User:Tohline/SSC/Synopsis StyleSheet

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 <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 \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>~0</math> <math>~=</math> <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, 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
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>~=</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>~=</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>~\leq</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>~=</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>~=</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>~=</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>~=</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>~0</math> <math>~=</math> <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.
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>~=</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>~\leq</math> <math>~ (4- 3\gamma) W_\mathrm{grav}+ 3^2 \gamma P_eV \, . </math>

See Also

© 2014 - 2021 by Joel E. Tohline |   H_Book Home   |   YouTube   | Appendices: | Equations | Variables | References | Ramblings | Images | myphys.lsu | ADS | 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