Spherically Symmetric Configurations Synopsis (Using Style Sheet)

Structure

Tabular Overview

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

① Detailed Force Balance

③ Free-Energy Identification of Equilibria

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

Hydrostatic Balance

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 \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>

Therefore, also,

<math>~R_0 ~\frac{\partial\mathfrak{G}}{\partial R}</math>	<math>~=</math>	<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>
	<math>~=</math>	<math>~\frac{R_0}{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>~W_\mathrm{grav} + 3(\gamma-1)U_\mathrm{int} - 3P_eV\, .</math>

② 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>

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, 2^nd-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

<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>
[P00], Vol. II, §3.7.1, p. 174, Eq. (3.145)

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>~R_0^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>	<math>~=</math>	<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>
	<math>~=</math>	<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>

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{\partial^2\mathfrak{G}}{\partial R^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>

Note the similarity with ⑥.

Alternatively, recalling that,

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

<math>~2S_\mathrm{therm} \, , </math>

the conditions for virial equilibrium and stability, may be written respectively as,

<math>~3P_e V</math>	<math>~=</math>	<math>~ 2S_\mathrm{therm}+ W_\mathrm{grav} </math>
<math>~\Rightarrow~~~ R^2 \biggl[\frac{\partial^2\mathfrak{G}}{\partial R^2}\biggr]_\mathrm{equil}</math>	<math>~=</math>	<math>~ 2W_\mathrm{grav} - 2(2-3\gamma)S_\mathrm{therm} + 2 \biggl[ 2S_\mathrm{therm}+ W_\mathrm{grav} \biggr] </math>
	<math>~=</math>	<math>~ 4W_\mathrm{grav} + 6\gamma S_\mathrm{therm} \, . </math>

⑤ 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>~ \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>

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

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

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

Bipolytropes

Stability Analysis: Applicable to Bipolytropic Configurations

④ Perturbation Theory

⑨ 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

<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>
[P00], Vol. II, §3.7.1, p. 174, Eq. (3.145)

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>.

As we have detailed in an accompanying discussion, the first derivative of the relevant free-energy expression is,

<math>~R ~\frac{\partial \mathfrak{G}}{\partial R}</math>

<math>~ 2S_\mathrm{tot} + W_\mathrm{tot} \, , </math>

where,

<math>~S_\mathrm{tot} \equiv S_\mathrm{core} + S_\mathrm{env}</math>

and

<math>~W_\mathrm{tot} \equiv W_\mathrm{core} + W_\mathrm{env} \, ;</math>

and the second derivative of that free-energy function is,

<math>~R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2}</math>

<math>~2\biggl[ W_\mathrm{tot} + (3\gamma_c - 2) S_\mathrm{core} + (3\gamma_e-2)S_\mathrm{env} \biggr] \, . </math>

This stability criterion may be rewritten as,

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

<math>~ 2[(3\gamma_c -4) S_\mathrm{core} + (3\gamma_e -4) S_\mathrm{env} ] \, . </math>

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,

<math>~\frac{S_\mathrm{core}}{S_\mathrm{env}}</math>

<math>~ \frac{(3\gamma_e - 4)}{(4 - 3\gamma_c)} \, . </math>

See the accompanying discussion.

If we make the reasonable assumption that, in equilibrium, the statements,

<math>~2S_\mathrm{core} = 3P_i V_\mathrm{core} - W_\mathrm{core}</math>

and

<math>~2S_\mathrm{env} = - 3P_i V_\mathrm{core} - W_\mathrm{env} \, ,</math>

hold separately, then we satisfy the virial equilibrium condition, namely,

<math>~2S_\mathrm{tot} + W_\mathrm{tot} \, ,</math>

and the second derivative of the relevant free-energy function can be rewritten as,

<math>~\biggl[ R^2 ~\frac{\partial^2 \mathfrak{G}}{\partial R^2} \biggr]_\mathrm{equil}</math>	<math>~=</math>	<math>~ 2(W_\mathrm{core} + W_\mathrm{env}) + (3\gamma_c - 2) (3P_i V_\mathrm{core} - W_\mathrm{core}) </math>
		<math>~ + (3\gamma_e-2)(-3P_i V_\mathrm{core} - W_\mathrm{env}) </math>
	<math>~=</math>	<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>

Note the similarity with ⑧ — temporarily, see this discussion.