SSC/SynopsisStyleSheet

Spherically Symmetric Configurations Synopsis (Using Style Sheet)

Structure

Tabular Overview

Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion — adiabatic index, $\gamma$

$dV=4\pi r^{2}dr$ and    $d{M}_r=\rho dV\Rightarrow M_r=4\pi \underset{0}{\overset{r}{\int }}\rho r^{2}dr$ $W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$ $=$ $-\underset{0}{\overset{R}{\int }}\left(\frac{G{M}_r}{r}\right)d{M}_r\propto R^{-1}$ $U_{\mathrm{i}\mathrm{n}\mathrm{t}}$ $=$ $\frac{1}{(\gamma -1)}\underset{0}{\overset{R}{\int }}4\pi r^{2}Pdr\propto R^{3-3\gamma }$
Equilibrium Structure
①   Detailed Force Balance	③   Free-Energy Identification of Equilibria
Given a barotropic equation of state, $P(\rho )$, solve the equation of Hydrostatic Balance $\frac{dP}{dr}=-\frac{G{M}_r\rho }{r^2}$ for the radial density distribution, $\rho (r)$.	The Free-Energy is, $\mathfrak{𝔊}$ $=$ $W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+U_{\mathrm{i}\mathrm{n}\mathrm{t}}+P_{e}V$   $=$ $-a(\frac{R}{R_0}{)}^{-1}+b(\frac{R}{R_0}{)}^{3-3\gamma }+c(\frac{R}{R_0}{)}^3.$ Therefore, also, $R_0\frac{\partial \mathfrak{𝔊}}{\partial R}$ $=$ $a(\frac{R}{R_0}{)}^{-2}+(3-3\gamma )b(\frac{R}{R_0}{)}^{2-3\gamma }+3c(\frac{R}{R_0}{)}^2$   $=$ $\frac{R_0}{R}[-W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}-3(\gamma -1)U_{\mathrm{i}\mathrm{n}\mathrm{t}}+3P_{e}V]$ 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, $0$ $=$ $W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+3(\gamma -1)U_{\mathrm{i}\mathrm{n}\mathrm{t}}-3P_{e}V.$
②   Virial Equilibrium
Multiply the hydrostatic-balance equation through by $rdV$ and integrate over the volume: $0$ $=$ $-\underset{0}{\overset{R}{\int }}r\left(\frac{dP}{dr}\right)dV-\underset{0}{\overset{R}{\int }}r\left(\frac{G{M}_r\rho }{r^2}\right)dV$   $=$ $-\underset{0}{\overset{R}{\int }}4\pi r^3\left(\frac{dP}{dr}\right)dr-\underset{0}{\overset{R}{\int }}\left(\frac{G{M}_r}{r}\right)d{M}_r$   $=$ $-\underset{0}{\overset{R}{\int }}\left[\frac{d}{dr}\right(4\pi r^{3}P)-12\pi r^{2}P]dr+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$   $=$ $\underset{0}{\overset{R}{\int }}3\left[4\pi r^{2}P\right]dr-\underset{0}{\overset{R}{\int }}[d(3PV)]+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$   $=$ $3(\gamma -1)U_{\mathrm{i}\mathrm{n}\mathrm{t}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}-[3PV{]}_0^R.$

Pointers to Relevant Chapters

⓪ 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

① 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

② 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
④   Perturbation Theory	⑦   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 $0$ $=$ $\frac{d}{dr}\left[r^4\gamma P\frac{dx}{dr}\right]+[{\omega }^2\rho r^4+(3\gamma -4)r^3\frac{dP}{dr}]x$ [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, $R_0^2\frac{{\partial }^2\mathfrak{𝔊}}{\partial R^2}$ $=$ $-2a(\frac{R}{R_0}{)}^{-3}+(3-3\gamma )(2-3\gamma )b(\frac{R}{R_0}{)}^{1-3\gamma }+6c\left(\frac{R}{R_0}\right)$   $=$ $\left(\frac{R_0}{R}{)}^2\right[2W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}-3(\gamma -1)(2-3\gamma )U_{\mathrm{i}\mathrm{n}\mathrm{t}}+6P_{e}V].$ 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, $3(\gamma -1)U_{\mathrm{i}\mathrm{n}\mathrm{t}}$ $=$ $3P_{e}V-W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$ $\Rightarrow R^2[\frac{{\partial }^2\mathfrak{𝔊}}{\partial R^2}{]}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}}$ $=$ $2W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}-(2-3\gamma )[3P_{e}V-W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}]+6P_{e}V$   $=$ $(4-3\gamma )W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+3^2\gamma P_{e}V.$ Note the similarity with ⑥. Alternatively, recalling that, $3(\gamma -1)U_{\mathrm{i}\mathrm{n}\mathrm{t}}$ $=$ $2S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}},$ the conditions for virial equilibrium and stability, may be written respectively as, $3P_{e}V$ $=$ $2S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}$ $\Rightarrow R^2[\frac{{\partial }^2\mathfrak{𝔊}}{\partial R^2}{]}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}}$ $=$ $2W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}-2(2-3\gamma )S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}+2[2S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}+W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}]$   $=$ $4W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+6\gamma S_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}.$
⑤   Variational Principle
Multiply the LAWE through by $4\pi xdr$, and integrate over the volume of the configuration gives the, Governing Variational Relation $0$ $=$ $\underset{0}{\overset{R}{\int }}4\pi r^4\gamma P(\frac{dx}{dr}{)}^{2}dr-\underset{0}{\overset{R}{\int }}4\pi (3\gamma -4)r^3{x}^2(\frac{dP}{dr})dr$     $-4\pi \left[r^4\gamma Px\right(\frac{dx}{dr}){]}_0^R-\underset{0}{\overset{R}{\int }}4\pi {\omega }^2\rho r^4{x}^{2}dr.$   $=$ $\underset{0}{\overset{R}{\int }}{x}^2(\frac{d\ln x}{d\ln r}{)}^2\gamma 4\pi r^{2}Pdr-\underset{0}{\overset{R}{\int }}(3\gamma -4)x^2(-\frac{G{M}_r}{r})4\pi \rho r^{2}dr$     $+\left[\gamma 4\pi r^{3}P{x}^2\right(-\frac{d\ln x}{d\ln r}){]}_0^R-\underset{0}{\overset{R}{\int }}4\pi {\omega }^2\rho r^4{x}^{2}dr.$ 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, ${\omega }^2$ $=$ $\frac{\gamma (\gamma -1)\underset{0}{\overset{R}{\int }}{x}^2(\frac{d\ln x}{d\ln r}{)}^{2}d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}-\underset{0}{\overset{R}{\int }}(3\gamma -4)x^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+3^2\gamma x^2{P}_{e}V}{\underset{0}{\overset{R}{\int }}{x}^2{r}^{2}d{M}_r}$
⑥   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, ${\omega }^2\underset{0}{\overset{R}{\int }}{r}^{2}d{M}_r$ $\le$ $(4-3\gamma )W_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}+3^2\gamma P_{e}V.$

Bipolytropes

Stability Analysis:   Applicable to Bipolytropic Configurations
⑧   Variational Principle	⑩   Free-Energy Analysis of Stability
Governing Variational Relation $\left(\frac{2\pi }{3}\right){\sigma }_c^2\underset{0}{\overset{R^{*}}{\int }}(x{r}^{*}{)}^{2}d{M}_r^{*}$ $=$ ${\gamma }_c({\gamma }_c-1)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{*}-(3{\gamma }_c-4)\underset{0}{\overset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$     $+{\gamma }_e({\gamma }_e-1)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^2(\frac{d\ln x}{d\ln r^{*}}{)}^{2}d{U}_{\mathrm{i}\mathrm{n}\mathrm{t}}^{*}-(3{\gamma }_e-4)\underset{r_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}}{\overset{R^{*}}{\int }}{x}^{2}d{W}_{\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{v}}^{*}$     $+3^2({\gamma }_c-{\gamma }_e)x_i^2{P}_i^{*}{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}^{*}.$	As we have detailed in an accompanying discussion, the first derivative of the relevant free-energy expression is, $R\frac{\partial \mathfrak{𝔊}}{\partial R}$ $=$ $2S_{\mathrm{t}\mathrm{o}\mathrm{t}}+W_{\mathrm{t}\mathrm{o}\mathrm{t}},$ where, $S_{\mathrm{t}\mathrm{o}\mathrm{t}}\equiv S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+S_{\mathrm{e}\mathrm{n}\mathrm{v}}$     and     $W_{\mathrm{t}\mathrm{o}\mathrm{t}}\equiv W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{e}\mathrm{n}\mathrm{v}};$ and the second derivative of that free-energy function is, $R^2\frac{{\partial }^2\mathfrak{𝔊}}{\partial R^2}$ $=$ $2[W_{\mathrm{t}\mathrm{o}\mathrm{t}}+(3{\gamma }_c-2)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(3{\gamma }_e-2)S_{\mathrm{e}\mathrm{n}\mathrm{v}}].$ This stability criterion may be rewritten as, $[R^2\frac{{\partial }^2\mathfrak{𝔊}}{\partial R^2}{]}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}}$ $=$ $2[(3{\gamma }_c-4)S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(3{\gamma }_e-4)S_{\mathrm{e}\mathrm{n}\mathrm{v}}].$ 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, $\frac{S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}}{S_{\mathrm{e}\mathrm{n}\mathrm{v}}}$ $=$ $\frac{(3{\gamma }_e-4)}{(4-3{\gamma }_c)}.$ See the accompanying discussion. If — based for example on ⑦ — we make the reasonable assumption that, in equilibrium, the statements, $2S_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}=3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}-W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}$     and     $2S_{\mathrm{e}\mathrm{n}\mathrm{v}}=-3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}-W_{\mathrm{e}\mathrm{n}\mathrm{v}},$ hold separately, then we satisfy the virial equilibrium condition, namely, $0$ $=$ $2S_{\mathrm{t}\mathrm{o}\mathrm{t}}+W_{\mathrm{t}\mathrm{o}\mathrm{t}},$ and the second derivative of the relevant free-energy function can be rewritten as, $[R^2\frac{{\partial }^2\mathfrak{𝔊}}{\partial R^2}{]}_{\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}\mathrm{l}}$ $=$ $2(W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+W_{\mathrm{e}\mathrm{n}\mathrm{v}})+(3{\gamma }_c-2)(3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}-W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}})$     $+(3{\gamma }_e-2)(-3P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}-W_{\mathrm{e}\mathrm{n}\mathrm{v}})$   $=$ $3^2{P}_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}({\gamma }_c-{\gamma }_e)+(4-3{\gamma }_c)W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(4-3{\gamma }_e)W_{\mathrm{e}\mathrm{n}\mathrm{v}}.$ Note the similarity with ⑨ — temporarily, see this discussion.
⑨   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, $\left(\frac{2\pi }{3}\right){\sigma }_c^2\underset{0}{\overset{R}{\int }}{r}^{2}d{M}_r$ $\le$ $(4-3{\gamma }_c)W_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}+(4-3{\gamma }_e)W_{\mathrm{e}\mathrm{n}\mathrm{v}}+3^2({\gamma }_c-{\gamma }_e)P_i{V}_{\mathrm{c}\mathrm{o}\mathrm{r}\mathrm{e}}.$

See Also