User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies
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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
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