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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