User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies

This page is intended to accompany and extend our discussion of Spherically Symmetric Configurations.

SUMMARY: The pair of simplified governing differential equations that must be solved in concert with one another to determine the equilibrium structure of spherically symmetric, self-gravitating configurations is:

Hydrostatic Balance

<math>\frac{1}{\rho}\frac{dP}{dr} =- \frac{d\Phi}{dr} </math> ,

Poisson Equation

<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math>

Solution Strategies

When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where <math>~\rho</math> is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,

<math>dm_r = \rho dr \oint dS = r^2 \rho dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\varphi = 4\pi r^2 \rho dr</math> ,

where we have pulled from the Wikipedia discussion of integration and differentiation in spherical coordinates to define the spherical surface element <math>dS</math>. Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,

<math>M_r \equiv \int_0^r dm_r = \int_0^4 4\pi r^2 \rho dr</math> .

We can also state that,

<math>\frac{dm_r}{dr} = 4\pi r^2 \rho </math> ,

which is often referred to as the differential relation that replaces the equation of continuity as a statement of mass conservation in spherically symmetric, static equilibrium structures.

Technique #1

Integrating the

© 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