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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, ~\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 Analysis

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 R^{-1} + bR^{3-3\gamma}+ cR^3 \, .

Therefore, also,

~\frac{d\mathfrak{G}}{dR}

~=

~aR^{-2} +(3-3\gamma)bR^{2-3\gamma} + 3cR^2

 

~=

~\frac{1}{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 \, .

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

~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

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,

~\frac{d^2 \mathfrak{G}}{dR^2}

~=

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

 

~=

~\frac{1}{R^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{d^2\mathfrak{G}}{dR^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 \, .

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

~\approx

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

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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