Spherically Symmetric Configurations Synopsis (Using Style Sheet)
Structure
Tabular Overview
Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion — adiabatic index, <math>~\gamma</math>
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<math>~dV = 4\pi r^2 dr</math>
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and
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<math>~dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>
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<math>~W_\mathrm{grav}</math>
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<math>~=</math>
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<math>~- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>
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<math>~U_\mathrm{int}</math>
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<math>~=</math>
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<math>~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>
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| Equilibrium Structure
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| ① Detailed Force Balance
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③ Free-Energy Identification of Equilibria
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Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of
Hydrostatic Balance
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<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>
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for the radial density distribution, <math>~\rho(r)</math>.
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The Free-Energy is,
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<math>~\mathfrak{G}</math>
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<math>~=</math>
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<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>
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<math>~=</math>
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<math>~-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 \, .</math>
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Therefore, also,
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<math>~R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math>
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<math>~=</math>
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<math>~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</math>
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<math>~=</math>
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<math>~\frac{R_0}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>
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Equilibrium configurations exist at extrema of the free-energy function, that is, they are identified by setting <math>~d\mathfrak{G}/dR = 0</math>. Hence, equilibria are defined by the condition,
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<math>~0</math>
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<math>~=</math>
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<math>~W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>
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| ② Virial Equilibrium
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Multiply the hydrostatic-balance equation through by <math>~rdV</math> and integrate over the volume:
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<math>~0</math>
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<math>~=</math>
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<math>~-\int_0^R r\biggl(\frac{dP}{dr}\biggr)dV - \int_0^R r\biggl(\frac{GM_r \rho}{r^2}\biggr)dV</math>
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<math>~=</math>
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<math>~-\int_0^R 4\pi r^3 \biggl(\frac{dP}{dr}\biggr) dr - \int_0^R \biggl(\frac{GM_r}{r}\biggr)dM_r</math>
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<math>~=</math>
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<math>~-\int_0^R\biggl[ \frac{d}{dr}\biggl( 4\pi r^3P \biggr) - 12\pi r^2 P\biggr] dr + W_\mathrm{grav}</math>
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<math>~=</math>
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<math>~\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>
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<math>~=</math>
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<math>~3(\gamma-1)U_\mathrm{int} + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .</math>
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Pointers to Relevant Chapters
⓪ Background Material:
① Detailed Force Balance:
② Virial Equilibrium:
Stability
Isolated & Pressure-Truncated Configurations
| Stability Analysis: Applicable to Isolated & Pressure-Truncated Configurations
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| ④ Perturbation Theory
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⑦ Free-Energy Analysis of Stability
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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
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<math>~0</math>
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<math>~=</math>
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<math>~
\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
</math>
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[P00], Vol. II, §3.7.1, p. 174, Eq. (3.145)
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to find one or more radially dependent, radial-displacement eigenvectors, <math>~x \equiv \delta r/r</math>, along with (the square of) the corresponding oscillation eigenfrequency, <math>~\omega^2</math>.
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The second derivative of the free-energy function is,
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<math>~R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
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<math>~=</math>
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<math>~
-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)
</math>
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<math>~=</math>
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<math>~\biggl(\frac{R_0}{R} \biggr)^2\biggl[
2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V
\biggr] \, .
</math>
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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,
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<math>~3(\gamma-1)U_\mathrm{int}</math>
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<math>~=</math>
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<math>~3P_e V - W_\mathrm{grav} </math>
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<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
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<math>~=</math>
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<math>~2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav} \biggr] + 6P_e V
</math>
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<math>~=</math>
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<math>~(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, .
</math>
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Note the similarity with ⑥.
Alternatively, recalling that,
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<math>~3(\gamma - 1)U_\mathrm{int}</math>
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<math>~=</math>
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<math>~2S_\mathrm{therm} \, ,
</math>
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the conditions for virial equilibrium and stability, may be written respectively as,
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<math>~3P_e V</math>
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<math>~=</math>
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<math>~ 2S_\mathrm{therm}+ W_\mathrm{grav} </math>
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<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>
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<math>~=</math>
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<math>~
2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav} \biggr]
</math>
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<math>~=</math>
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<math>~
4W_\mathrm{grav} + 6\gamma S_\mathrm{therm} \, .
</math>
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| ⑤ Variational Principle
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Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the,
Governing Variational Relation
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<math>~0</math>
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<math>~=</math>
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<math>~
\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
</math>
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<math>~
- 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 \, .
</math>
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<math>~=</math>
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<math>~
\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
</math>
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<math>~
+ \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 \, .
</math>
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Now, by setting <math>~(d\ln x/d\ln r)_{r=R} = -3</math>, we can ensure that the pressure fluctuation is zero and, hence, <math>~P = P_e</math> at the surface, in which case this relation becomes,
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<math>~\omega^2</math>
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<math>~=</math>
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<math>~
\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}
</math>
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| ⑥ Approximation: Homologous Expansion/Contraction
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If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
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<math>~\omega^2 \int_0^R r^2 dM_r</math>
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<math>~\leq</math>
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<math>~
(4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, .
</math>
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Bipolytropes
| Stability Analysis: Applicable to Bipolytropic Configurations
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| ⑧ Variational Principle
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⑩ Free-Energy Analysis of Stability
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Governing Variational Relation
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<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^{R^*} (x r^*)^2 dM_r^* </math>
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<math>~=</math>
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<math>~
\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}
</math>
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<math>~
+ ~\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}
</math>
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<math>~
+ ~3^2(\gamma_c - \gamma_e) x_i^2 P_i^* V_\mathrm{core}^* \, .
</math>
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As we have detailed in an accompanying discussion, the first derivative of the relevant free-energy expression is,
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<math>~R ~\frac{\partial \mathfrak{G}}{\partial R}</math>
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<math>~=</math>
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<math>~
2S_\mathrm{tot} + W_\mathrm{tot}
\, ,
</math>
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where,
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<math>~S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math>
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and
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<math>~W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math>
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and the second derivative of that free-energy function is,
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<math>~R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>
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<math>~=</math>
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<math>~2\biggl[
W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env}
\biggr] \, .
</math>
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This stability criterion may be rewritten as,
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<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
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<math>~=</math>
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<math>~
2[(3\gamma_c -4) S_\mathrm{core}
+ (3\gamma_e -4) S_\mathrm{env} ] \, .
</math>
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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,
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<math>~\frac{S_\mathrm{core}}{S_\mathrm{env}}</math>
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<math>~=</math>
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<math>~
\frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, .
</math>
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See the accompanying discussion.
If — based for example on ⑦ — we make the reasonable assumption that, in equilibrium, the statements,
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<math>~2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>
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and
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<math>~2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>
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hold separately, then we satisfy the virial equilibrium condition, namely,
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<math>~0</math>
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<math>~=</math>
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<math>~2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math>
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and the second derivative of the relevant free-energy function can be rewritten as,
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<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>
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<math>~=</math>
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<math>~
2(W_\mathrm{core} + W_\mathrm{env})
+ (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core})
</math>
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<math>~
+ (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env})
</math>
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<math>~=</math>
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<math>~
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} \, .
</math>
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Note the similarity with ⑨ — temporarily, see this discussion.
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| ⑨ Approximation: Homologous Expansion/Contraction
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If we guess that radial oscillations about the equilibrium state involve purely homologous expansion/contraction, then the radial-displacement eigenfunction is, <math>~x</math> = constant, and the governing variational relation gives,
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<math>~\biggl( \frac{2\pi}{3}\biggr)\sigma_c^2 \int_0^R r^2 dM_r</math>
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<math>~\leq</math>
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<math>~
(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} \, .
</math>
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See Also
