SSCpt2/SolutionStrategies - Revision history

Jet53man: /* Spherically Symmetric Configurations (Part II) */

2021-07-29T03:21:54Z

Spherically Symmetric Configurations (Part II)

← Older revision		Revision as of 23:21, 28 July 2021
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	=Spherically Symmetric Configurations (Part II)=		=Spherically Symmetric Configurations (Part II)=
	~~{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"~~
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	~~! style="height: 125px; width: 125px; background-color:white;" \|~~
	~~<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Solution<br />Strategies</b>]]</font>~~
	\|}

	<!-- [[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 [[SSCpt1/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,		Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [[SSCpt1/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,
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	==Solution Strategies==		==Solution Strategies==
			{\| 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#Equilibrium_Structures\|<b>Solution<br />Strategies</b>]]</font>
			\|}
	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 {{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 {{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,

174.64.8.247: /* Spherically Symmetric Configurations (Part II) */

2021-07-28T16:45:46Z

Spherically Symmetric Configurations (Part II)

← Older revision		Revision as of 12:45, 28 July 2021
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	=Spherically Symmetric Configurations (Part II)=		=Spherically Symmetric Configurations (Part II)=
			{\| 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#Equilibrium_Structures\|<b>Solution<br />Strategies</b>]]</font>
			\|}

	<!-- [[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 [[SSCpt1/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,		Equilibrium, spherically symmetric '''structures''' are obtained by searching for time-independent solutions to the [[SSCpt1/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,
			<br />
			<br />
	<div align="center">		<div align="center">
	<math>~\vec{v} = \hat{e}_r v_r = 0 \, .</math>		<math>~\vec{v} = \hat{e}_r v_r = 0 \, .</math>

Jet53man at 21:47, 26 July 2021

2021-07-26T21:47:53Z

← Older revision		Revision as of 17:47, 26 July 2021
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	==Solution Strategies==		==Solution Strategies==
	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 {{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">
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	<div align="center">		<div align="center">
	{{~~User:Tohline/~~Math/EQ_SSmassConservation01}}		{{Math/EQ_SSmassConservation01}}
	</div>		</div>

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	<div align="center">		<div align="center">
	{{~~User:Tohline/~~Math/EQ_SShydrostaticBalance01}}		{{Math/EQ_SShydrostaticBalance01}}
	</div>		</div>

	that is, a single governing integro-differential equation which depends only on the two unknown functions, {{~~User:Tohline/~~Math/VAR_Pressure01}} and {{~~User:Tohline/~~Math/VAR_Density01}} .		that is, a single governing integro-differential equation which depends only on the two unknown functions, {{Math/VAR_Pressure01}} and {{Math/VAR_Density01}} .


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	</div>		</div>

	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, {{Math/VAR_Enthalpy01}} and {{Math/VAR_Density01}}.

	<b>Numerical integration examples:</b>		<b>Numerical integration examples:</b>
	* Isothermal sphere —		* Isothermal sphere —
	** [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) tabulation]].		** [[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)].		** 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]].		** Outline of [[SSC/Structure/IsothermalSphere#Our_Numerical_Integration\|our numerical integration scheme]].
	* Spherical polytropes —		* Spherical polytropes —
	** [[~~User:Tohline/~~SSC/Structure/Polytropes#Tabulated_Properties\|Some tabulated global properties]].		** [[SSC/Structure/Polytropes#Tabulated_Properties\|Some tabulated global properties]].
	** Outline of [[~~User:Tohline/~~SSC/Structure/Polytropes#Straight_Numerical_Integration\|our numerical integration scheme]].		** Outline of [[SSC/Structure/Polytropes#Straight_Numerical_Integration\|our numerical integration scheme]].

	===Technique 3===		===Technique 3===
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	</div>		</div>

	This means that, throughout our configuration, the functions {{~~User:Tohline/~~Math/VAR_Enthalpy01}}<math>(\rho)~</math> and {{~~User:Tohline/~~Math/VAR_NewtonianPotential01}}<math>(\rho)~</math> must sum to a constant value, call it <math>~C_\mathrm{B}</math>. That is to say, the statement of hydrostatic balance reduces to the ''algebraic'' expression,		This means that, throughout our configuration, the functions {{Math/VAR_Enthalpy01}}<math>(\rho)~</math> and {{Math/VAR_NewtonianPotential01}}<math>(\rho)~</math> must sum to a constant value, call it <math>~C_\mathrm{B}</math>. That is to say, the statement of hydrostatic balance reduces to the ''algebraic'' expression,

	<div align="center">		<div align="center">
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	</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}} 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, {{Math/VAR_Enthalpy01}}, {{Math/VAR_Density01}}, and {{Math/VAR_NewtonianPotential01}} to one another.


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

	=See Also=		=See Also=

	* Part I of ''Spherically Symmetric Configurations'': [[~~User:Tohline/SphericallySymmetricConfigurations~~/PGE#Spherically_Symmetric_Configurations_.28Part_I.29\|Simplified Governing Equations]]		* Part I of ''Spherically Symmetric Configurations'': [[SSCpt1/PGE#Spherically_Symmetric_Configurations_.28Part_I.29\|Simplified Governing Equations]]

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Jet53man: Created page with "FORCETOC =Spherically Symmetric Configurations (Part II)=

SSCpt2/SolutionStrategies - Revision history

Jet53man: /* Spherically Symmetric Configurations (Part II) */

174.64.8.247: /* Spherically Symmetric Configurations (Part II) */

Jet53man at 21:47, 26 July 2021

Jet53man: Created page with "__FORCETOC__ =Spherically Symmetric Configurations (Part II)=

Jet53man: Created page with "FORCETOC =Spherically Symmetric Configurations (Part II)=