Difference between revisions of "User:Tohline/H Book"
(→Structure: Add chapter link to "polytropes") |
(→Spherically Symmetric Configurations: Add subsection to cover SSC Stability) |
||
| Line 27: | Line 27: | ||
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 [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, \varphi</math>) (see, for example, the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates]) 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... [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/PGE <more>] | 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 [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, \varphi</math>) (see, for example, the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates]) 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... [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/PGE <more>] | ||
===Structure=== | ===Structure:=== | ||
* <font color="darkblue">'''[http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies Solution Strategies]'''</font> | * <font color="darkblue">'''[http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies Solution Strategies]'''</font> | ||
| Line 36: | Line 36: | ||
** Isothermal sphere | ** Isothermal sphere | ||
===<font color="darkblue">'''[http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations | |||
===Stability:=== | |||
* <font color="darkblue">'''[http://www.vistrails.org/index.php/User:Tohline/SSC/Perturbations Solution Strategy]'''</font> | |||
* Example Solutions: | |||
** [http://www.vistrails.org/index.php?title=User:Tohline/SSC/UniformDensity Uniform-density sphere] | |||
** [http://www.vistrails.org/index.php/User:Tohline/SSC/Polytropes Polytropes] | |||
===<font color="darkblue">'''[http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations Dynamics]:'''</font>=== | |||
=Appendices= | =Appendices= | ||
{{LSU_HBook_footer}} | {{LSU_HBook_footer}} | ||
Revision as of 20:55, 6 February 2010
|
|---|
| | Tiled Menu | Tables of Content | Banner Video | Tohline Home Page | |
Preface from the original version of this HyperText Book (H_Book):
November 18, 1994
Much of our present, basic understanding of the structure, stability, and dynamical evolution of individual stars, short-period binary star systems, and the gaseous disks that are associated with numerous types of stellar systems (including galaxies) is derived from an examination of the behavior of a specific set of coupled, partial differential equations. These equations — most of which also are heavily utilized in studies of continuum flows in terrestrial environments — are thought to govern the underlying physics of all macroscopic "fluid" systems in astronomy. Although relatively simple in form, they prove to be very rich in nature... <more>
Context
- Virial Equations
Applications
Spherically Symmetric Configurations
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 (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>) — in spherical coordinates (<math>r, \theta, \varphi</math>) (see, for example, the Wikipedia discussion of integration and differentiation in spherical coordinates) 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... <more>
Structure:
- Solution Strategies
- Example Solutions:
- Uniform-density sphere
- Polytropes
- Zero-temperature White Dwarf
- Isothermal sphere
Stability:
- Solution Strategy
- Example Solutions:
Dynamics:
Appendices
|
|
|---|
|
© 2014 - 2021 by Joel E. Tohline |