SSCpt1/PGE

PGE for Spherically Symmetric Configurations

One-Dimensional PGEs

If the self-gravitating configuration that we wish to construct is spherically symmetric, then the coupled set of multidimensional, partial differential equations that serve as our principal governing equations can be simplified to a coupled set of one-dimensional, ordinary differential equations. This is accomplished by expressing each of the multidimensional spatial operators — gradient, divergence, and Laplacian — in spherical coordinates^† $(r,\theta ,\phi )$ then setting to zero all derivatives that are taken with respect to the angular coordinates $\theta$ and $\phi$. After making this simplification, our governing equations become,  
 
 

Equation of Continuity

$\frac{d\rho }{dt}+\rho \left[\frac{1}{r^2}\frac{d(r^2{v}_r)}{dr}\right]=0$

Euler Equation

$\frac{d{v}_r}{dt}=-\frac{1}{\rho }\frac{dP}{dr}-\frac{d\mathrm{\Phi }}{dr}$

Adiabatic Form of the
First Law of Thermodynamics

$\frac{dϵ}{dt}+P\frac{d}{dt}\left(\frac{1}{\rho }\right)=0$

Poisson Equation

$\frac{1}{r^2}\left[\frac{d}{dr}\right(r^2\frac{d\mathrm{\Phi }}{dr}\left)\right]=4\pi G\rho$

Footnotes

^†See, for example, the Wikipedia discussion of integration and differentiation in spherical coordinates.

See Also

• Part 2 of Spherically Symmetric Configurations: Structure — Solution Strategies
• Part 2 of Spherically Symmetric Configurations: Stability — Linearization of Governing Equations