The Structure, Stability, and Dynamics
of Self-Gravitating Systems

Joel E. Tohline

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Governing Equations for
Axisymmetric Objects in Simple Rotation

As soon as rotation is introduced into the problem, two general complications arise:

Furthermore, even in the case of spherically symmetric objects, it is rare that an equilibrium solution can be determined analytically (cf., our discussion herein of spherical polytropes). Hence, as a rule, the equilibrium properties of rotating, self-gravitating configurations are not describable in terms of analytical functions. (As is discussed in detail in an accompanying chapter, Maclaurin Spheroids are a notable exception to this rule and thereby play an extremely important role in our understanding of the structure and stability of rotating, self-gravitating objects.)

As a foundation for the discussion of the structure of axisymmetric objects sporting a variety of degrees of compressibility and an assortment of angular momentum distributions, we present here the set of equations that govern the equilibrium of rotating, self-gravitating fluid systems.

Constraints on Equilibrium Configurations

We seek to construct equilibrium, self-gravitating configurations that are axisymmetric and have simple rotation. Such structures are most easily described mathematically in cylindrical coordinates. (See the accompanying appendix for a variety of relevant expressions.) For example, to implement the "axisymmetric" constraint in cylindrical coordinates, we set j 0 in all spatial operators. By doing this, we derive the following

2-Dimensional, Axisymmetric Operators
(in Cylindrical Coordinates):

= evv + ezz [Equation III.F.1]
F = (1/v)v(v Fv) + zFz [Equation III.F.2]

2G = (1/v)v(v vG) + z(zG) [Equation III.F.3]
(v)F = ev [ vvvFv + vzzFv - vjFj/v ] +
ej [ vvvFj + vzzFj + vjFv/v ] +
ez [ vvvFz + vzzFz ] [Equation III.F.4]

Specification of the Velocity Flow-Field

The vector velocity in cylindrical coordinates quite generally takes the form:

v = ev (dv/dt) + ej (v dj/dt) + ez (dz/dt).

[Appendix Equation ?]

By the phrase "simple rotation," we mean that the only fluid motion in the equilibrium configuration is circular (axisymmetric) motion in the azimuthal direction, i.e.,

vv dv/dt = 0 and vz dz/dt = 0,

[Equations III.F.5]

w dj/dt = vj/v 0,

[Equation III.F.6]

and, furthermore, that the angular velocity w is only a function of the cylindrical radial coordinate, v. In summary, then, for all equilibrium axisymmetric objects considered throughout this H_Book, we will adopt the following

"Simple Rotation"
steady-state velocity flow-field:

v = ej (v w ),

[Equation III.F.7]


w = w(v).

Is "simple rotation" too simple?

It may very well be possible to construct equilibrium, axisymmetric models with steady-state flow fields different from the "simple rotation" prescribed here. On physical grounds, it has been argued that w should be uniform on cylinders -- i.e., w should not be a function of z -- in steady-state systems. But steady-state flows with vv 0 and vz 0 -- such as meridional circulation -- may exist for many different types of configurations. In practice, though, systems with more complicated flows have not been studied extensively. (Almost certainly because they are significantly more difficult to construct than models with simple rotation.) Without apology, we will restrict our discussions to configurations with "simple rotation," as just defined.

With simple rotation imposed, the convective operator [III.F.4] further reduces to the form:

(v)F = - ev [ w Fj ] + ej [ w Fv ] .

[Equation III.F.8]

Steady-State Condition

All equilibrium axisymmetric figures must be in steady-state as viewed by an observer in the inertial reference frame. Hence, in each of the principal governing equations that serve, as a group, to define the configuration's equilibrium structure, we must set t 0.

Euler's Equation

Specifically, in Euler's equation,

Dv = tv + (v)v (v)v ,

[Equation III.F.9]

and the imposition of simple rotation means, furthermore,

(v)v - ev [ v w2 ] .

[Equation III.F.10]

Hence, from Euler's equation we derive the following condition for

Hydrostatic Equilibrium
in rotating, axisymmetric objects:
ev [ v w2 ] = ( 1/r )P + F
= ev [ ( 1/r ) vP + vF ] + ez [ ( 1/r ) zP + zF ] .

[Equation III.F.11]

Continuity Equation and First Law of Thermodynamics

By setting t 0 and imposing simple rotation, the equation of continuity and the first law of thermodynamics both are automatically satisfied for any axisymmetric distribution of the mass density and specific entropy. Hence, these two principal governing equations need not be dealt with directly when constructing equilibrium, axisymmetric configurations.

The equation of continuity [I.B.3] and the first law of thermodynamics [I.C.14] both can be written in the following "conservative" forms:

tq + (q v) = 0,

[Equation III.F.12]

where q is a scalar physical variable. What does the variable q represent in the case of the equation of continuity? What does it represent in the case of the first law of thermodynamics?

Show that by setting t 0 and imposing simple rotation, the left-hand-side of this equation automatically goes to zero for any axisymmetric functional form of the variable q.

Poisson Equation

Finally, from the mathematical expression listed above [III.F.3] for the two-dimensional, axisymmetric Laplacian in cylindrical coordinates, we deduce directly that the gravitational potential in the expression for hydrostatic equilibrium can be determined from the

Poisson Equation
for axisymmetric configurations:

(1/v)v(v vF) + z(zF) = 4pGr .

[Equation III.F.13]


The equilibrium structure of self-gravitating, axisymmetric objects in simple rotation can be determined by solving simultaneously the following set of coupled partial differential equations in conjunction with a chosen barotropic equation of state:

vP = - r vF + r v w2,

[Equation III.F.14]

zP = - r zF,

[Equation III.F.15]

(1/v)v(v vF) + z(zF) = 4pGr .

[Equation III.F.13]

Employing a Barotropic Equation of State

In terms of the enthalpy, it becomes possible to rewrite the condition for hydrostatic equilibrium [III.F.11] in the following form:

ev [ v w2 ] = ( H + F ).

[Equation III.F.19]

A Centrifugal Potential

Furthermore, for configurations in simple rotation [III.F.7], it usually is possible to construct a scalar centrifugal potential y(v) such that,

ho2 y = - ev [ v w2 ],

[Equation III.F.20]

where ho is a normalization constant. Specifically, for any reasonable function w(v),

y - ho-2 [ v w2 ] dv

[Equation III.F.21]

satisfies this requirement.

Show that the centrifugal potential y(v) for any object that is rotating uniformly with angular velocity Wo is,

y = - v2 / 2,

[Equation III.F.22]

if ho = Wo.

With the centrifugal potential defined in this way [III.F.21], the statement of hydrostatic equilibrium for axisymmetric configurations in simple rotation [III.F.19] may be written,

( H + F + ho2 y ) = 0.

[Equation III.F.23]

In order that this condition be satisfied, it must be true that, throughout the equilibrium structure,

H + F + ho2 y = Co,

[Equation III.F.24]

where Co is a constant. It often is easier to construct rotating equilibrium configurations by demanding that this algebraic equation [III.F.24] be satisfied in conjunction with the axisymmetric Poisson equation than it is to construct rotating equilibria by solving the outlined set of coupled partial differential equations [III.F.13], [III.F.14], and [III.F.15] by a brute-force technique.

Deriving Solutions
via the Self-Consistent-Field Technique

Throughout much of this H_Book, we will rely upon a numerical "self-consistent-field" (SCF) technique to construct rotating equilibrium configurations having a variety of different degrees of compressibility and a variety of different geometries. As was mentioned briefly in the context of spherical polytropic configurations, through an SCF technique, one iterates back and forth between solutions to the two equations [III.F.13 & III.F.24] that govern the equilibrium structure of self-gravitating, axisymmetric objects in simple rotation until a satisfactory, "self-consistent" solution to both equations has been determined.

In particular, we will rely heavily upon the specific SCF technique that was developed by Hachisu in the mid-1980's. The accompanying pages describe in detail how an HSCF (Hachisu self-consistent-field) numerical algorithm is constructed, and provide a guide to the various www-accessible HSCF utilities that have been developed at LSU.

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