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,

<math>\nabla \rightarrow \frac{d}{dr}</math>
<math>\nabla\cdot \rightarrow \frac{1}{r}\frac{d}{dr}</math>
<math>\nabla \rightarrow \frac{d}{dr}</math>

Summaries

Structure	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.
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