User:Tohline/SphericallySymmetricConfigurations - Revision history

Tohline: /* Structure */

2012-11-07T04:39:10Z

Structure

← Older revision		Revision as of 04:39, 7 November 2012
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	(We recognize the first of these expressions as being the statement of [~~http://www.vistrails.org/index.php/~~User:Tohline/PGE/ConservingMomentum#Time-independent_Behavior hydrostatic balance] appropriate for spherically symmetric configurations.)		(We recognize the first of these expressions as being the statement of [[User:Tohline/PGE/ConservingMomentum#Time-independent_Behavior\|hydrostatic balance]] appropriate for spherically symmetric configurations.)

	We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions — {{User:Tohline/Math/VAR_Pressure01}}(r), {{User:Tohline/Math/VAR_Density01}}(r), and {{User:Tohline/Math/VAR_NewtonianPotential01}}(r). As has been outlined in our discussion of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-independent_Problems supplemental relations for time-independent problems], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}. (See below.)		We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions — {{User:Tohline/Math/VAR_Pressure01}}(r), {{User:Tohline/Math/VAR_Density01}}(r), and {{User:Tohline/Math/VAR_NewtonianPotential01}}(r). As has been outlined in our discussion of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-independent_Problems supplemental relations for time-independent problems], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}. (See below.)

	===Solution Strategies===		===Solution Strategies===
	In an [~~http://www.vistrails.org/index.php/~~User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies accompanying discussion], we outline three different techniques that have been used to solve the above pair of differential equations analytically, or to set them up for numerical solution once a supplemental barotropic relation has been identified. In summary, these techniques lead to the following governing relations:		In an [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies\|accompanying discussion]], we outline three different techniques that have been used to solve the above pair of differential equations analytically, or to set them up for numerical solution once a supplemental barotropic relation has been identified. In summary, these techniques lead to the following governing relations:

Tohline: /* Principal Governing Equations */

2012-11-07T04:36:50Z

Principal Governing Equations

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	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	==Principal Governing Equations==		==Principal Governing Equations==
	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\|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">		<div align="center">

Tohline: Move title image

2012-11-07T04:32:51Z

Move title image

← Older revision		Revision as of 04:32, 7 November 2012
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	<!-- __FORCETOC__ will force the creation of a Table of Contents -->		<!-- __FORCETOC__ will force the creation of a Table of Contents -->
	<!-- __NOTOC__ will force TOC off -->		<!-- __NOTOC__ will force TOC off -->
			=Spherically Symmetric Configurations=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	~~=Spherically Symmetric Configurations=~~

	==Principal Governing Equations==		==Principal Governing Equations==
	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 [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,

Tohline: /* Technique #3 */ Summarize technique #3

2010-02-01T20:07:46Z

Technique #3: Summarize technique #3

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	====Technique #3====		====Technique #3====
	~~Integrating~~ the		This technique does not immediately reduce the number of equations nor the number unknown variables, but it does change one of the governing differential equations into an algebraic relation. The resulting pair of equations are,

			<div align="center">
			<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> ,
			</div>
			and
			<div align="center">
			<math>H + \Phi = \mathrm{constant}</math> ;
			</div>
			the unknown functions are {{User:Tohline/Math/VAR_Enthalpy01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}}.

	==Stability & Dynamics==		==Stability & Dynamics==

Tohline: /* Technique #2 */ Summarize technique #2

2010-02-01T19:59:16Z

Technique #2: Summarize technique #2

← Older revision		Revision as of 19:59, 1 February 2010
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	====Technique #2====		====Technique #2====
	~~Integrating~~ the		This technique results in the following second-order differential equation which depends only on the two unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}} and {{User:Tohline/Math/VAR_Density01}}:
			<div align="center">
			<math>\frac{1}{r^2} \frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- 4\pi G \rho</math> .
			</div>

	====Technique #3====		====Technique #3====

Tohline: /* Technique #1 */ Summarize technique #1

2010-02-01T19:56:39Z

Technique #1: Summarize technique #1

← Older revision		Revision as of 19:56, 1 February 2010
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	====Technique #1====		====Technique #1====
	~~Integrating~~ the		This technique results in the following integro-differential equation which only depends on the two unknown functions, {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}:
			<div align="center">
			{{User:Tohline/Math/EQ_SShydrostaticBalance01}}
			</div>
			where,
			<div align="center">
			<math>M_r \equiv \int_0^r 4\pi r^2 \rho dr</math> .
			</div>

	====Technique #2====		====Technique #2====

Tohline: /* Solution Strategies */ Abbreviate "solution strategies" subsection because information has been moved to separate page

2010-02-01T16:31:29Z

Solution Strategies: Abbreviate "solution strategies" subsection because information has been moved to separate page

← Older revision		Revision as of 16:31, 1 February 2010
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	===Solution Strategies===		===Solution Strategies===
	In an [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies accompanying discussion], we outline three different techniques that have been used to solve the above pair of differential equations analytically, or to set them up for numerical solution once a supplemental barotropic relation has been identified.		In an [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies accompanying discussion], we outline three different techniques that have been used to solve the above pair of differential equations analytically, or to set them up for numerical solution once a supplemental barotropic relation has been identified. In summary, these techniques lead to the following governing relations:

	When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where {{User:Tohline/Math/VAR_Density01}} is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,
	~~<div align="center">~~
	~~<math>dm_r = \rho dr \oint dS = r^2 \rho dr \int_0^\pi \sin\theta d\theta \int_0^{2\pi} d\varphi = 4\pi r^2 \rho dr</math> ,~~
	~~</div>~~
	where we have pulled from the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates] to define the spherical surface element <math>dS</math>. Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,
	~~<div align="center">~~
	~~<math>M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .~~
	~~</div>~~
	~~We can also state that,~~
	~~<div align="center">~~
	~~<math>\frac{dm_r}{dr} = 4\pi r^2 \rho </math> .~~
	~~</div>~~
	~~This differential relation is often identified as a statement of mass conservation that replaces the equation of continuity for spherically symmetric, static equilibrium structures.~~

	====Technique #1====		====Technique #1====
			Integrating the


			====Technique #2====
			Integrating the


			====Technique #3====
	Integrating the		Integrating the

Tohline: /* Solution Strategies */

2010-02-01T16:25:02Z

Solution Strategies

← Older revision		Revision as of 16:25, 1 February 2010
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	===Solution Strategies===		===Solution Strategies===
			In an [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies accompanying discussion], we outline three different techniques that have been used to solve the above pair of differential equations analytically, or to set them up for numerical solution once a supplemental barotropic relation has been identified.

	When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where {{User:Tohline/Math/VAR_Density01}} is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,		When attempting to solve the identified pair of simplified governing differential equations, it will be useful to note that, in a spherically symmetric configuration (where {{User:Tohline/Math/VAR_Density01}} is not a function of <math>\theta</math> or <math>\varphi</math>), the differential mass <math>dm_r</math> that is enclosed within a spherical shell of thickness <math>dr</math> is,
	<div align="center">		<div align="center">

Tohline: /* Solution Strategies */ reword last sentence

2010-02-01T16:01:04Z

Solution Strategies: reword last sentence

← Older revision		Revision as of 16:01, 1 February 2010
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	where we have pulled from the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates] to define the spherical surface element <math>dS</math>. Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,		where we have pulled from the [http://en.wikipedia.org/wiki/Spherical_coordinate_system#Integration_and_differentiation_in_spherical_coordinates Wikipedia discussion of integration and differentiation in spherical coordinates] to define the spherical surface element <math>dS</math>. Integrating from the center of the spherical configuration (<math>r=0</math>) out to some finite radius <math>r</math> that is still inside the configuration gives the mass enclosed within that radius, <math>M_r</math>; specifically,
	<div align="center">		<div align="center">
	<math>M_r \equiv \int_0^r dm_r = \int_0^4 4\pi r^2 \rho dr</math> .		<math>M_r \equiv \int_0^r dm_r = \int_0^r 4\pi r^2 \rho dr</math> .
	</div>		</div>
	We can also state that,		We can also state that,
	<div align="center">		<div align="center">
	<math>\frac{dm_r}{dr} = 4\pi r^2 \rho </math> ,		<math>\frac{dm_r}{dr} = 4\pi r^2 \rho </math> .
	</div>		</div>
	~~which~~ is often ~~referred to~~ as ~~the differential relation~~ that replaces the equation of continuity ~~as a statement of mass conservation in~~ spherically symmetric, static equilibrium structures.		This differential relation is often identified as a statement of mass conservation that replaces the equation of continuity for spherically symmetric, static equilibrium structures.

	====Technique #1====		====Technique #1====

Tohline: /* Structure */ minor wording changes

2010-02-01T15:56:23Z

Structure: minor wording changes

← Older revision		Revision as of 15:56, 1 February 2010
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	(We recognize the first of these expressions as being the statement of [http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingMomentum#Time-independent_Behavior hydrostatic balance] for spherically symmetric configurations.)		(We recognize the first of these expressions as being the statement of [http://www.vistrails.org/index.php/User:Tohline/PGE/ConservingMomentum#Time-independent_Behavior hydrostatic balance] appropriate for spherically symmetric configurations.)

	We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions — {{User:Tohline/Math/VAR_Pressure01}}(r), {{User:Tohline/Math/VAR_Density01}}(r), and {{User:Tohline/Math/VAR_NewtonianPotential01}}(r). As has been outlined in our discussion of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-independent_Problems supplemental relations for time-independent problems], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}.		We need one supplemental relation to close this set of equations because there are two equations, but three unknown functions — {{User:Tohline/Math/VAR_Pressure01}}(r), {{User:Tohline/Math/VAR_Density01}}(r), and {{User:Tohline/Math/VAR_NewtonianPotential01}}(r). As has been outlined in our discussion of [http://www.vistrails.org/index.php/User:Tohline/SR#Time-independent_Problems supplemental relations for time-independent problems], in the context of this H_Book we will close this set of equations by specifying a structural, barotropic relationship between {{User:Tohline/Math/VAR_Pressure01}} and {{User:Tohline/Math/VAR_Density01}}. (See below.)

	===Solution Strategies===		===Solution Strategies===
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	<math>\frac{dm_r}{dr} = 4\pi r^2 \rho </math> ,		<math>\frac{dm_r}{dr} = 4\pi r^2 \rho </math> ,
	</div>		</div>
	which is often referred to as the differential relation that replaces the equation of continuity as a statement of mass conservation in spherically symmetric, static equilibrium structures.		which is often referred to as the differential relation that replaces the equation of continuity as a statement of mass conservation in spherically symmetric, static equilibrium structures.

	====Technique #1====		====Technique #1====