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</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 <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>		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>.

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