User:Tohline/SphericallySymmetricConfigurations/PGE - Revision history

Tohline at 20:17, 18 July 2015

2015-07-18T20:17:23Z

← Older revision		Revision as of 20:17, 18 July 2015
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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 />
	</div>		</div>

	<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].

	=See Also=		=See Also=
	* Part II of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|Solution Strategies]]		* Part II of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|Solution Strategies]]
	* Part II of ''Spherically Symmetric Configurations'': Stability — [[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 — [[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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Tohline: /* Spherically Symmetric Configurations (Part I) */ Move reference to wikipedia discussion to footnote "See Also" subsection

2015-07-18T20:16:42Z

Spherically Symmetric Configurations (Part I): Move reference to wikipedia discussion to footnote "See Also" subsection

← Older revision		Revision as of 20:16, 18 July 2015
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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\|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,		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 — gradient, divergence, and Laplacian — 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,

	<div align="center">		<div align="center">
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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 />
	</div>		</div>

			<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].

	=See Also=		=See Also=

Tohline: /* Spherically Symmetric Configurations (Part I) */

2012-11-07T18:54:55Z

Spherically Symmetric Configurations (Part I)

← Older revision		Revision as of 18:54, 7 November 2012
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	=Spherically Symmetric Configurations (Part I)=		=Spherically Symmetric Configurations (Part I)=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	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 — 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,		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 — 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,

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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 />
	</div>		</div>


	=See Also=		=See Also=

Tohline: /* See Also */ Add detailed pointer

2012-11-07T17:16:55Z

Tohline: /* See Also */ Add link to Part II of stability

2012-11-07T17:15:04Z

See Also: Add link to Part II of stability

← Older revision		Revision as of 17:15, 7 November 2012
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	=See Also=		=See Also=
	* Part II of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies\|Solution Strategies]]		* Part II of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies\|Solution Strategies]]
			* Part II of ''Spherically Symmetric Configurations'': Stability — [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Linearization of Governing Equations]]


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Tohline: Force table of contents

2012-11-07T16:49:26Z

Force table of contents

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	=Spherically Symmetric Configurations (Part I)=		=Spherically Symmetric Configurations (Part I)=

Tohline: Move title image

2012-11-07T04:20:02Z

Move title image

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			=Spherically Symmetric Configurations (Part I)=
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	~~=Spherically Symmetric Configurations (Part I)=~~

	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 — 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,		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 — 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,

Tohline: linnk modifications

2010-04-03T13:53:31Z

linnk modifications

← Older revision		Revision as of 13:53, 3 April 2010
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	=Spherically Symmetric Configurations (Part I)=		=Spherically Symmetric Configurations (Part I)=

	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,		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 — 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,

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	=See Also=		=See Also=
	* Part II of ''Spherically Symmetric Configurations'': Structure — [~~http://www.vistrails.org/index.php/~~User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies Solution Strategies]		* Part II of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies\|Solution Strategies]]



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Tohline: Add "see also" section

2010-02-01T21:01:28Z

Add "see also" section

← Older revision		Revision as of 21:01, 1 February 2010
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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 />
	</div>		</div>


			=See Also=
			* Part II of ''Spherically Symmetric Configurations'': Structure — [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies Solution Strategies]



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Tohline: Create "Part I" of SSC

2010-02-01T20:51:53Z

Create "Part I" of SSC

New page

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=Spherically Symmetric Configurations (Part I)=

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,

<div align="center">
'''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''' 

{{User:Tohline/Math/EQ_FirstLaw02}}

'''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> 
</div>

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	=See Also=		=See Also=
	* Part II of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies\|Solution Strategies]]		* Part II of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|Solution Strategies]]
	* Part II of ''Spherically Symmetric Configurations'': Stability — [[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 — [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Linearization of Governing Equations]]


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