# Spherically Symmetric Configurations Synopsis (Using Style Sheet)

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## Structure

### Tabular Overview

Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion — adiabatic index, $~\gamma$
 $~dV = 4\pi r^2 dr$ and $~dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr$ $~W_\mathrm{grav}$ $~=$ $~- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}$ $~U_\mathrm{int}$ $~=$ $~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}$
Equilibrium Structure
Detailed Force Balance    Free-Energy Identification of Equilibria
Given a barotropic equation of state, $~P(\rho)$, solve the equation of

Hydrostatic Balance

 $~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}$

for the radial density distribution, $~\rho(r)$.

The Free-Energy is,
 $~\mathfrak{G}$ $~=$ $~W_\mathrm{grav} + U_\mathrm{int} + P_eV$ $~=$ $~-a \biggl(\frac{R}{R_0}\biggr)^{-1} + b\biggl(\frac{R}{R_0}\biggr)^{3-3\gamma}+ c\biggl(\frac{R}{R_0}\biggr)^3 \, .$

Therefore, also,

 $~R_0 ~\frac{\partial\mathfrak{G}}{\partial R}$ $~=$ $~a\biggl(\frac{R}{R_0}\biggr)^{-2} +(3-3\gamma)b\biggl(\frac{R}{R_0}\biggr)^{2-3\gamma} + 3c\biggl(\frac{R}{R_0}\biggr)^2$ $~=$ $~\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]$

Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting $~d\mathfrak{G}/dR = 0$. Hence, equilibria are defined by the condition,

 $~0$ $~=$ $~W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .$
Virial Equilibrium

Multiply the hydrostatic-balance equation through by $~rdV$ and integrate over the volume:

 $~0$ $~=$ $~-\int_0^R r\biggl(\frac{dP}{dr}\biggr)dV - \int_0^R r\biggl(\frac{GM_r \rho}{r^2}\biggr)dV$ $~=$ $~-\int_0^R 4\pi r^3 \biggl(\frac{dP}{dr}\biggr) dr - \int_0^R \biggl(\frac{GM_r}{r}\biggr)dM_r$ $~=$ $~-\int_0^R\biggl[ \frac{d}{dr}\biggl( 4\pi r^3P \biggr) - 12\pi r^2 P\biggr] dr + W_\mathrm{grav}$ $~=$ $~\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}$ $~=$ $~3(\gamma-1)U_\mathrm{int} + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .$

### Pointers to Relevant Chapters

Background Material:

· Principal Governing Equations (PGEs) in most general form being considered throughout this H_Book PGEs in a form that is relevant to a study of the Structure, Stability, & Dynamics of spherically symmetric systems Supplemental relations — see, especially, barotropic equations of state

Detailed Force Balance:

· Derivation of the equation of Hydrostatic Balance, and a description of several standard strategies that are used to determine its solution — see, especially, what we refer to as Technique 1

Virial Equilibrium:

· Formal derivation of the multi-dimensional, 2nd-order tensor virial equations Scalar Virial Theorem, as appropriate for spherically symmetric configurations Generalization of scalar virial theorem to include the bounding effects of a hot, tenuous external medium

## Stability

### Isolated & Pressure-Truncated Configurations

Stability Analysis:   Applicable to Isolated & Pressure-Truncated Configurations
Perturbation Theory    Free-Energy Analysis of Stability

Given the radial profile of the density and pressure in the equilibrium configuration, solve the eigenvalue problem defined by the,

LAWE:   Linear Adiabatic Wave (or Radial Pulsation) Equation

 $~0$ $~=$ $~ \frac{d}{dr}\biggl[ r^4 \gamma P ~\frac{dx}{dr} \biggr] +\biggl[ \omega^2 \rho r^4 + (3\gamma - 4) r^3 \frac{dP}{dr} \biggr] x$ [P00], Vol. II, §3.7.1, p. 174, Eq. (3.145)

to find one or more radially dependent, radial-displacement eigenvectors, $~x \equiv \delta r/r$, along with (the square of) the corresponding oscillation eigenfrequency, $~\omega^2$.

The second derivative of the free-energy function is,

 $~R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}$ $~=$ $~ -2a\biggl(\frac{R}{R_0}\biggr)^{-3} + (3-3\gamma)(2-3\gamma)b \biggl(\frac{R}{R_0}\biggr)^{1-3\gamma} + 6c\biggl(\frac{R}{R_0}\biggr)$ $~=$ $~\biggl(\frac{R_0}{R} \biggr)^2\biggl[ 2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V \biggr] \, .$

Evaluating this second derivative for an equilibrium configuration — that is by calling upon the (virial) equilibrium condition to set the value of the internal energy — we have,

 $~3(\gamma-1)U_\mathrm{int}$ $~=$ $~3P_e V - W_\mathrm{grav}$ $~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}$ $~=$ $~2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav} \biggr] + 6P_e V$ $~=$ $~(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, .$

Note the similarity with .

Alternatively, recalling that,

 $~3(\gamma - 1)U_\mathrm{int}$ $~=$ $~2S_\mathrm{therm} \, ,$

the conditions for virial equilibrium and stability, may be written respectively as,

 $~3P_e V$ $~=$ $~ 2S_\mathrm{therm}+ W_\mathrm{grav}$ $~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}$ $~=$ $~ 2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav} \biggr]$ $~=$ $~ 4W_\mathrm{grav} + 6\gamma S_\mathrm{therm} \, .$

Variational Principle

Multiply the LAWE through by $~4\pi x dr$, and integrate over the volume of the configuration gives the,

Governing Variational Relation

 $~0$ $~=$ $~ \int_0^R 4\pi r^4 \gamma P \biggl(\frac{dx}{dr}\biggr)^2 dr - \int_0^R 4\pi (3\gamma - 4) r^3 x^2 \biggl( \frac{dP}{dr} \biggr) dr$ $~ - 4\pi \biggr[r^4 \gamma Px \biggl(\frac{dx}{dr}\biggr) \biggr]_0^R - \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .$ $~=$ $~ \int_0^R x^2 \biggl(\frac{d\ln x}{d\ln r}\biggr)^2 \gamma 4\pi r^2P dr - \int_0^R (3\gamma - 4)x^2 \biggl( - \frac{GM_r}{r} \biggr) 4\pi \rho r^2 dr$ $~ + \biggr[\gamma 4\pi r^3 Px^2 \biggl(-\frac{d\ln x}{d\ln r}\biggr) \biggr]_0^R - \int_0^R 4\pi \omega^2 \rho r^4 x^2 dr \, .$

Now, by setting $~(d\ln x/d\ln r)_{r=R} = -3$, we can ensure that the pressure fluctuation is zero and, hence, $~P = P_e$ at the surface, in which case this relation becomes,

 $~\omega^2$ $~=$ $~ \frac{\gamma (\gamma -1) \int_0^R x^2 \bigl(\frac{d\ln x}{d\ln r}\bigr)^2 dU_\mathrm{int} - \int_0^R (3\gamma - 4)x^2 dW_\mathrm{grav} + 3^2 \gamma x^2 P_eV}{ \int_0^R x^2 r^2 dM_r}$
Approximation:   Homologous Expansion/Contraction

If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, $~x$ = constant, and the governing variational relation gives,

 $~\omega^2 \int_0^R r^2 dM_r$ $~\leq$ $~ (4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, .$

### Bipolytropes

Stability Analysis:   Applicable to Bipolytropic Configurations
Variational Principle    Free-Energy Analysis of Stability

Governing Variational Relation

 $~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2 dM_r^*$ $~=$ $~ \gamma_c (\gamma_c-1) \int_0^{r^*_\mathrm{core}} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int} - (3\gamma_c - 4) \int_0^{r^*_\mathrm{core}} x^2 dW^*_\mathrm{grav}$ $~ + ~\gamma_e (\gamma_e-1) \int_{r^*_\mathrm{core}}^{R^*} x^2~\biggl( \frac{d\ln x}{d\ln r^*} \biggr)^2 dU^*_\mathrm{int} - (3\gamma_e - 4) \int_{r^*_\mathrm{core}}^{R^*} x^2 dW^*_\mathrm{grav}$ $~ + ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^* \, .$

As we have detailed in an accompanying discussion, the first derivative of the relevant free-energy expression is,

 $~R ~\frac{\partial \mathfrak{G}}{\partial R}$ $~=$ $~ 2S_\mathrm{tot} + W_\mathrm{tot} \, ,$

where,

 $~S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}$ and $~W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;$

and the second derivative of that free-energy function is,

 $~R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}$ $~=$ $~2\biggl[ W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env} \biggr] \, .$

This stability criterion may be rewritten as,

 $~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}$ $~=$ $~ 2[(3\gamma_c -4) S_\mathrm{core} + (3\gamma_e -4) S_\mathrm{env} ] \, .$

Hence, in bipolytropes, the marginally unstable equilibrium configuration (second derivative of free-energy set to zero) will be identified by the model that exhibits the ratio,

 $~\frac{S_\mathrm{core}}{S_\mathrm{env}}$ $~=$ $~ \frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, .$

See the accompanying discussion.

If — based for example on — we make the reasonable assumption that, in equilibrium, the statements,

 $~2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}$ and $~2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,$

hold separately, then we satisfy the virial equilibrium condition, namely,

 $~0$ $~=$ $~2S_\mathrm{tot} + W_\mathrm{tot} \, ,$

and the second derivative of the relevant free-energy function can be rewritten as,

 $~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}$ $~=$ $~ 2(W_\mathrm{core} + W_\mathrm{env}) + (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core})$ $~ + (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env})$ $~=$ $~ 3^2 P_i V_\mathrm{core}(\gamma_c - \gamma_e) + (4-3\gamma_c ) W_\mathrm{core} + (4-3\gamma_e)W_\mathrm{env} \, .$

Note the similarity with — temporarily, see this discussion.

Approximation:   Homologous Expansion/Contraction

If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, $~x$ = constant, and the governing variational relation gives,

 $~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R r^2 dM_r$ $~\leq$ $~ (4- 3\gamma_c) W_\mathrm{core}+ (4- 3\gamma_e) W_\mathrm{env}+ 3^2 (\gamma_c - \gamma_e) P_i V_\mathrm{core} \, .$