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## Spherically Symmetric Configurations

(Initially) Spherically Symmetric Configurations

### Equilibrium Structures

1D STRUCTURE

Hydrostatic
Balance
Equation
 ${\displaystyle {\frac {dP}{dr}}=-{\frac {GM_{r}\rho }{r^{2}}}}$
Solution
Strategies

Isothermal
Sphere
 ${\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}{\biggl (}\xi ^{2}{\frac {d\psi }{d\xi }}{\biggr )}=e^{-\psi }}$
via
Direct
Numerical
Integration

Isolated
Polytropes
Lane
(1870)
 ${\displaystyle {\frac {1}{\xi ^{2}}}{\frac {d}{d\xi }}{\biggl (}\xi ^{2}{\frac {d\Theta _{H}}{d\xi }}{\biggr )}=-\Theta _{H}^{n}}$
Known
Analytic
Solutions
via
Direct
Numerical
Integration
via
Self-Consistent
Field (SCF)
Technique

Free Energy
of
Bipolytropes

(nc, ne) = (5, 1)

### Stability Analysis

1D STABILITY

Three different approaches are used in the study of hydrodynamical stability of stars and other gravitating objects …

• The first approach is based on the use of the equations of small oscillations. In that case the problem is reduced to a search for the solution of the boundary-value problem of the Stourme-Liuville type for the linearised system of equations of small oscillations. The solutions consist of a set of eigenfrequencies and eigenfunctions. The following set of menu tiles include links to chapters where this approach has been applied to: (a) uniform-density configurations, (b) pressure-truncated isothermal spheres, (c) an isolated n = 3 polytrope, (d) pressure-truncated n = 5 configurations, and (e) bipolytropes having ${\displaystyle (n_{c},n_{e})=(1,5)}$.
• Second, one can derive a variational principle from the equations of small oscillations. Below, an appropriately labeled (purple) menu tile links to a chapter in which the foundation for this approach is developed. With the aid of the variational principle, the problem is reduced to the search of the best trial functions; this leads to approximate eigenvalues of oscillations. In spite of the simplifications introduced by the use of the variational principle and by not solving the equations of motion exactly, the problem still remains complicated … One menu tile, below, links to a chapter in which an analytic (exact) demonstration of the variational principle's utility is provided in the context pressure-truncated n = 5 polytropes.
• The third approach is what we have already referred to as a free-energy — and associated virial theorem — analysis. When this method is used, it is not necessary to use the equations of small oscillations but, instead, the functional expression for the total energy of the momentarily stationary (but not necessarily in equilibrium) star is sufficient. The condition that the first variation of the energy vanishes, determines the state of equilibrium of the star and the positiveness of a second variation indicates stability.

If one wants to know from a stability analysis the answer to only one question — whether the model is stable or not — then the most straightforward procedure is to use the third, static method … For the application of this method, one needs to construct only equilibrium, stationary models, with no further calculation. Generally the static analysis gives no information about the shape of the modes of oscillation, but, in the vicinity of critical points, where instability sets in, this method makes it possible to find the eigenfunction of the mode which becomes unstable at the critical point. Generally in what follows, this will be referred to as the "B-KB74 conjecture;" a menu tile carrying this label is linked to a chapter in which this approach is used to analyze the onset of a dynamical instability along the equilibrium sequence of pressure-truncated n = 5 polytropes.

--- Text in green taken directly from G. S. Bisnovatyi-Kogan & S. I. Blinnikov (1974); B-KB74, for short.

Pressure-Truncated
Isothermal
Spheres
 ${\displaystyle 0={\frac {d^{2}x}{d\xi ^{2}}}+{\biggl [}4-\xi {\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}{\frac {1}{\xi }}\cdot {\frac {dx}{d\xi }}+{\biggl [}{\biggl (}{\frac {\sigma _{c}^{2}}{6\gamma _{\mathrm {g} }}}{\biggr )}\xi ^{2}-\alpha \xi {\biggl (}{\frac {d\psi }{d\xi }}{\biggr )}{\biggr ]}{\frac {x}{\xi ^{2}}}}$ where:    ${\displaystyle \sigma _{c}^{2}\equiv {\frac {3\omega ^{2}}{2\pi G\rho _{c}}}}$     and,     ${\displaystyle \alpha \equiv {\biggl (}3-{\frac {4}{\gamma _{\mathrm {g} }}}{\biggr )}}$
via
Direct
Numerical
Integration

Yabushita's
Analytic Sol'n for
Marginally Unstable
Configurations
(1974)
 ${\displaystyle \sigma _{c}^{2}=0\,,~~~~\gamma _{\mathrm {g} }=1}$ and ${\displaystyle x=1-{\biggl (}{\frac {1}{\xi e^{-\psi }}}{\biggr )}{\frac {d\psi }{d\xi }}}$

Polytropes
 ${\displaystyle 0={\frac {d^{2}x}{d\xi ^{2}}}+{\biggl [}4-(n+1)Q{\biggr ]}{\frac {1}{\xi }}\cdot {\frac {dx}{d\xi }}+(n+1){\biggl [}{\biggl (}{\frac {\sigma _{c}^{2}}{6\gamma _{g}}}{\biggr )}{\frac {\xi ^{2}}{\theta }}-\alpha Q{\biggr ]}{\frac {x}{\xi ^{2}}}}$ where:    ${\displaystyle Q(\xi )\equiv -{\frac {d\ln \theta }{d\ln \xi }}\,,}$    ${\displaystyle \sigma _{c}^{2}\equiv {\frac {3\omega ^{2}}{2\pi G\rho _{c}}}\,,}$     and,     ${\displaystyle \alpha \equiv {\biggl (}3-{\frac {4}{\gamma _{\mathrm {g} }}}{\biggr )}}$
Isolated
n = 3
Polytrope

Our (2017)
Analytic Sol'n for
Marginally Unstable
Configurations
 ${\displaystyle ~\sigma _{c}^{2}=0\,,~~~~\gamma _{\mathrm {g} }=(n+1)/n}$ and ${\displaystyle ~x={\frac {3(n-1)}{2n}}{\biggl [}1+{\biggl (}{\frac {n-3}{n-1}}{\biggr )}{\biggl (}{\frac {1}{\xi \theta ^{n}}}{\biggr )}{\frac {d\theta }{d\xi }}{\biggr ]}}$

B-KB74
Conjecture
RE: Bipolytrope

(nc, ne) = (5, 1)

1D DYNAMICS

## Two-Dimensional Configurations (Axisymmetric)

(Initially) Axisymmetric Configurations

2D STRUCTURE

### Stability Analysis

2D STABILITY

#### Sheroidal & Spheroidal-Like

 The equilibrium models are calculated using the polytrope version (Bodenheimer & Ostriker 1973) of the Ostriker and Mark (1968) self-consistent field (SCF) code … the equilibrium models rotate on cylinders and are completely specified by ${\displaystyle ~n}$, the total angular momentum, and the specific angular momentum distribution ${\displaystyle ~j(m_{\varpi })}$. Here ${\displaystyle ~m_{\varpi }}$ is the mass contained within a cylinder of radius ${\displaystyle ~\varpi }$ centered on the rotation axis. The angular momentum distribution is prescribed in several ways: (1) imposing strict uniform rotation; (2) using the same ${\displaystyle ~j(m_{\varpi })}$ as that of a uniformly rotating spherical polybrope of index ${\displaystyle ~n^{'}}$ (see Bodenheimer and Ostriker 1973); and (3) using ${\displaystyle ~j(m_{\varpi })\propto m_{\varpi }}$, which we refer to as ${\displaystyle ~n^{'}=L}$, ${\displaystyle ~L}$ for "linear."

2D DYNAMICS

## Two-Dimensional Configurations (Nonaxisymmetric Disks)

Infinitesimally Thin, Nonaxisymmetric Disks

2D STRUCTURE

## Three-Dimensional Configurations

(Initially) Three-Dimensional Configurations

### Equilibrium Structures

3D STRUCTURE
 Special numerical techniques must be developed "to build three-dimensional compressible equilibrium models with complicated flows." To date … "techniques have only been developed to build compressible equilibrium models of nonaxisymmetric configurations for a few systems with simplified rotational profiles, e.g., rigidly rotating systems (Hachisu & Eriguchi 1984; Hachisu 1986), irrotational systems (Uryū & Eriguchi 1998), and configurations that are stationary in the inertial frame (Uryū & Eriguchi 1996)." — Drawn from §1 of 📚 S. Ou (2006, ApJ, Vol. 639, pp. 549 - 558)

#### Binary Systems

 Roche's problem is concerned with the equilibrium and the stability of rotating homogeneous masses which are, further, distorted by the constant tidal action of an attendant rigid spherical mass.

3D STABILITY

3D DYNAMICS