User:Tohline/SphericallySymmetricConfigurations

Spherically Symmetric Configurations

Principal Governing Equations

If we assume that our self-gravitating configurations are 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 (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>) — in spherical coordinates (<math>r, \theta, \phi</math>) then setting to zero all derivatives that are taken with respect to the angular coordinates <math>\theta</math> and <math>\phi</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>

Structure

Equilibrium, spherically symmetric structures are obtained by searching for time-independent solutions to the above set of simplified governing equations. The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_r v_r = 0</math>.

END one table and BEGIN another!

Stability & Dynamics

Stability

SUMMARY: The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically.

Dynamics

SUMMARY: The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically. The equilibrium structure of a self-gravitating, non-rotating, uniform-density sphere can be described analytically.

Appendices

© 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