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Spherically Symmetric Configurations (Part I)

Whitworth's (1981) Isothermal Free-Energy Surface
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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 ~(r, \theta, \varphi) then setting to zero all derivatives that are taken with respect to the angular coordinates ~\theta and ~\varphi. After making this simplification, our governing equations become,

Equation of Continuity

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


Euler Equation

\frac{dv_r}{dt} = - \frac{1}{\rho}\frac{dP}{dr} - \frac{d\Phi}{dr}


Adiabatic Form of the
First Law of Thermodynamics

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


Poisson Equation

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

See Also

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


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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