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Spherically Symmetric Configurations Synopsis

Whitworth's (1981) Isothermal Free-Energy Surface
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Spherically Symmetric Configurations that undergo Adiabatic Compression/Expansion — adiabatic index, <math>~\gamma</math>

<math>~dV = 4\pi r^2 dr</math>

and

   <math>~dM_r = \rho dV ~~~\Rightarrow ~~~M_r = 4\pi \int_0^r \rho r^2 dr</math>

<math>~W_\mathrm{grav}</math>

<math>~=</math>

<math>~- \int_0^R \biggl(\frac{GM_r}{r}\biggr) dM_r ~~ \propto ~~ R^{-1}</math>

<math>~U_\mathrm{int}</math>

<math>~=</math>

<math>~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr ~~ \propto ~~ R^{3-3\gamma}</math>

Equilibrium Structure

Detailed Force Balance

Free-Energy Analysis

Given a barotropic equation of state, <math>~P(\rho)</math>, solve the equation of

Hydrostatic Balance

LSU Key.png

<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>

for the radial density distribution, <math>~\rho(r)</math>.

The Free-Energy is,

<math>~\mathfrak{G}</math>

<math>~=</math>

<math>~W_\mathrm{grav} + U_\mathrm{int} + P_eV</math>

 

<math>~=</math>

<math>~-a R^{-1} + bR^{3-3\gamma}+ cR^3 \, .</math>

Therefore, also,

<math>~\frac{d\mathfrak{G}}{dR}</math>

<math>~=</math>

<math>~aR^{-2} +(3-3\gamma)bR^{2-3\gamma} + 3cR^2</math>

 

<math>~=</math>

<math>~\frac{1}{R}\biggl[ -W_\mathrm{grav} - 3(\gamma-1)U_\mathrm{int} + 3P_eV\biggr]</math>

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,

<math>~0</math>

<math>~=</math>

<math>~W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>

Virial Equilibrium

Multiply the hydrostatic-balance equation through by <math>~rdV</math> and integrate over the volume:

<math>~0</math>

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<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>

 

<math>~=</math>

<math>~\int_0^R 3\biggl[ 4\pi r^2 P \biggr]dr - \int_0^R \biggl[ d(3PV)\biggr] + W_\mathrm{grav}</math>

 

<math>~=</math>

<math>~3(\gamma-1)U_\mathrm{int} + W_\mathrm{grav} - \biggl[ 3PV \biggr]_0^R \, .</math>

Stability Analysis

Perturbation Theory

Free-Energy Analysis

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

<math>~0</math>

<math>~=</math>

<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>

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>.

The second derivative of the free-energy function is,

<math>~\frac{d^2 \mathfrak{G}}{dR^2}</math>

<math>~=</math>

<math>~ -2aR^{-3} + (3-3\gamma)(2-3\gamma)b R^{1-3\gamma} + 6cR </math>

 

<math>~=</math>

<math>~\frac{1}{R^2}\biggl[ 2W_\mathrm{grav} - 3(\gamma-1)(2-3\gamma)U_\mathrm{int} + 6P_e V \biggr] \, . </math>

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,

<math>~3(\gamma-1)U_\mathrm{int}</math>

<math>~=</math>

<math>~3P_e V - W_\mathrm{grav} </math>

<math>~\Rightarrow~~~ R^2 \biggl[\frac{d^2\mathfrak{G}}{dR^2}\biggr]_\mathrm{equil}</math>

<math>~=</math>

<math>~2W_\mathrm{grav} - (2-3\gamma)\biggl[3P_e V - W_\mathrm{grav} \biggr] + 6P_e V </math>

 

<math>~=</math>

<math>~(4-3\gamma)W_\mathrm{grav} + 3^2\gamma P_e V \, . </math>

Variational Principle

Multiply the LAWE through by <math>~4\pi x dr</math>, and integrate over the volume of the configuration gives the,

Governing Variational Relation

<math>~0</math>

<math>~=</math>

<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>

 

 

<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>

 

<math>~=</math>

<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>

 

 

<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>

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,

<math>~\omega^2</math>

<math>~=</math>

<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>

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, <math>~x</math> = constant, and the governing variational relation gives,

<math>~\omega^2 \int_0^R r^2 dM_r</math>

<math>~\approx</math>

<math>~ (4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, . </math>

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation