Difference between revisions of "User:Tohline/SphericallySymmetricConfigurations/PGE"

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=Spherically Symmetric Configurations (Part I)=
=Spherically Symmetric Configurations (Part I)=
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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 [[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 &#8212; gradient (<math>\nabla</math>), divergence (<math>\nabla\cdot</math>), and Laplacian (<math>\nabla^2</math>) &#8212; 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,
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 [[User:Tohline/PGE#Principal_Governing_Equations|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 &#8212; gradient, divergence, and Laplacian &#8212; in spherical coordinates<sup>&dagger;</sup> <math>~(r, \theta, \varphi)</math> 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,


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<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math><br />
<math>\frac{1}{r^2} \biggl[\frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) \biggr] = 4\pi G \rho </math><br />
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=See Also=
=See Also=
* Part II of ''Spherically Symmetric Configurations'':  Structure &#8212; [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Solution Strategies]]
* Part II of ''Spherically Symmetric Configurations'':  Structure &#8212; [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29|Solution Strategies]]
* Part II of ''Spherically Symmetric Configurations'':  Stability &#8212; [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Linearization of Governing Equations]]
* Part II of ''Spherically Symmetric Configurations'':  Stability &#8212; [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Linearization of Governing Equations]]
<sup>&dagger;</sup>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].




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Latest revision as of 20:17, 18 July 2015

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 <math>~(r, \theta, \varphi)</math> 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,

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>

See Also

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


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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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