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Spherically Symmetric Configurations Synopsis (Using Style Sheet)

Whitworth's (1981) Isothermal Free-Energy Surface
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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} \, .

See Also

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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