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Numerical simulation of viscous accretion disks

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Numerical simulation of viscous accretion disks
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Drimmel, Ronald Eugene, 1964-
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Accretion ( jstor )
Angular momentum ( jstor )
Artificial satellites ( jstor )
Average linear density ( jstor )
Mass ( jstor )
Particle density ( jstor )
Particle mass ( jstor )
Subroutines ( jstor )
Velocity ( jstor )
Viscosity ( jstor )
Accretion (Astrophysics) ( lcsh )
Astronomy thesis, Ph. D
Disks (Astrophysics) ( lcsh )
Dissertations, Academic -- Astronomy -- UF
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bibliography ( marcgt )
non-fiction ( marcgt )

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Thesis:
Thesis (Ph. D.)--University of Florida, 1995.
Bibliography:
Includes bibliographical references (leaves 277-280).
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Typescript.
General Note:
Vita.
Statement of Responsibility:
by Ronald Eugene Drimmel.

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NUMERICAL SIMULATION OF VISCOUS ACCRETION DISKS


By

RONALD EUGENE DRIMMEL










A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY


UNIVERSITY OF FLORIDA


1995




NUMERICAL SIMULATION OF VISCOUS ACCRETION DISKS
By
RONALD EUGENE DRIMMEL
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1995


For my parents,
for the poor, and to
Sacred Truth and Beauty.


ACKNOWLEDGMENTS
First, I must thank my advisor, Dr. James Hunter, Jr., who has always had high
expectations of me. He has not only given me guidance, but has taught me by example
many of the important qualities that a theorist should possess, including the qualities
of being careful, conservative, hard working, and honest. I thank him as well for his
conversation, and attempts to pass on to me some of his experience and physical intuition.
I also thank him as a teacher, especially for the lesson that one should take the time to
start from fundamental theory, and the importance of deriving even basic equations
before applying them to a problem. After my time in graduate school, I am certainly
the richer for our association. I must also thank the other members of my committee,
Drs. H. Kandrup, H. Smith, S. Dermott, and J. Klauder, for their helpful discussions,
questions, and recommendations.
Id like to thank Chad Davies, officemate extraordinaire, who offered encouragement
both as a fellow scientist and friend. I must also thank Clayton Heller for his helpful
input and questions on numerical aspects of this work, as well as Nikos Hiotelis, who
made available to the Astronomy Department the original N-body code, TREECOD,
and assisted me in getting started. To the many other fellow astronomers and students,
whose conversation I have also benefited from, I give thanks. This work was also made
possible by a NASA GSRP Fellowship, which gave me generous support during most
of this research, which also was supported in part by the University of Florida and the
IBM Corp., through their Research Computing Initiative at the Northeast Regional Data
Center of Florida, without which the numerical work would not have been possible.
iii


On a more personal and broader note, I would like to thank Jonathan Potter for his
friendship, and keeping me laughing when I needed it most, as well as all my other
friends, who have assisted me more than they will know. This includes Orlando Espin,
for teaching me that my love for astronomy and my love for the poor were not in
contradiction, but that I can serve all Gods children with and through my work as an
astronomer. Also on a spiritual note, and another friend who needs mention, is Mary
Flanagan, who gently guided me on my way. I have been blessed as well by the entire
community of Saint Augustine Catholic Church, which has provided sustaining strength,
healing, and abundant joy. I give gracious thanks as well to Teilhard de Chardin, for
helping me see the big picture. To my parents, who have always believed in me, I am
most grateful. Lastly, I must give ultimate thanks to my God, Master of the Universe and
Lover of Souls, who created all in beauty, and who blessed me with the eyes to see it.
IV


TABLE OF CONTENTS
ACKNOWLEDGMENTS iii
LIST OF TABLES vii
LIST OF FIGURES viii
ABSTRACT xi
CHAPTERS
1. INTRODUCTION 1
2. METHODS 6
Development of the Hydrodynamic Code 6
N-body Seed Code 6
SPH 10
Variable Smoothing Length and Neighbor Searching 14
Hydrodynamic Equations 16
Integration Method 19
Gravitational Softening 20
Accretion 23
Summary of Code Parameters 25
Tests 26
Maclaurin Disk 26
Two-Dimensional Shocks 27
Analysis of Disks 30
Evaluation of Effective Shear Viscosity 30
Evaluation of Modes and Their Frequencies 33
3. INITIALIZATION OF DISKS 39
Two-dimensional Disks 40
Maclaurin Disk 40
Exponential Disks 43
Inner Disks 51
4. SIMULATION OF TWO-DIMENSIONAL DISKS 56
Parameters 56
Evolution of Disks 60
General Features 60
Modal Evolution 62
Accretion 65
Tests of Code Parameters 71
V


5. FORMATION OF SATELLITES 84
Orbital Evolution 85
Formation Conditions 90
Encounter Model 95
6. SUMMARY AND CONCLUSIONS 108
Results 108
Code Development 108
Modal Evolution Ill
Accretion 112
Formation of Satellites 113
Future Work 115
A.T2DSPH 119
B. Algorithm for T2DSPH 262
C. PROGRAM EXPD 263
BIBLIOGRAPHY 277
BIOGRAPHICAL SKETCH 281
VI


LIST OF TABLES
Table 2-1: Code Parameters 25
Table 41: Models 58
Table 4-2: Models at T=28 dynamical times 61
Table 4-3: Accretion Rates (x 103) 67
Table 4-4: Normalized density amplitude growth rates and saturation levels 69
Table 5-1: Satellites 85
vii


LIST OF FIGURES
Figure 2-1: Illustration of a two-dimensional hierarchical tree cell structure with 17
particles. There are two levels of subcells below the root cell 7
Figure 2-2: Surface density and velocity profile of the Maclaurin disk after 26
dynamical times 27
Figure 2-3: Surface density of three two-dimensional accretion shocks. The bottom
accretion shocks surface density is offset from the top by -2, and the
middle by -1, from the top accretion shock 29
Figure 2-4: Normalized power spectrum of modes of model A2 (see Chapter 4) at
Time = 21 dynamical times 35
Figure 2-5: Dynamic power spectrum of the normalized density for mode = 2 of
model A2 36
Figure 2-6: Maximum of the power spectrum, in modes = 1 4, of the normalized
density in model A2 37
Figure 2-7: The average Lombe periodogram of normalized density temporal
fluctuations along Cartesian axes for model A2 38
Figure 3-1: Initial distribution of particles for the Maclaurin Disk 41
Figure 3-2: The initial Toomre Q parameter (solid line) and tangential velocity
(dotted line) with respect to radius for a Mc/Mq = 3 disk 46
Figure 3-3: Initial distribution of particles in an exponential disk 48
Figure 3-4: Initial density profile of an exponential disk with Mq = 0.25 and a scale
length of r3 0.25. Each point corresponds to the estimated density at
each particle position, evaluated with the SPH method, while the solid
line is the intended density profile 49
Figure 4-1: Particle distribution of Mc/Md = 3 models at Time = 10.0 dynamical
times 73
viii


Figure 4-2: Particle distribution of Mc/Mo 3 models at Time = 29.5 dynamical
times 74
Figure 4-3: Particle distribution of Mc/Md = 1 models at Time = 10.0 dynamical
times 75
Figure 4-4: Particle distribution of MJMq = 1 models at Time = 30.0 dynamical
times 76
Figure 4-5: Maximum power in modes 1 through 4 for model B2 77
Figure 4-6: Resonances in an accretion disk with Mc/Md = 3. Solid lines correspond
to the corotation resonance, the dotted lines to the Inner Lindblad
resonance, and the dashed lines to the Outer Lindblad resonance. ... 78
Figure 4-7: Mass accretion, as a fraction of initial disk mass, for models A1 through
A3 79
Figure 4-8: Mass accretion, as a fraction of initial disk mass, for B models 79
Figure 4-9: Mass accretion, as a fraction of initial disk mass, for C models 80
Figure 410: Mass accretion, as a fraction of initial disk mass, for D models 80
Figure 4-11: Mass accretion in models A2, A4, and A5, as a fraction of the initial
disk mass 81
Figure 4-12: Constant mass accretion rates for Mc/Aip = 3 accretion disks 82
Figure 4-13: Constant mass accretion rates for Mc/Md = 1 accretion disks 82
Figure 4-14: Mass accretion of the encounter model (solid line) compared to the
mass accretion of model D2 (dotted line) 83
Figure 415: Mass accretion, as a fraction of initial disk mass, for model D2 is
shown with three test models 83
Figure 5-1: Evolutionary sequence of model D2 showing formation of satellite. ... 96
IX


Figure 5-2: Evolutionary sequence of model D2 continued 97
Figure 5-3: Final configuration of model D2, showing the position of the satellite at
earlier times 98
Figure 5^4: Radial density profile and particle distribution of satellite D2-1 99
Figure 5-5: Final configuration of model B2, showing the position of the satellites at
earlier times 100
Figure 5-6: Final configuration of model B3, showing the position of the satellites at
earlier times 101
Figure 5-7: Final configuration of model Bl, showing the position of the satellites at
earlier times 102
Figure 5-8: Final configuration of model Dl, showing the positions of the satellites
at earlier times. Satellite 4, which has been reabsorbed by the disk, is
not shown 103
Figure 5-9: Encounter model 104
Figure 5-10: Encounter model continued 105
Figure 5-11: Encounter model continued 106
Figure 5-12: Encounter model at Time = 16.0. Note the three satellites that have
formed in the tidal tail 107


Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
NUMERICAL SIMULATION OF VISCOUS ACCRETION DISKS
By
RONALD EUGENE DRIMMEL
August, 1995
Chairman: James H. Hunter, Jr.
Major Department: Astronomy
A two- and three-dimensional numerical hydrodynamics FORTRAN code is devel
oped to model viscous accretion disks, employing the method of smoothed particle hy
drodynamics. The effective shear viscosity present in the code is evaluated. Using a
polytropic equation of state, models of self-gravitating accretion disks are evolved with
central mass to disk ratios of 1 and 3, with ratios of specific heats of 2 and 5/3, and
with an artificial viscosity parameter of 1, 0.5, 0.25. From these models it is found that
a characteristic mass accretion rate, constant with time, is maintained by the accretion
disks, and that mass accretion is inversely proportional to the strength of the local shear
viscous force. This is a consequence of the effect of local viscous forces upon the global
non-axisymmetric modes that primarily drive accretion in these disks. In addition, the
formation of satellites is observed in models which also develop a dominating m=l mode.
The relevance of these satellites to planet formation is speculated.
XI


CHAPTER 1
INTRODUCTION
Accretion disks form an important class of astrophysical objects, occurring on many
scales in a variety of contexts. In quasars and AGNs they are believed to be the central
driving engine responsible for the extreme luminosities generated by these objects. In
a more common manifestation, they are believed to be a natural product of the star
formation process (Shu et al., 1987), and to be the progenitors of potential planetary
systems. As a structure that preprocesses infalling material before it accretes onto the
central protostar, the accretion disk is referred to as a protostellar disk. Later, when mass
accretion has virtually ended and the disk mass is small compared to the young star at its
center, it is designated as a protoplanetary disk. This work, though it addresses accretion
disks in general, is focused on accretion disks with the general characteristics of massive
protostellar disks: the self-gravity of the disk will be important, and the central object,
representing the protostar, will be small compared with the spatial dimensions of the disk.
Though the existence of protostellar disks was first theoretically argued as a natural
consequence of the conservation of angular momentum, the observational evidence for
these objects has greatly improved. Initially the observational evidence was indirect.
Infrared (IR) excesses were observed around young stellar objects, particularly T Tauri
stars, though the star itself was unobscured by the material radiating at lower temperatures
(Rydgren & Zak, 1987). Ultraviolet (UV) emission lines are associated with these objects
as well. Other indirect evidence is the angular momentum regulation of T Tauri stars


9
with IR excesses (Edwards et ai, 1993). The most plausible, and simple, of explanations
for these observations was that the material responsible for the excess IR emissions was
in a flat, disk-like structure, and that material accreting from the disk onto the star was
producing the UV emission. Many of these inferred disks have little mass, and are still
thought to be examples of disks at a later stage in evolution.
More massive disks were inferred from young stellar objects with much larger IR
excesses (Hillenbrand et al., 1992). The flat-top IR spectra of these objects were
initially explained by massive disks that were generating their own luminosity through
viscous processes (Adams et al., 1988). However, more recently the spectra of these
objects were shown as more probably resulting from a dusty, infalling envelope (Whitney
& Hartmann, 1993; Hartmann, 1993). Hence, to this point, the indirect evidence for
large gaseous disks around young stars has provided enticing, but not decisive, evidence.
However, with new and improved instrumentation, such as the JCMT-CSO submillimeter
interferometer and the serviced Hubble Space Telescope, direct evidence of gaseous disks
around young stellar objects has been observed. (Koemer et al., 1993; Lay et al., 1993)
Numerically many have shown, as per expectations (Lin & Pringle, 1990), that
massive disks do form from a collapsing cloud possessing angular momentum equivalent
to that observed in molecular cloud cores (Bodenheimer et al., 1990; York et al., 1993;
York et al., 1995). In such a self-gravitating disk, nonaxisymmetric modes play a crucial
role, as they effectively transport angular momentum, thereby driving the evolution of the
disk. In addition, ubiquitous shear viscous forces are an important transport mechanism in
a gaseous disk; nonaxisymmetric modes may dominate when present, but viscous forces


3
will always be present to some degree. In general the resulting angular momentum
transport will be outwards, causing the disk to become more extended, while the bulk
of the mass flows inward, resulting in accretion onto the central object (Lynden-Bell &
Pringle, 1974). The strength of the viscous force has been inferred from the lifetimes of
the disks themselves, and is often termed as anomalous viscosity, as the strength of the
viscous forces are much stronger than can be explained by the kinetic molecular viscosity
of the gas itself. The most common type of viscosity that has been suggested is turbulent
viscosity, though several mechanisms for inducing the turbulence have been suggested.
These mechanisms include convection induced by thermal gradients, which requires an
optically thick disk, and a magneto-rotational instability, which requires the presence of
a weak magnetic field (Balbus & Hawley, 1991; Hawley & Balbus, 1991). In this work
I do not attempt to shed light on the mechanism responsible for the turbulence, but will
nonetheless assume that turbulence is responsible for the shear viscosity in the accretion
disk. These two mechanisms of angular momentum transfer, the global nonaxisymmetric
modes and the local viscous processes, do not act independently, but affect one another.
The theory of accretion disks, motivated in large degree by their occurrence in some
binary systems, particularly cataclysmic variables, has been most fully developed for
the thin Keplerian disk (Pringle, 1981). In this regime the self-gravity of the disk is
neglected, and only pressure supports the vertical structure. In contrast, the theory of
self-gravitating accretion disks is less developed. This is in part due to the nonlinearity
of the problem, which dictates that numerical studies are necessary to understand these
disks. Early numerical work on self-gravitating accretion disks began with N-body


4
modeling (Cassen et al. 1981; Tomley et ai, 1991). Papaloizou and Savonije (1991)
analytically investigate the instabilities that exist in two-dimensional self-gravitating
disks, and numerically follow the evolution of these unstable modes. Their numerical
method was to solve the hydrodynamic equations on a polar grid, which necessitated
the unphysical assumption of rigid inner and outer boundary conditions. They confirm
the importance of the nonaxisymmetric modes in redistributing the mass of the bounded
disk, though the subsequent evolution they observe characterizes real disks only in a broad
sense. In addition, their numerical code has an indeterminate amount of shear viscosity,
which also contributes to the angular momentum transport and mass redistribution that
they observe in their models.
Only recently have self-gravitating three-dimensional disks been investigated through
numerical simulation. The stability of three-dimensional tori has been investigated
both analytically and numerically (Papaloizou & Pringle, 1984; Papaloizou & Pringle,
1985; Zurek & Benz, 1986; Tohline & Woodward, 1992). Most recently Laughlin and
Bodenheimer (1994) have investigated the initial evolution of a disk that is formed from
a previous calculation of a collapsing gas cloud. In this later work ring modes that
had formed in a two dimensional collapse calculation were shown to be unstable in a
three-dimensional hydrodynamic simulation.
In this work I extend the numerical work of Papaloizou and Savonije (1991) by
providing a model that more closely resembles that of an astrophysical accretion disk,
but that is also restricted to two dimensions. In particular, the hydrodynamic evolution
of the disk is followed with a smoothed particle hydrodynamics (SPH) code, which


5
approximates a gas with a finite number of particles (Benz, 1990; Monaghan, 1992).
The evolution in phase space of these particles is determined by the Lagrangian form
of the hydrodynamic equations. As this method is not restricted to a fixed grid with
rigid boundaries, the disk is allowed to expand naturally in radius, and an inner boundary
condition allows accretion onto the central gravitating object. For the sake of continuity
with Papaloizou and Savonije (1991), the disks are initially given an exponential density
profile, though unlike them the disks are perturbed by introducing noise in the density
profile. In this way all unstable modes will be excited. The primary disk parameters
which are varied are the star to disk mass ratio, the ratio of specific heats, and the effective
shear viscosity of the disks. Like Papaloizou and Savonije (1991), a polytropic equation
of state is used to describe the gas. In particular, the effect of the shear viscosity on the
evolution of self-gravitating disks is investigated.
The following chapter describes the numerical method employed to model accretion
disks and its implementation in a FORTRAN code named T2DSPH. A listing of the
code itself is provided in Appendix 1. Tests of the code are also described, including
an evaluation of the effective shear viscosity that is present. Chapter 3 gives the details
of the initialization of the models that are evolved, an important and nontrivial aspect
of SPH. In Chapter 4 the initial parameters of the accretion disk models are first briefly
described, followed by a presentation of the results of the modeling. The evolution of the
nonaxisymmetric modes and the mass accretion observed in the models are discussed.
Chapter 5 addresses the formation of satellites that occur in some of the models, and
Chapter 6 summarizes the results and speculates on their significance.


CHAPTER 2
METHODS
Development of the Hydrodynamic Code
In this chapter the development of the numerical hydrodynamic code, named
T2DSPH, is described. A listing of the code is given in Appendix A, and an algo
rithm of the code is given in Appendix B. Test simulations are also described, including
the evaluation of the effective shear viscosity present in the method. In addition, methods
for analysing the disks are discussed in the last section.
N-body Seed Code
The code developed to evolve accretion disks is itself evolved from a version
of Hernquists N-body code called TREECOD (1987). This code uses a method of
calculating the gravitational forces on a system of particles, due to the collective influence
of the entire system of particles, using a method termed the hierarchical tree method. At
any particular time a system of N particles is spatially described, in Cartesian coordinates,
by a set of cells hierarchically organized. The root cell encompasses the entire system;
then succeeding levels of subcells are created. In two dimensions each cell will have four
subcells, while in three dimensions each cell has eight subcells. This is accomplished
by simply dividing a cell in half in each dimension (see Figure 2-1). The number of
particles in each cell is found, and successive levels of cells are created until subcells
have either one or no particles in it. Cells with a single particle can be considered the
6


7
leaves of the tree. The spatial aspect of the tree is expressed by allocating to memory the
position and dimension of each cell. Hence, the tree structure completely describes the
spatial structure of a system of particles. Further, by recording for each cell the pointer
to its parent cell, this tree may be ascended or descended.
Figure 2-1: Illustration of a two-dimensional hierarchical tree cell structure
with 17 particles. There are two levels of subcells below the root cell.
Moving from leaves to root, the mass and the center of mass of each cell is quickly
determined; in describing the spatial distribution of a system of N massive particles, the
mass distribution of the system is also described. This hierarchical tree structure can
now be utilized to calculate the gravitational forces on any single particle, and allows the
implementation of an advantageous approximation: the number of gravitation terms is
made significantly less than the N-1 terms a direct sum over all the other particles would
yield, by treating cells as gravitating particles. In other words, a single gravitational term
due to a given cell, its mass taken to be at the position of its center of mass, is used to


8
represent the collective gravitational influence of all the particles within the cell. Using
this approach, the gravitational force of a system of N particles on one of its members
can be approximated with a sum of Nt terms due to a set of Nt cells, where Nt N.
The choice of cells used in the gravitational force calculation will in general be
different for each particle, and the choice of which cells to use will determine the accuracy
of the approximation. To determine the set of cells used for a given particle, the tree is
descended from root to leaves, with the descent continuing to the next level of subcells
until the cell subtends an angle, as viewed from at the particle, which is less than
a specified tolerance angle, 9. In effect this method will treat the influence of nearby
neighbors as a direct sum, but simplify the influence of particles at a distance by grouping
them into larger cells. This approximation is then based on the philosophy that the local
gravitational field is not sensitive to the detailed spatial distribution of particles at a
distance. The accuracy of the approximation is increased further by including higher
order terms in a multipole expansion of the mass distribution within the cells. Taking
the mass to be at the center of mass, as described above, we already have the monopole
expansion term. The next higher order correction term is the quadropole term; since the
center of mass is used as the expansion center the dipole term is zero.
Hemquist has found that for a system with 4096 particles that a tolerance angle of
8 < 0.8 radians gives a relative error in the ith component of the acceleration, A(6ai)/i,
of less than 1%, where he defines the mean absolute deviation from the mean error,
(2.1)


9
N
and the absolute average acceleration.
He define the mean error as
direct
jo
(2.2)
1
a, aij,
(2.3)
where 8al} = a-Jee a^irect is the difference, in the ith component of the acceleration,
as calculated by the tree hierarchical and direct summation methods for particle j. This
was measured by Hernquist (1987) for a Plummer sphere, which has a density profile of
p{r) =
3 M
(2.4)
^ (r> + rS)5/2
where M is the mass of the system and ro is the scale length. For a typical two-
dimensional exponential disk, I measured an average relative error in the accelerations,
defined as
1 N i
>=i
(2.5)
For both an axisymmetric and a nonaxisymmetric disk, with a tolerance angle 9 0.8,
I found the average relative error to be of the order of \%.
The computational time required to sample the gravitational force field of an N-body
system at N points is proportional to the number of gravitational terms to be calculated.
Therefore the computational time of the direct sum method is of the order of N2, whereas
Hernquist finds that the tree hierarchical system is of the order of Mog(A0. Models of
accretion disks presented in this work employ approximately 5000 to 10000 particles;


10
for these numbers of particles the increase in efficiency, as compared to using a direct
summation method, is over a thousand fold. Nevertheless, there are other methods
for estimating the gravitational field of an N-body system that are more computationally
efficient, such as grid methods that use fast Fourier techniques to evaluate the gravitational
field on a set of grid points. The gravitational force upon a single particle is then
interpolated from the nearest points. While such methods are faster, the tree hierarchical
method possesses several attractive advantages. Grid methods are limited in resolution
to the grid size, whereas the tree methods local resolution is limited by the local particle
distribution itself, or the gravitational softening parameter, since its approximation is only
applied to particles at a distance. In addition, tree methods are not limited to the size or
geometry of a grid, as the root cell is resized at every step and empty cells do not take up
memory. Hence tree hierarchical methods conserve mass exactly. These characteristics
make tree methods ideal for modeling violent encounters, as well as models that have
a broad range in particle densities.
SPH
Using this N-body code as a skeleton, a two-dimensional hydrodynamic code was
developed, called T2DSPH (Appendix A). The numerical hydrodynamic method chosen
to model accretion disks is known as smoothed particle hydrodynamics (SPH), and in
developing the code I followed much of the guidance given by Hemquist and Katzs
own combination of SPH and the tree hierarchical method (1989). A three-dimensional
version of this hydrodynamics code was then developed and dubbed T3DSPH.


11
In essence, SPH is an interpolation method that allows any [continuous] function to
be expressed in terms of its value at a set of disordered points the particles, (Monaghan
1992). A finite number of particles is used to describe the density and bulk velocity field
of a fluid; hydrodynamical quantities needed to calculate the acceleration at the points
are estimated; and the accelerations are then applied to the particles using an appropriate
integrator. SPH, then, is a Lagrangian technique of solving the hydrodynamical equations,
employing a finite number of particles to approximate the fluid. A great advantage of such
a method is that it is quite general, being easily amendable to any choice of equation
of state. Also, a Lagrangian approach has an advantage over an Eulerian technique,
which must employ a grid, in that no symmetry is imposed or assumed, and further, the
simulation is not confined to a box of finite size. Reviews of this technique, which has
been popular in recent years to model a variety of astrophysical problems, are provided by
Monaghan (1992) and Benz (1990). The following discussion of the mathematical basis
of SPH follows the discussion given both in Benz (1990), and Hemquist and Katz (1989).
In SPH, quantities are estimated using the smoothed estimate,
(2.6)
where W is a smoothing kernel and h a smoothing length. The kernel W is spherically
symmetric and normalized:
(2.7)
If the kernel VP is a function strongly peaked about r = 0 then the estimate (/) can be


12
written as a Taylor expansion:
(/(r)) = /(r) + c(V2f)h2 + 0(/i3) ,
(2.8)
where the coefficient c = f u~h3W(u)du, (u = |r v'\/h) is independent of h. The
term proportional to h has vanished because W is an even function in r. Hence it is said
that the smoothed estimate of the function / is second order accurate in A. If the quantity
/ is known on a finite set of points described by a point distribution,
N
*(r) = X^r-rj),
j=i
(2.9)
then the integral expression for the estimate (/) (equation 2.3) can be written as a sum:
N
rmfMw
*}>
(2.10)
using the relation
(4g)=m+0(ft2).
(2.11)
B(r)' (B( r))
I have introduced the abbreviation WtJ = VV^dr, r;|,h), and written (nj) in the form
p(rj)/mj, where rrij is the particle mass and p(r;) is the density at the particle j. Using
equation (2.7), the density is estimated using
N
p(rj) = J2mjWxj-
(2.12)
Because of the form of the estimate [equation (2.7)], it is most convenient to estimate
quantities with the general form pA:
N
{pA)i = mjAjWij.
(2.13)


13
This convention is considered a golden rule of SPH implementation by Monaghan
(1992).
Various smoothing kernels can be used; in this work the spherically symmetric kernel
1 6u~ + 6u3, if 0 < u < 1/2
2(1 -u)3, if 1/2 < u < 1
(2.14)
otherwise,
is used, where u = r/h. This kernel is a fourth order basis spline, normalized for three
dimensions. In two dimensions normalization yields the leading factor of 10/77r/r. I
should also note that the traditional way of writing the above kernel is with h > 2h,
so that W is nonzero for r < 2h rather than h, but the above form is more convenient
numerically. The kernel above is the one most widely used in the SPH community, and
has the advantage that it defines a local interpolation since, being nonzero only when
r¡j = |r, ry | < h, neighboring particles alone are used to make the estimate. Higher
order basis splines were tested but did not yield significantly better estimates of a given
function for equivalent smoothing lengths, and gave worse results when derivatives were
estimated.
One of the powerful features of SPH is the ease with which the gradients of quantities
can be estimated:
(2.15)
(For the sake of brevity I will in general be writing /, for /(rt), and sums are to be
understood as being over neighboring particles only.) Note that the kernel is symmetric
(Wij = Wji), while its gradient is antisymmetric (Vr, WtJ = Vrj W]t). The divergence
of a vector quantity, such as velocity, can take a similar form. However, if the estimate


14
of a divergence is to be used in an equation of motion, it must be written in a form
that is antisymmetric in i and j in order to conserve angular momentum when particles
interact. Thus, for the quantity V v, the identity pV v = V pv v-V/j is used to
derive the following estimate:
(2.16)
where vtJ v, vy (see Monaghan 1992).
Variable Smoothing Length and Neighbor Searching
So far I have given the SPH formalism in the case where the smoothing length h
is a constant. In general, however, particles are given individual smoothing lengths hu
which are adjusted, both in space and time, so that the SPH particles have approximately
the same number of neighbors. This refinement will insure that quantities throughout
a model are estimated with comparable accuracy, and enables SPH to adjust the local
resolution as the local number density changes during the course of a models evolution;
denser regions will have finer resolution than diffuse regions. However, the form of the
kernel WtJ, with its dependence on h, is now uncertain. Again, to insure that angular
momentum will be conserved, the kernel is rendered symmetric in i and j by defining
the smoothing length h = htJ = (/i, + hj)/2. Neighboring particles j will now be
used that satisfy the condition rtJ < htJ. Another way to symmetrize the kernel is to
define WtJ = hi) + W(r¡j, hj))/2, which is the form used by Hemquist and
Katz (1989).


15
Determining the set of neighboring particles for each particle (those satisfying the
condition rl} < htJ) is known as neighbor searching, and can be one of the more time
intensive tasks in an SPH program. In the version of SPH developed for this work, the
neighbor lists are efficiently compiled using the already existing hierarchical tree data
structure used in the N-body portion of the code (see the beginning of this chapter).
First, the list of neighbors within h, is found by descending the tree, starting at the root,
and continuing to sublevels of those cells that intersect the neighborhood of particle i,
defined by ht. When the descent reaches a particle it is stored in the neighbor list of
particle i, provided it satisfies the condition rt] < ht]. At the same time, if r,; > hj for
neighbor j particle i is added to the list of neighbors of particle j, since these particles
will not be found when the neighbor search is done for particle j.
The adjustment of the smoothing lengths at each time step is adapted from the
procedure outlined by Steinmetz and Mller (1993), which uses a predicted smoothed
density, p*, to find a new smoothing length:
(2.17)
where ( is set to 1.3 and m is the average mass of the neighboring particles. The
predicted smoothed density is calculated by using
smooth
(2.18)
where
(2.19)


16
and
/ Or ..smooth v' t w r
(p-Y v). = mJPS7 (2.20)
j
All smoothed quantities, indicated by superscript, are estimated using the smoothing
length h* = h,/0.8.
This method maintains the number of neighbors, Ns, between 16 and 24 on the
average, except for the outer (diffuse) portions of the disks, where Ns can drop below 10.
In order to give reliable estimates the interparticle distance between neighbors should be
less than 0.5h, for the kernel used [equation (2.10)]. This requires that Ns > 13 in two
dimensions, and Ns > 34 in three dimensions. If there are large density contrasts in the
outer regions, the adjustment technique may also cause Ns for a particle to jump to some
large value. Therefore, as a practical matter, h, is adjusted if Ns is greater than 60 or less
than 10. As this adjustment takes place while the neighbor lists are being compiled, errors
in the neighbor lists of nearby particles will be introduced, which must be corrected.
Hydrodynamic Equations
To evolve the fluid, represented by a finite number of particles, the acceleration and
velocity of the fluid at each particle is applied to the particles themselves. The particles
can then be thought of as being parcels of fluid, or alternatively, as representative particles
moving with the fluid. In either case, such a systems dynamic evolution is described by
the Lagrangian form of the hydrodynamic equations:
dr,
dt
dv,
dt
vt
-YP, + a,visc V4>
Pi
(2.21)


17
where P, is the pressure, a-lsc is the artificial viscosity term, and V$, is the acceleration
due to the self gravity of the system, evaluated using the hierarchical tree method. The
subscript i indicates that the above set of equations is applied to each particle. The
acceleration on each particle is evaluated by estimating the terms on the right hand side
of the equation of motion using the SPH formalism. Care must be taken that each term
is in a form which is antisymmetric in i and j, so that the mutual forces of particles
will be equal in magnitude, but opposite in direction. In conjunction with the forces
being central, the antisymmetry of the force terms will insure that angular momentum is
conserved. Following Hernquist and Katz (1989),
-IvP, + a"se = -^mJ^2
PlPj
+ n vw,j,
(2.22)
where the artificial viscous term, taken from Monaghan (1992), is
-ac.jH.j+Pn2^
n., =
p>]
&Cij Htj
p>j
V,j Tij < 0
Vtj Tij > o
(2.23)
where
hvtJ Tij
Pij = y
rij + 72
(2.24)
In the above equations r:j=r, r;, and c,y = (c, + cy)/2 is the average of the sound
speed at i and j. The term rj2 0.01 /i2 is to prevent singularities. The viscosity
parameters a and f3 control the strength of the artificial viscous term. An artificial
viscous term is introduced into SPH to model the bulk viscosity necessary to reproduce
shocks. Without the dissipation that this term effects, there would be an interpenetration
of particles, the random kinetic energy would increase, and the particles would no longer
describe the bulk motion of a fluid. However, it also introduces an effective shear


18
viscosity. To reduce the effective shear viscosity Benz (1990) has introduced a switch in
the form of a multiplicative factor on mj, (ntJ mjfij). The factor ftJ is the average
of f and fj, where
l(V-v)l,
|(V-v)|, + |(Vxv)|, +0.001c,//¡.
(2.25)
and the curl of the velocity field is given by the estimate
(Vxv)i = |X! mjvO x (2.26)
'i
]
This factor effectively reduces the shear viscosity, but at the expense of not being able
to vary the effective shear. In order to use the parameter a to control the effective shear
viscosity, I do not apply the factor f,j to the a term in the numerator of II,y. In the
following subsection I present models of shocks using this hybrid artificial viscous term.
The set of hydrodynamical equations is closed with the inclusion of an equation of
state. Most simply a polytropic form can be used: P, = A'Sj, E being the surface
density. The index 7 is equivalent to the ratio of specific heats, and the constant K is
the square of the isothermal (7 = 1) sound speed. More generally, the ideal gas law
can be used in the form
P, = (7-l )piUi, (2.27)
where ut is the specific internal energy of particle i. The equation that governs the
evolution of the specific internal energy is the first law of thermodynamics, du
Pdplp2 + Tds, where all nonadiabatic effects are included in the change in specific
entropy, ds. The form of the thermal energy equation that is used in T2DSPH is the


19
same as in Hernquist and Katz (1989):
The first term of this equation is the adiabatic term, the second term is the non-adiabatic
viscous heating, and the final terms represent other non-adiabatic heating, H, and cooling
processes, C, not associated with viscosity. If the polytropic equation of state is used, then
the above equation is not needed. However, in this case, this equation is still integrated
for each particle with only the adiabatic term included. This is done so that the amount
of energy lost, due to the inclusion of an artificial viscosity with a adiabatic equation of
state, can be measured by measuring the change in the total thermal energy. (See further
discussion of this problem in section 2 of this chapter under adiabatic shock tests.)
Integration Method
Each particle has an intrinsic time step 8t determined by its velocity, acceleration
and a Courant-like condition:
(2.29)
MV- v|i+cj+1.2(acl+/?maxj|/i|)
where the parameter C is 0.3. The equations (2.17) are integrated using a time centered
leap frog integrator, in which the velocities and positions are alternately stepped forward
in time a half step out of synchronization. For an individual particle this can be written as
(2.30)


20
the superscripts being the time step index. If the equations for all the particles are
integrated simultaneously then the particle with the smallest time step will determine the
time step for the entire system. In accretion disks with large central masses, the range
in time steps can be large. To increase efficiency, multiple time steps are used, and
particles are assigned an individual time step that is a power of two subdivision of the
largest time step of the system:
At¡ Afsys max((,). (2.31)
Each particle, then, is assigned a time bin n¡ so that Af, < 6tt. Particles are always
allowed to move to a larger time bin (smaller time step), but may only move to a smaller
time bin if that time bin is time synchronized with the particles current time bin. Forces
are then calculated for each particle i only once per Af in order to step their velocities,
while positions for all particles are stepped every Afpos = Afsys/2nmiX+1. Advancing
the positions at every half step is necessary as forces need to be calculated at every half
step. Care is also taken that an estimated velocity v, is determined for all particles every
AfpOS > based on v, and the most recent estimate of a, as the viscous forces depend
on the local velocity field. For further details of implementing multiple time steps, see
Herquist and Katz (1989).
Gravitational Softening
Gravitational softening is a common, and necessary, convention employed in N-body
codes. Real gravitating systems often possess orders of magnitude more gravitating
particles than can be modeled directlythe computational requirements are simply


21
too great. However, such a system can be approximated by using a smaller number
of particles which interact via a softened gravity, which damps the effect of nearby
encounters. In modeling stellar systems, such as galaxies, the ratio of the number of
stars in a system being modeled over the number of gravitating particles employed in an
N-body code, is often as many orders of magnitudes as the number of particles. When
using an N-body code to model a gaseous system this difference between model and
reality becomes essentially infinite, as a finite number of particles is now representing
a continuous fluid.
The conventional method of softening is to use a form for the gravitational force
between two particles which possesses a small constant term in the denominator:
_ mimjrij
lJ ~ (r?-+£2)3/2
(2.32)
where G = 1 is assumed throughout this work, and e is the softening parameter. However,
an alternative form of gravitational softening can be derived within the SPH formalism.
The gravitational potential of a continuous mass distribution can be written as
V>(r)
f p{r')dr'
J I r r' |
(2.33)
Using the SPH estimate of the density [equation (2.9)] we have
Â¥>(r)
W(r' r j,e)dr'
r r
(2.34)
The integral Ij can be evaluated by noting that
V2Ij = \irW{r ij),
(2.35)


22
from Poissons equation. Since the operator (V-) = ^r^r2 in spherical coordinates we
have
r2VIj = J 47rff'(r rj, s)r2dr, (2.36)
or using VU; instead of Vr, and Vr = VitjVrUj, we have
= f
Viij J
47T£'iW(u)u2du,
(2.37)
where V represents Vr. From equation (2.29) we can write
Vy? = rrij'VIji
(2.38)
and using equation (2.31) we can write
3 J
Vy? = ^ trij -- < J W(u)u2du
>\7uJ,
(2.39)
where Vu_, = Uj/iij. Using the kernel
3/2(2/3 u2 + u3/2), if 0 < u < 1
W{u) =
7T£
1/4(2 u) ,
0,
if 1 < u < 2
otherwise
(2.40)
(equivalent to the kernel [equation (2.10)] with h > 2e), and evaluating the integral
in the previous equation, a polynomial expression for the gravitational acceleration on
particle i due to particle j can be written as atJ = -mjrtjg(rij), where
F [f ~ l^2 + 0 < u < 1
9ir) = \ pr[-i5 + f^3 ~ 3u4 + f^5 ~ |y6] 1 < u < 2 (2.41)
[ 1/r3 u > 2.
For the sake of completeness the contribution to the potential of a particle pair can be
written

( -f IW ~ u* + u5] + h o < u < i
/(r) = { ~Tsr 7 [h2 u3 + fouA Mu5] + l£ 1 < < 2 (2-42)
( 1/r u> 2.


23
The above expressions are given by Herquist and Katz (1989) without derivation,
citing Gingold and Monaghan (1978). This form of the softened gravity has the attractive
feature that it is equivalent to Newtonian gravity when a particle pairs separation is
greater than 2e.
Accretion
During the hydrodynamic simulation an inner region, within a radius Ra of the central
particle, is treated semi-analytically in order to circumvent the modeling of the central
object (protostar), to avoid small time steps, and to define a radius at which particles
are accreted onto the central object. The radius Ra ~ 0.05/?£>, and particles initially
within this radius are kept and redistributed within Ra at each time step, so as to insure a
continuous boundary at Ra, and thereby provide pressure support for the gas outside Ra.
If, instead, no density profile within Ra is prescribed, but particles are simply removed
from the simulation upon entering this inner region, the rate of accretion will be dependent
on the size of Ra. In real disks, on the other hand, the accretion rate is determined by
the combined action of any global non-axisymmetric modes present, and by viscous and
magnetic forces acting locally throughout the disk. Since the physical condition of the
disk determines the accretion rate, I wish the same to be true in the simulations.
While the outer particles in the disk (r > Ra) are evolved with SPH, inner particles
are given an axisymmetric distribution so as to insure that the density and velocity are
continuous across the boundary at Ra. This is done by first extrapolating from the outer
disk the density £ and the radial velocity Vr at Ra, which is accomplished by doing
a linear regression on ln(£), and VTt of particles in a small region around the inner


24
boundary. An estimate of V at Ra is made by assuming that the potential at Ra is due
to the central mass and an exponential disk, with a scale length 0.25Ra, in hydrostatic
equilibrium. Now it is left to specify E(r), V(r), and Vr(r) for the rest of the inner
region. One of two different density distributions were used: a massless polytropic disk
in a Keplerian potential, and an exponential disk. The initialization of these disks is
described in Chapter 3.
The particles in the inner disk are used as neighbors in the SPH estimations for the
particles located outside Ra to provide pressure support. To calculate the gravitational
forces, however, these particles are ignored; the inner disk and the central object are
treated as a single particle, with mass Mc = Mc(initial)+(accreted mass). The central
region remains centered on the central particle which moves only under the gravitational
influence of the outer disk, which is treated as a nongaseous particle during the simulation.
Particles outside Ra are allowed to accrete into this region, at which point they are
removed from the simulation and their masses are added onto Mc.
Treating the inner region in this semianalytical way introduces an inconsistency, in
that the gaseous mass within Ra (M,), determined from the density profile of equation
(2.39) or (3-40), is not the same as the mass initially within Ra. Not only that, but M,
will change as the boundary values of S and V change. However, the only purpose of
this inner disk is to provide a continuous inner boundary for the outer disk, rather than
being an attempt to physically model the disk within Ra.


25
Summary of Code Parameters
A list of the parameters which control the performance of T2DSPH is listed for
convenience in Table 2-1. The first parameter, N, determines how well a continuous
Table 2-1: Code Parameters
Parameter
Description
Value
N
Number of particles.
5131
£
Gravitational smoothing parameter
0.0272Rd
9
Tolerance angle
0.7 radians
C
Courant number
0.3
Ra
Radius of inner region
0.0544/?£)

Smoothing length adjustment parameter
1.3
a, (3
Artificial viscosity parameters
, 1.5
fluid is approximated by the numerical method. While a large number of particles results
in a better approximation, the cost is greater computational time per time step. The next
two parameters control aspects of the gravitational force calculations, while the Courant
number determines the size of the time integration steps. The fifth parameter defines the
radius about the central massive particle at which SPH particles are removed from the
simulation, and their masses added to the central particle. The parameter £ determines the
average number of neighbors, which should be greater than 13 for two dimensions, and
greater than 35 for three dimensions. The last two parameters are the artificial viscocity
parameters; ¡3 is not varied, while a is given one of three values to introduce varying
degrees of effective shear viscocity. In this sense a is treated as a physical parameter of
the gas. The nominal values used for the simulations presented in Chapter 4 and 5 are also


26
given. The units of length are expressed as fractions of the initial disk radius, /?£>. Tests
verifying that the chosen values result in accurate simulations are presented in Chapter 4.
Tests
To validate the accuracy of the hydrodynamics code, or at least to increase confidence
in its results, a variety of test models were run which can be compared to analytical
expectations. The first model presented is that of a stable, rotating Maclaurin disk. A
series of two-dimensional adiabatic shock fronts were also modeled to specifically test
the hybrid viscosity employed in T2DSPH (see previous section of this chapter).
Maclaurin Disk
A test to the code, using both SPH and self-gravity, is a simulation of a Maclaurin
disk, the two-dimensional counterpart to a Maclaurin spheroid, which represents a
polytropic stable solution to Poissons equation and hydrostatic equilibrium when 7=3.
The surface density profile is £(r) = S0^/l (r2/R2D), and is initialized by radially
stretching a regular grid of particles (see Chapter 3).
Figure 2-2 shows the surface density and velocity profile after 26 dynamical times
('Tdynamic = R-d/V{Rd))i the dotted line representing the analytical solution. The
particle velocities lie slightly below the analytical solution because the code employs
softened gravity. Because the disk is in solid body rotation the effective shear viscosity
of the code has no effect on the disk during the simulation, that is, angular momentum
transfer is not induced.


27
Figure 2-2: Surface density and velocity profile of the Maclaurin disk after 26 dynamical times.
Two-Dimensional Shocks
Two-dimensional adiabatic shocks, without self-gravity, were modeled as a means of
specifically testing the implementation of SPH in the program T2DSPH, and as a test of
the hybrid viscosity that is employed (see previous section). A two-dimensional sheet,
with an initial density equal to one, was initialized by placing particles on a regular
grid. Particles with a positive x coordinate are given an initial negative x component
velocity of Mach 1, and those with a negative x coordinate are given a Mach 1 velocity
in the opposite direction. This results in the formation of an accretion shock, having
two boundaries parallel to the y axis traveling away from each other. The sheet has a


28
finite extent, resulting in rarefaction waves traveling inward from the edges of the sheet.
However, for a time, the central region of the sheet will be uncontaminated by these
waves, allowing the simulation there to be compared with an analytical solution.
For all the tests presented here the polytropic equation of state was used, with 7 = 2.
From the jump conditions of an adiabatic shock, and using the polytropic equation of
state, a solution can be found for the shock velocity and post-shock density. However,
this will not be the desired solution. By employing an artificial viscosity, at the same
time that we use a polytropic equation of state, we have introduced an effective cooling,
because the kinetic energy dissipated by viscosity is not added to the system as heat.
This thermal energy would be added to the material if the specific thermal energies of
the particles, u¡, were evolved:
1 v'
= rnjUt]vtJ (2.43)
nonadiabatic .
In other words, in spite of our polytropic equation of state, we do not have a purely
adiabatic shock. This can be taken into account in our analytical solution by the addition
of a post-shock cooling term, Q (heat lost per unit mass), in the energy jump condition.
Therefore, using the polytropic equation of state, with K= 1, the jump conditions are:
(du,
nr
P\V\ = P 2^2,
P\V\+ p\ = P2v] + p],
(7 1)
7-1 1 2
Pi + =
7
(7-1)
7-1
P2
+ 7¡v2 + Qi
(2.44)


29
which results in the final solution:
where
-b+Jb2 + 4
(7 ~ l)(v72 Q) + ip]
7-1
Vo =
P\
Vo
P2 V0 + Vo
L 3-7 ,7-1^
b = u0 H g.
(2.45)
(2.46)
The solution above is given in the post-shock reference frame, with v0 and vo representing
the pre-shock velocity and the shock front velocity respectively.
1 -n-
0 o
i 1 r~
-1 1 1 1 1 1 r-
, I
_1 1 I I 1 L_
_l I 1 L_
-0.5
0
X
0.5
Figure 2-3: Surface density of three two-dimensional accretion
shocks. The bottom accretion shocks surface density is offset from
the top by -2, and the middle by -1, from the top accretion shock.


30
In Figure 2-3 three simulations are shown. The bottom two, with a = 0.25, 1, in
ascending order, and 0 0, do not use the viscosity switch. In the third simulation,
shown at the top of the figure, the hybrid viscosity is employed, with a 0.25, 3 = 1.5.
Although this solution is not as good as the first, it has much less effective shear viscosity.
Also, notice that there is no interpenetration of particles, as there is in the first case.
Analysis of Disks
Evaluation of Effective Shear Viscosity
In the SPH formalism shear viscous forces are not introduced directly, but instead
are a result of the artificial viscous term, avlsc, being evaluated over a finite region.
Hemquists form of the artificial viscous term illustrates this more explicitly; being
dependant on (V v) it formally introduces no shear force but is purely a bulk viscous
force. However, what is used numerically is an estimate of (V v), which is evaluated
by looking at particles within a finite region (neighborhood) around the point of interest.
In addition, the total force on a particle is a sum of symmetric forces between itself and
neighboring particles at other radii, and the tangential components of these forces transfer
angular momentum. These errors in the estimation formally vanish as the smoothing
length approaches zero.
This implicit introduction of the shear viscous forces in SPH causes an unfortunate
situation for those attempting to use this method to model astrophysical disks, where
large velocity gradients can exist and shear viscous forces play an important role. In
SPH both bulk and shear viscous forces are effectively coupled together, rather than


31
being independently parameterized. In an attempt to decouple the bulk viscosity needed
to describe shocks and the effective shear introduced by the artificial viscous term, an
alternate form for the artificial viscous term was introduced in section 2. Here I wish
to address how the effective shear viscosity scales with the viscous parameter a or any
other quantities. The functional form of the artificial viscous term suggests that, like
the Qi/ viscosity of accretion disk theory (Shakura & Sunyaev 1973), the effective shear
coefficient scales as otuch, c being the sound speed and h representing a characteristic
length. Artymowitz (1993) has suggested that for SPH av ~ 0.1a, with the characteristic
length being considered as the smoothing length. While the similar functional form of
the artificial viscous term and the viscosity of accretion disk theory is suggestive, it is no
guarantee that the implicit shear will behave similarly, being a by-product of estimation.
It is therefore desirable to determine not only the relative magnitudes of a and au, but
also whether the effective shear scales in the way that the artificial term suggests.
To investigate the effective shear associated with the artificial viscosity three models
were run with a = 1, 0.5, 0.25. An exponential disk was initialized, with the velocity field
now consistent with assuming that the disk is in a Keplerian potential with no pressure
support. These disks were evolved with SPH consistent with these assumptions, and they
remain axisymmetric in the absence of self-gravity. Since no pressure support is required
it was found to be advantageous to exclude the inner disk, and to simply remove particles
from the simulation when they approached within Ra of the gravitating particle located at
the origin. After a few dynamical times a radial velocity flow is established throughout
most of the disk as a result of the effective shear viscosity.


32
From standard accretion disk theory the radial velocity profile of an axisymmetric
thin disk, with no radial pressure support, is given by
(2.47)
where ft is the angular velocity, and the primes denoting radial derivatives (Pringle 1981).
If the shear coefficient u is not a constant, then
r
(2.48)
o
For a Keplerian potential ft1 = GM/r3, which upon substitution results in
r
(2.49)
o
If in SPH the shear coefficient, //, scales as avch, then a = v/ch will be constant
over the disk, and can be estimated by evaluating the above expression for v. The SPH
particles of the evolved disks are used to find the surface density and radial velocity
of annuli centered on the origin, and the integral in equation (2.48) is estimated by the
sum over the annuli
(2.50)
where Ar is the width of the annulli and is equal to .05Rq. Now at each radius r, the
shear viscous parameter au can be estimated by using
(2.51)
remembering that the sound speed c = \/2A'E when 7 = 2, and the smoothing length
h = 1.3-y/mp/E, where mp is the mass of each SPH particle and assuming that the


33
smoothed density L* = £. Using this method on the three models described above
yielded approximately constant with radius, with values of 0.1, 0.05,0 .02 .01
for each model respectively. Much of the uncertainty arises from a time variability that
is probably due to radial oscillations. The fact that olv is found to be approximately
constant over the disk confirms that the effective shear viscosity scales the same way as
the artificial (bulk) viscosity.
Evaluation of Modes and Their Frequencies
To follow the evolution of the nonaxisymmetric modes a method similar to that of
Papaloizou and Savonije (1991, hereafter PS) is adapted; a polar grid, concentric with
the center of mass of the system, is imposed on the disk with a radial extent from 0.1
to 1. The density is evaluated at 64 points, equally spaced in azimuth, for each of 25
equally spaced radii, and then normalized by dividing the density at each point by the
average density at each radius. The modes are identified by doing a Fourier analysis of
the normalized density in azimuth at each radius, and plotting the power spectrum as a
function of mode number and radius (Figure 2-4). Unlike PS, the density is not taken to
be the mass/area of individual cells defined by the grid, but instead the SPH formalism
is used to estimate the density at each grid point:
(2.52)
where the sum is over all particles having the grid point k within their smoothing length
hi. This analysis can be done at each time that the position and smoothing lengths of the
particles are output by the SPH code, which is at every 0.5 dynamical times (see Chapter


34
3 on choice of units). Another useful way of presenting the results of this analysis is with
a contour plot of the dynamic spectrum: the power in a particular mode as a function
of radius and time (Figure 2-5). The evolution of a particular disk can be more simply
characterized by considering the maximum power, with respect to radius, of each mode
as a function of time (Figure 2-6).
To find the frequencies that are present, the SPF1 code outputs the estimated density
at 100 points along both the x and y axes of a Cartesian grid, with its origin at the center
of mass of the system, at every third system time step. After the simulation a Fourier
analysis is done in time, at each radius, for a specified time interval. Contour plots of
the power, as a function of radius and frequency, can be displayed separately for each
axis, or averaged together (Figure 2-7). Provided that there are a limited number of
modes, the contour plots of the modes and of the frequencies can be used together to
match frequencies with particular modes.


Radius
35
Figure 2-4: Normalized power spectrum of modes of
model A2 (see Chapter 4) at Time = 21 dynamical times.


Radius
36
INT=0.05, BOT=0.05
Figure 2-5: Dynamic power spectrum of the normalized density for mode = 2 of model A2.


Power
37
Figure 2-6: Maximum of the power spectrum, in
modes = 1 4, of the normalized density in model A2.


Angular Frequency
38
B0T=13.2012=99.9% conf. Time = 20.0 to 30.0
Figure 2-7: The average Lombe periodogram of normalized
density temporal fluctuations along Cartesian axes for model A2.


CHAPTER 3
INITIALIZATION OF DISKS
In this chapter the initialization of the disks numerically evolved with SPH is detailed.
Initializing a disk for an SPH program involves finding a distribution of points in phase
space which suitably describes the particular density and velocity field of the disk. Except
for perturbations, all initial disks used in this work are axisymmetric and in radial
hydrostatic equilibrium. Global parameters describing a disk are the disk mass Mp
and the radius of the disk Rp. With the exception of the Maclaurin disk, and in the spirit
of modeling protostellar accretion disks, there will also be a central particle with mass
Mc, giving a total mass for the protostar/disk system of Mp = Mc + Mp. Treating the
central object as a point mass is equivalent to assuming it is several magnitudes smaller
in size than the disk radius. This assumption is valid for protostellar disks that have radii
approximately 10 1000AU. Parameters of the gas itself are the ratio of specific heats,
7, and the isothermal sound speed K. Unless stated otherwise, dimensionless units of
mass, length, and time will be used for which Mp = Rp = G = l. In this case the unit
of time then becomes the dynamical time: Tp = Rp/V(Rp) = (R3D/GMp) The
models may then be scaled to the desired dimensions. For instance, if Mp = 1M0 and
Rp = 100AU then the dynamical time is 159 years.
39


40
Two-dimensional Disks
Maclaurin Disk
A two-dimensional disk has already been presented in Chapter 2 as one of the
tests to the SPH code: the Maclaurin disk. This is a two-dimensional version of the
Maclaurin spheroid, which represents a polytropic stable solution to Poissons equation
and hydrostatic equilibrium when 7=3. The surface density profile of the Maclaurin
disk is
(3.1)
and is initialized by radially stretching a regular grid of particles. The central density
S0 is given by 3Md/2xR2d.
This method of constructing an axisymmetric disk I call the stretched grid method,
and is equivalent to transforming the radial coordinate, R, of a particle in a uniform disk
to the new radial coordinate r via a coordinate transform function: r = A(R)R. To apply
this method the transform function, or stretching factor, A(R), must be found which will
transform a uniform disk into a Maclaurin disk. The desired function can be found by
considering an invariant of the transform, the mass within a given radius. That is, the
mass within a given radius of the uniform disk, MdR2/Rp, is equivalent to the mass
within r = AR of the Maclaurin disk, which is
3/2'
(3.2)


41
Setting the above expression equal to MqR2/R2D, replacing r with AR, and solving for
A gives the desired transform function:
A{R) =
Rp
R
1-1-
R2
WD
2/3
1/2
(3.3)
Giving each particle in the uniform disk with radial coordinate Rt < Rp a new radial
coordinate r, = A(R,)Rt, and discarding any remaining particles, transforms a uniform
grid into a Maclaurin disk (see Fig 3-1).
-1 -0.5 0 0.5 1
x
Figure 3-1: Initial distribution of particles for the Maclaurin Disk.
This method, though quite general, was found to be of limited usefulness when it
was used to construct disks that possess stronger central mass concentrations, such as the


42
polytropic Keplerian and exponential disks described below. The undesirable feature of
the disks built with this method is the presence of artifacts of the grid from which the disk
was stretched: the direction to the nearest neighbor is not isotropic on the average, the
particles instead appearing to be planted in rows, and the edge of the disk is corrugated,
or scalloped. These artifacts are not severe in the Maclaurin disk shown in Fig 3-1, but
they become much more pronounced in centrally condensed disks.
Once the particles have been spatially distributed their velocities are determined.
The initial tangential velocity of each particle V, in the Maclaurin disk is kV0(r,),
V0(r¡) = CrJRo being the initial tangential velocity required if the disk were supported
purely by rotation, with C2 = SttGMd/ARd- The rotation parameter k = V/V0 = 0.4 is
assumed to be a constant with respect to radius, and gauges the importance of rotational
support. Maclaurin disks with k < 0.5 will be stable against secular instabilities (Binney
and Tremaine, 1987). The tangential velocity V0 was found from the gravitational force
on each particle, evaluated using a direct sum over all other particles, rather than using
the analytical formula for V0, because of the softened form of gravity employed. From
the requirement of radial hydrostatic equilibrium,
\_dP_
77)7
V2
V*
(3.4)
which follows from the radial component of the equation of motion in cylindrical
coordinates, and the assumptions of axisymmetry and no radial motion, the constant K in
the equation of state is found. It is determined to be a function of the disk parameters:
v 9 Md
(3.5)


43
The above expression is found from assuming unsoftened Newtonian gravity, whereas
the code that will evolve the initial conditions uses softened gravity. Because the value
of the softening parameter is small, the error introduced by this inconsistency is also
small. Using k = 0.4 and Md Rq 1 gives us K=2.8939.
Exponential Disks
Exponential disks were chosen to simulate accreting protostellar disks in large part
due to the previous theoretical and numerical work that has been done with such disks
(Papaloizou and Savoneji, 1991). This is the practical reason. That such a choice is
reasonable physically is somewhat bom out by the numerical result that the exponential
profile is preserved over most of the disk for the majority of the duration of the numerical
evolution, at least in the case where the central star to disk mass ratio was equal to three.
The models with equivalent star and disk masses differ significantly in that a m=l mode
completely dominates the mass distribution of the disk and erases all axisymmetry. In any
event, angular momentum transport and mass accretion reinforce the centrally condensed
mass distribution.
To represent the mass distribution of these disks, particles representing the gas, all
of equal mass, are placed in concentric rings around the central particle at the origin.
The surface density profile,
E(r) = S0 exp (r/ra),
(3.6)
is achieved by appropriately spacing concentric rings, from the inside out, so that the
correct interparticle separation, (m/Sjr))1^2, is nearly imposed. The scale length rs is


44
set to 0.25 in all models, while the ratio of the central to disk mass Mc/Md is either 3
or 1. The central density is then given by the expression
(3.7)
Before the disk is built the number of total desired particles, Ntot, is specified. The
gaseous particles are then assigned a particle mass of mp = Mo/Ntot- The first ring
of gaseous particles consists of six particles placed at equal angular intervals around
the central particle at a distance 0.5 x r(6mp), one half the radius within which a mass
of 6mp resides in. Here, and in what follows, the radius r(mr) is found by using the
Newton-Raphson method, with the expression for the mass that lies interior to a given
radius for the exponential disk:
m(r) = £02xr2 1 e T^T{r/rs + 1)
(3.8)
For subsequent rings, the ring radius rn is determined by finding the number of particles
in the ring, Nn, that minimizes the quantity
(3.9)
where r2 = (r12 + r'2_x)/2, with each primed radii being the outer radius of the annulus
about the nth ring. This is an attempt to equate the distance between ring n and n + 1
with the distance between particles in ring n. To calculate the quantity above, the radius
of an nth ring with Nn particles is found from r'n_l, and the radius r'n within which
n
there are Nr = Nj particles. This iterative procedure is continued until Nr > Ntot- If
i=i
Nr > Ntot then Ntot is set equal to Nr, the particle mass mp is reinitialized, and the disk


45
rebuilt. For an exponential disk this procedure will converge on a value of Ntot which
will result in a disk being built with NT = Ntot at the outer most ring. In the case where
5000 particles are initially requested, the above algorithm results in an exponential disk
with 5130 particles. Though the above procedure for building an axisymmetric disk with
particles on concentric rings is in principle quite general, requiring only that the desired
r
function m(r) = 2tt J T,(u)udu be specified, it is not known whether the iteration will
o
converge for other density profiles.
An initial circular velocity V is assigned for each ring of particles according to the
equation of radial hydrostatic equilibrium, which leads to
V'2 = 7/vrS7-2^ + v- = -7/v-s7-1 + v;2, (3.10)
dr r3
where V02/r for each ring is calculated by finding the radial gravitational acceleration for
each particle, through a direct summation, and averaging the radial components of the
acceleration of the particles in each ring. In doing this calculation softened gravity is used.
The value of the constant K, the square of the isothermal sound speed, is determined by
assuming radial hydrostatic equilibrium throughout the disk and specifying a minimum
value for Toomres stability parameter Q = kc/ttGT,, where /c = + 2^%-)1/2 is
the epicyclic frequency. Toomres parameter is a local stability parameter, derived from
the linearized hydrodynamic equations of a local region of a rotating gas sheet, which
describes the stability of a two-dimensional disk (Toomre 1964). When Q > 1 the disk is
stable against axisymmetric perturbations, but the disk generally remains unstable to non-
axisymmetric modes for values of 1 < Q < 3. The epicyclic frequency, k, is numerically


46
evaluated for each ring n with the expression
1 (VoV+i) v;2(rni))
Kn =
+ 2
rk (rn + l 1)
Using the expression c1 = 7A'E7-1, Toomres stability parameter can be written as
(7/x st-3)1/2
(3.11)
Q
tcG
(3.12)
An expression for the constant K can be found from specifying Qmin using the previous
equation:
K -
(Q min71^7)
(3.13)
7minjt (/cJ/Sj 7)
For all the two-dimensional exponential disks, Qmin is set to 1.15. In Figure 3-2 is shown
the initial velocity profile and the value of the Toomre Q parameter with respect to radius.
Figure 3-2: The initial Toomre Q parameter (solid line) and tangential
velocity (dotted line) with respect to radius for a MJMd = 3 disk.


47
A density perturbation is then seeded into the disk by adding independent Gaussian
noise to the x and y coordinates of each particle, the displacement having a standard
deviation of 0.02 of the interparticle distance. This perturbation is added after the
tangential velocity V is determined for all of the rings. The subroutine which generates
the Gaussian noise is taken from Numerical Recipes (Press et al., 1992). This introduces
a density fluctuation with an rms amplitude 0.03 times the average density. The purpose
of introducing the noise is twofold. The first is to excite all possible modes, so as to
observe which modes grow fastest and those which tend to dominate the disk. This was
the primary reason for the introduction of noise. The second, more serendipitous, is to
avoid a flaw in the initializing method. If the SPH particles were distributed on concentric
rings, without noise, then the regularity of their distribution will suppress some possible
responses while enhancing others. For instance, it was found from evolving a disk with no
noise, but with an m=2 perturbation (see below), that the initial growing mode developing
early in the disks evolution, while particles still lay on concentric rings, led to the rings
being distorted, and themselves becoming structures that excited a response. The folded
and compressed portions of the rings formed shock wakes to the arms.
The FORTRAN code which generates the initial positions and velocities of the
particles that describe the exponential disk is listed in Appendix C. The initial distribution
of points for the exponential disk is shown in Figure 3-3, while the initial radial profile
of the density is shown in Figure 3-4.


48
1
0.5
>- 0
-0.5
-1
Figure 3-3: Initial distribution of particles in an exponential disk.
In one model an m=2 density perturbation is introduced of the form E'(r, £0(r)p(r)cos (rmp), where p(r) = Ap[i:(r rs)/(RD rs)], with Ap being 0.01, and E0
representing the unperturbed density given by equation (3.5). This perturbation is induced
by displacing the particles along the circumference of their respective ring. To find the
displacements which will effect the desired density perturbation in a particular ring, I
consider the perturbed density in the form E'(fc,.r) = Acos(kx), where A = E0(r)p(r),
k = m/r, and x =

49
Figure 3-4: Initial density profile of an exponential disk with Mq = 0.25 and a scale length of
rs = 0.25. Each point corresponds to the estimated density at each particle position,
evaluated with the SPH method, while the solid line is the intended density profile.
as being the perturbed density of a one-dimensional sound wave at time equal zero,
which is more generally written as S'(x,t) = Acos(kx wt). Linearization of the one
dimensional equation of continuity and motion gives the wave equation
dv'
cl as'
(3.14)
dt S0 dx 1
where c0 is the sound speed equal to w/k. Using this equation and the given form of
the perturbed density, I find the perturbed velocity v' = c0p{r)cos(kx wt). From this
perturbed velocity a displacement can be found:
p(r)
Ao:
= J v dt =
k
-sin(kx wt).
(3.15)


50
The desired displacements are those at t = 0, at which time an equation for the perturbed
particle position x is formed:
x = x o
P{r)
k
sin(kx),
(3.16)
x0 being the unperturbed position, and Ax = x x0. This equation is solved for each
particle using the Newton-Raphson method.
To find the associated perturbed velocities, the velocities of the one-dimensional
sound wave employed above are not used. While the above derivation is sufficient for
finding the displacements, it is an oversimplification with regards to the gas dynamics
of the disk. Instead, the equation of radial hydrostatic equilibrium is used to find
the tangential velocity V{r, expression
V2(r,f)=V--i KZ
1 p(r)cos(m cot
?r(r rs)
1 rs
(3.17)
Another model that was evolved includes an encountering particle, with a mass of 0.5,
that approaches the central particle with a minimum distance of roughly 1.0. Otherwise
the disk and central particle are initially identical to model D2. The initial separation
between the encountering particle and central particle is about 8.772. The energy of the
encountering particle is specified to be
1 GMme
h T
2 a
(3.18)
where M is the mass of the central object and disk (1.0), me is the mass of the
encountering particle (0.5), and a is the separation. With this energy the relative velocity


51
of the particle is v = y/3GM/a. A common parameter to describe an encounter is the
impact parameter b. Instead of specifying this parameter I have specified the minimum
separation amin = 1.0. The impact parameter can then be determined as
6 =
v^'
(3.19)
with
^max
E +
GM rrif
0
(3.20)
min /
The energy of the encountering particle can also be parameterized in terms of the velocity
of the particle if removed to infinity, V0 = J£¡E. The specific angular momentum of
the encountering particle is then bV0. Using this, the components of the velocity can then
be determined: vt = bV0/a, and v\ v2 Vj, being the velocity tangential and parallel
to the separation vector respectively.
Inner Disks
As described in Chapter 2, a small inner region about the central particle is described
analytically, rather than being evolved with SPH. Two different inner disks are used.
If the disk mass within Ra is much less than Mc, it may be considered as a massless
Keplerian disk. Assuming axisymmetry in two dimensions, radial hydrostatic equilibrium,
and using a polytropic equation of state with 7 = 2, the density distribution within Ra
may be written
£M = 1^(1-*>)(}-£)+£(*.) (3.21)
It has also been assumed that k = V/V0 = V(Ra)/V0(Ra) is a constant throughout the
inner disk, and V0 is the circular rotation without pressure support. The inner particles are


52
repositioned on concentric rings to describe this density distribution, with their masses
being reassigned so that m.p is equal to Min divided by the number of inner particles,
where .V/¡n is the mass interior to Ra as determined from S(r) in the equation above.
Using Vo = y/GMc/r for the inner disk, the tangential velocities Vi = kV0{ri) are
determined. The radial velocity is specified by assuming that the mass accretion rate
m = 2irT,(r)Vr(r) is constant throughout the inner disk. Since the values of £, V, and
Vr at the boundary are determined at each time step, the inner disk must be reinitialized
at each time step as well.
The inner disk described above is specifically for 7 = 2; for other values of 7 an
alternate model of the inner disk must be used. The second inner disk that is used is
one with an exponential density profile,
E(r) = E0e-r/rs. (3.22)
The scale length is found from the linear regression used to estimate E(/?a), and the
central density is deduced from T,(Ra) and the scale length, rs. Again, as a fixed number
of particles is used to model this inner disk, their individual masses are being changed
during the run, and as in the first inner disk described, they are placed on concentric
rings. The velocity profile for the inner disk is found from assuming radial hydrostatic
equilibrium and using V0 = y/GMc/r + Ar, the parameter A being estimated at Ra. As
in the previous disk, the radial velocity profile is determined by assuming a constant
accretion rate m throughout the inner disk.
It has been stated above for both of the inner disks that the particles are distributed in
concentric rings. I now wish to describe more carefully the algorithm used to accomplish


53
this. In order to assure a continuous particle distribution at the boundary r = Ra,
the inner disks are built from the outside in. The first ring radius rj is the radius
at which the interparticle distance (mp/H(ri))^ is equal to 2(Ra ri), and is found
using the Newton-Raphson method and the appropriate density profile (equation 3.17
or 3.18). Particles placed in this and subsequent rings n, will describe the desired mass
distribution subject to the condition that the mass in each ring, Am, is equal to the mass
in an annulus, centered on the ring radius rn, with a width Ar equal to the interparticle
distance. Hence, for the first ring, the mass contained within An = (mp/E(ri))1/2 is
equal to Ami = m(Ra) rn(Ra An). [In general m(r) will signify the mass within
radius r.] The number of particles to be placed in the ring is determined by the mass in
the annulus, Ami, and the particle mass, mp. However, since the number of particles
iV placed in a ring must be an integer, and Am/mp is in general not an integer, a
small adjustment in rn and Ar must be made. For the first and subsequent rings, Nn is
found by rounding Amn/mp to the nearest integer value. Then the mass in the ring is
redetermined: Am = Nnmp. For the first ring a new Ari is redetermined by finding
the radius ry/ for which m(r\f) = m(Ra) Ami, using the Newton-Raphson method
and the appropriate expression for m{r\j), and then setting Ari = Ra r\j- Finally,
ri = Ra Ari/2. For the remaining rings the same procedure is used to redetermine
Ar from the new Amn, excepting that, instead of Ra, r = rn_i Arn_i/2 is used.


54
For claritys sake, I restate the algorithm for finding the radii of the concentric rings:
1.Using the Newton-Raphson method, find rn which satisfies the condition
2{rL r) = (mp/£(rn))1/2, where rL = Ra for n = 1, but rL = rn_i Arn_!/2
otherwise.
2. Arn = 2(r r)
3. Amn = m(r) m(r An)
4. Find the number of particles in ring n, Nn, by rounding Amn/mp to the nearest
integer value.
5. Redetermine the mass in ring n: Am = Nnmp.
6. Using the Newton-Raphson method, determine the radius for which m(r^/) =
m(rL) Amn.
7. Set Arn = rL r\j.
8. Finally, rn = rL Arn/2.
9. Proceed to next ring.
The building of the disk is discontinued when r\¡ < Arn/2.
In order to use the above algorithm, the expressions for the mass with a given radius
for the two inner disks must be known. For the inner polytropic Keplerian disk:
(3.23)
where k here is V/V0 and shouldnt be confused with the index. The expression for
m(r) for the exponential disk is given above in equation (3.7). For the polytropic disk
the above expression can be inverted to find r(m), the radius within which there is mass


55
77?. This will allow us to circumvent the use of the Newton-Raphson method in step 6)
above, and instead use the expression:
2 mT
B + sjB2 + 4(ttS(/?a) B/2R^'
(3.24)
where B = ttGMc[ 1 k~)/I\, and mr = m(Ri) Atti^.


CHAPTER 4
SIMULATION OF TWO-DIMENSIONAL DISKS
Parameters
To gain insight into the general behaviour and evolution of protostellar disks, a suite
of models of accretion disks were simulated. Three physical parameters were varied:
the central object to disk mass ratio, MJMd, the ratio of specific heats, 7, and the
artificial viscosity parameter a. As I am particularly interested in the early stages of the
evolution of protostellar disks when the disk is self-gravitating, values of Mc/Md of 3
and 1 were chosen.
The second physical parameter that is adjusted, 7, affects the hydrodynamical char
acter of the gas. In particular, as it appears as an exponent in the polytropic equation of
state, 7 determines the compressibility of the gas. While in three dimensions a value of
7 < 4/3 is unstable to gravitational collapse, in two dimensions 7 < 3/2 is unstable to
collapse. This is one example of how gravity is more effective in two dimensions than
in three. Values of 7 = 2 and 5/3 were chosen for modeling; 7 = 2 corresponds to a
gas with two degrees of freedom, while 7 = 5/3 to a gas with three degrees of free
dom. The value 7 = 2 was chosen because previous theoretical and numerical work on
two-dimensional disks have used this value (Papaloizou and Savonije, 1991). However,
though convenient for solving the hydrodynamical equations, 7 = 2 results in a stiff equa
tion of state. Therefore 7 = 5/3 was also used in the simulations. This value of 7 allows
the gas to be more compressible, though it is still stable against gravitational collapse.
56


57
Of particular interest in this study is the effect of viscosity on the evolution of self-
gravitating accretion disks, and the parameter used to vary the effective viscosity in SPH
is the artificial viscosity parameter a. Values of q = 1, 0.5, 0.25 were chosen, which
were shown in Chapter 2 to correspond roughly to effective shear viscosity parameter
values of av .1, .05, and .02. On the basis of time scale arguments applied to
real astrophysical disks, av is expected to be less than 0.1, where values between 0.04
and 0.002 are most commonly argued for. As mentioned in Chapter 1, these values are
much greater than those from molecular viscosity alone. Due to the as yet unresolved
question of what mechanism effects this anomalous viscosity, the value of a represents
the greatest unknown in the physics of accretion disks.
Varying the three parameters Mc/M£>, 7, and a as described above gives twelve
possible combinations, all of which are modeled. Table 1 summarizes the models
discussed in this chapter, as well as the time at which each simulation was terminated,
in dynamical times (see Chapter 3 for definition of units). All of the MJMq = 1 disks
terminate due to computational errors associated with adjusting the smoothing length
during the neighbor searching phase of a time step, and the finite size of the arrays in
which the lists of neighbors are stored. That is, the program has difficulty in adjusting the
smoothing length so that particles have more than ten, but more than sixty, neighbors.
This problem arises in disks in which an extreme density gradient borders a region
where the surface density, and hence also the number density of particles, is very low.
Such conditions arise in the disks with a mass ratio of one. In contrast, most of the
Mc/Md = 3 models ran to the specified time without difficulty. Their simulations


58
were terminated after they had run for a time comparable with the Mc/Md 1 disks.
Model A3 is the sole model with a mass ratio of three that ended due to the type of
difficulty mentioned above. In any event, the lengths of all the simulations were sufficient
to observe important properties of accretion disks, which are noted below and in the
following chapter; continuing the simulations much further, after a significant number of
particles already have been removed due to accretion, would be of questionable value.
Table 4-1: Models
Model:
mc/md
7
Q
Inner Disk
Tf
A1
3
2
0.25
Keplerian
33.10
A2
0.5
it
32.11
A3
1
ii
33.01
A4
3
2
0.5
Exponential
24.01
B1
1
2
0.25
Exponential
38.81
B2
0.5
38.51
B3
1
31.01
Cl
3
5/3
0.25
Exponential
32.61
C2
0.5
29.97
C3
1
29.55
D1
1
5/3
0.25
Exponential
33.01
D2
0.5
39.00
D3
1
32.63
While the model parameters discussed above describe the physical state of the initial
conditions, another set of parameters which may affect the simulations is that of the
code parameters, which determine details of the numerical simulation method. These
parameters are listed in Table 2-1, with their nominal values. Ideally, if the values


59
chosen for the code parameters are sufficient to yield accurate simulations, the resulting
simulations will be insensitive to variations in the code parameters. To test whether
the simulations are accurate, several test models have been numerically evolved, each
with the value of a single code parameter changed from its nominal value, so as to
yield a more numerically accurate simulation. If the nominal value is sufficient, then
the test simulation will show little or no change from the model run with nominal code
parameters. A number of such tests are done with model D2 for the parameters 0, C, and
N. The comparison between the tests and model D2 are given at the end of this chapter.
The gravitational smoothing parameter, e, is not varied for testing, because its range
is restricted by requiring that it be smaller than the accretion radius, Ra, and that it must
be larger than the local inter-particle separation. This last condition must be satisfied
if N particles are to behave as a system with a much greater number of particles, or
even a continuous medium (N * oo), as is the case here. Because e is of single value,
independent of the local number density, the latter condition cannot be satisfied in the
outer, diffuse, portions of the disks. The local inter-particle separation in two dimensions
is rc-1/2, with n representing the local number density. An average inter-particle distance
is defined as -1/2 = ^ttRq/N, which is equal to 0.025 for Rq = 1 and N = 5131.
The condition e > -1'2 is satisfied with e = 0.027, but £ < n-1/2 for r > 0.587 when
N = 5131 and rs = 0.25. When N is increased to 10093 particles the condition fails
for r > 0.757.
The other restriction, that 2e < Ra, is imposed so that all particles experience
a Keplerian, unsoftened, gravitational potential from the central object. Therefore


60
increasing e is possible if R is also increased. However, this is not desired, as the
inner region around the central particle, which is not hydrodynamically modeled, should
be a small percentage of the disk.
Evolution of Disks
General Features
Since the evolution of the models is terminated at different times, it is more useful
to compare the evolution of the models at a time that all the models attain. In Table
4-2 are tabulated the total accreted mass, the central object to disk mass ratio, and the
angular momentum (AM) that has been transported beyond the initial outer radius of the
disk, after twenty eight dynamical times. The units of mass and angular momentum are
the dimensionless units described in the previous chapter.
To qualitatively show both the effect of the viscosity parameter a and the ratio of
specific heats 7, the models are shown after ten dynamical times in Figures 4-1 and 4-3,
and after thirty dynamical times in Figures 4-2 and 4-4. For both the Mc/Mp = 1 and
3 models, increasing the viscosity effectively damps the amplitude of the modes. This
is even more noticeable at early times, where viscosity also suppresses the growth of
the modes. Of particular of interest is that the viscosity has suppressed the m=l mode
in the models having MJMq = 1. The mode is weakly visible in model B3 in Figure
4-4, but not visible in model D3, and a dominating m=l mode remains absent in this
last model until the simulation is terminated at T=32. This later model represents the
one exception in the modal evolution of the Mc/Md = 1 disks, which otherwise become
dominated by a slowly growing m=l mode. This mode is of a tidal nature, resulting


61
from the gravitational interaction between the massive central object and the disk, and
has been predicted by others (Savonije et al., 1992).
Table 4-2: Models at T=28 dynamical times
Accreted
Transported
Models
Q
mass (10'2)
mc/md
AM (10-2)
Al
0.25
3.504
3.790
2.501
A2
0.5
2.851
3.645
2.923
A3
1.0
2.042
3.477
2.562
B1
0.25
15.26
1.985
10.68
B2
0.5
14.23
1.895
9.877
B3
1.0
11.69
1.697
11.67
Cl
0.25
2.856
3.646
2.844
C2
0.5
1.808
3.430
2.808
C3
1.0
2.125
3.493
2.440
Dl
0.25
16.34
2.084
14.15
D2
0.5
13.47
1.833
9.461
D3
1.0
7.466
1.422
8.323
The effects of varying the ratio of specific heats are more subtle. At early times,
when the modes are growing, the models with smaller 7 show slightly more density
contrast between arm and inter-arm regions. At the later times, such as those shown in
the Figures 4-3 and 4-4, this trend is not apparent in the a = 0.25 and 0.5 disks, and is
reversed in the a = 1 disks. Another effect that is more obvious in the a = 0.25 and
0.5 disks is that those with 7 = 5/3 tend to show greater density variations along the
arms of the non-axisymmetric modes. At times this density variability will develop into
clumping in the outer portions of the arms, such as that seen in model Dl.


62
A more detailed description of the evolution of the modes, and of the mass accretion
observed in these disks are given in the next two sections. In the Mc/Md = 1 disks
another distinctive feature is seen, and that is the formation of small satellites in the outer
portions of disks. This is discussed in the following chapter.
Modal Evolution
The evolution of the simulated disks is typified by the presence of many transient
modes. In none of the Mc/Md = 3 disks does a single mode dominate the disk for more
than a few dynamical times. In all disks the growth of the non-axisymmetric modes
follows the general pattern that modes with higher mode number show faster growth
rates. However, as the amplitude in all modes increase, the modes of higher mode
number will peak and then decrease, while the modes of lower mode number continue to
grow. Hence, as the amplitude of the density fluctuations increases during the early part
of the evolution, different modes achieve temporary dominance. The maximum amplitude
is attained usually by an m=2 or m=3 mode. When the maximum power in a mode has
diminished it will usually not remain suppressed, but will later increase again to become
the mode with maximum power. In other words, the modes will initially grow at different
rates, but eventually the amplitude of all the modes seem to fluctuate at about the same
amplitude. The evolution of the maximum power of one model is shown in Figure 2-6.
The same behavior is initially seen in the Mc/M= 1 disks, but a slowly growing
m=l mode eventually dominates most of these disks, with one exception. Unlike the
other models, the dominating m=l mode is not a spiral wave, but instead is the result
of the central object and disk rotating about their center of mass. That is, the central


63
object, representing the star, is no longer centered on the mass distribution of the system.
Another characteristic of the modal evolution of these disks is that an m=2 mode will
dominate before this m=l mode becomes the dominant feature. Figure 4-5 shows a plot
of the maximum power reached by modes 1 through 4, between a radius of 0.1 and 1.0,
in the simulation of model B2.
It should be noted that characterizing the evolution by the maximum power present
in each mode may have limitations. Because the Fourier analysis was done on the
normalized density, the outer portions of the disk will primarily be represented, for this
part of the disk possesses the greatest density contrasts.
This transient behavior, as well as the number of modes present, does not allow the
frequencies to be unambiguously identified with a particular mode. The presence of many
initial transient modes is not surprising, as many modes will be excited by the perturbation
that has been seeded into the initial conditions. However, the persistence of this transient
behavior in a dissipative system is unexpected. In addition, the details of the evolution
of the modes for any given disk seem to be sensitive to initial conditions, in the sense
that a change in a computational parameter, such as the tolerance angle or the number of
particles, will cause the disks evolution to diverge in detail from a corresponding model
that is otherwise the same. This chaotic behavior brings into question the practicality of
describing in detail the complex evolution of the modes that are observed, though general
features of the evolution have been noted. (This same sensitivity to initial conditions is
seen in meteorology, where dissipation also is present.)


64
This behavior can be explained in two ways; the fluctuations are either reflecting
the physical behavior of such a system, or they are due to the failure of the numerical
method. If the behavior is physical then two processes may be occurring either separately,
or in conjunction. One process is the interaction between modes, where energy is being
transferred between modes continuously. In the disks that are modeled here, interaction
between the modes is enhanced because of the proximity of the corotation of a mode m
with the outer Lindblad resonance of the m 1 mode, and the inner Lindblad resonance
of the m+1 mode, in the frequency radius domain. In Figure 4-6 the location of the
resonances are shown. In such a system energy is easily transferred from one mode to
another. The other possible process is that the power in a mode is fluctuating due to
the presence of two modes with different frequencies, but the same mode number, which
would result in the oscillation of the amplitude of that mode.
Alternatively, the cause of the fluctuation in power may be due to a failure of the
SPH method. An example would be if the method could not describe a shock front
properly once it had developed a large amplitude. In this case the post-shock oscillations
could destroy a wave mode that has developed a shock front. There may be additional
problems in the outer portions of the disks where the number density of SPH particles can
become too low to describe regions of low density. It is in this part of the disk that the
highest normalized density amplitudes appear. Gaps in the particle distribution appear
between the arms of the non-axisymmetric modes; these gaps are low density regions
that are under-represented. To see if the transient character of the modes resulted from
such problems, models were run with 10093 particles. This increased the resolution of


65
the method, as the smoothing length h is inversely proportional to the square root of the
number density. While the under-represented regions of the disk were further out in the
disk than in the models with 5130 particles, the same transient behavior of the modes
was observed, though differing in detail. This suggests that the transient behaviour of
modes is an inherent effect found in such disks, rather than being a numerical effect.
Accretion
The rate of accretion onto the central object is determined by the rate of angular
momentum transport effected in the disk by shear viscosity and the non-axisymmetric
modes. In most of the models simulated, the mass accretion rate typically begins at a
large value, and within 2x dynamical times approaches a constant rate, which is more
sensitive to the viscosity parameter than to the ratio of specific heats. In some cases a
constant accretion rate is immediately realized (models A3 and C3). The mass accretion
is shown in Figures 4-7 through 4-11 for all models. The constant accretion rates in the
models were measured by fitting a line to the appropriate section of the mass accretion
curves, as shown by one example in the latter figure. Not surprisingly, the models with
Mc/Md = 1 show higher constant accretion rates than the Mc/Md = 3 disks. Although
it is not clear why constant accretion rates should be an endemic feature of accretion
disks, they are seen to some degree in most of the simulated disks, with only models
A1 and Cl showing a small continual decrease of their accretion rate with time. For
these two cases the accretion rates measured are the asymptotic accretion rate for most
of the simulations.


66
Interestingly, though a constant accretion rate is seen in almost all models, it is not
necessarily maintained throughout the run; in some cases the accretion rates change to
new constant values, sometimes with almost discontinuous abruptness. Also worthy of
note is the insensitivity of the initial accretion rates of models D1 and D2 to viscosity
(a = 0.25 and 0.5), as the mass accretion of these models is nearly equal. It is after new
constant accretion rates are re-established that the rates of these two models diverge. This
behavior is also observed in models Cl and C2, and therefore seems to be characteristic of
the Mc/Md = 1 models with lower values of a. With the exception of model D3, all of
the Mc/Mq = 1 models show at least one accretion rate shift, as well as the Mc/Md = 3
models A3 and C3, which both have a viscosity parameter equal to one. Two models
show a second accretion rate change: models A3 and Dl. For the Mc/Md = 1 models,
the first accretion rate change takes place at a transition point in the modal evolution of
the disk. The maximum power in the m=2 mode peaks preceding the change in accretion
rate, and the m=l mode becomes the dominant mode in the disk. The second accretion
rate change seen in model Dl takes place when the power in all modes with m > 1
abruptly falls to 0.2, one third the value of the m=l mode.
The exception to this behavior seen in the MJMq = 1 models is model D3, which
does not develop a dominant m=l mode. In this model the accretion rate changes very
slowly until a constant accretion rate is established. Unlike the MJMd = 1 models, the
constant accretion rate changes observed in models A3 and C3 are not clearly matched
with identifiable events in their modal evolution. In Table 3 the accretion rates, and the
times that they are attained, are given for all of the models in dimensionless units. To


67
convert the accretion rates to physical units, multiply the rates by the conversion factor
(Mt/Tq), the total mass divided by the dynamical time in the desired units. For instance,
using Mj IMq and the dynamical time Tq = 159 years, the accretion rates are found
to range between 2.36 x 10_6A//yr and 4.31 x 10~5.V//yr.
Table 4-3: Accretion Rates (x 10 3)
Model
mi
rho
to
m 3
3
A1
1.00
NA
A2
0.813
8
A3
1.13
0
0.688
8
0.375
20
A4
0.670
10
B1
3.88
7
6.11
18
B2
4.00
7
5.00
20
B3
3.00
9
7.32
22.5
Cl
0.813
NA
C2
0.531
7
C3
0.875
0
0.375
20.5
D1
4.25
6
6.86
18
4.63
27
D2
4.51
4
3.05
26
D3
2.13
17
More surprising is the dependence of the mass accretion on viscosity, for the mass
accretion increases as a decreases. This stated dependence is evident in Figures 4-7
through 4-10, though it is not established in the Mc/Mq = 1 disks with lower viscosity
parameter values until the constant accretion rate changes. The inverse dependence of
mass accretion upon shear viscosity is reflected in the mass accretion rates, which are
plotted in Figures 4-12 and 4-13. For the Mc/Md 3 models the trend is established


68
after the secondary accretion rates have been attained in the a = 1 disks, which are the
rates that coincide temporally with the mass accretion rates observed in the models with
smaller viscosity parameters. It is also the secondary accretion rates in the Mc/Md 1
models that show the same trend with shear viscosity, with the exception of the secondary
accretion rate of model B3.
This counter intuitive relationship between viscosity and mass accretion occurs
due to the combined actions of the global non-axisymmetric modes and local viscous
processes, which are not independent of one another. When both are present the viscosity
damps the more efficient mechanism of angular momentum transport provided by the
non-axisymmetric modes. The numerical experiments show that lower effective shear
viscosity allows the non-axisymmetric modes to attain greater strength, which in turn
become more effective at transporting angular momentum. This damping action of the
viscosity is not only qualitatively apparent, but is also evidenced by the rms amplitudes
of the density fluctuations.
In all the models, the total rms amplitude of the normalized density initially increases
exponentially with time. Most models then undergo a period of linear growth until a
saturation level is reached by the model; model D3 attains a saturation level immediately
after the initial exponential growth. Two models, D1 and D3, also show a second
saturation level being attained later in the simulation. In addition to this general behavior
of the normalized amplitudes, erratic fluctuations in its magnitude are apparent that take
place on a time scale of about two dynamical times. The fluctuations grow in amplitude
until the saturation level is reached, and are not obviously present until the period of


69
linear growth begins. The shear viscosity suppresses the rms amplitude of the density
fluctuations, causing lower saturation levels to be attained. Also, the viscosity generally
enhances the rate of the initial exponential growth.
Table 4-4: Normalized density amplitude growth rates and saturation levels.
Model
Viscosity (a)
Growth Rate
(at R = 0.55)
Growth Rate
(at R = 1)
Saturation
level(s)
A1
0.25
0.7
1.7
0.7
A2
0.5
1.3
1.8
0.55
A3
1
1.7
2.1
0.5
A4
0.5
1.3
1.8
0.5
B1
0.25
1.1
1.5
1.1
B2
0.5
0.9
1.5
0.9
B3
1
1.4
2.1
0.8
Cl
0.25
0.7
2.0
0.9
C2
0.5
0.9
2.1
0.8
C3
1
1.5
2.6
Not attained
0.6)
D1
0.25
1.0
1.8
0.65
D2
0.5
1.0
2.2
1.2 (0.7)
D3
1
1.2
3.0
0.3 (0.45)
The growth rates and saturation levels of the rms amplitude of the normalized density
fluctuations also vary with radius. The saturation level of the rms amplitude increases
with radius, as do the growth rates. Table 4 shows the growth rates for the various
models at the radial distances of 0.55 and 1.0 from the center of mass of the system. The
saturation level given corresponds to the smaller radius of 0.55.


70
The method of treating the central region and accretion in the hydrodynamic code
was intended to let the properties of the disk drive accretion, rather than be determined
by the imposed inner boundary condition around the central object. To test the degree to
which I am successful, models were run with two different inner disks (models A2 and
A4), and another with a larger Ra (model A5). Both types of inner disks, an exponential
disk and a massless Keplerian disk, provide pressure support at the inner boundary by
maintaining a continuous surface density and velocity field across the boundary (see
Chapter 3 for further details). Models A2 and A4 are identical except for the treatment
of the inner region, and the mass accretion in the two models is nearly the same (see
Figure 4-11). However, since the massless Keplerian disk results in the density gradient
being discontinuous at the boundary, the inner exponential disk was preferred. To what
degree the size of Ra alone influences accretion is tested by model A5, which is identical
to model A4 excepting the size of the inner region. The value of Ra for model A5,
0.0635, deviates from the nominal value of 0.0544. As can be seen in Figure 4-11, the
size of the inner accretion region does not significantly affect the mass accretion.
The initialization of an encounter model is described in chapter 3, which includes an
encountering particle with a mass of 0.5, and a disk that is otherwise equivalent to model
D2. In the resulting simulation it is found that the encounter takes place approximately
after ten dynamical times. The time sequence of this encounter is shown in the following
chapter, in Figures 5-9 through 5-12. Beside robbing a significant fraction of mass from
the disk, the encountering particle also greatly enhances the mass accretion onto the
central object. This is shown in Figure 4-14.


71
Tests of Code Parameters
As discussed at the beginning of this chapter, models with differing values of code
parameters, but equivalent model parameters, should be compared to confirm that the
choice of code parameters yield consistent results. Since it was impractical to run such a
series of tests for all combinations of the physical parameters, a single model was chosen:
model D2. The test models are designated TI, T2, and T3. In model T1 the tolerance
angle, 9, is given a value of 0.5, resulting in a gravitational approximation closer to a
direct summation of the gravitational terms due to N particles. In model T2 the Courant
parameter, C, is equal to 0.2, resulting in smaller integration time steps. In model T3
the number of particles used in the simulation is increased to 10093 particles. The mass
accretion of these three test models are shown with the mass accretion of model D2 in
Figure 4-15.
Model Tl, with a smaller tolerance angle than that of D2, is nearly equivalent to
model D2, indicating that the gravitational forces are being calculated to a sufficient
degree of accuracy with 6 = 0.7. Model T2 initially has the same mass accretion as D2,
but then deviates after twenty dynamical times. This may indicate that the hydrodynamic
calculations require smaller time steps to yield sufficiently accurate results.
Model T3, which possesses 10093 particles, differs most dramatically from model
D2. However, these two models are not physically equivalent for several reasons. By
increasing the number of particles by nearly a factor of two the local smoothing length is
decreased by approximately a factor of 2-1/2. This effectively changes the local viscosity
of the disk. An additional difference in model D2 and T3 is also seen when the modal


72
evolution of model T3 is inspected: a m=l mode does not eventually dominate the disk
of model T3 as it does model D2. This doubtless accounts for the smaller mass accretion
rate in model T3 in spite of the fact that, with a smaller smoothing lengths, the effective
local shear viscosity is smaller than in model D2. As mentioned above, weaker shear
viscosity typically leads to larger accretion rates. The reason that model T3 does not
develope a dominating m=l mode is most likely due to the fact that its initial density
perturbation also differs from that of model D2. This is because the density perturbation
was initially seeded into the disk by displacing the particles from the positions required
to describe a smooth exponential disk. These displacements are random in direction and
Gaussian in magnitude, with a standard deviation that is a fraction of the local interparticle
spacing. Because model T3 has a smaller interparticle spacing than model D2 the initial
noise in density in the two models are not the same. Hence, as models T3 and D2 are
not equivalent physically, these two models can not be directly compared.


0.5 a=0.25
73
7=2 7=5/3
Mc/Md=3
Figure 4-1: Particle distribution of Mc/Md = 3 models at Time = 10.0 dynamical times.


= 0.5 a-0.25
74
7=2
7=5/3
Mc/Md=3
Figure 4-2: Particle distribution of MJMd = 3 models at Time = 29.5 dynamical times


=0.5 ot=0.25
75
7=2
7=5/3
Figure 4-3: Particle distribution of MJMd = 1 models at Time = 10.0 dynamical times


76
Mc/Md=1
Figure 4-4: Particle distribution of MJMd = 1 models at Time = 30.0 dynamical times


Maximum Power
77
Figure 4-5: Maximum power in modes 1 through 4 for model B2.


angular frequency
78
Figure 4-6: Resonances in an accretion disk with Mc/Md = 3. Solid lines
correspond to the corotation resonance, the dotted lines to the Inner
Lindblad resonance, and the dashed lines to the Outer Lindblad resonance.


79
Figure 4-7: Mass accretion, as a fraction of initial disk mass, for models A1 through A3.
Figure 4-8: Mass accretion, as a fraction of initial disk mass, for B models.


80
Figure 49: Mass accretion, as a fraction of initial disk mass, for C models.
Figure 4-10: Mass accretion, as a fraction of initial disk mass, for D models.


81
Figure 4-11: Mass accretion in models A2, A4, and A5, as a fraction of the initial disk mass.


82
a
Figure 4-12: Constant mass accretion rates for Mc/Md = 3 accretion disks.
Figure 4-13: Constant mass accretion rates for Mc/Md = 1 accretion disks.


83
Figure 4-14: Mass accretion of the encounter model (solid line)
compared to the mass accretion of model D2 (dotted line).
Figure 4-15: Mass accretion, as a fraction of initial disk
mass, for model D2 is shown with three test models.


CHAPTER 5
FORMATION OF SATELLITES
The greatest difference between disks with higher and lower star to disk mass ratios
is that all disks that develop a dominating m=l mode also form satellites. This includes
all disks with Mc/Md = L with the exception of model D3. By way of example, the
later evolutionary sequence of model D2 is shown in Figure 5-1 and Figure 5-2. Figure
5-3 shows the final configuration of the disk, with the path of the satellite shown since
its formation. The satellite forms from a clump of gas that is recognizable as a distinct
structure at time = 25. At this time it is part of a spiral arm of a m=l mode that extends
to large radii. The mass in the initial clump of gas is 0.013A/j\ and at time = 39 the
satellite has a mass of 0.028Mj. Also, as the mass of the satellite increases, and the
combined mass of the star and the disk interior to the companion decreases, their mean
separation decreases. A closer view of the satellite is shown in Figure 5-4, which also
shows its radial density profile.
Other models form multiple satellites, which can interact with one another. Table 2
provides an overview of the formation and evolution of the satellites, giving the initial
separation R¡ between the mass and the central object disk center of mass, the initial
mass Mj, the time of formation T¡, the final separation Rf, and the final mass of the
satellite, Mf. The final time, Tf, corresponds to either the end of the simulation, or to
when the satellite was destroyed.
84


85
Table 5-1: Satellites
Model
Satellite
Rt
M¡
T¡
Rf
Mf
Tf
(x 10'2)
(x lO2)
B1
1
1.96
0.448
31.5
3.25
0.536
38.8
2
1.57
0.409
22.5
4.72
0.692
38.8
3
2.01
0.370
23.0
4.57
0.419
38.8
B2
1
2.23
0.546
24.0
3.72
1.277
38.5
2
1.68
0.975
24.0
0.81
4.045
37.5*
B3
1
1.33
1.374
21.0
1.48
4.776
31.0
2
1.41
0.760
24.0
2.26
1.72
31.0
D1
1
1.21
0.770
18.5
2.31
1.920
33.0
2
1.17
1.150
17.5
0.98
2.174
33.0
3
1.30
0.741
22.0
1.65
0.955
33.0
4
1.17
1.053
19.0
0.67
1.803
24.5*
D2
1
1.50
1.257
25.0
1.24
2.797
39.0
* Indicated satellites are reabsorbed into the disk.
Orbital Evolution
In the absence of close encounters with other satellites, the orbital evolution can be
understood by considering the simpler system of two point masses in circular motion
about a common center of mass. In such a two body system the separation of the two
masses is determined by the masses and the orbital angular momentum, given explicitly by
L~ m
G (mirni)2
(5.1)
where L is the orbital angular momentum about the center of mass, and m = m\ + m2
is the total mass. The above equation shows how the equivalent separation between


86
two bodies can decrease, assuming they remain on circular orbits. If the total mass
and orbital angular momentum remain constant, and mass is transferred from the more
massive body (the primary), m\, to its less massive companion (the secondary), mo,
then the separation will decrease as l/(m\7iio). The separation will also decrease if
the orbital angular momentum decreases. A change in the orbital angular momentum
can occur by a variety of mechanisms: a) external torques on the system, such as an
outer satellite, b) internal torques, which can transfer orbital angular momentum into,
or out of, the spin angular momentum associated with an extended asymmetrical mass
distribution about one of the bodies, c) mass loss from the system, which will carry away
orbital angular momentum with it, and conversely, d) mass accretion onto the system.
While mass loss from the system decreases the orbital angular momentum, it may cause
the separation to increase. From the above equation, and assuming that the mass loss
process does not alter the velocities of the two masses, and that the secondarys mass
remains constant, the condition for increasing the separation with mass loss from the
primary is > ^(l + VTT) ~ 2.56.
To understand the orbital evolution of the formed satellites, the above model is applied
as an approximation. The mass of each of the satellites, and the mass within the minimum
separation between the satellite and central object, is found. These masses are taken to
be the masses of the primary and secondary respectively. Hence, the primary mass mi
consists of the central object and the mass in the disk, as well as any inner satellites, that
lie within the minimum separation. In addition, the orbital angular momentum associated
with these two masses, with respect to their center of mass, is found. Using the above


87
formula (equation 5.1), the equivalent separation, a, of a system in circular motion, with
the same orbital angular momentum and masses, was found. Often the formed satellites
exhibit substantial eccentricity, which results in the actual separation oscillating about the
equivalent separation. All of the above mechanisms for altering the separation between
the disk-star and the satellite are observed to some degree. Mass is commonly lost
from the disk as mass is transported outward with angular momentum. As the primary
mass of the disk-star far exceeds the mass of the satellite, this mass loss results in the
separation increasing, if the mass of the satellite remains constant. Interaction with the
non-axisymmetric modes of the disk, which act as a reservoir of angular momentum,
is also important. Though this model is an approximation it gives sufficient insight to
understand the orbital evolution of the satellites.
In the case of model D2, where only a single satellite is present, the decrease in the
separation is primarily due to a change in the mass ratio of the primary and the satellite. In
model B2 (Figure 5-5) two satellites form almost simultaneously at the same polar angle.
The outer satellite then spirals outward, while the inner satellite spirals inward. Initially
this can be understood as resulting from the interaction of the two satellites. However,
while the outer satellites orbital angular momentum rises sharply soon after formation,
a comparable drop in the inner satellites orbital angular momentum is not observed; the
orbital angular momentum of the inner satellite initially remains constant. Therefore any
angular momentum transfered from the inner to the outer satellite must be offset by an
equal transfer of angular momentum from the disk to the inner satellite, or else angular
momentum is being transfered from the disk directly to the outer satellite. Later the


88
angular momentum of the inner satellite increases dramatically, due to interaction with
the non-axisymmetric modes in the disk. However, this increase of the inner satellites
orbital angular momentum is not enough to counter the effects of mass transfer from the
disk to the satellite, and it continues to spiral inward to eventually be reabsorbed into
the disk. At the same time the outer satellite continues to slowly gain orbital angular
momentum, even while its own mass, and the mass interior to it remains constant. The
source of this orbital angular momentum must be from the disk and satellite interior to
it, as no external torques are present.
In model B3 two satellites also form (Figure 5-6). However, due to a different
configuration of the satellites, they do not strongly interact. The innermost satellite
remains at nearly a constant separation for four dynamical times, then gently increases
to its maximum peak value at T=28.5, and then diminishes to its final value. Though the
separation does not vary greatly, the mass and orbital angular momentum show abrupt
increases, making steps to higher values. These steps are correlated with collisions of
the satellite with arms of a mode that extends out to the orbital radius of the satellite and
overtakes it. Between these collisions the separation between the satellite and primary
continues to decrease due to mass loss from the primary. In contrast to the inner satellite,
the outer satellites separation grows continuously after its formation. Its mass and orbital
angular momentum also remain constant for four dynamical times, after which both
increase as it encounters material moving outward from the disk. The continual change
in the separation is again due to the decrease of the primarys mass.


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