User:Tohline/SSC/Synopsis

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Spherically Symmetric Configurations Synopsis

Whitworth's (1981) Isothermal Free-Energy Surface
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For 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

Globalization

Free-Energy Analysis

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

Hydrostatic Balance

LSU Key.png

<math>~\frac{dP}{dr} = - \frac{GM_r \rho}{r^2}</math>

for the radial density distribution, <math>~\rho(r)</math>.

 

 


See Also

Whitworth's (1981) Isothermal Free-Energy Surface

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