Social Stars: Modeling the Interactive Lives of Stars in Dense Clusters and Binary Systems in the

465 Pages·2016·17.12 MB·English

UC Santa Cruz
UC Santa Cruz Electronic Theses and Dissertations
Title
Social Stars: Modeling the Interactive Lives of Stars in Dense Clusters and Binary Systems in
the Era of Time Domain Astronomy
Permalink
https://escholarship.org/uc/item/8p01x7b9
Author
MacLeod, Morgan Elowe
Publication Date
2016
License
https://creativecommons.org/licenses/by-nc-nd/4.0/ 4.0

Peer reviewed|Thesis/dissertation
eScholarship.org Powered by the California Digital Library
University of California
UNIVERSITY OF CALIFORNIA
SANTA CRUZ
SOCIAL STARS: MODELING THE INTERACTIVE LIVES OF
STARS IN DENSE CLUSTERS AND BINARY SYSTEMS IN THE
ERA OF TIME DOMAIN ASTRONOMY
A dissertation submitted in partial satisfaction of the
requirements for the degree of
DOCTOR OF PHILOSOPHY
in
ASTRONOMY AND ASTROPHYSICS
by
Morgan Elowe MacLeod
June 2016
The Dissertation of Morgan Elowe
MacLeod
is approved:
Professor Enrico Ramirez-Ruiz, Chair
Professor Douglas N. C. Lin
Professor Gregory Laughlin
Tyrus Miller
Vice Provost and Dean of Graduate Studies
Copyright ⃝c by
Morgan Elowe MacLeod
2016
Table of Contents
List of Figures viii
List of Tables xi
Abstract xii
Acknowledgments xiv
1 Introduction 1
1.1 Stars Around (and Eaten By) Massive Black Holes . . . . . . . . . . . . 2
1.2 Star-Eat-Star Universe: Common Envelope Episodes . . . . . . . . . . . 4
1.3 A Computational Modeling Ecosystem . . . . . . . . . . . . . . . . . . . 6
1.4 Outline of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 The Tidal Disruption of Giant Stars and Their Contribution to the
Flaring Supermassive Black Hole Population 9
2.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Stellar Evolution in the Context of Tidal Disruption . . . . . . . . . . . 13
2.3.1 Stellar evolution models . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 Post-MS stellar evolution . . . . . . . . . . . . . . . . . . . . . . 15
2.3.3 Tidal disruption basics applied to evolved stars . . . . . . . . . . 17
2.4 Hydrodynamics of giant disruption . . . . . . . . . . . . . . . . . . . . . 19
2.4.1 Initial models for hydrodynamic simulations . . . . . . . . . . . . 19
2.4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.3 Pericenter passage and mass removal . . . . . . . . . . . . . . . . 22
2.4.4 Evolution of unbound material . . . . . . . . . . . . . . . . . . . 27
2.4.5 Debris fallback and AGN flaring . . . . . . . . . . . . . . . . . . 31
2.5 Loss cone theory and rates of giant disruption . . . . . . . . . . . . . . . 35
2.5.1 Simplified nuclear cluster model . . . . . . . . . . . . . . . . . . 37
2.5.2 Impact of the structural diversity of observed galactic centers . . 46
2.5.3 The stellar diet of SMBHs . . . . . . . . . . . . . . . . . . . . . . 50
iii
2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.6.1 A limiting SMBH mass scale for tidal disruption flares . . . . . . 54
2.6.2 Contribution of giant star disruptions to local low-luminosity AGN 56
2.6.3 Detection of giant star tidal disruption flares . . . . . . . . . . . 58
2.6.4 Caveats and prospects . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Spoon-Feeding Giant Stars to Supermassive Black Holes: Episodic
Mass Transfer From Evolving Stars and Their Contribution to the
Quiescent Activity of Galactic Nuclei 65
3.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.3 Mass transfer from evolving stars . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Episodic flares over many pericenter passages . . . . . . . . . . . . . . . 75
3.4.1 Mass Stripping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.2 Return to the Black Hole . . . . . . . . . . . . . . . . . . . . . . 78
3.5 Estimating the population of mass-transferring stars . . . . . . . . . . . 82
3.6 Significance to the duty cycle of tidally-fed SMBHs . . . . . . . . . . . 88
3.6.1 Tidal disruption flares . . . . . . . . . . . . . . . . . . . . . . . . 89
3.6.2 Episodic mass transfer from evolving stars: Spoon-feeding . . . . 91
3.6.3 Diffusion to the loss cone . . . . . . . . . . . . . . . . . . . . . . 93
3.6.4 Comparison to stellar wind feeding . . . . . . . . . . . . . . . . . 94
3.6.5 Implications for low-luminosity active galactic nuclei . . . . . . . 96
3.7 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Illuminating Massive Black Holes With White Dwarfs: Orbital Dy-
namics and High Energy Transients from Tidal Interactions 101
4.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.3 Phenomenology of White Dwarf Tidal Interactions . . . . . . . . . . . . 105
4.3.1 Near-parabolic orbit tidal disruption . . . . . . . . . . . . . . . . 106
4.3.2 Tidal stripping in an eccentric orbit . . . . . . . . . . . . . . . . 108
4.3.3 Roche-lobe overflow from a circular orbit . . . . . . . . . . . . . 112
4.4 Stellar Clusters Surrounding MBHs . . . . . . . . . . . . . . . . . . . . . 113
4.4.1 A Simple Cluster Model . . . . . . . . . . . . . . . . . . . . . . . 114
4.4.2 Orbital Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.4.3 Scattering to the Loss Cone . . . . . . . . . . . . . . . . . . . . . 119
4.5 WD Capture and Inspiral . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.5.1 Binary Splitting and WD Capture . . . . . . . . . . . . . . . . . 122
4.5.2 Modeling the Evolution of Captured WD orbits . . . . . . . . . . 125
4.5.3 Distributions at the Onset of Mass Transfer . . . . . . . . . . . . 128
4.6 Detecting High Energy Signatures of White Dwarf Disruption . . . . . . 130
4.6.1 Dissipation and Emission Mechanisms . . . . . . . . . . . . . . . 132
4.6.2 Event Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
iv
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.7.1 The MBH mass function . . . . . . . . . . . . . . . . . . . . . . . 136
4.7.2 Ultra-long GRBs as WD Tidal Disruptions? . . . . . . . . . . . . 137
4.7.3 Prospects for simultaneous Electromagnetic and Gravitational Wave
Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
4.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5 Optical Thermonuclear Transients From Tidal Compression of White
Dwarfs as Tracers of the Low End of the Massive Black Hole Mass
Function 144
5.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.3 Hydrodynamic Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.4 Optical Thermonuclear Transients . . . . . . . . . . . . . . . . . . . . . 153
5.4.1 Light Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
5.4.2 Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
5.5 Multi-wavelength Accretion Counterparts . . . . . . . . . . . . . . . . . 167
5.5.1 Debris Fallback, Circularization, and Accretion . . . . . . . . . . 167
5.5.2 Accretion Disk and Thermal Emission . . . . . . . . . . . . . . . 171
5.5.3 Jet Production and Signatures . . . . . . . . . . . . . . . . . . . 172
5.6 Event Rates and Detectability of WD Tidal Disruption Signatures . . . 177
5.6.1 Specific Event Rate . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.6.2 The MBH Mass Function: Estimating the Volumetric Event Rate 179
5.6.3 Detecting Thermonuclear Transients with LSST . . . . . . . . . 180
5.6.4 Detecting Beamed Emission with High-Energy Monitors . . . . . 183
5.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
5.7.1 A Diversity of Thermonuclear Transients fromWD Tidal Disruption185
5.7.2 Are ULGRBs WD Tidal Disruptions? . . . . . . . . . . . . . . . 187
5.7.3 Strategies for Identifying WD Tidal Disruptions . . . . . . . . . 189
5.7.4 Uncovering the Mass Distribution of Low-Mass MBHs . . . . . . 190
5.8 Summary and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 191
6 The Close Stellar Companions to Intermediate Mass Black Holes 194
6.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
6.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.3 Direct N-body Simulations of Clusters Hosting IMBHs . . . . . . . . . . 199
6.3.1 Simulation Method . . . . . . . . . . . . . . . . . . . . . . . . . . 199
6.3.2 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . 202
6.3.3 Methodological Comparison to Previous Work . . . . . . . . . . 203
6.3.4 Cluster Global Evolution . . . . . . . . . . . . . . . . . . . . . . 205
6.4 Close Companions to the IMBH . . . . . . . . . . . . . . . . . . . . . . . 208
6.4.1 Stars Bound to the IMBH . . . . . . . . . . . . . . . . . . . . . . 208
6.4.2 Companion Capture & Orbital Hardening . . . . . . . . . . . . . 209
v
6.4.3 Companion Orbital Properties . . . . . . . . . . . . . . . . . . . 211
6.4.4 Companion Stellar Properties . . . . . . . . . . . . . . . . . . . . 214
6.4.5 Termination of Close Partnerships . . . . . . . . . . . . . . . . . 218
6.5 Dependence on Cluster Properties . . . . . . . . . . . . . . . . . . . . . 224
6.5.1 Companion Objects in Cluster Evolution . . . . . . . . . . . . . 226
6.5.2 Companion Demographics . . . . . . . . . . . . . . . . . . . . . . 227
6.5.3 Companion Dynamics . . . . . . . . . . . . . . . . . . . . . . . . 230
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
6.6.1 IMBH Companions in Old Clusters . . . . . . . . . . . . . . . . . 234
6.6.2 Dynamical Mechanisms for Repeated Tidal Disruption Flares by
IMBHs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
6.6.3 Revealing IMBHs in GCs . . . . . . . . . . . . . . . . . . . . . . 238
6.6.4 Extensions and Future Work . . . . . . . . . . . . . . . . . . . . 243
6.7 Summary & Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
7 Asymmetric Accretion Flows within a Common Envelope 248
7.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
7.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
7.3 Stellar Properties, Typical Scales and Gradients . . . . . . . . . . . . . . 252
7.3.1 Characteristic scales . . . . . . . . . . . . . . . . . . . . . . . . . 253
7.3.2 MESA Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 255
7.4 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261
7.4.1 Previous Numerical Studies of HLA . . . . . . . . . . . . . . . . 264
7.4.2 Numerical Approach and Simulation Setup . . . . . . . . . . . . 266
7.4.3 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . . . 269
7.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
7.5.1 Flow Morphology . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
7.5.2 Effects of Sink Size . . . . . . . . . . . . . . . . . . . . . . . . . . 275
7.5.3 Accretion of Mass and Angular Momentum . . . . . . . . . . . . 278
7.5.4 Disk Formation? . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
7.5.5 Drag . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
7.6.1 Cooling, Accretion, and Feedback from Embedded Objects . . . 292
7.6.2 Mass Accumulation during CE Inspiral . . . . . . . . . . . . . . 295
7.6.3 Loss of CE Symmetry during Inspiral . . . . . . . . . . . . . . . 296
7.6.4 The End of Dynamical Inspiral: Envelope Spin-Up and Heating . 298
7.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
7.8 Numerical Details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
7.8.1 Resolution Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
7.8.2 Fitting Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
vi
8 On the Accretion-Fed Growth of Neutron Stars During Common En-
velope 308
8.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308
8.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
8.3 Characteristic Conditions in NS Accretion . . . . . . . . . . . . . . . . . 311
8.3.1 Hoyle-Lyttleton Accretion within a CE . . . . . . . . . . . . . . 311
8.3.2 Microphysics and Hypercritical Accretion . . . . . . . . . . . . . 312
8.3.3 Limits on the Accretion Rate due to Flow Asymmetry . . . . . 313
8.4 Inspiral and Accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
8.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
8.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
9 The Onset of a Common Envelope Episode: Lessons from the Remark-
able M31 2015 Luminous Red Nova Outburst 325
9.1 Chapter Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
9.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
9.3 Outburst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330
9.3.1 Summary of Observations . . . . . . . . . . . . . . . . . . . . . . 330
9.3.2 Modeled Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 331
9.3.3 Ejecta Mass and Outburst Energetics . . . . . . . . . . . . . . . 334
9.4 Pre-Outburst Source . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
9.4.1 Stellar Evolution Models . . . . . . . . . . . . . . . . . . . . . . . 340
9.4.2 Source Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 340
9.5 System and Transient Together . . . . . . . . . . . . . . . . . . . . . . . 346
9.5.1 Combined Requirements . . . . . . . . . . . . . . . . . . . . . . . 346
9.5.2 Interpretation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
9.6 From Binary System to Transient Outburst . . . . . . . . . . . . . . . . 349
9.6.1 Pathways to the Onset of Common Envelope . . . . . . . . . . . 351
9.6.2 Constraining M2 in M31LRN 2015 . . . . . . . . . . . . . . . . . 357
9.7 Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.7.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
9.7.2 Comparison to Other Stellar-Merger Transients . . . . . . . . . . 364
9.7.3 Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
9.8 Mass Lost from L2 Prior to Contact in Roche Lobe Overflow Mergers . 369
10 Conclusion 372
vii
List of Figures
2.1 Stellar evolutionary stages . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Giant star structural profiles . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Simulation slices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4 Mass loss with varying impact parameter . . . . . . . . . . . . . . . . . 24
2.5 Mass loss as a function of β . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Temporal evolution of tidal tail material . . . . . . . . . . . . . . . . . . 29
2.7 Fallback rate of bound material . . . . . . . . . . . . . . . . . . . . . . . 32
2.8 Comparison of fallback accretion . . . . . . . . . . . . . . . . . . . . . . 33
2.9 Encounters with β = 1 rescaled to different black hole masses . . . . . . 36
2.10 Loss cone flux of solar-type stars . . . . . . . . . . . . . . . . . . . . . . 43
2.11 Scaling of the tidal disruption rate with increasing tidal radius . . . . . 46
2.12 Scaling of the tidal disruption rate in a sample of 41 early type galaxies 48
2.13 The stellar ingestion flaring diet of SMBHs . . . . . . . . . . . . . . . . 51
2.14 Tidal disruption flare production across the SMBH mass spectrum . . . 53
2.15 Duty cycle of tidal disruption flaring . . . . . . . . . . . . . . . . . . . . 55
2.16 Contributions to thermal emission . . . . . . . . . . . . . . . . . . . . . 59
2.17 Power-law slope of transient light curves . . . . . . . . . . . . . . . . . . 60
3.1 Mass stripping from a star in an eccentric orbit around a SMBH . . . . 72
3.2 Episodic mass transfer from a giant star to the SMBH . . . . . . . . . . 79
3.3 Profile of a repeating flaring episode . . . . . . . . . . . . . . . . . . . . 81
3.4 Phase space of stars that can evolve to transfer mass to the SMBH . . . 87
3.5 Monte Carlo realizations of the duty cycle of tidally fed black hole activity 90
4.1 Accretion-powered flares that result from tidal interactions . . . . . . . 106
4.2 The properties of tidal disruptions of WDs . . . . . . . . . . . . . . . . . 109
4.3 Characteristic scales for WD-MBH interactions in stellar cusps . . . . . 117
4.4 Fraction of loss cone flux in the large-scatter limit . . . . . . . . . . . . 121
4.5 Phase space of encounters between WDs and MBHs . . . . . . . . . . . 126
4.6 Orbital distributions of WDs captured from split binaries . . . . . . . . 129
4.7 Rates of different interaction channels per galaxy . . . . . . . . . . . . . 133
4.8 Transient luminosity versus duration . . . . . . . . . . . . . . . . . . . . 141
viii
5.1 The sequence of events of a deep-passing WD tidal disruption event . . 148
5.2 Visualization of the unbound remnant . . . . . . . . . . . . . . . . . . . 152
5.3 Bolometric light curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.4 Multiband light curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
5.5 Differential color evolution . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.6 Model thermonuclear transients in width-luminosity phase space . . . . 160
5.7 Spectra at t = 20 days compared to other supernovae . . . . . . . . . . . 162
5.8 Time series of spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
5.9 Spectra at t = 20 days from two angles . . . . . . . . . . . . . . . . . . . 165
5.10 Normalized model spectra near the SiII 6355 feature . . . . . . . . . . . 166
5.11 Peak timescales and accretion rates . . . . . . . . . . . . . . . . . . . . . 170
5.12 Mutli-wavelength light curve . . . . . . . . . . . . . . . . . . . . . . . . 176
5.13 Thermonuclear transients captured by a magnitude limited transient op-
tical survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.14 Effect of the slope of the MBH mass function on detection rate . . . . . 192
6.1 Characteristic radii from the IMBH . . . . . . . . . . . . . . . . . . . . . 207
6.2 Orbital properties of closest companions to the IMBH . . . . . . . . . . 212
6.3 Residence time of closest-partner status . . . . . . . . . . . . . . . . . . 215
6.4 Stellar properties of closest companions to the IMBH . . . . . . . . . . . 217
6.5 Interaction channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
6.6 Characteristic radii and close encounters . . . . . . . . . . . . . . . . . . 222
6.7 Characteristic radii with varying simulation parameters . . . . . . . . . 225
6.8 Exchange fraction of close encounters . . . . . . . . . . . . . . . . . . . . 227
6.9 Companion demographics in semi-major axis . . . . . . . . . . . . . . . 228
6.10 IMBH companion properties with varying simulation parameters . . . . 231
6.11 Late-time evolution of clusters . . . . . . . . . . . . . . . . . . . . . . . 235
6.12 Stellar-wind fed accretion . . . . . . . . . . . . . . . . . . . . . . . . . . 239
7.1 Density gradients encountered by embedded objects . . . . . . . . . . . 256
7.2 Time evolution of characteristic flow parameters for a 1M⊙ giant star . 257
7.3 Characteristic mach numbers and density gradients . . . . . . . . . . . . 262
7.4 Typical flow parameters plotted for CE scenarios . . . . . . . . . . . . . 263
7.5 Comparison of flow morphologies: density . . . . . . . . . . . . . . . . . 272
7.6 Comparison of flow morphologies: mach . . . . . . . . . . . . . . . . . . 273
7.7 Sink size and vorticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276
7.8 Mass and angular momentum accretion timeseries . . . . . . . . . . . . 280
7.9 Specific angular momentum of accreted material . . . . . . . . . . . . . 281
7.10 Accretion of mass and angular momentum summary . . . . . . . . . . . 282
7.11 Rotation imposed by an upstream density gradient . . . . . . . . . . . . 285
7.12 The role of equation of state . . . . . . . . . . . . . . . . . . . . . . . . . 288
7.13 Drag force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
7.14 The ratio of accretion luminosity to drag luminosity . . . . . . . . . . . 294
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