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</math>
<math>~U_\mathrm{int}</math>	<math>~=</math>	<math>~\frac{1}{(\gamma -1)} \int_0^R 4\pi r^2 P dr</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

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>

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>.

User:Tohline/SSC/Synopsis