The Evolutionary Pathways of Tidal Disruption Events:from Stars to Debris Streams, Accretion Disks, andRelativistic Jets
by
E. R. Coughlin
B.S., Lehigh University, 2011
M.S., University of Colorado at Boulder, 2013

A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillmentof the requirements for the degree ofDoctor of Philosophy
Department of Astrophysical and Planetary Sciences
2016This thesis entitled:
The Evolutionary Pathways of Tidal Disruption Events: from Stars to Debris Streams, Accretion
Disks, and Relativistic Jets
written by E. R. Coughlin
has been approved for the Department of Astrophysical and Planetary Sciences

Prof. Mitchell C. Begelman

Prof. Philip J. Armitage

Prof. Andrew J. Hamilton

Prof. Mark Rast

Prof. Oliver DeWolfe

Date
The final copy of this thesis has been examined by the signatories, and we find that both thecontent and the form meet acceptable presentation standards of scholarly work in the above
mentioned discipline.

Coughlin, E. R. (Ph.D., Astrophysics)
The Evolutionary Pathways of Tidal Disruption Events: from Stars to Debris Streams, Accretion
Disks, and Relativistic Jets
Thesis directed by Prof. Mitchell C. Begelman

Tidal disruption events, which occur when a star is destroyed by the gravitational field of asupermassive black hole, are unique probes of the inner regions of galaxies. In this thesis we explore
various stages of the tidal disruption process, in an attempt to relate the observable signatures oftidal disruption events to the properties of the disrupted star and the black hole. We use numerical
techniques to study the long-term evolution of the debris streams produced from tidal disruption
events, showing that they can be gravitationally unstable and, as a result of the instability, fragment into small-scale, localized clumps. The implications of this finding are discussed, and weinvestigate how the thermodynamic properties of the gas comprising the stream affect the natureof the instability. We derive an analytic model for the structure of tidally-disrupted, stellar debris
streams, and we compare the predictions of our model to numerical results. We present a model forthe accretion disk that forms from a tidal disruption event when the accretion rate surpasses the
Eddington limit of the supermassive black hole, showing that these disks are puffed up into quasispherical envelopes that are threaded by bipolar, relativistic jets. We compare the predictions of
this model to observations of the jetted tidal disruption event Swift J1644+57. Finally, we derive,from the relativistic Boltzmann equation, the general relativistic equations of radiation hydrodynamics in the viscous limit, which characterize the interaction between radiation and matter whenchanges in the fluid over the photon mean free path are small. Our results demonstrate that, incontrast to previous works, a radiation-dominated fluid does in fact possess a finite bulk viscosityand a correction to the comoving energy density. Using the general relativistic equations of radiation hydrodynamics in the viscous limit, we present two models to describe the interaction between
a relativistic jet launched during a tidal disruption event and its surroundings. These models show

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that regions of very large shear that arise between the fast-moving outflow and the surrounding
envelope possess fewer scatterers and a harder photon spectrum, meaning that observers looking“down the barrel of the jet” infer vastly different properties of the outflow than those who lookoff-axis.

Dedication

This thesis is dedicated to my wonderful wife Sarah and all of her sass.

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Acknowledgements

First I would like to thank my advisor, Mitch Begelman, for taking me on as one of his
students. In addition to providing crucial insights at various stages of the projects contained inthis thesis, he has been thoroughly supportive of my research endeavors and, especially during myfirst year as a graduate student, provided me the time necessary to learn the subject matter thatcomprises my work. By giving me the (guided) freedom to pursue my own lines of reasoning andthought he has been fundamental to my growth as a scientist.
I would also like to thank Chris Nixon and Phil Armitage, with whom I have had numerous
enlightening conversations about astrophysics. They have also been invaluable to the numericalaspects of my research, and they contributed significantly to the improvement of my disc golf gamethrough the countless rounds we played at Valmont.
Without the love and support of my parents, Kathleen Harley and Robert Coughlin, noneof this would have been possible, and to them both I am eternally grateful. By allowing me topursue my dreams they have made this thesis a reality. My grandparents Richard and Joan Huber(Gingop and Ganga) have been inspirational to much of my work; they serve as two role models
against whom I measure my success as a human being. My in-laws, Robert and Tina Berger, havebeen incredibly loving and kind to me since the day we first met (c. March 2006) and have always
supported me (and haven’t complained too openly about the fact that I stole their daughter andtook her to Colorado). I’d also like to thank my stepdad, Glenn Harley, for making the 27-hour
drive out to Colorado with me, encouraging me in my academic pursuits, and being a great rolemodel.

vii
The friends I met at the University of Colorado made graduate school a fun and enjoyable
experience. I had innumerable great times with my housemates, Chris Fowler, Katy Goodrich,
Jen Kulow, and Eddy Barratt (and, at an earlier epoch, Matt McJunkin and Seth Jacobson). I
thank Greg Salvesen and Susanna Kohler for enlivening the office, generating fruitful discussions,and adorning my wall with pictures of youthful Mitch. I am grateful to Grant Buckingham, Devin
Silvia, Jordan Mirocha, and Marek Slipski for the great times we had playing ultimate.
Finally, last but in no way least, I thank my wife Sarah Coughlin for not only tolerating me
since we met in high school, but for being the supportive, encouraging partner I needed throughout
the completion of this thesis. During my time in graduate school she never once doubted me (even
if I did) and kept me sane. Without her none of this would have been possible.

viii

Contents

Chapter
1 Introduction

1

2 Gravitationally Unstable Streams

11

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.2

Self-gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.3

Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.5

Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3 Post-periapsis Pancakes: Sustenance for Self-gravity in Tidal Disruption Events

25

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.2

The impulse approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3

3.4

3.2.1

Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.2.2

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.3.1

Simulation setup and initial conditions . . . . . . . . . . . . . . . . . . . . . . 37

3.3.2

Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.1

3.5

Is the pancake necessary? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

Implications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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3.6

3.5.1

Fragmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5.2

Fallback rate features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.5.3

Clump fates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.5.4

Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4 On the Structure of Tidally-disrupted Stellar Debris Streams

68

4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.2

Velocity distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.3

4.2.1

Self-similar velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.2

Radial positions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Stream width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.3.1

Quasi-hydrostatic width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3.2

Shear dominated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.3

Approximate, full solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4.4

Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.5

Density scalings and fragmentation conditions . . . . . . . . . . . . . . . . . . . . . . 91

4.6

4.7

4.5.1

Marginally bound material . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.5.2

Unbound material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.5.3

Bound material and overall scalings . . . . . . . . . . . . . . . . . . . . . . . 95

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.6.1

The neglect of angular momentum . . . . . . . . . . . . . . . . . . . . . . . . 98

4.6.2

Comparison with Kochanek (1994) . . . . . . . . . . . . . . . . . . . . . . . . 99

4.6.3

To fragment or not to fragment? . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.6.4

A more realistic entropy prescription . . . . . . . . . . . . . . . . . . . . . . . 106

Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

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5 Hyperaccretion During Tidal Disruption Events: Weakly Bound Debris Disks and Jets

113

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2

Zero-Bernoulli accretion model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.1

Gyrentropic flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.2.2

Self-similar solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.3

ZEBRA models of TDE debris disks . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.4

Jet properties and temporal evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.5

5.4.1

Inner regions of the accretion disk . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4.2

Accretion rate and jet power . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

5.4.3

Time-dependent analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

5.4.4

Swift J1644+57 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Discussion and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6 The General Relativistic Equations of Radiation Hydrodynamics in the Viscous Limit

164

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.2

Relativistic Boltzmann equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

6.3

Diffusion approach to the transport equation . . . . . . . . . . . . . . . . . . . . . . 171

6.4

Relativistic, diffusive equations of radiation hydrodynamics . . . . . . . . . . . . . . 177

6.5

Perturbation analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182

6.6

Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

7 Viscous boundary layers of radiation-dominated, relativistic jets. I. The two-stream model195
7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195

7.2

Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198

7.3

Two-stream boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
7.3.1

Basic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.3.2

Boundary layer equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

7.3.3

Self-similar approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

xi
7.3.4

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

7.4

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

7.5

Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

8 Viscous boundary layers of radiation-dominated, relativistic jets. II. The free-streaming jet
model

221

8.1

Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

8.2

Jetted boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225

8.3

8.2.1

Self-similar approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

8.2.2

Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
8.3.1

Comparative notes between the two-stream model (Chapter 7) and the freestreaming jet model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

8.4

Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250

9 Conclusions

253

Bibliography

256

Appendix
A Non-self-similar ZEBRA solutions

264

B Non-zero Bernoulli parameter

267

xii

Tables

Table

xiii

FiguresFigure1.1

A diagram illustrating the tidal disruption process. The dashed line gives the orbitof the center of mass of the star, and the arrows indicate the center of mass velocity.

1.2

3

A schematic of the self-intersection process that occurs during tidal disruption events.
Here α is the angle between the orientation of the pericenter of the incoming debris
and that of the outgoing debris, equal to α ' 10◦ for typical parameters, which is
the result of the general relativistic advance of periapsis. . . . . . . . . . . . . . . . .

2.1

6

The star at the time of disruption (top, left), where the colors indicate the column
density and distances are measured in units of tidal radii (' 7 × 1012 cm = 100R ),0.835 hours after periapsis (top, right; the top two figures coincide with the bottomleft panels of Figure 4 a) and b), respectively, of Lodato et al. 2009.), and 5.38 days
(middle, left), 15 days (middle, right), 22.4 days (bottom, left) and 1 month (bottom,
right) after disruption. The red, dashed curves on the bottom two rows indicate theanalytic predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.2

The computed density of the stream (solid, black curve), the analytic prediction (red,solid curve; equation 3.10), and the density at which self-gravity becomes important(red, dashed curve; equation 2.1) as functions of r (spherical distance from the hole)at 22.4 days from the point of disruption. Points above the red, dashed curve are
self-gravitating. The numerical solution extends to smaller radii because the star at
the time of disruption is not spherical (see also Lodato et al. 2009). . . . . . . . . . . 18

xiv
2.3

The fallback rate in solar masses per year calculated from the simulation (black, solid
curve) and the analytic prediction (red, dashed curve) as functions of time in years.
The temporal shift between the numerical solution and the analytic prediction isfrom the stellar distortion at the time of disruption (see text for further details). The
variability at late times is due to the accretion of clumps formed in the self-gravitating
stream. The variability starts about 2 years after disruption; however, this timescaledepends upon the details of the TDE and its environment, and therefore may be
shorter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.4

The density two years after disruption and, in the insets, close-ups of the fragmentsthat forms along the stream.

2.5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

The ratio h/RJ , h being the smoothing length and RJ the Jeans radius, as a functionof density at a time when the first clumps start to form. . . . . . . . . . . . . . . . . 23

3.1

The initial configuration of the star under the impulse approximation when β ' 1(this figure is not drawn to scale). The dashed curve traces out the orbit of thecenter of mass, which is assumed to be parabolic. The Cartesian coordinates areindicated by the diagram immediately below the black hole (which is indicated by
the black circle), z being out of the plane in a right-handed sense. The sphericalpolar coordinates are labeled r and φ on the diagram, and θ is measured out of theplane of the orbit from the z-axis (for the above figure that focuses on the x-y plane,θ = π/2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

xv
3.2

The streams of debris formed from the tidal disruption of a solar-type star by a
Mh = 106 M

hole situated at the origin. Each color represents a different time,
3/2 √the earliest (blue points closest to the origin) being at t = 100 rt / GMh ' 1.84
3/2 √days from disruption, the latest (yellow points) at t = 2200 rt / GMh ' 40.6 days
3/2 √from disruption. The time in between neighboring streams is 300 rt / GMh ' 5.53
days. The black hole (not drawn to scale) is indicated by the black circle near theorigin. While the radial positions of the gas parcels match well those from numericalanalyses (see Figure 1 of Chapter 2), the width obtained from equations (3.1) –(3.3) is significantly overestimated (the numerical solutions, had we shown them,would have amounted to lines plotted overtop of the streams in Figure 2). Thisfinding suggests that self-gravity is important for keeping the stream confined in thetransverse direction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3

Four snapshots of the in-plane evolution of the gas parcels comprising the edge of thestar at the time of disruption; for these figures we chose a 106 M hole and a solar-likestar. The particles comprising the front of the star at the time of disruption havebeen colored blue, while the back has been colored orange. The arrows indicate thedirection of motion of the center of mass. The bottom, left-hand panel shows that,at a time of roughly an hour after disruption, the front and back of the stream merge
and thereafter trade places. The impulse approximation thus leads to a caustic –where the debris streams form a two-dimensional surface – which occurs roughly anhour after disruption.

3.4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

The star at the time of disruption for an adiabatic index of γ = 1.5 (top, left), γ = 5/3(top, right), γ = 1.8 (bottom, left) and γ = 2 (bottom, right). The configuration has
clearly been altered from its original, spherical shape, showing that the tidal force
does not act exactly as an impulse as was assumed in Section 2. The central density
is also higher for smaller γ, which is predicted from the original stellar profile. . . . . 39

xvi
3.5

The stream at a time of 2.53 days from pericenter for an adiabatic index of γ = 1.5(top, left), γ = 5/3 (top, right), γ = 1.8 (bottom, left) and γ = 2 (bottom, right).
The stream thickness decreases dramatically and the fans become less pronounced
as γ increases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6Four snapshots of the average stream density (the average density of all particles at
a given radius r) as a function of r for the γ = 5/3 run. Initially the density remains
smooth throughout the stream; however, by about a day after the disruption, thedensity structure has developed a more complicated nature, consisting of a central
peak that is narrower than is predicted analytically and two shoulders. . . . . . . . . 43

3.7

The average density (same as Figure 6) as a function of r for γ = 1.5 (top, left
panel), γ = 5/3 (top, right panel), γ = 1.8 (bottom, left panel), and γ = 2 (bottom,
right panel) at a time of 2.53 days after disruption (see Figure 3.5 for the shapeof the streams at this time). The black, solid curves give the numerical solutions,while the red, dashed curves show the analytic predictions. It is apparent that larger
adiabatic indices correspond to an enhanced amount of variability in the density
along the stream, while a smaller adiabatic index results in more extended wings(this is also apparent from the tidal fans in the edges of the streams in Figure 3.5.) . 44

3.8

The maximum density along the stream as a function of time with γ = 5/3 (n =1.5); the numerical solution is given by the black, solid curve, and the analyticalsolution (equation 3.18) is given by the red, dashed curve. A time of zero here
corresponds to the time at which the star reaches the tidal radius. The time atwhich the numerically-obtained density starts to decrease is slightly earlier than theanalytic one, suggesting that the time at which the star is “disrupted” is actually
pre-periapsis. The first bump in the numerical solution, which occurs after a coupleof hours, indicates where the pancake starts to augment the maximum density. At
late times, both solutions follow the approximate power-law decline ρ ∝ t−1.8 .

. . . 45

xvii
3.9

The maximum density as a function of time for γ = 1.5 (black, solid curve), γ = 5/3(red, dashed curve), γ = 1.8 (green, long-dashed curve), and γ = 2 (blue, dottedcurve). It is apparent that the initial perturbation induced by the pancake is induced
sooner for larger γ, and the oscillation timescale of the perturbation is shorter forlarger γ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3.10 The particle distributions from an N -body simulation, where the initial conditions
were taken from the γ = 5/3 run at periapsis, at 37 minutes (left panel), 50 minutes(middle panel), and 62 minutes (right panel) post-disruption. The red particles
comprised the back of the star at the time of disruption, while the black constituted
the front of the star. This Figure demonstrates that a caustic – where the front andback of the stream merge to form an infinitely thin line – still occurs in the presenceof realistic initial conditions. However, as mentioned in the text, the outermost layersof the star that are stripped earlier (and therefore violate the frozen-in condition)
do not undergo this compression; this is evidenced from the fact that the “fans”
present in this figure undergo no distortion in terms of their color. The central panel
corresponds to the point at which the width of the stream has reached a minimum,
the half-width being H ' 0.1 rt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.11 The average density as a function of r for the unaltered, γ = 5/3 run (solid, black
curve) and the run that avoids the caustic (dashed, red curve), both at a time of 2.53
days (the same time as in Figure 3.7). This Figure shows that the pancake amplifies
the anomalous density structures along the stream, effectively enhancing the abilityof self-gravity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.12 The stream from the γ = 2 run (top panel) and a closeup view of the stream (bottom
panel), showing the clumps that have formed throughout the majority of the stream,
both at a time of 5.69 days after disruption. . . . . . . . . . . . . . . . . . . . . . . . 56

xviii3.13 The fallback rate computed for the γ = 1.5 (black curve) and γ = 5/3 (red curve)
runs. The purple curve is the canonical t−5/3 fallback rate for reference. It is apparent
that the return time of the most bound material is earlier for smaller γ, which isrelated to the amount of distortion imparted to the star at the time of disruption. At
late times, the accretion of clumps that have formed in the γ = 5/3 stream causes the
fallback rate to deviate significantly from the mean (the small amount of deviation
present in the γ = 1.5 run is numerical noise).

. . . . . . . . . . . . . . . . . . . . . 58

3.14 The fallback rate for the γ = 1.8 run (green, solid curve) and the analytic prediction(blue, dashed curve). We see that the numerical solution is larger than the analytic
one by an order of magnitude, and the fallback of bound clumps causes significant
deviation from the average, t−5/3 fallback at times greater than about 6 months from
disruption.
4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

The solution to equation (4.4) passing through the critical point f (2/3) = 1 (blue,
solid curve). The approximate solution given by f = −1 + 1/ξ, which matches theasymptotic limits of the true solution, is shown by the red, dashed, curve.

4.2. . . . . 72

A comparison between the self-similar (red, dashed curves) and numerical (black
points) radial velocity profiles, both normalized to the speed of light, for the debris
stream produced by the disruption of a solar-like star by a 106 M

black hole. The

numerical solution was taken from the simulation in Chapter 2. The different curves
represent the stream at different times, with each curve corresponding to, from left
to right, 11, 30, 60, 120, 180, 300, 451, and 601 days since disruption.

. . . . . . . . 74

xix
4.3

A log-log plot of the difference in velocity normalized to the speed of light, i.e.,∆v/c ≡ (vn − vss )/c, where vn is the numerically-obtained radial velocity and vss
is the analytic solution (equation 4.2), for the same times chosen in Figure 4.2 (i.e.,the dark blue set of points is 11 days after disruption, the light yellow set of points
is 601 days after disruption, and each set of points in between corresponds to theappropriate curve in Figure 4.2). As discussed in Section 6.1, the small discrepancies
shown here can plausibly be attributed to the neglect of angular momentum in the
self-similar solution. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.4

The analytic solution for the density when self-gravity dominates the shear of theblack hole (equation 4.45), shown in the left panel, and when the shear dominates
the self-gravity of the stream (equation 4.46), shown in the right panel, as functionsof r for the disruption of a solar-like star by a 106 M SMBH. We assumed isentropic
gas with γ = 5/3 and entropy calculated for solar-like parameters (S = 2.48 × 1014
[cgs]; Hansen et al. 2004). The different colors correspond to different times, with
3/2 √the earliest time being 2500 × rt / 2GMh ' 32 days after disruption (left-most,
3/2 √dark blue curve), the latest being 47500 × rt / 2GMh ' 620 days after disruption(right-most, light blue curve), and each curve differing from the previous one by
3/2 √7500 × rt / 2GMh ' 98 days. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.5

The numerical solution for the density when self-gravity dominates the shear of theblack hole (the solution to equation 4.41 with H given by equation (4.30), ∂r/∂µcalculated numerically, and ∂M/∂µ calculated numerically from equation (4.43) withn = 1.5), shown in the left panel, and when the shear dominates the self-gravity ofthe stream (the solution to equation 4.41 with H given by equation (4.33), ∂r/∂µcalculated numerically, and ∂M/∂µ calculated numerically from equation (4.43) withn = 1.5), for the same parameters and times chosen in Figure 5.3.

. . . . . . . . . . 86

xx
4.6

The numerical solution for the density that adopts the piecewise behavior for H(equation 4.34) so that the density is shear dominated in regions where ρ < Mh /(2πr3 )and self-gravity dominated where ρ > Mh /(2πr3 ), for the same parameters and times
chosen in Figure 5.3. The yellow, dashed line shows the curve Mh /(2πr3 ). . . . . . . 89

4.7

A particle plot of the density in the stream ρ in units of g / cm3 as a functionof r in tidal radii at a time of 57 days post-disruption. The red points are fromthe simulation used in Chapter 2, the black points are from an identical simulation
but with 107 , as opposed to 106 , particles, and the green, dashed curve shows thefunction Mh /(2πr3 ) – the approximate dividing line between the shear and selfgravity dominated solutions.

4.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

The quantity q = d ln(ρr3 )/d ln t as a function of µ when γ = 5/3, Mh = 106 M , theprogenitor is solar, and the gas is isentropic; the flow is assumed to be self-gravity
dominated in the left-hand panel and shear-dominated in the right-hand panel. The
3/2 √different curves show different times, with the blue curve at 1000 × rt
2GMh ' 13
3/2 √days after disruption, the brown curve at 11000 × rt
2GMh ' 143 days after
3/2 √disruption, and each curve differs from the next closest curve by 2000×rt
2GMh '
26 days (e.g., the yellow curve is 26 days after the blue curve, the green curve is 26
days after the yellow curve, etc.). The range of the y-axis was set to the same values
for each plot for ease of comparison. Fluid elements in the stream follow verticallines from one curve to the next. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.9

A schematic of the debris stream, not drawn to scale, returning to the black hole
under the approximations set out in our analysis (left drawing) and a more realistic distribution that accounts for the finite angular momentum of the hole (rightdrawing). This shows that, as the material gets very close to the hole, the tidal
compression becomes a tidal shear that serves to decrease the density. . . . . . . . . 100

xxi
4.10 A log-log plot of the stream width, H, in units of solar radii at the marginallybound location along the stream as a function of Lagrangian distance (i.e., moving
with the marginally-bound portion of the stream) from the SMBH. The black, solid
curve shows the solution from the same simulation performed in Chapter 2 (i.e.,a solar-like star destroyed by a 106 M

SMBH) but with 107 particles, while thered, dashed curve shows the analytic prediction H/R ∝ (r/rt )1/2 (the constant ofproportionality was set to 0.33).
5.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

The value of q obtained for γ = 4/3 (blue, solid) and γ = 5/3 (red, dashed) as wevary the left-hand side of equation (5.19). . . . . . . . . . . . . . . . . . . . . . . . . 127

5.2

The self-consistent model (equation (5.22)) for the angular momentum (blue, solid),
the self-similar model (red, dashed), and the psuedo-Newtonian distribution (black,dotted). Here we’ve set q = 1.5, γ = 4/3, the abscissa is in units of Schwarzschild√radii, and the angular momentum is normalized by GMh rs . . . . . . . . . . . . . . 131

5.3

Solutions for the density in the non-self-similar (blue, solid) and self-similar (red,dashed) limits. Here q = 1.5, γ = 4/3, the x-axis is in units of Schwarzschild radii,
and the density is measured in units such that ρ0 (rs /r0 )−q = 1, i.e., the red, dashedcurve is simply (r/rs )−2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.4

The mass contained within r for the angular momentum given in equation (5.22)(blue, solid) and that for the self-similar model (red, dashed). The parameters are
the same as those in Figure 5.3, with the same normalization for the density. . . . . 134

5.5

The angular momentum contained within r for the self-consistent model (blue, solid)
and the self-similar solution (red, dashed). The parameters and normalization are
the same as those in Figure 5.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.6

A schematic of the star at the time of disruption to illustrate the geometry. Notethat r = ηR∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

xxii
5.7

The solution for q(t) with a solar progenitor (M∗ = 1 M , R∗ = 1 R , γ∗ = 5/3), aradiation pressure-dominated gas (γ = 4/3), Mh = 105 M , y = 1, δ = 0.05, χ = 5,
and three different q0 , indicated by the legend. As one can see, the initial conditions
quickly become irrelevant to the long-term behavior of the solutions. . . . . . . . . . 144

5.8

q(t) for a radiation-dominated gas (γ = 4/3) and a variety of black hole masses.
Here y = 0.5, δ = 0.05, χ = 5, q0 = 3/2, γ∗ = 5/3, M∗ = 1M , and R∗ = 1R . The
legend displays the black hole mass in units of solar masses. We see that initiallyq falls off very rapidly, which is a consequence of initial conditions. However, theinitial conditions stop having a major effect early in the evolution of the system, andq then decreases less rapidly. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

5.9

The mass contained in the ZEBRA as a function of time for the same parameters as
those in Figure 5.8. The mass quickly increases initially, owing to the fact that the
fallback rate exceeds the accretion rate. However, as both rates decrease for latertimes, the mass levels off to a nearly constant value. . . . . . . . . . . . . . . . . . . 146

5.10 The jet luminosity, normalized to the Eddington luminosity of the black hole, forthe same parameters as those in Figure 5.8 with an efficiency  = 0.1. Initiallythe luminosity is very super-Eddington, decaying to only mildly super-Eddington atlater times. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
5.11 The black hole accretion rate, given by equation (5.26), in solar masses per year for
q0 = 1 (blue, solid curve) and for q0 = 2 (red, dashed curve) to illustrate where thesolutions have converged; the parameters are the same as those in Figure 5.8 and theblack hole has a mass Mh = 105 M . We have also plotted the fallback rate, shownby the black, dotted curve, to illustrate how the two accretion processes compare. It
is apparent from this figure that they match each other closely, and that after about
4 years, the black hole accretion rate exceeds the fallback rate. . . . . . . . . . . . . 149
5.12 The effective temperature as a function of time for different black hole masses. The
fiducial parameters are the same as those in Figure 5.8. . . . . . . . . . . . . . . . . 151

xxiii5.13 The black hole accretion rate (blue, solid curve) and the fallback rate (red, dashedcurve) for M0 = 0.1 × M /2, Mh = 105 M , m = 2, and otherwise the same parameters as in Figure 5.8, plotted on a log-log scale. The accretion rate follows ashallower power-law than the fallback rate, causing the former to exceed the latterfor times greater than about 0.5 years. . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.14 The power-law of the accretion rate, which we defined as macc , as a function of thepower-law index of the fallback rate is shown by the blue, solid curve. The value ofmacc is determined by performing a best-fit to the black hole accretion rate betweent = 0.2 and 1.5 years, during which time the accretion rate, for all values of m, iswell-described by a power-law. We have also plotted m for comparative purposes(red, dashed curve). We see that macc is always less than m. For small values of m,
the relationship between macc and m is roughly an offset, linear one. As m becomeslarger, however, macc displays a more nonlinear behavior, and the difference between
the two power-law indices becomes larger.
7.1. . . . . . . . . . . . . . . . . . . . . . . 155

The function fξ , which is the normalized z-component of the four velocity, in terms√of the parameter α̃ ∝ y/ z, for µ = 1 and a number of jet Lorentz factors, as
indicated by the legend. As we can see, the thickness of the velocity boundary layer,in terms of α̃, is nearly independent of Γj .

7.2

. . . . . . . . . . . . . . . . . . . . . . . 208

The variation of the normalized density, given by g, as a function of α̃ for the same
parameters as those chosen for Figure 7.1. The density remains below its asymptotic
value over a slightly larger range of α̃ for higher Γj , and the decrease in density within
the boundary layer is consistent with the findings of Arav & Begelman (1992). The
density formally equals zero at α̃ = 0; however, because g approaches zero at a very
slow rate (recall g(ξ) ∝ −1/ ln ξ), it appears from the Figure, which only samples afinite number of points around α̃ = 0, that the density remains positive and larger
for smaller Γj .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

xxiv
7.3

The solution to equation (7.27), h, which is the normalized number density of photons, for the same set of parameters as those chosen in Figure 7.1. The number
density of photons is seen to roughly track the number density of scatterers. Because the energy density of the radiation remains constant across the layer, theenergy per photon increases in the boundary layer. . . . . . . . . . . . . . . . . . . . 210

7.4

The solution for the normalized z-component of the four-velocity (fξ ) for Γj = 25
and three different values of µ, as indicated by the legend. The velocity profile does
not differ much as µ increases beyond 1, but for smaller values of µ the boundary
layer widens noticeably. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

7.5

The function g, which is the rest-frame number density of scatterers, for the same
parameters as in Figure 7.4. For smaller values of µ, the density is significantly
reduced from its asymptotic value over a larger range in α. . . . . . . . . . . . . . . 213

7.6

The solution to equation (7.27), which gives the number density of photons observedin the comoving frame, for the same set of parameters as in Figure 7.4. The radiation
number density roughly follows that of the scatterers. . . . . . . . . . . . . . . . . . 214

8.1

The behavior of fξ , which is the normalized z-component of the four-velocity, with
ξ for ζ = 4 (blue, solid curve), ζ = 10 (purple, dot-dashed curve), and ζ = 25 (red,dashed curve). As expected, the curves are all coincident when ξ  1 and ξ  1,with a non-self-similar transition (one that depends on ζ) in between those two limits.233

8.2

The function g, which is approximately the inverse of the lab-frame density, plottedwith respect to ξ for the same set of ζ chosen in Figure 8.1. As was true for fξ , g
is approximately self-similar close to and far from the jet, with the deviation fromself-similarity in the trans-relativistic region being apparent but small. . . . . . . . . 234

xxv
8.3

The normalized z-component of the four-velocity (fξ ) for µ = 1 and ζ = 4, 10, and25 (the solid, blue curve, the dot-dashed, purple curve, and the dashed, red curve,
respectively), which, for z ' z0 , correspond to Γj = 4, 10, and 25. We see that thewidth of the boundary layer is nearly unchanged as we alter the value of ζ. . . . . . 238

8.4

The normalized fluid-frame density of scatterers for the same set of parameters usedin Figure 8.3. For all solutions the number density of scatterers approaches zero as
we near the center of the jet. We see that the average number density of scatterers
within the boundary layer is lower for larger Lorentz factors. . . . . . . . . . . . . . 239

8.5

The normalized number density of photons for the same set of ζ used in Figure 8.3.

8.6

The photon number density closely follows that of the scatterers. . . . . . . . . . . . 240p
The y-component of the three-velocity normalized by 2/(9ρ00 κz0 ) (see equation(8.42)) for the same set of parameters used in Figure 8.3. For positive α̃, eachsolution is initially positive and reaches a relative maximum before approaching anegative constant, which shows that the flow expands outwards near the center ofthe jet and entrains ambient material far from the axis. . . . . . . . . . . . . . . . . 241

8.7

The function fξ , which is the normalized z-component of the four-velocity, for ζ = 10
and µ = e0 /ρ00 = 0.1, 1, and 10, which correspond to the blue, solid curve, the purple,
dot-dashed curve, and the red, dashed curve, respectively. Increasing the value ofµ, we see, has little effect on the solution, while decreasing µ drastically widens theboundary layer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

8.8

The normalized density of scatterers for the same values of µ chosen in Figure 8.7.
The mean value of the density decreases as µ decreases. . . . . . . . . . . . . . . . . 244

8.9

The normalized number density of photons for the same set of µ chosen in Figure8.7. This figure demonstrates, as we saw in Figure 8.5, that the density of photons
tracks that of the scatterers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

xxvi
8.10 The y-component of the three-velocity, normalized byp
2/(9ρ00 κz0 ) ∼ δ, for thesame set of µ chosen in Figure 8.7. We see that the relative maximum increases forsmaller µ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

Chapter 1

Introduction

Of the numerous predictions of Einstein’s general theory of relativity, perhaps the mostprofound is that of a black hole – a region of spacetime so severely warped by the presence ofmass that not even light can propagate freely. Initially regarded as mathematical curiosities thatwere of little physical relevance, black holes are now considered fundamental to many seeminglydistinct aspects of astronomy and play central roles in the evolution of the Universe. Furthermore,a supermassive black hole (SMBH), with a mass in excess of ∼ 105 M , is now thought to reside at
the center of nearly every galaxy.
Some of these galactic SMBHs reveal themselves through the intense emission of radiation across many wavelengths and thereby fall into the class of active galactic nuclei (AGN; e.g.,
Krawczynski & Treister 2013). While these AGN come in a variety of observational flavors – quasars(e.g., Warren & Hewett 1990), Seyfert galaxies (e.g., Maiolino & Rieke 1995), and radio galaxies(e.g., Begelman et al. 1984), to name a few – the underlying physical mechanism generating theluminous output is thought to be the same and arises from the accretion of material onto a SMBH.
However, galaxies harboring AGN are in the minority (at least as we see them today), and the central regions of most galaxies remain quiescent and give no indication of the gravitational monsterthat lies within.
When studying these quiescent galaxies, astronomers are usually forced to rely on empiricallaws such as the M −σ (Ferrarese & Merritt, 2000; Gebhardt et al., 2000) and Magorrian (Magorrian
et al., 1998) relations to infer information about the black hole at the galactic center. These laws

2
relate the SMBH mass to the velocity dispersion (M-σ relation) or the mass (Magorrian relation) of
the galactic bulge. One important caveat is that, while these laws seem to yield tight and reliablecorrelations over a fairly wide range in black hole mass, it is unclear whether they can be applied toevery galaxy. Indeed, a more direct probe of the central region of a quiescent galaxy would not only
yield useful information about the properties of the black hole at its center, it would also generate
an independent constraint on the validity of the M − σ and Magorrian relations.
Tidal disruption events (TDEs) provide one such probe by igniting brief periods of AGN-like
activity in otherwise-quiescent galaxies, thereby illuminating their central SMBHs. These eventsoccur when a star is tidally shredded by the SMBH at the center of a galaxy. Specifically, when a starof mass M∗ and R∗ is far away from a SMBH of mass Mh (but still within the sphere of influence ofthe hole), the star can effectively be treated as a point mass and the magnitude of the gravitational
field it experiences is simply fg = GMh /r2 , where r is the instantaneous distance between thecenter of mass of the star and the SMBH. However, the finite stellar radius becomes important as
the star nears the SMBH, with the side of the star closer to the hole feeling a stronger force than
the side that is farther away. This differential force across the stellar diameter – the tidal force –scales as ft = 2GMh R∗ /r3 and stretches the star along r, perturbing its original, assumed-sphericalstructure. If the distance between the center of mass of the star and SMBH continues to decrease,the magnitude of the tidal force can actually exceed the self-gravity of the star, fsg ' GM∗ /R∗2 ,resulting in its destruction. The radius at which the stellar self-gravity equals the tidal force, thetidal radius, is given by rt ' R∗ (Mh /M∗ )1/3 , and defines the approximate location at which thestar will be destroyed by the gravitational field of the SMBH (the numerical factor, of order unity,that enters this equation depends on the precise properties of the star; Guillochon & Ramirez-Ruiz2013). This process (and the complex gravitational and thermodynamical processes that ensue; seebelow) is known as a tidal disruption event, and the reader is referred to Figure 1.1 for a roughschematic that illustrates the interaction.
TDEs have been studied for decades, with the earliest investigations attempting to elucidate
the rates at which they should occur. The pioneering work of Frank & Rees (1976), who studied

3

rtFigure 1.1: A diagram illustrating the tidal disruption process. The dashed line gives the orbit of
the center of mass of the star, and the arrows indicate the center of mass velocity.

4
the process by which stars are scattered into the loss cone of orbits that plunge within the tidal
sphere of the SMBH at the center of a galaxy, demonstrated that this rate should be approximately
10−4 – 10−5 disruptions per galaxy per year. Since then, others have refined these estimates byperforming numerical simulations and accounting for uncertainties concerning the low-mass end of
the SMBH distribution (Brockamp et al., 2011; Stone & Metzger, 2014). The tidal disruption rate ofa supermassive black hole binary has also been calculated, and could be as high as 10−1 per galaxy
per year (Chen et al., 2009); such a high rate then has notable consequences for future surveys,
such as the Large Synoptic Survey Telescope (LSST), that are predicted to observe thousands tomillions of galaxies (Ivezic et al., 2008).
In addition to calculating the rates of TDEs, much work has been done on the tidal disruption
process itself and the fate of the disrupted debris. Early studies showed that the tidal potentialof the SMBH generates a range in the binding energy (i.e., the sum of the kinetic and potential
energies) of the disrupted material (Lacy et al., 1982). If the center of mass of the star is on aparabolic orbit, which should be true to a high degree of accuracy considering that most stars
scattered into the loss cone come from very far distances from the SMBH (Frank & Rees, 1976),
then this range in energies implies that roughly half of the material is bound to the black hole(negative binding energy) while the other half is unbound (positive binding energy) (Rees, 1988).
Thus, half of the disrupted debris will recede to large (but finite) distances from the black hole
before eventually returning to the point of disruption, while the other half will escape from theblack hole at some terminal velocity, which scales approximately as v∞ ' vesc (Mh /M∗ )1/6 ; herepvesc = 2GM∗ /R∗ is the escape velocity from the surface of the tidally-disrupted star. For thefiducial encounter between a solar-like star (i.e., one with a solar mass and a solar radius) and a
106 M SMBH, the most bound material takes about a month to return to the point of disruption,
while the escape velocity of the most unbound debris is on the order of a few percent the speed oflight. Furthermore, while the initial stage of the fallback of material to the SMBH may depend on
the specific details of the encounter (Lodato et al., 2009), asymptotically the rate at which debris
returns to pericenter should scale as t−5/3 , t being the time since disruption (Phinney 1989, but

5
see Guillochon & Ramirez-Ruiz 2013), and this feature has been seen in all TDEs observed so far(Komossa, 2015).
Despite the fact that the star is “tidally shredded” by the SMBH, the disruption processitself is relatively gentle (see, however, Carter & Luminet 1983, Bicknell & Gingold 1983, andGuillochon et al. 2009 for the scenario in which the star plunges deep inside the tidal radius, where
the encounter can be more violent). After passing through the tidal radius, the star is stretched
into a long thin debris stream (Chapter 2). In general, the gas parcels comprising this stream
evolve under the gravitational influence of the black hole in the radial direction, with self-gravity
being important for confining the stream in its transverse extent (Kochanek 1994, Chapters 2 – 4).
Because of the tidal potential mentioned above, half of this stream escapes from the influence ofthe black hole.
So far, little if any of the disruptive encounter presents observable evidence of its occurrence(but see Kasen & Ramirez-Ruiz (2010) for a discussion of recombination transients that may provide
weak optical signals on the order of weeks after disruption, and Guillochon et al. (2015) and Chenet al. (2015) for an analysis of the debris remnants that may mimic supernova remnants). However,the bound portion of the debris stream – the half of the disrupted star that eventually returns to
the point of disruption – can generate a substantial luminous flare across a number of wavelengths.
In particular, as material returns to the SMBH, it is compressed vertically and horizontally; thiscompression can then generate a shock that heats the gas and dissipates kinetic energy (Rees, 1988;
Evans & Kochanek, 1989). Furthermore, after material passes through pericenter for a second time,
its orbit is rotated through a general relativistic precession angle that amounts to approximately 10◦
for a solar-like star disrupted by a 106 M

SMBH (this also assumes that the pericenter distanceof the star was equal to the tidal radius at the time of disruption; Rees 1988). As this debris
recedes from the black hole for a second time, it will thus impact the portion of the debris stream
still making its way to the black hole (see Figure 1.2). This intersection will also serve to deplete
the kinetic energy of the debris at the expense of heating it viscously (see Piran et al. 2015 for a
discussion of the observable signals generated at this stage of the TDE).

6

outgoing
streamincoming
streamαFigure 1.2: A schematic of the self-intersection process that occurs during tidal disruption events.
Here α is the angle between the orientation of the pericenter of the incoming debris and that of
the outgoing debris, equal to α ' 10◦ for typical parameters, which is the result of the generalrelativistic advance of periapsis.

7
Through non-ideal interactions such as these, the bound debris from a TDE will generally
dissipate its ordered kinetic energy, ultimately resulting in material moving closer to the SMBH.
If the gas continues to lose its energy efficiently, then material will be rapidly funneled onto the
SMBH, releasing a large amount of radiation in the process. Through such a sequence of events, thebound portion of the debris stream should give rise to an intense period of accretion onto the blackhole, continuing as material from the debris stream returns to the pericenter of the original star andforming an accretion disk around the SMBH. Early predictions generally found that the spectra ofthese accretion disks should peak in the optical/UV (Cannizzo et al., 1990; Ulmer, 1999; Lodato &Rossi, 2011), with the lightcurve tracking the initial fallback rate for years before transitioning to a
viscous-like accretion phase (Lodato & Rossi, 2011). These predictions are generally in agreementwith observations (Komossa, 2015), though there have been a few surprises (see below).
An intriguing feature of the early stages of the accretion process concerns the rate at whichmaterial is consumed by the SMBH. Specifically, if the black hole mass is less than roughly 107 M(and the disrupted star is solar-like), then, assuming that the rate at which material returns topericenter closely mimics the accretion rate onto the black hole, one can show in a straightforwardmanner that the accretion rate onto the SMBH is super-Eddington for months to years (Strubbe &Quataert 2009, 2011, Chapter 5). During this super-Eddington phase, when the isotropic radiationpressure associated with the accretion luminosity exceeds the gravitational pull of the black hole,material may be blown off in a large-scale wind (Strubbe & Quataert, 2009), the disk may “spread”
to account for the high luminous output and redistribution of angular momentum (Shen & Matzner,2014), or the debris may be heated to the point where the accretion disk becomes quasi-spherical(Loeb & Ulmer 1997, Chapter 5).
One of the most profound aspects of tidal disruption events, from theoretical and observational standpoints alike, is that they can be responsible for launching collimated, fast-moving (i.e.,on the order of the speed of light) outflows of gas. While these relativistic jets are encountered inother seemingly-distinct astrophysical phenomena including, but not limited to, radio galaxies (e.g.,
Begelman et al. 1984), gamma-ray bursts (GRBs; e.g., Woosley 1993; MacFadyen & Woosley 1999;

8
Woosley & Bloom 2006), and Herbig-Haro objects (Reipurth & Bally, 2001), they were completely
unexpected from TDEs (but see the prescient analysis of Giannios & Metzger 2011) prior to thediscovery of the event Swift J1644+57 (hereafter J1644; Bloom et al. 2011; Burrows et al. 2011;
Cannizzo et al. 2011; Levan et al. 2011; Zauderer et al. 2011). When it originally triggered theburst alert telescope (BAT) onboard the Swift satellite, J1644 was a source of hard X-rays and soft
gamma-rays that was thought to be an ordinary GRB. However, continued monitoring with the
X-ray telescope (XRT) also onboard the Swift satellite showed that its level of activity continued
well past the normal threshold of GRB duration, with emission measured by the XRT extending
beyond 500 days after the initial detection (compared to the standard 1-10 second duration oflong GRBs). Observations in the radio with very long baseline interferometry (VLBI) also showedthat J1644 was simultaneously a strong source of radio emission, with radio observations yielding
a non-zero detection long after the cessation of the X-ray emission (see Zauderer et al. 2013 for the
long-term evolution of the radio emission). The X-ray lightcurve also exhibited an approximately
power-law decay, LX ∝ t−α , with 1 . α . 2.5.
Ultimately, the extremely long duration, power-law decline of the X-ray emission (with powerlaw index consistent with the theoretically-expected value of 5/3), proximity to the nucleus of an
identified host galaxy, and coincident (spatially and temporally) radio emission led observers andtheorists alike to interpret J1644 as the first-observed, jetted TDE. The luminosity of the source
was unequivocally super-Eddington by orders of magnitude (Zauderer et al., 2011), which has been
the invoked explanation for the presence of the relativistic jet (though the mechanism responsible
for launching and collimating the jet remains elusive and polemical; see Tchekhovskoy et al. 2014
and Chapters 7 and 8). Since the discovery of J1644, two other sources – Swift J2058+05 (Cenko
et al., 2012) and Swift J2111-28 (Brown et al., 2015) – have also been found that are qualitatively(and quantitatively, to a lesser extent) similar to J1644. It thus appears that relativistic jets makean appearance in a small subclass of TDEs, specifically those that are super-Eddington in nature.
Following the order-of-events picture of TDEs generated above, this thesis is organized to tell
the evolutionary history of a tidal disruption event – from the formation of the tidally-disrupted

9debris stream, to the development and evolution of the accretion disk, and finally to the behaviorof the relativistic jets that can be launched from these systems. Perhaps not surprisingly, thischronological order was not the order in which the research was done, with the projects on thedebris streams (Chapters 2 – 4) done most recently and the project on the accretion disks (Chapter5) done first. However, to preserve the readability of the thesis, we follow the history that makes
sense from a nature-oriented, as opposed to an academic, standpoint. We note, however, that
opting to do will result in earlier parts of the thesis referencing later parts.
The first three chapters of this thesis describe the dynamical evolution of the debris streams
generated from TDEs. In particular, Chapter 2 presents the findings of an analytical and numericalinvestigation that considers the self-gravity of tidally-disrupted debris streams, showing that thesestreams can be gravitationally unstable and form small-scale, gravitationally-bound “clumps.” In
Chapter 3 we investigate how the gravitational stability of these streams is affected by the thermodynamic properties of the gas, and we show that stiffer equations of state result in decreased
stability. Chapter 4 gives a predominantly analytic treatment of these debris streams, deriving
approximate expressions for their velocity structure, density profile, and stability criteria, and theresults are compared to the simulations performed in Chapter 2.
In Chapter 5 we consider the next stage of the tidal disruption process, and develop a newmodel to describe the structure of the accretion disks that form during the super-Eddington phaseof accretion that ensues from some TDEs. This ZEro-BeRnoulli Accretion, or ZEBRA, picture
postulates that the debris forming the disk absorbs energy liberated during the accretion processuntil it conforms to a weakly-bound (zero-Bernoulli) state. Using the zero-Bernoulli prescription,
we find self-similar expressions for the density, pressure, and angular momentum profiles of the
disk. We make comparisons to the event Swift J1644 and we show that our model is consistent
with observations if the black hole that powered this event has a mass of ∼ 105 M .
Finally, the last three chapters describe the evolution of the jets launched from these systems,
the energy and momentum of these jets dominated by the radiation released during the accretion
process. Chapter 6 presents a derivation of the general relativistic equations of radiation hydrody-

10
namics in the viscous limit, which describe how anisotropies in a radiation field transfer energy andmomentum between a relativistic jet and its surroundings. We show, in contrast to previous works,that the radiation-viscous stress tensor contains a non-zero bulk viscosity and a correction to the
comoving energy density. In Chapters 7 and 8 we use these equations in a boundary layer form,
the boundary layer assumption stating that the transition between the jet and its surroundings isconfined to a thin layer (on the order of the square root of the photon mean free path), and derive
two models for the structure of the jets launched from tidal disruption events. These models showthat regions of large shear that develop between the fast-moving outflow and the static ZEBRA envelope are characterized by fewer scatterers and, hence, a lower optical depth and a harder photon
spectrum.
Chapter 2 was published in original form in the Astrophysical Journal Letters, Volume 808,
Issue 1, Letter 11 (6 pp.). Chapter 3 was published in original form in the Monthly Notices of
the Royal Astronomical Society, Volume 455, Issue 4, pages 3612 – 3627. Chapter 4 was publishedin original form in the Monthly Notices of the Royal Astronomical Society, Volume 459, Issue 3,
pages 3089 – 3103. Chapter 5 was published in original form in the Astrophysical Journal, Volume781, Issue 2, Article Id. 82 (17 pp.). Chapter 6 was published in original form in the AstrophysicalJournal, Volume 797, Article Id. 103 (13 pp.). Chapter 7 was published in original form in the
Astrophysical Journal, Volume 809, Issue 1, Article Id. 1 (9 pp.). Chapter 8 was published inoriginal form in the Astrophysical Journal, Volume 809, Issue 1, Article Id. 2 (10 pp.).

Chapter 2

Gravitationally Unstable Streams

2.1

Introduction
When a star comes within a supermassive black hole’s (SMBH) tidal radius rt ' R∗ (Mh /M∗ )1/3 ,where R∗ and M∗ are the stellar radius and mass, respectively, and Mh is the black hole mass, thetidal force exerted by the hole across the star is sufficient to overcome its self-gravity, resulting in
its destruction. Early studies of these tidal disruption events (TDEs) demonstrated that roughly
half of the disrupted stellar debris is bound to the black hole (Lacy et al., 1982; Rees, 1988),
meaning that it will eventually return to the tidal radius, dissipate energy through shocks (Evans& Kochanek, 1989; Kochanek, 1994; Guillochon et al., 2014b), and form an accretion disk. The
resulting accretion power is then capable of producing a highly luminous event, and many havealready been detected (Komossa & Greiner, 1999; Gezari et al., 2008; Burrows et al., 2011; Cenko
et al., 2012; Bogdanović et al., 2014).
Considerations of the star at the time of disruption show that the rate at which the bound,tidally-stripped material returns to the hole decreases as t−5/3 (Phinney, 1989), and early simulations supported this estimate (Evans & Kochanek, 1989). More recently, authors investigatedthe consequences of the stellar composition on the rate of return (Lodato et al., 2009), illustratingthat the early fallback stages have a more complex temporal behavior that depend on the density
stratification. Guillochon & Ramirez-Ruiz (2013) showed that the pericenter distance can havedramatic effects on the rate of return of debris. General relativistic Lense-Thirring (Stone & Loeb,
2012) and apsidal (Bonnerot et al., 2016) precession and recompression shocks (Guillochon et al.,12
2014b) have also been investigated.
Most of the recent work on TDEs has been performed with the aid of numerical simulations.
For example, Guillochon et al. (2014b) used a grid code (see also Shiokawa et al. 2015) to follow
the evolution of the tidally disrupted debris from the initial encounter with the black hole to its
eventual return to pericenter, while Bonnerot et al. (2016) used a smoothed particle hydrodynamics(SPH) code to achieve the same feat (see also Hayasaki et al. 2015). In these cases, however, theset of parameters used to model the tidal disruption was somewhat unphysical – Guillochon et al.
(2014b) used a 103 M black hole when following the return of the debris to the pericenter radius,
while Bonnerot et al. (2016) used a more tightly bound stellar progenitor (e = 0.95 for their mostparabolic simulation, e being the stellar eccentricity). Both of these choices were made to reducethe computational cost at the expense of reality. Instead, a more likely mass for an SMBH is 105 –108 M , and the infalling star is initially so far from the hole that its orbit is effectively parabolic(e ' 1).
A full TDE with realistic parameters has not yet been modeled because of the extreme setof spatial and temporal scales. In spite of these difficulties, we present the first simulation that has
resolved the full duration of a TDE with a solar progenitor on a parabolic orbit, the periapsis ofwhich coincides with the tidal radius, and a 106 M

SMBH. The simulation runs from well before

the initial tidal interaction to long after the most bound debris returns to pericenter, corresponding
to roughly ten years after disruption. This enables us to calculate explicitly the rate of return oftidally stripped debris to the black hole, demonstrating the effects of self-gravity which dominatethat of the hole when the material is near apoapsis. In section 2 we provide analytic formulae
for the shape and density distribution of the stream as it recedes from the hole, and we develop
arguments which suggest the dominance of self-gravity over the tidal field of the hole at large radii.
In section 3 we describe the initial conditions and numerical method used to simulate the encounter.
Section 4 presents the results of the simulations and their analysis. We conclude and discuss further
implications of our findings in section 5.

13

2.2

Self-gravity
The forces acting on the debris are self-gravity, the tidal shear across the width of the stream,and pressure. If we consider a small volume within the stream of length δR and mass δM , thenthe self-gravity of the stream dominates the tidal field if

ρ0 &

3 Mh
,
2π r3

(2.1)

where ρ0 = 3δM/(4πδR3 ).
We can calculate when this condition is satisfied by approximating the stream as a circularcylinder of width H. In this case, the differential amount of mass dM contained in the stream is

dM = πH 2 ρ dx,(2.2)

where x is the length measured along its center. If the specific energy distribution of the gas isfrozen in at disruption, then the curve traced out by x is a function solely of the position of a given
gas parcel when the star is at pericenter, which we will denote by Rp = µR∗ , with −1 ≤ µ ≤ 1.
Modeling the star as a polytrope (Hansen et al., 2004), the differential amount of mass containedin dµ is (Chapter 5)
R ξ1
n
dM
1
µξ1 Θ(ξ) ξ dξ
= M∗ ξ1 R ξ1,
dµ2
Θ(ξ)n ξ 2 dξ

(2.3)

0

where n is the polytropic index of the gas, Θ(ξ) is the solution to the Lane-Emden equation andξ1 is the first root of Θ(ξ).
We can parametrize x by the functions r(µ) and θ(µ), where r and θ are solutions to theequations of motion of a test particle in the Newtonian gravity of the hole:

r=

`2
1
,
GMh 1 + e cos θ

r2 θ̇ = `.

(2.4)

(2.5)

14
Here ` and e are the specific angular momentum and eccentricity, respectively, of the gas parcel
with position µ, and are given by`=p
2GMh rt 1 − µ q −1/3 ,(2.6)

e = 1 − 2µ q −1/3 ,(2.7)

where q ≡ Mh /M∗ . We have assumed here that the gas parcels initially move with the center ofmass of the star and β = rt /rp = 1. By numerical integration of equation (2.5) we can determine
θ(µ), and r(µ) is then found by using that result in equation (2.4).
p
The infinitesimal distance along the curve is dx = dµ (r0 )2 + r2 (θ0 )2 , where primes now
denote differentiation with respect to µ. Using this result and equation (2.3) in equation (2.2) gives

ρ=

R ξ1

n
µξ1 Θ(ξ) ξdξp.
R
2πH 2 (r0 )2 + r2 (θ0 )2 0ξ1 Θ(ξ)n ξ 2 dξM ∗ ξ1(2.8)

Using this equation in condition (2.1) gives the range in µ over which self-gravity dominates thetidal shear.
The pressure of the material can also resist self-gravity. However, even if a small overdensity ofpradius δR and mass δM is pressure-supported, it must satisfy the Jeans condition δR . cs π/(ρ G).
If δR exceeds this value, any perturbation will cause material to clump around δM .
Finally, the local velocity gradient can potentially stabilize the stream. The divergence ofthe velocity decreases the density within the stream on a timescale τdiv ' 1/(∇ · v). If τdiv & τf f ,√where τf f ' δR3/2 / GδM is the local free-fall time, self-gravity will overcome the stabilizing effectof the divergence of the flow.
By balancing the tidal shear and self-gravity in the transverse direction, Kochanek (1994)
found that H ∝ r1/4 for a γ = 5/3 equation of state. We will use this when considering how thedensity (equation 3.10) varies with r (see Figure 2.2).

15

2.3

Simulation setup
To test our analytical reasoning above and the canonical t−5/3 fallback rate, we now employ

three dimensional hydrodynamic simulations. We use the SPH code phantom (Price & Federrath,2010; Lodato & Price, 2010) to simulate the tidal encounter. phantom was designed for runninghigh resolution hydrodynamic simulations with complex geometries (Nixon et al., 2012a,b; Martin
et al., 2014b,a; Nealon et al., 2015), and is therefore well suited to studying TDEs.
For our simulations we model the black hole as a Newtonian sink particle situated at theorigin. Self-gravity of the stellar material is modeled using a k-D tree alongside an opening angle criterion, calculating directly the forces arising over short distances (Barnes & Hut, 1986).
The opening angle for the simulations presented here was 0.5 (we ran an additional simulation
with a more accurate opening angle of 0.3 and found negligible differences with the run presentedhere). The gas retains a polytropic equation of state, i.e., only adiabatic energy gains and losses
are included (we also performed simulations with shock heating and, with our parameters, foundnegligible differences; see also Lodato et al. 2009).
We initialized our runs by placing 106 particles in the configuration of a γ = 5/3 polytrope;
this is achieved by first placing the particles on a close-packed sphere. This configuration is thenstretched to achieve the correct polytropic density distribution, the resulting polytrope then placed
at a distance of 10 rt from the hole with its center of mass on a parabolic orbit, its location at√
periapsis equal to the tidal radius (equations (2.4) and (2.5) with ` = 2GMh rt and e = 1). The
distance of 10 rt allows the initial configuration time to relax before interacting with the black hole.
Here we are primarily interested in exploring the effects of self-gravity on the stream evolution,and so we include self-gravity at all stages of the simulation. We are also interested in comparing
the theoretical rate of return of bound material (not necessarily equal to the accretion rate onto
the black hole), Ṁfb ∝ t−5/3 , with that inferred from the simulation. Therefore, once the disrupted
debris is beyond periapsis, the accretion radius of the black hole, initially well inside the tidal
radius, is extended to racc ' 3 rt (' 120Rg ). Once the bound material returns to the point of16
disruption, it is removed from the simulation.

2.4

Results
The top, left panel of Figure 2.1 shows the density of the debris stream at periapsis, while

the top, right is 0.84 hours after disruption; these reproduce Figure 4 of Lodato et al. (2009). The
bottom four show, from left to right and top to bottom, 5.38 days, 15 days, 22.4 days and one monthafter disruption. The red curves indicate the analytic predictions; the fact that they provide goodfits to the data means that the particles approximately trace out Keplerian orbits in the potentialof the black hole.
The black points in Figure 2.2 show the density distribution along the stream at t = 22.4
days after disruption; the solid, red curve gives the analytic prediction (equation (3.10) with
H = R (r/rt )1/4 ) and the dashed, red curve is the density at which the self-gravity of the debris and the tidal field of the hole are equal along the stream (the right-hand side of equation (2.1);
self-gravity dominates for points above this curve). The wings present in the simulated values,
which were also found by other authors (Lodato et al., 2009), are due to the fact that the shape ofthe star at the time of disruption is not spherical (see the left-most panel of Figure 2.1).
Figure 2.3 shows the rate of return of material to pericenter, Ṁf b (t), from the simulation(solid, black curve) and the analytic solution (red, dashed curve). The analytic estimate was deter√
mined by using equation (2.3) and the fact that µ(t) = (t/T )−2/3 , where T = 2πMh /(M∗ GMh )(R∗ /2)3/2(Chapter 5). The time at which material returns to pericenter is slightly earlier than that predicted
analytically, a consequence of the tidal distortion at the time of disruption (see Figure 2.3). The
asymptotic scaling Ṁf b ∝ t−5/3 is closely followed by the numerical solution at late times, reaffirming that the gas parcels follow approximately Keplerian motion. In our simulation, half of thebound material is accreted by about one year – shortly after the peak in the fallback rate.
Figure 2.3 also shows that the fallback rate quasi-periodically deviates from the t−5/3 law atlate times. This scatter is due to the fact that the stream fragments, forming small gravitationallybound clumps that enter the accretion radius at discrete times, and is not simply numerical noise.

2.05

t = 0.835 h

1017log column density [g/cm2]

11

0.01

11

log column density [g/cm2]

t=0

0
10
y / rty / rt

2

9
-0.01
9

-0.02

1.95

8

1.02

-0.05
4

t = 5.38 d

25

2

0
x / rt0.05
4
t = 15 d50
2log column density [g/cm2]

1
x / rtlog column density [g/cm2]

0.98

8

y / rty / rt

20

15
0

0
0

10

5
-70

-2

-60

-150

x / rt-2-100
x / rtt = 22.4 d

100

2

4
log column density [g/cm2]

4

150

200

t = 31 d

2100

log column density [g/cm2]-80

y / rty / rt

50

0

0

0

0

-50
-100
-200

-100
x / rt

0

-2
-300

-200

-100

0

-2

x / rtFigure 2.1: The star at the time of disruption (top, left), where the colors indicate the column
density and distances are measured in units of tidal radii (' 7 × 1012 cm = 100R ), 0.835 hours
after periapsis (top, right; the top two figures coincide with the bottom-left panels of Figure 4 a)and b), respectively, of Lodato et al. 2009.), and 5.38 days (middle, left), 15 days (middle, right),
22.4 days (bottom, left) and 1 month (bottom, right) after disruption. The red, dashed curves on
the bottom two rows indicate the analytic predictions.

18

10-3

t = 22.4 d

10-4
10-5

[g / cm3]

10-6
10-7
10-8
10-9
10-10
10-11
10-12

50

100

150

200

250

r / rtFigure 2.2: The computed density of the stream (solid, black curve), the analytic prediction (red,solid curve; equation 3.10), and the density at which self-gravity becomes important (red, dashedcurve; equation 2.1) as functions of r (spherical distance from the hole) at 22.4 days from the pointof disruption. Points above the red, dashed curve are self-gravitating. The numerical solution
extends to smaller radii because the star at the time of disruption is not spherical (see also Lodato
et al. 2009).

19

1

Numerical solution

dM/dt [MSun / yr]

Analytical solution

10-1

10-2

10-3
10-1

1

10

t [yr]

Figure 2.3: The fallback rate in solar masses per year calculated from the simulation (black, solid
curve) and the analytic prediction (red, dashed curve) as functions of time in years. The temporal
shift between the numerical solution and the analytic prediction is from the stellar distortion at
the time of disruption (see text for further details). The variability at late times is due to theaccretion of clumps formed in the self-gravitating stream. The variability starts about 2 years after
disruption; however, this timescale depends upon the details of the TDE and its environment, andtherefore may be shorter.

8

112

4000

97

192

114

98

116

193

t = 2 yr

110

96

191

6
-2601

-2600

-2096

-2094

-2092

-2090

y / rt

-2098

95

-2602

log column density [g/cm2]

20

-1978

-1977

-1976

-1975

4
2000

2

0

-6000

-4000

-2000

0

0

x / rtFigure 2.4: The density two years after disruption and, in the insets, close-ups of the fragmentsthat forms along the stream.

21
To emphasize this point, Figure 2.4 shows the stream two years after disruption and insets that
focus on clumps near the center of the stream.

2.5

Discussion and conclusions
We have presented the results of a TDE in which a solar mass star (with its pericenter at

the tidal radius and its center of mass on a parabolic orbit) was disrupted by a 106 M

blackhole. Contrary to past investigations, we resolved the full duration of the TDE – from the initial
encounter between the undisturbed star and the hole to long after the most bound, tidally-stripped
debris has returned to periapsis. When finished, more than 90% of the bound material was accreted.
Crucially, we are the first to follow this long-term evolution for a solar mass star on a parabolic
orbit around an SMBH.
The two main aims of this paper are (1) to determine the effects of self-gravity on the stream
far from the black hole (see Figures 2.3 and 2.4), and (2) to compare the rate of return of boundmaterial to the analytic estimates (Figure 2.3). The fiducial rate Ṁf b ∝ t−5/3 was determinedby assuming that the specific energy distribution of the material was frozen in at the time ofdisruption. However, the self-gravity of the material, which is important at large distances fromthe hole (Section 2), has the potential to alter the specific energy distribution (see also Guillochon &Ramirez-Ruiz 2013). It is therefore necessary to resolve the full disruption process with self-gravity
included at every step to determine the true rate of return.
Our simulations demonstrate that the fallback rate of debris closely mimics the theoreticallypredicted one, the biggest discrepancy arising from the return time of the most tightly boundmaterial. Therefore, the material, in agreement with intuition, follows approximately Keplerian
orbits about the hole. Interestingly, however, we also found that the fallback rate at late times
tends to over and under-estimate the t−5/3 law, a feature that is due to the fragmentation of thestream into gravitationally-bound clumps. When one of these clumps is accreted, the fallback rate
spikes above its average value, while the rate is reduced when the lower density debris between
clumps returns to periapsis.

22
We have shown analytically and numerically (see equation (2.1) and Figure 2.2) that the
local self-gravity of the stream dominates the tidal field of the hole at large radii. We also find thatthe pressure within the sphere of the center inset of Figure 2.4 is p ' 103 dyn/cm2 . Conversely, thepressure necessary for maintaining hydrostatic equilibrium is peq ' GM ρ/R ' 4πGρ2 R2 /3, where
the radius of the clump in Figure 2.4 is R ' 0.3 rt (= 30R ) and the density is ρ ' 10−6 g/cm3 ,which gives peq ' 106 dyn/cm2 . Finally, the computed divergence of the velocity just outside
the clump gives a divergence timescale of τdiv ' 1/(∇ · v) ' 107 s, while the infall timescale is√τf f ' 1/ Gρ ' 4 × 106 s. This shows that gas pressure and the local shear of the stream are also
incapable of overcoming self-gravity.
The tidal field of the hole, gas pressure, and the local velocity shear are all incapable ofsupporting the clump in Figure 2.4 against its own self-gravity. This finding suggests that theoriginal stream was gravitationally unstable, and small perturbations resulted in its fragmentation.
For the results presented here, these perturbations are induced by the discreteness of the numericalmethod. To substantiate this claim, we ran the simulation with 104 and 105 particles and foundthat clumps formed sooner with decreasing particle number. However, our pressure smoothinglength, equal to the gravitational smoothing length, was always at least as small as the Jeanspradius RJ = cs π/(Gρ) (see Figure 2.5); the collapse was therefore always resolved (Truelove
et al., 1997; Bate & Burkert, 1997). Additionally, the first clumps in the simulation presented
here form around a month after disruption; at this time, the smoothing length at the center of thestream is roughly 0.1 times the width of the stream, meaning that the forces are clearly resolved at
this time (see also Figure 5, which shows that the collapse itself is also resolved). Finally, we notethat our SPH simulations are not susceptible to artificial fragmentation (Hubber et al., 2006), andour results demonstrate that any physical perturbation imposed on a stream that satisfies (2.1) is
unstable to collapse.
To further investigate the instability of the stream, we performed additional runs with 105 and106 particles and seeded the initial distribution of debris with a small, but resolved, perturbation. In
these cases, clumps formed at almost the exact same time with nearly identical masses. Therefore,

23

Figure 2.5: The ratio h/RJ , h being the smoothing length and RJ the Jeans radius, as a functionof density at a time when the first clumps start to form.

24
a resolved noise field imposed on top of the otherwise smooth distribution of stellar debris results inconverged fragmentation. This finding further supports the fact that the stream is gravitationallyunstable, and any small physical perturbation to the distribution of the debris will cause it tofragment. We will present a more complete analytical and numerical analysis of the gravitational
stability of the stream in a future publication.
Real, fragmentation-inducing perturbations could be caused by a number of physical processes. For example, the interaction of the debris with any material surrounding the hole, thedensity distribution of which is neither necessarily smooth nor homogeneous, could cause local deformities within the stream. In light of this, the variability in the light curve of the event Swift
J1644+57 (Burrows et al., 2011; Levan et al., 2011; Zauderer et al., 2011), the recently-observed,
jetted TDE, could be interpreted as abrupt changes to the accretion rate induced by the fallbackof bound clumps. This notion is supported by the lack of any ultraviolet and optical emission fromthe event, which is indicative of a large amount of dust present in the circumnuclear environment(Burrows et al., 2011); interactions between the stream and this natal environment could haveinitiated early fragmentation of the stream, causing variability in the fallback rate on a timescalecommensurate with observations.
Finally, we note that while the mass contained in the fragments in Figure 2.4 is δM '0.005M , there is a range of clump masses. Therefore, we suggest that the object G2 – the clumpof material observed near the galactic center (Burkert et al., 2012) – and other such clouds, could
have been produced by the tidal disruption of a star in the recent past. Guillochon et al. (2014a)
reached a similar conclusion; their clump, however, was formed by fluid instabilities generated
through the interaction of a stream of debris with an ambient medium. Here, on the other hand,self-gravity causes the stream to fragment.

Chapter 3

Post-periapsis Pancakes: Sustenance for Self-gravity in Tidal Disruption Events

3.1

Introduction
A supermassive black hole of mass Mh can tidally destroy a star of mass M∗ and radius R∗

if the star comes within the tidal radius rt ' R∗ (Mh /M∗ )1/3 of the hole. In this scenario, called atidal disruption event (TDE), the star is shredded into a stream of debris. The properties of thedebris and its ultimate fate have been studied for decades, both analytically and numerically, andthe observational predictions generated from these studies have been tested.
Early analyses of TDEs showed that, due to the differential gravitational potential of theblack hole, half of the disrupted debris that was closer to the hole at the time of disruption isbound to the black hole, while the other half is unbound (Lacy et al., 1982; Rees, 1988). The
half that is bound will eventually return to the black hole, circularize, and form an accretion disk.
The properties and observational signatures of this accretion disk have been investigated by many
authors (e.g., Cannizzo et al. 1990; Loeb & Ulmer 1997; Strubbe & Quataert 2009, 2011; Lodato& Rossi 2011; Guillochon et al. 2014b; Shen & Matzner 2014). The power radiated during theaccretion process is enough to generate a highly luminous event, and some of these events havealready been observed (Bade et al., 1996; Komossa & Greiner, 1999; Halpern et al., 2004; Levanet al., 2011; Cenko et al., 2012; Bogdanović et al., 2014; Komossa, 2015).
Phinney (1989) showed analytically that the rate at which the debris returns to the black holedecreases with time as Ṁf b ∝ t−5/3 . This feature, coupled with the longevity of the signature, isthe observational “smoking gun” of a TDE. Many of the recently-observed TDE candidates exhibit

26
a lightcurve that decreases in a manner commensurate with this power-law rate (Bloom et al., 2011;
Zauderer et al., 2011; Cenko et al., 2012; Gezari et al., 2012; Bogdanović et al., 2014; Brown et al.,2015).
To investigate the complex hydrodynamical interactions that take place during TDEs, many
authors have resorted to numerical simulations. Early smoothed-particle hydrodynamics (SPH)
calculations supported the analytic estimates of Rees (1988) and Phinney (1989), showing that the
distribution of specific energies calculated not long after the time of disruption generates a fallback
rate that scales as Ṁf b ∝ t−5/3 (Evans & Kochanek, 1989). More recently, Lodato et al. (2009)
elucidated the effects of the structure of the progenitor star on the disruption process, demonstratingthat the early stages of the fallback depend on the properties of the star. Guillochon & RamirezRuiz (2013) investigated how the impact parameter β ≡ rt /rp , rp being the pericenter distance ofthe stellar progenitor, alters the nature of the event, and found that shallower impact parameters
often result in the survival of a bound stellar core. Finally, Hayasaki et al. (2013), Bonnerot et al.
(2016), Hayasaki et al. (2015), and Shiokawa et al. (2015) have looked into the effects of generalrelativity on the stream, showing how apsidal and Lense-Thirring precession can alter the formationof the disk that forms when the tidally-disrupted debris returns to pericenter.
In Chapter 2 we demonstrated that, when a solar-like star with a γ = 5/3 adiabatic equationof state is disrupted by a 106 M

hole, self-gravity can be important for determining the stream

properties during its late evolution (see also Kochanek 1994 and Guillochon et al. 2014b for a
discussion of self-gravity). In particular, they showed that the tidal influence of the black holebecomes sub-dominant to the self-gravity of the debris, which results in the late-time fragmentationof the stream into gravitationally-bound clumps. These clumps then return to the original pericenterat discrete times, causing the fallback rate of the material to fluctuate about the t−5/3 average.
An important question arising from the results of Chapter 2 is: when is the self-gravitational
nature of the stream revived post-disruption? As the tidal shear and the self-gravity of the starequal one another at the tidal radius, one might suspect that the self-gravity of the debris is mostinfluential at late times. Indeed, it is during this late evolution that we found in Chapter 2 that27
the stream gravitationally fragments. As we will show here, however, the self-gravity of the debris
can affect the stream evolution soon after the star passes through periapsis (on the order of hours
for the tidal disruption of a solar-like star by a 106 Mhole). We find that the star experiencescompressive forces in the orbital plane, which lead to the formation of a post-disruption pancake,similar to the one found by Carter & Luminet (1982) but oriented orthogonal to the orbital plane ofthe progenitor. This in-plane recompression then augments the importance of self-gravity, resulting
in perturbations on top of the stream that can induce early recollapse. (We note that we will beconsidering TDEs in which the star is completely destroyed, and hence these results should not
be confused with those of Guillochon & Ramirez-Ruiz 2013 who, in certain cases, found surviving
stellar cores for impact parameters less than one.)In Section 2 we present an analytical analysis of the stream under the impulse approximation,
which assumes that the star maintains hydrostatic balance until it reaches the tidal radius. We
demonstrate that, even when the pericenter distance and the tidal radius are approximately coincident, a caustic – a location where the orbits of the gas parcels comprising the stream collapse toa two-dimensional surface – occurs shortly after the star is disrupted. Section 3 presents numericalsimulations that demonstrate the effects of this caustic, and specifically shows how it can modify
the density structure of the stream for times long after disruption. We present a discussion of theresults of our simulations in Section 4 and consider the astrophysical implications of our findings
in Section 5. We conclude and summarize in Section 6.

3.2

The impulse approximation
Many authors (e.g., Carter & Luminet 1983; Rees 1988; Lodato et al. 2009; Stone et al. 2013)

have considered the disruption process from a simplified, analytic standpoint. While an analytic
approach almost certainly misses many of the intricacies associated with the realistic problem,it has the advantage of being able to characterize the bulk processes that take place during theinteraction. Furthermore, it is able to elucidate the manner in which those processes dependon the properties of the progenitor star and the black hole, which provide useful observational

28
diagnostics. Here we discuss the impulse approximation, which assumes that the star is able tomaintain hydrostatic balance until it reaches the tidal radius and it is thereafter disrupted, i.e., thepressure and self-gravity of the material are negligible after the star has passed through the tidal
radius.
Carter & Luminet (1982, 1983) considered the case where the pericenter distance of the star,
rp , is well inside the tidal radius of the hole (their affine star model; see also Stone et al. 2013
for an alternative approach to analytically modeling this scenario). For these high-β encounters,
where β ≡ rt /rp is the impact parameter, the impulse approximation can be applied early on in
the tidal disruption process. Because of the component of the tidal force that acts orthogonally tothe orbital plane of the star, the gas parcels comprising the top and bottom of the stellar envelope
undergo effective freefall, forming an infinitely thin plane, or caustic, at the pericenter radius (thelocation of the caustic is actually slightly after the pericenter, only equaling the pericenter distancefor β → ∞; Bicknell & Gingold 1983). This “pancaking” effect was then thought to be capableof igniting thermonuclear fusion via the triple-α process, resulting in the detonation of the star.
However, studies showed that the shocks near pericenter resulted in lower densities and pressuresin the stellar core than those predicted by Carter & Luminet (1982), meaning that the triple-α
process is unlikely to be initiated in these encounters (though some fusion via the CNO cycle may
occur; Bicknell & Gingold 1983).
On the other hand, when β ' 1, the star can retain its unperturbed structure for much longer.
In this case, one can approximate the star as being spherical, with every gas parcel moving withthe center of mass, until the pericenter is reached. Here we will focus on this case, not only because
it has not been treated as thoroughly as the β  1 scenario, but also because it has interesting
consequences for the disrupted material soon after the pericenter distance is reached. Later in thispaper, we will relax the assumptions made by this model with three dimensional hydrodynamic
simulations.

29
3.2.1

Equations

Once the star passes through the tidal radius, the pressure and self-gravity of the gas parcels
are assumed negligible, implying that they follow Keplerian orbits in the potential of the black
hole. The equations of motion that describe these orbits are given byr2 sin2 θ φ̇ = `,r4 θ̇2 +`2= k2 ,
sin2 θ
1 2 k2GMṙ + 2 −
= ,
2
r
r

(3.1)

(3.2)

(3.3)

where dots denote differentiation with respect to time, r(t), θ(t), and φ(t) are the respective radial,
polar, and azimuthal coordinates of the gas parcel under consideration, and M is the mass of theblack hole. Here `, k, and  are constants of integration, the first two being projections of thespecific angular momentum, while the last is the specific energy.
Setting the impact parameter to β ≡ rt /rp = 1, the point at which equations (3.1) – (3.3)
become valid occurs when the star reaches pericenter. We will let the orbit of the stellar progenitor
be confined to the xy-plane, with the periapsis on the positive-x axis and the location of the blackhole at the origin. The center of mass of the star will also trace out a parabolic orbit. The impulse
approximation then means that the star retains its unperturbed (assumed-spherical) structure until
it reaches pericenter, so that the initial conditions we will use for equations (3.1) – (3.3) will bethose depicted by Figure 3.1. Note that the entire star initially shares the velocity of the center ofmass, which is along the positive-y axis at pericenter.
With the setup given by Figure 3.1 in mind, we will define the initial position of a given fluid
element that comprises the star by the coordinates (ri , θi , φi ). Since the entire star moves withp
the center of mass, the velocity of every fluid element is given by żi = ẋi = 0, ẏi = 2GMh /rt .

30

r
v

rt

ϕ

R*

y

x

Figure 3.1: The initial configuration of the star under the impulse approximation when β ' 1 (thisfigure is not drawn to scale). The dashed curve traces out the orbit of the center of mass, whichis assumed to be parabolic. The Cartesian coordinates are indicated by the diagram immediately
below the black hole (which is indicated by the black circle), z being out of the plane in a righthanded sense. The spherical-polar coordinates are labeled r and φ on the diagram, and θ is measuredout of the plane of the orbit from the z-axis (for the above figure that focuses on the x-y plane,θ = π/2).

31
Transforming these conditions into spherical coordinates via the transformations z = r cos θ, y =r sin θ sin φ, x = r sin θ cos φ, we find

r
ṙi =

1
θ̇i =
ri

2GMh
sin θi sin φi ,
rt

r

1
φ̇i =
ri

2GMh
cos θi sin φi ,rtr

2GMh cos φi,
rt sin θi

(3.4)

(3.5)

(3.6)

and using these expressions in equations (3.1) – (3.3) givesr` = rir
k = ri

2GMh
rt

2GMh
sin θi cos φi ,
rt

(3.7)

q
cos2 φi + cos2 θi sin2 φi ,

(3.8)

GMh
=
rt



rt
1−
.
ri

(3.9)

Equation (3.9) shows that gas parcels with initial positions inside the tidal radius are bound ( < 0),while those outside are unbound ( > 0), which is what we expect.
In addition to its position, we will also be interested in the density of the stream. As wasdemonstrated in Chapter 2, the density structure can be determined by considering the star at thetime of disruption and assuming that the specific energies of the gas parcels are frozen in thereafter.
Making the additional assumption that the stream is a circular cylinder of cross-sectional radius H,
then we can show that the azimuthally-averaged density along the stream varies as (see Chapter 2)R ξ1nµξ1 Θ(ξ) ξdξp,ρ=R
2πH 2 (r0 )2 + r2 (φ0 )2 0ξ1 Θ(ξ)n ξ 2 dξM ∗ ξ1(3.10)

32
where M∗ is the mass of the disrupted star, n = 1/(γ − 1) is the polytropic index of the gas, Θ(ξ) is
the solution to the Lane-Emden equation and ξ1 is the first root of Θ(ξ) (Hansen et al., 2004). Hereµ is the dimensionless position of a gas parcel from the center of the star at the time of disruption,i.e., µ = Rp /R∗ , where Rp is the radial position of the gas parcel. Primes on the functions r and φdenote differentiation with respect to µ. We will return to the question of what determines H in
Section 3.2.

3.2.2

SolutionsWith equations (3.7) – (3.9) and the initial positions of the gas parcels, we can numericallyintegrate equations (3.1) – (3.3) to determine the temporal evolution of the debris stream.
Figure 3.2 shows the solution to equations (3.1) – (3.3) with the relevant initial conditions
for a TDE between a solar-type star and a 106 M hole. The first time (closest set of blue points)
is 1.84 days after disruption, while the longest, yellow set of points is 40.6 days after disruption,and coincides roughly with the time at which the most bound material has returned to pericenter.
Intermediate streams are shown at intervals of 5.53 days. We find overall good qualitative andquantitative agreement between the radial positions of these solutions and the solution to the fullproblem – making no assumption about the negligible nature of pressure and self-gravity – obtainedusing numerical simulations (see, e.g., the red curves in Figure 1 of Chapter 2).
However, we find disagreement between the width of the stream obtained from equations(3.1) – (3.3) and that from the simulations, the former being significantly wider than the latter.
This discrepancy is due to the fact that self-gravity plays a crucial role in determining the widthof the stream (Chapter 2). In other words, the H that appears in equation (3.10) is not simply
determined by the free expansion of the parcels in the gravitational potential of the hole (seeequation (3.17) below, which shows how H depends on the density of the stream in the limit thathydrostatic balance is upheld in the transverse direction).
The approximate point at which the self-gravity of the stream becomes important can, however, be gleaned from the solutions to equations (3.1) – (3.3). Figure 3.3 shows the evolution of

33

Figure 3.2: The streams of debris formed from the tidal disruption of a solar-type star by a
Mh = 106 M hole situated at the origin. Each color represents a different time, the earliest
3/2 √(blue points closest to the origin) being at t = 100 rt / GMh ' 1.84 days from disruption,
3/2 √the latest (yellow points) at t = 2200 rt / GMh ' 40.6 days from disruption. The time in
3/2 √between neighboring streams is 300 rt / GMh ' 5.53 days. The black hole (not drawn to scale)is indicated by the black circle near the origin. While the radial positions of the gas parcels match
well those from numerical analyses (see Figure 1 of Chapter 2), the width obtained from equations(3.1) – (3.3) is significantly overestimated (the numerical solutions, had we shown them, would
have amounted to lines plotted overtop of the streams in Figure 2). This finding suggests that
self-gravity is important for keeping the stream confined in the transverse direction.

34

Figure 3.3: Four snapshots of the in-plane evolution of the gas parcels comprising the edge of thestar at the time of disruption; for these figures we chose a 106 M hole and a solar-like star. The
particles comprising the front of the star at the time of disruption have been colored blue, while theback has been colored orange. The arrows indicate the direction of motion of the center of mass.
The bottom, left-hand panel shows that, at a time of roughly an hour after disruption, the front
and back of the stream merge and thereafter trade places. The impulse approximation thus leads
to a caustic – where the debris streams form a two-dimensional surface – which occurs roughly anhour after disruption.

35
the in-plane edge of the stream at four different times for the disruption of a solar-type star by a
106 M hole. The front of the stream (the fluid parcels comprising the leading edge of the polytropeat the time of disruption) has been colored blue, the back has been colored orange, and the arrow
indicates the instantaneous direction of motion of the center of mass. This figure demonstrates
that, roughly an hour after disruption, the leading and trailing edges of the stream form a caustic– a point where the two-dimensional, in-plane surface of the stream collapses to a one-dimensionalline – and thereafter trade places, the front becoming the back and the back becoming the front.
The tidal stream thus exhibits a “perpendicular pancake” shortly after disruption, the perpendicular aspect referring to the fact that the orientation of the pancake is orthogonal to theorbital plane of the debris. This pancake is analogous to but distinct from the one found by Carter& Luminet (1982), who noted that the top and bottom of the star flatten to a point of infinite
density at the tidal radius for high-β encounters. Here, however, the compressive motions occur inthe orbital plane.
The existence of the pancake encountered here can ultimately be attributed to the initialconditions: from Figure 3.1, it is apparent that the parcels along the line passing through thecenter of the star and perpendicular to the orbital plane all have their periapses at φ = 0. Thoseconstituting the leading edge of the star, however, have already passed through their periapses,
while the periapses of the parcels comprising the back of the star have not yet been reached. Fromthe conservation of angular momentum (3.1), the front of the star is therefore decelerating at thetime of disruption while the back is accelerating, which causes the two to cross at a certain location.
Specifically, if we differentiate equation (3.1) with respect to time, set θ = π/2 and use equation(3.4), we findφ̈i = −

4GMh
sin φi cos φi ,ri2 rt(3.11)

which shows that gas parcels with φi > 0 are decelerating in the φ direction, while those withφi < 0 are accelerating. Investigating this equation further, we see that the differential acceleration

36
across the star at the time of disruption is∆φ̈ ' −

4GMh∆φ,ri2 rt(3.12)

where ∆φ is the angle subtended by the star. Geometrically ∆φ ' 2R∗ /rt , which yields, after
setting ri ' rt ,32πGρ∗|∆φ̈| '
3



Mh
M∗

−1/3
,

(3.13)

where ρ∗ = 3M∗ /(4πR∗ 3 ) is the average stellar density. This expression shows that the changein acceleration from the front of the star to the back depends primarily on the properties of theprogenitor, though the inverse dependence on black hole mass shows that the effect should be
amplified for smaller-mass SMBHs.
During a realistic β ' 1 tidal encounter, the star will not retain perfect spherical symmetry
until reaching its pericenter. In particular, the outer, low-density material comprising the envelope
will be more easily stripped, resulting in an elongated, ellipsoidal configuration. However, thehigher-density core will be able to better maintain its structure. Therefore, while considering theentire star as spherical and moving with the center of mass at the time of disruption is likely too
simplistic for the physical problem, those initial conditions are perhaps reasonable for the central
regions.
Furthermore, the non-zero pressure of the gas will prevent the development of a true caustic.
On the contrary, the convergence of the Keplerian orbits will increase the pressure and density
until it reaches an approximate equilibrium. However, the stretching of the stream in the radial
direction will cause the density to decrease, which will likewise result in a more drastic lowering of
the pressure if the gas follows an adiabatic equation of state. The ability of the pressure to resist
the caustic will thus decrease with time, making it possible for the perpendicular pancake to alter
the nature of the debris stream.
The precise time at which the caustic occurs as it has been presented here depends only on

37
the gravitational field of the black hole. In reality, the self-gravity of the steam would serve toalter the precise nature of the pancake. However, we expect that self-gravity would only serve toenhance the focusing of the orbits and potentially generate the caustic at a slightly earlier time.
In the next section we present simulations that address the complexity of the full problem. As
we will see, the numerical solutions do exhibit interesting behavior near the time at which equations(3.1) – (3.3) predict the existence of a caustic, and this behavior is imprinted on the stream formuch later times.

3.3

Numerical simulations
To test whether or not the caustic discussed in the previous section affects realistic β ' 1tidal encounters, we now employ numerical simulations that allow the star to evolve in the tidal
field of the hole pre-periapsis and include the effects of pressure and self-gravity at all times.

3.3.1

Simulation setup and initial conditions

We use the SPH code phantom (Price & Federrath, 2010; Lodato & Price, 2010) to simulate
the tidal disruption of a solar-type star (one with a solar mass and a solar radius) by a 106 M black
hole. phantom is a highly efficient code and is especially useful for astrophysical problems involving
complex geometries and a large range of spatial and temporal scales. For other applications of thiscode, see, e.g., Nixon et al. (2012a,b); Martin et al. (2014b,a); Nealon et al. (2015).
In our simulations the star is initially assumed to be a polytrope with polytropic index γ(Hansen et al., 2004). The correct, polytropic density profile is obtained by first placing 106 particles
in a close-packed sphere, then stretching that sphere to obtain a good approximation to the exact
solution.
We place the polytrope at a distance of 10 rt from the hole, with the center of mass on aparabolic orbit. The distance at periapsis is equal to the tidal radius (β = 1). Every gas parcel
composing the star initially moves with the center of mass when the star is at 10 rt , and the lengthof time taken to traverse the distance to the hole is sufficient to allow the polytrope to relax. The

38
adiabatic index of the gas is always equal to the initial, polytropic index of the star.
Self-gravity is included at all stages of the TDE, and is employed via a k-D tree (Gafton& Rosswog, 2011) alongside an opening angle criterion, the latter employing a direct summation
method for the gravitational forces between neighboring particles (Price & Monaghan, 2007). The
simulations presented here used an opening angle of 0.5 (we have run simulations with smaller
opening angles and found negligible differences; see Chapter 2). Shock heating was not included
for the runs presented here, though we have done tests in which it was included and found only
negligible differences. We also do not account for non-adiabatic cooling; the gas therefore retains
its polytropic equation of state throughout the TDE.
We ran four different simulations, each identical to the next except in the adiabatic index
used for the gas. Specifically, we chose γ = 1.5, 5/3, 1.8, and 2, and thus our parameter spaceagrees with that chosen by Lodato et al. (2009) except for γ = 2. While adiabatic indices greaterthan 5/3 are difficult to realize physically in stellar progenitors (though they may be appropriate
for planets; Faber et al. 2005; Li et al. 2002), we included these cases to highlight the presence ofthe caustic and to compare to Lodato et al. (2009).

3.3.2

Results

Figure 3.4 shows the star at the time of disruption, with each panel corresponding to adifferent adiabatic index. As was commented upon in Section 2, the fact that the tidal force does
not act impulsively means that the polytrope is already distorted when it reaches its periapsis, andthis distortion is apparent from the figure. We also see that the central density is higher for lower
γ, which is a general feature of polytropes.
Figure 3.5 shows the disrupted stream 2.53 days after disruption for the four different adiabatic indices. In this case it is evident that a larger adiabatic index corresponds to a thinner,
denser stream. This result may seem counterintuitive, as one might expect the higher-density coreof the lower-γ polytropes to result in a denser stream. However, if one assumes that pressure andself-gravity are the two dominant terms controlling the width of the stream, which is a reasonable= 1.5
11
0.01

0.02

log column density [g/cm2]

0.02

= 5/3
11
0.01

10

log column density [g/cm2]

39

10y / rt0y / rt

0

9

9

-0.01

-0.01

8

8
-0.02

1
x / rt

1.02

0.98

0.02
= 1.8
11
0.01

1
x / rt1.02

0.02

log column density [g/cm2]

0.98

=2
11
0.01

10

log column density [g/cm2]

-0.02

10

y / rt0y / rt

0

9
-0.01

9
-0.01

8
-0.02

8
-0.02

0.98

1
x / rt

1.02

0.98

1
x / rt1.02

Figure 3.4: The star at the time of disruption for an adiabatic index of γ = 1.5 (top, left), γ = 5/3(top, right), γ = 1.8 (bottom, left) and γ = 2 (bottom, right). The configuration has clearly beenaltered from its original, spherical shape, showing that the tidal force does not act exactly as animpulse as was assumed in Section 2. The central density is also higher for smaller γ, which ispredicted from the original stellar profile.

8

8
15

y / rty / rt15= 5/3

log column density [g/cm2]

t = 2.53 days

γ = 1.5

log column density [g/cm2]

40

6

6

10

10

4

4

-35

-45

= 1.8

8

-35

=2

8
15

y / rty / rt15-40
x / rtlog column density [g/cm2]-40
x / rtlog column density [g/cm2]

-45

6
10

6
10

4
-45

-40
x / rt

-35

4
-45

-40
x / rt-35

Figure 3.5: The stream at a time of 2.53 days from pericenter for an adiabatic index of γ = 1.5(top, left), γ = 5/3 (top, right), γ = 1.8 (bottom, left) and γ = 2 (bottom, right). The stream
thickness decreases dramatically and the fans become less pronounced as γ increases.

41
assumption because of the nature of the perpendicular pancake, then the transverse structure of
the stream is governed by the equation of hydrostatic equilibrium:

∂φsg
1 ∂p
=−
,
ρ ∂s
∂s

(3.14)

where φsg is the gravitational potential due to the self-gravity of the debris and s is the transverse
distance from the center of the stream. Furthermore, if the variation in the self-gravitational
potential along the radial direction of the stream is small, which is a good approximation toward
the center of the stream owing to its approximately symmetric nature and only breaks down when
we approach its radial extremities, then the Poisson equation reads

1 ∂
s ∂s



∂φsg
s
= 4πGρ.
∂s(3.15)

Using this equation in conjunction with equation (3.14), we find that the equation of hydrostatic
equilibrium becomes

1 ∂
s ∂s



s ∂p
ρ ∂s


= −4πGρ.

(3.16)

With the polytropic equation of state p ∝ ργ , dimensional analysis of this equation shows that thecross-sectional radius of the stream varies asγ−2H∝ρ 2 ,

(3.17)

where here ρ is the density at the center of the stream. The precise constant of proportionality
depends on the entropy of the gas and the numerical solution to equation (3.16).
It is ultimately the scaling given by equation (3.17) that tends to outweigh the presence of ahigher-density core for smaller γ. Also, if we use this expression for H in equation (3.10), then thedensity along the stream varies asρ = ρmR ξ1nµξ1 Θ(ξ) ξdξpR(u0 )2 + u2 (φ0 )2 0ξ1 Θ(ξ)n ξ 2 dξ

1

!n
,

(3.18)

42
where ρm is a normalization constant, chosen such that the density equals the correct, central stellar
density at the time of disruption, and u ≡ r/rt .
The density along the stream already exhibits a number of interesting features well before 2.53
days. To exemplify this point, Figure 3.6 shows the average radial density (i.e., the average densityof all particles at a given radius r) along the stream for the γ = 5/3 run at times of t = 6.14, 9.57,
13.0, and 19.8 hours after disruption. Initially the density distribution along the curve is smooth,and matches well the distribution obtained if the original polytrope is stretched in one dimension(equation 3.18). However, at later times the density adopts a more intricate structure, exhibitinga sharper peak at the center of the stream and “shoulders,” evident from the bottom-right panelof Figure 3.6, that are not predicted analytically.
Figure 3.7 shows the average density along the stream for the four different adiabatic indicesat 2.53 days after disruption (the black curves are the numerical solutions, while the red, dashedcurves give the analytic estimate that results from equation 3.18). This Figure demonstrates thatthe small-scale density fluctuations that develop along the stream at later times are intensified
for larger γ. It is evident that lower adiabatic indices show relatively smooth variations in thedensity, and retain an approximately symmetric structure about the center of the stream. For larger
polytropic indices, however, the scale at which perturbations develop along the stream decreases
and the perturbations themselves become more erratic in amplitude and position. It is also clearthat a smaller adiabatic index results in more material at smaller and larger radii than would be
predicted analytically, and these “fans” are also apparent from Figure 3.5. This results from thefact that polytropes with smaller γ have lower-density envelopes, those envelopes being more easily
stripped at early times.
To determine when the density of the debris stream starts to exhibit the anomalous, smallscale structure that is apparent in Figures 3.6 and 3.7, Figure 3.8 shows the maximum density along
the stream as a function of time; the black, solid curve indicates the numerical solution, while thered, dashed curve gives the analytic prediction (equation 3.18). Aside from slightly over-predicting
its magnitude, the analytic solution matches the numerical one well, which shows that the stream

43

10-3
t = 6.14 h

t = 9.57 h

10-3[g / cm3][g / cm3]

10-4

10-4

10-5

8.5

8.6

8.7

8.8

11.6

11.8

r / rt1212.2

r / rtt = 13.0 h

t = 19.8 h

10-4[g / cm3][g / cm3]

10-4

10-5
10-5

14.5

15
r / rt

19

19.5

20

20.5

r / rtFigure 3.6: Four snapshots of the average stream density (the average density of all particles at
a given radius r) as a function of r for the γ = 5/3 run. Initially the density remains smooththroughout the stream; however, by about a day after the disruption, the density structure hasdeveloped a more complicated nature, consisting of a central peak that is narrower than is predicted
analytically and two shoulders.

44

10-3
10-4

= 1.5

= 5/3

t = 2.53 days

[g /cm3][g /cm3]

10-4
10-5

10-5
10-6

38

40

42

44

46

40

42

r / rt
10-2

46

44

46

10-1
= 1.8

=2

10-2

[g /cm3]

10-3[g /cm3]44
r / rt

10-4

10-3

10-4

10-5

10-5

10-6
40

42

44
r / rt46

40

42
r / rtFigure 3.7: The average density (same as Figure 6) as a function of r for γ = 1.5 (top, left panel),γ = 5/3 (top, right panel), γ = 1.8 (bottom, left panel), and γ = 2 (bottom, right panel) at a timeof 2.53 days after disruption (see Figure 3.5 for the shape of the streams at this time). The black,solid curves give the numerical solutions, while the red, dashed curves show the analytic predictions.
It is apparent that larger adiabatic indices correspond to an enhanced amount of variability in thedensity along the stream, while a smaller adiabatic index results in more extended wings (this isalso apparent from the tidal fans in the edges of the streams in Figure 3.5.)

45

10
Numerical
Analytical3max [g / cm ]

1

10-1

10-2

10-1

1

10
t [hr]

Figure 3.8: The maximum density along the stream as a function of time with γ = 5/3 (n = 1.5);
the numerical solution is given by the black, solid curve, and the analytical solution (equation 3.18)
is given by the red, dashed curve. A time of zero here corresponds to the time at which the starreaches the tidal radius. The time at which the numerically-obtained density starts to decrease isslightly earlier than the analytic one, suggesting that the time at which the star is “disrupted” isactually pre-periapsis. The first bump in the numerical solution, which occurs after a couple ofhours, indicates where the pancake starts to augment the maximum density. At late times, both
solutions follow the approximate power-law decline ρ ∝ t−1.8 .

46

10

= 1.5
= 5/3
= 1.8

1

3
max [g / cm ]

=2

10-1

10-2

10-3

10-4

10-1

1
t [hr]

10

Figure 3.9: The maximum density as a function of time for γ = 1.5 (black, solid curve), γ = 5/3(red, dashed curve), γ = 1.8 (green, long-dashed curve), and γ = 2 (blue, dotted curve). It isapparent that the initial perturbation induced by the pancake is induced sooner for larger γ, andthe oscillation timescale of the perturbation is shorter for larger γ.

47approximately maintains hydrostatic balance for all times during the disruption in the transverse
direction. Note that this result contrasts the findings of Kochanek (1994), who assumed that thestream was in free expansion until three dynamical times post-disruption, which is roughly 1.5
hours for the disruption of a solar-type star by a 106 M

hole (however, the assumption of free

expansion may hold in the limit of β  1). This plot also demonstrates that the first perturbation
to the density appears at a couple hours after disruption, resulting in a “ripple” that over- andunder-estimates the average value. The perturbations induced on the stream therefore behave ascompression-rarefaction waves.
Figure 3.9 shows the maximum density along the stream for the four different adiabatic
indices. It is evident that the first bump in the density occurs slightly sooner for larger γ, appearing
at around an hour for γ = 2, and that the temporal frequency of the perturbations increases as γ
increases. The average maximum density also falls off as a power-law for late times, which agrees
with the analytic prediction (Figure 3.8), with the power-law index being shallower for larger γ. In
particular, if we set ρmax ∝ t−mγ , we find m1.5 = 2.4, m5/3 ' 1.8, m1.8 ' 1.5, and m2 ' 1.2.

3.4

Discussion
We saw in the previous subsection that the impulse approximation – assuming that the starretains its spherical, undisturbed structure until it reaches the tidal radius – does a reasonable jobof fitting the numerically-obtained density profile of the tidally-disrupted debris stream when β =rt /rp = 1 (Figure 3.7). This agreement demonstrates that the stream width is set by hydrostaticbalance, while the length is determined by the radial positions of the gas parcels orbiting in thepotential of the black hole. However, at times corresponding to a few hours after disruption, thedensity profile begins to exhibit anomalous, small-scale structure that is not predicted analytically,
with important ramifications for the late-time evolution of the stream (Figures 3.7 –3.9).
This behavior was also noted by Lodato et al. (2009), who commented on the existence ofthe shoulders present in the density profile (see their Figure 7; they were interested in the behaviorof dm/d ∝ ρH 2 , the distribution of mass in energy space, as this yields information about the

48
fallback rate). Since they renormalized their specific energy distribution to match the peak, they
did not notice the sharper structure exhibited by the density in the central portion of the stream.
They argued that these shoulders arose from shock compression within the stream.
However, we find it unlikely that shocks alone can account for these anomalous features. For
one, shocks occur primarily in the outermost regions of the envelope at the time of disruption. The
majority of the material involved in the shocks is therefore confined to the tidal tails of the debris
stream (the fans at the edges of the streams in Figure 3.5; see Figure 8 of Lodato et al. 2009),
comprising only a small fraction of the total amount of mass contained in the stream. However,the perturbations occur throughout the majority of the stream, affecting a much larger fraction of
the material. The time at which the fluctuations begin to appear is also hours after the disruption,
well after the shocks that occur at pericenter. Furthermore, we have run additional simulations
that include shock heating; in these cases, the density profiles we find are nearly identical to thosepresented here, indicating that the amount of material that shocks significantly is small.
On the contrary, we find that a more reasonable origin for the anomalous structure present in
the numerical solutions is the combination of self-gravity and the “perpendicular pancake” discussed
in Section 2.2 – where in-plane compression of the star causes the front and back edges of the star toconverge to a one-dimensional line, or caustic (see Figure 3.3). This interpretation is supported by
the temporal coincidence of the ripples present in Figure 3.7 and the analytic prediction of when thecaustic arises, both occurring on the order of hours after disruption. We also note that the majorityof the stream, not just the central maximum, seems to be undergoing an increase in density when
the first perturbation occurs. This can be seen from Figure 3.8, which shows that the first increasein the density for the γ = 5/3 run starts to appear around a few hours after disruption. However,
the top, left panel of Figure 3.6 shows that at a time of roughly six hours after disruption, long after
the first perturbation has started to augment the maximum in the density, the entire stream stillretains a smooth density distribution that is well-matched by the analytic prediction. Indeed, thesharper peak and shoulders do not seem to appear until around 10 hours after disruption, whichis the top, right-hand panel of Figure 3.6. This indicates that the first increase in the maximum

49
density is occurring over the entire stream, not just in the central region where the maximum
occurs (for which Figure 3.8 applies), and the density everywhere is being incremented by the same
factor.
In further support of the interpretation that the caustic occurs in the simulations and enhances the density perturbations, recall that the existence of the caustic is ultimately related tothe initial conditions at the time of disruption: because every gas parcel is moving with the centerof mass of the star, the parcels comprising the back edge of the star have not yet reached theirperiapses, while those comprising the front have already passed through theirs. This configuration
then causes the back to accelerate and the front to decelerate, resulting in their eventual merger.
In a realistic TDE, the star does not retain perfect spherical symmetry all the way until the tidal
radius (Figure 3.4). In particular, the less dense, outer regions of the envelope will be stripped
earlier, causing them to violate the condition that they move with the center of mass. The denser,
central regions, however, may better retain their unperturbed structure, resulting in a pancake that
occurs mainly in the center of the stream.
To test this hypothesis, we ran a simulation in which we took the output of the γ = 5/3
phantom run when the star reached pericenter and evolved the particles solely in the gravitational
field of the hole, neglecting pressure and self-gravity. Figure 3.10 shows the distribution of particles
at a time of 37 minutes (left panel), 50 minutes (middle panel), and 62 minutes (right panel) postdisruption, the red particles being those that composed the back of the star at pericenter, the black
particles the front. This Figure shows that, at roughly an hour after disruption, the front and backedges of the stream switch places, with the point of maximum compression occurring in the middle
panel. Specifically, the half-width of the center of the stream at 50 minutes is roughly H ' 0.1rt ,which is only five times the value when the star is at pericenter. This Figure confirms that thecaustic still exists with realistic initial conditions. However, as predicted, the fans at the edge ofthe stream do not undergo a similar amount of compression and retain their original colors, whichis due to the fact that they were not moving with the center of mass at the time of disruption (i.e.,they were already stripped from the star; this is also supported by the fact that the fans extend

50

Figure 3.10: The particle distributions from an N -body simulation, where the initial conditions
were taken from the γ = 5/3 run at periapsis, at 37 minutes (left panel), 50 minutes (middle panel),and 62 minutes (right panel) post-disruption. The red particles comprised the back of the star at
the time of disruption, while the black constituted the front of the star. This Figure demonstrates
that a caustic – where the front and back of the stream merge to form an infinitely thin line –still occurs in the presence of realistic initial conditions. However, as mentioned in the text, theoutermost layers of the star that are stripped earlier (and therefore violate the frozen-in condition)
do not undergo this compression; this is evidenced from the fact that the “fans” present in thisfigure undergo no distortion in terms of their color. The central panel corresponds to the point at
which the width of the stream has reached a minimum, the half-width being H ' 0.1 rt .

51
farther in radially than the analytic solutions predict, which is apparent in Figure 3.7).
The preceding arguments illustrate that it is likely the caustic discussed in Section 2.2 thataugments the importance of self-gravity and generates the density fluctuations in the stream. Interestingly, Figures 3.8 and 3.9 show that this perpendicular pancake does not simply increase thedensity, but instead generates a compression-rarefaction wave. This is due to the fact that theincrease in the density likewise generates an increase in the pressure, which resists the compression.
Eventually, the continued squeezing of the stream results in the material being overpressured in
the transverse direction, which causes the stream to “bounce.”
The sharper peak that develops in the center of the stream arises from the self-gravity ofthe debris. In particular, the compression in the transverse direction augments the central density
to the point where material can be drawn in gravitationally in the radial direction, which createsthe more massive central peak and the two dips on either side of that peak in Figure 3.6. The
two shoulders that develop are regions of the stream that have not been gravitationally drainedof material by the central peak and are slightly denser than one would predict analytically due
to the pancake. More structure develops at late times, and local maxima are imprinted due tothe oscillation of the stream, ultimately due to the self-gravitating nature of the debris (see also
Figure 3 of Kochanek (1994), who found oscillations in the stream width and height due to pressure
and self-gravity). The points at which the density sharply drops off are the fans present in Figure
3.5, and have thus not been affected by the caustic (note from Figure 3.7 that the rate at which
the density falls off with radius in these regions parallels the analytic one, which confirms thisinterpretation).
From Figure 3.7, it is apparent that larger adiabatic indices result in more drastic fluctuations
that are induced by the caustic. The reason for this scaling is likely two-fold, the first being that,for the same physical radius R∗ , polytropes with larger adiabatic indices have flatter density profiles(note that this is not true in the dimensionless space spanned by ξ). Therefore, since the density
throughout the envelope differs from that of the core only when we are near the surface of thestar, polytropes with higher adiabatic indices can better retain their structure until they reach52
periapsis. This then results in more of the stream experiencing the effects of the caustic, whichcorrespondingly results in a more drastic increase in the density along the majority of the stream.
This is supported by Figure 3.7, which shows that the shoulders extend farther from the center ofthe stream as γ increases.
The second reason is that the stream is thinner for larger γ, which is evident from Figure 3.5.
Since the equilibrium width of the stream increases as γ decreases, the pancake is less effective incompressing the stream and correspondingly increasing the density to the point where self-gravity
can amplify the perturbations. Additionally, this scaling with H causes the average density of thestream to decrease less rapidly with time for larger γ (Figure 3.9). The overdensities within thestream are therefore more dense in an absolute sense, which increases the ability of the self-gravityof the debris to counteract the tidal shear imposed by the black hole.

3.4.1

Is the pancake necessary?

Figures 3.6 – 3.9 show that self-gravity can drastically modify the density profile of the disrupted debris stream from a TDE, causing a sharper peak near the center, small-scale fluctuations,and “shoulders,” all of which are not predicted analytically. These effects are long-lived, altering
the structure of the debris stream for days to months post-disruption (see also section 5). In addition, we saw in Section 2 that a caustic – where the front and the back of the stream intersect toform a two-dimensional plane – occurs not long after the disruption of the star under the impulse
approximation. Figure 3.10 shows that, even in a realistic TDE where the frozen-in assumption
does not apply, the orbits of the gas parcels near the center of the star converge to form thispost-periapsis pancake. Therefore, the self-gravity of the stream is augmented by the dynamical
focusing of the gas parcels in the transverse direction.
Because the numerical method treats the full complexity of the problem, including pressure,self-gravity, and the influence of the SMBH, the simulations presented here have not isolated theeffects of self-gravity and the pancaking of the orbits. Is it possible that the latter is actuallyunimportant, with the majority of the variation in the density of the stream due solely to the53
self-gravity of the debris?
To answer this question, recall that the pancake arises from the fact that, under the impulse
approximation, the front of the star is decelerating at the time of disruption while the back is
accelerating. Equivalently, the requirement that the entire star move with the center of massmeans that the gas parcels comprising the front of the star have already passed through theirpericenters, while those comprising the back have not yet passed through theirs. Therefore, toavoid the caustic but still maintain a realistic distribution of specific energies (half bound, half
unbound), one can simply impose that the initial velocities of the gas parcels satisfy ṙi = 0, θ̇i = 0,
and ri2 sin2 θi φ̇2i = 2GMh /rt . Thus, if the star had these (albeit contrived) initial conditions, thepost-disruption evolution would be unaffected by the caustic.
To examine the isolated effects of self-gravity, we used the output of the phantom runs when
the star was at pericenter (Figure 3.4) and modified the instantaneous velocities to reflect the initialconditions that avoid the caustic, i.e., we set ṙi = 0, θ̇i = 0, and ri2 sin2 θi φ̇2i = 2GMh /rt for allof the particles. What we generally found was that the anomalous features of the density profile
were still present, i.e., shoulders still formed and a more concentrated peak developed. However,
the magnitude of each of these features was significantly reduced; in particular, the shoulders
were much less pronounced, the central density peak was less sharp, and the density fluctuations
were less concentrated. The overall magnitude of the density was also down by a factor of a few,
and the increase in the density that occurred over the entire stream (see discussion above) was
not observed in the modified runs (see Figure 3.11, which illustrates these points). Finally, themorphology of the streams also differed, having larger widths and more extended fans in the cases
where the pancake did not occur.
These tests show that, in general, the anomalous features arise from the self gravity ofthe debris modifying the radial density distribution throughout the stream. However, as was
suggested in the previous subsection, the post-periapsis pancake is quite important for magnifying
and sustaining the self-gravitating nature of the stream.

54

10-3

[g / cm3]

10-4

10-5

40

45
r [rt]

Figure 3.11: The average density as a function of r for the unaltered, γ = 5/3 run (solid, black
curve) and the run that avoids the caustic (dashed, red curve), both at a time of 2.53 days (thesame time as in Figure 3.7). This Figure shows that the pancake amplifies the anomalous density
structures along the stream, effectively enhancing the ability of self-gravity.

55

3.5

Implications
We have demonstrated above that a caustic, or a “perpendicular pancake,” augments theimportance of self-gravity in the debris stream from a TDE. In particular, we found that thispancake and self-gravity cause density perturbations that are not predicted analytically (Figure3.7). In this section we briefly discuss several implications of our findings.

3.5.1

Fragmentation

One of the most profound implications is that these perturbations can result in the gravitational fragmentation of the stream. For γ = 2, the overdensities present in the stream at a time of
2.53 days are already self-gravitating and starting to collapse into small-scale, gravitationally-boundclumps (see Figure 3.12). For γ = 1.8, the stream also fragments, but not significantly until a timeof a couple weeks after disruption. The γ = 5/3 run also collapses at late times, but the time atwhich fragmentation occurs depends on the resolution of the simulation. As was commented uponin Chapter 2, this suggests that the stream itself is gravitationally unstable, but the perturbations
induced by the pancake and self-gravity are not sufficient to drive the fragmentation. This finding
also suggests that the limiting adiabatic index at which fragmentation occurs is closer to γ = 5/3
than γ = 2, as indicated by previous studies of compact object mergers (Lee & Ramirez-Ruiz 2007,in particular their Figure 23; see also our discussion below regarding the origin of this marginal
stability). We have run the γ = 1.5 simulation presented here out to nearly 10 years and have notfound recollapse, suggesting that the density profile of the stream is gravitationally stable.
In the γ = 1.8 run, the first clump forms near the center of the stream around a time of fivedays after disruption, with smaller-mass clumps forming at later times at distances progressively
farther from the central portion of the stream. By about two months after disruption, the clump
formation becomes less vigorous, and the clump masses saturate at approximately constant values
with an average clump mass of M̄c ' 0.55MJ , where MJ ' 9.54 × 10−4 M is the mass of Jupiter.
The maximum clump mass, however, is Mc,max ' 1.5MJ , showing that the clumps span a large

12

25

t = 5.69 days

10
20

log column density [g/cm2]56

y / rt

8

15
6

10
4
-75

-70

-65

-60

12
17.5

10

17

log column density [g/cm2]x / rty / rt

8

16.5
6

16

4

-70.5

-70

-69.5
x / rt-69-68.5

Figure 3.12: The stream from the γ = 2 run (top panel) and a closeup view of the stream (bottom
panel), showing the clumps that have formed throughout the majority of the stream, both at atime of 5.69 days after disruption.

57
range in mass.
On the other hand, the first clumps form at a time of around three days after disruption forthe γ = 2 run, and instead of forming one clump in the center of the stream, between five and ten
form around the same time at approximately evenly-spaced intervals along the stream (this agrees
with the findings of Lee & Ramirez-Ruiz 2007 and other studies of the tidal tails produced during
compact object mergers where very stiff equations of state were used). Fragmentation ceases withan average clump mass of M̄c ' 2.6MJ around two weeks after disruption, and the maximum clump
mass in this case is Mc,max ' 37MJ .
Since the γ = 5/3 run collapsed at late times but due to the small-scale numerical noise
inherent in the simulation, additional, resolved perturbations are required to study true fragmentation in this case. This marginal instability of the stream is likely due to the fact that the maximum
density in the stream drops off as ρ ∝ t−1.8 (see Figure 3.8), whereas the “density” of the blackhole scales as ρ ∝ 1/r3 ∝ 1/t2 , the last proportionality resulting from the fact that the orbits ofthe gas parcels initially follow r ∝ t2/3 . The decline in the density for the γ = 5/3 case is thus
barely above that of the black hole, meaning that the stream self-gravity only outweighs the tidal
shear by a small margin. Additionally, since ρ ∝ t−2.4 for γ = 1.5, we do not expect fragmentation
to occur in this case, and this is consistent with what is observed from the simulation.

3.5.2

Fallback rate features

When one of the of these clumps returns to pericenter, the fallback rate can spike above theaverage, t−5/3 decline by a significant fraction, as is apparent from Figure 3.13 for times greaterthan a few years, and from Figure 3.14 for times greater than a few months (the small amount of
scatter present for the γ = 1.5 fallback curve is due to numerical noise). If the tidally-disrupted
debris has already formed an accretion disk, the interception of one of these clumps by the disk cansignificantly augment the accretion rate onto the hole (though this is only true during the earlystages of the fallback, when the viscous time is short compared to the infall time; Cannizzo et al.
1990).

58

= 1.5
1

dM/dt [Msun / yr]

= 5/3

10-1

10-2

10-1

1

10

t [yr]

Figure 3.13: The fallback rate computed for the γ = 1.5 (black curve) and γ = 5/3 (red curve)
runs. The purple curve is the canonical t−5/3 fallback rate for reference. It is apparent that thereturn time of the most bound material is earlier for smaller γ, which is related to the amount ofdistortion imparted to the star at the time of disruption. At late times, the accretion of clumps
that have formed in the γ = 5/3 stream causes the fallback rate to deviate significantly from themean (the small amount of deviation present in the γ = 1.5 run is numerical noise).

59

= 1.8
Analytic

dM/dt [Msun / yr]

1

10-1

10-2

10-1

0.2

0.5
t [yr]

Figure 3.14: The fallback rate for the γ = 1.8 run (green, solid curve) and the analytic prediction(blue, dashed curve). We see that the numerical solution is larger than the analytic one by an orderof magnitude, and the fallback of bound clumps causes significant deviation from the average, t−5/3
fallback at times greater than about 6 months from disruption.

60
Figure 3.13 shows that, shortly after reaching their peaks, the fallback rates for both theγ = 1.5 and γ = 5/3 runs fall below the canonically-assumed t−5/3 power-law. There is then aperiod during which the rate is slightly shallower than the 5/3 rate; for the 5/3 run, this latter
period lasts from a few months until about a year, after which the rate resumes the t−5/3 decay.
For the γ = 1.5 run, however, the power-law is less steep than the 5/3 law after nearly ten years
from the disruption. This variable fallback rate is due to the accretion of various parts of thestream: the rate drops below 5/3 when the dip between the first shoulder and the central peak of
the stream (Figure 3.7) is accreted. The rate then becomes shallower than the 5/3 law when thedenser, central regions are accreted. This variation in the fallback rate means that observed TDEs
may not follow the t−5/3 law for much later times than previously suspected.
Figure 3.14 demonstrates that the peak fallback rate is significantly higher than the analytic
prediction (this is also true for the γ = 1.5 and γ = 5/3 runs), where the latter was calculated by√using the energy-period relation, which gives µ(t) = (t/T )−2/3 , T = 2πMh /(M∗ GMh )(R∗ /2)3/2
being the period of the most tightly bound debris, and the frozen-in condition (see Chapter 5 formore details). This increase in the fallback rate arises from the fact that the pancake has increased
the density above what would be predicted analytically, as is apparent from Figure 3.7. This meansthat the accretion rate onto the black hole is much higher than thought previously, making it more
likely that the TDE will result in a phase of super-Eddington accretion. Indeed, if we assume an
efficiency of  = 0.1 and Ṁacc = Ṁf b where Ṁacc is the accretion rate onto the black hole, thenthe peak accretion rate for the γ = 1.8 run in Figure 3.14 corresponds to an accretion luminosityof Lacc ' 80LEdd , compared to the analytic estimate of Lacc ' 8LEdd . Since the degree to which
the fallback rate is super-Eddington is inversely proportional to the black hole mass, we see that
more TDEs could be accompanied by a jetted-outflow phase like that seen for Swift J1644+57(Zauderer et al., 2011).

61
3.5.3

Clump fates

If an accretion disk has not yet formed, the clumps that are bound to the black hole can returnto the original pericenter distance. Since their densities will likely be lower than that of the stellar
progenitor, the tidal disruption radii of the clumps will be outside the tidal radius of the original
star. The returning clumps will therefore be “redisrupted” before reaching their pericenters, leading
to complicated interactions between the streams of incoming and outgoing debris that could avoid
the “dark year for tidal disruption events” suggested by Guillochon et al. (2015). Also, dependingon the magnitudes of general-relativistic apsidal and Lense-Thirring precession, these redisruptions
may tend to isotropize the accretion process, leading to a more symmetric inflow. This symmetricinflow may then lead to super-Eddington accretion luminosities, puffing up the accretion disk andpotentially leading to the production of jets (Chapter 5).
The clumps that form in the unbound portion of the stream will make their way out of thesphere of influence of the central SMBH and into the galaxy. In particular, if we recall that thepescape velocity of the most unbound material is vesc ' 2GM∗ /R∗ (Mh /M∗ )1/6 , then we find thatthe unbound clumps leave the sphere of influence of the black hole on a timescale of−2
tesc ∼ 10 σ100



Mh
106 M

2/3 

M?
M

−1/6 

R?
R

1/2
yrs,

(3.19)

Although their long-term evolution is uncertain and depends on the specific properties of the gas(e.g., heating and cooling rates due to ionizations and recombinations), these unbound clumps
could condense into planetary mass objects and brown dwarfs, producing a new class of hypervelocity objects that eventually leaves the host galaxy. Since the clump formation is most vigorous
for adiabatic indices γ & 5/3, those adiabatic indices being somewhat unphysical for real stellar
progenitors, it may seem as though the production of unbound objects is largely inhibited forrealistic TDEs; however, if cooling can significantly decrease the entropy (see below), then thenumber of clumps could be significantly augmented. Therefore, if there are between 10−4 and 10−5
disruptions per galaxy per year, the number of hypervelocity, low-mass objects could significantly
outweigh the number of hypervelocity stars. We plan to perform a more in depth analysis of the62detailed properties of the clumps in a future paper.
We also recall that the marginally bound material recedes to very large distances before
returning to the black hole. Therefore, similar to the unbound material, the clumps that form inthis region of the stream may have time to collapse into much denser objects (e.g., planets). These
objects may then be able to survive their plummet back into the tidal region of the black hole(though interactions with the surrounding stellar population may alter their pericenter distances
to be larger than the original tidal radius), forming a class of low-mass objects that remain boundto the black hole. Since they would still be very weakly bound, their orbital periods would be
anywhere from tens to thousands of years. Furthermore, if the clumps in this region do not become
overly dense, they may form weakly-bound clouds that are consistent with those observed near the
Galactic Center (e.g., the cloud G2; Burkert et al. 2012; Gillessen et al. 2012; Guillochon et al.
2014a).

3.5.4

EntropyIn these simulations, the gas maintained approximately constant entropy throughout theentire disruption process. In reality, the gas energy equation will be modified by losses due toradiative cooling and cooling or heating (depending on the optical depth of the stream) due torecombinations, which could significantly alter the equation of state of the gas and affect thenature of the caustic. A more realistic equation of state might therefore be of the form

p = S(r, t)ργ ,(3.20)

where S(r, t) is related to the entropy of the gas that is, in general, a function of both space andtime. When S(r, t) is a constant, it is apparent from this expression that, for the same change
in density, a smaller adiabatic index results in a correspondingly smaller decrease in the pressure.
This scaling then results in a larger cross-sectional radius of the stream, which ultimately enables
the debris to better resist the pancake and fragmentation for smaller adiabatic indices. We see,however, that if the entropy decreases with time, then the pressure could decrease faster than

63
would be predicted by an isentropic equation of state. Therefore, if cooling is efficient enough tosignificantly reduce the entropy of the gas, the pancake could induce fragmentation for γ less than
5/3. This result is particularly apparent if we use equation (3.20) in equation (3.16), which showsthat the cross-sectional radius scales asγ−2

H ∝ S 1/2 ρ 2 ,(3.21)

and using this relation in equation (3.10) yieldsρ = ρad (r, t)

S
S0

−n
.

(3.22)

Here ρad is the density one obtains for an adiabatic equation of state, given by equation (3.18), S0
is the original entropy of the gas at the time of stellar disruption, and we recall that n = 1/(γ − 1).
We see that a decrease in the entropy has a more pronounced effect for smaller γ, meaning that
efficient cooling would more easily result in recollapse for softer equations of state. In particular,
since ρad ∝ t−2.4 for γ = 1.5 (n = 2), we would only need S ∝ t−0.2 to bring the power-law toρ ∝ t−2 , which would make the stream marginally unstable to gravitational collapse.

3.6

Summary and conclusions
We have shown that a caustic – a surface where the orbits of the gas parcels comprising

the stream of tidally-disrupted debris formally attain infinite density – results from the impulse
approximation applied to β = rt /rp ' 1 tidal encounters. This pancake is analogous to the one
discovered by Carter & Luminet (1982); however, in this case the pancake occurs post-periapsis(on the order of an hour after the star reaches pericenter for the disruption of a solar-type star by
a 106 Mhole), and the compression occurs in the plane of the orbit of the stream, which causes

the orientation of the pancake to be perpendicular to the plane of the orbit (see Figure 3.3).
In a realistic TDE, the pressure of the gas will prevent the existence of a true caustic. To
test the effects of pressure in resisting the pancake, we simulated four tidal encounters between a64solar type star (R∗ = R

and M∗ = M ) and a 106 M

hole with the pericenter of the center ofmass of the parabolic, stellar orbit at the tidal radius (β = 1). The simulations differed only in the
adiabatic index of the gas, being γ = 1.5, 5/3, 1.8 and 2, making our parameter space close to thatchosen by Lodato et al. (2009).
A few hours after disruption, the density of the streams of debris produced by the disruption
exhibit anomalous behavior, showing compression-rarefaction oscillations not accounted for by the
analytic model (see Figures 3.6 – 3.9). We interpret these features as arising from the combinationof the perpendicular pancake and self-gravity, not only because of the temporal coincidence of thetwo phenomena, but also because the majority of the stream seems to be undergoing a systematicincrease in the density at the start of the first compression. This can be seen by noting that thefirst increase in the maximum stream density starts at a time of roughly an hour after disruption,
yet the stream seems to retain its stretched-polytropic structure, predicted analytically, after six
hours post-disruption (compare Figures 3.8 and 3.6). This suggests that a large portion of thestream is being compressed simultaneously and by the same factor, which is predicted for the
pancake; furthermore, this systematic increase in the density was not observed in the test runsthat avoided the caustic (see section 4.1). By using the periapsis velocities and positions of thegas parcels generated from the phantom runs as the initial conditions for an N-body simulation,
we also showed that the orbits of the central portions of the stream do tend to form a caustic(Figure 3.10). This finding is consistent with the fact that the dense, central portions of the starlikely retain their structure better until reaching pericenter, thus creating the conditions necessary
to form a post-periapsis pancake. On the other hand, the outer, less-dense regions of the envelope
are stripped from the star sooner, violating the impulse criterion that they move with the centerof mass until reaching pericenter, and thus avoiding the caustic.
The self-gravity of the stream, supplemented by the caustic, induces fluctuations on top of
the otherwise-smooth, analytically-predicted density profile, as evidenced in Figure 3.7. The factthat the analytic predictions match the numerical solutions well means that the stream width ispredominantly set by the balance between pressure and self-gravity, and does not undergo any

65episode of free expansion immediately after pericenter passage as expected previously (Kochanek,
1994). The effects of the caustic are long-lived, and the density profile of the stream evolves fora considerable amount of time after the initial perturbations are imposed. The variations inducedby the caustic and self-gravity drive deviations from the canonically-assumed t−5/3 fallback rate,
the power-law being first steeper and then shallower than 5/3, which can be seen from Figure 3.13.
The peak in the accretion rate is also higher than would be predicted analytically (Figure 3.14).
Remarkably, the combination of the caustic and self-gravity can cause the stream to fragment
into small-scale, gravitationally-bound clumps if the adiabatic index is high enough. Specifically,for γ = 2 and γ = 1.8, we find that the stream collapses at a time of a few days and a couple ofweeks, respectively. After a relatively short time — about two weeks for γ = 2 and two months for
γ = 1.8 — the clump formation stops and the masses of the clumps saturate. For γ = 2 the average
clump mass is M̄c ' 2.6MJ , while that for the γ = 1.8 run is M̄c ' 0.55MJ , MJ ' 9.54 × 10−4 M
being the mass of Jupiter. In both of these cases, however, the maximum clump mass is an orderof magnitude above the average, showing that there is a large range of clump masses. For γ = 5/3,
the stream does collapse, but the instability is started by small scale noise and so future simulations
with realistic perturbations are required (see also Chapter 2). For γ = 1.5, we find no fragmentationout to a simulated time of ten years post-disruption, suggesting that the stream is gravitationallystable.
The formation of these clumps has a number of interesting repercussions. For one, if an
accretion disk has already formed from the tidally-stripped debris, it can intercept one of theinfalling clumps and, especially if the clump mass is on the larger side (& 1MJ ) of the distribution,
significantly augment the accretion rate onto the black hole if the viscous timescale in disk is short(see Figure 3.14). Such periodic increases would be seen as variability in the lightcurve of the TDE,
consistent with that observed for Swift J1644+57 (Burrows et al., 2011; Levan et al., 2011; Zauderer
et al., 2011), and also for the events Swift J2058+05 (Cenko et al., 2012) and Swift J1112.2-82(Brown et al., 2015). If an accretion disk has not yet formed, these clumps can be “redisrupted,”creating complicated interactions between the incoming and outgoing debris streams. This would

66
then tend to isotropize the accretion process onto the hole and and cause increased variability inthe lightcurve of the TDE. The clumps that form in the marginally bound material may have time
to condense into more compact objects, such as planets and brown dwarfs, that can survive theireventual return to pericenter, allowing them to remain bound to the hole. The clumps formed in
the marginally bound segment of the stream may also form less dense clouds, the likes of which
are observed near our own Galactic Center (e.g., G2; Gillessen et al. 2012; Burkert et al. 2012).
Finally, the unbound clumps may form a new class of low-mass, hypervelocity objects that maketheir way out of the host galaxy on timescales of millions of years.
Our results are based on the encounter between a solar-type star and a 106 M

black hole.

In reality, the properties of the star and black hole undergoing a tidal encounter may differ fromthe fiducial parameters chosen here. However, the existence of the in-plane pancake, and theobservational consequences derived therefrom, depends only on the fact that the pericenter distancebe comparable to the tidal radius. In particular, if β  1, the star will be disrupted well before
reaching periapsis, while if β  1 the star will only be partially disrupted, as noted by Guillochon& Ramirez-Ruiz (2013). Interestingly, Guillochon & Ramirez-Ruiz (2013) also found that if 0.75 .
β . 0.85 for a γ = 5/3 equation of state, the star was initially completely destroyed by a 106 M
black hole; however, at a time greater than 104 seconds post-disruption, the central portion of thestream recollapsed into a single, massive core, with the outer extremities of the stream remaining
as tidal tails. We suggest that the perpendicular pancake pointed out here may have contributed
to this recollapse, and we plan to further investigate this possibility.
The origin of the pancake can be seen directly from equation (3.4), which shows that the gas
parcels comprising the front of the star at the time of disruption are decelerating, while those at theback are accelerating; this results in the eventual merger of the in-plane edges of the stream. The
differential acceleration across the star is given by equation (3.13), which shows that the magnitudeof the pancake is primarily affected by the properties of the progenitor star. However, the inverse
scaling with the black hole mass, although weak, implies that smaller mass black holes lead to alarger differential acceleration and, hence, stronger pancakes.

67
The pancake alone can augment the self-gravity to the point where the stream gravitationally fragments in the cases where γ = 1.8 and 2, and this result is ultimately related to the factthat larger adiabatic indices result in a decreased resistance to the compression. However, a nonadiabatic equation of state could alter these results quite dramatically. In particular, any coolingwould decrease the equilibrium width of the stream, enabling the pancake to leave a much more
pronounced effect on the debris. The effects of a time-dependent entropy are also increased forsmaller γ, as is apparent from equation (3.22), meaning that even streams with very low adiabatic indices could collapse if the gas-energy equation were evolved self-consistently. We plan toinvestigate alternative equations of state in a future paper.

Chapter 4

On the Structure of Tidally-disrupted Stellar Debris Streams

4.1

Introduction
When a star comes within a supermassive black hole’s (SMBH) tidal radius rt = R∗ (Mh /M∗ )1/3 ,where R∗ is the stellar radius and Mh and M∗ are the black hole and stellar masses, respectively,
the tidal shear due to the hole overcomes the self-gravity of the star. The tidal force subsequently
tears the star apart, with half of the torn stellar debris bound to the black hole, the other half
unbound (Lacy et al., 1982; Rees, 1988).
These tidal disruption events (TDEs) have been studied analytically, numerically, and observationally for nearly forty years. The earliest analytic studies showed that the rate of return of thebound material, Ṁf b , should scale roughly as Ṁf b ∝ t−5/3 at late times (Phinney, 1989), and earlynumerical simulations supported this scaling (Evans & Kochanek, 1989). Dozens of putative TDEs
have now been discovered (e.g., Komossa & Greiner 1999; Gezari et al. 2008; Burrows et al. 2011;
Cenko et al. 2012; Bogdanović et al. 2014; Miller et al. 2015; see Komossa 2015 for a review of theobservational status of TDEs), and the rates at which they have been discovered show tentative
agreement with early estimates of the rate at which TDEs occur (10−4 − 10−5 per galaxy per year;
Frank & Rees 1976; Stone & Metzger 2014).
In Chapter 2 we demonstrated numerically that the streams of tidally-stripped debris were
gravitationally unstable when the adiabatic index of the gas was set to γ = 5/3, resulting intheir fragmentation into bound clumps. These clumps, when accreted at discrete times, caused
the late-time fallback rate to fluctuate about the t−5/3 average (the minima being induced by the

69
accretion of lower-density material in between clumps). In Chapter 3 we then performed a morecomprehensive numerical study of TDEs in which the pericenter distance was comparable to thetidal radius, varying the polytropic index of the gas. They found that, for stiffer equations of state(γ & 5/3), collapse was induced sooner and was enhanced by a “post-periapsis pancake” that takes
place soon after the disruption.
Here we construct a model of the stream of debris produced from a TDE in an attempt
to characterize its general properties. We first demonstrate the existence of a simple, self-similarsolution for the radial velocity profile along the stream in Section 2, and we compare it to the resultsof past simulations and use it to determine the stream position as a function of radial distance andtime. In Section 3 we show how the stream width varies with density and radial position andvaries between its hydrostatically-equilibrated value and self-similar expansion. Section 4 provides
approximate, analytic forms for the density along the stream and we compare these expressions tonumerically-computed values. We analyze the rough scaling of the density with time in Section 5,
and we also demonstrate that there is a critical adiabatic index at which the stream fragments,
equal to γc = 5/3 for an isentropic equation of state. Section 6 provides a discussion of our findings
and considers the results of including other effects in our analytic treatment, and we summarize
and conclude in Section 7.

4.2

Velocity distribution

4.2.1

Self-similar velocity profile

As was noted in Chapter 3, assuming that the tidal disruption takes place impulsively provides
a very good approximation for the radial positions of the gas parcels from a TDE when the impact
parameter of the star is β = rt /rp = 1, where rp is the pericenter distance of the star. This impulse
approximation states that the star is able to maintain hydrostatic equilibrium until it reaches thetidal radius, at which point it is “destroyed,” meaning that the self-gravity and pressure of theresultant gas become negligible at this location and for any moment in time thereafter. Therefore,

70
each gas parcel of the tidally-stripped debris moves solely under the influence of the gravitational
field of the black hole.
When one applies the impulse approximation to the SMBHs we are considering here, theorbital eccentricities of the stream occupy a very narrow range of values centered around one.
Therefore, the velocity along the stream is very nearly radial for times not long after disruption.
Furthermore, as was also demonstrated in Chapter 3, the self-gravity of the stream keeps it narrowly
confined in the φ and θ directions. As a first approximation we can thus let v(r, θ, φ, t) ' vr (r, t)r̂,
where v is the velocity vector of the material in the stream and r̂ is the unit vector in the radial direction. If we further neglect the influence of pressure gradients and self-gravity in the r-momentum
equation, both of which should be small in comparison to the gravitational field exerted by thehole, then the radial component of the momentum equation is

∂vr
∂vr
GMh
+ vr
=− 2 ,
∂t
∂r
r

(4.1)

where Mh is the mass of the SMBH.
As we noted above, the eccentricities of the orbits are very nearly one for the entire stream.
Furthermore, the relevant timescale at any given distance from the hole is set by the dynamical√time, τd = r3/2 / 2GMh , at that radius. Therefore, a reasonable assumption for the radial velocity
is that it varies self-similarly according to

r
vr =

2GMh
f (ξ),
r

(4.2)

where
√
ξ=

2GMh tr3/2(4.3)

is time normalized to the dynamical time at radius r. If we insert this ansatz into equation (5.1),

71
we find the following self-similar equation to be solved for f :f0 =f2 − 1,2 − 3ξf(4.4)

where a prime denotes differentiation with respect to ξ.
In addition to this equation, one must also impose a boundary condition on f in order toobtain the full solution for the velocity profile. To do so, recall that the specific energies of thestream are narrowly distributed about zero and are

=


GMh 2
vr2 GMh−=f −1 ,
2
r
r

(4.5)

which shows that f is narrowly distributed about 1 initially. Investigating equation (4.4), thelocation at which f = 1 will correspond to f 0 = 0 if ξ 6= 2/3; however, if f 0 = 0, then it is easy tosee by taking more derivatives of equation (4.4) that every higher derivative of f is also zero, i.e.,f (ξ) = 1 is the solution if f 0 = 0 when f = 1. Since we we require the energies of the gas parcels
to be distributed about, but not exactly equal to, zero, we see that the point f = 1 must coincidewith ξ = 2/3. Therefore, our boundary condition on f is f (2/3) = 1, which we note corresponds
to a critical point in equation (4.4)1 .
In further support of the fact that f (2/3) = 1 is the only boundary condition that can
describe the debris stream, we note that equation (4.4) can be integrated exactly to give
C +f



p
p
f 2 − 1 − ln f + f 2 − 1(f 2 − 1)3/2

= ξ,

(4.6)

where C is a constant of integration. This equation cannot be solved analytically to isolate f (ξ);
however, if we set f = 1, the left-hand side of this equation is singular unless C = 0. The only
solution for vr that smoothly passes through marginally-bound portion of the stream must therefore
have C = 0. If we adopt this value of C and use L’Hospital’s rule to determine the limit of f → 1,
we find, as we expect, that ξ → 2/3.
1

The derivative at this point for the non-trivial solution, it can be shown, is f 0 (2/3) = −5/2.

72

Figure 4.1: The solution to equation (4.4) passing through the critical point f (2/3) = 1 (blue, solid
curve). The approximate solution given by f = −1 + 1/ξ, which matches the asymptotic limits ofthe true solution, is shown by the red, dashed, curve.

73
Figure 4.1 shows this critical solution by the blue, solid curve. For contrast, the red, dashedcurve shows the approximate solution f ' −1 + 1/ξ, which aids in seeing the asymptotic behaviorof the true function.
In order to assess the validity of the self-similar solution in matching the radial velocity profileof tidally-disrupted debris streams, Figure 4.2 compares the self-similar solution for vr (Equation
4.2) to the numerical simulation of Chapter 2. In this simulation a solar-type star (one with a solar
mass and solar radius) was destroyed by a 106 M

SMBH; the pericenter distance of the stellar

progenitor was equal to the tidal radius, the gas followed an adiabatic, γ = 5/3 equation of state,
and pressure and self-gravity were included at all stages of the simulation (see Chapter 2 for moredetails). The different curves in Figure 4.2 correspond to the stream of debris at different times,
with the earliest (farthest left on the plot) being roughly two weeks after disruption, the latest(occupying the largest radial extent) at almost two years (the Figure caption gives the precise
times). This Figure shows that the analytic, self-similar solution agrees extraordinarily well with
the numerical solution. The inset, which is a closeup of the velocity profiles at a time of 601 days
after disruption, quantifies the agreement to better than 1 part in 10−4 – 10−5 .
Figure 4.3 shows the difference in velocity, ∆v, between the numerically-obtained radial
velocity profile and the analytic velocity profile, i.e., ∆v = vn −vss , vn being the numerical solution,
vss the analytic solution (equation 4.2), normalized to the speed of light. Each set of points
corresponds to the time chosen in Figure 4.2. The small differences illustrated by this plot may be
due to the neglect of angular momentum in the self-similar solution (see Section 6.1).

4.2.2

Radial positions

With the function f completely determined, the radial positions of the gas parcels comprising
the stream can be found by solving the differential equation

r
ṙi =

2GMh
f (ξi ),
ri

(4.7)

74

vr / c

0.02

0

-0.02

2000

4000
r / rtFigure 4.2: A comparison between the self-similar (red, dashed curves) and numerical (black points)
radial velocity profiles, both normalized to the speed of light, for the debris stream produced by
the disruption of a solar-like star by a 106 M black hole. The numerical solution was taken fromthe simulation in Chapter 2. The different curves represent the stream at different times, witheach curve corresponding to, from left to right, 11, 30, 60, 120, 180, 300, 451, and 601 days sincedisruption.

75

Figure 4.3: A log-log plot of the difference in velocity normalized to the speed of light, i.e., ∆v/c ≡(vn − vss )/c, where vn is the numerically-obtained radial velocity and vss is the analytic solution(equation 4.2), for the same times chosen in Figure 4.2 (i.e., the dark blue set of points is 11 days
after disruption, the light yellow set of points is 601 days after disruption, and each set of points
in between corresponds to the appropriate curve in Figure 4.2). As discussed in Section 6.1, thesmall discrepancies shown here can plausibly be attributed to the neglect of angular momentum inthe self-similar solution.

76
where ri (t) is the radial position of gas parcel i, ξi =√3/2

2GMh t/ri

is the self-similar variable ofgas parcel i, and dots denote differentiation with respect to time.
Because f is not analytic, this equation cannot be integrated exactly to give ri (t). However,
we note that there is an exact solution for the marginally-bound material:

 p
2/3
3
rm =
2GMh t
.
2

(4.8)

p
As a check, differentiating this equation with respect to time gives ṙi = 2GMh /r, which, from√
3/2
equation (4.7), demands f (ξm ) = 1. On the other hand, ξm = 2GMh t/rm = 2/3, and since ourboundary condition is f (2/3) = 1, we see that equation (4.8) does indeed solve equation (4.7).
To determine the remaining positions of the gas parcels, we will apply an argument similar
to the impulse approximation: at some time tm , the stream is closely confined to the position of
the marginally-bound segment of the stream, with its radial extent equal to the diameter of the
disrupted star. We will thus letri (tm ) = rm (tm ) + δri ,(4.9)

where δri = µi R∗  rm (tm ) and |µi | < 1. Furthermore, we know that the tidal force of the blackhole imparts a spread of energies to the debris of (Lacy et al., 1982; Rees, 1988)

i =GMh R∗µi ,
rt2(4.10)

where µi is the dimensionless distance of gas parcel i from the center of mass of the star at thetime of disruption. However, from equation (4.5) we have that the energies of the gas parcels attime tm are

i =


GMh 2
f −1 ,
ri(4.11)

where the entire right-hand side is evaluated at tm , and since the stream is narrowly confined to77
the position of the marginally-bound parcel, we have

5
f '1−
2



2
ξ−
.
3

(4.12)

This equation is just a Taylor expansion of f about the point ξ = 2/3, and we used the fact thatf 0 (2/3) = −5/2 (see footnote above). We therefore have that5GMhi = −
ri



2
ξ−
,
3

(4.13)

and using equations (4.9) and (4.10) and keeping only first order terms in δri , we find that thepositions of the gas parcels satisfy

δri =

2
1 rm
R∗ µi .
5 rt2

(4.14)

Setting δri = µi R∗ , we see that the time at which the energies are “frozen in” corresponds to√
rm = 5rt in the self-similar approximation. This shows that, if we adopt our self-similar function
for the velocities, we cannot impose the impulse approximation exactly, which would amount tosetting the center of the star at rt at t = 0 with the radial extent of the stream equal to 2R∗ . Thisoccurs because, by specifying the positions of the gas parcels at some time t0 , we automaticallyknow their velocities via equation (4.7) and, consequently, their energies. Therefore, we do not
expect that the energies will automatically correspond to the correct values (equation 4.10) by√
enforcing ri (t0 ) = rt + µi R∗ , and indeed we see that we must have ri (t0 ) = 5rt + µi R∗ to correctly
match the energies.
√
With the initial conditions ri = rm (tm ) + µi R∗ , rm (tm ) = 5rt , and tm given by equation√(4.8) with rm = 5rt , we can numerically integrate equation (4.7) to solve for the positions of thegas parcels as a function of time. However, we can obtain an approximate expression for the initial
evolution of the debris stream by lettingr(µ, t) ' rm (t) + µr1 (t) + O(µ2 ),(4.15)

78
where we have let µi → µ become a continuous variable and r1 is a small correction (on the orderof R∗ /rt ) to the marginally-bound orbit rm (t) (note that this is just a Taylor expansion of r(µ, t)
about µ, truncated at first order). If we now insert equation (4.15) into equation (4.7), use equation(4.12) for the function f , and keep only terms to first order in µr1 , performing a bit of algebra
reveals that

ṙ1
4
=
.
r1
3t

(4.16)

This shows that the correction to the positions of the material from the marginally-bound orbit,
during the initial evolution of the debris stream, scales as r1 ∝ t4/3 . Using this expression for r1 inequation (4.15), this also shows that∂r
= r1 (t) ∝ t4/3 .
∂µ µ=0(4.17)

Note that this result is exact, as all of the higher-order µ corrections are zero at µ = 0. We will
use this result in Section 3.2.
Finally, while equation (4.15) is useful for the early evolution of the debris, it breaks down
once the orbits of the material start to deviate from ri ∝ t2/3 . We can obtain an approximate
expression for the positions of the gas parcels, one that is roughly valid over the whole stream, byrecalling that the function f is reasonably well-approximated by the function

1f ' −1 + ,ξ

(4.18)

as depicted in Figure 4.1. If we use this approximate form for f , then we can show that equation(4.7) can be solved for ri (t) to give

 p
2/3
ri (t) = 3 2GMh t + Ai t3/2,

(4.19)

where Ai is a constant of integration particular to gas parcel i. We can determine these constantsby, as above, requiring that the length of the stream is narrowly confined, initially, about the

79
marginally-bound position, i.e., ri (tm ) = rm (tm ) + δri , with δri = µi R∗ . Doing so gives√
9 3 (2GMh )3/4 R∗
Ai =
µi .
7/4
2
rm

(4.20)

Unfortunately, the value of rm cannot be determined by requiring that the energies match thosefrom the impulse approximation; this is because the marginally-bound orbit from equation (4.19) –where r ∝ t2/3 – occurs at ξ = 1/3, while from equations (4.5) and (4.12) it occurs where ξ = 1/2.
This discrepancy means that the energies do not correlate with the properties of the orbits, andarises ultimately from the fact that the boundary condition f (2/3) = 1 is not satisfied by equation√(4.18). Therefore, in this case we will simply let rm = 5rt – the same value found above when thecorrect form of f is used – and note that the positions of the gas parcels from equation (4.19) will
differ from the true values because of the erroneous energies. We thus find that the approximate
positions of the gas parcels are

 p
2/3
3/2
ri (t) = 3 2GMh t + χµi t,(4.21)√
9 3 (2GMh )3/4 R∗
χ≡
.
7/4
2
rm

(4.22)

where we defined

4.3

Stream width
The previous section exploited the fact that, to a very good approximation, the radial posi-tions of the gas parcels can be found by considering them as non-interacting test particles in thegravitational field of the hole. However, this will not necessarily be the case for the transverse
structure of the stream, as self-gravity and pressure can be important at both early and late times(Kochanek 1994, Chapters 2 and 3). If we account for self-gravity and pressure, the transverse

80
momentum equation becomes∂vs∂vs∂vs 1 ∂p
+ vr
+ vs
+
= gsg + gM,⊥ ,
∂t
∂r
∂s
ρ ∂s

(4.23)

where s is the cylindrical distance from the stream center, vs is the velocity in the s-direction, gsg
is the force due to self-gravity in the s direction, and gM,⊥ is the gravitational force arising fromthe black hole in the s-direction. Poisson’s equation can be written∇ · gsg = −4πGρ.

(4.24)

If we assume that the radial dependence of gsg is small in comparison with its s-dependence, whichshould be a good approximation owing to the degree of symmetry of the stream about its center ofmass and breaks down only near its radial extremities, then this equation can be integrated using
Gauss’ law to give

gsg ' −2πGρs.

(4.25)The perpendicular component of the gravitational field of the hole can also be written down asgM,⊥ ' −

GMh s
.
r3

(4.26)

Substituting these equations into equation (4.23) then gives∂vs∂vs 1 ∂p
GMh s
∂vs
+ vr
+ vs
+
= −2πGρs −
.
∂t
∂r
∂s
ρ ∂s
r3

4.3.1

(4.27)

Quasi-hydrostatic width

During the early evolution of the debris, the pressure and self-gravity of the material arehigh enough that the stream remains approximately in gravitational equilibrium in the transverse
direction (approximately because the density is evolving with time and the planar motions can81
cause the stream to be slightly over-pressured, resulting in oscillations; Chapter 3). In this limit
we can balance the pressure and self-gravity terms in equation (4.27), which givesp,2πρ2 G(4.28)

p = S(r, t)ργ ,(4.29)

2
Heq'

Here we will assume that the pressure varies as

where S(r, t) is related to the entropy of the gas. A perfect polytrope is obtained by setting
S(r, t) to a constant; however, the stream of debris will cool radiatively as it evolves, and will do soincreasingly as the material becomes optically thin to the radiation released during recombinations.
Also, recombinations can heat the gas when it is still optically thick (Kasen & Ramirez-Ruiz, 2010),
and shocks can likewise serve to heat the material if the star is on a deeply-plunging orbit. To
model these thermodynamic aspects of the problem, we will therefore permit S(r, t) to be a functionof r and t that could, in principle, be determined from the gas energy equation if all of the effects
controlling the internal energy of the gas were known. The equilibrium width is thus

2
Heq
=

Sργ−2
,
2πG

(4.30)

which is the same scaling that we found in Chapter 3.

4.3.2

Shear dominatedDepending on the adiabatic index of the gas or the amount of heating and cooling, the density
will fall off at such a rate that the tidal shear term on the right-hand side of equation (4.27) will
overcome the self-gravity (Chapter 3). The pressure gradient will also be insufficient to balance
the tidal compression when this happens, meaning that the advective terms on the left-hand side82
of equation (4.27) must become non-negligible. In this limit equation (4.27) becomes

∂vs
∂vs
∂vs
GMh s
+ vr
+ vs
=− 3 .
∂t∂r∂sr(4.31)We expect that the solution for vs will again scale self-similarly, as the dynamical time is the only
relevant timescale and only the gravitational field of the hole serves to alter the velocity vs . We
find that indeed there is a self-similar solution, and it is given by

s
vs =
r

r

2GMh
f (ξ),
r(4.32)

where f (ξ) is the same function that appears in the distribution of the radial velocity. Sincepvs = dsi /dt, where si is the transverse position of gas parcel i, and 2GMh /rf (ξ) = dri /dt, wefind that the transverse extent of the stream evolves as

Hi (t) =Hi,0ri (t),
ri,0(4.33)

where Hi,0 is the width of the stream at time t0 . When the stream is dominated by the tidal shearof the hole, this equation shows that the width of the stream simply scales in proportion to its
radial position and the entire evolution proceeds self-similarly.

4.3.3

Approximate, full solution

The above two restricted solutions demonstrate that, when the self-gravity of the stream
dominates over the tidal shear of the black hole, the width of the stream is set by hydrostaticbalance. On the other hand, when the tidal force of the black hole overwhelms self-gravity, H ∝ rand the entire stream evolves self-similarly.
Along the full radial extent of the debris, the extremities will typically be shear-dominated,while the density in the central region is high enough to dominate over the shear of the hole
(Kochanek 1994, Chapter 3). We therefore expect the scaling of H to vary between the self-gravity
dominated limit and the shear-dominated limit as we traverse the extent of the stream. In between

83
those two limits, both the shear terms and the self-gravity terms are important in equation (4.27),
and as a result we do not expect a simple solution for the velocity vs .
We can, however, obtain an approximate solution that interpolates between the self-gravity
and shear-dominated extremes by simply letting H be piece-wise defined about the point where
2πρ = Mh /r3 . We thus have

2

H =



γ−2


 Sρ
2πG

h
for 2πρ ≥ M
r3


2


 Hr20 r2hfor 2πρ ≤ M
r3

0

.

(4.34)

To ensure the continuity of the solution, H0 and r0 are given by the self-gravitating solution where
2πρ = Mh /r3 . In the next section we will see how ρ is related to H.

4.4

Density
In addition to knowing the geometry of the stream, we would also like to be able to infer its

density. To do so, we note that the mass contained in some segment of the stream isZ
M=

ρ(V, t)dV,(4.35)V (t)where V (t) is the volume of the stream segment that is, in general, time-dependent. The total masscontained in the segment is, however, time-independent, so the differential equation to be solvedfor ρ is

d
dt

Z


ρ(V, t) dV= 0.

(4.36)V (t)When the limits of integration in this equation are functions of time, using the fundamental
theorem of calculus results in the continuity equation. However, if we can transform to a specific
set of coordinates in which the volume element is time-independent, then the resulting equation
can be immediately inverted to solve for ρ. In particular, if we write the new set of coordinates

84
collectively as T , then the equation for ρ is simply

ρ

∂V
∂M
=
,
∂T
∂T

(4.37)

where ∂V /∂T is the Jacobian of the transformation between the physical (V ) and time-independent(T ) coordinates. Here ∂M/∂T is the differential amount of mass contained in the volume elementdT .
We can achieve the transformation V → T by choosing our physical coordinates as (r, s, ϕ),where r is the radial position of the center of the stream, s is cylindrical distance measured perpendicular from r, and ϕ is the angle swept out by s as it revolves around r. In terms of thesecoordinates, the mass contained in any segment of the stream isZM=ρ s ds dφ dr.

(4.38)

If we further define the H-averaged density asZ2

πH ρ̄ ≡

ρ s ds dφ,(4.39)

then the mass is given by

Z r1 (t)M =πH 2 ρ dr,(4.40)

r0 (t)

where we dropped the bar for ease of notation. By now transforming from r → µ, where µ is thedimensionless original position of the stream, we find (see also equation 2 of Chapter 2)

πH 2 ρ

∂r
∂M
=
.
∂µ
∂µ

(4.41)In order to make more progress with this equation, we must determine the quantities H(ρ, r),∂r/∂µ and ∂M/∂µ. As we saw in the previous section, H depends on ρ and r in a non-trivial way,
and will behave differently depending on whether the stream is shear dominated or self-gravity
dominated. However, we noted that a full, approximate solution can be obtained by assuming that85
H switches between its hydrostatic equilibrium value and self-similar expansion at the point where
2πρ = Mh /r3 , which gives equation (4.34) for H.
The most rigorous way of calculating ∂r/∂µ is to do so numerically, i.e., calculate the positionsof the gas parcels comprising the stream, r(µ, t), via equation (4.7) for a large number of µ andt, and at a specific time t interpolate over all of the µ to determine ∂r/∂µ. However, we can alsouse equation (4.21), which gives an approximate solution for the positions of the gas parcels asfunctions of µ and t, to obtain an analytic expression. Doing so gives∂r2= χt3/2 r−1/2 ,∂µ3(4.42)

which we note is approximately valid over the entire extent of the stream (this expression, however,gives ∂r/∂µ|µ=0 ∝ t7/6 , in contrast to the exact value of ∂r/∂µ|µ=0 ∝ t4/3 ).
Finally, the function ∂M/∂µ can be determined by using the impulse approximation andconsidering the star at the time of disruption. This assumption then gives (see Lodato et al.
(2009), Chapter 3 and Chapter 5 for details)
R ξ1
n
∂M
1
|µ|ξ1 Θ(ξ) ξdξ,= M∗ ξ1 R ξ1∂µ2
Θ(ξ)n ξ 2 dξ

(4.43)

0where Θ(ξ) is the solution to the Lane-Emden equation and ξ is the dimensionless radius of thepolytrope (Hansen et al., 2004). A slight difficulty arises from the fact that, if the energies of thegas parcels using our self-similar prescription are to match those from the true impulse approximation (the one that accounts for the finite angular momentum of the material), then the “time ofdisruption” – the point at which the energies are frozen in – corresponds to when the marginally√bound segment of the stream is at a distance of rm = 5rt , not at rt (see Section 2.2). This meansthat using equation (4.43) with the self-similar solution will overestimate the density, as we expect√that the stream will have stretched by some factor by the time it reaches the distance 5rt andcorrespondingly decreased the density.
Since the functions Θ(ξ) are only analytic for a select few values of n, equation (4.43) cannot,

86

10-8
10-9
ρ [g / cm3 ]ρ [g / cm3 ]

10-5

10-7

10-9

10-10
10-11
10-12
10-13

10-11
50

100

500

1000

5000

50

100

r / rt

500

1000

5000

r / rtFigure 4.4: The analytic solution for the density when self-gravity dominates the shear of the blackhole (equation 4.45), shown in the left panel, and when the shear dominates the self-gravity of thestream (equation 4.46), shown in the right panel, as functions of r for the disruption of a solarlike star by a 106 M SMBH. We assumed isentropic gas with γ = 5/3 and entropy calculated forsolar-like parameters (S = 2.48 × 1014 [cgs]; Hansen et al. 2004). The different colors correspond to
3/2 √different times, with the earliest time being 2500 × rt / 2GMh ' 32 days after disruption (left3/2 √most, dark blue curve), the latest being 47500 × rt / 2GMh ' 620 days after disruption (right3/2 √most, light blue curve), and each curve differing from the previous one by 7500×rt / 2GMh ' 98
days.

10-7
10-4
10-9

10-6

ρ [g / cm3 ]ρ [g / cm3 ]

10-5

10-7
10-8
10-9

10-11

10-13

10-10
50

100

500
r / rt

1000

5000

50

100

500

1000

5000

r / rtFigure 4.5: The numerical solution for the density when self-gravity dominates the shear of the blackhole (the solution to equation 4.41 with H given by equation (4.30), ∂r/∂µ calculated numerically,and ∂M/∂µ calculated numerically from equation (4.43) with n = 1.5), shown in the left panel,
and when the shear dominates the self-gravity of the stream (the solution to equation 4.41 with H
given by equation (4.33), ∂r/∂µ calculated numerically, and ∂M/∂µ calculated numerically fromequation (4.43) with n = 1.5), for the same parameters and times chosen in Figure 5.3.

87
in general, be simplified further. However, we can obtain an approximate solution by letting
Θ(ξ) ' 1, which is valid in the inner regions of all polytropes. We can therefore write∂M3' M ∗ 1 − µ2 .
∂µ
4(4.44)

Because H switches between its equilibrium value and self-similar expansion along the lengthof the stream, equation (4.41), even with the approximate, analytic expressions for ∂r/∂µ and∂M/∂µ, cannot be rearranged to write ρ in closed-form over the entire extent of the stream.
However, in the limit that self-gravity or the tidal shear dominates, we can solve the equation
exactly. Inserting equations (4.42) and (4.44) into equation (4.41) and inverting the function (4.19)
to solve for µ(r, t), we find that these limits correspond to

r1/2
ργ−1
=
B
sg 3/2sg

r3
1− 2 3χ tr−3/2
ρsh = Bsh 3/2tr3
1− 2 3
χ t

t

√
2 !
3 2GMh t
1−
r3/2(4.45)√
2 !
3 2GMh t
1−
r3/2(4.46)



where ρsg and ρsh refer to the density in the self-gravity and shear-dominated cases, respectively,
and for compactness we defined

Bsg ≡

9GM∗
4Sχ

(4.47)

and

45M∗
Bsh =
8πχ



Mh
M∗

2/3
.

For the shear-dominated solution, we let H0 = R∗ when r =(4.48)√

5rt , which is consistent with theimpulse approximation under our self-similar prescription as we saw in Section 2.2.
Figure 5.3 shows these two solutions, the left panel the self-gravity dominated case, the rightpanel being shear dominated, when Mh = 106 M , R∗ = 1R , M∗ = 1M , γ = 5/3, and the gas
is isentropic with S = 2.48 × 1014 [cgs], which is the value appropriate for a sun-like star (Hansen

88
et al., 2004). The different colors correspond to different times since disruption, with the earliest
3/2 √time being 2500 × rt / 2GMh ' 32 days after disruption (left-most, dark blue curve), the latest
3/2 √being 47500 × rt / 2GMh ' 620 days after disruption (right-most, light blue curve), and each
3/2 √curve differing from the previous one by 7500 × rt / 2GMh ' 98 days.We see that the density
structure differs substantially between the two analytic cases, retaining an “inverted” profile when
the stream is self-gravitating (the density increases as a function of r), but quickly assuming adensity profile that decreases monotonically for the shear-dominated case.
Figure 4.5 shows the numerical solution to equation (4.41) when self-gravity dominates (lefthand panel) and when shear dominates (right-hand panel). For these solutions, we calculated ∂r/∂µdirectly by numerically integrating equation (4.7) for a large number of µ (20000 points) and t,∂M/∂µ was computed numerically from equation (4.43) with n = 1.5 (γ = 5/3), and the physicalparameters were the same as those in Figure 5.3. We see that the positions of the gas parcels disagree
between the analytic solutions – equations (4.45) and (4.46) – and the numerical solutions that usedthe exact, directly computed forms for ∂r/∂µ and ∂M/∂µ (compare for example the locations at
which the density of the bound material goes to zero between the left-hand panels of Figures 5.3
and 4.5). In particular, the return time of the most bound material is vastly overestimated becauseof the incorrect scaling of the energies in the analytic case. However, the overall scaling of thedensity with r is upheld.
The solution that adopts a piecewise-continuous form for H (equation 4.34) is shown in Figure
4.6. The different curves correspond to the solution at the same times as Figures 5.3 and 4.5, andthe dashed line shows the curve Mh /(2πr3 ). We see that these solutions possess properties of boththe left and right-hand panels of Figure 4.5: when the stream is self-gravity dominated, the density
retains an inverted profile and increases as we move outward in specific energy. However, when thebound material nears the black hole and becomes shear-dominated, the tidal compression servesto decrease the stream width and produces an “up-turn” in the density profile as we approach the
hole.
Interestingly, the density profile of the simulation in Chapter 2 does not exhibit this up-turn

89

Figure 4.6: The numerical solution for the density that adopts the piecewise behavior for H (equation 4.34) so that the density is shear dominated in regions where ρ < Mh /(2πr3 ) and self-gravity
dominated where ρ > Mh /(2πr3 ), for the same parameters and times chosen in Figure 5.3. The
yellow, dashed line shows the curve Mh /(2πr3 ).

90Figure 4.7: A particle plot of the density in the stream ρ in units of g / cm3 as a function of r
in tidal radii at a time of 57 days post-disruption. The red points are from the simulation usedin Chapter 2, the black points are from an identical simulation but with 107 , as opposed to 106 ,particles, and the green, dashed curve shows the function Mh /(2πr3 ) – the approximate dividing
line between the shear and self-gravity dominated solutions.

91
in the density as the material approaches the black hole. This anomalous behavior is demonstrated
by the red points in Figure 4.7, which is a particle plot of density versus r from the simulationin Chapter 2. The black points in this Figure, on the other hand, are from a simulation that
is identical to that from Chapter 2 except that, instead of 106 particles, 107 SPH particles wereused. The green, dashed line shows the curve Mh /(2πr3 ). The fact that the black points show
the upturn in the density as we approach the black hole, consistent with what we expect from theanalytic arguments presented here, shows that the disagreement between the simulation in Chapter
2 and our predictions arises from the resolution of the simulation. Since the increase in the density
arises from the tidal compression exerted by the black hole, which results from the differential
gravitational force acting across the diameter of the stream, we see that the width of the incoming
debris stream is underresolved when only 106 particles are used.

4.5

Density scalings and fragmentation conditions
The most accurate way of determining how the density of the stellar debris stream produced

from a TDE varies with r and t is to solve equation (4.41) numerically. However, this direct way
doesn’t yield an obvious trend for how the density roughly scales with distance and time. Even fromequations (4.45) and (4.46), which give approximate expressions for ρ in the self-gravity and sheardominated limits, respectively, it is not overly apparent how the density qualitatively behaves as
the stream recedes from the black hole. We can, however, discern how certain parts of the stream
behave by returning to equation (4.41) and focusing only on specific energy, or specific µ, ranges.

4.5.1

Marginally bound material

Recall that the marginally-bound segment of the stream corresponds to µ = 0, follows r ∝t2/3 , and, from Section 2.2, satisfies ∂r/∂µ ∝ t4/3 . Returning to equation (4.41), we thus findH 2 ρm ∝ t−4/3 ,(4.49)

92
where ρm refers to the density at the marginally-bound part of the stream. If we now assume thatthe stream is self-gravity dominated, then using equation (4.30) for H showsρm ∝ S41− 3(γ−1)
− γ−1

t

.

(4.50)

If we now assume that the gas is isentropic, then, for adiabatic indices of γ = 1.5, 5/3, 1.8, and 2,
we find that ρm ∝ t−8/3 , t−2 , t−5/3 , and t−4/3 , respectively, which is in good agreement with thescalings found in Chapter 3.
Equation (4.50) holds when the self-gravity of the debris dominates over the tidal field of
the hole; however, if the density of the debris falls off at a rate steeper than t−2 , then even if it
starts out as being self-gravitating, the tidal shear of the black hole will eventually dominate theself-gravity of the debris. As we saw above, the critical γ at which self-gravity and the tidal field of
the hole balance one another at the marginally-bound portion of the stream, which we will denote
γc,m , is given by

5
γc,m = .
3(4.51)

When the tidal shear dominates the self-gravity of the stream, the ability of the debris to collapse
locally and fragment is seriously hindered. Therefore, we expect γc,m to represent the critical adiabatic index at which fragmentation is marginally possible: for γ > γc,m , the stream can fragment,
while for γ < γc,m , the stream is stable to fragmentation. This interpretation is supported by the
results of Chapter 3, where we found that the stream fragmented vigorously for γ = 1.8 and 2, fragmented at late times and due to numerical noise for γ = 5/3, and did not exhibit any fragmentationfor γ = 1.5 (see also Section 6 for a further discussion of this point).
When the tidal shear of the black hole dominates the stream self-gravity, then using the factthat H 2 ∝ r2 shows that the isentropic density scales asρm ∝ t−8/3 .

(4.52)

93
This demonstrates that the density of the stream will not necessarily fall off as a single power-law
throughout its entire evolution. For example, if the stream starts out self-gravitating and has anadiabatic index of γ = 1.5, then it will initially follow ρm ∝ t−2.66 before eventually transitioning
to the shallower power-law ρm ∝ t−2.22 .
If we let the entropy scale as S ∝ t−p , which provides some insight into the ability of heating
and cooling to affect the evolution of the stream density, then it is simple to show that the criticaladiabatic index along the marginally-bound portion of the stream becomes

γc,m =

5 p
− .
3 2

(4.53)

This illustrates that cooling does not need to be very efficient to significantly alter the fragmentation
properties of the stream. Alternatively, if shocks play a significant role in heating the gas, then theability of the stream to fragment will be correspondingly diminished. On the other hand, once thegas temperature drops to about 104 K, the gas can start to recombine and heat the debris, stalling
its temperature at 104 K and resulting in an adiabatic index of order unity. However, once theneutral fraction becomes significant, the optical depth decreases dramatically and recombinations
will serve to cool the gas, corresponding to a larger effective γ (Kasen & Ramirez-Ruiz, 2010).

4.5.2

Unbound material

For the material in the unbound portion of the stream, the gas parcels follow r ∝ t2/3 initially,
but eventually transition to r ∝ t. We therefore have ∂r/∂µ ∝ t when the material recedes to large
distances from the hole, and we find that equation (4.41) becomesH 2 ρu ∝ t−1 ,(4.54)

94
where ρu refers to the density in the unbound portion of the stream. If we now focus on the
self-gravity dominated limit, using equation (4.30) showsρu ∝ t−1/(γ−1) .

(4.55)

From this expression we see that the density of the unbound material falls off at a shallower rate
than the marginally-bound debris. Specifically, for γ = 5/3 we find ρu ∝ t−3/2 as compared toρmb ∝ t−2 . Furthermore, since the density of the black hole falls off as Mh ∝ t−3 for the unbounddebris as opposed to Mh ∝ t−2 for the marginally-bound debris, the critical adiabatic index atwhich the black hole density equals the stream density is correspondingly smaller. In particular, if
we set ρu ∝ t−3 , we find that the critical adiabatic index in the unbound portion of the stream is
γc,u = 4/3.
When the tidal shear of the hole dominates the self-gravity, using equation (4.33) in equation(4.54) givesρu ∝ t−3 .

(4.56)

As was true for the marginally-bound case, this shows that the density of the unbound material
will not necessarily fall off as a single power-law throughout its evolution if it passes from being
self-gravity to being shear-dominated.
Furthermore, because the energies of the debris are greater than zero by only a small amount,the initial evolution of even the unbound debris will still follow approximately ru ∝ t2/3 , and only
at some time ttr will the orbits transition to ru ∝ t. If we rewrite equation (4.15) as
 p
2/3
3
R∗ µ
r(µ, t) '2GMh t+ 4/3 t4/3 ,2tm(4.57)

where the second term in this equation comes from equation (4.16) with the requirement that

95
ri (tm ) = rm (tm ) + µi R∗ , then we see that the time at which this transition occurs is approximately
√
3/2
5 5 q
R∗
√
ttr '
.
3 µ3/2 2GMh

(4.58)

Thus, if the unbound stream is self-gravitating, then at ttr we expect the density to transition fromρu ∝ t−2 to ρu ∝ t−1.5 . Equivalently, if we are following an unbound fluid parcel with a narrow
range of specific energies, then we expect the density to transition from ρu ∝ r−3 to ρu ∝ r−1.5
at this time. For the disruption of a solar-type star by a 106 M

SMBH, this time corresponds to

ttr ' 48 days post-disruption.

4.5.3

Bound material and overall scalings

The above two subsections show that the marginally-bound and unbound material behave
quite differently in terms of their asymptotic scalings. As was true for the unbound material, theinitial evolution of the bound material will follow approximately rb ∝ t2/3 because the specificenergies, while being negative, are very close to zero. Since the bound material inevitably fallsback to the origin no matter how close the energy is to zero, though, there is unfortunately nosimple way of discerning the asymptotic behavior of this segment of the stream.
One quantity that gives some insight into its behavior, however, is the number q ≡ d ln(ρr3 )/d ln t
as a function of µ. When taking this logarithmic derivative, the quantities ρ and r are considered
functions of µ and t and µ is a constant. Therefore, q gives the power-law index of the ratio ofthe density along the stream relative to the density associated with the black hole tidal field as afunction of time and as we move along with a fluid element.
Figure 4.8 shows a plot of q(µ) for γ = 5/3 and the disruption of a solar-type star by a
106 M

SMBH (S was set to a constant). The left-hand panel gives the solution when the flow

is dominated by self-gravity, while the right-hand panel shows the case when the flow is shear
dominated. The different curves give different times since disruption, with the blue curve at ' 13
days after disruption, the brown curve at ' 143 days after disruption, and each curve differingfrom the next closest curve by ' 26 days. As we mentioned, the quantity q characterizes the

96

1

0

0

q

q

1

-1

-1

-2

-2

-3
-1.0

-0.5

0.0
μ

0.5

1.0

-3
-1.0

-0.5

0.0

0.5

1.0

μ

Figure 4.8: The quantity q = d ln(ρr3 )/d ln t as a function of µ when γ = 5/3, Mh = 106 M , theprogenitor is solar, and the gas is isentropic; the flow is assumed to be self-gravity dominated in theleft-hand panel and shear-dominated in the right-hand panel. The different curves show different
3/2 √times, with the blue curve at 1000 × rt
2GMh ' 13 days after disruption, the brown curve at
3/2 √11000 × rt
2GMh ' 143 days after disruption, and each curve differs from the next closest curve
3/2 √by 2000 × rt
2GMh ' 26 days (e.g., the yellow curve is 26 days after the blue curve, the green
curve is 26 days after the yellow curve, etc.). The range of the y-axis was set to the same values foreach plot for ease of comparison. Fluid elements in the stream follow vertical lines from one curveto the next.

97
instantaneous power law of the product ρr3 , i.e., the ratio of the density in the stream to that of
the black hole scales instantaneously as ρr3 ∝ tq .
These plots show us what to expect when we consider lines of constant µ – when we follow
along a single fluid element. When µ > 0, the specific energy of the gas parcel is greater than zero,and thus initially follow r ∝ t2/3 but eventually transition to r ∝ t for late times. We would thus
expect that, at early times, the ratio of the density of the stream to that of the black hole should
follow ρ r3 ∝ 1 (ρ r3 ∝ t−2/3 ) for self-gravity dominated (shear dominated) streams, while at latetimes it should follow ρ r3 ∝ t1.5 (ρ r3 ∝ 1) for self-gravity dominated (shear dominated) streams.
This behavior is exactly shown in the left and right-hand panels of Figure 4.8, and specifically therate at which this transition occurs. Likewise, if the material is exactly at µ = 0, it will always
follow ρ r3 ∝ 1 (ρ r3 ∝ t−2/3 ) for self-gravity (shear)-dominated streams, and this is seen in the
Figure.
Figure 4.8 also shows how the density of the bound material, fluid elements with µ < 0, evolves
with respect to the density of the SMBH. As we mentioned, there is no asymptotic behavior for thebound material because it inevitably falls back to the black hole. However, we see that the ratioof the stream density to the black hole density always scales as some power of t that is less than 1,
and in fact this power-law decreases as time increases. We thus expect that all material bound tothe black hole, if it follows a γ = 5/3 equation of state, will eventually become shear-dominated.

4.6

Discussion
In the preceding sections we developed pseudo-analytical expressions for the velocity profile ofthe debris, the positions of the gas parcels comprising the debris stream, and the density of the debris
stream. Here we discuss how the angular momentum of the debris might alter these expressions,
make comparisons to the previous work of Kochanek (1994) on tidally-disrupted debris streams,
consider the fate of the debris that satisfies the critical adiabatic index condition, and explore how
the entropy of the gas should realistically behave.

98
4.6.1

The neglect of angular momentum

The solutions we have given here rely on the fact that the angular momentum of the material
is small enough to be ignored when considering the evolution of the debris stream. Since the φpcomponent of the velocity is approximately given by rφ̇ ' `/r, where ` ' GMh rp is the specificangular momentum of the material and rp ' rt is the pericenter distance of the star, we find thatvφ /vr ' (r/rp )−1/2 . Thus, the neglect of angular momentum only breaks down for small radii,
and should be increasingly accurate as the pericenter distance of the progenitor gets smaller. This
argument also agrees with Figure 4.3, which shows that the agreement between the self-similarsolution and the numerical solution gets better as we proceed to larger distances from the hole(and, as a consequence, the ratio of the angular velocity to the radial velocity gets smaller).
One consequence of accounting for the finite angular momentum of the material is that theposition of the center of the stream (in terms of its width) is not purely radial, but instead sweeps
out some angle (see also Guillochon et al. 2015). Thus, the extent of the stream is not exactly(∂r/∂µ)dr, but is instead given by

∂r
dµ →
∂µ

s

∂r
∂µ

2


+ r2

∂φ
∂µ

2
dµ,

(4.59)

where φ is the in-plane angular position of the center of the stream; this is the equation employedin Chapters 2 and 3. Therefore, the overall length of the stream is slightly underestimated byignoring the angular momentum of the gas.
A more important consequence of the angular momentum of the material is that the orientation of the tidal force of the SMBH will change as the bound material returns to the origin.
Specifically, by assuming that the motion of the gas parcels was radial with a transverse extent s,
the tidal field of the hole only served to compress material in the transverse direction (viz. equation
4.26) and increase the density of the stream. However, once the orientation of the cross-sectional
width of the stream H rotates to the point where it is aligned with the radial displacement of thecenter of the stream, the tidal compression will transition to a tidal shear that results in a decrease

99of the density.
Figure 4.9 shows a rough drawing, not to scale, of the cross-section of the returning debris
stream that illustrates this point: the left side shows the current assumption about the natureof H versus the position of the center of the stream, while the right side gives a more accurate
picture that accounts for the finite angular momentum of the material. From this schematic we
see that, under the current, radial approximation for the position of the center of the stream (interms of the transverse width), the tidal field of the black hole will only compress the stream andconsequently enhance the density near the black hole. In actuality, however, the small amount ofangular momentum possessed by the debris causes it to miss the black hole, changing the tidal
compression of the hole into a tidal shear that actually decreases the density.

4.6.2

Comparison with Kochanek (1994)

Kochanek (1994) also performed an analysis of the debris streams produced from TDEs, in
an attempt to discern the nature of the self-intersection that ensues when the bound material isswung through its general relativistic apsidal precession angle (roughly ten degrees for a solar-likestar disrupted by a 106 M

SMBH; Rees 1988). Here we make a brief comparison between his

findings and ours.
One important result of Kochanek (1994) was that, if the impact parameter β ≡ rt /rp , where
rp is the pericenter distance of the center of mass of the progenitor star, is large enough, then selfgravity will not be important for determining the evolution of the stream. Here we find a similar
trend: from Figure 4.8, shear-dominated streams always follow ρ r3 ∝ t−q with q < 0. Thus, if the
impact parameter is high enough such that the density everywhere satisfies 2πρ < Mh /r3 , then thestream density will never be able to surmount the tidal shear of the SMBH (although the unboundmaterial does asymptotically follow ρ ∝ r−3 ).
Kochanek (1994) also found that the width of the stream varies as H ∝ r1/4 for a γ =5/3 equation of state. This result ultimately comes from the fact that the mass per unit lengthΛ ≡ ∂M/∂r is assumed to be Λ ∝ r−1 (see also Guillochon et al. 2014b for the expression when

100

2H

v
v

2H

Figure 4.9: A schematic of the debris stream, not drawn to scale, returning to the black hole under
the approximations set out in our analysis (left drawing) and a more realistic distribution that
accounts for the finite angular momentum of the hole (right drawing). This shows that, as thematerial gets very close to the hole, the tidal compression becomes a tidal shear that serves todecrease the density.

101
general γ are used alongside this prescription). However, we found from our analysis that the early
evolution of the stream – when all of the gas parcels are following approximately r ∝ t2/3 orbits –is characterized by

∂M
Λ=
=
∂r



∂r
∂µ

−1

∂M
∝ t−4/3 ∝ r−2 ,∂µ(4.60)

where the last relation follows from the fact that r ∝ t2/3 . Since H 2 ρ ∝ dM/dr, and ρ ∝ t−2 ∝ r−3
during this phase, we thus findH ∝ r1/2 ,(4.61)

which disagrees with the scaling H ∝ r1/4 found by Kochanek (1994). For general γ, using equations(4.30) and (4.50) gives2−γ

H ∝ r γ−1 .

(4.62)

Figure 4.10 shows H/R at the marginally-bound portion of the stream as a function of r/rt ,with the solid, black curve from the simulation described in Chapter 2 but using 107 particles, andthe red, dashed curve from the analytic prediction H/R ∝ (r/rt )1/2 (the proportionality constant
was set to 0.33, which provides a good by-eye fit). The width was calculated from the simulation
by first determining the number of particles that had specific energies within a small range centeredaround zero; for this plot, the absolute value of the maximum energy was 10−6 c2 (as compared tothe energy of the most energetic particles of ∼ 2 × 10−4 c2 ; see equation 4.10), which amounted
to roughly 105 particles and a total mass of δm ' .01M

contained within the zero-energy bin.

The maximum and minimum radii, rmax and rmin , respectively, within that bin were computed,
giving a radial extent δr = rmax − rmin of the marginally-bound portion of the stream. We thencalculated the quantity dM/dr ' δM/δr, determined the average density, and used equation (4.41)
to calculate the width. This Figure shows that the analytic prediction of H ∝ r1/2 fits the data well.
The disagreement between the prediction and the numerical results at early times is likely because

102

10

H [RSun]

5

2

1

10

102
r / rt103

Figure 4.10: A log-log plot of the stream width, H, in units of solar radii at the marginally-boundlocation along the stream as a function of Lagrangian distance (i.e., moving with the marginallybound portion of the stream) from the SMBH. The black, solid curve shows the solution from thesame simulation performed in Chapter 2 (i.e., a solar-like star destroyed by a 106 M SMBH) but
with 107 particles, while the red, dashed curve shows the analytic prediction H/R ∝ (r/rt )1/2
(the constant of proportionality was set to 0.33).

103
self-gravity is still acting to modify the specific energies in the radial direction. This interpretation
is supported by the fact that the number of particles in the marginally-bound energy bin goes from∼ 8.2 × 104 to ∼ 1.08 × 105 during that initial transient phase, after which point it remains almost
exactly constant.
Once the orbits of the unbound material transition to r ∝ t, which occurs around 50 days
for the disruption of a solar-like star by a 106 M

SMBH, we recover the ∂M/∂r ∝ r−1 scaling

assumed by Kochanek (1994). Using the fact that ρ ∝ t−3/2 ∝ r−3/2 during this phase, the relation
H 2 ρ ∝ ∂M/∂r yields H ∝ r1/4 , which is in agreement with Kochanek (1994).
Kochanek (1994) also discussed how ionizations and recombinations, viscosity, and an ambient
medium affect the evolution of the debris stream. We will return to these points in Section 6.4,
where we reconsider the entropy of the gas.

4.6.3

To fragment or not to fragment?

When the density of the black hole ρh ∼ Mh /r3 falls off at a rate that is steeper than thedensity of the debris stream, we expect that the self-gravity of the stream in the radial direction will
eventually overcome the shear of the black hole. Equivalently, if ρ & ρh , the dynamical timescale√τd ∼ r3/2 / GMh , which is the time over which material is sheared in the radial direction, will√
be longer than the free-fall timescale τf f ' 1/ 4πGρ. Thus, in this case the material is able toaggregate faster than the rate at which it is being torn apart, and we expect the stream to be
gravitationally unstable. Conversely, when ρ falls off more steeply than r−3 , the ordering of thetimescales reverses, and overdensities are sheared apart faster than they can collapse. We thus
expect the stream to be gravitationally stable when ρ falls off faster than r−3 .
These arguments are supported by Chapter 2, where we investigated, from a numerical perspective, how the evolution of the debris stream from a TDE depends on the adiabatic index of
the gas. We found that, when γ = 1.8 and 2, the stream fragmented early on in its evolution; from
our analysis ρm ∝ t−5/3 and t−4/3 when γ = 1.8 and 2, and thus we expect the stream to fragmentin these cases. On the other hand, in Chapter 3 we found that the stream never fragmented –104even after ten years post-disruption – when the adiabatic index of the gas was 1.5. From equations(4.50) and (4.52), the density should fall off as ρ ∝ t−8/3 for γ = 1.5, which is significantly steeper
than t−2 and we therefore expect, consistent with the numerical findings, that the stream is stable
to fragmentation in this case.
In Chapters 2 and 3 we also ran simulations in which the adiabatic index of the gas was
set to γ = 5/3. In this case, we found that the stream still fragmented, but the time at whichfragmentation occurred depended on the resolution of the simulation. In particular, if the numberof particles was increased, the stream fragmented later and vice versa. This finding then suggeststhat the stream is gravitationally unstable, but that the noise inherent in the numerical method isseeding the instability.
Is this finding and interpretation consistent with our analysis here? As we saw above, γ = 5/3
marks the critical adiabatic index (for an isentropic equation of state) where ρ ∝ t−2 during theearly evolution of the debris. In this case, then, the freefall and collapse timescales are proportional
to one another, and it is unclear whether or not the stream should fragment.
To answer this question, we note that a precisely analogous situation is encountered when one
considers Jeans collapse in the early Universe: as primordial overdensities form and try to collapse,
they are continuously stretched apart by the fact that the background density scales as ρ ∝ t−2(see, e.g., Coles & Lucchin 1995, p. 202). In this case, then, the freefall and dynamical timescales
are comparable, and the fate of the overdensities becomes unclear. However, one can show that
there are still growing, unstable modes, but instead of evolving as exponentials they only scale as
power-laws in time. To understand the origin of this result, we can argue somewhat heuristicallythat the relationδρ+δρ˙ + ',
τf f

(4.63)

where δρ+ is the growing mode of the instability and τf f is the free-fall timescale, holds moregenerally than just when τf f is time-independent (and, as a consequence, the instability grows

105pGρ0 t20 , where t0 is a characteristic timescale and ρ0 is√
2
a characteristic density, then it follows immediately that δρ+ ∝ t Gρ0 t0 – the overdensities growexponentially). Thus, if we write τf f ' t/

as power-laws.
For a star being disrupted by a SMBH, the characteristic timescale is the dynamical time
3/2 √at the tidal radius, so t0 ' rt / GMh , while the characteristic density ρ0 is the original stellar
density. We thus expect that when γ = 5/3, the growing modes should scale roughly as√δρ+ ∝ t α ,(4.64)

where α = ρ∗ rt3 /Mh is the ratio of the stellar density, ρ∗ , to the density of the SMBH at the timeof disruption, Mh /rt3 .
These arguments illustrate that when the density of the stream falls off as ∼ 1/r3 – which
occurs for the early evolution of the marginally-bound material when β ∼ 1 and γ = 5/3, andalso occurs for the shear-dominated portion of the unbound stream for any γ – there are growing
unstable modes. A more rigorous approach would be to perform a perturbation analysis on thefluid equations, and when this is done for Jeans collapse in the early Universe, the results of theheuristic argument are upheld (Coles & Lucchin, 1995). Since the problem is identical here interms of the scalings of the variables, it follows that the heuristic argument is also valid, and forcompleteness we have checked that equation (4.64) is recovered if one performs a perturbation
analysis. (Technically, the result is that ρ ∝ tN√

α , where N is a number of order unity.)

We thus find that the results of Chapters 2 and 3 are consistent with our analytical arguments:
when γ = 5/3, the stream is unstable to fragmentation. However, because the dynamical timescaleand collapse timescale are proportional in this case, the overdensities grow as power-laws insteadof exponentials. This considerably weaker growth rate means that resolving the instability is much
more difficult, and the noise in the density that is induced by the numerical method becomes thedominant contributor to the fluctuations that seed the instability. However, in the presence of a
sufficiently dense ambient medium, the shear between the stream and that medium would provide

106
a physical origin for the perturbations (Bonnerot et al., 2015).
Before moving on, we note that the simulations of Chapters 2 and 3 found that boundportions of the stream, specifically those very close to the marginally-bound orbit, also exhibited
fragmentation. Based on Figure 4.8, we would not expect this, as ρ r3 ∝ tq with q less than zerofor the bound material. However, our treatment here of the radial velocity ignored the effects
of self-gravity in the radial direction; it can be seen from Figure 4.7, though, that the denser
material towards the marginally bound segment of the stream generates a gravitational force on
the bound material that serves to counteract the gravitational pull of the SMBH. This additional
force will serve to reduce the amount of shear present within the flow, correspondingly lengthening
the dynamical time and conceivably allowing some parts of the bound material to collapse.

4.6.4

A more realistic entropy prescription

Most of the examples we considered here assumed that the constant-entropy nature of thepolytrope was preserved throughout the evolution of the debris. However, our equation (4.30)
permits a radially and temporally dependent entropy, which, through equation (4.41), would serveto alter the nature of the stream. In particular, if the entropy decreases due to cooling, the density
is correspondingly increased, making it easier for the stream to overcome the γ = 5/3 marginallyunstable condition (equation 4.53); likewise, if the internal energy of the gas increases, the widthof the stream grows and results in the gravitational stability of the stream.
When the stream initially starts to expand, any cooling is initially controlled by the fact thatthe surface radiates approximately as a blackbody:dL = σTef f (r, t)4 dA,(4.65)

where σ = 5.67 × 10−5 [cgs], dA is the area of the emitting surface, and Tef f is the effective
temperature that is, in general, dependent on time and position along the stream. The assumption

107
of a blackbody is justified by the fact that the optical depth across the width of the stream is −5/2
 −5/3rtτ ' ρκH ' κρ∗ R∗' κρ∗ R∗,rtt0

(4.66)

where κ is the Rosseland mean opacity and κρ∗ R∗ & 2 × 1010 for solar-like progenitors (R∗ = R
and ρ∗ ' 1 g / cm3 ). The stream is therefore very optically thick when the stream is still highly
ionized, which justifies the assumption of local thermodynamic equilibrium (but see below for whenrecombinations start to change the opacity). The area of the stream is simply dA ' 2πHdr, dr being
the infinitesimal distance along the stream. Since the total amount of internal energy containedin the stream is comparable to the amount contained in the original star, Eint ' GM∗2 /R∗ ' 1048
ergs for a solar-like progenitor, any radiation escaping from the photosphere should decrease theinternal energy on roughly the Kelvin-Helmholtz timescale – thousands to millions of years if the
disrupted star is roughly solar. Thus, the early evolution of the debris stream should be very nearly
adiabatic.
However, recombinations will start to occur as the temperature of the debris reaches roughly
104 K. As recognized by Kochanek (1994) and Kasen & Ramirez-Ruiz (2010), this could haveimportant consequences for the evolution of the debris stream, heating the material when it is
optically thick and cooling it as it becomes optically thin. If the stream follows a γ = 5/3 adiabatic
equation of state and is self-gravity dominated, then we can show, by returning to equation (4.41),
using equations (4.30), (4.57) and (4.43), and the ideal gas law, that temperature evolves as

T ' 2.35

GM∗ m
kB R∗



t
tm

−4/3
,

(4.67)

where the factor of 2.35 resulted from evaluating equation (4.43) numerically for n=1.5 (γ = 5/3)
at µ = 0, m ' 1.67 × 10−24 g is the mean molecular mass in the stream, kB = 1.38 × 10−16 [cgs]√√is Boltzmann’s constant, and tm = ( 5rt )3/2 / 2GMh ' 1.05 h for a solar-like progenitor and a

108
106 M SMBH. Solving this equation for trec , the time at which T = 104 , gives

trec ' 27 days

(4.68)

after disruption. This number could be slightly sooner or later depending on the mean molecularweight (we adopted m = mh = 1.67 × 10−24 g) and the initial temperature of the star.
Equation (4.68) should really be interpreted as an average value over the entire extent of
the stream. In particular, since the density, and hence the temperature, is maximized near themarginally-bound portion of the stream (see Figure 5.3), recombinations will start to occur at alater time than they will toward the radial extremities of the stream where the temperature iscorrespondingly lower. Likewise, the stream is not an infinitely thin line, but has a finite width H;
near H the gas temperature is lower, generating an earlier recombination time.
Initially – when the gas is still largely ionized – the optical depth is quite high, as evidenced
from equation (4.66), meaning that recombinations will heat the gas and keep it at a roughly
constant temperature of 104 K. Once recombinations start to occur, however, the decrease in theionized fraction will correspondingly lower the opacity of the gas and allow more of the stream tocool, generating a “cooling front” that moves inward from H (Kasen & Ramirez-Ruiz, 2010). Thus,
while recombinations will initially heat the gas and serve to increase the specific entropy S, at latertimes the optical thinness of the stream means that continued recombinations will actually serve
to decrease the specific entropy.
The local shear within the stream could also serve to heat the gas. If we prescribe thecoefficient of dynamic viscosity by η, then viscous heating will modify the gas energy equation via(Chapter 6)

p
γ−1



∂
∂
ln S + vr
ln S
∂t
∂r




'η

∂vr
∂r

2
,

(4.69)

where p is the gas pressure and S = p/ργ is the entropy as we defined it for a polytrope. We
see that the right-hand side of this equation falls off approximately as ∝ 1/r3 (assuming that the

109
temporal and spatial dependence of η is not too large), and consequently the viscous heating of thegas should not play a large role in the overall evolution of the debris stream.
Finally, the background radiation field present in the circumnuclear medium will also alter
the thermodynamical evolution of the stream. If the stream is optically thin, then the entire stream
would be maintained at the temperature of the radiation field Tb (in the most extreme case, Tb
would be the temperature of the cosmic microwave background). When the stream is still opticallythick, on the other hand, the radiation will be reprocessed by a thin, outer sheath (Kochanek,
1994).
When the recombinations or radiation from background sources start to heat the gas, it will
become over-pressured with respect to self-gravity. In this case, if we return to equation (4.27), we
see that the cross-sectional width of the stream varies as

H ' cs t,(4.70)

where

r
cs '

p
ρ

(4.71)

is the adiabatic sound speed. If we now use this expression for H in equation (4.41) and again
assume that the evolution proceeds adiabatically (p = Sργ , where this S is a constant that is larger
than the value before heating started), then we find

γ 2



ρ t ∝

∂r
∂µ

−1

∂M
.
∂µ

(4.72)

While the stream is still in the marginally-bound phase, ∂r/∂µ ∝ t−4/3 , and we thus find

ρ∝t

10
− 3γ

.

In this case, we see that γ = 5/3 still marks the critical γ at which ρ ∝ t−2 ∝ r−3 .

(4.73)

110

4.7

Summary and conclusions
In this paper we have presented a semi-analytic analysis of the debris streams produced from

TDEs, showing that there is a simple, self-similar prescription for the radial velocity of the debris.
pSpecifically, we found that if one assumes that the radial velocity varies as vr = 2GMh /rf (ξ),√
where f is a function of the self-similar variable ξ = 2GMh t/r3/2 , then the function f is given by
the solution to the equationf0 =f2 − 12 − 3f ξ(4.74)

with the boundary condition f (2/3) = 1. Figure 4.2 compares this velocity profile to the numericalsolution of Chapter 2, and shows that the two agree very well (see also Figure 4.3).
We also computed the cross-sectional radius of the stream, H, as a function of time anddensity. We found that, when the density and pressure are high enough, the stream width varies
in a quasi-hydrostatic manner as H 2 ∝ ργ−2 . However, if the shear of the black hole dominates
self-gravity and pressure, then the entire stream evolves self-similarly with H ∝ r.
The self-similar velocity profile and the solution for the cross-sectional radius of the stream
allowed us, in conjunction with equation (4.41), to solve for the density of the debris stream, ρ, asa function of space and time. Our approximate, analytical expressions (equations 4.45 and 4.46)
were shown to agree well with the more exact, numerical solutions that used the full forms for thequantities ∂r/∂µ and ∂M/∂µ (see Figures 5.3 and 4.5). An inverted density profile – where thedensity increases as a function of r – is actualized when the material is self-gravitating, while thedensity decreases monotonically when the shear of the black hole dominates the stream self-gravity.
We analyzed the general scaling of ρ with time and distance from the black hole, and showed
that, during the initial stage of evolution when the orbits of the gas parcels comprising the debris
stream follow r ∝ t2/3 , the portions of the stream that are self-gravitating satisfy ρ ∝ t−4/(3(γ−1)) .
This scaling demonstrates that γ = 5/3 marks the critical adiabatic index above which self-gravity
always dominates the tidal shear of the black hole, below which the self-gravity of the stream

111
becomes negligible. On the other hand, the unbound portion of the stream eventually follows
r ∝ t, which shows ρ ∝ t−1/(γ−1) and the critical adiabatic index is γ = 4/3. Finally, when thestream falls below the self-gravitating limit (when 2πρ < Mh /r3 ), the entire stream behaves selfsimilarly and ρ ∝ r−2 (∂r/∂µ)−1 . During this self-similar regime, the density falls off as r−4 whenr ∝ t2/3 (marginally-bound material, early evolution of the entire stream) and r−3 when r ∝ t(unbound material, late evolution).
Finally, we showed that the finite angular momentum of the debris, which was ignored in our
self-similar models, should have a small but noticeable effect as material falls back to the black hole.
Comparisons between our results and those of Kochanek (1994) were made, demonstrating that we
agreed on many aspects of the evolution of the debris stream but disagreed on some; in particular,
our finding that the width of the stream scales as H ∝ r1/2 for a γ = 5/3, self-gravitating stream
contrasts the scaling H ∝ r1/4 found by Kochanek (1994). We showed that when γ = 5/3, thestream of debris produced from a TDE should be gravitationally unstable; however, because thedensity of the stream scales as ρ ∝ r−3 for early times, which is identical to that of the black hole,√

the perturbations grow only as power-laws in time (specifically, δρ+ ∝ t α , where α ' ρ∗ rt3 /Mh
is the ratio of the density of the stellar progenitor to the density of the black hole at the time ofdisruption). We also considered how the entropy of the debris stream might evolve more realistically,
given that the gas will start to recombine after a certain time and that the ambient medium may
serve to heat the gas.
Our investigations assumed that the density of the circumnuclear medium does not affectthe propagation of the debris stream produced from a TDE. Because the original stellar density is
so much higher than the densities near the centers of most galaxies, this assumption is justified,with the bulk dynamics of the material largely unaffected (Kochanek, 1994). However, the KelvinHelmholtz instability could be important for modifying the density profile of some portions of the
returning debris stream (Bonnerot et al., 2015), and could also provide a physical length scale
over which density perturbations occur that cause the stream to fragment. Also, once the stream
propagates out to very large distances from the SMBH, the stream density will become comparable

112
to that of the ambient medium, and drag effects will start to alter its motion (Guillochon et al.,2015).
In order to make comparisons with past simulations, we only computed the explicit velocity
profile (the one that depends on both r and t) and compared it to the numerical solution for thecase when a solar-like star is destroyed by a 106 M

SMBH and the pericenter distance of theprogenitor star was equal to the tidal radius. However, we expect that this self-similar solutionis accurate over a very wide range of parameters (SMBH mass, stellar mass, impact parameter β,etc.) because it depends only on the local velocity and the local dynamical time, and this concern
only for local quantities should be true for most TDEs (i.e., we expect that the specifics of theinitial conditions of the encounter are rapidly forgotten as the stream recedes from the black hole).
Even for encounters in which the star is only partially destroyed or recollapses to form a bound core
(Guillochon & Ramirez-Ruiz, 2013), this self-similar velocity profile should hold for the disrupted
debris and only break down once we get very near to the surviving core.
The analysis performed here was done with the specific application of stars disrupted bysupermassive black holes in mind. However, we note that many of the results we obtained canbe applied to other tidally-interacting systems. In particular, the self-similar velocity profile andthe physical properties of the debris stream (the width and density distributions) described here
are also applicable to compact object mergers (e.g., Lee & Ramirez-Ruiz 2007), the disruptions ofplanets by stars (e.g., Faber et al. 2005), and conceivably the tidal tails generated from galacticinteractions (e.g., Knierman et al. 2012).

Chapter 5Hyperaccretion During Tidal Disruption Events: Weakly Bound Debris Disks
and Jets

5.1

Introduction
Tidal disruption events (TDEs), encounters between a star and a massive (& 105 M ) blackhole in which the star passes within the tidal radius of the hole rt = R∗ (Mh /M∗ )1/3 , where R∗ is thestellar radius, M∗ is its mass, and Mh is the mass of the SMBH, have interested the astronomicalcommunity for decades. Initial studies of TDEs focused on their potential for generating theluminosities observed in active galactic nuclei (Frank & Rees, 1976; Frank, 1978). While this pursuit
fell by the wayside (but see Milosavljević et al. 2006), TDEs continue to be useful for determining thepresence of black holes within galactic centers (Lacy et al., 1982; Rees, 1988). Many investigations,both computational and analytical, have been undertaken over the last forty years to elucidate thedynamics of the interaction between a star and a black hole and the luminosities associated withthe resultant TDE (Carter & Luminet, 1983; Evans & Kochanek, 1989; Loeb & Ulmer, 1997; Kimet al., 1999; Tchekhovskoy et al., 2014; Guillochon & Ramirez-Ruiz, 2013).
The earliest studies of the physics of the disruption noted that, due to the difference inthe gravitational potential across the star, nearly half of the progenitor mass is ejected from thesystem on hyperbolic orbits (Lacy et al., 1982; Carter & Luminet, 1982; Hills, 1988). The otherhalf remains bound to the black hole, with the orbits initially Keplerian (meaning that pressure
forces have not yet altered particle trajectories). After a few revolutions of the innermost material,
which occurs shortly after the disruption, hydrodynamical effects begin to modify the flow. These114
interactions result in the heating of the debris.
One heating agent arises from the pressure distribution of the tidally-disrupted star, whichcauses some of the orbits to possess a substantial inclination angle to the orbital plane of thecenter of mass of the star. When the material on these orbits intersects the mid-plane of the disk,
occurring roughly at periapsis, shock heating will simultaneously increase the internal energy of thegas and damp the inclination angle of the orbit. Because the innermost orbit has specific energy
E ≈ GMh R∗ /rt2 = GMh /2Ri , we see that Ri ≈ (R∗ /2)(Mh /M∗ )2/3 is its semi-major axis. For asolar-mass star and a million-solar-mass hole, the advance of periapsis at this orbit can amount todegrees. Because the innermost debris has the highest velocity within the disk, after one revolution
it will impact the slower-moving, outer material, generating a shock. The shock heating further
increases the thermal energy of the material and circularizes the orbits. Finally, material continues
to fall back at a rate that can greatly exceed the Eddington limit for some time (Evans & Kochanek,
1989; Strubbe & Quataert, 2011). This accretion stage pumps a significant amount of energy into
the flow.
Many authors have modeled the disks of bound material produced by TDEs (e.g., Cannizzo
et al. 1990; Cannizzo 1992) using the standard α parametrization of the viscosity and considering
the disk to be radiatively efficient and geometrically thin (Shakura & Sunyaev, 1973). However, theprocesses outlined in the previous paragraph add a large amount of thermal energy to the debris,which consequently “puffs up” the disk in the vertical direction. The super-Eddington accretion
also means that radiative diffusion is not effective in cooling the disk. We therefore believe that thethin disk model is incapable of describing the bulk properties of the flow during the super-Eddington
phase.
Loeb & Ulmer (1997) invoked the low specific angular momentum of the debris to enable
them to describe the gas as roughly spherical; they then assumed that the black hole accretes at
a rate to match its Eddington luminosity. To make this model self-consistent, they assumed thatthe spherical envelope, with an isentropic equation of state and a steep density profile ρ ∼ r−3 ,surrounded a rotating inner accretion flow, which would have to have a much flatter (and possible

115
even inverted) density profile to obtain the required accretion rate. The boundary between theseregions was assumed to lie at roughly the tidal disruption radius, where the mean specific angularmomentum of the debris is approximately Keplerian. While this model requires substantial angularmomentum transport within the inner accretion flow, it offers no explanation as to why angularmomentum should not be transferred to the outer envelope. If this happened, the boundary between
the two zones would move to smaller radii, requiring the density profile of the inner flow to become
even flatter (or more inverted), while the effect of rotation on the outer envelope would remain
small. It is difficult to see how such a configuration could continue to regulate its accretion rate
to remain at the Eddington limit. On the contrary, it seems likely that the energy generation
rate would become supercritical, with the envelope absorbing the excess energy until it became
unbound.
In an alternate approach, Strubbe & Quataert (2009) forced the material to conform to a“slim disk,” (Abramowicz et al., 1988) and supposed that an outflow carried away unbound debris.
However, a slim disk also necessitates that the rate at which matter reaches the black hole is only
mildly super-Eddington, and therefore a fair amount of mass must be contained at large radii – inthis case, far outside the tidal radius. The assumption of nearly-Keplerian orbits, built into theslim disk model, then implies that the total angular momentum required to support the flow islarger than the angular momentum available.
The past twenty years have seen the emergence of direct observational evidence to support
the existence of TDEs and their associated accretion disks (Piro et al., 1988). ROSAT discovered
the first potential candidates for TDEs in the X-ray (Bade et al., 1996; Komossa & Greiner, 1999).
Despite the fairly small set of statistics, astronomers used the ROSAT data to tentatively validate
the rate of 10−5 events per galaxy per year (Donley et al., 2002). Chandra, GALEX, and XMMNewton surveys have followed up on the events discovered by ROSAT, demonstrating that theluminosity-decay relation scales roughly as t−5/3 , as expected from early theoretical studies of TDEs
(Phinney 1989, but see Lodato et al. (2009) and Guillochon & Ramirez-Ruiz (2013) for arguments
against this scaling; also see section 4); they also found a few new potential candidates (Komossa

116
et al., 2004; Halpern et al., 2004; Gezari et al., 2008). Most recently, a flurry of analyses has
followed the discovery of the γ-ray, X-ray, and radio transient Swift J164449.3+573451 (hereafter
J1644+57), which is popularly believed to be the result of a TDE (Burrows et al., 2011; Bloom
et al., 2011; Cannizzo et al., 2011). These studies, in particular the X-ray and radio observations,
not only demonstrated the existence of a roughly t−5/3 power-law decline of the undoubtedly superEddington luminosity, but also confirmed the novel association of a relativistic jet with the TDE(Zauderer et al., 2011; Tchekhovskoy et al., 2014).
The combination of super-Eddington luminosity and a powerful jet suggests that accretion
onto the black hole is not self-regulated, in contrast to previous models. Here we adopt a different approach to modeling the super-Eddington accretion phase of the geometrically thick disks
produced by TDEs. We assume that the structure of the flow is regulated by its ratio of angularmomentum to mass, which is quite sub-Keplerian between the vicinity of the black hole and thephotospheric radius. Such a flow loses the ability to regulate its accretion luminosity, and absorbs
energy liberated near the black hole until it becomes very weakly bound. Instead of blowing itself
apart, however, we conjecture that these marginally bound envelopes can persist, with the excessaccretion energy emerging as a jet through the narrow rotational funnel. We propose that such amodel is consistent with the existence of a jet in Swift J1644+57; it also may be relevant (withgas self-gravity included) to the formation of gamma-ray burst jets in collapsars and to the rapid
growth of black holes inside quasi-stars.
In section 2 we illustrate the model that describes the fallback disks in super-Eddington TDEs
and promote reasons as to why this model is appropriate. In section 3 we use the results of section
2 to analyze a disk whose parameters (mass, angular momentum, etc.) are those of a typical TDE
and we show that the internal structure of the disk depends only on those bulk parameters. In
section 4 we consider the inner regions of the disk, where general relativity is important, discuss
the properties of the jet, delineate the temporal evolution of the disk properties, and compare ourmodel directly to the case of Swift J1644+57. We conclude and review the results in section 5.

117

5.2

Zero-Bernoulli accretion model
The disk of stellar material created by a super-Eddington TDE should be thick in the sensethat its scale height is some substantial fraction of its radial extent. Narayan & Yi (1994) wereamong the first to discover self-similar solutions for the vertically-averaged density, pressure, and angular momentum in which the velocity distribution was proportional to Keplerian. One interesting
consequence of their models was that the Bernoulli parameter, given by B = Ω2 r2 /2 − GMh /r + H,
where Ω is the angular velocity and H is the enthalpy, was shown to be greater than zero. Because
the Bernoulli function is a measure of the specific energy of the gas, this result implies that any
parcel of gas given an initial kick away from the SMBH would have energy at infinity.
The latter point motivated Blandford & Begelman (1999) to describe a flow consisting ofan advection dominated accretion flow (with Ṁ a function of radius) and a pressure-driven wind,calling these states “adiabatic inflow-outflow solutions” (ADIOS). The inflowing gas maintains anegative Bernoulli parameter by transferring mass, angular momentum, and energy to the wind.
The model, however, requires some unspecified mechanism – presumably some dissipative process– to drive the outflow. Moreover, the inflow zone shares the characteristic of slim disks thatthe specific angular momentum must be very close to Keplerian – more than 87 percent of the
Keplerian value for the case of γ = 4/3. To avoid a highly super-Eddington accretion luminosity,
this angular momentum distribution must extend to radii far beyond the tidal disruption radius.
The gas returning to the vicinity of the black hole following a TDE, however, has too little angularmomentum to permit this.
Thus, the gas distribution during the super-Eddington phase of a TDE is likely to resemble
the quasi-spherical envelope of the Loeb & Ulmer (1997) model, but without the ability to regulate
its accretion luminosity to a value close to the Eddington limit. The shock heating of different
parts of the disk and the energy input of the accreting black hole both raise the internal energy ofthe material, with turbulence, convection and internal shocks distributing that energy fairly evenly
throughout the disk. Eventually there will come a point in time where the Bernoulli parameter

118
approaches zero, leaving a marginally-bound, highly-inflated envelope.
Any further augmentation of the energy would start to unbind material. The question iswhether this unbound material is launched from a wide range of radii or from close to the blackhole, where the energy is injected. In the case of an ADIOS, the large angular momentum containedin the flow allows the system to maintain a disklike geometry, with a large “free” surface along
which a wind can develop. But in the present case, for which B approaches zero, the disk closes
up to a vanishingly narrow funnel, leaving the outer, quasi-spherical surface as the only plausible
location for the development of a wide-angle wind.
However, the injection of energy from the accreting black hole occurs deep in the interior of
the envelope, where timescales are much shorter than those throughout the bulk of the flow. The
accretion energy, pumped into the gas at a rate that is highly supercritical, is thus unlikely to beable to be efficiently advected to the outer regions where a wind could regulate the super-Eddington
luminosity. The only viable exhaust route left for the excess energy is then along the poles, where
the surface of the inflated envelope closes. We thus propose that, at this point in the evolution of
the fallback disk, where the accretion luminosity augments the binding energy of the envelope tothe point where a wind would develop if there were a free surface, a jet carries away the excessenergy.
In the situations we are considering, the mass of the black hole dominates the total massof the system. We can therefore approximate the gravitational potential by φ = −GMh /r, where
Mh is the mass of the SMBH and r is the radial distance from the hole (we are neglecting any
contribution from post-Newtonian gravity; see section 4.1 for a discussion of relativistic effects). In
spherical coordinates with this potential and the Bernoulli parameter equal to zero, the momentum
equations and the Bernoulli equation are, respectively,

1 ∂p
GMh `2 csc2 θ=− 2 +
,
ρ ∂r
r
r3

(5.1)

119

−

1 ∂p
`2 cot θ csc2 θ
,
=
ρ ∂θ
r2

(5.2)

GMh `2 csc2 θ
γ p
+
+
= 0,
r
2r2
γ−1ρ

(5.3)

where ` is the specific angular momentum of the gas. In the final line we used a specific form
for the enthalpy and assumed that the azimuthal velocity is much greater than the poloidal orradial velocity. Here γ is the adiabatic index of the gas, generally between 4/3 and 5/3 dependingon the relative contributions from radiation pressure and gas pressure. For most of what follows
we will assume that γ ≈ 4/3, as radiation pressure dominates the support of TDE debris against
gravity during the super-Eddington phase. This fluid description is appropriate to a ZEro-BeRnoulli
Accretion (ZEBRA) flow.

5.2.1

Gyrentropic flow

In Blandford & Begelman (2004), the authors described ADIOS disks as marginally stable
to the Høiland criteria. This assumption, verified numerically (Stone et al. 1999), demandedthat the surfaces of constant Bernoulli parameter, angular momentum, and entropy all coincide;these surfaces are termed gyrentropes. While the Høiland criteria determine a disks’s stability toconvection in the absence of magnetic fields, even a vanishingly small poloidal field can completely
destabilize a differentially rotating disk that is stable to those criteria (Balbus & Hawley, 1991,
1992; Stone & Norman, 1994). We will now show, however, that the zero-Bernoulli assumption
ensures the gyrentropicity of the flow, even in the presence of the magnetorotational instability(MRI).
One can show that ∇B = H∇ ln S + Ω∇`, where H is the enthalpy, S is the entropy, and Ω isthe angular velocity of the gas (Blandford & Begelman, 2004). Thus, since B ≈ 0, ∇S ∝ −∇`. This
relationship implies that surfaces of constant S are also those of constant `, which must then be
surfaces of constant B. This type of disk is therefore also gyrentropic, the constancy of B being the120
only assumption which led to that conclusion. Thus, while MRI may invalidate the assumption of
gyrentropicity on the grounds of the Høiland criteria, a constant-Bernoulli disk retains gyrentropic
flow (assuming that the magnetic energy density is not large enough to substantially alter the
dynamical equilibrium).

5.2.2

Self-similar solutions

From an analysis of equations (5.1), (5.2), and (5.3), one can show that the general solutionof `(r, θ) could have any functional form that depends on r and θ only through the combinationr sin2 θ (see Appendix A, notably equation (A.10)). When the envelope subtends a large range inradii, however, we expect the solution to have a roughly self-similar structure between the innerand outer boundaries of the disk. Blandford & Begelman (2004) derive the gyrentropic solutions
for arbitrary Bernoulli parameter B(θ)/r; the ZEBRA solutions are the special case with B = 0.
We will simply quote their findings here, and adapt our notation to one which is consistent with
theirs. For the density, pressure, and specific angular momentum (squared), respectively, we findρ(r, θ) = ρ0rr0

GMh ρ0p(r, θ) = β
r

−q



r
r0

(sin2 θ)α ,−q(sin2 θ)α ,

(5.4)

(5.5)

` 2 (r, θ) = aGMh r sin2 θ,(5.6)

q ≡ 3/2 − n,

(5.7)

where

α=

1 − q(γ − 1)
,
γ−1

(5.8)

121

β=

γ−1
,
1+γ − q(γ − 1)

(5.9)

a=2

1 − q(γ − 1),
1 + γ − q(γ − 1)

(5.10)

r0 is some characteristic inner radius and ρ0 is the density at that radius (and at the disk midplane).
The parameter n is defined by Blandford and Begelman so that the accretion rate is proportional
to rn ; mass-conserving accretion has n = 0.
One interesting aspect of these solutions is that n, and therefore q, which describes how
steeply the density and pressure fall off as functions of r, is not specified a priori, which introducesanother degree of freedom into the models. In general, however, we require that the exponent ofsin2 θ remain positive, ensuring that the density and pressure do not go to infinity at the poles.
We also expect that the energy produced in the disk should be a decreasing function of radius.
From the energy equation, we know that the luminosity is given by L ∼ Ṁ vr2 , and assuming thatthe power is produced by gas in regions with velocity appropriate to that for free-fall, we find thatL ∝ rn−1 . These two restrictions then impose that 3/2 − 1/(γ − 1) < n < 1, which translates to
1/2 < q < 1/(γ − 1). We see that, since the exponent of sin2 θ is always greater than or equal
to zero in our self-similar expressions, the density goes to zero only exactly at the poles. These
solutions thus represent quasi-spherical envelopes. The angular momentum distribution is modified
from that of Keplerian by the factor a, which is always less than unity for permissible values of n.
Blandford & Begelman (2004) noted the additional degree of freedom contained in theirsolutions. They then went on to describe the physical scenarios appropriate to different values ofn. In particular, different n give rise to larger or lesser amounts of outflow, accretion rates, energy
generation rates, etc. For our present considerations, however, a wind is unnecessary. The questionof the value of n therefore merits some careful consideration. In the next section we will see how theproperties of the disrupted star and the black hole in a TDE determine this as-yet-undeterminedparameter in our analysis. Interestingly, the ZEBRA models admit a wider range of n-values than

122
the range (0 < n < 1) consistent with ADIOS models.

5.3

ZEBRA models of TDE debris disks
The structure and evolution of a ZEBRA model for a TDE are governed by the total mass andangular momentum of the envelope, which change as matter falls back and is accreted or expelledRin a jet. The total angular momentum and mass of the fallback disk are, respectively, L = ` ρ dV
R
and M = ρ dV , where dV is an infinitesimal volume element and the integral is taken over thewhole fluid. Using the formalism and notation of the previous section, these can be written√ZZ
4πρ0 aGMh π/2 R −q+5/2
L =r(sin2 θ)α+1 dr dθ,
r0−q
0
r0
4πρ0
M = −qr0

Z π/2 Z R
0

r−q+2 (sin2 θ)α+1/2 dr dθ,(5.11)(5.12)

r0

where r0 is simultaneously the radius at which we specify the density and the inner radius of thedisk, and R denotes the outer radius. Due to the influence of the black hole, we expect r0 to beon the order of the Schwarzschild radius (or, more precisely, the location of the innermost stable
circular orbit (Bardeen et al., 1972)), and so we will write r0 = χ 2GMh /c2 , with χ a pure numberof order a few. To determine the outer radius, we compare the ability of the disk to transportenergy via advection to its ability to transport energy via radiative diffusion.
Although the photosphere of the envelope may be radiating at close to the Eddington limit,
the amount of energy generated in the interior of the disk will generally be much greater than thatable to be carried via diffusion; specifically, the luminosity carried into the polar regions exceeds
the Eddington limit by a factor of order ln(R/r0 ) (Jaroszynski et al., 1980; Paczyńsky & Wiita,
1980; Sikora, 1981). The dominant mode of energy transport will therefore be turbulent advection.
The advective flux can be written Fa = ypv, where p is the pressure, v is the local sound speed, andy is a number less than or of order one that describes the efficiency of advection (since we are reallyconcerned with the flux of enthalpy, which is 4p for a radiation-dominated gas, y could conceivably

123
be greater than 1). Since p ∼ ρv 2 and the advective luminosity is La ∼ 4πr2 Fa , we have thatLa ∼ 4πyr2 ρv 3 . When the saturated advective luminosity becomes roughly equal to the Eddington
limit, radiative diffusion will become the dominant mode of energy transport, allowing the disk tocool and become thin. Symbolically we have 4πymax r2 ρv 3 ∼ 4πGcMh /κ, where κ is the relevant
opacity. In this case we will use the opacity for electron scattering, given by κ ≈ 0.34 cm2 /g forcosmological abundances. This definition is equivalent to that which defines the trapping radius– the point in the flow at which the diffusion timescale equals the advective timescale (Begelman,
1978). Fluid interior to this radius entrains photons, rendering them incapable of escaping.
In addition to having a magnitude, the advective flux has a directionality. Writing Fa = Fa n̂,
the advective luminosity is obtained by integrating the dot-product of this vector over an area.
Because we are concerned with the energy escaping from the hole, the relevant area is the twosphere, and hence the only component of the flux relevant to the luminosity is that in the r̂-direction.
The quantity n̂ · r̂ will, in general, depend on θ, and in fact we expect it to be less than one asmuch of the flux is transported into the polar regions. Because we are unaware of the specificsof the directional dependence of the flux, we will simply incorporate those uncertainties into ourRRefficiency factor y, letting y p v n̂ · r̂dS ≡ ȳ p v dS, where S is the two-sphere and ȳ is an effective
efficiency. Performing the integrations, we find that the outer radius is given by2c Γ(α + 3/2)
r−q
√0
R −q+1/2 = √.
κ π Γ(α + 1) ρ0 yβ aGMh(5.13)The Γ-functions resulted from our integration of the angular dependence of p over the two-sphere,√
and for simplicity we replaced ȳ with y. Numerically we find that 1 < (2/ π)(Γ(α+3/2)/Γ(α+1)) <2 over permissible values of α, so that its inclusion in our expression does not significantly alter our
results. We will include the Γ-functions here, however, because they will simplify (visually) someof the relationships we will describe in later sections. To offer some insight into the meaning of the124previous expression, note that it may be writtenR'(1v0 q−1/2
r0 ,
)
c/τ0

(5.14)

where τ0 is the optical depth and v0 is the local Keplerian velocity, both evaluated at r0 . From
this form of the equation, it is evident that photons must be trapped at r0 , namely the inequalityv0 > c/τ0 must hold, to ensure that our assumption about the radiative inefficiency of the flow be
upheld.
Recall that 1/2 < q < 1/(γ − 1), a restriction that resulted from requiring the density to
be finite at all angles and the energy generation rate to increase inwards. The adiabatic index of
the gas will generally be between 4/3, and 5/3, meaning that 1/(γ − 1) < 3, and consequently
1/2 < q < 3. Returning to equations (5.11) and (5.12), we see that this range of q will always leave
the lower bound on the radial integration, namely r0 , relatively unimportant (unless q is exactly 3,a case that we will have to consider separately) if R  r0 , an assumption that we can check. With
these considerations, we find for the total angular momentum and mass√
2π 3/2 ρ0 aGM Γ(α + 3/2) R −q+7/2
L =,Γ(α + 2) −q + 7/2
r0−q
2π 3/2 ρ0 Γ(α + 1) R −q+3.
r0−q Γ(α + 3/2) −q + 3

M =(5.15)(5.16)

Solving for the radius of the disk in terms of the mass of the disk and the mass of the black hole,
we find

R=



2/5
√
yκβ a(3 − q) p
M GMh
4πc

14

' 9 × 10

MM

2/5 

Mh
106 M(5.17)

1/5
cm.

(5.18)

This relation yields R ≈ 103 rs , rs being the Schwarzschild radius of the black hole, for Mh = 106 M
and M = 1M . Because we expect that r0 ≈ f ew × rs , we see that neglecting the lower bound in

125
the integrations of M and L was justified.
Our goal is to use the total mass and angular momentum, calculable from initial conditions,to determine q. This value will then inform us of how a larger progenitor star, a larger black hole,or more angular momentum will influence how steeply the density or pressure falls off with distancefrom the hole. By performing a bit of algebra, we can rearrange equations (5.13), (5.15), and (5.16)
to yieldf (M , L , Mh ) ≡



yκ
4πc

1/6

√
Γ(α + 1)5/6 Γ(α + 2)5/6 (7/2 − q)5/6
M GMh
=
.
3−q
L 5/6
β 1/6 a1/2 Γ(α + 3/2)5/3(5.19)The left-hand side of this expression, denoted f (M , L , Mh ), depends only on the total mass ofthe disk, the total angular momentum of the disk, and the black hole mass (in addition to a few
physical constants; note that its dependence on y, the parameter we introduced to describe theefficiency of convection, is to the 1/6th power, and therefore only affects our answers very weakly).
The right-hand side, on the other hand, is only a function of q, which we could in principle invert toisolate q itself. The gross properties of the progenitor star and the black hole therefore determine
the density, pressure, and angular momentum profiles of the fallback disks associated with superEddington TDEs.
In order to calculate q for a given TDE, we need to parametrize the total mass and angularmomentum in terms of those values appropriate to a certain event, both of which will depend on
the progenitor star. In order to be tidally disrupted, the star must pass within the tidal radiusrt ≈ R∗ (Mh /M∗ )1/3 of the black hole, where R∗ is the stellar radius and M∗ is its mass (the precise
point of disruption clearly depends on the details of the stellar composition, rotation, and othercomplications, but numerical results indicate that the true location does not vary from that given
by more than a factor of ∼ 1.5 for realistic interiors; Ivanov & Novikov 2001). Due to the tidal
force on the star and the tidal potential, nearly half of the stellar debris is ejected from the blackhole on hyperbolic orbits (Lacy et al., 1982). The other half remains bound to the SMBH. The
initial mass of the disk should therefore be on the order of M ≈ M∗ /2, though the actual amount

126
should be slightly less than this when we account for material that has already been accreted andthe still-raining-down debris outside R (see section 4). At the tidal radius, conservation of energypdictates that the star has a velocity of v∗ = 2GMh /rt , and hence the disk material has a totalpangular momentum of L ≈ M∗ GMh R∗ /2(Mh /M∗ )1/6 (again, this is a slight overestimate). By
parametrizing the mass and angular momentum as such, equation (5.19) becomes
11/36

5y

1/6

M∗

1/18

M6

=
5/12

R∗

Γ(α + 1)5/6 Γ(α + 2)5/6 (7/2 − q)5/6.
3−q
β 1/6 a1/2 Γ(α + 3/2)5/3(5.20)

Here M∗ is the progenitor’s mass in units of solar masses, R∗ is its radius in units of solar radii,
and M6 is the black hole mass in units of 106 M . Interestingly, the left-hand side is virtuallyindependent of the black hole mass, meaning that the density and pressure distributions of TDE
fallback disks are almost exclusively determined by the progenitor star.
Figure 5.1 illustrates the value of q obtained as we vary the function f (M , L , Mh ). As we
increase f (M , L , Mh ), the value of q approaches q → 1/(γ − 1). By analyzing equation (7.15),
we can show that α = 0 for this value of q; recalling that ρ ∝ (sin2 θ)α , we see that the flow isspherically symmetric. This result makes sense when we realize that, in order for f (M , L , Mh ) toapproach large values, the angular momentum must be very small.
The value of q rapidly decreases as we decrease f . Recalling our lower limit on q, namelythat q > 1/2, we see that there is a lower limit on the value of the left-hand side of (5.19) which
is, after further investigation of Figure 5.1, nearly independent of the adiabatic index. Numerically
we find

y

1/6 M

√

GM

L 5/6

&





179 for γ = 5/3,(5.21)

163 for γ = 4/3

where we have used the opacity for Thomson scattering and the units are cgs. If one violates theselower bounds, our model ceases to describe the disk adequately. We can show that, if the inequality
is not satisfied, then R < Rc , where Rc = L 2 /(GMh M 2 ) is the circularization radius, which is

127

2.5
2.0

q

1.5
1.0
Γ=43
0.5
0.0

Γ=53

-0.5
2

4

6

8

10

f HM, L, Mh L
Figure 5.1: The value of q obtained for γ = 4/3 (blue, solid) and γ = 5/3 (red, dashed) as we vary
the left-hand side of equation (5.19).

128
obtained by balancing gravity and the centrifugal force. When f (M , L , Mh ), defined in equation(5.19), falls below this critical value, pressure forces play a negligible role in the dynamics of thesystem, and the disk becomes thin. It is therefore no surprise that our model breaks down in thislimit.
The total mass and angular momentum of the disk thus determine the large scale propertiesof the envelope, which in turn determine the density and pressure profiles. Extrapolating these
profiles to the region of the disk near the black hole, we can estimate conditions in the vicinityof the innermost stable circular orbit, which then quantify the accretion rate and rate of energy
generation. Because any further absorption of energy would lead to a positive Bernoulli parameterand unbind the envelope (see Appendix B for notions concerning non-zero Bernoulli parameter),
we conjecture that the accretion energy must escape through the funnel in the form of a fast jet.
We would like to be able to say something about the properties of this jet. Also, the accretionof the black hole and the continual fallback of material outside the envelope (at a rate roughly proportional to t−5/3 for later times) are changing the mass and angular momentum of the system; thevalues of q resulting from this section should therefore be interpreted as initial, or bulk, parameters.
By modeling the mass and angular momentum of the disk in a time-dependent manner, we will beable to gain some insight into possible observational diagnostics one could use to infer the presenceof a ZEBRA flow.

5.4

Jet properties and temporal evolution

5.4.1

Inner regions of the accretion disk

In the above analysis we assumed that ignoring the inner regions of the accretion disk, wheregeneral relativistic effects become important, was permissible. In those regions, however, we knowthat the angular momentum must exceed its Keplerian value, i.e., that with φ = −GMh /r, toaccount for the stronger gravitational acceleration. This excess of angular momentum at smaller
radii and its interplay with the pressure gradient could, in principle, significantly alter the flow at

129larger radii and change our results.
Models that investigated the inner regions of thick disks around black holes, termed “Polishdoughnuts”, were developed in the late 1970’s and early 80’s, and research along these lines continues to the present day (Abramowicz et al., 1978; Kozlowski et al., 1978; Paczyńsky & Wiita,
1980; Jaroszynski et al., 1980; Paczynski & Abramowicz, 1982; Komissarov, 2006; Qian et al., 2009;
Abramowicz & Fragile, 2013; Pugliese et al., 2013). In these models, authors assume ad hoc forms
for both the specific entropy and angular momentum, and from these functional forms one may
infer the pressure and density from the relativistic energy and momentum conservation equations.
The portion of the ZEBRA envelope closest to the black hole should in many respects resemble
a Polish doughnut, with the extra constraint that the flow has zero Bernoulli function. Insteadof pursuing the lines followed by many authors in examining the consequences of the relativistic
conservation equations, we will follow a slightly different route which incorporates our model.
To analyze the specifics of the flow near the black hole and its impact on the outer regionsof the envelope, we will restrict our attention to the case where the space-time metric is that of
Schwarzschild. With this assumption, we then replace the standard point-mass potential withthe “pseudo-Newtonian” potential of Paczyńsky & Wiita (1980), so φ → −GMh /(r − rs ), wherers = 2GMh /c2 is the Schwarzschild radius. While this potential tends to produce inaccurate
numbers for some quantities (Tejeda & Rosswog, 2013), its prediction of the innermost stable
circular orbit and the marginally bound orbit suffice for our treatment.
By manipulating the momentum and Bernoulli equations with this potential, we can showthat the most general form of the angular momentum must satisfy `2 (r, θ) = `2 (φ r2 sin2 θ), i.e.,
the angular momentum is only a function of the combination φ r2 sin2 θ (note that this result is
consistent with equation (A.10) in which a specific basis set for the functions is used). In the selfsimilar limit, we showed that `2 = aφ r2 sin2 θ. However, in addition to approaching the self-similarvalue in the r → ∞ limit, the angular momentum must also match that of the psuedo-Newtonian
distribution (that with the Paczyński-Wiita potential but without a pressure gradient) at someinner radius where the pressure gradient goes to zero. Because the self-similar solution will not130necessarily satisfy the second condition, we must search for non-self-similar distributions. The
specific form we will adopt is `2 = A + D GMh r2 /(r − rs ), where A and D are constants and we are
restricting our attention to the equatorial plane. In general the angular momentum distribution
could be more complicated. However, it must monotonically increase with radius throughout – adecrease in the specific angular momentum with radius is highly unstable to convection (Goldreich& Schubert, 1967; Seguin, 1975), unless it is accompanied by a strong increase in entropy, which isunlikely. It must also approach the self-similar solution in the asymptotic limit. The previous form
is the simplest that satisfies both of these criteria.
Requiring that the angular momentum approach its self-similar value for large r yields D = a,where a is given by equation (5.10). With our specific form for the angular momentum, we canmanipulate the momentum equations to find exact expressions for both the density and the pressure.
Setting the pressure gradient equal to zero at some radius rm where `2 (rm ) = `2P N (rm ), where `2P N =2GMh r(r/(r − rs ))2 is the pseudo-Newtonian angular momentum, yields A = 8GMh rs (1 − a/2)
and rm = 2 rs , which is the marginally bound orbit. Our solution for the self-consistent angularmomentum is thusar2
` = GMh 4rs (2 − a) +.
r − rs2(5.22)

Figure 5.2 illustrates three different angular momentum distributions for a given set of parameters: that given in equation (5.22), the self-similar solution, and the pseudo-Newtonian distribution. As we can see, the self-consistent model flattens out in the inner region to exceed both thepseudo-Newtonian distribution and the self-similar angular momentum.
It may seem like this excess of specific angular momentum could alter significantly our estimates of q, R, and other properties of the envelope. However, the two conserved quantities
in a tidal disruption event are the total angular momentum and total mass, and it is not clear
how much these differ from those in the self-similar limit. Therefore, to answer whether or not
this modified potential truly affects our results, we must also determine how the density varies in

131

{SC
8
{SS
6
{

{PN

4

2

0

10

20

30

40

50

r
rs
Figure 5.2: The self-consistent model (equation (5.22)) for the angular momentum (blue, solid), theself-similar model (red, dashed), and the psuedo-Newtonian distribution (black, dotted). Here we’ve
set q = 1.5, γ =√4/3, the abscissa is in units of Schwarzschild radii, and the angular momentum isnormalized by GMh rs .

132
Rr
the self-consistent limit. After evaluating the density, we can form the integrals
4πr2 ρ dr and
Rr
4πr2 ` ρ dr to obtain the enclosed mass and angular momentum as functions of r, respectively(the lower bound will not significantly affect the result in either case). By comparing these functions to their analogs in the self-similar limit, we can assess how significantly the pseudo-Newtonian
potential affects our conclusions.
Figure 5.3 displays the density for both the self-consistent (using equation (5.22) for the angular momentum) and self-similar solutions. At small radii strong gravity reduces the density fromthat in the self-similar limit, while at large radii they are indistinguishable. Figure 5.4 illustrates
the total mass contained within r for the self-similar and non-self-similar models, while Figure 5.5
shows the total angular momentum contained within r for both models. The self-consistent model
predicts an increased amount of mass and angular momentum at larger radii, which is due to aslight increase in its density, relative to the self-similar solution, at intermediate radii. This meansthat the use of the Pascyński-Wiita potential makes a more compact ZEBRA (in other words, wewould enclose the same amount of mass and angular momentum, fixed by the TDE, at a smaller
radius). However, we do not believe that this alteration will change our results much, as the physics
is largely dictated by the ratio of the total angular momentum to mass, which is conserved fromthe TDE. Thus, while relativity can alter significantly the behavior of the density, pressure, andangular momentum at small radii, its effects on the bulk properties of the ZEBRA are minimal.
We therefore expect that its inclusion in our models will not significantly change the results.
If the Bernoulli parameter were very small (compared to GMh /r) all the way to the blackhole, the gas would release little energy in the form of a jet. However, we expect that this is notrealistic, and that fluctuations in the inner part of the flow will lead to the root-mean-squared valueof the binding energy being some significant fraction of that for the innermost stable circular orbit(ISCO) (= 6GMh /c2 for a Schwarzschild black hole). To quantify this statement, recall that, for a

133

0.005
Self-consistent

0.004

Self-similar

Ρ

0.003

0.002

0.001

0.000
10

20

30

40

50

r
rs
Figure 5.3: Solutions for the density in the non-self-similar (blue, solid) and self-similar (red,dashed) limits. Here q = 1.5, γ = 4/3, the x-axis is in units of Schwarzschild radii, and the density
is measured in units such that ρ0 (rs /r0 )−q = 1, i.e., the red, dashed curve is simply (r/rs )−2.2 .

134

8000

Self-consistent

6000
MHrL

Self-similar

4000

2000

0
0

100

200

300

400

500

r
rs
Figure 5.4: The mass contained within r for the angular momentum given in equation (5.22) (blue,
solid) and that for the self-similar model (red, dashed). The parameters are the same as those in
Figure 5.3, with the same normalization for the density.

135

120 000
Self-consistent

100 000
80 000
LHrL

Self-similar
60 00040 000
20 000
0
0

100

200

300

400

500

r
rs
Figure 5.5: The angular momentum contained within r for the self-consistent model (blue, solid)
and the self-similar solution (red, dashed). The parameters and normalization are the same as
those in Figure 5.3.

136
non-steady flow, an additional term must be added to the Bernoulli function:

Z
B→B+

∂v
· dr,
∂t

(5.23)

where the integral is taken over the flow line of the fluid element. In terms of scalings, this term ison the order

Z

∂v
∆v r
· dr ∼.
∂tτd(5.24)

Here ∆v is the change in the velocity over the dynamical timescale, τd , and r is the radius at which
we are considering the (predominantly circular) flow line. Using τd ∼ r/v and ∆v ∼ v, we find∆v r/τd ∼ v 2 (there is also a term, of order ∼ v 2 , that arises from the inclusion of the v · ∇v
term in the momentum equation; however, because it is of the same order as the time derivative, itsuffices to consider only this term). At the ISCO and regions interior to that radius, this fluctuation
term can reach substantial fractions of c2 . Because the change in velocity can be either positive or
negative, the average will cancel out over the whole flow. The root-mean-squared of the additionalterm will not cancel, however, and will lead to fluctuations that lead to both positive and negative
Bernoulli parameter. The fluctuations to the negative side give rise to bound flows, which thenrelease that energy in the form of a very energetic jet.
As in the case of ADIOS models (Blandford & Begelman, 2004), the mechanism responsible
for this energy dissipation is not specified — its existence and nature will have to be verified later.
Once this energy is injected, however, its escape will most likely be in the form of a jet due tothe rotational funnel. Inside the radius at which the material becomes bound, our previous model
breaks down. Because this region constitutes a minute fraction of the disk, we do not expect thatits presence will have much of an impact on our results.

137
5.4.2

Accretion rate and jet power

We derived the density, pressure, and angular momentum distributions for ZEBRA envelopes
under the assumption that the poloidal and radial velocities were significantly less than that in theazimuthal direction. While this proposition is upheld in the bulk of the flow, it must break down
in regions near the SMBH, specifically in those regions where relativity prevents the existence ofstable circular orbits. It follows that the radial velocity in this region should be appropriate to thatpof gravitational free-fall, or vr ∼ GMh /r.
To make contact with the self-similar region of the flow, we will take r0 to be the radius atpwhich gravitational infall becomes substantial, and we have vr0 = δ GMh /r0 , where δ is a number
less than one. With this assumption, the mass accretion rate onto the black hole should be on theorder of that in the spherically symmetric regime, or Ṁacc = 4πr02 ρvr0 . Plugging in our expressions
for relevant quantities, we findṀacc = 4πδρ0p3/2
GMh r0 .

(5.25)

We can solve for ρ0 in terms of the inner radius, the total mass of the disk and the mass of theblack hole. Doing so and putting the expression into the equation for Ṁacc yields

2




− (3−q)pp52GMh 3/2−q
yσT
M GMh
×
M GMh
h(q),
2
c
4πcmp

(5.26) 2√ − 5 (3−q)
2 Γ(α + 3/2)(3 − q)h(q) ≡ √,(3 − q)β a
Γ(α + 1)π(5.27)

3/2−qṀacc = δ χ
wherewhich is a function only of q and γ. We have also parametrized the inner radius in terms of the
Schwarzschild radius, viz. r0 = χ 2GMh /c2 .
The jet luminosity is given by Lj = Ṁacc c2 , where  is the accretion efficiency of the black
hole. To arrive at this result it was necessary to introduce a number of factors that relate to our138
uncertainty of the details of the flow, namely , the radiative efficiency, χ, the inner edge of thedisk, δ, the fraction of free-fall of the velocity, and y, the advective efficiency. However, we expectthat , δ, and y are somewhere in the range of 0.01 − 1.0, and we know that χ should be on theorder of a few (strictly, this value and the radiative efficiency depend on the spin of the hole and its
orientation relative to the disk). Therefore, although we have a number of unknowns, their range
in parameter space is rather small. Explicitly we find for the jet power2− (3−q)pp52GMh 3/2−q
σT
Lj = µ M c GMh
M GMh
h(q),
×
2
c
4πcmp
2

(5.28)

where we set µ ≡  δ χ3/2−q y −2(3−q)/5 for compactness. For a solar progenitor, a million-solar-mass
black hole, y = 0.5, δ = 0.05, and  = 0.1, we find Lj ≈ 5 × 1047 erg s−1 ≈ 4 × 103 LEdd for the jetluminosity, where LEdd = 4πGcMh mp /σT is the Eddington luminosity of the black hole assuming
ionized hydrogen (here we have solved equation (5.20) to determine the value of q, which, for thesenumbers, is q ≈ 2.4).

5.4.3

Time-dependent analysis

In order to be tidally disrupted, the stellar progenitor must pass within a pericenter distanceof rp = x rt , where rt = R∗ (Mh /M∗ )1/3 is the tidal radius and x is a number that is less than or
about one. Here we will restrict our attention to the case where x = 1. Our motivation for doing
so is that other authors have shown, using hydrodynamical simulations, that the complexities ofthe encounter for smaller and larger x render an analytical treatment insufficient for describing thephysics of the TDE (Guillochon & Ramirez-Ruiz, 2013). Because running a numerical simulation
to determine the exact feeding rate to the ZEBRA is outside the scope of this paper, we will only
consider those disruptions which occur exactly at the tidal disruption radius and maintain that theanalytical approach is accurate enough for our purposes.
After the star is disrupted, the most tightly bound material is placed on an orbit with semimajor axis Ri = (R∗ /2)(Mh /M∗ )2/3 (see section 1 for a derivation). Because the point of disruption
occurs at the tidal disruption radius, the eccentricity of this orbit is very large. Other less-bound139
gas parcels (those with larger semi-major axes) are thus on nearly-parabolic orbits as they recedefrom the hole. The initial configuration of the tidally-stripped material that is going to fall backis therefore a thin, highly-elliptical disk, confined roughly to the plane occupied by the disrupted
star.
When the innermost gas undergoes one complete orbit, shock heating and other effects (seeIntroduction) begin to circularize the orbits and alter the structure of the debris disk. After acertain amount of time has passed, on the order of a few orbits of the innermost material, theheating causes the disk to puff up into a spheroid of radius R0 – this is the ZEBRA. In the previous
sections we assumed, for simplicity, that the entire mass that is bound to the black hole (nearly
half the stellar progenitor) comprised the ZEBRA. However, because tidally-stripped material iscontinually falling back onto the accretion region, it is not clear that this assumption is valid.
To determine how much mass is contained in the initial ZEBRA and the rate at which material
is falling back onto the accretion region, we will pursue a line of analysis similar to that in Lodato
et al. (2009), and consider the star at the time of disruption. At this point in time, the center ofmass is at the tidal radius, and, assuming the star is on a parabolic orbit, the binding energy of thecenter of mass is zero. Denote the position of a gas parcel contained in the star by rg = rt − ηR∗ ;η = 1 is the edge of the star closest to the hole, η = −1 is that farthest from the hole, and we are
restricting our attention to the plane of the orbit. The specific gravitational energy of a gas parcel
is then given by

p =

GMh
GMh
−
rt
rt − ηR∗(5.29)

GMh R∗
η,rt2(5.30)'−

where in the final line we approximated the tidal radius as being much greater than the stellar
radius, valid for the supermassive black holes we are considering. After disruption, the gas parcels
fly apart, cooling adiabatically and occupying roughly Keplerian orbits. The energy-period relation

140
for Keplerian orbits yields the semi-major axes of these orbits:

R∗
Rp =
2η



Mh
M∗

√

2/3
=

GMh
2π

2/3

t2/3 ,(5.31)

where t is the fallback time. Note that, by inserting η = 1, this expression reproduces the correct
orbit for the most tightly bound debris.
The rate at which material returns to pericenter is found by using the chain rule, specificallydM/dt = (dM/dη)(dη/dt). From equation (5.31), we can readily determine dη/dt. To calculate
dM/dη, we will assume that the stellar progenitor is well-approximated by a polytropic equationof state; a number of authors have shown that the equation of state of the star has importantconsequences for the rate of return of material, and so it is not adequate simply to consider aconstant-density profile (Lodato et al., 2009; MacLeod et al., 2012; Bogdanović et al., 2014; Guillochon & Ramirez-Ruiz, 2013). In this case, ρ(R) = λ θ 1/(γ∗ −1) , where λ is the density at the centerof the star, γ∗ is the polytropic index of the gas that comprises the star, and θ(R) is the solution to the Lane-Emden equation. R is spherical distance measured from the center of the stellar
object. We will parametrize the location of a gas parcel within the star in terms of the variables
R, r and z, where r is the distance from the center of the star in the plane of the orbit and z is
the distance perpendicular from the plane of the orbit (see Figure 5.6 for clarification). Using thefact that dM = ρ dV , where dV = 2πz dz dr is the volume element, and making simple geometricsubstitutions, we can show that

dM
= 2π
dη

Z R∗
ρ R dR(5.32)

R ξ1
1
M∗ ξ1 ηξ1 θ γ∗ −1 ξ dξ=,
2 R ξ1 θ γ∗1−1 ξ 2 dξ

(5.33)

ηR∗

0

where ξ is the dimensionless radius defined through the Lane-Emden equation and ξ1 is the firstroot of θ(ξ) (see, e.g., Hansen et al. 2004). Putting everything together, we obtain for the mass

141

R*

R

z

To Holer

Figure 5.6: A schematic of the star at the time of disruption to illustrate the geometry. Note thatr = ηR∗ .

142
rate of returndM
M∗ R∗ ξ1
=
dt
6



2πMh
√
M∗ GMh

2/3

R ξ1

1
γ∗ −1

ξ dξ
ηξ θ
= Ṁf b .
t−5/3 × R 1 1
ξ1 γ −1 2
∗
θ
ξ
dξ
0

(5.34)

This expression gives the rate at which material is drained from the tidally-disrupted debris cloud
and added to the ZEBRA. Note that equation (5.34) only holds for η < 1, or for times t > tr , where


tr =

R∗
2

3/2

2πMh
√
M∗ GMh

(5.35)

is the time taken for the innermost material to undergo one complete orbit. Noting that the original
mass contained in the bound tidally-disrupted material is roughly M∗ /2, we can write an expression
for the remaining mass that is still raining down onto the ZEBRA after a time t:

M∗−
Mf b (t) =
2

Z t

dM 0
dt ,
0
tr dt

(5.36)

where dM/dt0 is given by expression (5.34) with t → t0 .
The jet and black hole are also extracting angular momentum from the disk. However, it isnecessary for the disk material to transport a large amount of its angular momentum outwards,
via viscous or magnetic processes (neither of which we have attempted to include in this model) inorder to be accreted by the black hole. The angular momentum of the disk material is thus nearly
unaffected by the presence of the hole. However, as we noted previously, natal stellar material
is still falling back onto the envelope. As we have assumed that hydrodynamic effects have only
influenced the particles in the region of the ZEBRA envelope, this material still approximately
retains its specific angular momentum from the time of disruption, adding this angular momentum
to the disk as it falls back. Using equation (5.36) , we find for the total angular momentum as afunction of time


 

p
Mh 1/6 M∗
L = 2GMh R∗− Mf b (t)M∗2(5.37)

143
After the ZEBRA has inflated to a radius R0 , the envelope will not only lose mass to theblack hole at the rate described by equation (5.26), but it will also gain mass at the expense of thetidally-stripped debris that is still falling back. We can solve for M in terms of other quantities
by rearranging equation (5.19), and our differential equation for q is then M˙ = dM/dt − Ṁacc ,where dM/dt is given by equation (5.34) and Ṁacc by equation (5.26). To solve this differential
equation, we need an initial value for q. This initial condition can be determined by assuming that
the ZEBRA takes some time to inflate, at which point it has some mass and angular momentum,
which in turn yield an initial q. However, the time to inflate depends sensitively on the heating
rates and other physical processes, for which we do not have a reliable model. We also expect thatany knowledge of the initial conditions should be lost after a certain amount of time, and thatthey should only reflect a transient initial behavior. We will therefore leave the initial value ofq, which we will denote q0 , as an unspecified parameter, and only when systems with different q0
converge on a single solution will we consider our models accurate. After numerically integrating
the differential equation for q(t), we can go on to compute M (t), L (t) and Lj (t).
Figure 5.7 demonstrates how q(t) changes as we alter q0 for a solar progenitor, a 105 M blackhole, and a number of other input values. As expected, the initial conditions strongly influence thebehavior of q(t) for early times, but, after about 0.1 years, which is about 2.5 tr for this configuration,
the different values become indistinguishable. We can thus say with confidence that after this time
our models represent the fully-inflated ZEBRA. This timescale, namely a few revolutions of theinnermost material, is also consistent with our expectations concerning the amount of time needed
for the shock heating to add enough energy to the system. Figure 5.8 shows q(t) for various blackhole masses and a set of fiducial parameters. The initial conditions cause q(t) to decrease rapidly.
However, the knowledge of such initial conditions is quickly lost, and the system settles into a statein which q(t) decreases less rapidly. Figure 5.9 shows the mass contained in the envelope as afunction of time. Because the fallback rate exceeds the accretion rate, the mass initially increases
rapidly. For later times, the accretion rate and the fallback rate both drop significantly enough toleave a roughly constant mass. Figure 5.10 plots the jet luminosity; as we can see, the luminosity

144

1.70
1.65
q0 =1

q

1.60
1.55

32

1.50

2

1.45
1.40
1.35
1.30
0.08

0.10

0.12

0.14

0.16

0.18

0.20

t @yrD
Figure 5.7: The solution for q(t) with a solar progenitor (M∗ = 1 M , R∗ = 1 R , γ∗ = 5/3),
a radiation pressure-dominated gas (γ = 4/3), Mh = 105 M , y = 1, δ = 0.05, χ = 5, and threedifferent q0 , indicated by the legend. As one can see, the initial conditions quickly become irrelevantto the long-term behavior of the solutions.

145

1.4
1.2

q

1.0
0.8
0.6

Mh =105 M

0.4

5´105 M

0.2

106 M

0.0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

t @yrD
Figure 5.8: q(t) for a radiation-dominated gas (γ = 4/3) and a variety of black hole masses. Herey = 0.5, δ = 0.05, χ = 5, q0 = 3/2, γ∗ = 5/3, M∗ = 1M , and R∗ = 1R . The legend displays theblack hole mass in units of solar masses. We see that initially q falls off very rapidly, which is aconsequence of initial conditions. However, the initial conditions stop having a major effect earlyin the evolution of the system, and q then decreases less rapidly.

146

0.16
0.14
0.12

M

M

0.10
0.08

Mh =105 M

0.06

5´105 M

0.04

106 M

0.02
0.0

0.5

1.0

1.5

2.0

2.5

3.0

t @yrD
Figure 5.9: The mass contained in the ZEBRA as a function of time for the same parameters as
those in Figure 5.8. The mass quickly increases initially, owing to the fact that the fallback rate
exceeds the accretion rate. However, as both rates decrease for later times, the mass levels off to anearly constant value.

147

200

Mh =105 M
150

LEdd

Lj

5´105 M100

106 M

50

0
0.0

0.5

1.0

1.5

2.0

2.5

3.0

t @yrD
Figure 5.10: The jet luminosity, normalized to the Eddington luminosity of the black hole, for thesame parameters as those in Figure 5.8 with an efficiency  = 0.1. Initially the luminosity is very
super-Eddington, decaying to only mildly super-Eddington at later times.

148
predicted by our model is super-Eddington for a significant amount of time, though the amountof time for which that statement is true decreases as the black hole mass increases. The fact thatthe jet power is supercritical is a good consistency check on our model. The time at which theaccretion rate becomes sub-Eddington is roughly the same time at which q = 0.5, where our model
begins to break down.
One of the popularly-cited hallmarks of a tidal disruption event is that the accretion luminosity is proportional to t−5/3 . However, this result only holds for a constant-density star. Ourmodels account for the mass distribution in the original progenitor, placing more mass on orbits
with larger semi-major axes, and they are also consistent with the initial rise in the fallback rate.
Our accretion rate also takes into account fluid interactions, meaning that our black hole accretionrate, and consequently the jet luminosity, does not necessarily mimic exactly the mass fallback rate.
In Figure 5.11 we plot the accretion rate onto the black hole for two different q0 (we chose twodifferent q0 to demonstrate when the solutions converge), given by equation (5.26), and the fallback
rate onto the accretion region, which is the solution to equation (5.34), for the same set of fiducial
parameters as those in Figure 5.8 and the black hole has a mass Mh = 105 M . It is apparent fromthe figure that the accretion rate onto the black hole follows the fallback rate rather tightly, butthere exist notable differences. The first difference is that the accretion rate is less than the fallback
rate for the times shown; this finding is consistent with Figure 5.9, as the mass is increasing for
all times shown. The second is that there exists a temporal lag between the qualitative features
shared by the two rates; the most salient example of this characteristic is the difference in the time
taken to reach the maximum, which is evident in the figure. Specifically, the fallback rate reaches
its maximum at t ≈ 0.094 years, while the black hole accretion rate peaks at t ≈ 0.11 years. The
third difference is that the accretion rate of the black hole follows a less-steep power law than the
fallback rate for later times. By fitting the fallback rate as Ṁf b ∝ t−mf b between 1 and 2 years, wefind that the power law is mf b ≈ 1.63; by inspecting equation (5.34), we expect that the fallback
power law should asymptotically approach mf b = 5/3. By fitting the accretion rate for the sameamount of time and by the power-law form Ṁacc ∝ t−macc , we find that macc ≈ 1.49. As one can

149

5.00
M acc Hq0 =1L

M @M yr-1 D

1.00
0.50

M acc Hq0 =2L

0.10

M fb

0.05

0.01

0.1

0.2

0.5

1.0

2.0

5.0

t @yrD
Figure 5.11: The black hole accretion rate, given by equation (5.26), in solar masses per year for
q0 = 1 (blue, solid curve) and for q0 = 2 (red, dashed curve) to illustrate where the solutions
have converged; the parameters are the same as those in Figure 5.8 and the black hole has a mass
Mh = 105 M . We have also plotted the fallback rate, shown by the black, dotted curve, to illustrate
how the two accretion processes compare. It is apparent from this figure that they match each otherclosely, and that after about 4 years, the black hole accretion rate exceeds the fallback rate.

150
see in the figure, at a time of about 4 years the black hole accretion rate exceeds the fallback rate,
corresponding to a decrease in the mass contained in the ZEBRA. This result makes sense, as we
expect accretion to occur even if there is no fallback of material onto the envelope.
Another property of the envelope that we can calculate is its effective temperature. As wehave argued in section 3, the surface should occur roughly at the trapping radius, and so thetemperature is given byT=

GcMh mpσT σSB R 2

1/4 

4' 6.5 × 10

Mh
106 M

1/4

R
1014 cm(5.38)

−1/2
K,(5.39)

where σSB = 5.67 × 10−5 cgs is the Stefan-Boltzmann constant. ZEBRA envelopes produced by
TDEs thus tend to peak in the far-UV or soft X-ray band. We can also solve for the effective
temperature as a function of time, as shown in Figure 5.12.
Because R is proportional to M 2/5 (see equation (5.17)), the temperature decreases initiallyas mass is gained from the fallback of tidally-stripped material. However, for later times when theblack hole accretion rate and the fallback rate both decrease substantially, the temperature remains
nearly constant.
Owing to the fact that the photons are trapped interior to R, we expect there to be ahigh degree of coupling between the particles comprising the ZEBRA envelope and the photons
produced at the photosphere. The spectrum should therefore be very well-matched by a blackbody
distribution. Depending on the temperature of the envelope and the composition of the disrupted
star, however, there may also be present a number of absorption and emission features. With thesetemperatures, electron scattering may also produce a color-corrected spectrum.

5.4.3.1

Power-law fallback rate

As one can see in Figure 5.11, the black hole accretion rate closely matches the fallback rate

151

65 000
Mh =105 M

T @KD

60 00055 0005´105 M

50 000106 M

45 00040 00035 00030 000
0.0

0.5

1.0

1.5

2.0

2.5

3.0

t @yrD
Figure 5.12: The effective temperature as a function of time for different black hole masses. The
fiducial parameters are the same as those in Figure 5.8.

152of the tidally-disrupted material. An interesting question is whether this close equality is always
true, or if it just happens to be the case for the specific analytic model that we chose. To analyze
the effects of altering the fallback rate, we will let Ṁf b scale as a power-law, specificallyṀf b = M0 (m − 1) t−m ,(5.40)

where the proportionality constant has been chosen to be consistent with the fact that at tr , theorbital period of the innermost material, Mf b = M0 , where M0 is the mass of the material that has
yet to be accreted (note, however, that we are not considering the equation accurate until muchlater than tr , as the fallback rate must first peak and then decline to the power-law decay). Fromthe initial work of Phinney (1989), it was thought that m should always be on the order of 5/3.
Our Figure (5.11) also indicates that this scaling holds for later times. More recently, however, it
has been shown that this power-law decay may not be followed, even for times much later than thatat which the peak fallback occurs. In particular, Guillochon & Ramirez-Ruiz (2013) demonstrated
that variations in the impact parameter, which we defined as x, can lead to partial disruptions
that, owing to the continued gravitational influence of the surviving stellar core, cause m to deviatesignificantly from 5/3. We will therefore leave this quantity as a variable and inquire as to theeffects of its variation on the black hole accretion rate.
One might expect that M0 = M∗ /2, as the TDE leaves roughly half of the progenitor boundto the black hole. However, partial disruptions, which result from grazing encounters with the blackhole, leave an intact stellar remnant. In these instances, the total mass bound to the hole is always
less than M∗ /2. We will therefore leave M0 as a free variable, typically on the order of a fractionof M∗ /2.
As we argued previously, the material that comprises the ZEBRA must lose its angularmomentum before being accreted by the black hole. By following the same line of reasoning, we

153
can show that the angular momentum contained in the envelope is given by  1−m p
Mh 1/6t
L = M0 2GMh R∗
1−
.
M∗
tr

(5.41)

With this expression, we can go through the same analysis as in the previous subsection andnumerically solve for q(t) and all other time-dependent quantities. However, instead of reproducing
all of the plots in the previous subsection for different values of m, we will concentrate on how theaccretion rate onto the black hole compares to the fallback rate, which will inform us of the way inwhich the jet luminosity relates to the fallback rate.
Guillochon & Ramirez-Ruiz (2013) demonstrated that m varies between roughly 1.5 and
2.2, depending on the value of the impact parameter (see their Figure 7). They also showedthat steeper fallback rates follow from shallower impact parameters, as a result of the gravitational
influence of the surviving stellar remnant. Consequently, the values of m and M0 are not completely
independent. We can show, however, that there exists a nearly-linear scaling between M0 and themagnitude of the black hole accretion rate. Since the fallback rate is also linear in M0 , we will
simply consider M0 a constant and note that the true value of the accretion rate for a given m may
be higher or lower.
Figure 5.13 illustrates, on a log-log scale, the accretion rate and the fallback rate for M0 =0.1 × M /2, Mh = 105 M , m = 2, and the parameters adopted in Figure 5.8. We see that theaccretion rate also follows a power-law decline, but one that is shallower than that for the fallback.
Defining Ṁacc ∝ t−macc , we find, in this case, that macc ≈ 1.70.
For other power-law fallback rates, a qualitatively similar behavior is exhibited by the accretion rate. In particular, the rate at which mass is accreted by the hole falls off as a power-law, but
one that is less steep than the rate at which material impacts the ZEBRA. To illustrate how thevalue of m affects macc , Figure 5.14 shows the value of macc given the value of m. To determine
macc , we have performed a best-fit over the timescale of t = 0.2 – 1.5 years, during which time
all of the accretion rates follow power-law decays. As one can see, the relationship between the154

M @M yr-1 D

0.1

M acc
M fb

0.01

0.001

0.1

0.2

0.5

1.0

2.0

t @yrD
Figure 5.13: The black hole accretion rate (blue, solid curve) and the fallback rate (red, dashedcurve) for M0 = 0.1×M /2, Mh = 105 M , m = 2, and otherwise the same parameters as in Figure
5.8, plotted on a log-log scale. The accretion rate follows a shallower power-law than the fallback
rate, causing the former to exceed the latter for times greater than about 0.5 years.

155

2.2
macc
2.0
m

macc

1.8

1.6

1.4

1.2
1.4

1.6

1.8

2.0

2.2

m
Figure 5.14: The power-law of the accretion rate, which we defined as macc , as a function of
the power-law index of the fallback rate is shown by the blue, solid curve. The value of macc isdetermined by performing a best-fit to the black hole accretion rate between t = 0.2 and 1.5 years,
during which time the accretion rate, for all values of m, is well-described by a power-law. We havealso plotted m for comparative purposes (red, dashed curve). We see that macc is always less thanm. For small values of m, the relationship between macc and m is roughly an offset, linear one. Asm becomes larger, however, macc displays a more nonlinear behavior, and the difference between
the two power-law indices becomes larger.

156
two power-laws is approximately linear for low values of m, becoming increasingly nonlinear as mincreases. Thus, while the difference between the two is approximately ∆m ≡ m − macc ≈ 0.15 form = 1.4, the disparity becomes ∆m ≈ 0.50 for m = 2.2.

5.4.4

Swift J1644+57

The object Swift J1644+57 was found as both a source of X-rays and γ-rays by the Swiftsatellite, and thought initially to be a gamma-ray burst (GRB) (Markwardt et al., 2011). However,the variability and longevity of the source soon proved that such an association was unlikely,
and the proximity of the event to the nucleus of a host galaxy at redshift z = 0.354 led to thebelief that the event was triggered by a TDE (Levan et al., 2011). By modeling the spatially andtemporally coincident radio emission by the interaction of fast-moving ejecta with the circumnuclear
environment, it was demonstrated that a mildly relativistic jet was likely generated during the TDE(Zauderer et al., 2011; Metzger et al., 2012). Here we investigate the consistency of our ZEBRA
models with the observations of Swift J1644+57.
The peak, isotropic X-ray luminosity of Swift J1644+57 reached 4 × 1048 erg s−1 , withan average value appropriate to ∼ few × 1047 erg s−1 (Burrows et al., 2011). When corrected for
beaming effects, we recover a true luminosity of LX ∼ 1045 − 1046 erg s−1 for a jet opening angle of5◦ (Bloom et al., 2011). The mechanism responsible for the generation of the X-rays is still unclear,though its origin is consistent with inverse Compton scattering of photons near the launching pointof the jet (Markoff et al., 2005; Bloom et al., 2011). It is difficult to constrain directly the massof the black hole that resides in the host galaxy, but empirical galaxy luminosity relations imply
105 M . Mh . 106 M (Saxton et al. (2012) and references therein). The energy generation rateis therefore highly super-Eddington (even if we increase the upper limit of the black hole mass to
107 M ).
The prompt evolution of the X-ray emission was highly chaotic. However, after about 10 days
from the initial trigger, the flux followed a decline that was well-approximated by a power-law (seeFigure 1 in Tchekhovskoy et al. 2014). Therefore, if our model is to adequately describe the157
evolution of Swift J1644+57, the first constraint it must satisfy is that the X-ray flux follow a
power-law decline for later times. The exact value of the power-law index is uncertain owing to thelarge degree of intrinsic scatter in the X-ray data, but it is consistent with the range predicted in
Guillochon & Ramirez-Ruiz (2013) (also see Tchekhovskoy et al. 2014).
The second constraint on our model comes from the fact that the X-ray flux of Swift
J1644+57 dropped precipitously after about 500 days, most likely indicating the shut-off of thejet (Zauderer et al., 2013). Because our model requires the accretion rate to be super-Eddington
during the jetted phase, our jet luminosity should be roughly the Eddington limit of the hole att ≈ 500 days. However, because the initial behavior of the X-ray flux was highly chaotic, it isunclear at what point from the time of disruption the Swift satellite began observing the source,
and the timeline of 500 days is therefore ambiguous.
For the analytical models described in the previous subsection, the luminosity of the jet
depends not only on the black hole mass, but also on a number of other parameters which describe
the details of the flow, e.g., y, δ, etc. However, the results are largely insensitive to those parameters,
and hence we will adopt the values that produced Figures 5.8 – 5.10. Moreover, if the impact
parameter differs significantly from x = 1, making the power-law model the most valid descriptionof the fallback process, the values of m and M0 must also be incorporated into the model.
The flux we observe is altered by the Lorentz factor and opening angle of the jet, bothof which are uncertain and could change with time. However, if we assume that both of thesequantities are constant, the flux we observe and the accretion luminosity of the hole are linearly
related. Thus, while the magnitude of the observed flux cannot be determined exactly with ourmodel (without performing a more in-depth analysis of the jet), its qualitative appearance can be
reproduced. The value of M0 , which is a relevant quantity if the TDE occurs outside the tidal
radius, also affects the magnitude of the jet luminosity. The time of disruption is also unknown, as
the initial chaotic behavior of the event is not predicted by any model, making the time at which
the observed flux reached a maximum incalculable. However, since there is a steady decline after
a timescale on the order of days with no recurring rise, it is probable that the maximum fallback

158
rate occurred somewhere near in time to the triggering event.
With all of these considerations, we will restrict our attention to times greater than roughly
15 days after the trigger, where the observed flux approximately follows a power law. Given thisrestriction and our uncertainty in the exact value of the intrinsic flux, the first constraint on ourmodel is that the power-law index of the jet luminosity should be between 1.5 and 2.2. The second
constraint is that the accretion luminosity produced by the hole should be near the Eddington
limit of the hole after about 500 days from the time of the triggering. The luminosity of the jet
must also change by about an order of magnitude during this length of time, evident in Figure 1
of Tchekhovskoy et al. (2014).
If we adopt the model discussed in the beginning of section 4.3, which places the periapsisof the disrupted star exactly at the tidal radius, and we choose the set of fiducial parameters that
produced Figure 5.8, the only free parameter left is the mass of the black hole. Because the powerlaw index for later times is around 1.5, there will exist a qualitatively good fit between the modeland the data. Our models predict that smaller black holes produce a greater change in the jetluminosity over the duration of the super-Eddington event, and an order-of-magnitude change inthe luminosity requires a black hole of mass Mh ' 105 M . For black holes with masses in thisrange, the accretion luminosity of the hole is on the order of its Eddington limit around 500 days
after the maximum fallback. This prescription is thus broadly consistent with Swift J1644+57.
If the tidal disruption occurs at a distance such that the star is only partially destroyed, thepower-law rate of return is the most appropriate method by which we can analyze the fallback
onto the ZEBRA. Since the power-law associated with the accretion luminosity is always between
about 1.5 and 1.7, there will exist qualitatively good agreement between the observations of Swift
J1644+57 and the ZEBRA prediction. For these models, the change in flux being an order of magnitude again requires that Mh ' 105 M , consistent with the description that places the periapsis of
the star at the tidal disruption radius. For these fallback rates, the accretion luminosity is near the
Eddington limit of the hole after 500 days from the maximum, though the precise number depends
on M0 .

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5.5

Discussion and conclusions
We have outlined a novel approach to describing the super-Eddington accretion disks gen-erated during tidal disruption events. Following Loeb & Ulmer (1997), we used the low specificangular momentum of the tidally-disrupted material to place the material in a nearly-sphericallysymmetric configuration around the hole. However, instead of forcing a strictly spherical envelope
to enclose a thick disk, which we believe to be unstable due to the absorption of energy and transfer of angular momentum, we self-consistently account for the distribution of angular momentum
throughout the material.
In our models, the accretion energy released by the black hole and shock heating pump asignificant amount of energy into the debris, puffing up the disk. We encounter a point where theBernoulli parameter approaches zero, leaving a quasi-spherical envelope that is marginally bound.
Further energy input would unbind the material, most likely resulting in a wind (see Appendix
B). Instead of creating a wind, we posit that the accretion energy of the black hole is instead
redirected to the poles, resulting in the formation of a jet that serves as the exhaust route for theexcess energy. The resulting configuration is a zero-Bernoulli accretion (ZEBRA) flow, threaded
by a bipolar jet. This type of object is specifically relevant to the recently observed X-ray transient
Swift J1644+57.
The creation of the jet is a natural consequence of the fact that ZEBRA envelopes are closed
up all the way to the poles, leaving no disk surface from which one could launch a wind and remove
the super-Eddington accretion luminosity. Because the liberated gravitational energy would haveto propagate through the entire system to be released at the photosphere, which is not possible
owing to the supercritical nature of its generation rate, it is forced to exit along the poles. Anotherconsequence of the super-Eddington luminosity is the inability of the flow to cool efficiently, forcing
it to maintain its zero-Bernoulli nature. The supercritical accretion luminosity thus forms thecornerstone of the consistency of our model.
Following the analysis of Blandford & Begelman (2004), we demonstrated the existence of

160
self-similar solutions to the momentum and Bernoulli equations with B = 0, which form a particularsubset of the gyrentropic flows discussed by those authors. The gyrentropic nature of our flows is a
result only of our assumption of the globally-zero Bernoulli parameter, independent of any stability
considerations, such as the presence or absence of MRI. The ZEBRA flows were shown to close up
only exactly at the poles, indicating the quasi-spherical nature of the envelopes.
We showed that there exists an unspecified parameter, denoted by q (linearly related to theparameter n of Blandford & Begelman 2004), which characterizes the radial gradients of the densityand pressure and the sub-Keplerian nature of the flow. For TDEs, the total mass and angularmomentum of the progenitor star, coupled to our specification of the trapping radius as the edgeof the envelope, determine the value of q. For low specific angular momentum, the gradients ofthe density and pressure increase, approaching the isentropic value of a non-rotating star as theangular momentum goes to zero.
ZEBRA envelopes have a radial extent of hundreds to thousands of Schwarzschild radii,
validating our neglect of general relativistic effects over the bulk of the flow. However, the excess
of angular momentum at small radii to account for the relativistic gravitational field could playa significant role in our determination of the gross properties of the configuration. By using thepseudo-Newtonian potential of Paczyńsky & Wiita (1980), we demonstrated that, while the specificangular momentum, pressure, and density can deviate significantly from their Newtonian values
in regions close to the hole, the total mass and angular momentum of the envelope are largely
unaffected.
These models apply to the super-Eddington phase of accretion, namely when the flow is
unable to cool via radiative losses. We were able to predict the accretion and jet luminosities
associated with ZEBRA flows, and found that, indeed, the rates are highly supercritical, providing
a self-consistency check on our assumptions. Another aspect of our flows that is asserted a priori isthat the Bernoulli parameter is precisely zero, which we know must break down close to the hole.
The implications of a non-zero, but constant, B are addressed in Appendix B. The results derived
in the previous sections are shown to be insensitive to this assumption provided that |B| < GMh /r.

161
Because of the very high accretion rates, an appreciable amount of mass is lost on a dynamically relevant timescale. This consideration allowed us to determine the time-dependent natureof the properties of the accretion flow and the jet. By using an analytic model closely followingthat of Lodato et al. (2009) to describe the fallback rate of tidally-stripped material, it was shownthat the jet luminosity roughly follows the rate at which material returns to pericenter, but witha few notable differences. In addition to using the model for which the pericenter distance of the
disrupted star equals the tidal radius, we investigated the consequences of letting the fallback onto
the ZEBRA scale as a power-law. This model serves as a proxy for the late evolution of TDEs
for which the pericenter of the stellar progenitor lies outside the tidal radius. In these cases, theaccretion rate onto the hole also follows a power-law decline, but with a power-law index that isless steep than that of the fallback rate. We also demonstrated that ZEBRA envelopes producedby TDEs should have approximately-constant effective temperatures of T ' 5 × 104 K, placing thepeak of their bolometric luminosities in the far UV to soft X-ray.
We compared our models with the observed properties of the transient X-ray source Swift
J1644+57. Because of the uncertainties in the opening angle of the jet, its Lorentz factor, andthe time at which the X-ray flux reached its peak magnitude, we were unable to place many
direct constraints on our model. However, we found broad consistency with our models and theobservations if the black hole has a mass on the order of 105 M , assuming a disrupted star of solar
type and a constant jet Lorentz factor and opening angle.
The existence of a ZEBRA is contingent on the availability of an exhaust route for the excessenergy produced in the accretion process. In this account we have presupposed the existence ofa jet as this conduit, and we demonstrated its consistency with the source Swift J1644+57. We
have foregone, however, any explicit analysis concerning its generation or its interaction with the
ZEBRA flow. We also neglected any changes in the Lorentz factor or the beaming angle of the jet,both of which would have observable effects on the X-ray luminosity. These aspects of the problem
will be addressed in a future paper.
In addition to tidal disruption events, ZEBRA flows may manifest themselves in other as-

162
trophysical systems. One such application is to failed supernovae, or collapsars (Woosley, 1993;
MacFadyen & Woosley, 1999). In this model, a highly evolved, rotating star undergoes a type IIsupernova. The core collapses directly to a black hole, the remaining stellar material creating an
accretion disk and producing a jet. The internal shocks within the jet provide one mechanism capable of producing the gamma rays we observe in long gamma-ray bursts (GRB). Outflows farther
from the poles are thought to unbind the envelope and produce the supernova signature observedin many long GRBs (Woosley & Bloom, 2006). However, there have been a few cases in which weobserve a GRB devoid of any supernova afterglow (Fynbo et al., 2006), even though the location of
the GRB should have provided no impediment (e.g., dust extinction or light contamination) to ourobservation of the afterglow. It is possible that, in these instances, the outflows away from the poles
were not sufficient to unbind the envelope, leaving it intact above the black hole. This environmentis precisely that in which a ZEBRA flow would arise, as a wind is unable to be created due to thepresence of the overlying stellar material. As more energy is pumped into the material, the entire
mass of the progenitor may come to an approximate equilibrium described by our B = 0 prescription. Another application would be in the deep interior of a quasi-star, a giant proto-galactic gas
cloud supported by black hole accretion (Begelman et al., 2006b, 2008). Because the black holeaccretes at the Eddington limit of the total quasi-star, whose mass far exceeds that of the hole, theaccretion rate is highly supercritical. The overlying gas prevents the generation of a wind or any
other exhaust mechanism, making a ZEBRA flow the most appropriate description of the fluid.
In both of the previous examples, the accretion rates and other physical processes create anatural environment for ZEBRAs. However, they both differ from tidal disruption events in thegravitational role played by the black hole: in TDEs, the black hole dominates the mass of thesystem, and so the gravitational potential is given by that of a point mass. In a failed supernova,
the black hole generated by the collapse is on the order of the mass of the overlying material.
Thus, as we move away from the hole into the ZEBRA envelope, there will come a point where theenclosed mass roughly equals that of the hole. The point-mass prescription then becomes invalid.
For a quasi-star, the black hole constitutes only a small fraction of the total mass, and hence the163
self-gravitating nature of the flow must be considered in order to adequately describe the propertiesof the ZEBRA.
The fluid and Bernoulli equations with an arbitrary gravitational potential may be written
down in a straightforward manner, Poisson’s equation being the extra constraint that closes thesystem. An analysis of these relations, in which we self-consistently include both the angularmomentum of the gas and its self-gravitating nature, will be deferred to a later paper.

Chapter 6

The General Relativistic Equations of Radiation Hydrodynamics in the Viscous
Limit

6.1

Introduction
Radiation contributes substantially to the dynamics of many astrophysical systems. The

most natural way to analyze the mechanics of such systems is through the formalism of radiationhydrodynamics, wherein one treats the radiation as a fluid that interacts with matter. With thecovariant derivative denoted ∇µ , the equations of radiation hydrodynamics can be generalized toinclude both gravitational fields and relativistic motions by writing them in the manifestly covariantform∇µ (T µν + Rµν ) = 0,

(6.1)

where Greek indices range from 0 to 3, repeated upper and lower indices imply summation,

T µν = w0 U µ U ν + p0 g µν

(6.2)

is the energy-momentum tensor of the massive constituents of the fluid (w0 is the enthalpy in thefluid rest-frame, U µ is the four-velocity, p0 is the pressure in the fluid rest-frame, and g µν is theinverse of the metric associated with the background geometry) and

Rµν =

Z

kµ kν f

d3 k
k0

(6.3)

165
is the energy-momentum tensor of the radiation (see, e.g., Mihalas & Mihalas 1984 for a morethorough discussion of the origin of this tensor). In equation (6.3), k µ is the four-momentumof a photon, f is the distribution function of the radiation that describes the density of quantain both momentum and position space, and d3 k/k 0 is the Lorentz-invariant phase-space volume.
Because f is a scalar, such a “moment” formalism, i.e., proceeding by taking integrals over the
distribution function, is a natural way to analyze the dynamics of the radiation in a covariant
fashion. Note, however, that equation (6.3) is only valid in a locally-flat frame – one in whichgµν = ηµν , ηµν being the Minkowski metric – as otherwise one must include factors that dependon the metric (see, e.g., Debbasch & van Leeuwen 2009a). To ensure that the radiation energymomentum tensor (and derivatives thereof) transforms correctly at all points in the spacetime underconsideration, one can explicitly insert the metric-dependent factors that enter into the phase-spacevolume element. Another, equally valid manner by which one can obtain the general-relativistic
form of Rµν , however, is by evaluating the integrals in equation (6.3) and writing the results ina manifestly-covariant form, guaranteeing their frame independence. Because we will be working
directly with the distribution function, the latter route is the simpler one to follow and is the onethat we will pursue in our ensuing analyses (see sections 3 and 4).
For many astrophysical applications of the equations of radiation hydrodynamics, the medium
under consideration is optically thick, meaning that the radiative flux observed at any location
within the fluid is very nearly zero. This means, equivalently, that a photon is scattered a large
number of times as it propagates through the medium, and that the radiation field as seen by amoving fluid element within the medium is approximately isotropic. It has been known for sometime, however, that the finite mean free path of a photon leads to the presence of viscous-like terms
in the radiation energy-momentum tensor (Thomas, 1930). The viscous nature of some fluids,
especially in applications for which photons dominate the pressure of a system, can therefore be
attributed in part to their interactions with radiation.
Eckart (1940) analyzed the origin of generic, viscous effects in relativistic fluids, whether due
to radiation or to other phenomena (see also Landau & Lifshitz 1959). He exploited the fact that166
the four-velocity of the fluid, U µ , and the projection tensor, Πµν = U µ U ν + g µν , are time-likeand space-like tensors, respectively, that can be used to decompose any arbitrary tensor. Eckart
realized that each of the terms in his decomposition had a physical interpretation, related to the
dynamic viscosity or heat conduction of the fluid, which allowed him to postulate a form for theviscous stress tensor of a relativistic fluid that reduced correctly to its non-relativistic counterpart.
Thomas (1930) (and others after him, e.g., Lindquist 1966; Castor 1972; Buchler 1979; Munier
1986; Chen & Spiegel 2000) used the special relativistic Boltzmann equation, which describeschanges to the distribution function due to particle-particle collisions, to derive the correction
to the radiation energy-momentum tensor for the special case of Thomson scattering. Eckart’s
approach, on the other hand, was more phenomenological in nature, employing thermodynamic
arguments and an understanding of the Newtonian limit of the viscous stress tensor to derive its
relativistic generalization. Weinberg (1971), with the intent of using the results to analyze entropy
production in the early universe, compared the two approaches and showed that one of Eckart’s
assumptions, namely that the viscous stress tensor be trace free, led to the incorrect conclusion thatthe bulk viscosity of the fluid vanish (see also Misner & Sharp (1965), where a similar approach isused to evaluate radiative effects during core-collapse supernovae). In his analysis, the part of therelativistic stress tensor expressing viscosity and heat conduction, a slightly generalized version ofEckart’s, was of the form

∆T

µν



2
α
= −η Π Π∇σ Uρ + ∇ρ Uσ − gσρ ∇α U
− ζ Πµν ∇α U α3− χ Πνσ U µ + Πµσ U ν
T U ρ ∇ρ Uσ + ∇σ T , (6.4)µσνρ

where η, ζ, and χ are the coefficients of dynamic viscosity, bulk viscosity, and heat conduction,
respectively, and T is the temperature of the gas. In the special-relativistic limit, ∇µ = ∂µ is justthe partial derivative, but is the covariant derivative if curvilinear coordinates are being used (orif the fluid is in a gravitational field). His comparison between the two theories enabled him tocalculate η, ζ, and χ for a radiating fluid, confirming the notion that small-scale anisotropies in the167radiation field, i.e., on the order of the mean free path of a photon, generate viscous-like effects.
Although they yield similar results, the fluid/thermodynamic approach of Eckart (1940),
and correspondingly that of Weinberg (1971), is fundamentally different from the kinetic theory
approach of Thomas (1930). The first difference arises from the fact that the former is a single-fluid
analysis, meaning that ∆T µν , as given by equation (6.4), is the viscous correction to “the fluid.”
The second disparity comes about because Eckart defines thermodynamic quantities in terms of
the energy-momentum tensor of the fluid; for example, Eckart (1940) and Weinberg (1971) bothdefine the comoving energy density as e0 = Uµ Uν T µν , meaning that there is no correction to theobserved energy. It is for this reason that there is no term proportional to U µ U ν in equation (6.4);
however, the Eckart decomposition for an arbitrary tensor, one that has no restriction imposed
upon it concerning the comoving energy density, will have such a term.
On the other hand, the kinetic theory approach considers the radiation and the matter to be
two separate, interacting fluids – the same viewpoint that underlies all of radiation hydrodynamics– meaning that the viscous terms are understood as corrections to the radiation energy-momentum
tensor, rather than that of the matter. Also, the kinetic theory description uses the Boltzmann
equation to determine the distribution function, the viscous stress tensor then being written in termsof integrals over that function, viz., equation (6.3). There is thus no need to define bulk physicalparameters, such as the temperature T or the comoving energy density e0 , in terms of the stresstensor. Indeed, one could verify the validity of Eckart’s assumption that ∆e0 = Uµ Uν ∆T µν = 0 if
one knew the distribution function.
Our goal here is to analyze the Boltzmann equation and thereby evaluate the stress tensor
for a relativistic, radiating fluid in the limit that the finite mean free path of the radiation provides
the source of the viscosity. In section 2 we present the general relativistic Boltzmann equation, theformalism of general relativity being necessary because of the fact that scattering is handled mosteasily in the comoving (accelerating) frame of the fluid. In section 3 we restrict our attention to thecase where the scattering is dominated by Thomson scattering and we solve the resultant equation
for the distribution function to first order in the mean free path. Section 4 presents the equations

168
of radiation hydrodynamics in the viscous limit and we demonstrate that the stress tensor departs
from equation (6.4) in a few important ways, the first being that our equation has an additionalcorrection to the comoving energy density, and the second being that our result is independent ofthermodynamic considerations, such as the assignment of a temperature, and is therefore applicable
to non-equilibrium radiation fields. Our coefficient of bulk viscosity also differs from that derived
by Weinberg (1971). We perform a Fourier analysis of the perturbed equations in section 5 andshow that they are indeed stable to perturbations of the fluid on scales larger than the mean freepath of the radiation. A discussion of future applications, some comments on current radiation
magnetohydrodynamic codes and conclusions are presented in section 6.

6.2

Relativistic Boltzmann equation
The distribution function for the radiation must satisfy a transfer equation – some statementof the conservation of photon number. As mentioned in the introduction, a natural choice for thisequation is the Boltzmann equation, which describes the changes to the distribution function owing
to emission, absorption, and scattering processes. However, because photons are massless, we mustuse a relativistic version of the Boltzmann equation; an obvious generalization to the relativistic
regime is∂f∂f∂f+ v i i + v̇ i i = δf coll
∂t
∂x
∂v

→ kµ

∂f
∂f
+ k̇ µ µ = δf coll ,
∂xµ
∂k

(6.5)

where Latin indices adopt the range 1 – 3, Greek indices range from 0 – 3, repeated upper and lower
indices imply summation, and k µ is the four-momentum of a photon. The right-hand sides representchanges to the distribution function through interactions with the surrounding medium. Equation(6.5) is the relativistic Boltzmann equation often encountered in the literature (e.g., Mihalas &Mihalas 1984). There are, however, two important subtleties associated with equation (6.5) that

169
are often not mentioned explicitly.
The first issue is that the sum over k̇ µ on the left-hand side of (6.5) must incorporate theconstraint that the four-momentum of the photon lie on the null cone, viz., kµ k µ = 0 (this problem
vanishes in flat space because photons travel in straight lines, and hence k̇ µ = 0), which raises thequestion of whether or not the sum should occur over all four components of the momentum or justthree. Furthermore, if we take the latter route, which three should we choose? Comparing equation(6.5) to the non-relativistic expression suggests taking the three spatial components; relativistic
covariance, however, demands that the time component should not be treated differently from thespatial momenta.
The second subtlety stems from the fact that, in general relativity, there are two different
momenta from which we can choose to describe the system: the covariant, kµ , and the contravariant,k µ , components, related by kµ = gµν k ν where gµν is the metric. The distribution function treats
the spatial variables, xµ , as independent of the momentum. Therefore, should we consider xµ andk µ as independent coordinates, or xµ and kµ ? It is apparent that, depending on which one wechoose, the results will differ as the metric gµν is a function of the spatial coordinates.
In our ensuing treatment of equation (6.5), we will be analyzing interactions in the fluid frame– the one comoving with a given fluid parcel. Such a frame will, in general, be accelerating, meaningthat k̇ µ 6= 0 and there will be an acceleration-induced metric gµν , forcing us to confront each of thepreviously-raised questions. Recently, Debbasch & van Leeuwen (2009a,b) addressed these issues
directly by returning to the most general definition of the distribution function, a sum of Dirac delta
functions in position and momentum space. They demonstrated (Debbasch & van Leeuwen, 2009a)that one may consider either the covariant, spatial components ki or the contravariant components
k i as the momentum independent from xµ . However, as we mentioned previously, the Boltzmann
equations that result from these choices are not identical, meaning that one must also define twodifferent distribution functions, one dependent on the covariant components and the other on thecontravariant components, to proceed unambiguously. Even though the routes were shown to beequivalent, one must be careful to use the components appropriate to a given Boltzmann equation.

170
In Debbasch & van Leeuwen (2009b), the authors showed that equation (6.5) is correct if 1)
the sum involving k̇ i in the second term is performed over the three spatial components, 2) the
distribution function f is considered to be a function of the contravariant, spatial components k i ,and 3) any appearance of k0 (or k 0 ) is replaced by k0 (k i ). The dependence of k0 on the spatial
momenta can be determined by solving the equation kµ k µ = 0 for k0 . As we stated, the frame ofinterest is the comoving frame of the fluid, and we will denote the components of any tensor in this0

frame with a prime on the index; e.g., k i is the ith component of the momentum in the comoving
frame. With this convention, the relativistic Boltzmann equation becomes

0

kµ

∂f
i0
µ0 ν 0 ∂f= δf coll ,0 − Γµ0 ν 0 k k
µ
∂x
∂k i0

(6.6)

where we have used the geodesic equation,

0

0

0

0

k̇ i + Γiµ0 ν 0 k µ k ν = 0,

(6.7)



∂µ0 gν 0 α0 + ∂ν 0 gµ0 α0 − ∂α0 gµ0 ν 0

(6.8)

0

to replace k̇ i , and

1 0 00
Γiµ0 ν 0 = g i α
2are the Christoffel symbols associated with the metric gµ0 ν 0 which, for our purposes, is induced by
the acceleration associated with the comoving frame.
Equation (6.6) was provided by Lindquist (1966) and Castor (1972). Since they were considering spherically symmetric flows, they opted to change the form of equation (6.6) by using an
orthonormal tetrad adapted to spherical coordinates. For our purposes, however, we will not be in
the position to take advantage of any specific coordinate symmetries, so equation (6.6) will suffice.
Finally, it should also be mentioned that the right-hand side is evaluated at a specific location– the point in space and time where the collision occurs. The left-hand side, therefore, must also
be evaluated at that spacetime point.

171

6.3

Diffusion approach to the transport equation
Here we will first make the approximation that the temperature of the gas is high enoughsuch that all species, considered to comprise a single, massive fluid, are ionized, meaning thatthe right-hand side of equation (6.6) incorporates effects due only to scattering. In this case, thecollisional term can be written (Hsieh & Spiegel, 1976)

δf coll = n

0

Z
R(k

µ0

Z
3 0
µ0
µ0 µ0 d ki
0, ki )f (x , ki ) 00 − nki

d3 kf0µ0 µ0µ0 µ0
R(kf , k )f (x , k ) 00 ,
kf

(6.9)

where R is the redistribution function (not to be confused with the stress-energy tensor) and n0 is
the rest-frame number density of scatterers. The first term represents scatterings into the state k µ000from any initial momentum state kiµ , while the second embodies scatterings out of state k µ into0

any other state kfµ . As long as photon wavelengths are long compared to the Compton wavelengthand the gas is non-relativistic in the rest frame of the fluid, two suppositions we will make here,
the redistribution function (in the comoving frame of the fluid) is that appropriate to Thomson
scattering:

R(k

µ0

0
3σT
, kiµ ) =

16π



0
0
x0 x0
y0 y0
z0 z0 2
1 + (k̂ k̂i + k̂ k̂i + k̂ k̂i ) δ(k 0 − ki0 ),(6.10)

where σT is the Thomson cross section (or, more generally, the cross section relevant to the scat0

0

0

0

0

0

terer). The equation for R(kfµ , k µ ) is identical to that for R(k µ , kiµ ) but with kiµ → kfµ . Inserting
equation (6.10) into equation (6.9) and the result of that substitution into equation (6.6) gives our
final form for the transfer equation.
It should be noted that the integrals in equation (6.9) are performed under the restrictionthat the photon four-momenta lie on the null cone. Since collisions occur instantaneously at fixedlocations in space, we can approximate the metric to be locally that of flat space, i.e., gµ0 ν 0 = ηµ0 ν 0 ,p
00
and we can use the null cone condition (kµ0 k µ = 0) to write k 0 = |k0 | = (k x0 )2 + (k y0 )2 + (k z 0 )2 .
0

Every appearance of k 0 in equation (6.9) can thus be replaced by |k0 |.
Our goal here is to discern how the radiation field responds to gradients in the flow velocity.

172
In the next section, we will use the equations of radiation hydrodynamics to deduce how the flow
couples to changes in the radiation field, thereby completing the picture. To achieve this goal we
will adopt a diffusion approximation, asserting that the distribution function may be written as
f ' f0 + f1 , where f1 is a small correction to f0 . The “smallness” of f1 is encoded in the mean
free path ∼ (n0 σT )−1 and the gradients of the flow velocity, v, across the mean free path, meaningthat f1 ∼ f0 dv/dτ , where τ ∼ n0 σT x is the optical depth.
Because the right-hand side of equation (6.6) is proportional to the optical depth (per unit
length), the consistency of our diffusion approach demands that δf0 coll = 0. It can be verified
that the right-hand side vanishes for any function f0 that depends only on the magnitude of the
momentum, meaning that f0 is isotropic in the comoving frame. This result is consistent with the
expectation that, in an optically-thick medium, the flux of radiation observed in a frame comoving
with a fluid parcel is very nearly zero. Since the collision operator acts at a point in spacetime,
however, the zeroth-order distribution function can also contain any other secular variation in space000

and time, meaning that its most general form is f0 (k µ , xµ ) = f0 (|k0 |, xµ ). The first-order transfer
equation, to be solved for f1 , is then

kµ

0

∂f0
i0
µ0 ν 0 ∂f0= δf1 coll .
0 − Γµ0 ν 0 k k
µ
∂x
∂k i0

(6.11)

To make more progress on this relation, we must determine the metric associated with thecomoving frame. To do so, we will first assume a planar configuration of the fluid, with neither
velocity nor variation in the x direction. We will then use the fact that the coordinate transformation to move into the comoving frame of a given fluid parcel is a local Lorentz transformation, theinverse of which is given by
Z t0
t=
0

Z t0
z=
0

Γ dt00 +Z z00Γvz dz 00 +Z y0

Γvy dy 00 ,





Z y0
Z z0 
vz vy
vz2
00
dz +
Γ − 1 dy 00 ,Γvz dt +1+ 2 Γ−1
2
v
v
0
0
00

(6.12)

0(6.13)

173

Z t0

00

Z z0

Γvy dt +

y=
0

0





Z y0 
vy2
vz vy
00
1+ 2 Γ−1
Γ − 1 dz +dy 00 ,v2v
0x = x0 ,(6.14)(6.15)

where v 2 = vz2 +vy2 , Γ = (1−v 2 )−1/2 is the Lorentz factor (not the Christoffel symbol), and we have,
without loss of generality, chosen the origin of the primed coordinate system to coincide with thatof the lab frame. The integrals are necessary here because the velocities are all dependent on thecoordinates t0 , z 0 , and y 0 , but because we will ultimately be evaluating our expressions at the origin,
R t0
one would obtain the same answer by letting 0 Γ dt00 = Γ t0 , etc. Double-primed coordinates are
simply dummy variables where for each integrand we let t0 → t00 , etc. The line element, which inflat space is given by

ds2 = −dt2 + dz 2 + dy 2 + dx2 ,(6.16)

is invariant with respect to our choice of coordinates; the metric can therefore be determined bydifferentiating equations (6.12) – (6.15), inserting the results into equation (6.16) and groupingterms (see Castor 1972 for a similar, but non-relativistic, approach).
00Calculating the Christoffel symbols, using the chain rule to determine ∂f /∂k i and ∂f /∂xµ ,
inserting the expressions into equation (6.6) and evaluating the result at the origin (as this is the location of the fluid parcel – where the scattering occurs and where spacetime is locally Minkowskian),
we find

kµ

0

kµ

0

∂f0
i0
µ0 σ 0 ∂f0=
0 − Γµ0 σ 0 k k
µ
∂x
∂k i0



∂f0
Γ2 ∂f0
z0 2
y0 2
0 z0
0 y0
y0 z0
,
−
A
(k
)
+
A
(k
)
+
A
|k
|k
+
A
|k
|k
+
A
k
k
2
3
4
5
1
∂xµ0 |k0 | v 2 ∂|k0 |(6.17)

174
00where, by using the definitions of the Christoffel symbols and using U µ = δ µ00 , it can be shownthatΓ2
A1 = ∇z 0 Uz 0 ,v2(6.18)

Γ2
A2 = ∇y0 Uy0 ,v2(6.19)

Γ2
A3 = ∇00 Uz 0 ,v2(6.20)

Γ2
A4 = ∇00 Uy0 ,v2(6.21)

Γ2
A5 = ∇ y 0 U z 0 + ∇ z 0 U y 0 .
v2(6.22)

0

We broke up the derivative ∂/∂xµ into two separate components: ∂f0 /∂|k0 |, taken such that allq0
appearances of xµ not contained in the definition of |k0 | = gi0 j 0 k i0 k j 0 , through the metric, are0

kept constant, and ∂f0 /∂xµ |k0 | , taken such that all spatial coordinates that appear through |k0 |are held fixed. Note that we must use the relativistically-correct version of the magnitude of
the photon momentum, i.e., one involving the metric, because we are taking derivatives of the
distribution function before evaluating the result at the location of the fluid parcel. Thus, eventhough spacetime is flat exactly at the point of interest, deviations exist at neighboring locations –the derivative requiring that we evaluate the distribution function at those locations. Though we
did not explicitly denote it, the derivatives in equations (6.18) – (6.22) are to be evaluated at theorigin.
00

The right-hand side of equation (6.6) involves an integral over f1 (recall that R(k µ , kiµ ) is
given by equation (6.10)), and the most direct means of evaluating f1 would be to expand both theleft-hand side of the transfer equation and the function f1 in terms of spherical harmonics. Instead175of pursuing this route, however, we will make the educated guess

0

0

0

0

0

0

f1 = B1 (k z )2 + B2 (k y )2 + B3 k z + B4 k y + B5 k z k y ,(6.23)

0where the B’s are functions of xµ and |k0 | (since the k’s are just linear combinations of sphericalharmonics, this method yields the same result as proceeding in the more rigorous fashion of expanding the functions in terms of spherical harmonics). Inserting this ansatz into δf1 coll , performing
the integrals and comparing powers of k on both sides, we findB1 =10 1 1 ∂f0∇ z 0 Uz 0 ,9 n0 σT |k0 | ∂|k0 |(6.24)

B2 =

10 1 1 ∂f0∇ y 0 Uy 0 ,9 n0 σT |k0 | ∂|k0 |(6.25)

1
B3 = 0
n σT




∂f0
1 ∂f0
∇00 Uz 0 − 0,∂|k0 ||k | ∂z 0 |k0 |(6.26)

1n0 σ T




∂f0
1 ∂f0
0
0
∇
,
U
−
0 y
∂|k0 ||k0 | ∂y 0 |k0 |(6.27)


10 1 1 ∂f0
B5 =∇ y 0 Uz 0 + ∇ z 0 Uy 0 .
9 n0 σT |k0 | ∂|k|(6.28)B4 =In addition, however, we find that there are isotropic terms, i.e., only dependent on |k0 |, that arise000from the integrations over (k z )2 and (k y )2 ; it is also apparent that k 0 ∂f0 /∂t0 , the first term inthe sum in equation (6.17), represents an isotropic contribution to the left-hand side. Since thecollision integral is zero for any isotropic term, however, these additional terms cannot be accounted
for with our correction to the distribution function. (Equivalently, there are terms proportional
to the Y00 spherical harmonic, a constant, that cannot be balanced by adding more terms to the
distribution function.) We are therefore forced to equate these extraneous collision terms and176
the time derivative, yielding an extra constraint that the zeroth-order distribution function mustsatisfy:

1
∂f0
∂f0
0
.
∇i0 U i |k0 |
= 0
0
3
∂|k |
∂t |k0 |(6.29)

What does this condition mean physically? As an illustrative example, let us take the case where
the zeroth-order distribution function is given by that for blackbody radiation:

C

f0 =
e

|k0 |
T

,

(6.30)−1

where C is a constant, the value of which is unimportant, and we have taken Boltzmann’s constant
to be one. With this form for f0 , equation (6.29) becomes1∂T0= − T ∇ i0 U i .
0∂t3(6.31)

This, however, is just the gas energy equation for an isentropic, relativistic gas in the frame comoving
with the fluid. Equation (6.29), therefore, is equivalent to the statement that the zeroth-order
distribution function be isentropic.
The first-order correction to the distribution function, using equations (6.24) – (6.28), is thus
10 1 ∂f0
z0 2
y0 2
y0 z0
ρ κf1 =(k ) ∇z 0 Uz 0 + (k ) ∇y0 Uy0 + k k (∇y0 Uz 0 + ∇z 0 Uy0 )9 |k0 | ∂|k0 |




∂f0
1
z0
z 0 ∂f0y 0 ∂f0y0+
k ∇00 Uz 0 + k ∇00 Uy0 − 0 k+k(6.32)∂|k0 ||k |∂z 0 |k0 |∂y 0 |k0 |0where κ = n0 σT /ρ0 is the opacity. Since we used a relativistic approach to derive this expression,
we should be able to write it in a covariant fashion. This is indeed the case, the covariant formbeing

ρ0 κf1 =10 1 ∂f0 µ0 ν 0 σ0 α0∂f0 ν 0 σ0 µ0
1 µ0 ν 0
0 Uν 0 −∇ΠΠΠ 0 k ∇µ0 ,|k0 | f0 , (6.33)
0k U
0 Π α0 k k ∇µ0 Uν 0 +µσ
σ
9 |k0 | ∂|k0 |∂|k0 ||k0 | ν

177
0 0

0

0

0 0

where Πµ ν = U µ U ν + g µ ν is the projection tensor introduced by Eckart (1940) (see section 1).
0

We introduced the quantity ∇µ0 ,|k0 | to signify the derivative with respect to coordinate xµ holding|k0 | constant. This is an important distinction if we want to calculate derivatives of f1 as we mustq0
use the relativistically-correct definition |k0 | = gi0 j 0 k i0 k j 0 and gi0 j 0 depends on xµ .
The preceding analysis only considered a single fluid element. However, the location of ourorigin, tantamount to the position of the fluid element under consideration, is arbitrary, meaningthat equation (6.33) is applicable to the entire fluid. Also, even though we only considered planarflows, we can easily generalize the approach to include three-dimensional motion and variations,and (6.33) still holds. Greek indices therefore range from 0 to 3 in equation (6.33).
In the next section we will use equation (6.33) and integrals thereof to write the equationsof radiation hydrodynamics, equation (6.1), in terms of the four-velocity of the fluid, the massdensity, and the radiation energy density, to which we will add the continuity equation for the
scatterers to close the system. How do we reconcile these with equation (6.29), which seems to be
an additional constraint? Recall that equation (6.29) was derived by equating the “extra,” isotropic
terms arising from both the collision integral and the derivatives of f0 . If we were to attempt toderive the second order correction to the distribution function, f2 , we would doubtless encountermore isotropic terms arising from the derivatives of f1 and collision integrals of f2 , this time to
first order in the optical depth, which we would have to add to equation (6.29). Equation (6.29)
therefore only captures effects to zeroth order in the mean free path. Although it constrains thespatial and temporal derivatives of f0 appearing in equation (6.33), it has no effect on the formof this equation. However, the gas energy equation (see equation 6.50) should reduce to equation(6.29) in the limit that ρ0 κ → ∞.

6.4

Relativistic, diffusive equations of radiation hydrodynamics
The equations of radiation hydrodynamics, valid in any frame, are given by equation (6.1).

Because we now have the distribution function to the requisite order in the optical depth, we cansimplify those equations by writing the radiation energy-momentum tensor as Rµν = R0µν + R1µν ,178where each Rµν is given by equation (6.3) with the appropriate distribution function. We will
derive each of these tensors in the comoving frame but write them in a manifestly covariant form,
the results then being applicable in any coordinate system.
The comoving, isotropic energy-momentum tensor is given by

0 0
R0µ ν =

Z00k µ k ν f0

d3 k 0,k 00

which we can show is equivalent to

0 0
1
0
0
0 0
R0µ ν = e0 U µ U ν + e0 Πµ ν ,3(6.34)

where

0

Z ∞

e = 4πf0 |k0 |3 d|k0 |(6.35)

0is the isotropic radiation energy density. Since the projection tensor is orthogonal to the fourvelocity, a time-like vector that selects the energy component of a tensor, it is reasonable to associate
e0 /3 with the pressure, or momentum density, exhibited by the fluid. With this association, equation(6.34), not surprisingly, demonstrates that radiation acts like a gas with a relativistic equation ofstate, i.e., one with an adiabatic index of 4/3.
The correction to the energy-momentum tensor,

0 0
R1µ ν =

Z00k µ k ν f1

d3 k 0,k 00(6.36)

will have a number of terms, as evidenced by equation (6.33). Integrating by parts and performing
0 0some simple manipulations, we can show that R1µ ν is given by



2
α0
∇σ0 Uρ0 + ∇ρ0 Uσ0 − gσ0 ρ0 ∇α0 U
9 3
9 15
3



10 4 0 µ0 ν 01α0ν 0 µ0 σ 00 ρ00µ0 ν 0 σ 0−e Π ∇ α0 U −U Π+U Π
4e U ∇ρ0 Uσ0 + ∇σ0 e . (6.37)
9 9
3

0 0
10 4 0 µ0 ν 0
10 4 0 µ0 σ0 ν 0 ρ00
ρ κR1µ ν = −e U U ∇ α0 U α −
eΠ Π

0

179
Written in this manner, it is evident that R1µν transforms like a tensor. It should also be noted thateach term in the tensor contains a derivative of a quantity, either the fluid velocity or the energyof the radiation field, with respect to the optical depth, which is what we expected.
Equation (6.37) differs from equation (6.4), the viscous stress tensor proposed by Weinberg(1971), in two notable ways. The first is that we have not postulated the existence of a temperature; instead we just used the fact that the zeroth-order distribution function is isotropic in the
comoving frame to leave the comoving energy density, given by equation (6.35), as an unknown.
The temperature has in fact been a difficult quantity to define in past treatments (see Weinberg’s
discussion of the reconciliation between the results of Thomas (1930) and Eckart’s general form fora relativistic viscous stress tensor; see also Lima & Waga 1990), and it is reassuring to find thatthe physics is perfectly well-described without invoking such a quantity.
The second difference is contained in the presence of the first term of our stress tensor,
proportional to U µ U ν , which shows that there is a correction to the comoving radiation energy
density – defined to be zero by Eckart (1940) and Weinberg (1971) – given by∆e0 = −

10 4 e00∇α0 U α .
0
9 3ρκ(6.38)

This expression can be understood as follows: imagine that we take a spherical volume of fluid0

and contract it by some amount, i.e., such that ∇α0 U α is negative. Because of the nature of
the Thomson cross section, any radiation intersected by the contracting fluid will be scatteredpreferentially in the direction of motion. Therefore, the radiation energy in this volume will be
increased owing to the in-scattering of photons, which is reflected in equation (6.38). Furthermore,
0 0

if we recall that the term in the stress tensor proportional to Πµ ν can be interpreted as the pressure
exerted by the radiation on a fluid element, we see that∆p0 = −

10 4 e010∇µ0 U µ = ∆e0 .
9 9 ρ0 κ
3(6.39)

This demonstrates that the change in pressure is 1/3 the change in energy, which is what we expect

180– the relativistic nature of the photon gas is preserved independent of the manner in which we
expand the distribution function. It can also be shown that the trace of R1µν is zero, which isanother familiar property of a relativistic gas.
Comparing the other terms in equation (6.37) to the form of an arbitrary viscous stress tensor
given by (6.4), we find

η=

8 e0
27 ρ0 κ(6.40)

for the coefficient of dynamic viscosity, which agrees with the findings of Loeb & Laor (1992).
Thomas (1930) used an incorrect form for the Thomson cross section, so Weinberg (1971) did nothave the factor of 10/9. The proportionality to e0 is sensible, as the viscous effect is mediatedby radiation; therefore, a higher radiation energy density permits a higher transfer of energy andmomentum to neighboring fluid elements. The viscous effect is also proportional to the mean freepath of the radiation, which is also a reasonable result: smaller mean free paths mean that theobserved velocity difference across a mean free path is smaller for a given shear, implying lesstransfer of momentum per scattering. The coefficient of bulk viscosity is found to be1ζ = η,
3(6.41)and is not zero, as predicted by Eckart (1940) and Weinberg (1971) for a radiation-dominated fluid.
Because we did not introduce a temperature, we cannot define a coefficient of heat conduction ina manner analogous to that of Weinberg (1971). However, for the case where the zeroth-order
distribution function is that of blackbody radiation, it can be verified that χ = 4aT 3 /3, which
agrees with his findings.
We can also calculate the correction to the flux of photons, where the flux four-vector is given
by

µ

F =

Z

kµ f

d3 k
.
k0

(6.42)

181
As expected, the zeroth-order flux only has a non-vanishing number density in the comoving frame,
given by0
F00 ≡ N 0 = 4πZ

f0 |k0 |2 d|k|,(6.43)

which can be written covariantly as00F0µ = N 0 U µ .

(6.44)

As for the energy-momentum tensor, we can use equation (6.33) and integrate by parts to write
the correction to the flux vector in terms of N 0 . We find

0
10
1 0 0
0
0
ρ κF1µ = − N 0 U µ ∇α0 U α − Πµ σ

0

9

3



0

α0

3N U ∇α0 Uσ0 + ∇σ0 N

0
.

(6.45)We see that this expression yields0∆N 0 = −Uµ0 F1µ = −10 N 00∇ α0 U α0
9 ρκ

(6.46)

as the correction to the comoving number density of photons, a result in contrast to the analysis ofEckart (1940), who defined ∆N 0 to be zero. However, equation (6.46) has a similar interpretation
to equation (6.38): by noting that a contracting gas preferentially scatters photons in the direction of
motion of the scatterers, one would expect an increased amount of radiation within that contracting
volume. We also find∆e0
4 ∆N 0
=
,
0
e
3 N0

(6.47)

reaffirming the notion that radiation behaves as a relativistic gas and demonstrating that it is these
extra photons, ∆N 0 , that add to the energy of the contracting fluid.
For the case of a cold gas, where T µν = ρ0 U µ U ν , we will, for completeness, write down the

182
full set of equations:0∇µ ρ U


∇µ

µ


= 0,

(6.48)

 0





4 0
e µσ νβ
10 1
8
α
µ ν
α
ρ + e 1−∇α U
U U − ∇µ 0 Π Π∇σ Uβ + ∇β Uσ + gβσ ∇α U3
9 ρ0 κ
27
ρκ


 0 
e
1
1 µν
β
0
µσ ν
νσ µ
0
= 0. (6.49)
4U ∇β Uσ + ∂σ ln e
+ g ∂µ e − ∇µ 0 Π U + Π U
3
3
ρκ
0

The first of these is just the continuity equation. We will also derive the gas energy equation,
obtained by contracting equation (6.49) with the four velocity, which gives

 0



1 0
4 10
e µ
8 e0
µ
α
α
∇µ (e U ) + e ∇µ U =∇µ 0 U ∇α U +∇σ Uβ + ∇β Uσ + gσβ ∇α U Πµσ ∇µ U β
33 9ρκ27 ρ0 κ




 0 
1 µσ
1 e0
e
β
0
β
0
µ
σ
σ
µ
+ Π ∇µ 0 4U ∇β Uσ + ∂σ ln e+
4U ∇β Uσ + ∂σ ln e
2U ∇µ U + U ∇µ U .
3ρκ
3 ρ0 κ
0µ(6.50)The left-hand side is just the change in energy for an adiabatic, γ = 4/3 gas, where γ is the adiabatic
index. Note that if e0 ∼ T 4 and we let ρ0 κ → ∞, the left-hand side equals equation (6.29). The
right-hand side therefore represents the energy added to the radiation during interactions with thescatterers (see section 6 for a discussion concerning the entropy generated by this heat addition).

6.5

Perturbation analysis
Hiscock & Lindblom (1985) showed that the general viscous tensor proposed by Eckart (1940)is unstable to small perturbations in a fluid. An interesting question is whether or not these
instabilities appear in our set of equations.
To answer this question, consider an equilibrium solution (ρ0 , e0 , U µ ) where all of the variables
are constants in space and time, the fluid is motionless and the space is flat. On top of thisequilibrium solution we will impose perturbations on our variables (δρ0 , δe0 , δU µ ) small enough

183
such that their products are negligible. With this configuration, the zeroth-order fluid equations
are trivially satisfied. The first-order perturbations to the energy-momentum tensors areδT µν = U µ U ν δρ0 + ρ0 (U ν δU µ + U µ δU ν ),(6.51)

441
δR0µν = U µ U ν δe0 + e0 (U ν δU µ + U µ δU ν ) + g µν δe0 ,3
3
3(6.52)


e0 µ ν
2
8 e0 µσ νραα∇σ δUρ + ∇ρ δUσ − gσρ ∇α δU
U U ∇α δU −Π Π
9 3 ρ0 κ
27 ρ0 κ
3

 0

0
1
e ρ
1
10 4 e µν
α
µ νσ
ν µσ
0
Π ∇α δU −U Π +U Π
4 0 U ∇ρ δUσ + 0 ∇σ δe . (6.53)−
9 9 ρ0 κ
3
ρκ
ρκ

10 4
δR1µν = −We assumed here that the perturbations are small enough such that gravitational corrections canbe ignored, i.e., δgµν = 0. The first-order conservation equations that must be satisfied are now∇µ (δT µν + δR0µν ) = −∇µ δR1µν .

(6.54)

These four equations must also be coupled to the mass continuity equation, the first-order correction
to which is∇µ (U µ δρ0 + ρ0 δU µ ) = 0.

(6.55)The normalization of the four-velocity, Uµ U µ = −1, demonstrates that δU 0 = 0.
For the present analysis we will restrict our attention to planar flows, such that δU x ≡ 0
and any perturbations in the x direction are exactly zero. In this case, only the ν = 0, y, andz components of equation (6.54) are non-trivial. Carrying out the derivatives, we find that they
become, respectively,∂ 0 4 04
10 e0 ∂
1 1
δe + e ∇i δU i = (1 + ) 0∇i δU i +∇i ∇i δe0 ,∂t
3
3
9 ρ κ ∂t3 ρ0 κ

(6.56)

184
4 e0 ∂ 2
4 0 ∂
8 e0
1 1 ∂ ∂ 0
i
z
i
z 1 ∂
0
+
∇
∇
δU
+2∇
∇
δU
(ρ + e ) δU +δe =
δU z + 0
δe , (6.57)
i
i
z
0
0
2
3 ∂t
3 ∂z
27 ρ κ
3 ρ κ ∂t3 ρ κ ∂t ∂z04
4 e0 ∂ 2
∂
1 ∂ 0
8 e0
1 1 ∂ ∂ 0iyi
(ρ0 + e0 ) δU y +
+ 0
∇
∇
δU
+2∇
∇
δU
δe =
δU y + 0
δe . (6.58)
i
i y
0
3 ∂t
3 ∂y
27 ρ κ
3 ρ κ ∂t2
3 ρ κ ∂t ∂y
We will now impose the restriction that each of our perturbations varies periodically asµδξ = ξ0 eikµ x ,(6.59)

where kµ = (−ω, ki ) and xµ = (t, xi ) (this approach is equivalent to taking the Fourier transform of
the equations). Before substituting these expressions into equations (6.56) – (6.58), however, firstrecall that δR1µν < δR0µν , as required by our ordering scheme adopted when solving the Boltzmann
equation perturbatively. Investigating the correction to the comoving energy density, this inequality
impliese0∇i δU i . δe0 ,ρ0 κor, in terms of the wavenumber,ki δU i
δe0
.
,
ρ0 κ
e0
where we introduced the absolute value signs because of the presence of the imaginary unit. Sinceδe0 ∼ e0 , δU i ∼ 1 and ki ∼ 1/λ, where λ is the wavelength of the perturbation, this inequality
becomes λρ0 κ & 1. We see, therefore, that the optical depth across one wavelength must be
greater than one in order for this perturbation analysis to hold. Equivalently, the wavelength ofthe perturbation, or the scale over which the perturbation acts, must be larger than the mean freepath of a photon. If we induce changes on spatial scales smaller than the mean free path, we will
violate the assumption that the mean free path is a good “smallness” parameter for describing
interactions between the fluid and the radiation.

185
Investigating the other terms in δR1µν , we find that the inequalitye0 ρU ∇ρ δU µ . U µ δe0ρ0 κmust also be upheld, which can be rearranged to give

ω
. 1.
ρ0 κ

(6.60)

Recalling that 1/ω is proportional to the period of oscillation of the perturbation, this inequality
means that the oscillation time scale must be longer than the light-crossing time over the mean
free path.
With the previous inequalities in mind, we will now insert our Fourier modes into equations(6.56) – (6.58). The resulting three algebraic relations can be written as

M µν δξ ν = 0,

(6.61)

where


δe0









δξ ν =  δU z 




δU y

(6.62)

and



1 1 2
 −iω + 3 ρ0 κ k

4
3



e0
ie0 − 19
9 ρ0 κ ω


kz

4
3



19 e0
0
ie − 9 ρ0 κ ω k y





 





0
0
µ
1
ω
4
8
e
16
e
z
0
0
2
z
2
z
y
,

M ν =  3 i − ρ0 κ k −iω(ρ + 3 e ) + 27 ρ0 κ k̃ + 2(k )kk027 ρ κ
 


 


1
ω
16 e0 z y
2 + 2(k y )2y
0 + 4 e0 ) + 8 e0
i
−
k
k
k
−iω(ρ
k̃
0
0
0
3
ρκ
27 ρ κ
3
27 ρ κ
(6.63)
where k 2 ≡ (k y )2 + (k z )2 and k̃ 2 ≡ k 2 + 9ω 2 /2.
If the perturbations δξ ν are to be non-trivial, we demand det(M µν ) = 0, which results ina dispersion relation that gives w(k i ). Before setting the entire determinant equal to zero, recall

186
that physically-meaningful frequencies and wavenumbers (from the standpoint that they can be
described as viscous corrections to the energy momentum tensor) satisfy w/(ρ0 κ) ∼ k i /(ρ0 κ) . 1.
Therefore, a first approximation to the dispersion relation can be obtained by setting all appearancesof 1/(ρ0 κ) to zero in the matrix given by (6.63). Doing so and taking the determinant, we finddet(M µν ) = iω


 


4 0
4 0
4 0 2
0
0
2
ρ + e
ω ρ + e − e k = 0,33
9the solutions to which are clearlyω = 0,

2
ω=±
3

s

e0
k.
ρ0 + 43 e0(6.64)

The first of these represents a perturbation with a constant offset from the surrounding medium.
The second two are the sound waves that propagate through the optically-thick plasma; if e0  ρ0 ,p
we recover the familiar result for the sound speed of a radiation-dominated gas cs = 4p0 /3ρ0 ,√where p0 = 3e0 is the radiation pressure, while for e0  ρ0 , we find cs = c/ 3, which is the correct
ultrarelativistic speed of propagation.
We can show that the full determinant can be written
8 e0 24 00k̃ − iω ρ + e
27 ρ0 κ
3








 
1 k2
8 e0
4 0
4 0
19 ω
ω
2
2
0
×
−
iω
k̃
+
2k
−
iω
ρ
+
e
−
e
i
−
i
−
k2 ,
3 ρ0 κ
27 ρ0 κ3
9
9 ρ0 κρ0 κ

det(M µν ) =(6.65)
meaning that the dispersion relation is given by the solutions to





8 e0
9 2
4 0
2
0
k + ω − iω ρ + e = 0,
27 ρ0 κ
2
3



1 k2
− iω
3 ρ0 κ



(6.66)








8 e0
3 2
4 0
4 0
19 ω
ω
2
0
k
+
ω
−
iω
ρ
+
e
−
e
i
−
i
−
k 2 = 0, (6.67)
9 ρ0 κ
2
3
9
9 ρ0 κ
ρ0 κ

187
where in each of these we used the definition of k̃ 2 .
To solve these, recall that the perturbation frequencies permissible in our analysis are approximately given by equation (6.64) with small corrections of order 1/(ρ0 κ). If we set ρ0 κ = 0
in the preceding two polynomials, we see that the first reduces to ω = 0, while the second gives
the sound waves. In (6.66), we will therefore let ω = ω1 /(ρ0 κ) and keep only first order terms in
1/(ρ0 κ). Doing so, we find

ω1 ' −i8e0k2 .
27 ρ0 + 43 e0(6.68)

Since the perturbations scale as δξ ν ∼ e−iωt , ω1 is a decaying solution. The e-folding timescale ofconstant-offset perturbations is thus

27 ρ0 c2 + 43 e0 ρ0 κ
τd '
,
8
e0 c
k2

(6.69)

where we explicitly reintroduced the speed of light c, which shows that smaller-scale fluctuations
decay more rapidly.
To solve equation (6.67), we will let ω = ω± + ω1 /(ρ0 κ), where ω± is given by2
ω± = ±3se0k,
ρ0 + 34 e0(6.70)

the zeroth-order (in the mean free path) solution to the dispersion relation. Using this approximation and keeping only highest-order terms, we find

ω1 = −

32 0 00 2
ρ e + 32i (ρ0 )2 + 27
27 (e ) 2k .
6
(ρ0 + 34 e0 )2(6.71)The decay timescale for traveling waves is therefore on the same order as that for the constant-offset
perturbations.
When we solved these dispersion relations, we used the fact that ω/(ρ0 κ) must be small forour analysis to hold. If we had not taken this perturbative approach, however, we would have

188
obtained conflicting results. For example, the exact solution to equation (6.66) is
v

u

 e0
2 
u
ρ0 + 43 e0128
ρ0 κ kt
ω = i 8 e0
1± 1+.
81 ρ0 + 43 e0
3 ρ0 κ

(6.72)

If we take the root with the negative sign, the result reduces to equation (6.68) in the limit that we
keep only first-order corrections in k/(ρ0 κ); but, if we take the positive sign, we find that the solution
is a growing mode, showing that the fluid perturbations are unstable. However, since these growing
modes always have ω/(ρ0 κ) > 1, our treatment of the equations of radiation hydrodynamics, which
only considers fluid perturbations on timescales greater than the light-crossing time over one opticaldepth, is invalid.
One can understand the physical origin of these growing modes by imagining that we oscillate
a fluid parcel in such a way that its period is one-half of the light-crossing time over the mean freepath that separates it from its neighboring fluid element. In this case, by the time the informationfrom the neighboring fluid parcel returns back to the originally-perturbed parcel, the relative velocity between the two fluid elements will be in the same direction as the initial perturbation. The
viscous force will then serve to increase the amplitude of the oscillation of the fluid element and,taken over many optical depth light-crossing times, this effect will only be amplified, resulting in a
runaway process.
It is for this reason that the analysis of Hiscock & Lindblom (1985) resulted in the predictionof growing modes. If we take the positive root of equation (6.72) and use numbers for water atroom temperature, we find τ = 1/ω ' 10−35 s is the e-folding time of the perturbations. As pointed
out by Hiscock & Lindblom (1985), viscous heating would cause water to boil on an absurdly shorttimescale. This result is incorrect, however, because of the inability of the fluid to communicate
over such timescales; these high frequencies blatantly violate the assumption that the mean freepath of the radiation is sufficiently small to describe local fluid deformations.
For a general coefficient of dynamic viscosity η, this requirement can be translated to a statement similar to, “frequencies must satisfy the inequality ωη/e0 . 1, where e0 is the locally-observed

189
energy density of the fluid, in order for viscous effects to describe physically the perturbations thattake place in the medium.” It is also not surprising that in the limit of c → ∞, this result disappears
as the light-crossing time is zero, meaning that the Newtonian limit of these equations will never
suffer from such growing instabilities. We can see this explicitly by reinserting the factors of c into
equation (6.66), which reads





8 e0
9 ω2
iω 0 2 4 0
2k +−ρ c + e = 0.
27 ρ0 κ2 c2
c3
Ignoring the factor of ω 2 /c2 and solving reveals only decaying solutions.

6.6

Discussion and Conclusions
We have shown, in agreement with intuition and past efforts, that radiation behaves like aviscosity in the optically-thick limit. Our analysis is based on the general relativistic Boltzmann
equation, general relativistic corrections being necessary because the scattering, for which we usedthe differential Thomson cross section, is handled most easily in the local, accelerating frame of thefluid. The correction to the photon distribution function, given explicitly by equation (6.33), wasused to calculate the correction to the energy-momentum tensor of the radiation (equation (6.37)),
and many of the terms agreed with the predictions of Eckart (1940) and Weinberg (1971).
There are, however, a few differences between our form for the correction to the radiationenergy-momentum tensor and the general viscous stress tensor proposed by Weinberg (1971), thefirst being that ours does not depend on thermodynamic considerations such as the assignmentof a temperature to the zeroth-order distribution function. The temperature-independence of ourequations means that the isotropic radiation distribution need not correspond to that of blackbody
radiation. Indeed, because our approach treats the radiation and the scatterers as two interacting
media, it is not clear what a single temperature would mean. The second difference is that ourviscous tensor contains a correction to the comoving energy density of the radiation, which is
not predicted by previous approaches to deriving the viscous tensor for a relativistic fluid. We

190demonstrated that this term appears naturally from the manner in which radiation is scattered ina contracting (or expanding) fluid element. This correction is consistent with, and in fact expected
from, the notion that the radiation is a relativistic (adiabatic index of 4/3) gas. Furthermore, ourmodel predicts a correction to the number density of photons (equations (6.45) and (6.46)), thepresence of which corresponds exactly with the increase in the energy density of the fluid.
We performed a perturbation analysis on our equations, took the Fourier transform of theperturbed equations and calculated the dispersion relation (equations (6.66) and (6.67)). Because
our treatment is concerned with the limit in which the stress tensor is a small perturbation proportional to the mean free path of the radiation, the wavenumbers (k i ) and frequencies (ω) of theFourier modes must satisfy k i /(ρ0 κ) ∼ ω/(ρ0 κ) . 1. These inequalities mean, sensibly, that fluid
motions cannot alter the radiation field over scales smaller than the mean free path and at rates
faster than the light-crossing time over the mean free path. With these inequalities in mind, we
showed that the viscous terms cause the perturbations to decay exponentially, a familiar result.
Interestingly, if one solves the dispersion relations without regard to the inequality ω/(ρ0 κ) . 1,
one recovers growing mode solutions. This result sheds light on the results of Hiscock & Lindblom(1985), who demonstrated that the Eckart (1940) relativistic viscous stress tensor is unstable tosmall perturbations. Our approach shows that such instabilities only arise when one violates theassumption that the viscous nature of the fluid is a “good” approximation, i.e., when the light crossing time over the viscous length scale, defined in any problem by η/(c e0 ), η being the coefficient ofdynamic viscosity and e0 the locally-observed energy density of the fluid, is short enough to describe
local fluid deformations. The unstable nature of the modes also vanishes in the non-relativistic limit
as the light-crossing time over any distance is formally zero.
Weinberg (1971) calculated the bulk viscosity of a relativistic radiating fluid to be zero.
Our analysis, on the contrary, determined the coefficient to be ζ = η/3, which can be quite large
for radiation-dominated plasmas. It might therefore be expected that our model would predicta different rate of entropy production in the universe. However, if we use his expression for theentropy production rate (see the expression above equation (2.20) in Weinberg 1971) and confine

191
our analysis to the case in which zeroth-order distribution function can be described by that ofa blackbody, we find that our bulk viscosity and the correction to the comoving energy density
exactly cancel. We therefore find that Weinberg’s results concerning the entropy generation in theearly universe by small anisotropies in the radiation field are upheld.
The previously-developed equations are applicable to a range of physical scenarios, the cosmological evolution of the early universe being one such application (see, in addition to Weinberg
1971, Caderni & Fabbri 1977; Harrison 1977; Tauber 1978; Johri & Sudharsan 1988; Hu & Dodelson
2002; Khatri et al. 2012). Another application would be in analyzing the physics of the boundary
layers established between radiation-dominated jets and their environments (Arav & Begelman,
1992), as viscous dissipation due to radiation could be important – especially in the case where themotion of the outflow is relativistic (Walker, 1990). Such an analysis could shed light on the peculiar
event Swift J1644+57, thought to be the first-observed, jetted, super-Eddington tidal disruption
event (TDE) (Burrows et al., 2011; Bloom et al., 2011; Cannizzo et al., 2011; Zauderer et al., 2011).
If the transient jet carried away the accretion luminosity generated during the gravitational infallof tidally-stripped debris (Chapter 5), the interaction between the jet and the overlying envelope
may be well-described by the equations radiation hydrodynamics in the viscous limit. Comparing
the theoretical results with observations could yield insight into the terminal Lorentz factor of theoutflow, the properties of the tidally-disrupted star, and the mass of the black hole residing in thecenter of the host galaxy.
Applying these equations to the outflows predicted by the collapsar model of gamma-ray
bursts (GRBs) (Woosley, 1993; MacFadyen & Woosley, 1999), wherein the core of a massive, dying
star collapses directly to a black hole from which a jet is launched, may also prove fruitful. If themechanism that accelerates the outflow is ultimately derived from the prompt accretion of material
onto the newly-formed black hole, i.e., if the fireball model correctly describes the dynamics (Rees& Meszaros, 1992), radiation would contribute significantly to the energetics of the jet. Even if
the energy is provided by the spin of the black hole (Blandford & Znajek, 1977) or the accretion
disk (Blandford & Payne, 1982), radiation could still play a large part in determining the dynamics

192
of the outflow. An accurate representation of the interaction of the jet with the overlying stellar
envelope and the circumstellar environment may therefore be obtained by employing the viscous
equations of radiation hydrodynamics. A comparison of the theoretical expectation gleaned fromsuch an analysis with the promptly-emitted gamma rays and the X-ray afterglow could yield new
information concerning the progenitors of GRBs and their surrounding environments.
Owing to their apparent complexity, the equations of radiation hydrodynamics, when applied
to supercritically accreting compact objects and their surroundings, are often solved numerically(Eggum et al., 1988; Okuda, 2002; Okuda et al., 2005; Ohsuga et al., 2005; Ohsuga, 2007). Simulations of such systems have also been extended to include magnetic fields (radiation magnetohydrodynamic; RMHD), the presence of which is potentially important not only for the dynamics ofthe gas but also for MRI-induced accretion and jet collimation (Turner et al., 2003; Ohsuga et al.,2009; Sa̧dowski et al., 2014; McKinney et al., 2014). In these simulations, strong gravity is incorporated either through the pseudo-Newtonian potential of Paczyńsky & Wiita (1980), or by using thecovariant set of equations and the Kerr geometry. In all of these numerical schemes, the moment
formalism is adopted (see the introduction) when solving for the properties of the radiation andincorporating the coupling of the radiation to the gas. In addition, a “closure” relation is adopted
that allows one to truncate the number of moments needed to close the system of equations at a
finite level. Given our preceding analysis, an interesting question is whether or not these closure
relations accurately capture relativistic radiation viscosity.
Flux-limited diffusion (Levermore & Pomraning, 1981), wherein the flux is proportional to
the gradient of the energy density, is one such closure scheme used by, e.g., Ohsuga et al. (2005) andOhsuga et al. (2009). This approximation allows one to interpolate between optically-thick and thinregimes, useful for an optically-thick disk that launches an optically-thin jet – the result observedin many of these simulations. An issue with this type of closure, especially when considering thepresent paper, is that the viscosity must be included separately and is not a direct result of theequations of radiation hydrodynamics. Ohsuga et al. (2005), for example, used a modified-alphaprescription for the coefficient of viscosity, setting the coefficient of dynamic viscosity to η ∼ p/ΩK ,193
where p is the total (gas plus radiation) pressure and ΩK is the Keplerian velocity. This form forthe viscosity clearly does not reduce to equation (6.40) in the radiation-dominated, viscous limit.
The modified-alpha prescription also does not incorporate relativistic effects, those due to strong
gravity or high velocities, both of which could be important in the vicinity of the black hole andin regions of high shear between the jet and the inflated disk. It is therefore unlikely that theflux-limited diffusion closure scheme accurately reproduces the effects of radiation viscosity in thehigh-optical depth limit.
The M1 closure scheme (Levermore, 1984) assumes that there exists some reference frame,
the radiation rest frame (not necessarily the same as fluid rest frame), in which the radiation flux
vanishes and the pressure is one third of the energy density (the Eddington approximation), andcan also interpolate between optically-thick and -thin regimes. This scheme has been extended toincorporate general relativistic effects (Sa̧dowski et al., 2013), and has been employed by Sa̧dowski
et al. (2014) and McKinney et al. (2014) to study the role of radiation on hyperaccreting black
hole-disk systems. Because the frame in which the Eddington approximation applies is not thefluid frame, there are small corrections to the energy-momentum tensor of the radiation in the fluid
rest-frame when the optical depth is large, which is what one expects. If one keeps the lowest-ordercorrections to the fluid-frame pressure tensor (see equation (34) of Sa̧dowski et al. (2013)), the result
0 0

0 0

is P i j ' δ i j e0 /3+O(F 2 ), where F is the fluid-frame energy flux, which demonstrates that, indeed,
the fluid-frame radiation pressure reduces to the Eddington approximation when the optical depth
is large. In the viscous limit, the rest-frame energy flux is proportional to gradients of the energy
density and the velocity over the mean free path of the radiation (see the (0, i) components of ourequation (6.37) or note that the diffusion limit is F ∼ ∇e0 ). The M1 scheme therefore indicatesthat the first-order deviation of the pressure, or the momentum flux, from the Eddington closure is
proportional to the square of the derivative of fluid quantities over the optical depth. This result,however, is in direct contrast with the (i, j) components of our equation (6.37) that demonstrate,consistent with the manner in which viscosity usually operates, that the lowest-order correction to
the momentum flux in the viscous limit is proportional to the gradient of fluid quantities over the194
optical depth, i.e., it is of the same order as the energy flux. The M1 closure therefore does notaccurately reproduce the viscous transport of momentum in the high-optical depth limit. Due tothe covariant manner in which the M1 scheme is incorporated and the self-consistent inclusion of
the radiation terms (i.e., no ad-hoc form for the viscosity), higher-order effects of the anisotropic
radiation field are likely well-reproduced in the interaction between the relativistic jetted outflows
and the inflated torus observed in the simulations of Sa̧dowski et al. (2014) and McKinney et al.
(2014).
Finally, Jiang et al. (2014) recently performed a non-relativistic, 3D, RMHD simulation of
the gas around a ∼ 6M

black hole. Contrary to the previously-mentioned authors, they did

not assume a closure relation to compute the moments of the radiation energy-momentum tensor
and instead determined the pressure of the radiation directly, offering an unbiased depiction of
the manner in which radiation transfers energy and momentum between neighboring gas parcels.
The non-relativistic nature of the simulation, however, may underestimate the viscous effects in thetransition between the mildly-relativistic outflow observed in their simulation and the disk material.
Having written down the equations of radiation hydrodynamics in the viscous limit, theauthors plan to pursue the applications of those equations to relativistic, radiative shear layers in
a companion paper. In the process, we plan to quantitatively assess how the differences between
our radiation-viscous fluid equations and those of Weinberg (1971) affect the physical evolution ofastronomical systems.

Chapter 7

Viscous boundary layers of radiation-dominated, relativistic jets. I. The
two-stream model

7.1

Introduction
Astrophysical jets almost certainly exist as aggregates of massive particles, magnetic fields,

and radiation. In certain scenarios, however, the contribution of radiation to the energetics of theoutflow far outweighs those of the particles and magnetic fields, meaning that one can essentiallyneglect the presence of the latter two entities.
One such scenario occurs in the collapsar model of long gamma-ray bursts (GRBs; Woosley
1993; MacFadyen & Woosley 1999). In this model, the core of a massive, evolved star collapses
directly (or with a short-lived neutron star phase) to a black hole during the infall stage of a typeII supernova. The energy released by the material accreting onto the black hole, and ultimately
observed as the gamma-ray emission, is collimated into bipolar jets – the jet formation confirmed
by energetics arguments (Waxman et al., 1998; Fruchter et al., 1999; Frail et al., 2001) and theobservations of breaks in the X-ray afterglow light curves (Panaitescu, 2007; Dado et al., 2008;
Racusin et al., 2009) – and is often sufficient to unbind the stellar envelope, resulting in a supernova(Galama et al. 1998; Bersier et al. 2004; Kamble et al. 2009; Levan et al. 2014, but see Fynbo et al.
2006). If one assumes that the mass of the remnant black hole is on the order of a few solar masses,
its accretion luminosity exceeds the Eddington limit by roughly ten orders of magnitude, meaningthat radiation pressure, even if the flux is nearly isotropic, is likely an important mechanism for
driving and sustaining the outflow (the fireball model; Rees & Meszaros 1992). Even if the jet is196
launched by magnetohydrodynamical mechanisms (Blandford & Znajek, 1977; Blandford & Payne,1982), radiation could still play a prominent role in determining the dynamics of the jet. Arguments
concerning the time necessary for the jet to break through the stellar envelope also seem to disfavor
Poynting-dominated jets (Brown et al. 2015; but see Mundell et al. 2013).
Jets produced during tidal disruption events (TDEs; Giannios & Metzger 2011) – when astar is destroyed by the tidal force of a supermassive black hole – could provide another class ofradiation-dominated outflow. After the star is tidally disrupted, roughly half of the shredded debris
remains bound to the black hole and returns to the tidal disruption radius. If the black hole has
a mass less than roughly 107 M , that rate of return can exceed the Eddington limit of the blackhole by orders of magnitude for a significant amount of time (on the order of days to months;
Evans & Kochanek 1989; Strubbe & Quataert 2009). Provided that this material can rapidly
accrete onto the black hole, which is likely the case due to the tidal dissipation of kinetic energy(Kochanek, 1994; Guillochon et al., 2014b) and relativistic precession effects (Rees, 1988; Evans& Kochanek, 1989), the energy released during the accretion process will also be supercritical. It
was during this supercritical phase that the event Swift J1644+57 was seen to have an associatedjetted outflow (Burrows et al., 2011; Bloom et al., 2011; Cannizzo et al., 2011; Zauderer et al.,2011) (the source Swift J2058+05 may provide another example of a jetted, super-Eddington
TDE; Cenko et al. 2012). Although the jet launching mechanism for this event is uncertain, themagnetic field of the tidally-disrupted star, assuming its flux is approximately conserved, is almost
certainly insufficient to power the outflow. Therefore, unless one invokes the existence of a fossil
magnetic field (Tchekhovskoy et al., 2014; Kelley et al., 2014), the radiation pressure associatedwith the accretion luminosity likely plays some role in powering the jet. At any rate, the radiation
released during the supercritical accretion process affects the dynamics of the collimated outflow
and contributes substantially to its overall energy and momentum.
Collapsar jets inject a significant amount of energy into the overlying stellar envelope as
they punch their way into the circumstellar medium, creating a pressurized “cocoon” of shockedmaterial with which the jet interacts (Morsony et al., 2007; Lazzati et al., 2007; López-Cámara197
et al., 2013), and the progenitors themselves – usually taken to be highly-evolved, Wolf-Rayet stars(Matzner, 2003; Woosley & Bloom, 2006) – are likely sustained primarily by radiation pressure.
The initial stages of collapsar jet propagation are therefore characterized by the transfer of energy
and momentum between two radiation-dominated fluids. Because of the high accretion rates andlow specific angular momentum, the fallback disks generated during the super-Eddington phaseof TDEs are likely optically and geometrically thick and radiation pressure-supported (Rees 1988;
Loeb & Ulmer 1997, Chapter 5). In the zero-Bernoulli accretion (ZEBRA) model of Chapter 5, forexample, the accreting material is inflated into a quasi-spherical envelope that surrounds the black
hole. The dynamics of the jets of supercritical TDEs are therefore also modulated by the presenceof a radiation pressure-supported, external medium.
Previous authors used the supersonic propagation of the outflow to model the interactionof the collapsar jet with the overlying envelope as an oblique shock-boundary layer structure(Bromberg & Levinson, 2007; Kohler et al., 2012; Kohler & Begelman, 2012, 2015). In thesemodels, the outflow is assumed to consist of a perfect, single fluid with a relativistic equation ofstate, and some have included the presence of magnetic fields (e.g., Kohler & Begelman 2012). A
more realistic picture, however, is obtained by considering the jet as a composite of the massive
scatterers present in the outflow and the radiation that accompanies it. Indeed, this approach
constitutes the underlying framework of radiation hydrodynamics. Furthermore, non-ideal, i.e.,
viscous, effects will tend to “smear” the discontinuities otherwise present in the system, resulting
in a more gradual transition of the fluid quantities between the jet and the environment.
In the limit that the mean free path of a photon is small, radiation acts like an effective
viscosity, with a coefficient of dynamic viscosity that depends both on the radiation pressure andthe density of scatterers (see section 2 of this paper), and transfers momentum and energy between
neighboring fluid elements (Weinberg, 1971; Loeb & Laor, 1992). In the radiation-dominatedinteraction between collapsar jets and TDE jets and their respective ambient media, the effects ofradiation viscosity should be quite large. Arav & Begelman (1992) considered the effects of radiation
viscosity on the evolution of boundary layers in the two-stream approximation (see section 3 of this198
paper); their treatment, however, was of a non-relativistic nature, meaning that their results have
limited applicability to GRB and TDE jets.
In this paper, the first of two, we consider the effects of radiation viscosity on the propagationof relativistic jets in radiation-rich environments. In section 2 we present the equations of radiationhydrodynamics in the viscous limit. Section 3 applies those equations to the two-stream problem,
wherein the jet and ambient medium are considered to be two separate, interacting fluids, and wecompare our results to the non-relativistic treatment of Arav & Begelman (1992). In section 4
we discuss the results of the analysis and comment upon the application of our models to the jets
produced by supercritical TDEs such as Swift J1644+57, GRBs, and other astrophysical systems.
In a second paper (Coughlin & Begelman, 2015) we present an alternate model, the free-streaming
jet solution, and compare it to the two-stream solution presented here.

7.2

Governing equations
As mentioned in the introduction, radiative forces behave like an effective viscosity in thepresence of shear, when the change in fluid quantities across the mean free path of a photon issmall. The precise means by which this viscous coupling manifests itself can be determined byanalyzing the Boltzmann equation.
In Chapter 6 we used the general relativistic Boltzmann equation for Thomson scattering todiscern the effects of radiation viscosity in the presence of both relativistic velocities and gravitational fields (i.e., in accelerating reference frames). Instead of reproducing their work here, we will
simply quote the equations of radiation hydrodynamics for a cold gas (gas pressure much less thanboth the gas rest mass density and the radiation pressure) that result from their analysis (see theirequation 49):


∇µ

 0





4 0
10 1
1 µν
8
e µσ νβαµ ν0α
ρ + e 1−∇α U
U U + g ∂µ e − ∇µ 0 Π Π∇σ Uβ +∇β Uσ +gβσ ∇α U3
9 ρ0 κ
3
27
ρκ
 0 


1
e
µσ ν
νσ µ
β
0
− ∇µ 0 Π U + Π U
4U ∇β Uσ + ∂σ ln e
= 0. (7.1)
3
ρκ
0

199
Here the speed of light has been set to one, Greek indices range from 0 – 3, ρ0 is the fluid restframe density of scatterers, e0 is the fluid rest-frame radiation energy density, κ is the scattering
opacity (in units of cm2 g−1 ), gµν is the metric of the spacetime, ∇µ is the covariant derivative,
U µ is the four-velocity of the flow, and Πµν = U µ U ν + g µν is the projection tensor. The Einsteinsummation convention has been adopted here, meaning that repeated upper and lower indices imply
summation. This equation shows that the coefficient of dynamic viscosity, η, for an optically-thick,radiation-dominated gas is

η=

8 e0
,
27 ρ0 κ(7.2)

which agrees with previous findings (Blandford et al., 1985; Loeb & Laor, 1992); note that thisspecific coefficient is only for the case when the gas and radiation interact through Thomson scattering. The gas energy equation, which will also be useful for us, can be obtained by contracting
equation (7.1) with Uν , which we can show becomes (see equation 50 of Chapter 6)




 0
1 0
4 10
e µ
8 e0
µ
α
α
∇µ (e U ) + e ∇µ U =∇µ 0 U ∇α U +∇σ Uβ + ∇β Uσ + gσβ ∇α U Πµσ ∇µ U β
33 9ρκ27 ρ0 κ
 0 




1 µσ
e
1 e0
β
0
β
0
µ
σ
σ
µ
+ Π ∇µ 0 4U ∇β Uσ + ∂σ ln e+
4U ∇β Uσ + ∂σ ln e
2U ∇µ U + U ∇µ U .
3ρκ
3 ρ0 κ
0µ(7.3)
To close the system, we require that the normalization of the four-velocity be upheld and thatparticle flux be conserved:

Uµ U µ = −1,0∇µ ρ U

µ

(7.4)


= 0.

(7.5)

Equations (7.1) and (7.3) – (7.5) constitute six linearly independent equations for the six unknowns
U µ , e0 , and ρ0 .

200
In addition to the energy density of the radiation, e0 , one can also calculate the number densityof photons by requiring that the number flux, F µ , be conserved. One can show (see Chapter 6)that the equation ∇µ F µ = 0 becomes, in the viscous limit,






1 10 0 µ
1 µσ
0 µ
σ
0 α
µ
0
∇µ N U = ∇µ 0,
N U ∇σ U + N U ∇α U + Π ∇σ N
ρκ 9
3

(7.6)

where N 0 is the rest-frame number density of photons. Once we solve the equations of radiationhydrodynamics for the four-velocity of the fluid and the mass density of scatterers, we can solve
equation (7.6) for the number flux of photons.
The goal of the next two sections is to apply equations (7.1) and (7.3) – (7.6) to the boundary
layers established between fast-moving jets and their environments. For a more thorough discussionof the nature of the equations of radiation hydrodynamics in the viscous limit, we refer the reader
to Chapter 6.

7.3

Two-stream boundary layer
Arav & Begelman (1992) considered the Blasius (1908) boundary layer problem, wherein one

analyzes the dynamics of viscous flow over a semi-infinite, rigid plate, with the viscous effects attributed to radiation. They showed, however, that the requirement that both velocity components
vanish on the lower plate, the no-slip condition, results in a divergent boundary layer thickness.
The authors then examined the more physical scenario of two interacting fluids, one moving at
some asymptotic velocity and the other asymptotically stationary, known as the two-stream approximation. In this case the no-slip condition no longer applies, and they were able to show thatthe boundary layer thickness remains finite.
The treatment of Arav & Begelman (1992) was non-relativistic, meaning that their results
have limited utility when one considers the boundary layers established between GRBs and superEddington TDEs and their ambient media. The enthalpy of the radiation, which is not ignorable
in radiation-dominated flows, was also not included in their momentum equation. Here we extend

201
their analysis to the relativistic regime and we include the radiation enthalpy.

7.3.1

Basic setup

We assume that the flow is plane-parallel with no variation in the x-direction. The covariant
derivatives in equations (7.1) and (7.3) – (7.5) can therefore be replaced by ordinary partial derivatives. Even though this simplification significantly reduces their complexity, the most compact
representation of the equations is still given by (7.1) and (7.3) – (7.5) with ∇µ → ∂/∂xµ , so we do
not write them again here.
The majority of the motion is along the z-axis, meaning vz  vy and vx ≡ 0. At some initialpoint y = z = 0, the “jet,” whose constant, asymptotic (y → ∞) velocity is denoted vj , encounters
the ambient medium, the asymptotic (y → −∞) velocity of which is zero. The asymptotic densitiesof the jet and the ambient medium, denoted ρ0j and ρ0a , respectively, are both taken to be constant.
The line y = 0 represents the surface that divides the jet and ambient material, and consistent
with any boundary layer analysis, we also assume that the gradient along y is much greater thanthat along z, so ∂/∂z  ∂/∂y.

7.3.2

Boundary layer equations

The complexity of equations (7.1) and (7.3) – (7.5) can be reduced by introducing the boundary layer thickness δy such that δ ∼ δy/δz is a small parameter when δz is chosen to be a typicallength scale in z. By keeping terms only to lowest order in δ, we will recover a set of reduced
boundary layer equations.
To determine how δ depends on asymptotic fluid quantities (it is the reciprocal of the squareroot of the Reynolds number in the classical Blasius boundary layer), we compare the lowest-orderterms in δ in the gas energy equation to the inviscid terms. By equating these terms we are requiringthat the viscous heating of the radiation contribute a non-negligible increase in the entropy of thefluid, but because the inviscid terms are proportional to the divergence of the four-velocity (seeequation 8.11), this equality can only be true when the gas is compressible, i.e., when the flow

202
velocity is supersonic. When the flow becomes very subsonic, the energy equation can be ignored
as the fluid is essentially incompressible. Making the substitutions ∂/∂y ∼ 1/δy, ∂/∂z ∼ 1/z,v ∼ vj and ρ0 ∼ ρ00 in equation (8.3), where ρ00 = ρ0j if we are in the jet (y > 0) or ρ0a if we are in
the ambient medium (y < 0), we find that the boundary layer thickness scales as

δ2 ∼

1

,
ρ00 κ z Γj vj(7.7)

where Γj = (1 − vj2 )−1/2 is the Lorentz factor of the jet. The boundary layer thickness therefore√
scales roughly as 1/ τ , where τ ∼ ρ00 κz is the fluid-frame optical depth along z.
We can now use our expression for δ (8.7) to keep only lowest-order terms in equations (7.1)
and (7.3) – (7.5). The resulting ν = y, ν = z, gas-energy and continuity boundary layer equations
are, respectively,

∂e0
= 0,
∂y

∂
∂xµ







4 0
8 ∂ e0 ∂ 
1 de0
0
µ
ρ + e U Γvz +−Γvz = 0,33 dz
27 ∂y ρ0 κ ∂y

(7.8)

(7.9)



∂
1 0 ∂U µ
8 e0 ∂Uµ ∂U µ
0 µ
e
U
+
e
=
,
∂xµ
3 ∂xµ
27 ρ0 κ ∂y ∂y(7.10)





∂ 0
∂ 0
ρ Γvz +
ρ Γvy = 0.
∂z∂y(7.11)

The first of these demonstrates, as in the non-relativistic limit, that the pressure is constant acrossthe boundary layer. For the remainder of this section we will assume that e0 (z) = e00 , i.e., that theradiation energy density is independent of z.

203
7.3.3

Self-similar approximation

The solution to the continuity equation (7.11) can be obtained by introducing the stream
function ψ through the definitions

ρ0 κΓvz =∂ψ,∂y(7.12)∂ψ,∂z(7.13)

ρ0 κΓvy = −which manifestly solves the continuity equation; we introduced a factor of κ, the opacity, to ensurethat ψ is dimensionless. These relations also demonstrate that vy ∼ δvz , which is what we expect:
the velocity in the direction perpendicular to the majority of the motion is reduced by a factor of
δ. Note, however, that the definition of the stream function now involves the density, which is notconstant in this analysis.
As is done in the standard Blasius treatment, we assume that the stream function varies
self-similarly as

r
ψ=

8 0
ρ κ Γj vj z 1/2 f (α),
27 0(7.14)wherey
α = y/δy = √
z

r

27 0
ρ κ Γj vj
8 0(7.15)

is our self-similar variable (we used equation (8.7) for the boundary layer thickness) and f is afunction to be determined from equations (8.12) and (8.11). We will also assume that the density
varies self-similarly asρ0 = ρ00 g(α),
where g is a second function.

(7.16)

204
One can use these definitions in equations (8.12) and (8.11) to derive a set of coupled,
nonlinear, ordinary differential equations for f and g. It is mathematically convenient, however, todefine a new self-similar variable ξ byZ α
ξ=g(α̃) dα̃,(7.17)

0where α̃ is a dummy variable of integration, and write the functions f and g in terms of thisvariable. This approach is similar to the one taken by Arav & Begelman (1992).
With this parametrization, the velocities areΓvz = Γj vj fξ ,1
ρ Γvy = √
2 z
0

r



8 ρ00 Γj vj
αfξ g − f ,27
κ(7.18)(7.19)

where a function with a subscript ξ denotes the derivative of that function with respect to ξ, i.e.,fξ = df /dξ, fξξ = d2 f /dξ 2 , etc. Substituting these relations and the self-similar scaling for ρ0 ,equation (7.16), into equations (8.12) and (8.11), we find the following self-similar equations for f
and g:

g fξ (fξξ )2
1
4− g + µ f fξξ + µ Γ2j vj2= µ g fξξξ ,2
3
1 + vj2 Γ2j (fξ )2(7.20)

g 2 (fξξ )2
3
gξ f = Γ2j vj2,
2
1 + Γ2j vj2 (fξ )2(7.21)

where we have defined µ as the ratio e00 /ρ00 . The term −2 µ f fξξ /3 and the last term on the lefthand side of equation (7.20) were absent in the treatment of Arav & Begelman (1992) because
they did not include the enthalpy of the radiation. In the non-relativistic, µ  1 limit, equation(7.20) reduces to the standard Blasius equation (by rescaling the self-similar variable) and g and f
decouple from one another, as was found by Arav & Begelman (1992).

205
Before proceeding further, recall that the two-stream problem separates the jet and ambientmaterial into two distinct media, the dividing line for our problem chosen to be y = 0, and that theasymptotic densities attained in these two media are ρ0j and ρ0a for the jet and ambient material,
respectively. Therefore, in equations (7.14), (7.15) and (7.16), the parameter ρ00 refers to either ρ0jor ρ0a depending on the sign of α and, hence, ξ. The functions f and g are thus piecewise defined
about y = 0, as are the self-similar variables α and ξ, with solutions for ξ > 0 corresponding to jet
quantities and those for ξ < 0 corresponding to the ambient medium. Equations (7.20) and (7.21)
should therefore be interpreted as a total of four equations for four functions: f defined in the jet,f defined in the ambient medium, g defined in the jet, and g defined in the ambient medium. Notethat µ can also differ between the two media, depending on the asymptotic density (but e00 must
be continuous across y = 0 because of equation (8.10)).
Keeping in mind this subtlety of equations (7.20) and (7.21), we must additionally determine
the boundary conditions satisfied by f and g. The first two conditions satisfied by f can be
determined by recalling that v → vj as y → ∞ and v → 0 as y → −∞. Investigating equation(7.18) and noting that ξ scales with y, these requirements translate tofξ (∞) = 1,fξ (−∞) = 0.

(7.22)We also require that the density approach its asymptotic values in the limits of y → ±∞. Fromequation (7.16), this givesg(±∞) = 1.

(7.23)

Now, note that if the jet and ambient materials are to retain their respective identities, thenthe flow along the surface of contact at y = 0 must remain parallel to that surface. In other words,
there must not be any mass flow across the boundary, i.e., this surface is a contact discontinuity,

206
which means that vy (0) = 0. From equation (7.19), this shows that f must satisfy

f (0) = 0.

(7.24)The other two boundary conditions can be determined by requiring that the normal andtransverse components of the energy-momentum tensor, the divergence of which we took to obtainequation (7.1), be continuous across the point of contact y = 0. We can show that these restrictions
demand that fξ and fξξ be continuous across y = 0, which closes the system.
We can also calculate the comoving-frame number density of photons N 0 by solving equation (7.6). Keeping only terms to lowest order in the boundary layer thickness, the equation ofconservation of photon number becomes





1 ∂
1 ∂N 00 µ∇µ F = ∇µ N U−= 0.
3 ∂y ρ0 κ ∂yµ(7.25)

As we did for the number density of scatterers, we will assume that the photon number density
varies self-similarly across the boundary layer as

N 0 = N00 h(ξ),(7.26)

where N00 is the asymptotic number density of the jet or the ambient medium. With this form forthe number density, we find that the equation for h becomes9
3 2 2 g (fξξ )2
Γj vj
h − f hξ = g hξξ .
2
2
2
2
4
1 + Γj vj (fξ )(7.27)

As for the functions f and g, h is really piecewise defined across the boundary y = 0, and so equation(7.27) is really two equations – one for the photon number density in the jet, and another for thatin the ambient medium. The requirement that the number density of photons asymptoticallyapproach its jet and ambient values givesh(±∞) = 1.

(7.28)

207
The y-component of the flux must also be continuous across the boundary, which, when writtenout, shows that the derivative of h must be continuous across ξ = 0, which yields the final twoboundary conditions.
Finally, the solutions to equations (7.20), (7.21), and (7.27) will be in terms of the parameter
ξ, which is itself a function of g (equation (8.40)). We would like the solutions to be in terms of theparameter α, which is directly related to the physical coordinates y and z (equation (7.15)). The
transformation can be achieved by differentiating equation (8.40), rearranging, and integrating to
yield

Z ξ
0

dξ˜
= α,
˜
g(ξ)

(7.29)

where ξ˜ is an integration variable and we set the integration constant to zero because we demandξ(α = 0) = 0. After solving equations (7.20) and (7.21) for g(ξ), we can numerically integrate andsolve equation (7.29) for ξ in terms of α.

7.3.4

Solutions

Equations (7.20), (7.21) and (7.27), together with the boundary conditions on f , g, and h,
govern the behavior of the velocity, number density of scatterers, and number density of photons
throughout the two-stream boundary layer. Here we present and analyze the solutions to thoseequations as we vary the asymptotic jet Lorentz factor Γj and the quantity µ = e00 /ρ00 .
One caveat with the definition of α is that it depends on Γj via equation (7.15). Therefore,
if we plot solutions with different Γj , we must be careful to incorporate this dependence so thatthe range of physical space we consider for each solution is the same. Because of this fact, in thefollowing figures we will plot our solutions as functions of the variableαyα̃ = p
=√
z
Γj vj

r

27 0
ρ κ.
8 0

(7.30)

208

Figure 7.1: The function fξ , which is the normalized z-component of the four velocity, in terms of√
the parameter α̃ ∝ y/ z, for µ = 1 and a number of jet Lorentz factors, as indicated by the legend.
As we can see, the thickness of the velocity boundary layer, in terms of α̃, is nearly independent of
Γj .

209

Figure 7.2: The variation of the normalized density, given by g, as a function of α̃ for the same
parameters as those chosen for Figure 7.1. The density remains below its asymptotic value over aslightly larger range of α̃ for higher Γj , and the decrease in density within the boundary layer isconsistent with the findings of Arav & Begelman (1992). The density formally equals zero at α̃ = 0;
however, because g approaches zero at a very slow rate (recall g(ξ) ∝ −1/ ln ξ), it appears fromthe Figure, which only samples a finite number of points around α̃ = 0, that the density remains
positive and larger for smaller Γj .

210

Figure 7.3: The solution to equation (7.27), h, which is the normalized number density of photons,
for the same set of parameters as those chosen in Figure 7.1. The number density of photons isseen to roughly track the number density of scatterers. Because the energy density of the radiation
remains constant across the layer, the energy per photon increases in the boundary layer.

211
Figure 7.1 shows the solution for the normalized z-component of the four-velocity (fξ ) as a√function of the self-similar variable α̃ ∝ y/ z for a number of jet Lorentz factors and µ = 1. We
see that the width of the boundary layer is nearly unchanged as we vary the Lorentz factor of theasymptotic jet. The value of fξ at the contact discontinuity is lower for larger Γj , resulting in agreater shear (∼ fξξ ) as one proceeds into the jet. The flattening of the velocity around α̃ = 0 arises
from the behavior of the density around this region and the function ξ(α) determined therefrom.
Figure 7.2 illustrates the manner in which the density varies over the boundary layer for thesame parameters as those chosen for Figure 7.1. Consistent with the findings of Arav & Begelman(1992), we find that the transition from the jet to the ambient medium carves out a region of low
density material. This behavior can be understood by noting that the shear in the flow causes
viscous heating of the fluid, which results in an increase in the specific entropy s0 . Since the specificentropy scales as s0 ∝ e0 /(ρ0 )4/3 for a radiation-dominated gas, an increase in the entropy at constantpressure corresponds to an decrease in the density of scatterers. As was found by Arav & Begelman(1992), the density equals zero at the origin, which can be gleaned from the asymptotic behaviorof equation (7.21): for ξ  1, we can let fξ ∼ fξξ ∼ ξ, and solving the resultant approximate
differential equation shows that g ∝ −1/ ln(ξ).
In Figure 7.3 we plot the solution to equation (7.27), the normalized, rest-frame number
density of photons, for the same set of Lorentz factors and µ = 1. By comparing this with Figure
7.2, we see that the photon number density roughly tracks that of the scatterers. However, thephoton number density remains below its asymptotic value significantly farther into the ambient
medium than does the particle density. Because the energy density of the radiation is constantacross the boundary layer, the decrease in photon density corresponds to a higher average energy
per photon increases within the boundary layer – a clear manifestation of viscous heating, as noted
by Arav & Begelman (1992).
Figures 7.4, 7.5, and 7.6 show the z-component of the four-velocity, the number density of
scatterers, and the photon number density, respectively, for Γj = 25 and µ = 0.1, 1, and 10. The
overall qualitative behavior of the solutions is similar to that depicted in Figures 7.1 – 7.3. More

212

Figure 7.4: The solution for the normalized z-component of the four-velocity (fξ ) for Γj = 25 andthree different values of µ, as indicated by the legend. The velocity profile does not differ much asµ increases beyond 1, but for smaller values of µ the boundary layer widens noticeably.

213

Figure 7.5: The function g, which is the rest-frame number density of scatterers, for the same
parameters as in Figure 7.4. For smaller values of µ, the density is significantly reduced from its
asymptotic value over a larger range in α.

214

Figure 7.6: The solution to equation (7.27), which gives the number density of photons observed in
the comoving frame, for the same set of parameters as in Figure 7.4. The radiation number density
roughly follows that of the scatterers.

215
specifically, however, we find that values of µ greater than unity cause the boundary layer thickness
to decrease, but not appreciably. On the contrary, a value of µ only marginally less than one
results in a significant widening of the boundary layer. This dependence is ultimately related tothe compressibility of the fluid and the relation between that compressibility and the sound speedof a radiation-dominated gas (see discussion below).

7.4

Discussion
The plots of the previous subsection demonstrate how radiation viscosity affects the boundary

between a fast-moving flow and its surroundings under the two-stream approximation. Our analysis
generalizes the treatment of Arav & Begelman (1992) by permitting the jet velocity to be relativistic
and by incorporating the enthalpy of the radiation in the momentum equation. Our results are
similar to those found by Arav & Begelman (1992) (compare, e.g., their Figure 3 to our Figures
7.1 and 7.2); there are, however, a few notable differences.
For one, our boundary layer thickness, given by equation (8.7), differs from that of Arav &Begelman (1992), who found δ 2 ∼ e0 /(ρ00 2 κ vj z). In addition to the Lorentz factor contained in ours,
their boundary layer thickness has an additional factor of e0 /ρ00 , meaning that, in the non-relativistic
limit, our results do not agree. This discrepancy arises from the fact that, while we compared thelowest-order terms in δ to the inviscid terms in the gas energy equation to obtain our boundary
layer thickness, they compared the viscous term to the inertial term – the one proportional to ρ0 –in the z-component of the momentum equation to obtain theirs. Because they ignored the enthalpyof the radiation, the inertial term was the only inviscid term present in the momentum equation,
making it the only one available to balance the viscous part. However, if one does not ignore theradiative contribution to the momentum equation, one can now equate the viscous terms to either
the term proportional to the mass density or the one proportional to the enthalpy. Because thedensity equals zero at α = 0, there is always some location at which the enthalpy term exceeds theinertial term, making it more reasonable to equate the former to the viscous contribution than thelatter. If one follows this route, one recovers our ordering for the boundary layer thickness.

216
The second difference is that their solutions depend on the square of the Mach number,
M 2 ≡ v 2 /c2s , where c2s = 4e0 /(9ρ00 ) is the non-relativistic sound speed. Our solutions, on the other
hand, depend on both the jet velocity vj and the ratio µ = e0 /ρ00 , which is proportional to the nonrelativistic sound speed. One reason for this difference arises from the discrepancy between our
definitions of the boundary layer thickness. Another is due to the fact that our solutions included
the enthalpy of the radiation; had Arav & Begelman (1992) included this term, factors of the soundspeed would have arisen in their z-momentum equation. Finally, our solutions also depend on thevalue of the jet velocity – not just the ratio of the jet speed to the sound speed – because we
included all relativistic effects, meaning that the speed of light now plays a role in determining theevolution of the system.
Our solutions show that, for fixed µ, the thickness of the boundary layer is approximately
independent of the jet Lorentz factor, which is due to the competition between the scaling ofp
the fiducial boundary layer thickness δ ∝ 1/ Γj (see equation (8.7)) and the viscous heating.
Specifically, a larger Γj results in a smaller δ and a greater shear; this shear increases the specificentropy s0 ∝ e0 /(ρ0 )4/3 and, since the pressure is a constant, this increase in the entropy implies adecrease in the density of scatterers which widens the boundary layer.
Figures 7.4, 7.5, and 7.6 show how the solutions for the velocity, density of scatterers, anddensity of photons vary for a fixed jet Lorentz factor (we chose Γj = 25) but for a variable µ.
Increasing µ relative to µ = 1 tends to slightly decrease the boundary layer thickness, while decreasing the value of µ relative to µ = 1 dramatically increases the thickness. This behavior arises
from the fact that the viscous heating, which decreases the density of scatterers and widens theboundary layer, is effective when the gas is compressible. The compressibility of the fluid, however,is only important when the flow velocity is supersonic, and we can show that the sound speed of
the gas is (Chapter 6)

2
cs =
3

r

µ
,
1 + 4µ/3(7.31)

217p
which, noting that e0 = 3p0 , where p0 is the radiation pressure, correctly reduces to cs ' 4p0 /(3ρ0 )√in the limit µ  1 and cs ' 1/ 3 in the limit µ  1. Thus, when µ  1, the location of the sonicpoint extends farther into the ambient medium, resulting in a widening of the boundary layer. On
the other hand, when µ  1, the sonic point approaches the jet, but only slightly due to the factthat the sound speed approaches a constant as µ → ∞. In fact, based on this reasoning, we expect
our solutions to be independent of µ in the large µ limit, which is indeed reflected in equation(7.20).
The densities of scatterers and photons both decrease within the boundary layer. This behavior has two interesting consequences. First, the lower number density of scatterers means thatthe optical depth is lower in the boundary layer. We are therefore able to see farther into themedium along lines of sight that probe regions of high shear. Second, because the energy density of
the radiation stays unchanged as we traverse the media, the average energy per photon increases,
resulting in a harder spectrum within the boundary layer.
Equation (8.10) shows that the radiation energy density, and hence the pressure, is constantacross the boundary layer, which is ultimately a statement of the causal connectedness of the jet.
This means, equivalently, that the boundary layer thickness δy can be traversed by a sound wave
in less time than it takes the jet material to cross the distance δz. Since the transverse sound speed
is related to the isotropic sound speed by cs⊥ = cs /Γj , we find that the boundary layer thickness δmust satisfy δ . cs /Γj . When this inequality is no longer satisfied, equation (8.10) no longer holds,
and we must include more terms in all of the boundary layer equations that account for changes in
the pressure.
The solution for g equals zero at α = 0 which, as we mentioned, can be determined by inspecting the asymptotic (ξ → 0) limit of equation (7.21). This feature was also found by Arav &Begelman (1992), and can be understood physically by noting that, when the jet and the ambient
medium initially interacted, the surface separating them served as a discontinuity in velocity, resulting in an infinite shear and entropy generation. Although the solutions presented here illustrate
the time-steady state of the system after radiation viscosity has smoothed out the discontinuity,

218
the infinite entropy along the contact discontinuity persists and drives the density to zero. Mathematically, this behavior is ultimately due to the fact that f (0) = 0, which itself came from therequirement that there be no mixing at the surface separating the jet and the external environment.
This boundary condition is necessary to ensure that the two media retain their respective identities and underlies the two-stream assumption, and it allows us to prescribe different asymptotic
properties of those media.

7.5

Summary and conclusions
We applied the relativistic equations of radiation hydrodynamics in the viscous limit to

the two-stream boundary layer, expanding on the past work of Arav & Begelman (1992). Theseequations are applicable as long as changes in the fluid quantities are small over the mean free pathof a photon.
An interesting feature of the solutions presented here is the depression in the number densityof scatterers within the boundary layer separating the jet and the ambient medium, which isconsistent with the findings of Arav & Begelman (1992). We also showed that the number densityof photons N 0 roughly tracks the density of scatterers, reaching a minimum towards, but not at,
the contact discontinuity. Therefore, observers viewing a relativistic outflow with lines of sightthat probe regions of high shear see farther into the outflow and they also see a higher energy perphoton, and hence a harder spectrum.
Our solutions show that the physical boundary layer thickness does not depend strongly on
the jet Lorentz factor Γj , which results from a competition between viscous heating and the scalingpof the fiducial boundary layer thickness δ ∼ 1/ Γj . The dependence of our solutions on µ arises
from the fact that the change in entropy of the flow is related to its compressibility, which is mostimportant where the outflow velocity is supersonic. Since the sound speed scales as cs ∼√µ whenµ . 1, the point at which the outflow becomes subsonic extends farther into the ambient medium
when µ is small, resulting in a widening of the boundary layer. When µ becomes larger than one,√however, the sound speed does not increase much, asymptotically approaching 1/ 3, meaning that219
the sonic point only slightly approaches the jet, yielding a marginal decrease in the boundary layer
thickness.
A number of assumptions about the nature of the jet and its surroundings are built in to
the two-stream solutions presented here. Specifically, we adopted a two-dimensional, plane-parallelgeometry for the flow and its surroundings, and we demanded that there be no pressure gradient(e0 (z) = e00 ) in the ambient medium. While these assumptions greatly enhanced the tractability ofthe problem, they somewhat hinder the astrophysical relevance of the solutions, as no systems likely
conform precisely to these restrictions. However, in local regions of an outflow, where the radiusof curvature is large and the pressure can be considered relatively constant, these solutions may be
actualized. Therefore, while the physics of an entire jet-disk system may be poorly modeled by thetwo-stream scenario, local patches of the outflow, where the geometry can be treated as locally flatand the pressure gradient is small, are likely well-described by the solutions presented here.
The two-stream solutions could be applied to relativistic, radiation dominated jets, such as
those that appear during super-Eddington TDEs, the event Swift J1644+57 being one such case.
The event Swift J2058+05, observed shortly after Swift J1644, is another source that has beeninterpreted as a jetted TDE (Cenko et al., 2012). A comparison between the models presented
here and these sources could provide valuable information concerning their progenitors and theproperties of the jets themselves. The application of these models to long GRBs could likewise
prove fruitful, potentially yielding, for example, information concerning the Lorentz factor of thejet, the stellar progenitor, and the direction of the line of sight between the observer and the source.
These models may also be applicable to jetted X-ray binary systems, or “microquasars”(Mirabel & Rodrı́guez, 1999; Fender et al., 2004, 2009). For example, Arav & Begelman (1993)
applied their non-relativistic, radiation-viscous solution (Arav & Begelman, 1992) to the source SS
433 (Fabrika, 2004; Begelman et al., 2006a). Since the jets of SS 433 are mildly relativistic, their
speeds being vj ' 0.26 c (Margon & Anderson, 1989), including the relativistic terms arising fromthe treatment presented here may place new constraints on the properties of those jets and thesurrounding envelope.

220
Finally, quasi-stars – protogalactic gas clouds supported by an accreting black hole – mayalso contain jets (Begelman et al., 2006a, 2008; Czerny et al., 2012). Since the power radiated by
the black hole at the center of a quasi-star supports the overlying gaseous envelope, the mass ofthe envelope greatly exceeding that of the hole, the black hole accretes supercritically by several
orders of magnitude. The jets launched from these systems are therefore radiation-dominated, andas they propagate through the quasi-star envelope, radiation-viscous effects likely dominate theinteraction between the two media. Applying the solutions presented here to these systems could
then yield information about the properties of these jets and the role they may have played during
the epoch of galaxy formation.
One drawback to these models and their physical application, however, is that they requirethat the jet and ambient medium be separated by a contact discontinuity, which results in the
non-physical vanishing of the density of scatterers at the interface. Furthermore, this prevents thejet from entraining ambient material; while this is not particularly problematic for the two-stream
problem, in which the jet is considered infinite in extent, realistic jets have a finite width and total
momentum, meaning that the entrainment of ambient material will cause a decrease in the outflow
velocity with z that cannot be captured with the two-stream treatment.
In a companion paper (Coughlin & Begelman, 2015), we investigate a different type of viscousboundary layer – the free-streaming jet model. This model treats the entire system, jet and ambient
medium, as a single fluid, which removes the need for a contact discontinuity and allows the density
to remain non-zero throughout the boundary layer. We also show that the entrainment of ambientmaterial causes an overall slowing of the outflow.

Chapter 8

Viscous boundary layers of radiation-dominated, relativistic jets. II. The
free-streaming jet model

Particles, magnetic fields, and radiation all contribute to the propulsion of relativistic jets,
though the relative contribution of each is still an open matter of debate. In certain situations,however, the mechanism responsible for launching the jet operates simultaneously with the releaseof a large amount of energy in the form of radiation, making it likely that photons dominate thebulk energetics. This scenario occurs, for example, during the super-Eddington phase of jetted
tidal disruption events (TDEs), such as Swift J1644+57 (Burrows et al., 2011; Zauderer et al.,2011) and Swift J2058+05 (Cenko et al., 2012). Radiation-dominated jets should also be present
in the collapsar picture of long gamma-ray bursts (GRBs; Rees & Meszaros 1992; Woosley 1993;
Meszaros & Rees 1993; MacFadyen & Woosley 1999; Piran 2004), where the energy released in theform of gamma-rays is ultimately derived from accretion onto a black hole, the associated accretion
luminosity exceeding the Eddington limit of the hole by more than ten orders of magnitude. In
both of these cases, the propagation of the radiation-dominated jet is modulated by the presenceof a radiation pressure-supported environment; for super-Eddington TDEs, this environment is in
the form of a highly inflated, quasi-spherical torus of fallback debris (Chapter 5), while a “cocoon”
of shocked jet material (Morsony et al., 2007; López-Cámara et al., 2013) and the overlying stellar
envelope itself (Matzner, 2003; Woosley & Bloom, 2006) serve as the confining medium for GRBs.
In the previous chapter (Chapter 7) we presented a model that describes the viscous interaction of a radiation-dominated, relativistic jet with its surrounding medium. In that analysis we222
treated the jet and its surroundings as two separate fluids, interacting with one another via small
anisotropies in the comoving radiation field that are explicitly accounted for in the equations ofradiation hydrodynamics in the viscous limit. This two-stream approximation, in agreement with
the non-relativistic analysis of Arav & Begelman (1992), demonstrates the manner in which theshear between the two fluids carves out a region of low density material within the boundary layer
between them. We also deduced the dependence of the boundary layer thickness on the asymptotic
properties of the jet and the ambient medium.
These models also show, however, that the contact discontinuity separating the jet and its
surroundings, necessary for maintaining their respective identities, results in the likely unphysicalvanishing of the density of scatterers along that surface of separation. The contact discontinuity
also prevents the jet from entraining ambient material; since the jet in the two-stream model
we considered had an infinite amount of momentum, any entrainment or lack thereof is formallyinconsequential to the evolution of the system. However, realistic jets – those with finite extent– will almost certainly engulf more material as they expand into their surroundings; because thetotal amount of momentum in the system, which is realistically finite, must be conserved, thatentrainment will then result in an overall deceleration of the outflow that cannot be captured in
the two-stream model.
In view of these unphysical properties of the two-stream treatment – the vanishing of themass density of scatterers along the contact discontinuity and the lack of entrainment – we present
here an alternative boundary layer scenario to describe the interaction of a relativistic, radiationdominated jet with its surroundings. This “free-streaming jet” model, which has a well-known
counterpart in the non-relativistic, incompressible limit (see Chapter 10, Section 12 of Kundu &Cohen 2008), assumes that the jet is injected through a narrow opening into a static, homogenous
medium, and that far enough from that opening the entire system can be modeled as a single fluidthat is independent of the details at the injection point. By considering the jet and the ambient
medium as one fluid we obviate the need for a contact discontinuity, which we demonstrate allows
the density of scatterers to remain finite throughout the flow and for the jet to entrain material.

223
In section 2 of this paper we present the equations of radiation hydrodynamics in the viscous
limit. Section 3 uses those equations to analyze the free-streaming jet model, and we demonstrate
the existence of approximate self-similar solutions in the limit that the interaction between the jet
and the ambient medium is concentrated in a thin boundary layer. In section 4 we discuss theimplications of our model and make comparisons to the two-stream scenario, and in section 5 weconclude and consider the application of the free-streaming jet model to super-Eddington TDEs,
GRBs, and other astronomical sources.

8.1

Governing equations
When changes in fluid quantities over the mean free path of a photon are small, radiation

behaves like an effective viscosity and transfers momentum and energy between neighboring fluid
elements. The precise form of the viscosity can be determined by investigating the general relativistic Boltzmann equation, which was done in Chapter 6 for the case where Thomson scattering
dominates the interactions between the photons and scatterers in the fluid rest frame. In this limit,
we found that the relativistic equations of radiation hydrodynamics for a cold gas (gas pressuremuch less than the gas rest mass density and radiation pressure) are (see equation (49) of Chapter
6; see also Chapter 7):


∇µ




 0


4 0
10 1
1 µν
8
e µσ νβαµ ν0α
ρ + e 1−∇α U
U U + g ∂µ e − ∇µ 0 Π Π∇σ Uβ +∇β Uσ +gβσ ∇α U3
9 ρ0 κ
3
27
ρκ
 0 


1
e
− ∇µ 0 Πµσ U ν + Πνσ U µ
4U β ∇β Uσ + ∂σ ln e0= 0. (8.1)
3ρκ0Here the speed of light has been set to one, Greek indices range from 0 – 3, ρ0 is the fluid rest frame
mass density of scatterers, e0 is the fluid rest frame radiation energy density, κ is the scattering
opacity (in units of cm2 g−1 ), gµν is the metric of the spacetime, ∇µ is the covariant derivative,
U µ is the four-velocity of the flow, and Πµν = U µ U ν + g µν is the projection tensor. The Einsteinsummation convention has been adopted here, meaning that repeated upper and lower indices imply
summation. This equation also shows, in agreement with previous findings (Blandford et al., 1985;

224
Loeb & Laor, 1992), that the coefficient of dynamic viscosity, η, for an optically-thick, radiationdominated gas is

η=

8 e0
.
27 ρ0 κ(8.2)

We will also write down the gas energy equation, obtained by contracting equation (8.1) with thefour-velocity Uν , which gives (see equation (50) of Chapter 6)

 0



4 10
e µ
8 e0
1 0
µ
α
α
∇µ 0 U ∇α U +∇σ Uβ + ∇β Uσ + gσβ ∇α U Πµσ ∇µ U β∇µ (e U ) + e ∇µ U =3
3 9ρκ
27 ρ0 κ
 0 




1
e
1 e0
β
0
µ
σ
σ
µ
+ Πµσ ∇µ 0 4U β ∇β Uσ + ∂σ ln e0 +
4U
∇
U
+
∂
ln
e
2U
∇
U
+
U
∇
U
.
σ
µ
µ
β σ
3
ρκ
3 ρ0 κ
0µ(8.3)
To close the system, we require that the normalization of the four-velocity be upheld and thatparticle flux be conserved:

Uµ U µ = −1,

(8.4)



∇µ ρ0 U µ = 0.

(8.5)

Equations (8.1) and (8.3) – (8.5) constitute six linearly independent equations for the six unknowns
U µ , e0 , and ρ0 .
In addition to the energy density of the radiation, e0 , one can also calculate the number
density of photons by requiring that the number flux, F µ , be conserved. One can show (Chapter6) that the equation ∇µ F µ = 0 becomes, in the viscous limit,






1 10 0 µ
1 µσ
0 µ
σ
0 α
µ
0
∇µ N U = ∇µ 0
N U ∇σ U + N U ∇α U + Π ∇σ N
,
ρκ 9
3

(8.6)

where N 0 is the rest-frame number density of photons. Once we solve the equations of radiationhydrodynamics for the four-velocity of the fluid and the mass density of scatterers, we can solve

225
equation (8.6) for the number flux of photons.
The goal of the next two sections is to apply equations (8.1) and (8.3) – (8.6) to the boundary layers established between fast-moving jets and their ambient media. For a more thoroughdiscussion of the nature of the equations of radiation hydrodynamics in the viscous limit, we refer
the reader to Chapter 6.

8.2

Jetted boundary layer
In this section we consider the problem where a narrow stream of material is continuously

injected into a plane-parallel, static, ambient medium, known as the free-streaming jet problem.
If the Reynolds number of the outflow is high, the transition between the stream of material andthe external environment will be confined to a thin layer, permitting the use of a boundary layer
approximation.
The basic setup is similar to that of the two-stream problem (see Chapter 7), with the motionof the 2-D, plane-parallel injected stream predominantly along the z-direction, the majority of thevariation along y, no variation or velocity in x, and the point of injection at y = z = 0. Now, however, there is no contact discontinuity between the stream and the ambient environment, meaningthat the entire system is considered a single fluid. We therefore have less freedom in prescribing
the asymptotic characteristics of the jet and the environment; however, this configuration permits
mixing between the two media, allowing the outflow to entrain material, which almost certainly
occurs in realistic jets.
We can reduce the complexity of equations (8.1) and (8.3) by assuming that the interaction
between the jet and the ambient medium takes place over a thin boundary layer of thickness δy,thin in the sense that δ ≡ δy/δz is a small number when δz is a typical length along the jet. In thiscase, δ scales in an identical fashion to that derived for the two-stream boundary layer, which canbe determined by comparing leading-order terms in the boundary layer thickness to the inviscid
terms in the gas energy equation. Setting the left-hand side of equation (8.3) to ∼ e0 Γj vj /z, we226find, as in Chapter 7,

δ2 ∼1
ρ0 κ z Γj vj

.

(8.7)

Here vj and Γj = (1−vj2 )−1/2 are the jet velocity and Lorentz factor, respectively, measured at somecharacteristic length along the jet axis z0 , and ρ0 is a characteristic density of scatterers throughout
the outflow. Due to the fact that the boundary layer thickness is the same, the boundary layer
equations governing the outflow are identical to those found in paper I, which can be compactly
written as






1 µν
8 ∂ e0 ∂U ν
4 0
µ ν
0
0
∇µ ρ + e U U + g ∂µ e =,3327 ∂y ρ0 κ ∂y0∇µ ρ U

µ

(8.8)


= 0.

(8.9)

In paper I we dealt with the ν = y and ν = z components of equation (8.8) and its contraction
with Uν – the gas energy equation (8.3). For the free-streaming jet problem, however, it will be
more convenient to deal with the ν = y component of equation (8.8), the gas energy equation,and the contraction of equation (8.8) with the projection tensor Πzν , which, as we will see, is therelativistic, viscous counterpart of the Bernoulli equation. Performing a few manipulations, we findthat these equations become, respectively,

∂e0
= 0,
∂y

(8.10) 
1
8 e0 ∂S 2∇µ e0 U µ + e0 ∇µ U µ =,3
27 ρ0 κ ∂y(8.11)





4
1 de0
8 ∂ e0 ∂S
ρ0 + e0 U µ ∇µ S + Γ
=
,
3
3 dz
27 ∂y ρ0 κ ∂y(8.12)

0∇µ ρ Uµ= 0.

(8.13)

227
where




p
2
2
S = ln Γvz + 1 + Γ vz
= arcsinh(Γvz ).

(8.14)

The first of these shows that the radiation energy density, and consequently the pressure, is constantacross the boundary layer. In the inviscid limit, equation (8.12) can be transformed to give

 

4 e0
U ∇µ Γ 1 +
= 0,
3 ρ0
µ

(8.15)

which, as we mentioned, is the relativistic generalization of the Bernoulli equation. We will also
assume that the ambient energy density is independent of z, i.e., that e0 (z) = e00 , with e00 a constant.
Because it will be convenient, we will change variables from y to τ , where τ is given byZ yτ≡ρ0 (ỹ, z) κ dỹ,(8.16)

0which is related to the optical depth across the boundary layer as measured from the axis. In termsof this variable, equations (8.11) and (8.12) become, respectively,

 
 
4 0 µ
1
8 0 ∂S 2e U ∇µ 0 = e κ
,
3
ρ
27
∂τ

(8.17)


4 e0
8
∂2S
U µ ∇µ S = e 0 κ 2 ,
0
3ρ
27
∂τ

(8.18)

1+

where we have used the assumption that e0 (z) = e00 . It should also be noted that the y-componentof the convective derivative is now with respect to τ , not y, i.e.,

U µ ∇µ = U z∂∂+ ρ0 κ U y .
∂z∂τIn the ensuing section we will seek self-similar solutions to these equations.

(8.19)

228
8.2.1

Self-similar approximation

We can immediately solve the continuity equation (8.13) by introducing the stream function
ψ via

Γvz =

∂ψ
,
∂τ

ρ0 κ Γvy = −

∂ψ
,
∂z

(8.20)(8.21)

where the factor of κ ensures that ψ remains dimensionless. In paper I we showed that there exist
self-similar solutions for ψ, the velocity, the comoving density of scatterers and comoving density of√photons in terms of the variable α = y/δy, where δy ∼ z. One difference between the two-stream
problem and this type of outflow, however, is that we expect the jet to expand into the ambient
medium, entraining material in the process. Therefore, as we look farther along the z-direction, theamount of inertia – predominantly in the form of radiation for the systems that we are considering– contained in the flow will increase. Owing to the conservation of momentum, the z-componentof the velocity should thus be a decreasing function of z.
In light of this observation, we will assume that the z-component of the four-velocity scalesasΓvz = Γj vj

z
z0

−m

df
df
≡ζ ,
dξ
dξ

(8.22)

where z0 is a characteristic length scale in the z-direction, m is a positive constant, for brevity
we defined ζ ≡ Γj vj (z/z0 )−m , and f is a function of the self-similar variable ξ ' τ /δτ ; δτ is thecharacteristic boundary layer thickness in terms of τ , which, from equations (8.7) and (8.16) in theprevious subsection, is given by δτ ' ρ0 κ δy, orδτ 2 '

ρ0 κz
.
Γj vj

(8.23)

229
We are most interested in the behavior of the properties of the outflow when the velocities
are relativistic, as this is the limit that is most applicable to sources of astronomical interest, andin order for the self-similar nature of our solutions to be upheld, the dependence of our self-similarfunctions on the bulk properties of the outflow, such as the z-dependence of the jet Lorentz factor
along the axis, should be minimal. A fully self-similar solution is likely impossible here, as the speedof light plays a role in setting a finite scale factor for our solutions and becomes problematic when
we try to connect the ultra- and non-relativistic regions of the outflow. We will show, however,that the non-self-similarity of our solutions only affects a small region of the outflow.
In paper I, we showed that the self-similarity of the comoving density of scatterers, ρ0 , wasapproximately satisfied, i.e., we could find solutions with ρ0 ∼ g(ξ). For the case at hand, then,this assumption implies that g is independent of ζ. This means, however, that the observer-framedensity of scatterers, given by ρ = Γρ0 ' ζ g, can be made arbitrarily large (as g is independent of
ζ in the self-similar limit), which is a nonsensical result.
Motivated by this reasoning, we conclude that the comoving density of scatterers is unlikely
to be independent of ζ. On the contrary, a more reasonable approximation is that the observerframe density of scatterers varies self-similarly. Investigating equations (8.17) and (8.18), we seethat the ρ0 -dependent quantity that enters both is the combination 1 + 4e0 /(3ρ0 ). Our expectationthat the observer-frame density varies self-similarly then prompts the assumption

1+

4 e0
4
= µ Γ g(ξ),
03ρ
3(8.24)

where µ ≡ e0 /ρ00 , ρ00 being the density of scatterers in the ambient medium. (Note that we could
have simply let ρ0 Γ = g(ξ), but equation (8.24), which is merely a change of variables from theinitial assignment ρ0 Γ = g(ξ), will allow equations (8.17) and (8.18) to be written in a more compact
form.)
We can determine the value of m, which controls how rapidly the flow decelerates due toentrainment, by integrating the momentum flux, Ṗ ' e0 Γ2 vz2 for a radiation-dominated system, over230
the entire boundary layer, and requiring that the result be independent of z. This is equivalent torequiring that the total momentum contained in the outflow be conserved. Integrating e0 Γ2 vz2 fromy = −∞ to ∞, using equations (8.22) and (8.24), taking the ultrarelativistic limit and changing
variables from y to ξ, we find


Ṗ '

z
z0

 1 −2m Z ∞ 
2

−∞

df
dξ

2
dξ.

(8.25)

Since the integral in this equation is a constant that is greater than zero, we find that m = 1/4 if
the momentum flux is conserved.
With this value of m, the z-component of the four-velocity scales asΓvz = Γj vj

z
z0

−1/4

df
,
dξ

(8.26)and the self-similar variable ξ is given bys
ξ=τ

9Γ2j vj2
2ρ00 κz0



z
z0

−3/4
.

(8.27)We then find from equation (8.20) that the stream function must satisfyrψ= 1/22 0z
ρ0 κz0f.
9
z0(8.28)

Equation (8.21) then gives the y-component of the four-velocity:sρ0 κΓvy = − 1/2  
2 ρ00 κ 1 z −1/2z∂ξ
f + z0fξ ,9 z0 2 z0z0∂z(8.29)

where, both here and in future equations, a subscripted ξ on the functions f and g denotes differentiation with respect to ξ and the number of subscripts indicates the number of derivatives, i.e.,fξ = df /dξ, fξξ = d2 f /dξ 2 , etc.

231
Inserting equations (8.24), (8.26) and (8.29) into equations (8.17) and (8.18), changing variables from τ to ξ and performing a bit of algebra, we find that they become, respectively,1− Γf gξ = ζ
2



∂S
∂ξ

2


∂2S
− vz 2 ,
∂ξ



1
∂2S
2
− g (fξ ) + 2f fξξ =
.
4
∂ξ 2

(8.30)(8.31)

Now, the solutions we seek for f and g should depend only on ξ, as otherwise our assumptionof the self-similarity of those functions breaks down. Because of the complicated dependence of
S on Γvz , we see that the solutions will not be self-similar for arbitrary Γj . However, in theultrarelativistic limit, for which Γ ' Γvz , we see that S ' ln(Γvz ), and it is apparent that thesolutions are indeed self-similar, i.e., the ζ dependence of equations (8.30) and (8.31) drops out.
As these equations are third order in f and first order in g, we require four boundary conditions to solve them numerically. Since Γvz is given by equation (8.26) and it is assumed thatΓvz = Γj vj (z/z0 )−1/4 along the axis, our first boundary condition is given by fξ (0) = 1. For thesecond, we note that the jet axis is the streamline along which the y-component of the velocity is
zero. From equation (8.29), then, we find f (0) = 0 (we will see that ∂ξ/∂z = 0 at ξ = 0). The thirdboundary condition is obtained by noting that the z-component of the velocity should approachzero as we proceed into the ambient medium, so that, from equation (8.26), we find fξ (∞) = 0.
Finally, the density of scatterers should approach that of the ambient medium far from the jet
center. Equation (8.24) then gives g(∞) = 1 + 3/(4µ).
As we noted, the truly self-similar limit of equations (8.30) and (8.31) is obtained by setting
S = ln(Γvz ), but we encounter an issue with this scaling when we consider the boundary conditions
on our flow in the ambient medium. In particular, we expect the outflow velocity to approach zerofor ξ  1; when the velocity becomes subrelativistic, however, ln(Γvz ) → −∞, when in actuality
the vz  1 limit of equation (8.14) is S ' ln(1 + Γvz ) ' Γvz ' 0. Thus, we see that taking theultrarelativistic limit of equations (8.30) and (8.31) will not result in the solutions matching the232correct boundary conditions far from the jet center.
To correct this problem, we will allow the function S to take on its full form and treat ζ
as a constant, which will break the self-similarity of our solutions. However, since our boundary
conditions do not depend on the value of ζ, the functions f and g themselves should be largely
independent of that parameter. We therefore expect the assumption of self-similarity to be upheldin the ultrarelativistic (ξ  1) and the non-relativistic (ξ  1) limits of our solutions, with small
deviations from self-similarity in the trans-relativistic regime of the outflow.
To exemplify this point, Figures 8.1 and 8.2 show, respectively, the variation of fξ – thenormalized z-component of the four-velocity – and g – approximately the inverse of the observerframe density – in terms of the parameter ξ for ζ = 4, 10, and 25. As we anticipated, the functions
for different ζ are indistinguishable for ξ  1 and ξ  1, meaning that the self-similarity isnearly exact in those regions. In between those limits, where the flow is trans-relativistic, thedeviations from self-similarity are apparent, though they remain small. Keeping the full form of
S in equations (8.30) and (8.31) therefore preserves well the self-similarity of our solutions andprovides a reasonable interpolation between the relativistic and non-relativistic regions of the flow.
The solution for g approaches infinity as we near the axis, which can be seen by investigating
the small-ξ behavior of equation (8.31). Specifically, letting f ' ξ, fξ ' 1, Γ ' ζ, and vz ' 1,
equation (8.31) can be written as1
ξgξ + g ' −2(fξξ,0 )2 ,2

(8.32)

where fξξ,0 is the second derivative of f evaluated at ξ = 0. This equation can be integrated, andwe findC
g ' √ − (fξξ,0 )2 ,
ξ

(8.33)

C being a constant of integration. Since it is roughly proportional to 1/ρ0 , the value of g cannot be
negative, meaning that C must be greater than zero. Therefore, the asymptotic, ξ  1 behavior

233

Figure 8.1: The behavior of fξ , which is the normalized z-component of the four-velocity, with ξfor ζ = 4 (blue, solid curve), ζ = 10 (purple, dot-dashed curve), and ζ = 25 (red, dashed curve).
As expected, the curves are all coincident when ξ  1 and ξ  1, with a non-self-similar transition(one that depends on ζ) in between those two limits.

234

Figure 8.2: The function g, which is approximately the inverse of the lab-frame density, plotted withrespect to ξ for the same set of ζ chosen in Figure 8.1. As was true for fξ , g is approximately selfsimilar close to and far from the jet, with the deviation from self-similarity in the trans-relativistic
region being apparent but small.

235√
of g is g ∼ 1/ ξ.
In addition to the mass density of scatterers and the velocity, we can also calculate the number
density of photons, N 0 , throughout the boundary layer. As we showed in Chapter 7, the equationof photon number conservation (8.6) becomes, to lowest order in the boundary layer thickness,





1 ∂
1 ∂N 00 µ=∇µ N U.
3 ∂y ρ0 κ ∂y(8.34)

Performing a few manipulations, this equation becomes

µ



U ∇µ

N0
ρ0


=

κ ∂2N 0.
3 ∂τ 2(8.35)

As was true for the density of scatterers, we expect the observer frame number density of photons
to vary approximately self-similarly, which will be true if we let

N0
N00
=
h(ξ),
ρ0
ρ00

(8.36)

where N00 is the number density of photons in the ambient medium and h is a dimensionless function.
Inserting this ansatz into equation (8.35) gives

1∂2− f hξ = ζ 2
3
∂ξ




h
.
3
Γg − 4µ(8.37)For the boundary conditions on h, we first require that the number density of photons approachthat of the ambient medium in the ξ  1 limit. Equation (8.36) then gives h(∞) = 1. For thesecond condition, return to equation (8.34), integrate both sides from y = −∞ to ∞, and requirethat the derivative of N vanish in both of those limits. Doing so yields

∂
∂z

Z ∞
−∞

∞

N 0 Γvz dy = −N 0 Γvy.

(8.38)−∞The right-hand side can be determined by returning to the continuity equation, integrating from∞y = −∞ to ∞ and performing a few manipulations to show that Γv∞ −∞ = −f∞ , where f∞ is236
the function f evaluated at infinity. Using the definition of N 0 in terms of h, we find that equation(8.38) becomesZ ∞h fξ dξ = 2 f∞ ,(8.39)−∞

which serves as our second boundary condition on h. This integral states that the increase in thenumber flux of photons occurs at a rate provided by the influx of material at infinity.
The equations derived in this section were all written in terms of the variable ξ, which isitself a function of g via equations (8.16) and (8.27). To write the solutions in terms of the physicalparameter y, we can return to equation (8.16), differentiate both sides with respect to y, rearrange
the resulting equation and integrate to yield

Z ξ
0


3
Γg −
dξ˜ = y4µs

 
9 Γ2j vj2 ρ00 κ z −3/4
≡ α,
2
z0
z0

(8.40)

˜ and fξ (ξ),˜ thiswhere ξ˜ is a dummy variable of integration. Once we calculate the functions g(ξ)
relation can be integrated and solved numerically to yield ξ(α). This expression also shows that



Z ξ

ζ 2 (fξ )2 g
3 ∂ξ
1
˜
p
Γg −
=
dξ − 3 α ,4µ ∂z
4z
1 + ζ 2 (fξ )20(8.41)

which we can use in equation (8.29) to give1
ρ0 κΓvy =
4

r





Z ξ
fξ
ζ 2 (fξ )2 g
2 ρ00 κ˜p3α −dξ − 2f
3
9 z
Γg − 4µ1 + ζ 2 (fξ )20(8.42)

Equation (8.41) also confirms that ∂ξ/∂z = 0 when ξ = 0, which we used in order to determine theboundary condition f (0) = 0.

8.2.2

Solutions

In this section we plot solutions for the outflow velocity, the density of scatterers and thedensity of photons for various values of µ and ζ. As was mentioned in the previous subsection,
the physical self-similar variable against which we would like to plot our solutions is given by α.

237However, as is apparent from equation (8.40), the definition of α depends on ζ. Therefore, if we arecomparing, for example, the outflow velocity of two systems with differing ζ, we must incorporate
the ζ dependence in α so that the range of physical space that we consider for each solution is thesame. For this reason, in this section we will plot our solutions as functions of the variableαα̃ ≡ = yζs 
9 ρ00 κ z −1/2.
2 z0 z0(8.43)

Figure 8.3 shows the solution for fξ , the normalized z-component of the four-velocity, forζ = 4, 10, and 25. Since ζ = Γj vj (z/z0 )−1/4 , these values of ζ scale approximately linearly with theLorentz factor until z  z0 . The outflow velocity is maximized at the origin and decays as we movefarther into the ambient medium. We see that the width of the boundary layer, loosely defined as
the value of α̃ at which fξ is some fraction of its central value, is nearly unchanged as we modify ζ.
The average value of the normalized z-component of the velocity is also slightly larger for larger ζ.
Figure 8.4 demonstrates how the normalized, fluid-frame mass density of scatterers varies
as we traverse the boundary layer for the same set of parameters chosen in Figure 8.3. Since g
approaches infinity as we near the origin, the mass density of scatterers, related to g by equation(8.24), equals zero at the origin for all of the solutions, meaning that the center of the jet is evacuatedof massive particles. The average comoving density of scatterers across the boundary layer is also
lower for larger ζ, in accordance with equation (8.24). In Figure 8.5 we plot the normalized number
density of photons for the same set of ζ. It is evident that the density of photons closely follows thedensity of scatterers throughout the boundary layer. We see, however, that the density of photons
stays above and below the density of scatterers as we move toward and away from the center of thejet, respectively. The photon density also remains finite at the center of the jet.
p
The y-component of the three-velocity normalized by 2/(9ρ00 κz0 ) ∼ δ is illustrated in Figure
8.6 for the same set of ζ used in Figure 8.3. For ξ > 0, each solution initially has a positive vy , whichshows that the jet material expands away from the axis. The y-component of the three-velocity
then reaches a relative maximum, one which increases slightly for larger Γj , before approaching

238

Figure 8.3: The normalized z-component of the four-velocity (fξ ) for µ = 1 and ζ = 4, 10, and25 (the solid, blue curve, the dot-dashed, purple curve, and the dashed, red curve, respectively),
which, for z ' z0 , correspond to Γj = 4, 10, and 25. We see that the width of the boundary layer
is nearly unchanged as we alter the value of ζ.

239

Figure 8.4: The normalized fluid-frame density of scatterers for the same set of parameters used in
Figure 8.3. For all solutions the number density of scatterers approaches zero as we near the centerof the jet. We see that the average number density of scatterers within the boundary layer is lower
for larger Lorentz factors.

240

Figure 8.5: The normalized number density of photons for the same set of ζ used in Figure 8.3.
The photon number density closely follows that of the scatterers.

241p
Figure 8.6: The y-component of the three-velocity normalized by 2/(9ρ00 κz0 ) (see equation (8.42))
for the same set of parameters used in Figure 8.3. For positive α̃, each solution is initially positiveand reaches a relative maximum before approaching a negative constant, which shows that the flow
expands outwards near the center of the jet and entrains ambient material far from the axis.

242
a negative, constant value. This behavior is then inverted for negative ξ. Because the transverse
velocity approaches a negative constant for ξ > 0 and a positive constant for ξ < 0, we see that thejet entrains material from the ambient medium.
Figures 8.7 – 8.10 illustrate how our solutions depend on µ. As is apparent, changing thevalue of µ does not drastically alter the qualitative aspects of the functions. We do see, however,that decreasing µ from 1 to 0.1 results in a large increase in the boundary layer thickness; on thecontrary, changing µ from 1 to 10 results in only a slight narrowing of its thickness. It is also
evident that a smaller µ compared to 1 results in a lower average value of the density throughout
the boundary layer and a larger peak in the transverse velocity vy .

8.3

Discussion
Figures 8.3 – 8.6 demonstrate that the width of the boundary layer is nearly independentof ζ, which is the result of the scaling of our boundary layer thickness δ. Specifically, note frompequation (8.7) that δ is given by δ ∼ 1/ ρ0 Γj . Since our ansatz posited that ρ0 Γ ' g, with g adimensionless function of order unity, and Γ ' Γj ' ζ for z ∼ z0 , the boundary layer thickness isroughly independent of ζ.
Figures 8.7 – 8.10 illustrate that a value of µ = 0.1 causes the boundary layer thickness toincrease dramatically compared to µ = 1, while setting µ = 10 causes only a slight narrowing of
the width compared to µ = 1. This dependence is due to the fact that the viscous heating, which
increases the specific entropy by decreasing the density of scatterers (at fixed pressure), is mostefficient when the flow is compressible, as is evident from the gas energy equation (8.11). The fluid
only becomes compressible, however, when the flow is supersonic, and we can show (Chapter 6)that the sound speed of a radiation-dominated gas is

2
cs =
3

r

µ
.
(8.44)
1 + 4µ/3√
When µ  1, the sound speed reduces to cs ∼ µ, and the location at which the flow becomes

243

Figure 8.7: The function fξ , which is the normalized z-component of the four-velocity, for ζ = 10
and µ = e0 /ρ00 = 0.1, 1, and 10, which correspond to the blue, solid curve, the purple, dot-dashedcurve, and the red, dashed curve, respectively. Increasing the value of µ, we see, has little effect on
the solution, while decreasing µ drastically widens the boundary layer.

244

Figure 8.8: The normalized density of scatterers for the same values of µ chosen in Figure 8.7. The
mean value of the density decreases as µ decreases.

245

Figure 8.9: The normalized number density of photons for the same set of µ chosen in Figure 8.7.
This figure demonstrates, as we saw in Figure 8.5, that the density of photons tracks that of thescatterers.

246

p
Figure 8.10: The y-component of the three-velocity, normalized by 2/(9ρ00 κz0 ) ∼ δ, for the same
set of µ chosen in Figure 8.7. We see that the relative maximum increases for smaller µ.

247transsonic extends farther into the ambient medium, widening the boundary layer. Conversely,√
when µ  1, the sound speed approaches a constant ' 1/ 3, which results in only a slightnarrowing of the boundary layer.
Equation (8.10) shows that the energy density of the radiation, and hence the pressure, isconstant across the jet, which is a statement of the causal connectedness of the boundary layer. In
order for this equation to remain valid, then, we require that the transverse sound crossing timeover the boundary layer thickness δy be less than the time it takes the fluid to traverse the distanceδz. Since the transverse sound speed is given c⊥ = cs /Γ, where cs is given by equation (8.44), thisrequirement yields the inequality δ . cs /(Γj vj ). Once this inequality is no longer satisfied, equation(8.10) does not hold, and we must include more terms in equations (8.10) – (8.11) to account for
the gradients in the energy density of the radiation.
As we noted in Section 3, the density approaches zero as we near the axis of the jet. Physically,
this effect arises from the fact that, for z  z0 , the Lorentz factor grows unbounded and theboundary layer thickness goes to zero. Therefore, the center of the jet originates along a curve of
infinite shear and, consequently, infinite entropy. Since the specific entropy scales as s0 ∝ e0 /(ρ0 )4/3
and e0 is constant across the boundary layer, we see that the density of scatterers must equal zeroat the center of the jet.
From Figures 8.6 and 8.10, we see that the y-component of the velocity initially causes theoutflow to expand into its surroundings. When |ξ| becomes large, however, the directionality of vy
reverses toward the jet. The outflow therefore entrains material from the ambient medium not only
by expanding in the transverse direction, but also by dragging material in from the environment.
Interestingly, the y-component of the velocity approaches a non-zero, constant value as wemove into the external medium. This behavior arises from the fact that the jet is removing material
from the system at a rate Ṁ ∼ ρ0 Γvz dy ∼ z 1/2 , showing that amount of mass excavated from theenvelope increases as we move along z. Therefore, in order to maintain a steady-state, we require
a constant influx of material at infinity that can resupply the amount lost due to the jet. This
interpretation is substantiated by integrating the continuity equation (8.13) from y = −∞ to248
y = ∞, which shows that the y-component of the velocity at infinity scales as vy,∞ ∼ −f∞ /2. The
factor of 1/2 arises from the fact that the mass loss rate scales as Ṁ ∼ ρ0 Γvz dy ∝ z 1/2 ; therefore,
if one could decrease the mass loss rate to one that was constant in z, then the y-component of
the velocity would vanish at infinity. Likewise, if one could create a scenario in which Ṁ scaled
as a negative power of z, then the value of vy would maintain an efflux of material at infinity tokeep the system from amassing inertia towards the center of the jet (thus violating the steady-state
assumption).
The free-streaming jet solutions analyzed in this paper are only valid when the Lorentz factor
is relativistic, i.e., when ζ > 1. When ζ  1, Γ ' 1, ρ0 ' ρ00 , and we can show that conservation of
momentum along the z-axis results in the scaling vj ∼ z −1/3 , δ ∼ z 2/3 , which agrees with the resultsof the incompressible, non-relativistic theory (in Chapter 10, Section 12 of Kundu & Cohen 2008,
see their discussion at the top of page 383). Therefore, once z & Γ4j z0 , the outflow will undergo a
non-self-similar transition from the solution presented here to its non-relativistic counterpart.
The free-streaming jet model has observational consequences. For example, the decrease in
the density along the jet axis means that observers looking down the barrel of the jet see farther
into the outflow. Because Γ ∝ z −1/4 and e = Γ2 e0 , where e is the lab-frame radiation energy density,
this means that those observers see a higher energy density of photons. Also, those same observers,
using the fact that the lab-frame radiation number density is N ∼ ΓN 0 , see an energy per photon ofe/N ∼ Γe0 /N 0 . Therefore, not only do the on-axis observers see a more Lorentz-boosted spectrum
because they can see deeper into the outflow, their spectrum is also hardened from the fact thatN 0 is minimized along the axis of the jet (where y = 0 in the figures of Section 3).

8.3.1

Comparative notes between the two-stream model (Chapter 7) and thefree-streaming jet model

Chapter 7 analyzed how the presence of a radiation pressure-supported envelope affected
the propagation of a radiation-dominated, relativistic jet through the two-stream approximation.
This approximation treats the jet and its surroundings as semi-infinite, separate fluids, a contact

249
discontinuity serving as the surface of separation between the two. In contrast, the free-streaming
jet solution presented here treats the entire system – the jet and its surroundings – as a single fluid.
There is thus no formal distinction between “jet material” and “ambient material,” meaning thatno contact discontinuity exists in the system.
By comparing Figure 1 of Chapter 7 and Figure 8.3 of the previous section, we see that thez-component of the four-velocity at a fixed z behaves similarly between the two models. Namely,
the solution starts at some “jet” velocity, which corresponds to y = ∞ in the two-stream modeland to y = 0 in the jet model, and smoothly transitions to a velocity of approximately zero over theextent of a few boundary layer thicknesses. However, the full spatial dependence of the velocity,
one that includes variation in the z-direction, differs drastically between the two models: while
the two-stream jet maintains a constant Γj along z, we found here that the z-component of the
four-velocity scales as Γvz ∝ z −1/4 , implying that the overall velocity of the jet slows as we lookfarther down z. This behavior arises from the fact that the free-streaming jet can entrain material,
this entrainment causing an increase in the inertia contained in the outflow and a resultant decrease
in its velocity.
The general behavior of the number densities of scatterers and photons between the models
is also similar, which can be understood by comparing Figures 2 and 3 of Chapter 7 to Figures 8.4
and 8.5 of Section 3, respectively. In particular, both densities decrease within the boundary layer
separating the outflow and its surroundings, asymptotically approaching their jet values as y → ∞in the two-stream model and as y → 0 in the free-streaming jet model. Likewise, each approaches
its ambient value as y → −∞ in the two-stream model and as y → ±∞ in the free-streaming jet
model. Although the gross properties of both are similar, one striking difference arises, however,in the behavior of the scatterer density: in the two-stream model, the existence of the contact
discontinuity causes the function g, and hence the density ρ0 , to vanish within the boundary layer(it vanishes specifically at y = 0 – the location of the contact discontinuity). Conversely, ρ0 remains
non-zero throughout the boundary layer that connects the jet and its surroundings in the model
presented here, and only as we near the center of the jet does the density of scatterers go to zero.

250
The presence of a contact discontinuity thus results in the likely non-physical vanishing of themassive particles within the boundary layer.
Finally, the scaling of the boundary layer thickness itself differs between the two models. In
the two-stream case, we found that δ ∝ z 1/2 (see equation (15) of Chapter 7). Therefore, as one
looks farther down the z-axis, the boundary layer that develops between the jet and the ambientenvironment extends into both media at a rate proportional to z 1/2 . Contrarily, equation (8.40)
shows that the free-streaming jet boundary layer expands into its surroundings as ∝ z 3/4 (though
this transitions to z 2/3 in the non-relativistic limit; see the discussion at the top of page 383 in
Chapter 10, Section 12 of Kundu & Cohen 2008). The free-streaming jet boundary layer thus
expands more rapidly than does the two-stream solution.

8.4

Summary and conclusions
Employing the equations of radiation hydrodynamics in the viscous limit, which are applicableas long as changes in fluid quantities are small over the mean free path of a photon, we analyzed
the dynamics of a relativistic, free-streaming jet under the boundary layer approximation. Thisapproximation, which should be upheld in jets with transverse optical depths substantially greaterthan one, states that variations in the properties of the outflow are confined to a thin layer of width
δ, and it allowed us to transform the full set of equations (8.1) – (8.5) into a set of greatly simplified
boundary layer equations (8.10) – (8.13).
Perhaps the biggest difference between the two-stream solutions, presented and analyzed in
paper I, and the free-streaming jet solutions presented here is in the distinction between the jet
and the ambient medium. In the former, the two are considered as distinct, interacting entities,
which allows one to specify separately their asymptotic properties. The latter approach, on theother hand, considers the whole configuration as a single fluid.
Because one has more freedom in specifying the properties of the outflow, the two-stream
solution has the added benefit of being able to treat scenarios in which the properties of the jet
and the ambient medium differ significantly. However, maintaining the distinction between the two

251
media necessitates the existence of a surface of contact between the two across which no fluid canflow, meaning that the jet cannot entrain ambient material. Furthermore, as was demonstrated in
Chapter 7, this boundary condition results in the density formally vanishing at the interface, which
is likely non-physical.
By treating the outflow and the environment as one fluid, we demonstrated that entrainment
does occur in the radiation-viscous, free-streaming jet solution, which causes the z-component of
the four-velocity of the jet to slow as ∼ z −1/4 (this power-law, however, may differ if the adopted
symmetry is azimuthal as opposed to planar; see below). These solutions also show that the
comoving densities of scatterers and photons decrease dramatically within the boundary layer,with the density of scatterers approaching zero as one nears the center of the jet. Therefore,
because observers that look “down the barrel of the jet” can see farther into the outflow, they see amore Lorentz-boosted energy density than those that view the outflow off-axis. They also observea higher energy per photon, given by e/N ' e0 Γo /N 0 , both because N 0 is minimized along the axis
and because Γo , the observed Lorentz factor, is larger. Such features are in qualitative agreementwith the event Swift J1644+57, where such an observer orientation is invoked to explain the X-ray
emission (Zauderer et al., 2011).
In addition to super-Eddington TDEs and long GRBs (as well as the relatively new classof “ultra-long” GRBs; Levan et al. 2014) – the two applications considered in the Introduction –the free-streaming jet model could be applied to other sources. As mentioned in Chapter 7, thismodel may also be relevant to microquasars (Fender et al., 2004) (particularly those that fall in
the class of ULX’s; King et al. 2001) such as the object SS 433 (Fabrika 2004; Begelman et al.
2006a; see Arav & Begelman (1993) for an application of the two-stream model to this source).
Additionally, a jetted quasi-star – a protogalactic gas cloud supported by a supercritically accreting
black hole (Begelman et al., 2006a, 2008; Czerny et al., 2012) – provides another situation in whicha radiation-dominated jet propagates alongside a radiation pressure-supported envelope.
The free-streaming jet solution presented here is limited to describing plane-parallel, twodimensional systems. One consequence of this assumption is that the entrainment of material,

252
which slows down the jet as Γ ∝ z −1/4 , happens effectively in one dimension. If, on the other
hand, one imposed azimuthal symmetry and described the system in terms of spherical coordinates(r, θ, φ), which is likely more relevant for realistic jetted systems, the entrainment would occur in
two dimensions. Other conditions being equal, the slowing of the jet along the axis would then be
more pronounced, i.e., the Lorentz factor would scale as Γ ∝ r−p with p > 1/4.
The other main assumption of the free-streaming jet model presented here is that the energy
density of the ambient medium is independent of z, i.e., e0 (z) = e00 . If a pressure gradient were
present, this force would tend to accelerate (or, in principle, decelerate) the jet material, offsetting
the power-law scaling Γ ∝ z −1/4 . In fact, one can imagine that if the pressure gradient were strong
enough, it could indeed reverse the overall slowing of the jet and cause the outflow to accelerate. If
the energy density scaled as e0 = e00 (z/z0 )−q , which is relevant for super-Eddington TDEs as longas the constant q satisfies q > 3/2 (Chapter 5), this situation could be actualized near the launch
point of the jet.
In an ensuing paper, we plan to compare more quantitatively the predictions made by the freestreaming jet model presented here and the observations of Swift J1644+57. We will also extend
our analysis to incorporate a spherical geometry – as azimuthal symmetry is almost certainly more
relevant for this and other systems than the planar symmetry adopted here – as well as radiallydependent ambient pressure and density profiles.

Chapter 9

Conclusions

Tidal disruption events provide information about the properties of the supermassive black
holes residing in the centers of galaxies, thereby constraining our models of galaxy growth andthe evolution of the Universe. However, our ability to correctly interpret that information hinges
on our understanding of the tidal disruption process. This thesis has hopefully conveyed to thereader that this process is incredibly intricate and encompasses a wealth of distinct, physical processes, including the formation of gas streams, the creation of accretion disks, and the launching ofrelativistic jets.
Sections 2 – 4 analyzed the formation and evolution of tidally-disrupted debris streams, using
a combination of analytical and numerical techniques. We showed that, once the stream recedes
to large distances from the black hole, the self-gravity of the stream in the radial direction can
result in its fragmentation into small-scale, gravitationally bound clumps. The time at which thestream fragments and the properties of the clumps (e.g., their masses and radii) depend on thethermodynamics of the gas, and we performed a suite of simulations to delineate that dependence.
Finally, we constructed an analytic model to describe these debris streams, showing that there is a
simple, self-similar function that characterizes the velocity profile of these streams and agrees very
well with our simulations. We also provided approximate expressions for the density distributionsof tidally-disrupted debris streams, and we showed that the effective adiabatic index of the gas
must be γ ≥ 5/3 if the stream is to be gravitationally unstable.
Section 5 presented a new model for the accretion disks that form during the super-Eddington

254
phase of tidal disruption events. This model assumes that the radiation liberated during the accretion process is trapped by the inflowing material, increasing the specific energy, or Bernoulli
parameter, of the accretion disk. Once the Bernoulli parameter approaches zero, the material
is only marginally-bound to the black hole, and any further augmentation of the specific energy
would effectively unbind the disk. Therefore, once this zero-Bernoulli stage is reached, we postulatethat the remaining accretion energy is funneled into two, bipolar jets that remove the accretion
energy from the system without destroying the debris disk, the resulting configuration being azero-Bernoulli accretion (ZEBRA) disk. We showed that the characteristics of the progenitor starand the supermassive black hole completely determine the properties of the resulting ZEBRA flow,
and the high pressure and sub-Keplerian angular momentum of the material cause these disks tobe “puffed up” into quasi-spherical envelopes with narrow, centrifugally-supported funnels that
self-consistently provide exhaust routes for the jets. We made comparisons between the ZEBRA
model and the jetted tidal disruption event Swift J1644+57, and, based on simple timing arguments, demonstrated that the black hole powering Swift J1644+57 must have a mass of ∼ 105 M(assuming the ZEBRA model correctly describes the accretion process).
Sections 6 – 8 investigated the nature of the relativistic jets launched from tidal disruption
events. These outflows are unique in that radiation can contribute overwhelmingly to their energy
and momentum, and it is thus the photons present in the jet that mediate the interaction between
the jet and the ZEBRA envelope. We developed the precise way by which this interaction takes
place when the photon mean free path is small, showing from the general relativistic Boltzmann
equation that small anisotropies in the comoving radiation field generate an effective viscosity thattransfers energy and momentum from the jet to its surroundings (and vice versa). We derived
the equations describing this viscous interaction – the general relativistic equations of radiationhydrodynamics in the viscous limit – and applied them to a boundary layer model of the jets
produced from tidal disruption events. The boundary layer assumption states that the transition
from the jet to the surrounding medium is confined to a thin transition region, which we showed ison the order of the square root of the photon mean free path. These boundary layer models showed255
that in the transition from the jet to the surrounding ZEBRA, where the shear is maximized, thenumber of scatterers decreases dramatically and the photon spectrum hardens. Thus, the angle
between the jet axis and the observer angle has important consequences for the inferred propertiesof the outflow.

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Appendix A

Non-self-similar ZEBRA solutions

Blandford & Begelman (2004) demonstrated that, if the angular momentum is distributed
in a quasi-Keplerian fashion, i.e. `2 ∝ GM r sin2 θ, then there exist self-similar solutions for thepressure and the density throughout the disk. Here we wish to demonstrate that the converse of thisstatement, namely “If the pressure and density fall off in a self-similar manner, then the angularmomentum is quasi-Keplerian,” is also true. In the process we will find the general solution for thedensity, pressure, and angular momentum distributions of ZEBRA flows in a Keplerian potential.
As a reminder, the momentum and Bernoulli equations governing the ZEBRA flow are−

1 ∂p
GMh `2 csc2 θ=− 2 +
,
ρ ∂r
r
r3

(A.1)

1 ∂p
`2 csc2 θ cot θ
=
,
ρ ∂θ
r2

(A.2)

GMh `2 csc2 θ
γ p
+
+
= 0.
2
r
2r
γ−1ρ

(A.3)

Now make the following auxiliary definitions:`2 csc2 θGMh=f (r, θ),
2
r
r

(A.4)

GMh
h(r, θ).
r(A.5)

p(r, θ) =265
Inserting these definitions into (A.1), (A.2), and (A.3), we find the following differential equation
for h:

1∂h
1 ∂hh(r, θ) + r
=
.
γ−1
∂r
2 ∂θ

(A.6)

The solution to this partial differential equation may be found most simply by separating variables.
Doing so, we obtainh(r, θ) = r

1
− γ−1

Z ∞

c(λ)(r sin2 θ)λ dλ.

(A.7)

0

Here λ is the arbitrary constant obtained from the separation of variables technique, and c(λ) is
the constant of integration which is, in general, a function of λ. The total solution, capable ofbeing matched to arbitrary boundary conditions, is then a sum of the eigensolutions appropriate
to a single λ (if λ takes on a discrete set of values, the integral becomes a sum and c(λ) → cλ ).
The range of λ has been chosen in hindsight to be consistent with the restriction that the squareof the angular momentum and the density both be positive.
Using (A.7), we can readily determine expressions for the density, pressure, and angularmomentum, which we find to beρ(r, θ) = r

1
− γ−1

Z ∞
(λ +
0

p(r, θ) = GMh r

γ
− γ−1

γ
)c(λ)(r sin2 θ)λ dλ,
γ−1

Z ∞

c(λ)(r sin2 θ)λ dλ,

(A.8)

(A.9)

0

R∞

λ c(λ)(r sin2 θ)λ dλ` (r, θ) = 2GMh r sin θ R ∞ 0.
γ2
λ0 (λ + γ−1 )c(λ)(r sin θ) dλ2

2(A.10)We are now in a position to prove the statement at the beginning of this appendix: if we require thedensity or pressure to vary self-similarly, then c(λ) = c0 δ(λ − λ0 ), where c0 is a constant independent266
of λ and δ(x) is the Dirac delta function. Inserting this relation for c(λ) into (A.10), we find

` 2 (r, θ) =2λ2γ GMh r sin θ,
λ + γ−1(A.11)

which agrees with the result in section 2.2 if we let λ = n −3/2 + 1/(γ −1). The angular momentum
distribution of a self-similar flow is therefore quasi-Keplerian. Furthermore, even if c(λ) is not adelta function, the functional dependence of the angular momentum is the Keplerian solution
multiplied by a ratio of integrals, with each of those integrals having the same leading power of r.
For this reason the dominant behavior of the angular momentum will always be Keplerian.

Appendix B

Non-zero Bernoulli parameter

One of the tenets upon which much of our previous analysis rests is that the Bernoulli
parameter is exactly zero. Here we would like to investigate the consequences of letting B become
negative, meaning that the disk is more than marginally bound; this situation may occur if the jetturns on before enough energy is pumped into the debris disk. We will also examine the case whereB > 0, and we will show that this regime is associated with a wind.
Assuming that we can still regard B as roughly constant, then ∇B = 0, and the gyrentropic
nature of the flow in the disk is preserved (see Section 2). The fluid and Bernoulli equations are
now

−

1 ∂p
GMh `2 csc2 θ=− 2 +
,
ρ ∂r
r
r3

(B.1)

1 ∂p
`2 cot θ csc2 θ
=
,
ρ ∂θ
r2

(B.2)

GMh `2 csc2 θ
γ p
+
+
= B,
2
r
2r
γ−1ρ

(B.3)

where we have written the Bernoulli parameter as B, with B a negative number. We can generalize
our analysis in the previous appendix to include non-zero B. The resultant self-similar solutions

268
for the angular momentum, density, and pressure are




GMh
` =a
+ B r2 sin2 θ,
r
2

 GMh
ρ = ρ0p = βρ0r +B
GMh
r0 + BGMh+Brα+ 1  2 αγ−1r(sin2 θ)α ,r02

 GMhr +B
GMhr0 + B

α+ 1  2 αγ−1r(sin2 θ)α .
r02

(B.4)

(B.5)

(B.6)

Here our notation is consistent with that in section 2, i.e. a, α, n, and β retain their original
definitions. We see that a disk with finite binding energy differs in its radial structure from thatconsidered previously only in regions where GMh /r ∼ |B|, with the angular dependence completely
unaltered. Therefore, our analysis in sections 2-4 concerning the properties of the disk is largely
incorrect only if GMh /|B| < R.
To determine when and if this inequality is satisfied, let us assume that the inverse is true,i.e. R < GMh /|B|, so that the results from the preceding sections are almost correct. Then we canapproximate the outer radius by the expression in section 3. We then find, in order for our neglectof the Bernoulli parameter to be permissible, that B must satisfy|B| < GMh

2/5 .

√√
κy
4πc M GMh β a(3 − q)(B.7)

For our current models, the right-hand side of (B.7) takes on values that are on the order of ≈ 10 17 .
To see if this number is consistent with our neglect of finite binding energy, we can further specify
the Bernoulli parameter by recalling the gravitational potential energy of the disk and its relation
to the star, which implies B = −δ(GM∗ /R∗ )(Mh /M∗ )1/3 , where δ is a numerical factor. Inserting
numbers into (B.7), we find that our assumptions in sections 2-4 are correct if δ . 1. As we haveargued, the shock heating and energy generation in the inner regions of the disk are thought to raise
the Bernoulli parameter, so that δ . 1 should be satisfied in nearly all cases. However, for larger

269
black hole masses or larger progenitors, the binding energy, and hence the Bernoulli parameter,
will increase, and the assumption of B ≈ 0 may start to break down.
By changing the sign of B, we obtain the solutions for positive-Bernoulli disks. As anticipated,these models yield finite pressure, density, and angular momentum at infinity, confirming our
suspicions that positive-Bernoulli disks produce winds. In fact, if B becomes too large, the density
again approaches a power law but with 2α replacing −q. In this limit we can show that, for qthat leave the density finite at the poles, all solutions predict an energy that increases as we goout in radius. These two physically meaningless conclusions lead us to the assertion that positive
B solutions do not describe wind-less disks, and hence are not appropriate to our modeling of thedebris disks produced by tidal disruption events.