Models of black hole disks with self-gravity and non-constant angular momentum

98 Pages·2014·3.53 MB·English

by  Christos Vourellis

Aristotle University of Thessaloniki
M.Sc. Thesis
Models of black hole disks with
self-gravity and non-constant angular
momentum
Author: Supervisor:
Christos Vourellis Prof. Nikolaos Stergioulas
A thesis submitted in fulfilment of the requirements
for the degree of Master in Science in Computational Physics
in the
Department of Physics
October 2014

ARISTOTLE UNIVERSITY OF THESSALONIKI
Abstract
Faculty of Sciences
Department of Physics
Master in Science in Computational Physics
Models of black hole disks with self-gravity and non-constant angular
momentum
by Christos Vourellis
We construct the first numerical models of general relativistic, self-gravitating accretion
tori around black holes with a non-constant specific angular momentum distribution.
The models are axisymmetric and stationary and the specific angular momentum in-
creases outwards as a power law. This generalizes previous cases, where only constant
specificangularmomentumwasconsideredorwherethemodelswerenotself-gravitating.
Thenon-constancyofspecificangularmomentumwasshowntostabilizethediskagainst
the axisymmetric runaway instability while self-gravity was neglected. We are thus par-
ticularly interested in models that are massive and form an inner cusp. As a first
application, we study the existence of equilibrium models in a restricted region of the
parameter space and show that self-gravitating massive tori lie on a different surface of
equilibrium models, that features an overlap in mass.
Acknowledgements
I want to thank my advisor for giving me the change to work in such an advanced and
competitive scientific field and for his constant encouragement and support throughout
the whole project. I also want to thank the professors of the masters program for their
efforts to teach me and my colleagues as much as they could ...
iii
Contents
Abstract ii
Acknowledgements iii
Contents iv
List of Figures vi
List of Tables ix
1 Introduction 1
2 Theoretical Background 3
2.1 Tori in equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Non Constant specific angular momentum . . . . . . . . . . . . . . . . . . 8
2.3 Self-gravitating equilibrium disks . . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Torus properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Numerical method and various tori cases. 18
3.1 Numerical framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Detailed approach to the constructed models . . . . . . . . . . . . . . . . 21
3.2.1 Unstable models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1.1 Totally unstable configurations. . . . . . . . . . . . . . . 21
3.2.1.2 Configurations between the totally unstable limit and
the cusp. . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.2.2 Models at the cusp limit . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.3 Beyond the cusp limit . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2.4 Models at the W = 0 limit. . . . . . . . . . . . . . . . . . . . . 40
peak
3.2.5 Approaching the upper limit . . . . . . . . . . . . . . . . . . . . . 44
3.2.6 The upper limit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4 The parameter space of equilibrium models 54
4.1 The subspace for the specific angular momentum parameters KAJS = 0.2
EOS
and rAJS = 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
out
4.2 The subspace for the specific angular momentum parameters KAJS = 0.2
EOS
and rAJS = 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
out
iv
Contents v
4.3 The subspace of the specific angular momentum parameters for KAJS =
EOS
0.3 and rAJS = 39 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
out
4.4 The subspace of the specific angular momentum parameters for KAJS =
EOS
0.3 and rAJS = 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
out
5 Conclusions and comments 80
List of Figures
3.1 The effective gravitational potential for an unstable torus. . . . . . . . . . 22
3.2 The isopotential contours of an unstable torus. . . . . . . . . . . . . . . . 23
3.3 The four metric functions of an unstable torus in comparison with the
ones in the AJS torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 The effective potential of a torus between the totally unstable and cusp
limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.5 Theisopotentialcontoursofatorusbetweenthetotallyunstableandcusp
limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.6 The four metric functions for a torus between the totally unstable and
the cusp limits in comparison with the ones in the AJS torus. . . . . . . . 30
3.7 The effective potential of a torus filling exactly its Roche lobe. . . . . . . 31
3.8 The isopotential contours of a torus filling exactly its Roche lobe. . . . . . 32
3.9 The four metric functions for a torus in the cusp limit in comparison with
the ones in the AJS torus. . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.10 The effective potential of a torus beyond the cusp limit. . . . . . . . . . . 36
3.11 The isopotential contours of a torus beyond the cusp limit. . . . . . . . . 37
3.12 The fourmetric functionsfor atorus beyondthe cusplimit incomparison
with the ones in the AJS torus. . . . . . . . . . . . . . . . . . . . . . . . . 39
3.13 The effective potential of a torus in the W = 0 limit. . . . . . . . . . . 40
peak
3.14 The isopotential countours of a torus in the W = 0 limit. . . . . . . . 41
peak
3.15 The four metric functions for a torus at the W = 0 limit.. . . . . . . . 43
peak
3.16 The effective potential of a torus approaching the upper limit. . . . . . . . 44
3.17 The isopotential contours of a torus approaching the upper limit. . . . . . 45
3.18 Thefourmetricfunctionsforatorusapproachingtheupperincomparison
with the ones in the AJS torus. . . . . . . . . . . . . . . . . . . . . . . . . 48
3.19 The effective potential of the torus just before the location of maximum
density reaches the outer edge of the torus. . . . . . . . . . . . . . . . . . 49
3.20 Theisopotentialcontoursofthetorusjustbeforethelocationofmaximum
density reaches the outer edge of the torus. . . . . . . . . . . . . . . . . . 50
3.21 The four metric functions for a torus just before the location of maximum
density reaches the outer edge of the torus. . . . . . . . . . . . . . . . . . 53
4.1 The subspace of the specific angular momentum parameters for KAJS =
EOS
0.2 and rAJS = 39 models. (a) for the initial trial values of the constant
out
k, (b) for the rescaled values. . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.2 The effective potential for the cases presented in Figures 4.1 for tori with
KAJS = 0.2 and rAJS = 39. . . . . . . . . . . . . . . . . . . . . . . . . . . 57
EOS out
vi
List of Figures vii
4.3 The dependence of the torus mass from the (input and rescaled) specific
angular momentum parameters for KAJS = 0.2 and rAJS = 39. (a) is for
EOS out
the initial trial values of the constant k, while (b) is for the rescaled values. 58
4.4 The contour plot of the three-dimensional data. The torus mass as a
function of the two specific angular momentum parameters for KAJS =
EOS
0.2 and rAJS = 39 models. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
out
4.5 The parametric plot showing the dependence of the specific angular mo-
mentum constant from the torus mass and the critical torus mass for
different values of slope q, for KAJS = 0.2 and rAJS = 39 . . . . . . . . . . 60
EOS out
4.6 The subspace of models for the specific angular momentum parameters
for KAJS = 0.2 and rAJS = 60. (a) is for the initial trial values of the
EOS out
constant k, while (b) is for the rescaled values. . . . . . . . . . . . . . . . 63
4.7 The effective potential for the cases presented in Figures 4.6 for tori with
KAJS = 0.2 and rAJS = 60. . . . . . . . . . . . . . . . . . . . . . . . . . . 64
EOS out
4.8 The dependence of the torus mass on the (input and rescaled) specific
angular momentum parameters for KAJS = 0.2 and rAJS = 60. (a) is for
EOS out
the initial trial values of the constant k, while (b) is for the rescaled values. 65
4.9 The torus mass as a function of the two specific angular momentum pa-
rameters for KAJS = 0.2 and rAJS = 60 models. . . . . . . . . . . . . . . . 66
EOS out
4.10 Parametric plot showing the dependence of the specific angular momen-
tum constant from the torus mass and the critical torus mass for different
values of slope q, for KAJS = 0.2 and rAJS = 60. . . . . . . . . . . . . . . 67
EOS out
4.11 The subspace of the specific angular momentum parameters for KAJS =
EOS
0.3 and rAJS = 39 models. (a) is for the initial trial values of the constant
out
k, while (b) is for the rescaled values.. . . . . . . . . . . . . . . . . . . . . 69
4.12 TheeffectivepotentialforthecasespresentedinFigures4.11fortoriwith
KAJS = 0.3 and rAJS = 39. . . . . . . . . . . . . . . . . . . . . . . . . . . 70
EOS out
4.13 The dependence of the torus mass from the (input and rescaled) specific
angular momentum parameters for KAJS = 0.3 and rAJS = 39. (a) is for
EOS out
the initial trial values of the constant k, while (b) is for the rescaled values. 71
4.14 Thecontourplotofthethree-dimensionaldataforKAJS = 0.3andrAJS =
EOS out
39. The torus mass is shown as a function of the two specific angular
momentum parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.15 Parametric plot showing the dependence of the specific angular momen-
tum constant from the torus mass and the critical torus mass for different
values of exponent q, for KAJS = 0.3 and rAJS = 39. . . . . . . . . . . . . 73
EOS out
4.16 The subspace of the specific angular momentum parameters for KAJS =
EOS
0.3 and rAJS = 60. (a) is for the initial trial values of the constant k,
out
while (b) is for the rescaled values. . . . . . . . . . . . . . . . . . . . . . . 75
4.17 TheeffectivepotentialforthecasespresentedinFigures4.16fortoriwith
KAJS = 0.3 and rAJS = 60. . . . . . . . . . . . . . . . . . . . . . . . . . . 76
EOS out
4.18 The dependence of the torus mass on the (input and rescaled) specific
angular momentum parameters for KAJS = 0.3 and rAJS = 60. (a) is for
EOS out
the initial trial values of the constant k, while (4.18b) is for the rescaled
values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.19 Contour plot of the three-dimensional data showing in figure 4.3. The
color scale corresponds to the torus mass. . . . . . . . . . . . . . . . . . . 78
List of Figures viii
4.20 Parametric plot showing the dependence of the specific angular momen-
tum constant from the torus mass and the critical torus mass M for
cr
overlap, for different values of exponent q, for KAJS = 0.3 and rAJS = 60. 79
EOS out
5.1 The changes in the outer radius for the higher values of the torus mass. . 82
5.2 The changes in the equation of state constant for the higher values of the
torus mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
List of Tables
3.1 Detailed information about the representative model of an unstable torus. 22
3.2 Detailed information about the torus between the totally unstable stable
and the cusp limits. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Detailed information about the torus in the cusp limit. . . . . . . . . . . . 33
3.4 Detailed information about the torus beyond the cusp limit. . . . . . . . . 37
3.5 Detailed information about the torus at the W = 0 limit. . . . . . . . 41
peak
3.6 Detailed information about the torus approaching the upper limit. . . . . 46
3.7 Detailedinformationaboutthetorusjustbeforethelocationofmaximum
density reaches the outer edge of the torus. . . . . . . . . . . . . . . . . . 51
5.1 The minimum, mean and maximum value of the torus mass which causes
the overlap in mass. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
ix