# User:Tohline/SphericallySymmetricConfigurations/PGE

# Spherically Symmetric Configurations (Part I)

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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^{†} <math>~(r, \theta, \varphi)</math> then setting to zero all derivatives that are taken with respect to the angular coordinates <math>~\theta</math> and <math>~\varphi</math>. After making this simplification, our governing equations become,

**Equation of Continuity**

<math>\frac{d\rho}{dt} + \rho \biggl[\frac{1}{r^2}\frac{d(r^2 v_r)}{dr} \biggr] = 0 </math>

**Euler Equation**

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

Adiabatic Form of the

**First Law of Thermodynamics**

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</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>

# See Also

- Part II of
*Spherically Symmetric Configurations*: Structure — Solution Strategies - Part II of
*Spherically Symmetric Configurations*: Stability — Linearization of Governing Equations

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

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