User:Tohline/SSC/Synopsis

Spherically Symmetric Configurations Synopsis

Spherically Symmetric Configurations
<math>~dV = 4\pi r^2 dr</math>    and     <math>~dM_r</math> <math>~=~</math> <math>\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>
Detailed Force Balance	Virial Equilibrium		Free-Energy Analysis
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>.	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 4\pi r^3 dP + W_\mathrm{grav}</math>   <math>~=</math> <math>~\int_0^R 4\pi \biggl[ 3r^2 P dr - d(r^3P)\biggr] + W_\mathrm{grav}</math>   <math>~=</math> <math>~\int_0^R 3\biggl[ 4\pi r^2 P dr \biggr] - \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>		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>

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