SSCpt1/PGE - Revision history

174.64.8.247: /* See Also */

2021-08-01T20:34:12Z

Jet53man: /* See Also */

2021-07-28T04:00:56Z

Jet53man: /* PGE for Spherically Symmetric Configurations */

2021-07-28T03:59:25Z

PGE for Spherically Symmetric Configurations

← Older revision		Revision as of 23:59, 27 July 2021
Line 5:		Line 5:
	{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"		{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
	\|-		\|-
	! style="height: 125px; width: 125px; background-color:~~white~~;" \|		! style="height: 125px; width: 125px; background-color:lightgreen;" \|
	<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations\|One-Dimensional<br />PGEs]]</font>		<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations\|One-Dimensional<br />PGEs]]</font>
	\|}		\|}

Jet53man: /* PGE for Spherically Symmetric Configurations */

2021-07-28T03:58:30Z

PGE for Spherically Symmetric Configurations

← Older revision		Revision as of 23:58, 27 July 2021
Line 3:		Line 3:

	=PGE for Spherically Symmetric Configurations=		=PGE for Spherically Symmetric Configurations=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:white;" \|
			<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations\|One-Dimensional<br />PGEs]]</font>
			\|}
	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 [[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,		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 [[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,
			<br />
			<br />
			<br />

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

Jet53man: /* PGE for Spherically Symmetric Configurations */

2021-07-12T19:01:04Z

PGE for Spherically Symmetric Configurations

← Older revision		Revision as of 15:01, 12 July 2021
Line 3:		Line 3:

	=PGE for Spherically Symmetric Configurations=		=PGE for 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 [[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,		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 [[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">

Jet53man at 18:59, 12 July 2021

2021-07-12T18:59:40Z

← Older revision		Revision as of 14:59, 12 July 2021
Line 3:		Line 3:

	=PGE for Spherically Symmetric Configurations=		=PGE for 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 [[~~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,		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 [[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">
Line 20:		Line 20:
	<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />		<font color="#770000">'''First Law of Thermodynamics'''</font></span><br />

	{{~~User~~:~~Tohline/~~Math/EQ_FirstLaw02}}		{{ Template:Math/EQ_FirstLaw02 }}


Line 28:		Line 28:
	</div>		</div>

	=~~See Also~~=		=Footnotes=
	* 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]]

	<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].		<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=
			<ul>
			<li>
			Part 2 of ''Spherically Symmetric Configurations'': Structure — [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|Solution Strategies]]
			</li>
			<li>
			Part 2 of ''Spherically Symmetric Configurations'': Stability — [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Linearization of Governing Equations]]
			</li>
			</ul>


	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: Created page with "FORCETOC =PGE for Spherically Symmetric Configurations= If the self-gravitating configuration that we wish to construct is spherica..."

2021-07-12T18:55:30Z

Created page with "__FORCETOC__  =PGE for Spherically Symmetric Configurations= If the self-gravitating configuration that we wish to construct is spherica..."

New page

__FORCETOC__


=PGE for 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 [[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&dagger; <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">
'''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>

=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'': Stability — [[User:Tohline/SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29|Linearization of Governing Equations]]

&dagger;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].

{{ SGFfooter }}

← Older revision		Revision as of 16:34, 1 August 2021
Line 48:		Line 48:
	Part 2 of ''Spherically Symmetric Configurations'': Stability — [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Linearization of Governing Equations]]		Part 2 of ''Spherically Symmetric Configurations'': Stability — [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Linearization of Governing Equations]]
	</li>		</li>
			<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
			<li>[[SSC/Index\|Spherically Symmetric Configurations (SSC) Index]]</li>
	</ul>		</ul>


	{{ SGFfooter }}		{{ SGFfooter }}

← Older revision		Revision as of 00:00, 28 July 2021
Line 43:		Line 43:
	<ul>		<ul>
	<li>		<li>
	Part 2 of ''Spherically Symmetric Configurations'': Structure — [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|Solution Strategies]]		Part 2 of ''Spherically Symmetric Configurations'': Structure — [[SSCpt2/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|Solution Strategies]]
	</li>		</li>
	<li>		<li>
	Part 2 of ''Spherically Symmetric Configurations'': Stability — [[~~User:Tohline/~~SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Linearization of Governing Equations]]		Part 2 of ''Spherically Symmetric Configurations'': Stability — [[SSC/Perturbations#Spherically_Symmetric_Configurations_.28Stability_.E2.80.94_Part_II.29\|Linearization of Governing Equations]]
	</li>		</li>
	</ul>		</ul>

SSCpt1/PGE - Revision history

174.64.8.247: /* See Also */

Jet53man: /* See Also */

Jet53man: /* PGE for Spherically Symmetric Configurations */

Jet53man: /* PGE for Spherically Symmetric Configurations */

Jet53man: /* PGE for Spherically Symmetric Configurations */

Jet53man at 18:59, 12 July 2021

Jet53man: Created page with "__FORCETOC__ =PGE for Spherically Symmetric Configurations= If the self-gravitating configuration that we wish to construct is spherica..."

Jet53man: Created page with "FORCETOC =PGE for Spherically Symmetric Configurations= If the self-gravitating configuration that we wish to construct is spherica..."