User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies - Revision history

Tohline: /* Technique 3 */

2017-04-10T21:58:58Z

Technique 3

← Older revision		Revision as of 21:58, 10 April 2017
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	giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}} to one another.		giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}} to one another.


			<b>Self-Consistent Field (SCF) Technique:</b>
			* Spherical polytropes —
			** [[User:Tohline/SSC/Structure/Polytropes#Tabulated_Properties\|Some tabulated global properties]].
			** Outline of [[User:Tohline/SSC/Structure/Polytropes#HSCF_Technique\|our implementation of the HSCF scheme]].

	=See Also=		=See Also=

Tohline: /* Technique 2 */

2017-04-10T21:55:57Z

Technique 2

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	that is, a single second-order governing differential equation which depends only on the two unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}} and {{User:Tohline/Math/VAR_Density01}}.		that is, a single second-order governing differential equation which depends only on the two unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}} and {{User:Tohline/Math/VAR_Density01}}.

	<b>~~Examples~~:</b>		<b>Numerical integration examples:</b>
	* [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|~~Isothermal sphere~~]] — ~~outline~~ of our numerical integration scheme.		* Isothermal sphere —
			** [[User:Tohline/SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) tabulation]].
			** Tabulation by [http://adsabs.harvard.edu/abs/1949ApJ...109..551C Chandrasekhar & Wares (1949, ApJ, 109, 551)].
			** Outline of [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|our numerical integration scheme]].
			* Spherical polytropes —
			** [[User:Tohline/SSC/Structure/Polytropes#Tabulated_Properties\|Some tabulated global properties]].
			** Outline of [[User:Tohline/SSC/Structure/Polytropes#Straight_Numerical_Integration\|our numerical integration scheme]].

	===Technique 3===		===Technique 3===

Tohline: /* Technique 2 */

2017-04-10T21:43:41Z

Technique 2

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	If we multiply this expression through by <math>~r^2</math> then differentiate it with respect to <math>~r</math>, we obtain,		If we multiply this expression through by <math>~r^2</math> then differentiate it with respect to <math>~r</math>, we obtain,
	<div align="center">		<div align="center">
	<math>\frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr)</math> ~~\, ,~~		<math>\frac{d}{dr}\biggl( r^2 \frac{dH}{dr} \biggr) =- \frac{d}{dr} \biggl( r^2 \frac{d\Phi}{dr} \biggr) \, ,</math>
	</div>		</div>

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	that is, a single second-order governing differential equation which depends only on the two unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}} and {{User:Tohline/Math/VAR_Density01}}.		that is, a single second-order governing differential equation which depends only on the two unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}} and {{User:Tohline/Math/VAR_Density01}}.

			<b>Examples:</b>
			* [[User:Tohline/SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|Isothermal sphere]] — outline of our numerical integration scheme.

	===Technique 3===		===Technique 3===

Tohline: Clean up text to match current standards of H_Book

2015-07-18T20:57:23Z

Clean up text to match current standards of H_Book

Show changes

Tohline: /* Technique 3 */ add few words to last sentence

2012-11-07T21:48:21Z

Technique 3: add few words to last sentence

← Older revision		Revision as of 21:48, 7 November 2012
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	<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> ,		<math>\frac{1}{r^2} \frac{d }{dr} \biggl( r^2 \frac{d \Phi}{dr} \biggr) = 4\pi G \rho </math> ,
	</div>		</div>
	giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}}		giving us two equations (one algebraic and the other a <math>2^\mathrm{nd}</math>-order ODE) that relate the three unknown functions, {{User:Tohline/Math/VAR_Enthalpy01}}, {{User:Tohline/Math/VAR_Density01}}, and {{User:Tohline/Math/VAR_NewtonianPotential01}} to one another.

	=See Also=		=See Also=

Tohline: /* Spherically Symmetric Configurations (Part II) */ make local links actually local

2012-11-07T21:40:43Z

Spherically Symmetric Configurations (Part II): make local links actually local

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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 [[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==

Tohline: /* Spherically Symmetric Configurations (Part II) */ Detailed link in first paragraph

2012-11-07T17:10:54Z

Spherically Symmetric Configurations (Part II): Detailed link in first paragraph

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	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	[[Image:LSU_Structure_still.gif\|74px\|left]]		[[Image:LSU_Structure_still.gif\|74px\|left]]
	Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [~~http://www.vistrails.org/index.php/~~User:Tohline/SphericallySymmetricConfigurations/PGE identified set of simplified governing equations]. The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_r v_r = 0</math>. After setting the radial velocity, <math>v_r</math>, and all time-derivatives to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>3^\mathrm{rd}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> (Euler) and <math>4^\mathrm{th}</math> give, respectively,		Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29\|identified set of simplified governing equations]]. The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_r v_r = 0</math>. After setting the radial velocity, <math>v_r</math>, and all time-derivatives to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>3^\mathrm{rd}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> (Euler) and <math>4^\mathrm{th}</math> give, respectively,

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

Tohline: /* See Also */ Correct detailed link

2012-11-07T17:07:56Z

Tohline: /* See Also */ Add more detailed link pointer

2012-11-07T17:06:08Z

See Also: Add more detailed link pointer

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	=See Also=		=See Also=
	* Part I of ''Spherically Symmetric Configurations'': [~~http://www.vistrails.org/index.php/~~User:Tohline/SphericallySymmetricConfigurations/~~PGE~~ Simplified Governing Equations]		* Part I of ''Spherically Symmetric Configurations'': [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Spherically_Symmetric_Configurations_.28Part_II.29\|Simplified Governing Equations]]

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

Tohline: Move title image

2012-11-04T14:39:21Z

Move title image

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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 (Part II)=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}

	~~=Spherically Symmetric Configurations (Structure — Part II)=~~
	[[Image:LSU_Structure_still.gif\|74px\|left]]		[[Image:LSU_Structure_still.gif\|74px\|left]]
	Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/PGE identified set of simplified governing equations]. The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_r v_r = 0</math>. After setting the radial velocity, <math>v_r</math>, and all time-derivatives to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>3^\mathrm{rd}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> (Euler) and <math>4^\mathrm{th}</math> give, respectively,		Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/PGE identified set of simplified governing equations]. The steady-state flow field that must be adopted to satisfy both a spherically symmetric geometry and the time-independent constraint is, <math>\vec{v} = \hat{e}_r v_r = 0</math>. After setting the radial velocity, <math>v_r</math>, and all time-derivatives to zero, we see that the <math>1^\mathrm{st}</math> (continuity) and <math>3^\mathrm{rd}</math> (first law of thermodynamics) equations are trivially satisfied while the <math>2^\mathrm{nd}</math> (Euler) and <math>4^\mathrm{th}</math> give, respectively,
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	=See Also=		=See Also=
	* Part I of ''Spherically Symmetric Configurations'': [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/PGE Simplified Governing Equations]		* Part I of ''Spherically Symmetric Configurations'': [http://www.vistrails.org/index.php/User:Tohline/SphericallySymmetricConfigurations/PGE Simplified Governing Equations]



	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}

← Older revision		Revision as of 17:07, 7 November 2012
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	=See Also=		=See Also=
	* Part I of ''Spherically Symmetric Configurations'': [[User:Tohline/SphericallySymmetricConfigurations/~~SolutionStrategies~~#Spherically_Symmetric_Configurations_.~~28Part_II~~.29\|Simplified Governing Equations]]		* Part I of ''Spherically Symmetric Configurations'': [[User:Tohline/SphericallySymmetricConfigurations/PGE#Spherically_Symmetric_Configurations_.28Part_I.29\|Simplified Governing Equations]]

	{{LSU_HBook_footer}}		{{LSU_HBook_footer}}