Difference between revisions of "User:Tohline/SphericallySymmetricConfigurations"
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=Spherically Symmetric | =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 [http://www.vistrails.org/index.php/User:Tohline/PGE 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, | |||
<div align="center"> | |||
<math>\nabla \rightarrow \frac{d}{dr}</math><br /> | |||
<math>\nabla\cdot \rightarrow \frac{1}{r}\frac{d}{dr}</math><br /> | |||
<math>\nabla \rightarrow \frac{d}{dr}</math> | |||
</div> | |||
==Summaries== | |||
<div align="right"> | <div align="right"> | ||
Revision as of 22:58, 31 January 2010
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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
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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. |
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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. |
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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
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