ 16
AU. This also provides a good fit to models with different inner hole sizes and hence I use this relation
in all the hydrodynamic models. Also, note the similarity of the relations for the identical ionization
parameter range, this implies that it is the direct field that is dominating in both cases. Therefore, I
can be confident that the simple model of X-ray heating outlined in Chapter 2 provides an accurate
description of the physical processes occurring in protoplanetary discs. Furthermore, I can combine the
3.1 Methods 55
two temperature ionization parameter relations to a range that will span all possible ranges expected
in photoevarporation models. Thus, at the most basic level I assume that this temperature ionization
parameter relation is independent of all physical parameters considered in this study (X-ray luminosity,
mass and disc structure). Since, these relations show that the gas temperature varies monotonically with
ionization parameter in the X-ray heated regime, it can be used to prescribe the gas temperature for
regions lying at columns of < 1022 cm−2 from the central source. For columns larger than 1022cm−2
Ercolano et al. (2009b) found that the gas and dust were thermally coupled. Hence, I use the dust
temperatures of D’alessio et al. (2001) to fix the gas temperatures for columns greater than 1022cm−2. In
order to check that our temperature parametrization is appropriate for the final converged flow solution,
this density field will be feed back into MOCASSIN to compare the temperatures obtained.
3.1.2 Radiation-Hydrodynamic Methods
In order to accurately determine the mass-loss rates and the kinematic and morphological structure of
the evaporative flow, I need to compute the temperature distribution in the flow and feed this into a
hydrodynamic code.
The ZEUS Code
The hydrodynamic codes I have chosen to use are the publicly available ZEUS set of hydrodynamical
codes. The ZEUS code was originally the brain child of Michael Norman in the 1970’s, as a hydrody-
namical scheme to study star formation, but has evolved significantly over the years to include extra
physics and optimizations to modern supercomputer architectures. However, the underlying hydrody-
namic algorithms are essentially the same. The method for intergrating the hydrodynamic equations are
described in detail by Stone & Norman (1992a,b) for the ZEUS-2D code and its extension to 3D and MPI
in Hayes et al. (2006) in the ZEUS-MP/2 code. Therefore, here I mainly summarise the basic outline
of the integration method. The ZEUS code employs a staggered finite difference grid. In this setup the
hydrodynamic equations are naturally differenced on the grid using central volume differencing, allow-
ing fluxes to be conserved to approximately machine precision. Vectors are stored and updated on cell
faces, while scalars are stored and updated at cell centres, and differencing is performed using a covariant
formalism. This differencing allows the code to switch between various coordinates systems efficiently,
with caretesian and cylindrical & spherical polars having been implemented.
The ZEUS code integrates the hydrodynamic equations using an operator splitting technique. The
equations are split into two parts, source terms (i.e. non-advective terms) and transport terms (advective
terms). At each time-step the source terms are used to update the hydrodynamic variables first, and then
the advective terms are calculated from these updated values, and used to calculate the fluxes to transport
across the grid. Then the new time-step is calculated using the Courant condition, where I choose a
Courant number of 0.5 to ensure numerical stability. I have modified the ZEUS suite of hydrodynamic
codes (Stone et al. 1992a,b; Hayes et al. 2006) (both ZEUS-2D and ZEUS-MP/2) to include the heating
from X-ray radiation, by writing a new module that calculates the gas temperature. This temperature is
56 Chapter 3. Numerical Models
Table 3.1: Primordial Disc models
Stellar Mass [M] log10(LX) [erg s−1] Code Radial range [AU]
0.7 28.3 ZEUS-2D [0.33,100]
0.7 28.8 ZEUS-2D [0.33,100]
0.7 29.3 ZEUS-2D [0.33,100]
0.7 29.8 ZEUS-2D [0.33,100]
0.7 30.3 ZEUS-2D [0.33,100]
0.7 30.8 ZEUS-2D [0.33,100]
0.1 28.3 ZEUS-MP/2 [0.03,46]
0.1 28.8 ZEUS-MP/2 [0.03,46]
0.1 29.3 ZEUS-MP/2 [0.03,46]
0.1 29.8 ZEUS-MP/2 [0.03,46]
0.1 30.3 ZEUS-MP/2 [0.03,46]
then used to update the internal energy density of gas (e) in the wind via the ideal equation of state:
e=
kB
µmh(γ−1)ρT (3.1)
The mean molecular weight is fixed in the flow, to a value of µ = 1.37125. The internal energy is only
required to update the pressure needed for evaluation of the momentum equation. The pressure is thus
determined again from the ideal gas law, via:
p= (γ−1)e (3.2)
and hence the procedure is independent of choice of γ . This thermal update is done during both the
source step and transport step in order to minimise the time to a converged solution.
I assume azimuthal and mid-plane symmetry and therefore adopt a 2D axis-symmetric grid. Since, I
expect a photoevaporative flow to be approximately spherical at large radius, I use a spherical grid and
neglect magnetic fields and self-gravity. The rotation option in ZEUS is enabled, which introduces the
necessary pseudo-forces arising from rotating frames. The inner and outer radial boundaries are set to
outflow, while the angular boundaries are given the appropriate symmetry boundary conditions.
3.2 Primordial Discs
In order to determine the flow structure for the disc for the majority of its standard viscously accreting
phase, I calculate hydrodynamic models for two masses and the range of X-ray luminosities observed
for young stars, the details of these simulations are summarised in Table 3.1.
3.2.1 Numerical Method
The grid is chosen to be in the range θ = [0,pi/2] and the radial range is shown in Table 3.1, I adopt a
radial range out to 100AU in the case of simulations around 0.7M stars, since previous calculations
around stars of this mass indicate that the majority of the mass-loss occurs inside 50AU (Alexander et al.
3.3 Inner Hole Flows 57
2004; Ercolano et al. 2008b,2009b). For the case of a 0.1M star no such previous calculation exists,
although the theoretical model in Chapter 2 indicates that I could simply scale the 0.7M result and
use an outer boundary of ∼ 15AU, I urge on the side of caution and use and outer boundary of 46AU.
The inner boundary is chosen to be well inside 0.1Rg, where the disc transitions from a bound X-ray
heated atmosphere to an X-ray driven wind, and is at small enough radii not to interfere with the flow.
Grid cells are logarithmically spaced in the radial direction, so that there is adequate resolution at small
radius: in particular I ensure that the onset of the flow, and the scale height of the bound X-ray ‘dark’ disc
is resolved. For calculations performed with the ZEUS-2D code I use 100 cells in the angular direction
and 250 in the radial direction. The calculations performed with the ZEUS-MP/2 included the added
complication that in order to use the distributed memory architecture of the DARWIN supercomputer
(for maxium efficiency and cost!) the number of grid cells needed to be divisible by multiples of 4.
Therefore, accounting for this and the increased radial range of the 0.1Msimulations I use 144 angular
cells and 288 logarithmic radial cells.
Initial Conditions
The vertically hydrostatic density and temperature structures generated by (D’alessio et al. 2001) are
taken as the underlying structures for both the 0.1 & 0.7 Mstar. The 0.7Mdisc structure is the same
as described previously for the calculation of the temperature-ionization parameter and the 0.1Mdisc
model is a 3×10−3M disc around a star with T∗ = 3000K and R= 0.8R . These disc structures were
then allowed to evolve hydrodynamically using the numerical method, with the gas temperatures being
updated at every time-step, until a ‘steady-state1’ disc/wind system was obtained. It should be noted that
the density structure of the X-ray ‘dark’ regions is also allowed to evolve hydrodynamically, i.e. it is not
reset to the original density structure at every time-step (compared with previous EUV hydrodynamic
simulations e.g. Alexander et al. (2008); Font et al. (2004) which - since they did not model the EUV
‘dark’ disc - reset the base density of the flow to its original value at each time-step). I emphasise that
the mass loss over the duration of the simulation is much less than the initial disc mass, and the obtained
wind structures are close to a steady state.
3.3 Inner Hole Flows
As was discussed previously, photoevaporation combined with viscous evolution results in a gap opening
in the primordial disc. The inner disc then drains onto the star leaving the outer disc exposed to direct
irradiation by the central star. The flow expected from such a system is of course different from a
primordial disc’s photoevaporative flow, and in this section I set out the methods used to calculated
mass-loss rates from discs with inner holes.
3.3.1 Numerical Method
In order to describe the variation of mass-loss rate with inner hole radii, I have run several models with
inner holes at different radii, out to a radius beyond which disc holes become hard to detect observation-
1Where the term ‘steady-state’ strictly means quasi-steady-state or more formally M˙w× tMdisc.
58 Chapter 3. Numerical Models
ally (i.e. at around ∼ 10− 100 AU depending on the stellar mass). I used a regularly space spherical
grid in order to maximise resolution while minimising computational effort (see Alexander et al. 2006a).
The grid was constructed with 400 cells in the radial direction and 200 cells in the angular direction,
again sufficient to resolve the onset of any flow and the scale height of the ‘dark’ disc at all radii. All
other ZEUS parameters were kept the same as above and table 3.2 summarises the input parameters used.
As starting points for the calculations, the hydrodynamic primordial disc models were cut a radius close
to the desired inner hole radius (Note: the inner hole radius will be set by dynamical balance with the
flow, hence the final inner hole radius will be slightly different to this value). Then the density profile
was smoothed over four pressure scale lengths at inner edge, where the angular velocity was adjusted to
match the new imposed radial pressure gradient. In most cases the model was then allowed to evolve
hydrodynamically until it attained a steady flow solution. However, as we shall see when we come to the
results, the simulations around the 0.1M star with the largest inner holes ∼ 5 AU violate the ‘quasi-
steady state’ regime discussed above due to vigorous mass-loss rates and low disc surface densities. This
is exactly the situation encountered by Alexander et al. (2006a), even though the time integrated mass-
loss rate over the simulation was comparable to the mass reservoir in the initial bound disc. One can
argue that since the sound crossing time-scale of the flow is much shorter than the expected erosion time
for the inner-edge of the disc, then the simulations may attain a very rough ‘steady-state’, which will
presumably vary on the disc’s dynamical time-scale. However, given I don’t expect the disc structure
to be important for determining the mass-loss rates then the effect of a disc structure that varies on it’s
dynamical time-scale to be small. Therefore, I can extract a run of the mass-loss rate as a function of
inner hole size as the inner edge of the disc is eroded to larger and larger radius as in Alexander et al.
(2006a). In addition, just to be sure that I can trust these mass-loss rates I take the wind/disc structure
from a time-step when the system has reached this rough steady state and perform a simulation where
the density of the X-ray ‘dark’ mid-plane is reset at every time step in order to attain an exact steady state
solution to compare with the rough steady state results.
3.4 Results
The systems were found to have converged after∼4-6 orbital time-scales at the outer boundary. The flow
structure after 10 orbital time-scales at the outer boundary is used in the analysis. All simulations yield a
well defined photoevaporative flow and bound disc, and find that the presence of a photoevaporative flow
above the bound disc, causes very few changes to the structure of the underlying bound discs calculated
by D’alessio et al. (2001), which is reassuring since these hydrostatic disc structures were calculated in
order to fit the dust observations of protoplanetary discs.
3.4.1 Primordial Discs
In Figure 3.3 I show the velocity and density structure for a X-ray photoevaporating disc around a
0.7M star with an X-ray luminosity of 2×1030 erg s−1, with the mass-loss rate for this system found to
be 1.4×10−8M yr−1, in good agreement with the analytic prediction derived in Chapter 2. As expected
that the mass-loss profile is broad extending over many 10’s of AU, but is roughly complete by 50-60AU,
I also find as expected that the sonic surface occurs at height well above the X-ray ‘dark’/‘bright’ tran-
3.4 Results 59
Table 3.2: Inner Hole Models
Mass [M] log10(LX) [erg s−1] Rin [AU] Code Grid range [AU]
0.7 30.3 8.3 ZEUS-2D [1.33,90]
0.7 30.3 9.7 ZEUS-2D [1.67,90]
0.7 30.3 11.7 ZEUS-2D [3.35,100]
0.7 30.3 14.2 ZEUS-2D [3.35,100]
0.7 30.3 17.7 ZEUS-2D [3.35,100]
0.7 30.3 21.1 ZEUS-2D [3.35,100]
0.7 30.3 30.5 ZEUS-2D [3.35,100]
0.7 30.3 64.5 ZEUS-2D [10,450]
0.7 30.3 86.9 ZEUS-2D [15,500]
0.7 28.8 5.4 ZEUS-2D [0.45,93]
0.7 28.8 10.5 ZEUS-2D [0.45,93]
0.7 28.8 19.6 ZEUS-2D [0.45,93]
0.7 28.8 62.3 ZEUS-2D [1.0,448]
0.7 29.8 5.7 ZEUS-2D [0.45,93]
0.7 29.8 11.2 ZEUS-2D [0.45,93]
0.7 29.8 20.3 ZEUS-2D [0.45,93]
0.7 29.8 65.0 ZEUS-2D [1.0,448]
0.7 30.8 6.0 ZEUS-2D [0.45,93]
0.7 30.8 12.1 ZEUS-2D [0.45,93]
0.7 30.8 22.7 ZEUS-2D [0.45,93]
0.7 30.8 71.8 ZEUS-2D [1.0,448]
0.1 30.3 0.6 ZEUS-MP/2 [0.05,20]
0.1 30.3 4.71 ZEUS-MP/2 [0.29,30]
0.1 29.3 0.7 ZEUS-MP/2 [0.05,20]
0.1 29.3 4.51 ZEUS-MP/2 [0.29,30]
sition indicating the assumption of an optically thin X-ray wind is a good one. In Figure 3.3 I show
the total mass-loss rates obtained for all primordial disc simulations compared to the theoretical model
calculated in Chapter 2, this figure indicates that the two major results of the model are correct: the
mass-loss rates scale approximately linearly with X-ray luminosity and they are roughly independent of
stellar mass, along with showing good agreement with the order of magnitude calculations performed
previously. One of the major starting points of the model is that the thermal structure of the flow should
be set by the mass of the central star and scale with the escape temperature, meaning that the thermal
structure of the flow should be invariant when considered in terms of radii scaled by Rg, this means that
the density structure should scale as M−2∗ for a fixed r/Rg, and the mass-loss profile should be fairly
self-similar when also considered in terms of R/Rg. In Figure 3.5 I compare the thermal, density and
velocity structures of two models with masses of 0.1M and 0.7M. This shows clearly that the thermal
structure of the flow is indeed invariant when considered in terms of r/Rg and the sonic surfaces roughly
follow the same loci, along with the density structure and flow topology roughly scaling as expected.
We can look in more detail at the self-similarity of the mass-loss profile by looking at cumulative mass-
loss rate as a function of cylindrical radius, which are shown in Figure 3.6, the model predicts that this
should be approximately self-similar and this is what is found for variations in stellar mass (when one
considers it in terms of R/Rg) and X-ray luminosity. The profile is very broad with approximately 50%
60 Chapter 3. Numerical Models
Radius (AU)
H
ei
gh
t (A
U)
 
 
0 10 20 30 40 50 60
0
5
10
15
20
25
30
35
40
45
50
D
en
si
ty
 ( g
 cm
−
3  
)
−22
−20
−18
−16
−14
−12
−10
Figure 3.3: The density structure of the flow solution for a primordial disc around a 0.7Mstar with
and X-ray luminosity of 2× 1030 erg s−1. The red line indicates the τ = 2/3 surface from the central
star, while the magenta line shows the sonic surface. The streamlines are plotted at 5% intervals of the
integrated mass-loss rate.
1028 1029 1030 1031
10−11
10−10
10−9
10−8
10−7
X−ray Luminosity [erg s−1]
M
as
s−
Lo
ss
 R
at
e 
[M
   y
r−1
]
⊙
Figure 3.4: Mass-loss rate plotted against X-ray luminosity, the open squares show the simulations results
for a 0.7Mstar, the filled circles show the simulation results for a 0.1Mstar. The solid line show the
numerical fit I will use for solar type stars in Chapter 4 to conduct a population synthesis study, the
dashed line shows the order of magnitude estimate obtained in Chapter 2.
3.4 Results 61
2
4
6
8
10
 
 
 
 
2 4 6 8 10
Te
m
pe
ra
tu
re
 [K
]
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
x 104
lo
g 1
0(D
en
sit
y) 
[g 
cm
−
3 ]
−22
−20
−18
−16
−14
−12
Radius [Rg]
H
ei
gh
t [R
g]
0 2 4 6 8
0
2
4
6
8
Figure 3.5: Figure showing the flow topology around a 0.1 (left panels) and 0.7 (right panels) M star
for an X-ray luminosity of 2×1029 erg s−1. The top panels show the density structure with streamlines
plotted at 5% intervals of the integrated mass-loss rate. The lower panels show the temperature structure
of the flow, where the solid line show the sonic surface.
of the mass-loss occurring inside 20AU for a 0.7Mstar and the remain 50% occurring outside and very
little mass-loss occurs outside 60-70AU. The profiles also become broader with increasing X-ray lumi-
nosity, although this broadening is very small when considered in terms of the entire radial range of the
photoevaporative flow.
3.4.2 Inner Hole Flows
The mass-loss rates are found to be approximately insensitive to the inner-hole size. Furthermore, as
expected they scale linearly with X-ray luminosity. In Figure 3.7 I show the mass-loss rates as a function
of inner hole size and X-ray luminosity for discs around a 0.7Mstar and in Figure 3.8 for a 0.1Mstar.
Figure 3.9 shows the flow topology from a disc around a 0.7Mstar with an X-ray luminosity of
2× 1030 erg s−1 and an inner hole at ∼18 AU, in all cases the first streamline is sub-sonically driven
radial inwards, at a radius of R ∼ Rhole/2 the gas has obtained an azimuthal velocity equal to the break
up velocity and gas escapes vertically passing through the sonic surface at a height of z ∼ Rhole and a
temperature around ∼ 6000K considerably higher than the escape temperature, accounting for the non-
spherical velocity divergence and magnitude of the centrifugal force. The model described in Chapter 2
demonstrated that no similarity solution exists and this is what is found, as the mass-loss profiles cannot
be described as function R/Rg. However, I do find that the solution is scalable in X-ray luminosity and
the profile is scalable in terms of escape temperature, demonstrating the basic ideas of the model.
Figure 3.10 shows the evolution of the ‘quasi steady-state’ simulation with time, during the first
300 years of evolution the photoevaporative flow slowly strips of gas from the inner edge of the disc
62 Chapter 3. Numerical Models
0 10 20 30 40 50 60 70 80
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radius [AU]
N
or
m
al
is
ed
 C
um
ul
at
ive
 M
as
s−
lo
ss
 R
at
e
 
 
LX − 28.3
LX − 28.8
LX − 29.3
LX − 29.8
LX − 30.3
LX − 30.8
Figure 3.6: Cumulative mass-loss profiles for discs around a 0.7Mstar with various X-ray luminosities,
(the profiles for 0.1Malso agree with those shown when considered in terms of R/Rg but are not shown
for clarity). The points represent the individual simulations, where the solid line shows a numerical fit
used later in Chapter 4
10 100
10−11
10−10
10−9
10−8
10−7
Radius [AU]
Ac
cr
et
io
n 
R
at
e 
[M
   y
r−1
]
1029 1030 1031
X−ray Luminosity [erg s−1]
⊙
Figure 3.7: Mass-loss rate as a function of inner hole radius (left panel), the points show the results
from the simulations (upright triangles - LX = 30.8; circles - results taken from Owen et al. 2010 for
LX = 30.3; squares - LX = 29.8; downward triangles - LX = 28.8) and the dashed line shows the fit used
throughout this work. The right panel shows the mass-loss rate as a function of X-ray luminosity (• -
≈ 5AU; × - ≈ 10AU; + - ≈ 20AU; ∗ - ≈ 70AU) with the dashed line showing the fit used in Chapter 4.
3.4 Results 63
0 2 4 6 8 10 12
10−9
10−8
Inner Hole Radius [AU]
M
as
s−
Lo
ss
 R
at
e 
[M
   y
r−1
]
⊙
Figure 3.8: Figure showing the mass-loss rate as a function of inner hole size for discs around a
0.1Mstar. The filled points are those for an X-ray luminosity of 2×1030 erg s−1 and the open points are
those for and X-ray luminosity of 2×1029 erg s−1. The crosses are the mass-loss rates calculated from
the eroded inner hole models, while the circles are the moving average calculated from these mass-loss
rates. The dashed line shows the mass-loss rates for primordial discs with the same X-ray luminosity.
Radius [AU]
H
ei
gh
t [A
U]
0 5 10 15 20 25 30 35 40 45 50
0
5
10
15
20
25
30
35
40
Figure 3.9: Flow structure from a disc with an inner hole of 17.7AU, around a 0.7Mstar with an X-ray
luminosity of 2× 1030 erg s−1. The streamlines are plotted at 5% intervals of the integrated mass-loss
rate, and the magenta line shows the sonic surface.
64 Chapter 3. Numerical Models
Time=39.3 years
H
ei
gh
t [A
U]
0
5
10
15
Time=88.4 years Time=176.8 years Time=275.1 years
Time=353.6 years
Radius [AU]
0 5 10 15 20
0
5
10
15
Time=392.9 years
5 10 15 20
Time=432.2 years
5 10 15 20
Time=491.2 years
 
 
5 10 15 20 25
lo
g 1
0(D
en
sit
y) 
[g 
cm
−
3 ]
−20
−19
−18
−17
−16
−15
−14
50 
km s−1
Figure 3.10: Time evolution of ‘quasi steady state’ simulation of a disc with an inner hole and an X-ray
luminosity of 2×1030erg s−1 around a 0.1Mstar. The colour map shows the density structure and the
arrows indicate the velocity structure. Each panel indicates a snapshot of the flow structure.
(note: that as there is no viscosity in the disc that the inner disc region is not resupplied with material and
the erosion time-scale is quicker than would be expected from a viscous disc model). As the hole grows
the layer of warm bound X-ray heated gas grows in size. Once the hole size grows to Rhole ∼ 10AU,
this region becomes large enough that it can clear the remaining disc, which is completely cleared on
the dynamical time-scale (∼100 years at 10AU). This was not seen in the simulations calculated for the
0.7Mstar, the reason for this is since I have moved to lower masses the column in the disc mid-plane
has become considerably lower and reached the point where the size of the X-ray heated bound disc has
become of order the radial pressure height of the disc. If I had the computational resources to calculate
inner hole models around solar type stars with very large radii we would have seen exactly the same
process occurring.
Understanding the Inner hole Scalings
Examination of the streamline topology for various inner hole radii (over a factor of ∼ 2) suggests that
the topology of the innermost streamline is approximately scale free (i.e with an over all radial scale that
varies with Rhole) and as a consequence the variation of temperature and velocity along this streamline (as
a function of distance along the streamline scaled to Rhole) is approximately the same in each simulation
(see Figure 3.11 ). This alludes to the possibility of constructing an approximate scaling argument even
though in Chapter 2 I demonstrated an exact similarity solution does not exist. Nevertheless, if we use
the empirical numerical result (i.e. that the flow topology of the innermost streamlines simply scales with
Rhole and that the velocity and temperature variation along the streamline is the same in scaled distance
units) then we can derive the origin of the found mass-loss scaling law. The constancy of temperature
at given l˜ implies constancy of ξ and hence a density scaling as R−2hole. Then the mass flux along the
streamline (at fixed l˜) scales with density (∝ R−2hole), velocity (constant) and area of streamline bundle
(∝ R2hole). Therefore, the mass-flux along this streamline is invariant of inner hole radius.
The origin of a scalable innermost streamline is likely to originate from the fact that unlike primordial
3.5 Numerical tests 65
1
2
3
H
ei
gh
t [R
/R
ho
le
]
0
2
4
6
8
10
x 105
So
un
d 
Sp
ee
d 
[cm
 s−
1 ]
0 0.5 1
1010
1011
1012
Length [Rhole]
D
en
si
ty
×
R
ho
le
2
 
[g 
cm
−
1 ]
0 0.5 1
0
2
4
6
8
Ve
lo
ci
ty
 [c
m 
s−1
]
Figure 3.11: Figure showing the streamline topology and density, sound speed and velocity along the
sub-sonic section of the streamline. The streamline topology is shown in the top left hand panel scaled in
terms of Rhole, with the sonic surface shown as the dotted line. Then bottom left panel show the density
× R2hole, the top right show the sound speed and the bottom right show the velocity, where all are scaled
in terms of l/Rhole. The black lines are for a hole of 14 AU, the blue line for a hole of 21AU and the blue
for an inner hole of 30 AU.
discs, where the thermal pressure is dominant in determining the streamline topology it is the centrifugal
force that dominates. The streamlines emerging from the inner edge of the disc move towards the cen-
tral star meaning (see Figure 3.9) the centrifugal force quickly becomes dominant. As the form of the
effective potential is scalable in terms of Rhole, in the limit of negligible pressure the streamline topology
would also be scalable in terms of Rhole, and this ‘strong’ body force limit is the most likely origin of the
scalable streamline topology. Provided the total mass-flux is dominated from the inner hole region - as
the simulations indicate - then the total mass-loss should scale in a similar fashion to the mass-flux on
the innermost stream-bundle, resulting in a total mass-loss rate that is approximately independent of the
inner hole radius.
3.5 Numerical tests
The photoevaporation model presented in Chapter 2 made some rather strong predictions regarding the
variation (or non-variation) of the mass-loss rate with physical parameters. While the predicted scalings
with X-ray luminosity and stellar mass can be tested using the simulations discussed above, it is important
to asses the prediction that the underlying disc structure and FUV heating will have almost no affect on
the derived results. This is obviously important since the surface density of a disc, optical properties
of the central star and FUV luminosity can vary by large factors over the millions of years of evolution
expected during the disc’s lifetime. In order to test the role of disc structure and FUV heating I restrict
66 Chapter 3. Numerical Models
the calculations to one stellar mass and X-ray luminosity, a simulation with M∗ =0.7M and LX =
2×1030erg s−1.
Variations in disc structure
Perhaps the easiest way to vary the disc’s underlying structure along with generating very different
thermal and density structure, is to instead of coupling the X-ray ‘dark’ disc to the dust temperature
allow it to evolve adiabatically. Obviously, such a situation has no physical basis but it easily generates
some different disc structures, which can be changed simply by varying the adiabatic index in the ZEUS
code (Note: as described above the X-ray heated region evolves independently of the choice of γ).
Such tests should clearly show up if the disc’s structure plays an important role in photoevaporation.
Therefore, I perform simulations with γ = 1.10, 1.33, 1.41, 1.67 using exactly the same parameters as
the 0.7M primordial disc calculations described above.
In order to consider some more realistic variations in disc structure, I have performed a further set
of simulations where the disc structure is varied in a more physical manner, namely due to variations
in the disc’s surface density and the (total) luminosity of the central star. Therefore, one simulation
has been performed with a disc mass that is 10% of the original calculation (i.e. Md = 2.6×10−3M).
Furthermore, in order to account for variations in the thermal structure of the disc due to luminosity
variations I have increased and decreased the dust temperature by a factor of
√
2. Since the X-ray
‘bright’/‘dark’ transition occurs in a region where the dust is optically thin, then this is just effectively a
change in the stellar luminosity by a factor of 4. These simulations are performed on the same grid range
as the original calculation, although I switch to the ZEUS-MP/2 code and adopt the more efficient grid
resolution of Nr×Nθ = 288×144.
Effects of FUV radiation
In Chapter 2 I argued that since X-ray heating can heat the gas to higher temperatures than the FUV, then
the mass-loss rate will be set by the X-ray physics. In order to test this, I perform some simulations that
roughly aim to include the effect of extra-heating below the X-ray dominated region. If FUV heating was
important to the calculation of photoevaporation rates then one would require a detailed calculation of X-
ray and FUV heating and cooling, along the lines of combing the X-ray algorithm with an FUV algorithm
(e.g. Richling & Yorke, 2000). While such a calculation would be necessary to predict line emission
from the warm atomic and molecular layer, the theoretical framework described above predicts it is
unimportant when photoevaporation rates are what is required. Therefore, a full X-ray + UV calculation
is unnecessary and beyond the scope of this thesis. Thus, in order to test the model predictions I simulate
the affect FUV heating by adding an extra heating source to the X-rays at the level expected from FUV
heating.
Therefore, I calculate the FUV flux throughout the disc/wind system in an identical manner to Rich-
ling & Yorke, (2000) including opacity only from dust extinction (e.g. I ignore opacity arising from the
ionization of Carbon and Sulphur -a second order effect- overestimating the flux and gas temperature). I
then turn the attenuated FUV flux into a temperature using a parametrisation of the external FUV photo-
evaporation calculations performed by Richling & Yorke (2000) and internal disc heating calculations of
3.5 Numerical tests 67
Radius [AU]
H
ei
gh
t [A
U]
 
 
−800 −600 −400 −200 0 200 400 600 800
−1000
−500
0
500
0
0.5
1
1.5
2
2.5
Figure 3.12: Comparison of gas temperature from the external photoevaporation calculation of Rich-
ling & Yorke, (model A) compared with the temperature given by the parametrisation shown above, the
colourmap shows the ratio of the two temperatures Tparam/Tactual, this shows the parameterisation repro-
duces the temperature to within a faction of ∼ 2. The white region shows those which are optically thick
to the direct FUV field and are therefore excluded from our comparison as I can neglect the diffuse FUV
field to first order in our problem (internal photoevaporation).
Gorti & Hollenbach (2008). The parametrisation:
TFUV = 1400K
(
fFUV
300 erg s−1 cm−2
)0.35( n
5×106cm−3
)−0.25
(3.3)
reproduces the temperature structure of the region directly irradiated by FUV in Richling & Yorke (2000)
calculated (labelled model A) as shown in Figure 3.12, and the temperature structure listed for the
‘launching point’ of the Gorti & Hollenbach (2008) calculation to within a factor of 2. I perform a
simulation with a very large FUV luminosity of 1×1031erg s−1 and choose a depleted (from ISM type
dust) value of the FUV extinction of 300 cm2 g−1 used by Richling & Yorke (2000) and a extinction that
is depleted by a further factor of 10. Furthermore, since the conclusions of Gorti et al. (2009) indicate
that disc structure affects the FUV mass-loss rates, where more depleted discs drive more efficient winds,
I calculate a model that would indicate if this effect was real and important. This model uses the mass
depleted disc described earlier (with a Md = 2.3×10−3 M), along with assuming the disc’s atmosphere
is very settled with a low dust to gas mass ratio, and therefore a FUV extinction of 2 cm2 g−1 (approxi-
mately half that of the settled disc described by D’alessio et al. 2001), in this case the mass fraction of
the disc penetrated by the FUV is likely to be an over-estimated.
At every point in the disc/wind system I choose the heating mechanism X-rays, FUV or dust that
produces the highest temperature. I emphasis this is no attempt to calculate the self-consistent thermal
structure of a disc heated by FUV/X-rays, but purely an investigation into the effect FUV type heating
will have on the photoevaporation. I demonstrating this further by running calculations in which I vary
68 Chapter 3. Numerical Models
the parametrisation, while this of course has no physical basis it is a simple way to demonstrate that are
our results are not being driven by our underlying choice of temperature parametrisation, but rather the
physics of the problem i.e. the flow being mediated by the conditions at the X-ray heated sonic surface.
To do this I calculate an extra 4 models for the model with an FUV extinction of 30 cm2 g−1, where I
vary the power law indices (in all four possible combinations) of the flux and density variation by factors
of one-half and two and find no significant differences from the calculations presented below.
3.5.1 Results
I have performed a large set of tests. Including those in which I have neglected certain physical processes
(e.g FUV irradiation), and those I have been required to make on order to make the numerical calculations
tractable (e.g. use of ionization parameter - temperature relation), along with the standard issues that exist
with numerical models (e.g. convergence), so that we can be confident that the solutions obtained are
accurate and are unaffected by the assumptions discussed above.
Numerical Convergence
With calculations that rely on numerical methods, such as the ones used in this work, we must always be
sure that the results obtained are the ones we set out to obtain, i.e. the issue of numerical convergence. In
the case of photoevaporation calculations, I must ensure that our flows are adequately spatially resolved.
In order to test this I have repeated one calculation with double resolution (the 0.7M, LX = 2×1030 erg
s−1 model) and find the flow is converged to> 5%. In order to calculate photoevaporation rates I require
that the flow is in steady state; while I physically motivated this by waiting until after 10 dynamical-
time scales at the order edge of the disc (the longest time-scale in the system) to extract information,
I go one step further to ensure a true steady-state. To do this I calculate the Bernoulli potential in the
rotating frame2 noting that this will be constant in a steady flow. For each streamline I determine the
enthalpy term by parametrizing the run of pressure and density as a baratropic relation (i.e. P(ρ)) along
the streamline, then integrating along the streamline. I note that the flow is not strictly baratropic or
irrotational and thus don’t expect the Bernoulli potential to the same for different streamlines. Figure
3.13 shows that the Bernoulli constant is conserved along the streamlines to within 3-4% for the 0.7M,
LX = 2×1030 erg s−1 indicating our simulations are extremely close to a true steady state.
Numerical Simplifications
In order to make any detailed calculation numerical tractable, simplifications must always be made. In
this case the main simplification was the coupling of the radiative transfer to the numerical fluid mechan-
ics. This was done by assuming that the thermal structure can be described in terms of local variables
only, (i.e. the ionization parameter and neutral column) and that the flow was in radiative equilibrium. In
order to be confident the parametrization of X-ray heating is correctly reproducing the temperature struc-
ture in the flow I have re-calculated the gas temperatures of the steady-state solution using MOCASSIN.
Figure 3.14 shows a comparison between the temperatures determined by the radiation-hydrodynamic
2This is also known as the Jacobi potential
3.5 Numerical tests 69
0 0.2 0.4 0.6 0.8 1
0
2
4
6
8
10
12
14 x 10
11
Relative Length Along Streamline
Ja
co
bi
 C
on
st
an
t (a
rbi
tra
ry 
un
its
)
Figure 3.13: Plot of the Bernoulli potential along various streamlines in the range 2-50AU, plotted against
the normalised streamline length. The variation in the mean value along each streamline is ∼ 3%
algorithm and those obtained using MOCASSIN. This shows that the errors are<15% for 50% of the grid
and errors <40% for 95% of the grid. In Figure 3.15 I show the comparison between the gas tempera-
tures determined using the algorithm and those obtained using MOCASSIN at various cuts in cylindrical
radii. It shows good agreement throughout the grid especially in the launch region of the flow (region
of large temperature gradient) and in the regions of the flow which are subsonic (red regions) where the
thermal pressure gradient can be a first order effect. Note that the disagreement between the temperatures
in the flow/disc transition region is an artifact of MOCASSIN, which imposes a hard transition between
the molecular and atomic zones (see discussion in Ercolano et al. 2009b), hence the errors quoted can be
considered an overestimate.
Throughout the modelling I assume that the gas is always in thermal equilibrium with the radiation.
This assumption can be tested by comparing the gained energy flux of the wind to the ionizing lumi-
nosity. Figure 3.16 shows a plot of specific energy for a primordial disc around a 0.7Mstar with an
X-ray luminosity of 2× 1030 erg s−1. Considering the streamlines around the median mass-loss value
(launching from 15-20AU) the total change in specific energy along these streamlines from launch (with
a negative specific energy of ∼ 1011 erg g−1) to exit from the grid (with a positive specific energy of
∼ 5× 1011 erg g−1) is ∼ 6× 1011 erg g−1. Given the grid boundary is far from the sonic surface this
can be considered close to the maximum gained energy flux of the wind. In order to compare with the
input energy rate (Ltot), I can compute the mechanical luminosity of the wind using Lmech ≈ M˙w×∆e/ρ ,
which evaluates to ∼ 1029 erg s−1. Thus it is clear that the energy gained in the wind is much less than
the input energy from the X-rays (2× 1030 erg s−1). For all flows (both the primordial and inner-hole
flows) I explicitly calculate the gained energy flux in the wind and find that its LX (>8%). Hence, the
assumption of thermal equilibrium is valid in this case and since the mass-loss rate scales linearly with
X-ray luminosity and is roughly self-similar then it is valid in all cases considered.
70 Chapter 3. Numerical Models
Figure 3.14: Histogram of cells with temperature errors less than a given value. Black solid line refers to
errors in cells in entire grid, red dashed line to errors from cells that are locally sub-sonic (in the rotating
frame).
Figure 3.15: Radial cuts showing temperature comparison for R=2,10,20AU. Solid line indicates tem-
peratures calculated by the ZEUS-2D algorithm, where the red solid line indicates where those cells
are locally subsonic (in the rotating frame) and in the photoevaporative flow. Blue stars represent the
temperatures determined by MOCASSIN.
3.5 Numerical tests 71
Radius (AU)
H
ei
gh
t (A
U)
 
 
10 20 30 40 50 60 70 80 90
10
20
30
40
50
60
70
80
90
lo
g 
|Sp
ec
ific
 En
erg
y| 
( e
rg 
g−
1  )
9
9.5
10
10.5
11
11.5
12
12.5
13
Figure 3.16: Figure showing the specific energy of gas particles throughout the grid. Streamlines are
plotted at 5% of the total mass-loss rate. The dark blue lane indicates the transition between a overall
negative specific energy (bound gas) and an overall positive specific energy (unbound gas). This clearly
shows that the total energy gained by gas launched into the wind is less than LX with a total efficiency
∼ 8% compared to the input X-ray energy.
Physical assumptions and model tests
Based on the theoretical understanding gained in Chapter 2, I will make some further assumptions about
the photoevaporation rates when I consider the disc’s evolution as a function of time. Namely, that the
photoevaporation rates are fixed by the X-ray luminosity rather than the evolving disc structure below,
and that I can neglect the EUV and FUV radiation fields. Given the theoretical description of the flow
predicts that the sonic surface controls the mass-flux, In Figure 3.17 I show the temperature at the sonic
surface as a function of cylindrical radii, compared to the temperature the simple model predicts from
that radius. Furthermore, I can understand why inside 1Rg the temperature at the sonic surface is below
the expected value and above at radii greater than 1Rg. As the maximum temperature to which the
X-rays may heat the gas is Tgas ∼ 2× 104 K, which is below the escape temperature inside 1Rg the
sonic surface occurs at a height of ∼ 1Rg irrespective of the radius it is launched from. While the
simple models assume that the centrifugal force is completely negligible by the sonic surface, Figure 3.5,
shows that this assumption becomes worse as the cylindrical radii increases, and explains why at radii
greater than 1Rg the sonic temperature is slightly higher than the predicted value. However, these results
indicate that the model described in Chapter 2 describes the basic physics necessary to understand X-
ray photoevparotion and its scalings with physical parameters. Figure 3.18 shows the density and
velocity structure of calculations for identical physical parameters but with varying disc structures, I
note that all models models where I vary the disc structure, either by forcing the discs to be adiabatic or
72 Chapter 3. Numerical Models
10−1 100 101
103
104
Radius [Rg]
Te
m
pe
ra
tu
re
 [K
]
Figure 3.17: The temperature at the sonic surface as a function of cylindrical radii (scaled by Rg) for 3
models with two different stellar masses (0.1 & 0.7M) and X-ray luminosities of (2×1029 & 2×1030
erg s−1). The open squares show the original 0.7M, 2×1030erg s−1 calculated by Owen et al. (2010),
the black crosses are those of the 0.1M, 2× 1029erg s−1 and the open circles are for the 0.7M, 2×
1030erg s−1 model calculated with at disc depleted in mass by a factor of 10 from the original calculation.
The black line shows the temperature expected from the order of magnitude calculations performed in
Chapter 2. I note all primordial disc calculations have sonic temperatures that follow the same loci.
Figure 3.18: Density and velocity structure for photoevaporative winds with same stellar parameters
(M∗ = 0.7M, LX = 2×1030 erg s−1) but varying disc structures. From left to right: Original calculation
by Owen et al. (2010); a reduction in dust temperature by
√
2; a reduction in the disc mass by a factor of
10. All figures shows very similar velocity and density structures in the wind.
3.5 Numerical tests 73
Radius [AU]
H
ei
gh
t [A
U]
 
 
0 2 4 6 8 10 12 14 16 18 20
0
5
10
15
lo
g 1
0(I
on
iza
tio
n F
rac
tio
n)
−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
Figure 3.19: Ionization structure of the X-ray wind around a 0.7Mstar with an X-ray luminosity of
2×1030erg s−1, the streamlines are equispaced in 2% of the cumulative mass-loss rate and the magenta
contour shows the region of 98% ionization fraction (i.e. roughly the penetration depth of EUV photons).
This shows that an X-ray driven wind is itself entirely optically thick to EUV photons and prevent them
from reaching the disc surface.
varying the D’alessio et al. disc structures used in the calculations show very similar behaviour and flow
topologies. Furthermore, the total mass-loss rates and profiles are fairly invariant to dramatic changes
in disc structure (including the adiabatic discs and more realistic variations) with total mass-loss rates
that range between 0.9− 1.7× 10−8M yr−1compared to the original 1.4× 10−8M yr−1for a X-ray
luminosity of 2×1030 erg s−1.
Impact of EUV radiation
As mention at the begining of this Chapter the input spectrum includes an EUV component. While the
initial radiative transfer calculations indicated that it was X-ray heating that dominates at the radii im-
portant for photoevaporation, the approximation that the heating is exclusively controlled by the X-rays
can be checked using radiative transfer calculations. Therefore, using MOCASSIN to determe the ioniza-
tion structure will highlight regions where the EUV is important. This result is shown in Figure 3.19,
which clearly indicates that the X-ray wind drives a flow that is itself optically thick to EUV photons.
Therefore, if the EUV was able to reach the disc then so would the X-rays. Thus the X-ray driven flow
will always prevent the EUV from driving a photoevaporative flow, and it is the X-rays that control the
heating and hence mass-loss rate from the surface of a disc iradiated with an EUV and X-ray spectrum.
3.5.2 Inclusion of ‘FUV’ heating
Our calculations using this extra ‘FUV-type’ heating does not affect the mass-loss rates obtained with a
value of 1.2×10−8M yr−1for the initial calculation with the same depleted FUV extinction coefficient
74 Chapter 3. Numerical Models
as Richling & Yorke (2000), a value of 1.1× 10−8M yr−1for the calculations with the very depleted
FUV extinction coefficient, and a value of 1.3× 10−8M yr−1for the mass depleted disc. In all sim-
ulations the sonic surface occurs in the X-ray heated regions and therefore confirms the simple ideas
presented in Chapter 2, as well as demonstrating that the photoevaporation rates are only set by the X-
ray physics alone and independent of extra FUV heating, as well as large changes in the underlying disc
structure. Figure 3.20 shows the density, velocity structure the maximum FUV penetration model (i.e.
the mass depeleted disc) including ‘FUV-type’ heating. This shows as, expected that the sonic surface
occurs in the X-ray heating region, while the FUV heated flow is confined to a layer between the bound
dust heated disc and the X-ray heated flow. Furthermore I find that even though the density and temper-
ature structure of the ‘FUV’ heated region varies between model runs, in all cases the thermal structure
of the sonic surface is as shown in Figure 3.17. Therefore, the FUV layer must be adjusting its structure
in order to ‘feed’ the X-ray heated region with the correct mass-flux, and satisfy the hydrodynamical
conditions at the sonic surface. Therefore, extra heating below the X-ray heated region has little affect
on the photoevaporation rates in the range R∼ 1−100AU .
The structure of the disc shown above does allow a possibility of driving an FUV flow at large radii.
Since the height of the extra heated region increase with radius, I cannot rule out a thermally driven
flow through this ‘wedge’ below the X-ray heated region. Such a flow could then escape at large radius,
and undergo its sonic transition when the temperature of the X-ray and FUV heated region become
comparable. While this will have no effect on the X-ray heated flow and hence photoevaporation rates
from this region, simple estimates based on the column of FUV heated gas at R∼ 100 suggest mass-loss
fluxes through this wedge of ∼ 10−8M yr−1are sustainable. While the simulations show no ‘wedge’
flow at small radius, gas does leave the grid sub-sonically in the FUV heated region. Due to the use
of outflow boundary conditions on this grid boundary, it is impossible to tell if this flow is real or an
artefact of outflow boundary conditions (which are only exact for super-sonic flow). Therefore, further
conjecture on the role or possibility of these FUV ‘wedge’ flows is not possible without a more detailed
investigation.
3.6 Discussion
As described below, in order to make these calculations feasible I have needed to make a series of
simplifications. In the previous section I described in detail the tests performed in order to ensure the
the model described in Chapter 2 provided an accurate description of the underlying physics of the
problem, as well as ascertaining the accuracy to which I may trust the numerical results presented in
this calculations. These test have shown that model provides an accurate description of the physics of
photoevaporation, namely that the sonic surface occurs in the X-ray heated region and that the X-ray
physics determines the mass-flux through the sonic surface. Therefore, variations in disc structure and
inclusion of extra heating mechanisms should not affect the mass-loss rates to first order and this is
indeed what is found. Dramatic variations in disc structure only resulted in a maximum variation of
∼ 35% in the mass-loss rates. Furthermore, I have shown that the numerical simplifications do not effect
the derived mass-loss rates and are confident that the results are accurate enough (i.e. error in the mass-
loss for an individual X-ray luminosity is much less than the expected spread in mass-loss rates for the
entire X-ray luminosity range) for the study disc evolution and I can proceed with freedom in following
3.6 Discussion 75
Radius [AU]
H
ei
gh
t [A
U]
 
 
0 10 20 30 40 50 60
0
10
20
30
40
50
60
lo
g 1
0(D
en
sit
y) 
[g 
cm
−
3 ]
−22
−20
−18
−16
−14
−12
50
km/s
Figure 3.20: Velocity and Density structures of winds including an extra ‘FUV’ like heating mechanism,
the arrows show the velocity in the wind and the black lines show regions where the extra heating mecha-
nism is dominating over the X-rays and dust and the majenta contour shows the sonic surface. The model
shown is the depleted disc with the extremely low FUV extinction allowing for maximum penetration of
the underlying disc.
the evolution of discs undergoing photoevaporation.
3.6.1 Final Clearing of the Disc
Figure 3.10 shows the evolution of the ‘quasi steady-state’ simulation with time, during the first 300years
of evolution the photoevaporative flow slowly strips of gas from the inner edge of the disc (I note again
that as there is no viscosity in the disc, so that the inner disc region is not resupplied with material, and
the erosion time-scale is quicker than would be expected from a viscous disc model).
Since the inner streamline topology for inner hole flows is fixed, there is no requirement that the total
column density along the mid-plane to the bound disc is 1022 cm−2, in fact the column in the flow scales
as R−1hole and is always much less than 10
22 cm−2. Therefore, the X-ray absorbing column is a warm,
bound X-ray heated layer (typically at T ∼ 200−400K) behind the flow, separated by an entropy discon-
tinuity (which can be thought of as a contact discontinuity) and can be identified in all the simulations
with inner holes in the disc. However, in the case of simulations around the 0.1M star this region
eventually dominates the dynamics. In these simulations, as the hole grows the layer of warm bound
X-ray heated gas grows in radial thickenss, ∆. Once the hole size grows to Rhole ∼ 10AU, this region
becomes large enough that it can push the remaining disc out to large radii, and is completely cleared
on the dynamical time-scale (∼100 years at 10AU). This was not seen in the simulations calculated for
the 0.7M star; the reason for this is since the disc has a lower mass and lower temperature for the low
mass case, the pressure in the disc’s mid-plane is considerably lower than the higher mass case. Thus the
density in the warm bound region is correspondingly lower, meaning that the warm bound region needs
a larger ∆ in order to maintain a column density of 1022 cm−2.
76 Chapter 3. Numerical Models
10
1022
1023
Inner Hole Radius [AU]
Co
lu
m
n 
[cm
−
2 ]
Figure 3.21: Column density from the central star to the edge of the grid along the disc’s mid-plane, as a
function of radius to the X-ray ‘dark’ disc.
For the disc to become dynamically unstable and be pushed out by this warm, bound X-ray heated
region, we simply require that the radial pressure gradient in this region grows to become similar in size
to gravity (i.e. ∇P/ρ ∼ GM∗/R2), which is simply stating HR ∼ R where HR is the radial pressure scale
length of the X-ray heated disc. This is a necessary, but not sufficient condition as the pressure gradient
in this X-ray bound region will be large whatever its size. When this X-ray warm, bound region is small it
must be in dynamical balance with the photoevaporative flow and the dust heated disc. The time-scale on
which it can obtain this equilibrium is the sound crossing time of the region tsc ≈ ∆/cs. When ∆/H 1,
this region can adjust to dynamical equilibrium quickly and the disc/flow system is stable, even though
the pressure in the region is significant. However, at the point when ∆ ∼ H this region cannot rapidly
obtain dynamical equilibrium. Therefore, this region will expand vertically on the same time-scale. This
vertical expansion can clearly be seen in the time series plots, and as it expand the mid-plane density
drops resulting in higher temperatures. Once, the temperature of the mid-plane is increasing on the
same time-scale as this expansion a runway process will occur. As the disc cannot adjust rapidly to
the increasing mid-pane temperature and will continue to expand, and increase its temperature further.
Eventually, a high pressure region develops at the edge of the dust heated disc that pushes the remaining
disc material away rapidly on the dynamical time-scale. The fact we have seen this process only in a
couple of simulations, with the same initial disc structure, mass, but different X-ray luminosities does
not allow us to generalise the situation to all masses and disc structures. However, plotting the column
from the central star to the edge of the simulation grid (which will be identical to the column to infinity,
since the disc’s column in dominated at small radii), shown in Figure 3.21 is instructive. This shows the
disc transitions from slow photoevaporative erosion, to rapid hydrodynamical clearing at a total column
of N ∼ 1023 cm−2, ten times larger than the penetration depth of the X-rays, this is indicative that this
3.6 Discussion 77
processes relates to the fraction of the disc penetrated by the X-rays. Since all the 0.7M disc possess
high mid-plane pressures ∆ < H in all cases, and there is no evidence of this process beginning in any
simulations. Furthermore, the simulations which differ by an order of magnitude in X-ray luminosity
show the same processes of clearing at the same time-scales, indicating that the process may be directly
insensitive to X-ray luminosity.
It is worth mentioning that while the simulations have allowed us to gain a macroscopic understanding
of this process, because the simulations were designed to study the photoevaporative process (which
occurs in optically thin X-ray heated gas) whereas this process occurs in the gas that is transitioning from
the optically thick to thin regime, a column cut-off will not accurately reproduce the thermal structure of
this region. Furthermore, as the gas is considerably cooler (hundreds to thousands rather than thousands
to tens of thousands K as in the photoevaporation calculations) I cannot neglect the affect molecular
cooling will have, along with the added affect of FUV heating (which may or may-not be important
depending on the dust properties at the mid-plane of the disc at large radii). A disc with dust that is
ISM like at the mid-plane will be heated by X-rays first; however, if the dust is significantly depleted
this process may begin earlier than the X-ray alone calculations predict as the FUV may penetrate larger
columns of dust depleted gas than the X-rays. Furthermore, the heated regions appear to be unstable to
large scale convective modes and therefore, full 3D calculations that include X-ray and FUV radiation
along with dust evolution would be required to accurate study the evolution of such a process and are
certainly beyond the scope of this work.
3.6.2 Comparison with previous work
Solutions to the photoevaporative flow problem have evolved over the years and while this work has
shown that it is predominately driven by the X-rays, it is worth discussing the differences between the
new solutions presented in this work and the previous calculations. Either with pure EUV or X-ray
radiation, or with Ercolano et al. (2009b) who considered the combined effect of X-EUV radiation.
The predicted mass loss rates due to pure EUV photoevaporation have been accurately determined as
1.1×10−10Myr−1 (Font et al. 2004) for a 0.7M star and an ionizing flux of 1041s−1. Also Alexander
et al. (2006a) found that when the inner hole first forms, the mass loss rate rises by about a factor ten and
then increases mildly as the hole grows according to R1/2hole.
As discussed in Chapter 1, previous estimates of X-ray photoevaporation rates to be found in the
literature are qualitatively different from the values presented in the present work since they are not
based on hydrodynamic calculations. Alexander et al. (2004) estimated mass-loss rates per unit area at
∼ 20 AU which were similar to those obtained in this work. Ercolano et al. (2009b) argued for similar
surface mass-loss rates as those of Alexander et al. (2004), but estimated a much larger launching regions
(i.e.over a region 5-50AU). They therefore concluded that X-ray photoevaporation would dominate over
EUV photoevaporation at all but the lowest of X-ray luminosities. The estimates of Gorti et al. (2009)
suggested much lower mass loss rates than the EUV case, implying that X-rays would only have an
indirect effect on disc dispersal. However, as discussed in Chapter 1 this result is a consequence of the
rather unusual and certainly unphysical method for estimated photoevaporative mass-loss rates.
The mass loss rate of 1.4×10−8Myr−1 that I obtain from our radiation hydrodynamical calcu-
lations can be compared to the estimate of Ercolano et al. (2009b) for the same input parameters
78 Chapter 3. Numerical Models
0 10 20 30 40 50 60 70
−5
−4.5
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
Radius (AU)
lo
g 
(S
urf
ac
e M
as
s−
Lo
ss
 R
ate
) (
arb
itra
ry 
un
its
)
Figure 3.22: The surface mass-loss rate for the X-EUV wind (solid black line) obtained in this work
compared to the EUV wind (dashed blue line) (Font et al. 2004). I have scaled the peaks of the profile to
be the same value in order to compare the relative broadness of the different profiles. Clearly this shows
the X-EUV profile is much broader than the EUV profile.
(model FS0H2Lx1 of ECD09) of 6.7×10−9 Myr−1 and the theoretical prediction in Chapter 2 of
8× 10−9M yr−1. This difference arises from the fact that Ercolano et al. (2009b) assumed that the
gas escaped at the sound speed at the point at which the internal energy of the gas was equivalent to the
escape potential. It is nevertheless reassuring that these values are of a similar order of magnitude. I also
find that the mass-loss occurs over a large radial range of 1-70AU, in contrast to the EUV case where it
is steeply peaked about one characteristic radius (for a 0.7M star this would be 6.2AU). Given the mass
loss rates obtained and the monotonic temperature-ionization parameter relations found (see Figure 3.2),
it is clear in this case that X-ray photoevaporation will be a dominant effect in disc evolution.
3.7 Conclusion
In this Chapter I have performed detailed radiation-hydrodynamic calculations, in order to determine
accurate photoevaporative mass-loss profiles and rates, along with testing the physical model presented
in Chapter 2. I have found that as described in Chapter 2 it is the sonic surface in the X-ray heated region
and the theory presented in Chapter 2 provides an accurate picture for understanding the underlying
physics involved in photoevaporation from discs around young stars. The main findings of this Chapter
are:
1. The algorithm described in this Chapter provides an accurate and computationally feasible method
for including X-ray heating of protoplanetary discs in a hydrodynamic scheme.
2. The derived mass-loss rates scale approximately linearly with X-ray luminosity and are indepen-
dent of the mass of the central star.
3.7 Conclusion 79
3. The mass-loss rates are insensitive to changes in disc structure and the inclusion of extra ‘FUV’
type heating as well as EUV and X-ray radiation fields and it is the X-ray properties at the sonic
surface alone that to first order determine the mass-loss rates.
4. The mass-loss profiles from primordial discs are very broad, including mass-loss over a large radial
range 0.1−10Rg, much broader than the previous EUV only calculations.
5. The mass-loss rates from discs with inner holes are approximately independent of inner hole size
and mass and scale linearly with X-ray luminosity.
6. Once the radial pressure scale height of a disc with an inner hole becomes optically thin to X-ray
irradiation the disc becomes dynamically unstable and is rapidly swept from inside out to large
radius by the thermal pressure gradient resulting from the X-ray heating inner rim of the disc.
In this and the previous Chapter I have set out the underlying physics of photoevaporation from
discs exposed to high energy radiation that includes X-rays, alone with performing accurate nu-
merical calculations that have determined the mass-loss profiles as well as total mass-loss rates.
I have confirmed that these are insensitive to the changes in disc structure that will occur during
the Myrs of disc evolution and therefore are ready to proceed to calculations of disc evolution that
include photoevaporation in the next Chapter.

4
Protoplanetary disc evolution and dispersal
IT is now time to turn our attention to the evolution of protoplanetary discs, which are undergoingthe dual action of viscosity and photoevaporation. As I have argued throughout Chapters 2 & 3, thephotoevaporative mass-loss from a primordial disc will be approximately independent of the current
disc structure, and set by the X-rays. Therefore, I use the mass-loss profiles derived from the numerical
calculations of X-ray photoevaporation described in the previous Chapter. In the introduction, I argued
that viscous evolution coupled with photoevaporation was one of the currently preferred ideas for the
destruction of protoplanetary discs, at its most basic level reproducing the two-time-scale phenomenol-
ogy and concept of inside out clearing indicated by observations of ‘transition’ discs. In this work, I
have been able to characterise the mass-loss from discs in the full range of parameter space observed.
Therefore, a natural question arises: is photoevaporation the dominant mechanism for disc destruction?
This is the question I set out to discuss in this Chapter. To do this I will follow the evolution of a
synthetic population of evolving discs, which can then be compared to the observations. Even though
the theory of photoevaporation can be extended to all masses, at the current time the observations of disc
evolution at lower masses are not complete enough to make a good comparison at this stage, and there
is still much debate as to their interpretation (see Ercolano et al. 2011a,b and Currie & Sicilia-Aguilar
2011, for contrasting views). Therefore, I shall restrict the discussion to those observations which are
best understood, i.e. solar type stars (using observations in the range 0.5-1.0M to give a roughly even
spread about the 0.7M photoevaporation models calculated earlier), and only discuss observations from
this range of stellar mass.
4.1 X-ray Luminosity Function
As described previously, the X-ray luminosity (LX ), is the crucial input for the photoevaporation model,
where I take the X-ray luminosity to span the range 0.1−10keV. Since the choice of LX fully describes
81
82 Chapter 4. Discs evolution and dispersal
the mass-loss rate due to photoevaporation in a given system, we must describe how the X-ray luminosity
of stars varies within a population of young stars.
It has been known for some time that there is a large scatter in the X-ray luminosity of T Tauri stars
for a given stellar mass (and bolometric luminosity). While there have been no long term studies into X-
ray variability around young stars, they are known to ‘flare’ both as CTTS (Feigelson & DeCampi 1981)
and WTTS (Walter & Kuhi 1984); however, the activity-rotation relationship is similar to other active
stars indicating that the scatter is intrinsic rather than solely arising from variability (Gu¨del et al. 2004).
Therefore, I have used the data for the Taurus cluster (Gu¨del et al. 2007) and the Orion Nebular cluster
(Preibisch et al. 2005) to build cumulative X-ray luminosity functions1 (XLFs) for all pre-main sequence
stars (including both CTTs and WTTs) in the 0.5 to 1 M range. The data for these two regions shows
good agreement at low luminosity, but they differ at high luminosities, in the sense that the Orion data
contains a higher fraction of sources with high LX . The difference is most probably not intrinsic to the X-
ray properties of these two regions, but due to the different treatment of strong flares in the two samples.
Strong flares were excluded in the Taurus data, in contrast to the Orion sample where luminosities were
averaged over the whole observational period. Strong flares are relatively rare (although not sufficient
to account for the different luminosity functions), and Albacete Colombo et al. (2007) found that non-
flaring sources have a median kbT = 2.1± 0.3KeV, compared to flaring sources with kbT = 3.8KeV.
Therefore, due to their higher X-ray temperature flares tend to emit most of their radiation in the hard
X-ray region (as shown by observations of objects in a ‘flare’ state compared to in a ‘quiescent’ state, e.g.
Imanishi et al. 2001), where the thermal impact is low. For this reason, the Taurus sample is the most
appropriate for use in the photoevaporation model, since it should provide a better approximation of the
quiescent, and therefore softer X-ray luminosity. Therefore, I adopt the Taurus XLF for the use in the
population synthesis model. I assume that the X-ray luminosity function remains invariant throughout
the stars’ pre-main sequence evolution. While it is known that the median of the stellar X-ray luminosity
function does in fact decrease with age due to stellar spin down (e.g. Hemplemann et al. 1995; Gu¨del
et al. 1997; Gu¨del 2004), the evolution of the X-ray luminosity during the disc dispersal phase is much
smaller (Gu¨del 2004) and a recent study of the evolution of X-ray luminosity as a function of time
(Ingelby et al. 2011) concluded that there was no evidence for a decrease in X-ray luminosity for the first
10Myr. Certainly, any evolution of the X-ray luminosity for ages up to several tens of Myrs is smaller
than the observed spread in X-ray luminosities at ∼ 1Myr. Furthermore, there is observational evidence
that the X-ray luminosities of CTTs and WTTs are systematically different, with WTTs in general being
more luminous than CTTs (e.g. Neuhauser et al 1995; Stelzer & Neuhauser 2001; Flaccomio et al. 2003;
Preibisch et al. 2005). This has lead to a discussion of X-ray emission being ‘disturbed’ by accretion, in
terms of either X-ray absorption in accretion columns (Gregory et al. 2007) or confinement of the X-ray
producing corona in accreting systems (Preibisch et al. 2005). As discussed in the introduction, Drake
et al. (2009) suggested that X-rays may modulate accretion through photoevaporation (something they
called ‘photoevaporation starved accretion’). Such a scenario may be able to account for the difference in
the observed X-ray luminosities of CTTs and WTTs since more luminous stars will lose their discs first,
and is something I will aim to test within the framework of this population synthesis model. Therefore, in
order to assess the effect of ‘photoevaporation starved accretion’ in explaining the X-ray observations I
1Where the X-ray luminosity is taken to be in the range 0.3-10keV for the Taurus sample and 0.5-8keV for the Orion
sample, consistent with my definition which includes both the ‘soft’ and ‘hard’ component.
4.2 Photoevaporating Viscous Discs 83
1028 1029 1030 1031 1032
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
X−ray Luminosity [erg s−1]
Cu
m
ul
at
ive
 P
ro
ba
bi
llit
y
Figure 4.1: Cumulative X-ray luminosity functions for the Taurus (black solid line) and Orion (red
dashed line) clusters for solar type stars with mass in the range [0.5,1.0] M, including both WTTs and
CTTs.
adopt the null stance: the X-ray luminosity of an individual young stellar object (YSO) remains constant
in time as individual disc models evolve from CTTs to WTTs.
4.2 Photoevaporating Viscous Discs
As shown in the introduction, the evolution of the surface density of a photoevaporating and viscously
evolving disc can be described in one dimension, using the formalism of Lynden-Bell & Pringle (1974):
∂Σ
∂ t
=
3
R
∂
∂R
[
R1/2
∂
∂R
(
Σν(R)R1/2
)]
− Σ˙w(R, t) (4.1)
where ν(R) describes the viscosity term and Σ˙w(R, t) represents the mass-loss due to photoevaporation
calculated by differentiating a functional fit2 to the cumulative mass-loss profile, shown as the black
line in Figure 3.6. In the introduction I argued that the combination of viscous evolution (first term on
the RHS of Equation 4.1) and photoevaporation (second term on the RHS of Equation 4.1) naturally
reproduces a two time-scale clearing, with clearing beginning when the accretion rate through the disc
becomes comparable to the mass-loss rate due to photoevaporation. While much of this thesis has been
dedicated to the detailed evaluation and understanding of the photoevaporation term Σ˙w, little attention
has been paid to the magnitude of the viscosity term. The magnitude, scaling, and in some regions of
the disc perhaps even the origin of the viscosity is still poorly understood. At first sights it might seem
2The form of the fits is shown in Appendix A
84 Chapter 4. Discs evolution and dispersal
like we are doomed to fail, since without this theoretical understanding how can we proceed? However,
since photoevaporation has little effect on disc evolution when the mass-loss rate is below the accretion
rate, the disc’s evolution is entirely controlled by the viscosity. Thus as we shall see below, if we restrict
ourselves to a single mass it is possible to use the current observations to construct a model dependent
description of the viscosity, which I can use to follow the evolution of the disc population.
4.2.1 Viscosity Law and Initial Conditions
In order to proceed in determining the magnitude of the viscosity I must first make some observationally
and theoretically motivated assumptions. As mention above, during the majority of the disc’s evolution
we can consider the effects of photoevaporation to be negligible. Hartmann et al. (1998) showed that the
observations of the accretion rate as a function of time were consistent with a viscosity law that scaled
linearly with radius, i.e. ν ∝ R. Therefore, currently a simple power law description is the best we can do
for the viscosity law, and as discussed in the introduction, observations indicate I should adopt a viscosity
law of the form ν = ν0R, where ν0 = αc2s/Ω, where I always evaluate α at 1AU. Furthermore, I adopt
the zero-time similarity solution of Lynden-Bell & Pringle (1974), for which the initial surface density
distribution takes the form:
Σ(R,0) =
Md(0)
2piRR1
exp(−R/R1) (4.2)
These initial conditions, together with the viscosity parameter α and the X-ray luminosity fully determine
the evolution of the disc through Equation 4.1.
In order to determine a model dependant viscosity parameter it is necessary to construct a ‘null model’
of viscous evolution without photoevaporation, that when combined with the observed X-ray luminosity
function can explain the observed decline in disc fractions with age. In effect I am asking whether
there is a universal set of disc parameters which can explain the variation in disc lifetime from cluster
to cluster, purely in terms of the observed spread in X-ray luminosity. As discussed above, the basic
principle of all photoevaporation models is that discs should evolve viscously, hardly noticing the effects
of photoevaporation, until the mass accretion rates in the discs have fallen to a value that is comparable
to the photoevaporation rate, at which point the remaining disc material is rapidly cleared.
Viscous evolution alone with ν ∝ R predicts that accretion rates evolve as:
M˙∗(t) = M˙∗(0)
(
1+
t
tν
)−3/2
(4.3)
where tν is the viscous time-scale at R1. This evolution should therefore be observed in discs before
photoevaporation sets in. We can equate the fraction of disc-bearing pre-main sequence stars at a given
time ( fd), with the fraction of stars in the X-ray luminosity function that have luminosities less than a
cut-off X-ray luminosity Lc( fd). In order for objects with X-ray luminosities equal to Lc to be about to
lose their discs, the viscous accretion rate must be equal to M˙w(Lc) at this point. Therefore, for each
cluster shown in Figure 4.3 (the sample compiled by Mamajek 2009), this method determines the lowest
possible accretion rate a star can have without loosing its disc by photoevaporation (at that time). Or
for a single set of initial conditions this method will extract the accretion evolution of a star that is
evolving under the action of viscous evolution alone, with no photoevaporation i.e. the ‘null’ model
allowing the determination of the viscous initial conditions. The result is shown in Figure 4.2, where
4.2 Photoevaporating Viscous Discs 85
0.1 1 10
10−10
10−9
10−8
10−7
Time [Myr]
Ac
cr
et
io
n 
R
at
e 
[M
   y
r−1
]
⊙
Figure 4.2: Accretion rates as a function of time for the ‘null’ disc model shown as the blue open circles.
The errors in accretion rate are estimated from the
√
N errors in the disc fractions. The solid line shows
a fit of Equation 4.3 used to determine suitable values of Md(0) and tν . The triangles show the results
for a X-ray luminosity function with a spread about the median 5 times larger (black ‘up’) and 5 times
smaller (red ‘down’) than observed.
each point represents the current accretion rate cut-off in a cluster implied by X-ray photoevaporation.
The results for the disc fractions are scaled assuming an initial close binary fraction of 14% (consistent
with the observational value of 10-20% Kraus et al. 2009), by considering that the 0.3Myr old cluster
NGC 2024 (Haisch et al. 2001a), which shows a disc fraction of 86%, is too young for any disc to have
been destroyed by photoevaporation or planet formation (e.g. Alexander & Armitage, 2009), but only
through binary interactions. The solid line in the plot represents a suitable fit of Equation 4.3 to the data,
from this fit we can extract an initial accretion rate of M˙∗(0) = 5× 10−8M yr−1and a viscous time of
tν = 7×105yr. From these two values we can calculate an initial disc mass of Md(0) = 0.07M, which
is similar to the canonical value (10% of the stellar mass) at which viscous angular momentum transport
takes over from self-gravity.
Along with giving us appropriate initial conditions for our disc population model, the above also
provides a stringent test of the hypothesis that X-rays are key to disc evolution and dispersal. If the
X-rays were not the dominant dispersal mechanism, there is no a priori reason to expect a ‘null’ model
(constructed only using knowledge of the X-ray luminosity function and observed disc fractions) to
reproduce a plausible evolution of the accretion rates seen in CTTs, both in terms of the time exponent
(in Equation 4.3) and its initial value. In fact, increasing or decreasing the spread about the median of
the Taurus X-ray luminosity function by a factor of∼5 or greater makes a fit of Equation 4.3 to the ‘null’
model difficult, as shown in Figure 4.2. Although this agreement could be fortuitous, it is reassuring that
the X-ray luminosity function, disc lifetimes and accretion histories are consistent with the hypothesis.
Furthermore, in order to uniquely specify the viscous evolution we must pick suitable values of α
86 Chapter 4. Discs evolution and dispersal
0 2 4 6 8 10 12 14 16 18 20
0
10
20
30
40
50
60
70
80
90
100
Time [Myr]
D
is
c 
Fr
ac
tio
n 
[%
]
Figure 4.3: Primordial disc fraction as a function of time (solid black line) from our XPE population
calculated using 500 disc models with the mass-loss rate determined by randomly sampling the Taurus
X-ray luminosity function, all discs evolve from a single set of initial conditions. The points are observed
disc fractions compiled by Mamajek (2009). Model disc fractions have been scaled to account for disc
destruction by close binary interaction. Note that the structure of the black line reflects the structure in
the X-ray luminosity function.
and R1, which in turn specifies ν0. While any combination of R1 and α that give the required viscous
time will reproduce the same ‘null’ viscous model (i.e. non-photoevaporating), a disc evolution model
that includes photoevaporation mildly breaks this degeneracy. By performing a fit (by eye) of a viscously
evolving photoevaporating disc population to the disc fractions used to derive the ‘null’ model I obtain
values of α = 2.5× 10−3 and R1 = 18AU, although due to the large scatter in the disc fractions and
only mild breaking of the α , R1 degeneracy, errors on these values are estimated to be around a factor
of 2-3. Figure 4.3 shows the result of this exercise compared to the disc fractions compiled by Mamajek
(2009), where the synthetic disc population used the single set of initial conditions: Md(0) = 0.07M,
α = 2.5×10−3 and R1 = 18AU, and shows excellent agreement with the observations.
4.3 A population of viscously evolving, photoevaporating discs
I have used initial conditions derived in the previous section to construct a population synthesis model
for the evolution of discs dominated by viscosity and photoevaporation. In order to produce a population
with sufficient statistical accuracy to compare to observations, I calculate the evolution of 500 discs,
where the X-ray luminosity is the only variable between each model, which is determined by a random
sampling of the Taurus X-ray luminosity function.
4.4 Results and Discussion 87
Computing the disc’s evolution
Since Equation 4.1 is a linear inhomogeneous diffusion equation, it is in principle easy to construct an
analytic solution to the problem using a Green’s function method. Such an approach involves indefinite
integrals of uncommon transcendental functions (which would need to evaluated numerically), and is
therefore much more uninstructive than would first be imagined. As such, it is much simpler to just solve
Equation 4.1 numerically from the outset, a method adopted by Clarke et al. (2001) and Alexander et al.
(2006b). I integrate the diffusion equation forward in time using a standard first order explicit scheme, on
a grid uniformly spaced in R1/2, with 1000 grid cells in the range [0.0025,2500] AU, sufficient to resolve
the disc’s evolution (e.g. Alexander et al. 2006b). In order to ensure strict flux conservation, I use central
differencing where fluxes are evaluated on cell boundaries and scalars are evaluated at cell centres. I use
zero torque boundary conditions (i.e. set Σ = 0 in the boundary cells), and in order to ensure numerical
stability in time, I use a time-step of:
∆t = min
[(
1
4
)
R∆R
3ν
]
(4.4)
which is half the time-step formally required for stability (von-Neuman time-step). Furthermore, since
the addition of the photoevaporative sink term can, in practice, allow the numerical solution to become
negative, I use a numerical base surface density of Σ = 1× 10−20 g cm−2. The numerical fits to the
photoevaporation profiles are shown in Chapeter 3 and I switch between the primordial and inner-hole
profile after the gap has opened and the mid-plane column of the inner disc reaches a level lower than
1022 cm−2.
I use this method to follow the evolution of each disc. The evolution of the surface density for the
median model (model with median LX ) is shown in Figure 4.4. All discs show qualitatively similar
behaviour to this disc. The disc evolves unaffected by photoevaporation for the majority of its life.
However, when photoevaporation becomes important it begins to flatten the surface density in the region
of R ∼ 10AU indicating photoevaporation is starving the inner disc of material. Once the accretion rate
is no longer capable of sustaining a flow all the way to the star, a gap rapidly forms. While the inner disc
drains, the inner edge of the outer disc is slowly eroded to larger and larger radii. Once the inner disc
has entirely drained the outer disc is exposed to direct photoevaporation and eroded to larger and larger
radii.
4.4 Results and Discussion
As mentioned above the qualitative evolution of all discs is identical: viscous evolution followed by gap
opening and inner disc draining, then outer disc clearing. While this phenomenological aim was the basis
of constructing the photoevaporation model, the calculation of a population synthesis allows much more
direct comparison with disc evolution indicators. In the previous section I showed conclusively that the
spread in X-ray luminosities (and hence photoevaporation rates) was sufficient to explain the spread in
disc lifetimes, without the need for large variations in initial conditions.
One of the other arguments for a spread in initial conditions is the spread in accretion rates seen for
a given age (e.g. Hartmann et al. 1998). In Figure 4.5 we show the predicted evolution of accretion rate
88 Chapter 4. Discs evolution and dispersal
0.1 1 10 100
10−6
10−4
10−2
100
102
104
106
Radius [AU]
Su
rfa
ce
 D
en
sit
y 
[g 
cm
−
2 ]
Figure 4.4: The evolution of the disc’s surface density during the disc clearing phase. The first line shows
the zero time surface density profile, the next shows the profile at 75% of the discs lifetime (∼ 3.5Myr)
and the remaining lines show the surface density at 1% steps in disc lifetime.
with time for individual disc models (each line shows the evolution of accretion rate with time for one
disc model, for a given value of the X-ray photoevaporation rate). This clearly shows that a spread in
accretion rates at late times > 1Myr does not necessary require a spread in initial conditions, as some
authors have suggested is necessary outside this framework (e.g. Armitage et al. 2003; Alexander &
Armitage, 2009). However, there is observationally a spread in accretion rates seen at early times<1Myr.
This variability cannot be fit with a single set of initial conditions, since photoevaporation has had no
time to act on the disc. The failure to match the spread in accretion rates at early times < 1Myr may
not be surprising, since at early times the angular momentum transport mechanism may be dominated
by self-gravity and accretion may be episodic (e.g. Lodato & Rice, 2005). In reality the zero time point
in our models corresponds to the point at which the transition from a self-gravitating to a viscous disc
occurs, as indicated by the determination of Md(0) = 0.1M∗. This may explain, why our initial accretion
rate is significantly lower than some of the accretion rates measured for the youngest objects (Hartmann
et al. 1998).
Furthermore, better agreement at early times could easily be obtained by assuming a range of initial
surface density profiles. Indeed our choice of a self-similar surface density distribution at zero time has
no physical basis, only convenience (in fact it would be extremely surprising for discs to be born with
the surface density distribution of the zero time similarity solution of Equation 4.1 given viscous angular
momentum transport is not a key process during the disc formation stage). I emphasise that any initial
surface density distribution with the same initial viscous parameters Md(0), tν and α will tend to the
same evolution after a few viscous times (e.g. Lynden-Bell & Pringle, 1974). For this reason, I do not
attempt to explain disc evolution at early times here, since we are mainly interested in the question of
disc dispersal, for which only the viscous evolution phase at > 1Myr is relevant.
4.4 Results and Discussion 89
Figure 4.5: Each solid line represents the evolution of accretion rate as a function of time for an individual
disc model undergoing X-ray photoevaporation. All models have the same initial conditions, the X-ray
luminosity is the only variable.
Certainly it is somewhat implausible to expect there to be no spread in initial conditions; however, we
can certainly rule out the large spreads in initial conditions (>1dex) as required by some other models.
Furthermore, the agreement between the disc fractions and spread in accretion rate is certainly a good
indication that the dominant dispersal mechanism is indeed photoevaporation. The accuracy of the ‘null’
model is certainly reassuring in supporting this hypothesis. However, the most direct tests of the pho-
toevaporation model will come from comparing X-ray luminosities to disc evolution parameters, since
if X-rays had no implications on disc evolution, then there is no explicit reason for X-ray luminosity to
correlate with disc evolution parameters.
4.4.1 Photoevaporation starved accretion
As discussed in the Introduction, Drake et al. (2009) suggested that coronal X-rays suppress the accretion
flow onto young solar-type stars through the driving of a photoevaporating wind. This photoevaporation
starved accretion phase can explain the tentative negative correlation between mass accretion rate and
stellar X-ray luminosity reported by Drake et al (2009). Moreover the reduction in disc lifetime in
strong X-ray sources can explain the observation that the X-ray luminosities of accreting T Tauri stars
are systematically lower than those of non-accretors.
The median photoevaporation model in Figure 4.4 clearly shows that there is indeed a phase in the
disc evolution (before the opening of the gap) where the effects of this ‘starving’ are apparent in the radial
90 Chapter 4. Discs evolution and dispersal
0
2
4
log(Surface Density) [g cm−3]
lo
g(S
urf
ac
e D
en
sit
y) 
[g 
cm
−
3 ]
0.1 1 10 100
10−11
10−10
10−9
10−8
Radius [AU]
Ac
cr
et
io
n 
R
at
e 
[M
   y
r−1
]
⊙
Figure 4.6: Surface density (top) and accretion rate (bottom) radial profiles. The solid line represents a
disc undergoing no photoevaporation and the dashed line represents the median disc model ≈ 0.5Myr
before a gap opens in the disc due to XPE.
dependence of the surface density, and hence the accretion rate. In Figure 4.6 we compare the accretion
rate and surface density profiles of the median disc model undergoing photoevaporation, 0.5Myr before
the gap opens against those of a disc which is only subject to viscous evolution. Inside 70 AU the
accretion rate drops before it reaches the star compared to the standard case where the accretion rate tends
to a constant value throughout the entire disc. The photoevaporation starved accretion lasts for∼20-30%
of the disc lifetime in the X-ray model, with significant consequences for global disc evolution.
The negative M˙-LX correlation reported by Drake et al (2009) is a simple consequence of the fact
that discs with higher X-ray luminosities produce more vigorous winds, causing the disc’s accretion
rates to be lower than those for discs with less vigorous winds. Thus, a negative M˙-LX correlation is
expected for clusters with a relatively narrow age range. However, this effect is counteracted by the fact
that discs with lower X-ray luminosities take longer to evolve, spending more time at lower accretion
rates compared to high luminosity objects. Therefore, if the X-ray luminosity was compared to accretion
rates for the entire population’s lifetime, in clusters with very large age spreads, a positive correlation
4.4 Results and Discussion 91
−10.5 −10 −9.5 −9 −8.5 −8 −7.5 −7
log(Accretion Rate) [M  yr−1]
 
 
29
29.5
30
30.5
31
 
 
 
 
−11 −10.5 −10 −9.5 −9 −8.5 −8 −7.5
28.5
29
29.5
30
30.5
lo
g(X
−ra
y L
um
ino
sit
y) 
[er
g s
−
1 ]
 
⊙
Figure 4.7: Synthetic observations of X-ray luminosity and accretion rates of clusters with a uniform age
spread, the synthetic observations are shown as points while a linear fit to the data is shown as the solid
line. The top left and right panels and the bottom left panels represent young clusters with uniform age
spreads of 0.5-3.5Myr, 1.5-4.5Myr and 4-8Myr, respectively. The bottom right panel represents the disc
population observed over the entire evolution.
might be expected. In Figure 4.7 I show plots of M˙-LX for synthetic ‘cluster’ members, selected to
give clusters with various ‘observationally sensible’ age ranges of 3-4Myr (e.g. Palla & Stahler 2000,
Haisch et al. 2001ab; as indicated by the luminosity spread in the H-R diagram), along with the plot
of the total disc population (bottom right panel). I note that the interpretation of luminosity spreads as
age spreads is currently under debate, with the spread in luminosities possibly arising from accretion
history (e.g. Ballesteros-Paredes & Hartmann 2007; Baraffe et al. 2009) and much smaller age spreads
(< 1Myr). While an large age spread will wash out any M˙∗−LX variation, I will see if the large age
spreads predicted from pure luminosity spreads are sufficient to remove the expected correlations, or it
persist as indicated by observations (Drake et al. 2009).
As expected, the photoevaporation model predicts a positive M˙-LX correlation for clusters with large
age spread and a negative correlation for clusters with a narrow age spread. As a comparison, the age
spread in Orion is roughly 2-3Myr (Haisch et al. 2001b), which explains the observation of a negative
correlation by Drake et al. (2009).
The observation of systematically higher X-ray luminosities of non-accreting TTs (WTTs) compared
to accreting TTs (CTTs) can also be explained by the model in terms of photoevaporation starved accre-
tion. Figure 4.8 shows the time evolution of the median X-ray luminosity of the CTTs (solid line) and
92 Chapter 4. Discs evolution and dispersal
0 2 4 6 8 10 12
1028
1029
1030
1031
Time [Myr]
X−
ra
y 
Lu
m
in
os
ity
 [e
rg 
s−1
]
Figure 4.8: Time evolution of the median X-ray luminosity in the synthetic cluster for CTTs (solid line)
and WTTs (dashed line). The dotted line represents the transition X-ray luminosity between a CTTs and
a WTTs as a function of time. The data points are taken from the compilation of Gu¨del et al. (2007) in
the Taurus cluster, filled circles refer to CTTs and filled squares refer to WTTs.
WTTs (dashed line) populations for our model compared to the data compiled by Gu¨del et al. (2007) for
the Taurus cluster (black circles and red squares). The dotted line represents the critical X-ray luminos-
ity as a function of time which in our model separates CTTs and WTTs, and corresponds to the X-ray
luminosity of objects which have just opened a gap and have begun the clearing phase. The number
of anomalous objects (i.e. CTTs above the line and WTTs below the line) is small at ages ? 1 Myr.
Given uncertainties in age determinations from isochrone fitting (particularly at < 1 Myr; Mayne &
Naylor 2008; Baraffe et al. 2009) the agreement between the median X-ray luminosities for the CTTs
and WTTs populations at Taurus’ median age of 2 Myr is very encouraging.
4.4.2 The nature of transition discs: accreting and non-accreting
The recent observations of a class of ‘transition’ discs with residual gas inside the inner dust radius and
a signature of accretion, has prompted some authors to question the viability of photoevaporation as the
formation mechanism for the inner hole in some sources (e.g. Cieza et al 2010). Previous EUV-driven
photoevaporation models (e.g. Alexander & Armitage 2009) indeed predicted that at the time of gap
opening the surface density of the gas in the inner disc and the accretion rates due to the inner disc drain-
ing onto the star should be undetectable. However, this is not the case when X-rays are included, mainly
due to the fact that the wind rates can be two orders of magnitude higher than the EUV-driven rates. This
means that at the time of gap opening the mass of the draining inner disc, and the accretion rate remains
4.4 Results and Discussion 93
1 10 100
10−4
10−3
10−2
10−1
100
101
Inner Hole Radius [AU]
Pr
ob
ab
ilty
 [%
]
Figure 4.9: Probability distribution function of accreting (solid line) and non-accreting (dashed line) tran-
sition discs with inner hole radius. They are both scaled so that the integrated sum of both distributions
is 100%.
detectable for a non-negligible amount of time. For the disc population generated in this work, Fig-
ure 4.9, shows that the accreting inner holes and non-accreting inner-holes (M˙ < 1×10−11M yr−1) are,
in general, equally likely out to a radius of Rin ∼ 5AU. Clearly as the disc is further photoevaporated and
its inner radius moves out, the accretion signatures onto the star become less evident and non-accreting
inner holes dominate at radii larger than 20 AU. The total integrated ratio out to 10AU (the radius probed
by 24µm emission around solar-type stars) is found to be 25% accreting and 75% non-accreting for the
entire population. Although, I caution that this is not equivalent to the observed fraction of accreting
to non-accreting objects in a individual cluster, where the cluster age should also be accounted for. In
young clusters, the transition disc population is dominated by high X-ray luminosity objects which give
rise to a considerably longer accreting inner hole phase. In contrast this ratio is much lower in old clus-
ters where the transition disc population is dominated by low X-ray luminosity objects that have very
short accreting inner-hole phases.
It is perhaps worth noting at this point that the detection of a ‘transition’ disc observationally is
made through observation of the dust continuum spectral energy distribution (SED). As mentioned in the
introduction, Alexander & Armitage (2007) examined the behaviour of dust in a photoevaporating disc
finding that, under the action of dust drag, the time-scale for dust grains to drain onto the star is of order
103yrs after the gap opens. This dust removal time-scale is approximately two order of magnitudes faster
than the gas draining time-scale. Therefore, an observer would certainly see a significant drop in opacity
in the inner disc immediately after a gap has opened, while the gas will still linger in the inner dust disc
for the duration of its viscous draining time-scale of ∼ 105yrs.
I have used the population synthesis model to investigate the accretion rate versus inner hole size
94 Chapter 4. Discs evolution and dispersal
Figure 4.10: Transition disc probability map in the M˙-Rin plane. We have applied an observational cut-
off of 10−11M yr−1in accretion rate and objects below the cut-off are all shown at the bottom of the
diagram. We show a representative sample of solar-type objects classified as ‘transition’ discs; data from
taken from Espaillat et al. (2007a,b,2008,2010) - Red Circles, Hughes et al. (2009,2010) - Red Squares,
Kim et al. (2009) - Red Diamonds, Calvet et al. (2005) - Black Diamonds, Merı´n et al. (2010) - Black
Squares and Cieza et al. (2010) - Black Triangles, where error-bars are given in the literature they are
shown. We also show two model tracks corresponding to a 1dex spread in X-ray luminosities about the
median model in magenta.
evolution for transition discs created by XPE under the assumption of immediate dust clearing at the
time of gap opening. Figure 4.10 shows the probability distribution of the disc models in the M˙-Rin
plane. The symbols represent a sample of observations of solar-type objects classified as ‘transition’
discs by Espaillat et al. (2007a,b,2008,2010) - Red Circles, Hughes et al. (2009,2010) - Red Squares,
Kim et al. (2009) - Red Diamonds, Calvet et al. (2005) - Black Diamonds, Merı´n et al. (2010) - Black
Squares and Cieza et al. (2010) - Black Triangles (Although Cieza et al. 2010 do not fit for the inner-
hole radius they list as transitional sources those discs that have a deficit of emission in the Spitzer IRAC
bands; therefore we conservatively estimate an inner hole radius of < 10AU for all their sources).
It is immediately clear from the figure that there is a population of large inner hole, strongly accreting
transition discs that cannot have been created by photoevaporation. Gap opening by a giant planet or
grain growth is perhaps the most plausible explanation for these objects. However, there is a significant
number of discs (≈ 17/28) with inner holes that are consistent with a photoevaporation origin.
Since the observations of ‘transition’ discs by their nature are rare, we are plagued by low number
statistics and observational uncertainties. A quick look at Figure 4.10 indicates that different selection
criterion applied by different authors on similar data sets probes different regions of this parameter space.
4.4 Results and Discussion 95
Inner Hole Radius [AU]
lo
g 1
0 
(A
cc
ret
ion
 R
ate
) [M
  y
r−1
]
1 10 100
−11
−10.5
−10
−9.5
−9
−8.5
−8
−7.5
−7
⊙
Figure 4.11: The inner hole radius and accretion rate of low-mass objects classified as ‘transistion’ discs
by various authors: cyan triangles (Brown et al. 2009); blue filled triangles & squares (Cieza et al. 2010);
black open squares (Ercolano et al. 2009); black filled squares (Espaillat et al. 2008,2010); black open
circles (Hughes et al. 2009, 2010); red filled circles (kim et al. 2009); magenta filled diamonds (Merin
et al. 2010); green filled star (Najita et al. 2010). Where error-bars or ranges are listed they are shown.
Any object classified as non-accreting is placed on the plot with an accretion rate of 10−11M yr−1. The
grey region represents the region of parameter space where the objects are consistent with gaps created
by photoevaporation. The blue and black dashed regions represent those regions for a 0.3 and 1 M star
respectfully, calculated using the similarity solutions (for mass dependence) described in Chapter 2, see
text below for details.
Therefore, in order to improve our low number statistics I have built an extended version of Figure 4.10,
shown in Figure 4.11 which includes the full mass spectrum of observations.
Using the photoevaporation theory developed in Chapter 2 we can fully describe the region of ‘tran-
sition’ disc parameter space in which transition discs with a photoevaporative origin would lie. Since
the radial scale of the photoevaporative flow scales linearly with mass, the radius at which the gap opens
will obviously scale linearly with mass too. Furthermore, the gap opens when the accreation rate falls
below the photoevaporation rate, then the stellar mass is not important in determining the accretion rates
of transition discs, other than through the implicit mass dependence on the photoevaporation rate through
the X-ray luminosities variation with stellar mass. In Figure 4.11 I show the region in which photoevap-
orating transition discs are expected (with individual regions for 1Mand 0.3Mstars shown separately),
over-plotted with all current observations3of discs classified as transition discs: cyan triangles (Brown
3Several other types of astrophysical objects are found to enter these samples (e.g. AGB star or debris disk) in these cases
we follow the definitions of the authors and only plot objects that are classified as protoplanetary discs with a gap/hole in its
dust disc.
96 Chapter 4. Discs evolution and dispersal
et al. 2009); blue filled triangles & squares (Cieza et al. 2010); black open squares (Ercolano et al.
2009); black filled squares (Espaillat et al. 2008,2010); black open circles (Hughes et al. 2009, 2010);
red filled circles (kim et al. 2009); magenta filled diamonds (Merin et al. 2010); green filled star (Najita
et al. 2010). I note that as Cieza et al. (2010) do not fit for an inner hole radius but rather list the
Spitzer band of the inner hole I split the sample into two: the objects with a λturnoff ≤ 5.8µm the inner
hole is conservatively set to be less than 5AU, for λturnoff = 8µm I again conservatively set an inner hole
of 10AU (since the inner-hole must occur in the temperature range λ = 8 – 24µm); I add that these
are safe overestimates based on simple temperature structures of protoplanetary discs. It is also worth
noting that at very low masses ∼ 0.1M Ercolano et al. (2009a) showed that current observations are
not good enough to determine the presence of holes smaller than ∼ 1 AU arising from a lack of contrast
between the disc and the stellar photosphere. The photoevaporation region is constructed by using the
fact that the photoevaporation profile is self-similar when the radius is scaled in terms of Rg. (i.e. if the
gap opens at 1AU around a 1M star it would open at 0.1AU around a 0.1M star) and I use an X-ray
luminosity scaling of LX ∝M
3/2
∗ (Priesch et al. 2005, Guedel et al. 2007) to extend the region calculated
in Figure 4.10 to the full mass range (M∗ > 1.5M), thus the left-hand slope of the grey region maps
the LX ∝ M
3/2
∗ and scales as R
3/2
hole. The right-hand slope of the grey region and steeper slopes of the
dashed triangles are obtained by fitting power laws to both sides of the photoevaporation region shown
in Figure 4.10 and scaling the radii linear with mass as the model predicts. Unsurprisingly, this again
shows that the observations can be separated into two groups, those with small inner-holes > 20AU and
small accretion rates> 10−8 M yr−1 that are consistent with a photoevaporative origin, and those discs
with large accretion rates and large holes that cannot have a photoevaporative origin.
In both cases I note the lack of objects with large holes and low accretion rates, this could be due to
observational biases, as the longest IR wavelength observation in most cases is the 24µm (MIPS1) Spitzer
band, which constitutes a radius of roughly 10−20AU for solar type stars. However, we can be fairly sure
that there is not a large number of undetected, heavily accreting objects (? 10−11−10−10)M yr−1 ob-
jects. Certainly Herschel’s sensitivity in the FIR, will probe this region of parameter space definitively
and should provide much better statistics of the speed of inner disc clearing, compared to outer disc
clearing.
One obvious consequence of an X-ray photoevaporation mechanism is that the properties of transition
discs should be correlated with the X-ray luminosity, something no other model of photoevaporation
or ‘transition’ disc origin would predict. In Figure 4.12, we show two such correlations namely the
inner hole radius (left panel) and accretion rate (right panel) against X-ray luminosity, considering only
accreting ‘transition’ discs (i.e. those with an accretion rate > 1×10−11M yr−1). The plots have been
generated by a random sampling (in time) of the accreting transition phase of each disc model several
times, and should therefore provide a reasonable estimate of both the general form of the correlation plus
the associated scatter. Clearly, since discs with higher X-ray luminosities open gaps earlier and at higher
accretion rates a strong positive correlation between LX and M˙ would be expected and is reproduced by
the models as shown.
Furthermore, I also recover a positive correlation between inner hole radius and X-ray luminosity for
accreting transition discs as shown in Figure 4.12. We can understand this by remembering that while the
inner disc is still draining, the inner hole radius is also being eroded outwards as shown in Figure 4.4. For
more X-ray luminous objects the region being eroded is more depleted due to photoevaporation starved
4.4 Results and Discussion 97
1 10
1030
1031
Radius [AU]
X−
ra
y 
Lu
m
in
os
ity
 [e
rg 
s−1
]
10−12 10−11 10−10 10−9 10−8
Accretion Rate [M   yr−1]⊙
Figure 4.12: Simulated observations of accreting ‘transition’ discs. The left hand panel shows X-ray
luminosity plotted against inner-hole radius. The right hand panel shows X-ray luminosity plotted against
accretion rate where an observational cut-off of 10−11M yr−1has been used.
accretion and the magnitude of the mass-loss is higher. Thus, the inner-hole radius is able to be eroded
to larger radius during the accreting phase. This explains the positive slope of the right hand extent of
the symbols in the left panel of Figure 4.12.
Consequences of a different X-ray luminosity function
The X-ray luminosity function is a crucial input into the XPE model. In this work I have used the Taurus
X-ray luminosity function as this best represents the quiescent X-ray flux the disc sees throughout its
lifetime. However, here I discuss the consequences of a different X-ray spectrum that may be incident
on the disc. If the incident spectrum the disc sees is systematically harder than the one used (through
attenuation of the X-ray spectrum by large neutral columns close to the star) or softer than the spectrum
used, the qualitative behaviour of the disc population would remain the same, since one can vary the ini-
tial condition to fit. Therefore, it is important to assess the qualitative changes to some of the predictions
relating to transition discs. As discussed previously it is only the soft X-rays (0.1-2keV) that have any
thermal impact, then the result of a changing spectrum can just be considered to be a change in the soft
X-ray luminosity incident on the disc. Therefore, overall mass-loss rates can simply be scaled to this
new harder/softer spectrum.
In Section 4.1 I argued that any X-ray luminosity function with a similar spread can have a ‘null’
model constructed to fit the observations of disc fraction. Thus, a harder or more attenuated spectrum
would result in lower-mass loss rates, which would require a lower initial accretion rate, compared to a
98 Chapter 4. Discs evolution and dispersal
softer spectrum, which would require a higher initial accretion rate to explain the observed disc fractions.
Provided this effect is systematic across all values of LX (i.e. it doesn’t change the spread of X-ray
luminosities) then the consequences for the predicted ‘transition’ disc population can be considered. For
the harder/more attenuated spectrum a lower initial accretion rate is required implying a lower initial
disc mass, and therefore a smaller population of accreting transition discs, with lower accretion rates and
smaller inner holes. The converse is true for a softer spectrum which requires a higher initial accretion
rate and hence larger initial disc mass, giving rise to a larger population of accreting transition discs, with
higher accretion rates and larger holes.
4.4.3 Final clearing of the disc
As discussed in Chapter 3, the numerical models have demonstrated a further phase in disc dispersal,
the rapid removal of disc material due to a ‘thermal sweeping’ period. However, without knowledge of
how it generalises to all masses, disc structures and X-ray luminosities I cannot accurately estimate its
implications for the evolution of the disc population. Without this knowledge I am unable to terminate
the evolution of the disc due to ‘thermal sweeping’ at the correct time, and instead stop the evolution
when Rhole=100 AU, when the IR indicators of a protoplanetary disc have almost disappeared and the
observational constrains on clearing through this region are sparse. While the total columns of these
‘relic’ discs are small (> 1023 cm−2) suggesting that many may have already gone through this ‘thermal
sweeping’ process, it is important to discuss the implications of their continued existence in case the
‘thermal sweeping’ does not occur until much later times around solar type stars.
Therefore, if we ignore the ‘thermal sweeping’ period we can estimate the remaining clearing time by
dividing the disc mass by the photoevaporation rate (since it is expected to be approximately independent
of inner hole radii). Such a calculation shows that in about 10% of cases (discs irradiated by a low X-
ray flux) the final clearing time-scales would exceed 10Myr, resulting in a population of long lived
discs with large inner holes. These objects will certainly survive for a long enough time for their dust
continuum emission to be observed. The spectral energy distribution (SED) of these cold massive discs
should be similar to that of young debris discs, where a debris disc model is normally considered to be a
single temperature belt of optically thin dust (thought to be constantly replenished by collisions between
planetesimals) at a given radius from the star (Wyatt 2008). This suggests that some of the sources that
are currently classified as debris discs could in fact be photoevaporation ‘relics’.
In order to asses this I have used the radiative transfer code of Whitney et al. (2003a,b) to calculate
the SED of a typical relic using the standard input disc structure and dust properties of the code. The
properties of these relics were taken from the end point of the lowest X-ray luminosity (and hence the
longest lived) model. While such a disc is likely to be settled in the dust, I computed three models: a
flat dust distribution, a fully mixed dust distribution and a disc where Hdust = 0.1H. Figure 4.13 shows a
plot of the fractional excess of the disc compared with the stellar photosphere at 24µm and 70µm, where
the models are compared with observations of objects classified as young debris discs around solar-type
stars (Wyatt 2008). It is clear from this figure that the relics with a degree of dust settling share the same
space in the excess-excess plot as sources currently classified as young debris discs. The fully-mixed
discs show a 70µm excess which is probably too large to be classified as a debris disc, although it is
extremely unlikely that any disc could survive for > 10Myr without undergoing dust settling. I also note
4.5 Conclusions 99
1 10
1
10
100
1,000
24µm Excess
70
µm
 E
xc
es
s
Figure 4.13: Plot of 70µm excess above the photosphere against 24µm excess for disc models with a
large inner hole (100 AU) and low X-ray luminosities and therefore a predicted remaining lifetime >
10Myr. The squares show discs close to edge-on and increasing 24µm excess represents the evolution
from a flat to fully mixed dust distribution. Similarly the diamonds show the disc models close to face-
on. Also shown as data points are observations of young solar-type stars (< 100Myr) with IR excesses
classified as debris discs, taken from (Wyatt 2008).
that the predicted 850µm emission from the evolved relics falls below the current detection limits at 50
pc (Andrews & Williams 2005), and thus these relics would not contradict previous sub-mm observations
of WTTs which show that most are devoid of emission out to 500 AU (Duvert et al 2000). Only ALMA
will be able to separate these large massive discs from canonical debris discs, and hence confirm or
dismiss the existence of this proposed class of objects.
However, I emphasise that such objects may not exist at all, as they may be cleared by the ‘thermal
sweeping’ process seen around lower mass stars in the previous Chapter. Certainly, further work on
this process around solar type stars is needed before any definitive statements on the existence or non-
existence of ‘relic’ discs is made.
4.5 Conclusions
I have coupled the numerical radiation-hydrodynamic models of X-ray photoevaporation perfomed in
Chapter 3 to a viscous evolution model to construct a population synthesis model, with which we have
studied the physical properties of primordial and transition discs. The initial conditions and viscosity law
are constrained using recent observations of disc fractions in nearby clusters (Mamajek 2009). The model
requires a viscosity coefficient of approximately α = 2.5×10−3 to match the observational constraints.
The main conclusions are as follows:
1. X-rays play a major role in the evolution and dispersal of discs around solar-type stars, driving
vigorous photoevaporative winds.
2. I have constructed a ‘null’ accretion disc model using only knowledge of the observed disc frac-
tions and X-ray luminosity functions under the assumption that discs are dispersed through X-ray
100 Chapter 4. Discs evolution and dispersal
photoevaporation. This ‘null’ model shows very good agreement with observed accretion rates
in YSOs as well as their evolution with time, providing further independent confirmation of the
viability of X-ray photoevaporation as a dominant dispersal mechanism.
3. X-rays suppress accretion by preventing accreting material from reaching the star, since this ma-
terial is removed through photoevaporation. This ‘photoevaporation starved accretion’ (Drake et
al 2009) produces a negative correlation between X-ray luminosity and accretion rate for clusters
with a relatively narrow age spread, in agreement with the observational correlation reported by
Drake et al. (2009) in the Orion data.
4. The model successfully reproduces the observation that WTTs are systematically brighter in X-
rays than CTTs. Whereas this has previously been interpreted as modification of X-ray emission
or detectability by the presence of discs, our models support Drake et al. (2009) in reversing the
causal link - i.e. that disc lifetimes are regulated by photoevaporation and hence disc-less stars are
on average those with higher LX .
5. A large fraction (? 50%) of observed transition discs can be easily explained by X-ray photoevap-
oration. There is however a population of strongly accreting transition discs with large inner holes
(> 20AU) that lie outside the M˙−Rin region predicted by the models, suggesting that alternative
mechanisms are responsible for their inner hole (e.g. binary interaction, grain-growth and/or planet
formation).
5
Detecting the wind I: theoretical spectra of
photoevaporating discs
IN order to provide more direct tests of the photoevaporation model I will discuss emission fromthe wind itself. In particular, I will discuss emission lines from the gas, which through resolvedspectroscopic measurements can provide direct evidence for a photoevaporative wind, and be used to
test the ionization and kinematic structure of the wind. Currently, observations of resolved gas emission
lines from the upper disc and photoevapoartive flow are sparse, with quality data only available for the
complicated ‘transition’ disc TW Hya. Therefore, in this Chapter I will discuss predictions for several
observable gas emission lines, along with an analysis of the observational data, particulararly for TW
Hya.
5.1 The status of current observations
While gas line emission has proved a successful test of the external photoevaporation model (e.g. Rich-
ling & Yorke 2000), these diagnostics have only recently begun to be used to constrain the intrinsic pho-
toevaporation model. Perhaps the earliest indications of a direct detection of a photoevaporative wind
were the observations by Hartigan et al. (1995) of the optical lines: OI 6300 & 5577A˚, SII 6731A˚ and
NII 6583A˚. The original aim of their work was to study jet emission from young stellar objects and
contained a large sample of 42 T Tauri stars. However, most objects showed line profiles with two com-
ponents. These multi-component line profiles were split into two regions: the high velocity component
(HVC - |v|> 60 km s−1) and the low velocity component (LVC - |v|< 60 km s−1). While the HVC was
found to be well described by jet models, it was the LVC that was not consistent with any of the availble
models and it is this component I discuss here. The LVC emission lines showed blue-shifts in the range
∼ 5±2km s−1, line widths of O10km s−1 and luminosities around 10−5L with a∼ 1dex spread, where
101
102 Chapter 5. Detecting the wind I
the LVC was best resolved (compared to the HVC) in the OI 6300A˚ line, and was always present when
the young star had an NIR excess (i.e. a primordial disc). Furthermore, a comparison of the line ratios
indicated that the LVC arises in a region with a significantly different temperature and density from the
HVC (or jet), and that the variation of the blue shift with the critical density indicates that the emitting
region is ‘accelerating’ towards regions of lower densities. In fact Hartigan et al. hypothesised that the
most likely origin is a (at least) partially ionized disc wind emanating from somewhere in the radial range
0.1-20AU. With the most startling fact was the one to one correspondence between an NIR excess and
LVC blue-shift, and this is what a photoevaporative wind would predict. However, Font et al. (2004)
showed conclusively that while an EUV photoevaporative wind is capable of producing suitable blue-
shifts, the predicted luminosities only reached ∼ 10−7L which is inconsistent with the observations.
This incompatibility is easy to understand, since to produce OI emission, neutral oxygen atoms need to
be present in the flow and EUV photons fully ionize the region. Furthermore, Gorti et al. (2011) recently
discussed OI emission within the framework of the FUV photoevaporation, and conclude that while the
observed line fluxes can be matched (through non-thermal OI emission, via OH dissociation), an FUV
flow is necessarily cooler (thus slower) and is unable to reproduce blue-shifts of ∼ 5km s−1 Therefore,
an EUV driven wind is simply too ionized and an FUV wind too slow to match the observations. While
the Hartigan et al. observations provided an excellent sample of line observations, the low resolution
and confusion with the HVC component mean that comparisons of line profiles are not possible with the
sample.
It was not until observations by Pascucci & Sterzik (2009) of the NeII 12.8µm from 4 young stars, that
direct evidence for photoevaporation was presented, with TW-Hya (which presents with an almost edge
on disc) being the closest showing a well resolved, blue shifted and non axi-symmetric profile. This was
followed up in detail in Pascucci et al. (2011) and was supplemented with observations of several optical
forbidden lines, including the OI 6300A˚. In this case the NeII line profile was also shown to be consistent
with an EUV photoevaporative origin (Alexander, 2008). The NeII line profile shows a clear blue shift
of 6± 1km s −1 and a FWHM of ∼ 14 km s−1, which is wider than expected from pure thermal and
Keplerian broadening indicating its production in an hydrodynamic disc wind. Unfortunately, TW Hya is
not a simple primordial disc but an accreting transition disc (which is consistent with a photoevaporative
origin), with an inner hole in the range 1−4AU (Calvet et al. 2002, Ratzka et al. 2007), and unlike the
disc bearing Hartigan et al. (1995) objects, the LVC components of the optical lines are not blue-shifted,
indicating they may not be tracing a flow.
In this Chapter I will use the results of radiative transfer post-processing of the numerical models
calculated in Chapter 3 to produce theoretical line profiles and luminosities, these can then be compared
to the current observations as well making predictions for future observations. While this method is
flexible to produce line-profiles and luminosities for any required line, I will restrict my discussion to
those lines which have attracted previous attention from observers, to either compare with current data
or make predictions for currently observable lines.
5.2 Radiative transfer post-processing
In order to calculate the line emission from the radiation-hydrodynamic models calculated in Chapter 3,
the solutions must again be run through a radiative transfer code. While the algorithm accurately deter-
5.2 Radiative transfer post-processing 103
mined the thermal structure, and thus dynamical structure of a photoevapoartive wind, the algorithm does
not calculate the ionization structure and level populations of the species we are interested in, which are
necessary for computing line fluxes and profiles. While the information calculated during the radiation-
hydrodynamic algorithm is sufficient to compute these quantities, it is easier to use a radiative transfer
post-processing of the temperature and density structures generated by the hydrodynamic calculations.
In order to do this, the structures are fed back into MOCASSIN, which then calculates the ionization
structure of the region and the level populations of all the species. Once MOCASSIN has obtained these
quantities it can then compute the emissivities of any desired line. The calculations were carried out
using the input spectrum shown in Chapter 3 (the same spectrum used to determine the temperature-
ionization parameter relation) The atomic database included opacity data from Verner et al. (1993) and
Verner & Yakovlev (1995), energy levels, collision strengths and transition probabilities from Version
5.2 of the CHIANTI database (Landi et al. 2006, and references therein) and the improved hydrogen and
helium free-bound continuous emission data of Ercolano & Storey (2006). However, there are several
missing quantities that are worth pointing out: excitation rates of Ne+ by collisions with neutrals are
not known, therefore theoretical calculations of this line, including this work, only account for collisions
with electrons. This is likely to result in the NeII 12.8µm line flux being underestimated by a factor
of >2 (Glassgold et al 2007). Furthermore, the collisions with neutral hydrogen and oxygen (which are
likely to dominate as in the case of the OI 6300A˚ line) are not included in the calculation for the 1S0 level
due to the unavailability of the collision rates. This will lead to underestimates for the OI 5577A˚ line,
although it will not affect the calculation of the OI 6300A˚ line. The following elemental abundances
are used in the calculations, given as number densities with respect to hydrogen: He/H = 0.1, C/H =
1.4×10−4, N/H = 8.32 × 10−5, O/H = 3.2 × 10−4, Ne/H = 1.2 × 10−4, Mg/H = 1.1 × 10−6, Si/H =
1.7 × 10−6, S/H = 2.8 × 10−5, these are solar abundances (Asplund et al. 2005) depleted according to
Savage & Sembach (1996). The dust model of D’alessio et al. (2001) used for the initial condition of the
disc is adopted for the final flow structure.
I perform this analysis for models of a 0.7 M star, where the X-ray luminosity spans the observed
range for T Tauri stars of this mass ( 2×1028 ≤LX ≤ 2×1030 erg s−1). Furthermore, I consider the disc
at different stages of dispersal, by considering discs with cleared inner holes of sizes (Rhole = 8.3 AU,
14.2 AU and 30.5 AU) for an X-ray luminosity of 2×1030 erg s−1. While the temperature and ionization
structure in the X-ray heated region is accurately known, as described in Chapters 2 & 3, it is expected
that the FUV is important between the X-ray heated region and dust heated region. The ionization
structure, temperature and level populations are highly dependent on chemistry occurring in the disc
in this region, something MOCASSIN is currently unable to treat. Therefore, I only consider the line
emission to be well constrained from within the X-ray heated region and do not discuss the predictions
from the model for emission that occurs in the upper regions of the bound disc.
5.2.1 Calculation of the line profiles
The two-dimensional maps of emissivities (calculated using the MOCASSIN post-processing described
above) and gas velocities (taken form the radiation-hydrodynamic calculation) were used to construct
a three-dimensional model of the disc and calculate the line-of-sight emission profiles for a number of
transitions. The emissivities are calculated using 120 radial points (NR = 120) and 1500 height points
104 Chapter 5. Detecting the wind I
(Nz = 1500). By assuming azimuthal symmetry about the z-axis, and reflection symmetry about the disc
mid-plane the resultant 3D grid has dimensions NR×2Nz×Nφ . I take a value of Nφ = 130 and checked
that this was large enough to fully resolve the profiles. For the chosen emission lines the disc atmosphere
is optically thin and therefore I assume that the line contribution of each parcel of gas can escape freely
along the line of sight, provided that this does not intercept the mid-plane (e.g. Alexander 2008, Schisano
et al. 2010). Although, when I consider discs with cleared inner holes, this cleared inner hole region is
taken to be optically thin to the line; therefore, lines of sight that intercept the bound disc’s mid-plane
are taken to be optically thick to the lines. This means that for discs with inner holes, certain lines of
sight will be able to ‘see’ both the blue shifted and red shifted components of the photoevaporative flow.
Furthermore, for almost edge-on inclinations escape columns are large enough for dust attenuation to
start playing a role. Therefore, comparisons with objects at high inclinations is not yet possible.
The line luminosity is then computed by including a Doppler broadening term in each cell. Hence
the line luminosity at a given velocity v is computed by evaluating the following integral by direct sum-
mation.
L(v) =
∫
V
d3r
`(r)√
2pivth(r)2
exp
(
− [v− vlos(r)]
2
2vth(r)2
)
(5.1)
where `(r) is the volume averaged power emitted at a point r, vlos(r) is the projected gas velocity along
the line of sight and vth is the local thermal velocity of the emitting atom given by
vth =
√
kbT (r)
µimH
(5.2)
with µi the mean molecular weight of the required atom. The resolution of the velocity array is 0.25 km
s−1. I have evaluated the profiles of the OI 6300A˚, the NeII 12.8µm, as these are the only lines for which
high resolution data is currently available for comparison.
5.3 Results
Tables 5.1 & 5.2 list the emission line luminosities for the models. I note only a small subset of all the
lines calculated are listed in the tables (a full set is listed in Ercolano & Owen 2010), however these
are the lines that have been currently observed. The tables are organised as follows: Table 5.1 lists the
luminosities for the primordial disc models with varying X-ray luminosity (log(LX /erg s−1) = 28.3, 29.3
and 30.3) while Table 5.2 show the results for the inner hole models irradiated by log(LX/erg s−1) = 30.3
with varying hole sizes.
A simple inspection of these tables immediately reveals that the integrated line luminosities are not
very sensitive (factors of 2-5 at most) to the size of the inner hole. However, inner hole models do
produce, in most cases, lines that are approximately a factor of two more luminous than those of a
primordial disc with the same X-ray luminosity. This is due to the different physical conditions in inner
hole regions, that are less dense and most importantly on average warmer than primordial disc winds.
In general, line luminosities are correlated with X-ray luminosity. However, the response of different
lines to changes in LX is different for different transitions, and depends on how sensitive to changes in
luminosity a line’s emitting region is, and on the dependence of a given emission line to temperature,
5.3 Results 105
Table 5.1: Primordial disc models with varying X-ray luminosity.
Species Wavelength Line Luminosity [L]
[A˚] log(LX )=28.3 29.3 30.3
OI 6302.030 2.83E-07 1.93E-06 1.25E-05
OI 5578.890 5.22E-09 7.83E-08 9.12E-07
NeII 128155.8 9.25E-08 5.51E-07 5.41E-06
NeIII 155545.2 1.01E-08 1.16E-07 1.06E-06
SII 4069.760 2.23E-07 1.64E-06 1.22E-05
SII 6732.690 2.08E-07 1.16E-06 1.08E-05
MgI 4573.180 6.11E-08 8.21E-07 1.08E-05
Table 5.2: Inner hole disc models with varying hole size.
Species Wavelength Line Luminosity [L]
[A˚] Rhole=8.3 AU 14.2 AU 30.5 AU
OI 6302.030 4.51E-05 4.48E-05 3.59E-05
OI 5578.890 1.47E-07 9.13E-08 5.16E-08
NeII 128155.8 1.21E-05 1.16E-05 1.04E-05
NeIII 155545.2 2.43E-06 2.90E-06 3.44E-06
SII 4069.760 2.43E-05 1.99E-05 1.49E-05
SII 6732.690 2.78E-05 3.07E-05 3.14E-05
MgI 4573.180 1.12E-06 9.13E-07 6.66E-07
density or ionisation structure. A number of factors are relevant in determining which region contributes
to the emission of a given line. The ion’s abundance is important; collisionally excited lines of NeII
and OI require Ne+ and O0, along with colliding particles such as electrons and/or neutral hydrogen.
Collisionally excited lines depend exponentially on the gas temperature through the Boltzmann factor
and therefore preferentially trace the hotter regions. In the case of collisionally excited lines in the IR,
this temperature dependence is weakened by the fact that the gas temperature is generally much higher
than the line excitation temperature, causing the Boltzmann term to tend to unity. Gas density also
influences the emissivity of collisionally excited lines through the critical density, above which the line
emission is suppressed.
It should also be noted that increasing the X-ray luminosity not only affects the ionisation rate of the
wind, but also affects its density. Higher luminosities drive a more vigorous (denser) wind, with a linear
correlation between LX and gas density in the wind as shown in Chapter 2.
5.3.1 Line profiles
Figure 5.1 shows the normalised profiles of the NeII fine structure line at 12.8µm and the OI forbidden
line at 6300 A˚ for the primordial disc model with log(LX/erg s−1) = 28.3 (left), 29.3 (middle) and 30.3
(right). Figure 5.2 shows the same line profiles for inner-hole models with log(LX/erg s−1) = 30.3 and
inner hole radii of 8.3 AU (left), 14.2 AU (middle) and 30.5 AU (right). Profiles are computed for ten
inclination angles from 0o to 90o in steps of 10o. The profiles were degraded to an instrumental resolution
of R = 100,000, typical of the best ground based facilities.
Figure 5.3 shows emissivity maps of the primordial disc (left panels) and inner-hole models (right
panels), with velocity vectors overlain. The models from top to bottom are for increasing X-ray lumi-
106 Chapter 5. Detecting the wind I
0
0.05
0.1
0.15
0.2
N
or
m
al
is
ed
 F
lu
x
 
 
−40 −20 0 20
0
0.02
0.04
0.06
0.08
0.1
0.12
−20 0 20
Velocity [km s−1]
−20 0 20 40
0o
10o
20o
30o
40o
50o
60o
70o
80o
90o
[NeII]
[OI]
Figure 5.1: Line profiles for (from the top) NeII 12.8µm, OI 6300A˚ and a generic hydrogen re-
combination line. The left, middle and right panels are for primordial disc models irradiated by
log(LX/erg s−1) = 28.3, 29.3 and 30.3, respectively. These line profiles are all normalised, such that∫ 40
−40L[v/(km s
−1)] dv = 1. The profiles are colour-coded according to their inclination angle, a colour
legend in given in the top-left panel.
nosities (left) and increasing inner hole radii (right). The magenta and yellow contours delineate the
region containing 85% of the emission for the NeII 12.8µm and the OI 6300A˚, respectively. The noisy
appearance of some of the emitting regions are due to Monte-Carlo fluctuations and the imposition of a
hard cut-off to draw the contours.
The OI 6300A˚ line (yellow contours) originates in the wind, or in the primordial disc case, the wind
and the very inner (hot and bound-R< 1 AU) region of the disc which is where the temperatures are high
enough to populate its upper level. The bound disc compared to wind contribution varies with the X-ray
luminosity, as the X-ray luminosity increases the density in the wind increases, and therefore so does the
wind’s contribution which dominates at the highest X-ray luminosity, However, at the lower luminosities
the line profiles shows increasing complexity with the multi-component nature of the profile. Therefore,
it is the contribution from the bound hot inner disc that produces the ‘winged’ profiles (resulting from the
high Keplerian broadening) for the OI 6300A˚ clearly seen in the lower luminosity primordial models.
The NeII mid-infrared (MIR) lines can arise in much cooler regions and therefore, while still predom-
inantly produced in the wind, they are not confined to the hot inner layers, but span over a larger region.
The NeII emitting region (magenta contours in Figure 5.3) and line profiles vary significantly with X-ray
luminosity, tracing the highest electron density regions since L(NeII)∝ n2e .
The reason why inner hole models produce very small or no blue-shifts is that at low inclinations
both the blue and red-shifted components of the wind are visible through the hole, resulting in only a
5.3 Results 107
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
Velocity [km s−1]
N
or
m
al
is
ed
 F
lu
x
 
 
−40 −20 0 20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
−20 0 20
Velocity [km s−1]
−20 0 20 40
0o
10o
20o
30o
40o
50o
60o
70o
80o
90o
[NeII]
[OI]
Figure 5.2: Line profiles for (from the top) NeII 12.8µm, OI 6300A˚ and a generic hydrogen recombina-
tion line. The left, middle and right panels are for inner-hole disc models irradiated by log(LX/erg s−1)
= 30.3, with inner holes of radius 8.3, 14.2 and 30.5, respectively. These line profiles are all normalised,
such that
∫ 40
−40L[v/(km s
−1)] dv = 1. The profiles are colour-coded according to their inclination angle,
a colour legend in given in the top-left panel.
small, often undetectable, blue-shift. At higher inclination the lines are Keplerian and no blue-shift is
expected anyway. I will discuss the significance of this result in the next section, when comparisons are
made with observations, but note at this point that this is only true if the winds are optically thin to the
lines in question.
The main profile characteristics are summarised in Tables 5.3 and 5.4. These include the velocity of
the peak(s), vpeak, the full-width-half-maxima (FWHM) and the profile “type”. Where the line profiles
have been classified as shown in Figure 5.4. Type “A” is a single peaked profile that could be well
approximated by a Gaussian. Type “B” is a single peaked Gaussian-like profile, but with large wings.
This is generally produced when there is both a wind and bound disc contribution to the line and/or the
emission region is spread over a large radial domain. Type “C” is a double peaked profile, in this case
the two values of vpeak are given. Type “D” are profiles where two clear peaks are not resolved, yet the
profile is broad, flat-topped or significantly asymmetric; in this case the value at the centre of the line (i.e.
the middle of the FWHM segment, horizontal dashed red line in Figure 5.4) is given; strictly speaking,
this is not the location of the ‘peak’ however I keep the label vpeak for simplicity.
5.3.2 NeII 12.8µm and OI 6300A˚ radial profiles
The model’s radial intensity profiles of the NeII 12.8µm and of the OI 6300A˚ line are shown in the left
and right panels of Figure 5.5. The top panels show the response of the radial intensity profiles of these
108 Chapter 5. Detecting the wind I
 
 
Radius [AU]
10 20 30 40 50
10
20
30
40
 
 
10
20
30
40
50
H
ei
gh
t [A
U]
10
20
30
40
Denstiy [g cm−3]
10 20 30 40 50 60
−22 −20 −18 −16 −14 −12 −10
10 km/s
Figure 5.3: Density maps showing the location of the 85% emission region of HI recombination lines
(black contour), OI 6300A˚ line (yellow contour) and NeII 12.8 µm line (magenta contour). The velocity
field is represented by the red arrows. Left: Primordial disc models irradiated by log(LX/erg s−1)= 28.3
(top), log(LX/erg s−1) = 29.3 (middle) and log(LX/erg s−1) = 30.3 (bottom). Right: Inner hole models
irradiated by log(LX/erg s−1) = 30.3 with inner hole radii of Rhole = 8.3 AU, 14.2 AU and 30.5 AU
lines to changes in the X-ray luminosity. Wider profiles are expected for lower X-ray luminosities since
these drive a wind that is less dense than in the higher X-ray luminosity case, meaning that the emission
region is more extended. The bottom panels of Figure 5.5 show the radial intensity profiles for discs
irradiated by the same X-ray luminosity (2×1030erg/sec), with the black line representing a primordial
disc and the red, green and blue lines representing discs with inner holes of size 8.3, 14.2 and 30.5 AU,
respectively. Both the NeII and OI lines show a very clear sequence with inner hole size suggesting that
spectro-astrometry (e.g. Pontoppidan et al 2009) may perhaps be used to infer the evolutionary states
of gaseous discs in the clearing phase. One clear testable fact is that the peak of the emission in all
inner-hole cases is at radii smaller than the dust inner radius of the disc. This is obvious by looking at
the structure of the wind and the location of the emission region (see Figure 5.3). So the presence of gas
in the inner hole of these objects is simply due to gas that is being photoevaporated from the inner edge
5.4 Discussion: The emission line spectra 109
−40 −20 0 20
0
0.2
0.4
0.6
0.8
1
N
or
m
al
is
ed
 F
lu
x
−20 0 20
Velocity [km s−1]
−20 0 20
−20 0 20 40
A                                        B                                       C                                       D
Figure 5.4: Representative shapes for the line profile shapes calculated from our models. Profile A (left)
can be well approximated by a Gaussian. Profile B is a Gaussian with large wings. Profile C is a double
peaked profile and profile D are profiles where two clear peaks are not resolved, yet the profile is broad,
flat-topped or significantly asymmetric. The blue vertical dots show the location of v/rmpeak as given in
Table 5.3 and the red horizontal dashed line shows the FWHM for the given profiles.
of the disc.
5.4 Discussion: The emission line spectra
5.4.1 Oxygen
The OI 6300A˚ forbidden line deserves special attention. As mentioned earlier, this line was detected by
Hartigan et al. (1995), blue-shifted by∼ 5 km s−1 for all T Tauri with an NIR disc. Detection of this line
in T Tauri stars has been reported by various other authors including White & Hillenbrand (2004) and
Mohanty et al (2005). The OI 6300A˚ line profiles reported by Hartigan et al. (1995) all showed a LVC,
(also seen also in other forbidden lines, such as OI 5577 A˚, and SII 6731 A˚), which was interpreted by
these authors as being produced in a slow moving disc wind. This was further demonstrated by the fact
that blue-shifts were largest for forbidden lines of lowest critical density, suggesting that the emitting
region accelerates away from the surface of the disc.
While Hartigan et al. did not ascertain the nature of the disc wind, a photoevaporative wind origin is a
natural solution. The EUV calculations of Font et al. (2004) failed to reproduce the observed OI 6300A˚
line luminosities; a result of the fact that an EUV-driven wind is, by construction, fully ionised. The
OI 6300A˚ line can only then be produced in a thin layer at the bottom of the photoevaporating (ionised)
layer. However, an X-ray driven photoevaporation process, yields a disc wind that is predominantly
neutral, thus allowing abundant production of the OI 6300A˚ line. Indeed the model predict luminosities
of a few 10−5 L for LX = 2×1030erg/s, roughly two orders of magnitude higher than the value predicted
by Font et al. (2004) and in the range of values reported by Hartigan et al. (1995). Figure 5.6 shows a
comparison of equivalent widths observed in the LVC of OI 6300A˚ by Hartigan et al. (black histogram)
with those predicted by the models (red and blue line). The red line represents the primordial disc models
irradiated by X-ray luminosities between 2×1028 and 2×1030 erg s−1 and the blue line corresponds to
inner hole sources with hole sizes between∼8 and∼30 AU irradiated by an X-ray luminosity of 2×1030
erg s−1. The higher equivalent widths shown on the histogram are lines where even the LVC velocity
range (−60 to 60 km s−1) is saturated by emission from a jet (the x-axis is truncated in this plot and
could go up to 7A˚). However, I note that the range of X-ray luminosity, mass and hole size that I have
110 Chapter 5. Detecting the wind I
0 10 20 30 40 50 60 70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radius [AU]
N
or
m
al
is
ed
 In
te
ns
ity
 
 
log10(LX)=28.3
log10(LX)=29.3
log10(LX)=30.3
0 10 20 30 40 50 60 70
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radius [AU]
N
or
m
al
is
ed
 In
te
ns
ity
 
 
log10(LX)=28.3
log10(LX)=29.3
log10(LX)=30.3
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radius [AU]
N
or
m
al
is
ed
 In
te
ns
ity
 
 
R=R
sub
R=8.3AU
R=14.2AU
R=30.5AU
0 10 20 30 40 50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Radius [AU]
N
or
m
al
is
ed
 In
te
ns
ity
 
 
R=R
sub
R=8.3AU
R=14.2AU
R=30.5AU
Figure 5.5: Radial intensity profiles of NeII 12.8µm (left panels) and OI 6300A˚ (right panel). The top
panels show how the profiles respond to variations in the X-ray luminosity in the case of a primordial disc,
while the bottom panels show the response to different hole sizes (Rhole). Rsub is the dust sublimation
(destruction) radius. All models shown in the bottom panels are irradiated by an X-ray luminosity of
2×1030erg/sec.
investigated is restricted, and detailed modelling of individual sources will provide the ultimate test for
an X-ray photoevaporative explanation to the OI problem.
The luminosity of the OI 5577 A˚ is under predicted by the models compared to the Hartigan et al.
(1995) observations. However, this is not a cause of concern, since this is most likely due to the fact that
collisions with neutral hydrogen (which are important for the OI 6300A˚ line) are not available for all
levels for the OI 5577 A˚ line as mentioned above, and it is at this stage not possible to predict the errors
in the calculated luminosities for this line.
5.4.2 Neon
Since the prediction by Glassgold et al (2007) and the first detections of the NeII 12.8 µm line in T Tauri
discs (Pascucci et al 2007; Espaillat et al 2007; Lahuis et al 2007; Herczeg et al 2007), this line has
received a lot of attention in the literature, both observationally (Gu¨del et al 2010; Pascucci & Sterzik
2009; Najita et al 2009; van Boekel et al 2009; Flaccomio et al 2009) and theoretically (Meijerink et al,
5.4 Discussion: The emission line spectra 111
Table 5.3: Line profiles from primordial discs irradiated by log(LX/erg s−1)= 28.3, 29.3 & 30.3. The
table lists the velocity of the peak, the full-width-half-maximum (FWHM) and the type of the profile (see
Figure 4 for details), for ten inclinations from 0o to 90o. The profiles were degraded to an instrumental
resolution of R = 100,000.
NeII 12.8µm
log(LX )= 28.3 erg/s log(LX )= 29.3 erg/s log(LX )= 30.3 erg/s
io vpeak FWHM Profile vpeak FWHM Profile vpeak FWHM Profile
[km/s] [km/s] [km/s] [km/s] [km/s] [km/s]
0 0 3.7 B -0.25 5.0 B -2.25 10.2 A
10 -0.25 6.0 A -0.75 10.8 B -3.25 10.6 A
20 -1 10.1 A -2.75 13.9 A -4.5 11.9 A
30 -1.25 13.6 D -3.75 15.4 A -5.25 13.4 A
40 -3/+1.75 15.9 C -4.25 16.6 A -5.5 15.0 A
50 -3.5/+2 17.6 C -4.75 17.6 A -5.5 16.5 A
60 -3.75/+2.5 18.9 C -2.25 18.6 D -5 17.8 A
70 -4/+3 19.8 C -4.75/+0.75 19.4 C -4.5 19.0 A
80 -4/+3.5 20.4 C -4.25/+1.75 20.0 C -3.75 19.8 A
90 ±3.75 20.8 C ±3.25 20.6 C ±2.5 22.6 C
OI 6300 A˚
log(LX )= 28.3 erg/s log(LX )= 29.3 erg/s log(LX )= 30.3 erg/s
io vpeak FWHM Profile vpeak FWHM Profile vpeak FWHM Profile
[km/s] [km/s] [km/s] [km/s] [km/s] [km/s]
0 -0.25 6.4 A -0.5 10.0 B -7.5 13.3 A
10 -4.75 14.0 A -3.75 15.0 B -7.75 13.0 A
20 -5.75 15.9 B -6 15.2 B -8 12.6 A
30 -5.5 16.0 B -7.25 14.6 B -8 12.6 A
40 -5.5 16.2 B -8 16.0 B -7.75 13.1 A
50 -5.25 17.0 B -7.5 18.0 B -7.25 13.9 A
60 -4.75 17.8 B -6.5 20.1 B -6 15.0 A
70 -4.25 18.5 B -5 21.7 B -4.5 16.0 A
80 -3.5 19.1 B -3.25 22.7 B -2.5 16.7 A
90 0 19.5 B 0 23.1 B 0 17.0 A
2008; Schisano et al. 2009; Alexander 2008; Hollenbach & Gorti 2009); it was originally proposed as a
probe of X-ray irradiation of T Tauri discs. The large scatter in the L(NeII) for stars of a given LX may
be due to differences in the irradiating spectrum or the disc structure (Schisano et al 2009), variability,
or for the higher L(NeII) cases to stellar jets (Hollenbach & Gorti 2009; Shang et al. 2010).
The L(NeII) calculated from the models are in the range of those observed and show a roughly linear
dependence with X-ray luminosity. However, what the models show is that this line is produced almost
exclusively in the photoevaporating wind, as clear from the blue-shifted profile for non edge-on inclina-
tions (Table 5.3 and Figure 5.1). It is also clear that a clean linear relationship with LX is not expected (or
found) given that the wind structure and the NeII emitting region change in response to different X-ray
luminosities. In this perspective it is easy to understand the scatter in the L(NeII)-LX relation noticed by
Gu¨del et al (2010) even for the non-jet sources.
The NeIII to NeII line ratio for our models is never lower than ∼0.1, while observations generally
indicate lower values, e.g. ∼0.06 (Lahuis et al 2007), ∼0.045 (Najita et al 2010), although in the case of
WL5 a value of∼0.25 is reported by Flaccomio et al (2009). Unfortunately few detections of this line are
112 Chapter 5. Detecting the wind I
Table 5.4: Line profiles from inner-hole discs irradiated by log(LX/erg s−1) = 30.3 erg/s, with inner hole
radii of 8.3, 14.2 and 30.5 AU. The table lists the velocity of the peak, the full-width-half-maximum
(FWHM) and the type of the profile (see Figure 4 for details), for ten inclinations from 0o to 90o. The
profiles were degraded to an instrumental resolution of R = 100,000.
NeII 12.8µm
Rhole = 8.3 AU Rhole = 14.2 AU Rhole = 30.5 AU
io vpeak FWHM Profile vpeak FWHM Profile vpeak FWHM Profile
[km/s] [km/s] [km/s] [km/s] [km/s] [km/s]
0 -1 15.0 A -0.5 13.3 A -0.5 10.9 A
10 -1.25 15.4 A -0.75 13.9 A -0.5 11.2 A
20 -2 16.7 A -1.25 15.2 A -0.5 12.1 A
30 -2.75 19.7 A -1.75 17.6 A -0.75 13.7 A
40 -1.5 23.3 D -1 20.8 D -1.5 15.9 A
50 -1.5 26.2 D -1 23.5 D -0.5 18.2 D
60 -1.5 28.4 D -1.25 25.8 D -0.75 20.0 D
70 -7.25/+2.5 29.8 C -5.75/+2.25 27.2 C -3/+1.5 21.3 C
80 -7/+4.25 30.2 C -5.75/+3.75 27.8 C -3.25/+2.25 21.7 C
90 ±5.75 30.1 C -5/+5 27.5 C ±3.25 21.3 C
OI 6300 A˚
Rhole = 8.3 AU Rhole = 14.2 AU Rhole = 30.5 AU
io vpeak FWHM Profile vpeak FWHM Profile vpeak FWHM Profile
[km/s] [km/s] [km/s] [km/s] [km/s] [km/s]
0 -0.25 17.1 A 0 14.7 A 0 13.5 A
10 -0.25 17.2 A 0 15.0 A 0 13.6 A
20 -1 18.6 A -0.25 16.2 A 0 14.2 A
30 -2 22.4 A -1 19.1 A -0.25 15.7 A
40 -0.75 26.9 D -0.5 23.3 D -0.75 18.5 A
50 -9/+5 30.1 C -6.5/+4.25 26.6 C -0.25 21.5 D
60 -11.25/+7.25 32.4 C -8.75/+6.25 28.9 C -5.25/+4 23.8 C
70 -12.25/+8.75 34.0 C -9.75/+7.75 30.5 C -6.25/+5.25 25.3 C
80 -12/+10.25 34.9 C -10/+8.5 31.4 C -6.75/+5.75 26.2 C
90 ±11.25 35.0 C ±9.5 31.6 C ±6.5 26.5 C
currently available and it is therefore hard to draw statistically significant conclusions. To make matters
worse this ratio is quite uncertain in theoretical models due to the lack of neutral collision rates, as men-
tioned above, and uncertainties in the charge exchange coefficients. In view of the current observational
and theoretical difficulties with this line ratio, its diagnostic power cannot yet be harnessed.
5.4.3 Sulphur
The models predict SII 6717A˚ and 6731A˚ lines in the range observed by Hartigan et al. (1995) for
the LVC. The ratio of the 6731A˚ to the 6717A˚, which is a useful density diagnostic, is approximately
two in the models, and it compares well with the value reported for most of the sources in Hartigan et
al. (1995). A ratio of two at ∼7000 K corresponds to an electron density of just below 104cm−3, again
consistent with a wind origin for this line. Furthermore, all of the detections for the LVCs of these lines in
Hartigan et al. (1995) where the source presents with an NIR disc show a small blue-shift, reinforcing the
conclusion that all discs show evidence of a wind which can be produced by photoevaporation. The OI
5.5 A direct comparison: TW-Hya 113
0 0.2 0.4 0.6 0.8 1
0
1
2
3
4
5
6
7
Wavelength [Angstrom]
Fr
eq
ue
nc
y
Figure 5.6: OI 6300 A˚ equivalent width histogram for the LVC sources in HEG95 compared to the
model predictions for primordial discs (red line) irradiated by X-ray luminosities between 2×1028 and
2×1030 erg/sec, and compared to inner hole sources (blue line) with hole sizes between∼8 and∼30 AU
irradiated by an X-ray luminosity of 2×1030 erg/sec. The error in the equivalent widths is around the
10-20% level.
to SII ratios also compare favourably, when the OI line luminosities observed are in the range predicted
by the model (i.e. < 10−4L). However, I note that this ratio can be as large as one order of magnitude
for the higher OI luminosities detected, these detections correspond to objects where the jet emission has
saturated the LVC and are associated with large equivalent widths in the LVC OI line (e.g. CW Tau, DQ
Tau and HN Tau).
5.4.4 Magnesium
The models predict a rather strong MgI line at 4573 A˚ (∼10−5L for LX = 2×1030erg s−1). This line
has been detected to date, in four young low mass stars, namely VY Tauri (M0; Herbig 1990), XZ Tau B
(M1; White & Ghez 2001) and very recently in the low-mass variable TWA 30 (Looper et al 2010) and
TW Hya (Pascucci et al. 2011). Detection of this line in M-stars is rendered possible by the much lower
stellar continuum in the B-band compared to higher mass stars and the strength of this line indicates that
it should be target for resolved spectroscopic measurements, providing complementary observations to
the OI 6300A˚ line, particularly around lower mass stars.
5.5 A direct comparison: TW-Hya
The recent observation of a blue-shifted NeII 12.8µm line profile by Pascucci & Sterzik (2009) in a few
well known YSOs with discs, including TW Hya, has been interpreted as the first direct evidence for
photoevaporation. Unlike the blue-shifted OI 6300A˚ line, the NeII 12.8µm profile for the pure EUV
wind (Alexander, 2008) is consistent with the observed profile from TW Hya. Furthermore, Pascucci et
al. (2011) supplemented the NeII observations with optical forbidden lines in Oxygen and Sulphur. The
114 Chapter 5. Detecting the wind I
EUV model can successfully reproduce the NeII line (Alexander, 2008), although as discussed above,
the EUV model cannot explain the luminosities of the OI line observed in all T Tauri stars with NIR discs
by Hartigan et al. (1995), and as described in Chapter 3 the EUV field in the presence of the X-rays, is
unable to reach the disc’s surface. Therefore, the TW Hya observations need to be considered within the
framework of the X-ray photoevaporation model, which as I have discussed in this thesis, best describes
the true photoevaporation flow from discs.
TW-Hya is a 0.7M star (Muzerolle et al. 2000) at a distance of 51± 4pc (Mamajek et al. 2005),
with an X-ray luminosity of 1.4×1030 erg s−1(e.g. Kastner et al. 1999). Therefore, the most appropriate
model to use for a first analysis is the 0.7M, LX = 2×1030 erg s−1 calculated in this work. Furthermore,
TW-Hya is a disc that presents at very low inclinations, with estimates in the range 4–8o, thus effects
from absorption along the line of sight through the disc’s atmosphere can be considered a minimal effect
(e.g. Schisano et al. 2010). The first comparison that can be performed is the measured line fluxes;
the observed line flux in NeII is 4.5± 0.3× 10−14 erg s−1 cm−2 compared to a predicted line flux of
6.6±1×10−14 erg s−1 cm−2 for the LX = 2×1030erg s−1, and therefore a simple scaling to the observed
X-ray luminosity of TW-Hya means that the predicted line luminosity is consistent with the observations.
In order to compare the theoretical line profiles to the observed profiles they are scaled to the same total
flux and I use the 6o model as it lies in the middle of the current estimates for the inclination of the
inner disc (Although at low inclinations the line profile is not a particularly strong function of inclination
and cannot be used to further constrain the inclination). This comparison is shown in Figure 5.7, which
shows excellent agreement with the observations. Thus it is clear that while the NeII line provides clear
evidence for a photoevaporative flow emerging from the disc, it is unable to constrain the mass-loss rate,
Since the line can be matched by a 10−10M yr−1 pure EUV flow, or a 10−8M yr−1 X-ray driven flow.
This can be understood since the NeII line traces electron density and the EUV flow and X-ray flow have
very similar electron density structures. The EUV flow has an ionization fraction of unity, while the X-
ray flow has an ionization fraction of approximatively ∼ 0.01 in the region of NeII emission. Therefore,
the two order of magnitude in mass-loss rate is counteracted by a two order of magnitude difference in
ionization fraction.
As both the EUV- and X-ray model produce a good match to the NeII data, it seems that this particular
diagnostic cannot be used to discriminate between them. Therefore other diagnostics are required such
as the OI 6300A˚ line discussed previously, that only the X-ray driven photoevaporation model which
predicts a largely neutral wind has been able to reproduce the observed luminosities and blue-shifts.
Towards this aim Pascucci et al. (2011) also considered the emission from optical forbidden lines in
OI, SII and MgI with non-detections presented in a few other optical lines (NII, OII). However, unlike
the observations of Hartigan et al. (1995) these optical lines show no blue-shifts, but with luminosities
(in solar units) in OI 6300A˚ of 1.1× 10−5, OI 5577A˚ of 1.5× 10−6, SII 4067A˚ of 1.4× 10−6, a SII
4077A˚ upper limit of< 4.8×10−7 and MgI 6.4×10−7. While the Oxygen line flux is roughly consistent
with the model predictions (note again the model prediction for the OI 5577A˚ line is a lower limit as
the model does not include the collision rates with neutrals), it over predicts the Sulphur and Magnesium
lines. Pascucci et al. (2011) suggested that the low luminosities could arise from a depletion of sulphur in
the disc, something suggested by Lamzin et al. (2004) who argued that refractory elements are depleted
in the TW Hya gas disc. This emphasises the problem that elemental abundances will always be an issue
of in these calculations leading to an additional sources of uncertainty. Or as we discuss below if there is
5.5 A direct comparison: TW-Hya 115
−40 −30 −20 −10 0 10 20 30 40
−0.01
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
Velocity [km s−1]
N
or
m
al
is
ed
 In
te
ns
ity
Figure 5.7: Theoretical NeII profile for primodial disc (solid) and preferred ‘gap’ disc (dashed) discussed
below, compared to the observed spectrum of TW Hya at six different PAs: green - 0o, magenta - 90o,
black - 180o, red - 270 o and grey - 333o.
a gap in gas disc that corresponds to the dominant emission region then these lines will be observed with
significantly lower luminosities.
However, the most curious aspect of these observations is the fact that they are not blue-shifted. This
is curious as the optical lines have higher Boltzmann factors and thus are preferential emitted from gas
that is hotter than the NeII emitting gas. This is clearly shown in Figure 5.3, where the OI emission traces
the hot ? 104K gas at small radii, whereas the NeII emission traces the flow at larger radii. This means
that while there is a photoevaporative flow emerging from the disc at radii ? 4AU, there is no overall
blue-shifted flow emerging from inside a few AU. Since TW Hya has been classified a ‘transition’ disc
(e.g. Calvet et al. 2002), this may provide an explanation for this observation as the inner disc is optically
thin to optical and IR photons.
As mentioned at the start of this Chapter, TW Hya is an accreting ‘transition’ disc consistent with a
116 Chapter 5. Detecting the wind I
photoevaporative origin. Thus there are several possible inner gas disc structures depending on which
mechanism created the inner dust hole. I will now discuss the implications of these structures for the
observed optical emission lines.
5.5.1 The structure of TW Hya’s inner gas disc
In the Introduction I outlined several mechanisms which are thought to give rise to an observed accreting
‘transition’ disc: photoevaporation, a planet embedded in the disc (Rice et al. 2003) and grain growth
(Dullemond & Dominik, 2005). For TW Hya to still be an accreting ‘transition’ disc there must be
an inner gas disc that is being accreted by the star. However, in the case of grain growth the gas disc
is unchanged from a primordial one, and the flow topology will be identical to a primordial disc. An
embedded planet creates a gap in the gas disc, where this gap has size that depends on the mass of
the planet. For a small planet the gap will be much smaller than the scale of the photoevaporative
flow (Org), and the photoevaporative flow will be unchanged from the primordial one. However, a
massive planet/binary companion could open a gap larger than the photoevaporative flow scale, and
thus create a gap in the flow itself. Therefore, like a transition disc created through grain growth, a
transition disc created by a small embedded planet will have a photoevaporative flow almost identical
to the photoevaporative flow from a standard primordial disc. Although, with an opacity hole inside the
inner edge of the outer gas disc, due to the removal of dust opacity. However, in the case of a gap opened
by photoevaporation it can be much larger, and in the case of TW Hya possible extend from ∼ 0.6AU
(∼ 0.1rg) to ∼ 2− 4 AU where the inner edge of the dust disc is found (e.g. Calvet et al. 2002, Razak
et al. 2007). Therefore, the photoevaporative flow structure may well be different to the primordial flow
near the gap. These scenarios are schematically shown in Figure 5.8, with the possible flow structures
and emission regions for the optical lines and the NeII line. The case of a standard primordial disc is
shown for instructive purposes at the top, where the OI emission arises from around 1AU, whereas the
NeII emission arises at slightly larger radii ∼ 5 AU. For a transition disc created by a small planet or
by grain growth, the flow structure and emission regions are unchanged, the only difference is the inner
2-4AU of the disc are optically thin in the line. In the final case there is a large gap in the disc, with an
inner optically thin (in dust and the line) gas disc extending out to∼ 0.1Rg and an outer gas and dust disc
starting at 2–4AU. In this case the OI emission only arises from the bound atmosphere of the inner disc
and the NeII emission arises from a photoevaporative flow being driven of the outer disc, and there is no
emission from within the gap.
In all cases the non blue-shifted optical lines can be understood, since TW Hya is at very low inclina-
tion then the line of sight will see both sides of the flow and hence the red-shifted flow will counterbalance
the blue-shifted flow leaving a total line profile that is centred on the stellar velocity.
In order to investigate these possibilities, I use the primordial disc model calculated earlier to estimate
the OI 6300A˚ line profiles from the type of disc structures expected. I note that since the NeII emission
is dominated at radii larger than the inner hole the line profile shown in Figure 5.7 is unaffected. When
the flow topology is primordial disc-like (grain growth or small planet); to represent a dust hole in the
gas disc the disc’s mid-plane is made optically thin to the line, thus any lines of sight that go through
this dust hole ‘see’ both sides of the wind. In the case of a photoevaporation created transition disc it is
much harder to develop a model; to do so properly would require a new hydrodynamic calculation, since
5.5 A direct comparison: TW-Hya 117
Figure 5.8: Schematic showing possible disc structures for TW Hya. The blue region represents the
OI emitting gas, and the green represents NeII emitting gas. The top disc represents a primordial gas
and dust structure. The middle disc shows an inner opacity drop in the dust disc due to grain growth
or a small gap due the presence of a small planet; however the gap in the gas disc is small compared to
the photoevaportive flow scale. The bottom disc shows a large gap in the gas disc, either carved out by
photoevaporation of the presence of a massive planet/binary companion.
A black filled disc indicates it is optically thick to the line emission, whereas a white disc represents that
it is optically thin to line emission.
118 Chapter 5. Detecting the wind I
−30 −20 −10 0 10 20
0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
N
or
m
al
is
ed
 In
te
ns
ity
−20 −10 0 10 20
Velocity [km s−1]
−20 −10 0 10 20 30
Figure 5.9: Comparisons of theoretical OI 6300A˚ line profiles to the observed emission, the blue points
represent the median OI 6300A˚ profile (Pascucci et al. 2011) and the lines represent the theoretical
profiles. The left-hand panel shows a theoretical profile for a primordial disc. The centre panel shows a
primordial gas disc with an opacity hole at 2AU (dashed) and 4AU (solid). The right-hand panel shows
a gap in the primordial disc from 0.6AU to 2AU (dashed) and 4AU (solid).
I have performed no calculations with an inner gas disc. However, a first approximation comes from
starting with the primordial flow structure and making the inner hole optically thin in the line as before,
along with removing the flow and disc from within the gas disc’s gap. This is a reasonable starting point
since in this case I expect the OI to come mostly from the bound inner gas disc, which is fairly accurately
described in the primordial disc calculation.
In Figure 5.9 I show these estimated line profiles compared with the observed profile and note the
other observed optical lines have an identical morphology in both the observations and models.
This clearly demonstrates that while a line centred on the stellar velocity can be accounted for by any
of the ‘transition’ disc models, the models that do this by ‘seeing’ both sides of the wind are too broad to
match the observed line. In fact the line is close to being explained by thermal and Keplerian broadening
alone, without any kinematic broadening in the wind. This is shown in the case of the simple model used
as an attempt to replicate the emission from a disc with a large gap in the gas disc. As mentioned earlier
the OI emission arises from two components: the hot inner bound disc and the inner region of the flow.
In the gap model the inner flow falls in this gap so that the bound hot inner disc is the only component
remaining. Therefore, this naturally gives rise to a component that is only broadened by thermal and
Keplerian components. Furthermore, this analysis shows that the opacity hole must extend to a radii
greater than 2AU otherwise the OI emitting component would show up as a detectable blue-shift in the
line profiles. The model with a gap between 0.6AU and 4AU is clearly the best fit, although it does not
reproduce the line wings perfectly.
While this analysis is simplistic, it is clear that the optical lines cannot arise from a region of the disc
that is photoevaporating. Therefore, either the flow is somehow suppressed at small radii or there is a
large gap in the disc such that there is no flow arising from the inner disc. Resolved observations that
determine the radial location of the NeII and optical lines should help unravel the reason why the NeII
emission is emitted from within a photoevaporative flow and the optical lines are not. Should spatially
resolved spectroscopy confirm this gap scenario, then observational confirmation of a massive companion
5.6 Conclusions 119
would determine whether this is a ‘transition’ disc that has been generated by planet formation, otherwise
a photoevaporation origin is the remaining viable alternative.
5.6 Conclusions
In this Chapter I have calculated the line emission from gas within a photoevaporative wind and compared
this with observations. The main findings are summarised below:
1. Line emission provides a useful diagnostic to probe a photoevaporative wind, which can be used
to test the photoevaporation model.
2. The X-ray photoevaporation model provides a natural solution to the presence of a blue-shifted
OI 6300A˚ line, observed by Hartigan et al.(1995). However, an FUV or EUV driven photoevapo-
rative flow is incapable of reproducing this observation.
3. The NeII line provides an excellent tracer of the photoevaporative wind, although it is unable to
place constrains on the mass-loss rates.
4. Line emission from ‘transition’ discs can be used to to probe the presence and structure of an inner
gas disc.
5. The ‘transition’ object TW Hya shows direct evidence for photoevaporation in the NeII line, how-
ever there is no evidence in the optical lines for a photoevaporative flow. This indicates that the
gas disc in TW Hya is not a standard primordial gas disc, and the dust’s opacity gap is indicative
of a gap in the gas disc.
6. The preliminary findings based on rough modelling of the OI line suggest that the inner disc of
TW Hya is truncated at the same radius in both gas and dust.
In this Chapter I have discussed emission lines as a way of investigating photoevaporative flows.
However, line emission is only a local probe of certain regions of the flow at specific density and tem-
perature regimes. Therefore, in the next Chapter I will consider a global tracer for a photoevaporative
flow.

6
Detecting the wind II: dust emission in edge-on discs
IN the previous two Chapters the tests of the photoevaporation model have either relied on indirectstatistical comparisons, or local tests of small scale regions of the photoevaporative flow. Whilstthese tests have proved invaluable in gathering information about the presence of photoevaporation
and its implications for disc evolution, it would be extremely useful if we had a large scale probe of the
entire flow.
In Chapter 5 I showed that line emission from the gas in the flow was locally confined to regions with
a specific density and temperature, which does not provide an overall picture of the global flow structure.
However, we know protoplanetary discs also contain a large reservoir of small dust grains, which can
be entrained in the photoevaporative flow. These, entrained dust grains could then be carried to large a
radius, enabling the global flow morphology and structure to be traced by the dust grains. Obviously, this
extra dust mass in the photoevaporative wind is going to have negligible impact on the SED of a young
star, as the SED is dominated by the dust properties at and just above the disc’s photosphere. However, in
the case that the disc is edge-on, the central star is obscured by the optically thick inner dust disc, allowing
low surface brightness imaging of scattered light from dust grains in the optically thin regions. Scattered
light imaging of edge on discs has been used to infer properties of the disc’s atmosphere (e.g. Watson et
al. 2007) and the presence of in-falling rotating envelopes around young (Class I) stars (Stark et al. 2006).
Therefore, it is certainly possible that dust entrained in a photoevaporative wind may also be a possible
target for scattered light imaging. With a proposal of this type the main issue is the possibility of actually
observing the faint emission from a tenuous photoevaporative wind above the disc’s atmosphere, and this
boils down to resolution and detection limits. Therefore, flows which are more luminous and occur on
larger scales will be easier to detect and resolve with current facilities. The scale of any photoevporative
flow clearly scales with mass through Rg ∝ M∗ and as we go to larger mass the scale of the flow will
become larger. In order to consider how the luminosity of dust in a photoevaporative region will vary
with mass, I assume that the luminosity of such a region is dominated by scattered light (which it will
121
122 Chapter 6. Detecting the wind II
be since the region will be optically thin). So the luminosity of this region Ls will scale as the number of
scattered photons, which is given by:
Ls ∝ L∗Ps (6.1)
where Ps is the photon scattering probability, which will scale as the size of the region over the photon
mean free path (Rg/`mfp), where `mfp is given by the density multiplied the scattering cross-section (or
size of the dust grain squared). Now using M˙w= n×A×cs, we see that the density is given by n∝ M˙wR−2g
and the luminosity of this region scales as:
Ls ∝ L∗M−1∗ M˙w(M∗)σ(M∗) (6.2)
Since the luminosity of pre-main sequence stars varies as a high power of stellar mass L∗ ∝M3−4∗ then the
luminosity of these regions will also increase as a high power of stellar mass. Herbig Ae/Be stars are more
luminous than their lower mass T Tauri counterparts and provide an opportunity to observe the expected
large scale morphology in resolved images. As mention above such analysis will be only useful for
discs observed at nearly edge-on inclinations. Several edge-on discs around young stars exhibit extended
emission above and below their mid-plane at NIR wavelengths (e.g. Padgett et al. 1999 & Perrin et al.
2006) and some have been further imaged through AO systems providing excellent spatial resolution,
as in the case of PDS 144N (Perrin et al. 2006). Therefore, edge-on discs around Herbig Ae/Be stars
provide the perfect opportunity to detect a disc wind on a global scale, through light scattered by the dust
grains that will inevitably be entrained by the wind.
In this Chapter I present a ‘proof of concept’ calculation to investigate the imprint of photoevaporation
on the extended emission observed around edge-on Herbig Ae/Be stars. I will compare our model to
current observations of edge-on discs, along with following the fate of crystalline grains in the wind as
they are transported to large radii.
6.1 Model
Considering the force balance on a dust grain, it is easy to show that small dust particles are entrained in
a photoevaporative wind, which carries them out to large distances. Takeuchi et al. (2005) showed that
the drag force on a grain is approximately:
Fd ≈ mdρwv
2
w
ρda
(6.3)
where md , ρd and a are, respectively; the mass, the density and the radius of a dust grain (assumed to
be spherical) and ρw and vw are the density and velocity of the wind. At large radius (z/R> 1) the flow
is approximately spherical, implying ρwvw falls off as 1/r2. Since gravity also falls off as 1/r2 and vw
increases monotonically with radius, then if a grain is still entrained when the centrifugal force falls away
(l ∼ Rlaunch), it will be entrained permanently, allowing dust to be carried to very large distance from the
star, since the drag force will dominate over gravity.
I build a simplified model to test the possibility of extended emission from dust grains due to a
photoevaporative wind, and I choose a set of assumptions that allows us to place an upper limit on
6.1 Model 123
this expected level of emission. Namely I assume that the entire dust population is able to reach the
launching surface of the wind via some turbulent mechanism and ignore the effects of settling and grain
growth. This simple model ignores the details of the bound regions of the disc and thus ignores any
emission produced by this region, including only the wind itself and its contributions to emission. This
simplification means the model will not be accurate near the mid-plane, where emission from dust in
the disc’s upper atmosphere dominates over the wind emission. Therefore, the model will be unable to
reproduce the observed intensities and optical depths seen at several scale heights above and below the
mid-plane in an edge-on disc; however, here we are interested in the extended emission, which in the
absence of an in-falling envelope will be dominated by the photoevaporative wind.
The method for constructing scattered light images can be split into three separate parts: (i) hydro-
dynamic calculations of the photoevaporative wind; (ii) calculation of the dust profile distribution and
crystallinity based on the hydrodynamic solution; (iii) radiative transfer modeling of the dust distribution.
6.1.1 Hydrodynamic photoevaporative wind
While the wind driving source for lower mass (T Tauri) stars is likely to be X-rays as described through-
out this thesis, the wind driving source around intermediate mass stars has not yet been thoroughly
investigated. X-ray photoevaporation may still occur to some degree; however, the lower LX/Lbol ratio
of Herbig Ae/Be compared to T Tauri stars (e.g. Albacate Colombo et al. 2007) and their higher EUV
fluxes, may reduce the role of X-rays in driving the wind. I adopt here the EUV driven wind of Hol-
lenbach et al (1994) and hydrodynamic solution of Font et al. (2004), which is a simple and scalable
hydrodynamic solution, allowing us to consider a wide range of parameter space, something not possible
with more complicated FUV models (Gorti & Hollenbach 2009) or X-ray models described in Chap-
ter 3. The simplified EUV treatment is suitable for the purpose of this work which aims at being the first
approach in studying the qualitative aspects of scattered light emission from a disc wind. To this end, I
have repeated the calculation of Font et al. (2004) and Alexander (2008) and I refer the reader to these
papers for a detailed description of the model setup, although the basics are summarised below.
6.1.2 Model Setup
As the mass-loss rates are not well known for Herbig Ae/Be stars, I have left the ionizing luminosity
(which sets the mass-loss rates) for an EUV driven wind (Hollenbach et al. 1994) as a free parame-
ter and consider the effect of changing mass-loss rates on the morphology, colour of the emission and
crystallinity distribution in the winds. I consider an EUV wind from a primordial disc around a 2.5M
star with a range in ionizing luminosity from 1041 to 1045 s−1 in steps of 1 dex. These luminosities
correspond to mass-loss rates in the range 10−10−10−8Myr−1 and are similar to values calculated by
Alexander et al. (2004). In order to calculate the images, I compute scattered light images at 1.6, 2.1
& 3.8µm with band passes corresponding to the H, K’ & L’ bands. In order to consider the relative
contributions of crystalline to amorphous dust I take the calculated crystallinity fractions in the disc of
Dullemond et al. (2006) for an evolved Herbig Ae/Be star. I further assume that this disc crystallinity
fraction is the same at the base of our photoevaporative wind, such that the crystallinity fraction along the
streamline can be calculated by following the thermal evolution of the dust entrained on that streamline.
124 Chapter 6. Detecting the wind II
Radius [rg]
H
ei
gh
t [r
g]
 
 
0 5 10 15 20 25
0
2
4
6
8
10
12
14
16
18
20
D
en
si
ty
 [n
g]
−4
−3.5
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5          20 km/s
Figure 6.1: EUV wind structure: the colour-map shows the density, while the arrows show the direction
and magnitude of the gas flow. Note that just after launch the flow is very close to being spherical.
Numerical EUV Photoevaporative Wind
In order to determine an accurate kinematic structure of the wind I must compute a numerical solution
to the problem. I use the ZEUS2D code (Stone & Norman 1992a) and employ a spherical grid with
θ = [0,pi/2], the radial grid cells are logarithmically spaced, such that I have adequate resolution at
small radius to resolve the onset of the flow. The calculation is an isothermal wind calculation with the
sound speed set to cs = 10km s−1 and the radius scaled to the length scale rg. I use a radial range of
r = [0.05,40]rg with Nr = 240 and Nθ = 50. The number density at the base (i.e. the density along the
θ = pi/2 axis) of the wind was calculated semi-analytically by Hollenbach et al. (1994) and scales as
R−3/2 for R< Rg and as R−5/2 outside Rg, as in Font et al. (2004) and Alexander et al. (2008) I use the
smooth base density profile suggested by Font et al. (2004) that varies between the two power laws:
n(R) = ng
[
2
(R/Rg)15/2+(R/Rg)25/2
]1/5
(6.4)
where ng is the density at Rg which was determined through the numerical calculations of Hollenbach et
al. (1994) to be:
ng ≈ 2.8×104
(
Φ∗
1041s−1
)1/2( M∗
1M
)−3/2
cm−3 (6.5)
where Φ∗ is the ionizing luminosity. I note that this hydrodynamic calculation does not include a cold
bound ‘disc’ component, since as discussed above the base density structure of the wind is known a
priori. This base density, along with Keplerian rotation is reset at every time-step and the model is al-
lowed to evolve to a steady state, launching a wind from the grid’s mid-plane (representing the disc’s
surface). As expected, I find excellent agreement with the results of the Font et al. (2004) and Alexan-
der (2008) calculations. In Figure 6.1 I show a plot of the converged wind structure showing that the
wind is approximately spherical once it has reached several scale heights, in agreement with our earlier
discussion.
6.1 Model 125
Radius [rg]
H
ei
gh
t [r
g]
 
 
0 5 10 15 20 25 30 35
5
10
15
20
25
30
35
M
ax
iu
m
 G
ra
in
 S
ize
 [µ
m
]
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
Figure 6.2: Colour map of the maximum size of entrained dust particles at a given position for Φ∗ =
1043s−1, I note that most dust comes from within 1-5rg.
6.1.3 Dust Distribution
In order to calculate the dust distribution in the wind, I must make some assumptions about the under-
lying dust distribution in the disc. I adopt a dust to gas mass ratio of 0.01, and a power law grain size
distribution of index -3.5 (MRN), with grain sizes ranging from amin=5×10−3µm to amax=1mm and I
adopt spherical grains with a density of 1g cm−3. Assuming that the dust is fully mixed within the disc up
to the transition between the cold bound disc and the hot EUV heated flow, I then calculate streamlines
from the base of the flow to the edge of the grid. For each streamline, I then compute the force balance
between the drag force (calculated from Equation 1), gravity and the centrifugal force. I take the dust
grain as entrained if the net force along the streamline is > 0. I then obtain the maximum grain size
entrained along the entire streamline (making sure to check that each grain can be entrained the entire
length of the streamline). In Figure 6.2 I show the obtained maximum grain size as a function of position
in the flow for an ionizing luminosity of Φ∗ = 1043s−1.
The dust density can then be computed from this maximum grain size under the assumption that the
dust all comes from the same underlying population described above. In Figure 6.3 we show the obtained
dust density, noting that the combination of a photoevaporative wind and a selective dust population
naturally reproduces a ‘wingnut’ structure.
6.1.4 Radiative Transfer
I calculate the radiative transfer as a two step process: The temperature structure of the dust is calculated
using a 3D grid calculated from the 2D dust density calculations, using a version modified of MOCASS-
INV2 to allow cell-to-cell variations of the grain size distribution; Then this temperature structure is used
126 Chapter 6. Detecting the wind II
Radius [AU]
H
ei
gh
t [A
U]
 
 
0 50 100 150 200 250 300
50
100
150
200
250
300
D
us
t D
en
si
ty
 [g
 cm
−
3 ]
−26
−25.5
−25
−24.5
−24
−23.5
−23
−22.5
−22
Figure 6.3: Dust mass density distribution in the wind model for the Φ∗ = 1043s−1 calculation.
to determine synthetic scattered light images .
I interpolate the results from Section 6.1.3 onto a 3D Cartesian grid with resolution NX ×NY ×NZ =
99×99×50 assuming azimuthal symmetry, using the range in the x and y axis of [-32,32]rg and in the
z axis of [0,32]rg. The input SED used to irradiate the wind is taken from the model set of Robitaille
et al. (2006), Model Number:3011099, corresponding to a 2.51M star with an effective temperature of
9480K and radius of 2.18R, surrounded by an optically thick disc extending into the dust destruction
radius. The irradiating SED is placed at the origin of the Cartesian grid; this approximation is valid in
this case where we are mainly interested in scattering of NIR photons on the much larger scale of the
wind i.e. on a scale much greater than the scale the disc emission arises from.
In order to calculate images of the dust distribution, I have used the simple ray tracing driver for
MOCASSIN, called MOCASSINTHINIMAGE. The standard Monte-Carlo radiative transfer techniques
used to calculate scattered light images in MOCASSIN (i.e. by capturing scattered photons along the
line of sight of interest) is inappropriate for use in this case. The wind structure is too optically thin to
obtain a converged image with a computationally feasible number of energy packets, since most packets
leave the grid without interacting with the distribution1. MOCASSINTHINIMAGE performs a ray tracing
calculation throughout the grid in both scattered and thermal emission, under the assumption that the
medium is optically thin (true of the wind always). I note again that since I have not explicitly included
the dust in the ‘bound’ region of the disc we will not calculate scattered light emission from this region. I
also assume the disc mid-plane is optically thick to all radiation, and hence only consider one hemisphere
of the wind.
1It is important to emphasise that this is not the case for the temperature calculation discussed above, where an interaction
is not required with a photon packet to include its contribution to the radiative energy density and hence temperature in that cell
(see Lucy 1999; Ercolano et al 2003).
6.2 Images 127
6.1.5 Calculations of Crystallinity Structure
The origin of crystalline dust grains detected in protoplanetary discs via infra-red (IR) spectroscopy (e.g.
Bouwman et al. 2001) and directly within our own solar system (e.g. Wooden et al. 1999) is still a matter
of discussion. The source of all dust in star formation - the ISM - is inferred to be entirely amorphous (e.g.
Kemper et al. 2005). Spectroscopy of discs (van Boekel et al. 2005; Apai et al. 2005) indicates a non-
negligible level of crystalline dust outside the crystalline radius, where grains are hot enough (T > 800K)
to be thermally annealed and converted into crystalline grains (e.g. Fabian et al. 2000; Bockelee´-Morvan
et al. 2002). This discovery has lead to the development of disc models with radial mixing to allow
crystalline grains that formed in the hot inner disc to be transported to larger radii (e.g. Morfill & Voelk
1984; Gail 2001, 2002; Wehrstedt & Gail 2002; Bockelee-Morvan et al. 2002; Dullemond et al. 2006;
Hughes & Armitage 2010). Another suggestion put forward by Shu et al. (1996), is that crystalline grains
could be carried outwards by an x-wind. Given that the crystallization radius of <1-2AU is smaller than
the minimum launch radius for a photoevaporative wind (> 2AU) around Herbig Ae/Be stars and lower
mass stars, the photoevaporative wind cannot be the direct source of the crystalline grains at larger radii,
but could work in combination with radial mixing to produce the observed enhancement. Furthermore,
crystallinity around edge-on discs can be used to probe the source of the dust in the extended emission,
since a disc origin will give rise to crystalline dust grains in the wind while an in-falling envelope will
contain only amorphous grains.
In order to consider the structure of the dust grains in the wind, I have built a very simple dust
crystallinity model. I assume that the grains are launched from a disc that is infinitesimally thin and has
the crystalline structure calculated by Dullemond et al. (2006), for a Herbig star with a mass of 2.5M
after 3.3Myr of evolution. Any dust that was amorphous and is heated to a temperature > 800K while
entrained in the wind is then assumed to immediately become totally crystalline, as the crystallization
time-scale is shorter than the flow time-scale (e.g. Dullemond et al. 2006). Any dust that starts off
crystalline or becomes crystalline at any point in the flow is assumed to be crystalline for the remainder of
the flow regardless of the temperature it reaches at any other stage (I find that no dust in the flow reaches
a temperature exceeding the evaporation temperature, which would destroy crystalline dust grains). In
order to calculate a radial profile of the fraction of dust in the wind that is crystalline/amorphous I must
apply some cut off on the flow, since the total mass of the flow is a function of time since it switched
on, rather than a converged quantity. Since much of the information on dust structure comes from the
10µm silicate feature, a temperature cut-off is most appropriate. Thus I calculate the mass-fraction of
crystalline grains out to a temperature of 100K; as such this should provide a reasonable estimate of
the observable crystallinity-fraction (anything colder is extremely unlikely to make much impact on the
10µm silicate feature). Furthermore, since each cylindrical radial region will be cut by many streamlines
with different crystallinity fractions, in order to calculate an average crystallinity fraction we perform a
number density weighted average out to the temperature cut-off.
6.2 Images
In Figure 6.4 I show calculated images of the 5 different models in the H, K’ & L’ bands; they all show a
‘wingnut’ morphology. The emission is strongest in the bluer bands with the H band being strongest and
128 Chapter 6. Detecting the wind II
H
ei
gh
t [A
U]
250
500
250
250
 
 
250
Radius [AU]
−500 −250 0 250
0
250
−250 0 250 −250 0 250 −250 0 250 500
3 4 5 6 7 8 9 10 log(Intensity)
[Arbitrary Units]
Figure 6.4: Synthetic images for disc models with irradiating fluxes Φ∗ = 1041,1042,1043,1044,1045erg
s−1, the far left hand column shows the image in the H band, the next column displays the K’ band and
the next column the L’ band. The far right column displays an RGB composite image (L’,K’ & H bands
respectively). The images are individually scaled so that there is a 5dex spread between the brightest
pixel and the darkest. All images assume the disc is edge-on (i.e. an inclination of 90o), therefore I block
out the star assuming it would be blocked out by the presence of an optically thick disc.
the L’ being the weakest.
As the mass-loss rate increases the emission becomes more extended and reaches larger radii, since
the density along a streamline falls to a given value at larger radius. I also find that all of the emission
is scattered light from the central star and inner disc, many orders of magnitude above any thermal
emission from the wind itself, in agreement with the assertion that the wind is optically thin. The fact
that the emission is dominated by scattered light allows us to understand the evolution of the colour
variation and emission morphology as the mass-loss rates change. For high mass-loss rates the emission
morphology follows the gas density distribution of the wind (i.e. essentially spherical with a gap on axis
in a region where the wind is negligible), while more pronounced ‘wingnut’ profiles are clearly visible
for lower values of mass-loss rate. This arises from the fact that scattering at a given wavelength is
dominated by dust grains of comparable sizes. Given that the bands we are interested in are in the range
1-5µm, then when there is large spatial variation in grains of this size, then a profile that differs markedly
from the gas density would be expected (e.g. the wingnut). However, for large enough mass-loss rates,
dust grains with sizes in the range 1-5µm can be entrained in the flow everywhere and the emission
then simply follows the gas density distribution. I find that all of the synthetic images are dominated
by blue light; this is somewhat expected since the smallest grains can always be entrained in the flow
and outnumber the larger grains, thus dominating the opacity for an MRN size distribution as the one
considered here. Furthermore, the negative slope with wavelength of the irradiating field, which in the
1-5µm range is dominated by emission from the disc, means that short wavelength photons are more
abundant, contributing to the blue appearance of the synthetic images.
6.3 Crystallinity Profile 129
Table 6.1: Model Luminosities
log(Φ∗) s−1 log(Luminosity) erg s−1 Hz−1
H K’ L’
41 15.8 15.4 14.0
42 17.4 17.2 16.2
43 18.3 18.2 17.5
44 18.9 18.8 18.2
45 19.4 19.3 18.7
6.3 Crystallinity Profile
We have used the wind profiles and dust temperature information from the radiative transfer calculation
(Section 6.1.4) to calculate crystallinity fractions in the wind. I have assessed the impact of the wind
on the observable crystallinity fraction for discs, where the inner disc is both obscured (edge-on) and
unobscured. In the unobscured case, to observe the disc’s crystallinity fraction the wind will also make
a contribution along the line of sight. I compare the disc’s emission to the wind’s emission at 10µm and
find that at all radii the disc dominates over the wind, meaning that the observed crystallinity fraction for
non edge-on discs is unaffected by the photoevaporative wind. However, when the disc is observed close
to edge-on, the disc’s contribution will be undetectable and the crystallinity fraction in the wind may be
studied.
It is important to note at this point, that the radial distribution of the crystallinity fraction in a wind
is likely to differ from the disc’s ‘native’ distribution. In Figure 6.5 we show radial profiles of the
crystallinity fraction obtained from a photoevaporative wind (solid black line) and compare it to the input
crystallinity distribution of the disc (dashed blue line). The figure shows that at small radii< 2.5AU there
is no crystallinity in the wind simply because there is no wind inside this radius to carry the dust grains to
large heights. Between 2.5-200AU there is a falling crystallinity fraction; however, it is always in excess
of the disc’s underlying fraction, beyond outside 200AU it continues to fall, even though the crystallinity
fraction in the disc has now risen above the level in the wind. The crystallinity fraction in the wind
remains the same for all ionizing luminosities and is insensitive to the temperature cut-off described in
Section 6.1.5 in the range 150-50K.
The observed wind crystallinity profile is a direct consequence of the streamline topology, which
implies that the grains in the wind at any given cylindrical radius must have originated in the disc at a
smaller radius and hence, in general, with a higher crystallinity fraction. The rise in the disc crystallinity
at a radii > 100AU is not translated into the wind population, since most of the dust in the wind is
entrained inside 5rg ≈ 100AU. The low wind rates at > 100AU, coupled with the geometric dilution of
the wind, mean that the crystallinity fraction in the wind continues to fall with radius. The insensitivity
of the crystallinity fraction to the mass-loss rate is simply arises from the fact that the number of dust
particles at a given point scales approximately as ndust ∝ Φ
1/2
∗ . This is the same scaling as the gas, since
it is largely unaffected by the change in maximum grain size as crystalline and non-crystalline grains are
entrained to the same extent.
130 Chapter 6. Detecting the wind II
1 10 100
0.1
1.0
Radius [AU]
Cr
ys
ta
llin
ity
 
Figure 6.5: Cylindrical radial profiles of the wind crystallinity fraction (solid line) calculated as a number
density weighted mean in the wind and the underlying disc distribution taken from Dullemond et al.
(2006) for an evolved 3.3Myr 2.5M Herbig star (dashed line).
6.4 Comparisons to Observed Edge-on Objects
There are several objects that show extended emission above and below the dark dust lane indicative of
an optically thick disc. All the objects in the sample presented by Padgett et al. (1999), which includes
the well known ‘Butterfly Star’, show extended scattered light emission above and below a dark dust
lane. However, the objects in the Padgett et al. (1999) sample are all relatively low-mass (Class I)
young stars, and the high fluxes detected in scattered light indicate that there is much more mass present
than would be expected in a photoevaporative wind. Furthermore the observed structure is much more
filamentary than our model predict. These observations clearly indicate that a photoevaporative wind is
too weak to have produced the observed extended emission. Padgett et al. (1999) suggested that these
objects are young (Class I) objects, with the extended emission arising from an in-falling envelope. Wolf
et al. (2003) and Stark et al. (2006) have successfully reproduced these observations using detailed
radiative transfer modelling that include the combination of a disc component, an in-falling envelope and
an outflow cavity.
However, PDS 144N (Perrin et al. 2006) is a more promising candidate for a photoevaporative wind
(shown in Figure 6.6) , with images showing a morphology very similar to the ‘wingnut’ shapes predicted
for the lower mass-loss rates Φ∗ ≤ 1043erg s−1. Furthermore adopting the Φ∗ = 1043erg s−1, which pre-
dicts a surface brightness of 0.01 Jy arcsec−2 in H at a radius and height of (R= 100,z= 100)AU, I can
derive a distance to PDS 144N. By comparing the observed image to our predicted image I find a distance
of 200-400pc, which is in the range of previous estimates2. As discussed in Perrin et al. (2006), it is
2Previous estimates range from 140 to 2000pc. Perrin et al. (2006) use a nearby A5 Herbig star (PDS 144S) to derive
a distance of 1000pc, taking the system to be a wide binary. However, such a distance places the system 300pc above the
6.4 Comparisons to Observed Edge-on Objects 131
Figure 6.6: Three colour plot of PDS 144 where the H,K’ and L’ band are shown as blue, green and red
respectively. The scale bar assumes a distance of 1000pc (Figure from Perrin et al. 2006).
highly unlikely that the morphology arises from foreground extinction, as the suggested explanation to
the morphology of the Padgett et al. (1999) sample. Perrin et al. (2006) indeed discuss photoevaporation
as a possible source for the extended emission. However, they oversimplify the photoevaporation model
by just considering the radial scale rg, ignoring the fact that, as discussed in Section 2, dust grains can
be carried to very large distances in the wind, and that at several flow scale heights from the disc the
streamlines are spherical. The models show that the use of rg in discussing the characteristic scales of
the flow in dust is a poor approximation, since, as shown in Figure 6.2, dust entrained at ∼ 1rg can be
easily carried to radii and heights of > 20rg, where the maximum grain size increases with height. The
reason for this morphology is streamlines at higher heights originate at smaller cylindrical radii where the
gas density is high enough to entrain large grains, and this produces a dust density that varies on scales
different to rg as shown in Figure 6.3. Perrin et al. (2006) present models of the object in which they
construct an in-falling envelope with a jet cavity surrounding an extended passive accretion disc. While
their model (Figure 6 of Perrin et al. 2006) can also reproduce extended emission, it cannot reproduce
the distinctive ‘wingnut’ morphology seen in the observations and in the model presented in this work
(see the top panels of Figure 6.4).
However both the model presented by Perrin et al. (2006) and the ones presented in this work fail
to reproduce one important aspect of the observations. The observed colour of PDS 144N is such that
the extended regions of the emission become dominated by the redder scattered light. This is opposite to
what is predicted by the Perrin et al. (2006) model which predicts a colour variation of red to blue with
height and the blue band dominating the emission at large height. This is due to a dust population whose
maximum grain size decreases with height (as you move from the disc population to the envelope - ISM
like - population). On the contrary, the photoevaporation model can reproduce the sign of the observed
Galactic plane. This would require the ejection of the wide binary system from a dense cluster, or formation in situ, both of
which are unlikely.
132 Chapter 6. Detecting the wind II
0 50 100 150 200 250 300 350 400 450 500
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Height [AU]
∆(
H−
X)
Figure 6.7: Change in colour (∆H-X) as a function of height above the mid-plane, at a cylindrical radius
of 100AU. Green line: X=K’ band colour; red line: X=L’ band colour. The model shown is theΦ∗= 1043
s−1 calculation.
colour variation i.e that the relative strength of the red light increases with height. This is demonstrated
in Figure 6.7 where we plot the change in H-K’ and H-L’ for the Φ∗ = 1043 s−1 model as a function of
height above the mid-plane for the Φ∗ = 1043 s−1, where there is variations in the grain population at
sizes of a few microns, which dominate the scattering opacity at NIR wavelengths. While our models
predict that the colours become redder with height above the disc (due to the fact that larger grains are
entrained at greater height, see Figure 6.2, and these scatter red light more efficiently), our models predict
scattered blue light to remain dominant at large heights. As discussed in Section 6.2, in our model blue
light dominates due to the spectral slope of the irradiating SED and the fact that small grains are present
in the wind at all heights. I estimate using Figure 3 of Perrin et al. (2006) a change in H-L’ (∆(H-L’))
of >1.5, at least three times larger than our simple model can predict. However, in order to allow the
redder bands to dominate over the bluer bands one needs to (i) increase the available disc emission at the
redder wavelengths so there is more original flux to scatter and/or (ii) remove the smaller grains which
are currently dominating the scattered light; (iii) increase the optical depth to the star/disc so the extended
region becomes optically thick, resulting in the reddening of photons as they pass through. Grain growth
in the disc provides a natural solution to both (i) and (ii). Dullemond & Dominik (2005) have shown that
the dust grains in a disc easily grow to ∼ 1µm reducing the opacity of the disc at shorter wavelengths
and resulting in a disc spectrum that falls less steeply or even rises to longer wavelengths in the 1-5µm
range. Furthermore, the removal of small grains from the disc will also result in the removal of small
grains from the wind, reducing the scattering efficiency at the shorter bluer radiation.
Future observations of the 10µm silicate feature may help disentangle the origin of the dust particles
in the extended region around PDS 144N since an in-falling envelope would be entirely composed of
amorphous ISM type grains, while, as shown by our models, a photoevaporative wind origin implies the
6.5 Conclusions 133
presence of crystalline silicate grains in the extended regions.
6.5 Conclusions
I have considered the fate of dust grains that can be entrained in a photoevaporative wind from the surface
of a disc surrounding a Herbig Ae/Be star. For a median mass-loss rate of ∼ 10−9Myr−1, I find that
grains up to radii of several microns can be entrained in the wind. I also show that, once entrained in the
wind, the dust grains will remain entrained and be carried out to very large radius (> 10rg).
1. The combination of a photoevaporative wind structure and a variable dust grain population re-
sulting from the variable drag force at the base of the wind, can naturally reproduce a ‘wingnut’
morphology of the dust density distribution in the wind.
2. The model is not applicable to the well known edge-on discs in the Padgett et al. (1999) sample,
which are too young and optically thick to be explained with a photoevaporative origin, but are
likely to arise from an in-falling envelope as shown in Wolf et al. (2003) and Stark et al. (2006).
3. Synthetic images show a wingnut morphology inferred from the dust density distributions, similar
to observations of the edge-on disc around PDS 144N (Perrin et al. 2006). However, the synthetic
images are dominated by emission from the smallest grains entrained in the flow, hence failing to
reproduce the colour gradient of the observations, which show redder emission at larger heights
above the disc. Grain growth in the disc, is a natural solution to the colour problem, and I estimate
severely depleted abundance of grains with radii smaller than 1µm.
4. I find that crystallinity fraction will remain unchanged when the disc’s photosphere can be ob-
served since the observable mass in the wind is much less than the observable mass in the disc.
5. When the disc photosphere is obscured (i.e. for edge-on discs) the crystallinity fraction in the wind
is significantly enhanced above the discs underlying crystallinity fraction.
The observations of edge-on discs around more massive stars, could provide a new opportunity to
study the structure of photoevaporative flows arising from the discs of forming stars. Where an
observed ‘wingnut’ morphology and colour gradient are indicators of dust entrained in such a flow.
Finally, I suggest that detection of crystalline grains in extended emission around an edge on disc
is indicative of a photoevaporative wind and argues against an envelope origin for the extended
emission.

7
Summary and Outlook
IN this thesis I have considered photoevaporation from protoplanetary discs as a dispersal mecha-nism. This is motivated by the need to identify the dominant dispersal mechanism (if in fact there isa dominant one) for protoplanetary discs, as observations indicated that disc lifetimes are too short,
and discs transition from a primordial to cleared state too rapidly to be explained by pure viscous effects.
Currently the two ideas for disc dispersal are the removal of disc material to form planets or the removal
of disc material in a photoevaporative wind. It is now observationally established that both these pro-
cesses must be occurring for at least some forming young stars. The main question this thesis attempts
to address is: Can photoevaporation provide a dominant dispersal mechanism around all young stars?
In order to study the photoevaporative flow, I have used a combination of simple scaling laws and
numerical methods, and Chapters 2 & 3 outline the attempts to consider photoevaporation from real
protoplanetary discs that are exposed to the full spectrum of ionizing radiation. This analysis has shown
that it is likely to be the X-rays that are responsible for driving and setting the photoevaportive mass-loss
rates, and that the mass-loss rates scale linearly with X-ray luminosity and do not explicitly depend on
the mass of the central star. Furthermore, given the wind topologies that arise from primordial discs, this
mass-loss rate is set at the X-ray wind’s sonic surface, where the flow is close to being spherical. This
condition places powerful constraints on the role of disc structure and the FUV flux on the derived mass-
loss rates. Since in a spherical flow the temperature at the sonic surface is just given by the Parker wind
result, which purely depends on stellar mass and position, then the sound speed at the sonic surface is
fixed. A fixed temperature at a given radius also fixes the density in X-ray heating, thus the mass-flux is
set at the sonic surface, irrespective of the underlying disc structure. Therefore, the underlying structure
of a protoplanetary disc must adjust itself to feed the X-ray driven photoevaporative flow at the correct
mass-loss rate, rather than the mass-loss rate adjusting itself to changes in the disc’s structure.
Once the accretion rate through the disc has fallen below the mass-loss rate in the wind, photoevapo-
ration opens a gap in the disc. This separates the disc into an inner and outer disc, where the inner disc
135
136 Chapter 7. Summary & Outlook
rapidly drains onto the central star exposing the outer disc to high energy radiation. The topology of the
wind then changes and the outer disc is eroded to larger and larger radii, where this efficient clearing
arises because mass-loss rate is approximately independent of the distance to the outer disc. Finally, at
some point the mid-plane column of the disc drops to a point where a sufficient mass fraction of the disc
is heated up and it is dynamically destroyed. Thus photoevaporation driven by the X-rays does provide a
dispersal mechanism that is capable of clearing the entire disc in a suitable time-scale.
In Chapter 4 the photoevaporation model developed was used to follow the evolution of an entire
population of viscously evolving discs, in order to asses whether photoevaporation can be a dominant
dispersal mechanism. To do this I follow the evolution of simple 1D disc models that are under the action
of viscosity and photoevaporation, where our currently uncertainty of the viscosity is parametrised over,
based on observations. This analysis shows that the observed spread in X-ray luminosity is sufficient to
reproduce the observed spread in current observables, e.g. accretion rate and disc lifetime, while repro-
ducing the required overall evolution with theoretically sensible viscosity parameters. Furthermore, the
model is able to account for a large fraction of so called ‘transition’ discs and the observed correlations
between X-ray luminosity and disc evolution diagnostics.
In Chapters 5 & 6 I have derived direct observable consequences of a photoevaporative wind, in both
gas and dust diagnostics. The gas line emission provides a powerful probe of the photoevaporative flow
and currently offers the only direct evidence for photoevaporation occurring from discs. Furthermore,
an X-ray driven photoevaporative wind provides the only explanation for the ∼ 5 km s−1 component of
the OI 6300A˚ line observed by Hartigan et al. (1995), and the other line emission currently compares
favourably with available observations. Chapter 6 discusses the observable implications of the small dust
particles that a photevaportive wind may entrain. I use higher mass Herbig Ae/Be stars as a test bed for
this idea, where the dust in the wind could be imaged provided the disc is close to edge-on, giving rise
to a ‘wingnut’ morphology. This analysis suggests that dust entrained in a photoevaporative flow may be
the origin of the unexplained extended emission around the young Herbig star PDS 144N.
7.1 Outstanding Issues
While the model presented in this thesis has satisfied its original aim - to develop a dominant dispersal
mechanism for all protoplanetery discs - there are still several issues that remain unresolved. Perhaps
the biggest source of uncertainly is the value of the X-ray luminosity the disc actually sees. Accretion
columns, magnetic disc winds and jets all provide a source of neutral attenuation along the line of sight
between the star’s corona and the outer disc. In the work presented here I have assumed that this neutral
column has a negligible affect on the X-ray flux between the star and disc, and is valid up-to neutral
columns in the range 1020−1021cm−2. With neutral columns larger than this, the effective X-ray lumi-
nosity the disc will ‘see’ will be significantly lower that the values used in this work. While this will not
affect the wind structure and derived mass-loss rates, a scaling between the observed X-ray luminosity
function and the X-ray luminosity function seen by the disc may need to be included. Gorti et al. (2009)
argue that this neutral column may in fact be a function of accretion rate with very large columns ∼ 1022
cm−2 at the earliest times, something that is suggestively indicated by the observations of Gu¨del et al.
(2010). Therefore, while I have assumed that the X-ray luminosity is fixed on the time-scales of disc
evolution (since the intrinsic X-ray luminosity of the star does not vary with time), the X-ray luminosity
7.2 Implications and future direction 137
that reaches the disc may vary over time, as the neutral column close to the star starts off at a large value
but falls over time. Observationally it may appear that the affect of neutral columns close to the central
star maybe important, with Guedel et al. (2010) estimating neutral columns in the jets/magnetic winds
and accretion columns around young stars of> 1023 cm−2, thus modelling is required to asses how these
neutral columns between the star and observer translate into neutral columns between the star and disc.
Certainly if the majority of the neutral column does scale with accretion rate, then it may not be that big
an issue, since photoevaporation only begins to become important for disc dispersal when the accretion
rate to falls to values just above the photoevaporation rate (i.e. when the neutral columns may have fallen
to values that allow the X-rays to fully reach the disc).
Therefore, while the model presented in this thesis provides the first starting point of the maximum
effect of photoevaporation, it may need to be calibrated in order to account for the evolving neutral
columns close to the star, simply by adjusting the X-ray luminosity the disc receives compared to the
intrinsic X-ray luminosity as a function of time.
As alluded to in Chapter 3, the inclusion of an extra source of FUV heating is unlikely to affect
the X-ray dominated flow in the inner region. This does not completely rule out an FUV driven flow
underneath, that undergoes a sonic transition at very large radius > 100AU. While this is unlikely to
impinge greatly on the evolutionary models presented in this thesis, it may have an impact on the outer
regions of the disc. While a full inclusion of FUV heating was not possible within the framework of
the radiation-hydrodynamic algorithm presented here, it is in principle simple to do. A standalone FUV
algorithm does already exist and is presented in Richling & Yorke (2000); a coupling of this algorithm
with the X-ray algorithm (and possibly a better treatment of the dust temperatures) would be required.
Also the possibility of such a flow will depend on the interaction of such an FUV flow with the outer disc
and X-ray wind above and will require simulations with a larger dynamic range than those used here,
out to a least several hundred AU. Until these calculations are performed the photoevaporation problem
around T Tauri stars cannot claim to have been solved, although the theory and tests performed in this
thesis show that the flow from the inner regions and hence the disc’s evolution is likely to be controlled
by the X-ray radiation field.
7.2 Implications and future direction
There are several interesting implications that photoevaporation may have on the protoplanetary disc.
The radial size distribution of dust grains expected for protoplanetary discs implies that most of the
mass resides in the largest particles. Such dust grains will not be entrained in the photoevaporative
wind, therefore photoevaporation preferentially removes gas from the disc leading to an increase of the
disc’s dust to gas mass ratio. At high gas to dust mass ratios the streaming instability (e.g. Youdin &
Johansen 2007) has been invoked as a possible route to planetesimal formation. However, the required
dust to gas mass ratio is much higher (0.03− 0.1) than the value expected in the disc at formation
(0.01). Photoevaporation has often been invoked (e.g. Throop & Bally 2005) as a leading candidate
for increasing the dust to gas ratio, often without much explanation. In fact a zeroth order calculation
only yields a 3/2 increase: if we assume that the disc evolves viscously until M˙∗ = M˙w when the disc is
instantaneously cleared, and further assume no dust mass is lost in the wind then the final disc gas mass
is 2M˙∗τd (from Equation 1.21), but as no dust mass is lost in the wind then the final dust mass must be
138 Chapter 7. Summary & Outlook
0 2 4 6 8 10
10−2
10−1
Time [Myr]
D
us
t t
o 
G
as
 m
as
s 
ra
tio
Figure 7.1: The Evolution of the global dust to gas ratio of individual discs with time (assumed to be
0.01 at t = 0), taken from the disc population calculated in Chapter 4 shown as black lines under the
assumption of no dust mass lost in the wind and no migration. The blue line shows the value expected
from a simple zeroth order calculation described in the text.
3X0M˙∗τd where X0 is the initial dust to gas ratio. Therefore the final dust to gas ratio is 3/2X0.
A better but still simplistic calculation is shown in Figure 7.1 for the disc population shown calculated
in Chapter 4. This shows that the global dust to gas mass ratio reaches values much higher than the value
implied by a factor of 3/2 increase. The origin of the rapid removal of gas material just before clearing is
yet to be understood, but it certainly implies that this avenue of photoevaporation-triggered planetesimal
formation is worth exploring. Furthermore, the high global dust to gas mass ratios imply that locally it
may start to exceed unity. At this point the back reaction of the dust’s angular momentum on the gas
cannot be neglected, and may in fact suppress migration of dust particles. Therefore, any calculation
that aims to follow the dust evolution may need to account for this effect. Certainly a two-dimensional
calculation that incorporates dust migration, settling and photoevaporation may provide an interesting
avenue of future research within the planet/planetismal formation field.
A
Numerical Fits to the Mass-loss Rates
IN this appendix, I provide fits to the total mass-loss rates and mass-loss profiles (Σ˙w) obtained fromthe numerical calculations performed in Chapter 3. The fits are based on the results provided byan online ‘function finder’ service, provided at http://zunzun.com, and their functional form does
not represent the results of an analytic calculation and should be treated in such a way. Furthermore,
these fits were performed in such a way as to described the cumulative mass-loss rates accurately, which
is important for the global viscous evolution. As such the fits are least accurate at small radius, should
sensitivity at these radii become important, more accurate fits to this region can be provided upon request.
The surface mass-loss profiles Σ˙w given in Equations A.2 & A.5 are provided in normalised form and
must be scaled so that
∫
2piRΣ˙wdR yields the required total mass-loss rate, given in Equations A.1 & A.4.
A.1 Primordial Discs
The total mass-loss rate variation with X-ray luminosity is shown in Figure 3.4, where the solid line is
described by:
M˙w = 6.25×10−9
(
M∗
1 M
)−0.068( LX
1030erg s−1
)1.14
M yr−1 (A.1)
The mass-loss profile, derived from the solid line in Figure 3.6 is described by:
Σ˙w(x> 0.7) = 10(a1 log10(x)
6+b1 log10(x)
5+c1 log10(x)
4+d1 log10(x)
3+e1 log10(x)
2+ f1 log10(x)+g1)
×
(
6a1 log(x)5
x2 log(10)7
+
5b1 log(x)4
x2 log(10)6
+
4c1 log(x)3
x2 log(10)5
+
3d1 log(x)2
x2 log(10)4
+
2e1 log(x)
x2 log(10)3
+
f1
x2 log(10)2
)
× exp
[
−
( x
100
)10]
(A.2)
139
140 Appendix A. Numerical Fits to the Mass-loss Rates
where a1 = 0.15138, b1 = −1.2182, c1 = 3.4046, d1 = −3.5717, e1 = −0.32762, f1 = 3.6064, g1 =
−2.4918 and:
x= 0.85
(
R
AU
) (
M∗
1 M
)−1
(A.3)
where Σ˙w(x< 0.7) = 0. Logarithms of the form log(x) are taken taken using the natural base.
A.2 Discs with Inner holes
As described previously, the total mass-loss rate is approximately independent of inner hole radius, which
is approximately given by:
M˙w = 4.8×10−9
(
M∗
1 M
)−0.148( LX
1030erg s−1
)1.14
M yr−1 (A.4)
The mass-loss profile, derived from the solid line in Figure 3.7 is described by:
Σ˙w(y) =
[
a2b2 exp(b2y)
R
+
c2d2 exp(d2y)
R
+
e2 f2 exp( f2y)
R
]
×exp
[
−
( y
57
)10]
(A.5)
where a2 = −0.438226, b2 = −0.10658387, c2 = 0.5699464, d2 = 0.010732277, e2 = −0.131809597,
f2 =−1.32285709 and:
y= 0.95(R−Rhole)
(
M∗
1 M
)−1
(A.6)
where Σ˙w(y< 0) = 0.


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