SSC/Virial/PolytropesEmbedded/FirstEffortAgain - Revision history

Jet53man: /* Role of Structural Form Factors */

2021-10-12T21:31:08Z

Role of Structural Form Factors

Jet53man: /* Isolated Nonrotating Adiabatic Configuration */

2021-10-12T21:26:33Z

Isolated Nonrotating Adiabatic Configuration

← Older revision		Revision as of 17:26, 12 October 2021
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	We see that, for <math>~\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass. Conversely, for <math>~\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius. For <math>~\gamma_g > 2</math> or <math>~\gamma_g < 4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii. However, for <math>~\gamma_g</math> in the range, <math>~2 > \gamma_g > 4/3</math>, configurations with larger mass have smaller equilibrium radii. (Note that the related result for [[~~User:Tohline/~~SSC/Virial/Isothermal#Virial_Equilibrium_of_Isothermal_Spheres\|isothermal configurations]] can be obtained by setting <math>~\gamma_g = 1</math> in this adiabatic solution, because <math>~K = c_s^2</math> when <math>~\gamma_g = 1</math>.)		We see that, for <math>~\gamma_g=2</math>, the equilibrium radius depends only on the specific entropy of the gas and is independent of the configuration's mass. Conversely, for <math>~\gamma_g = 4/3</math>, the mass of the configuration is independent of the radius. For <math>~\gamma_g > 2</math> or <math>~\gamma_g < 4/3</math>, configurations with larger mass (but the same specific entropy) have larger equilibrium radii. However, for <math>~\gamma_g</math> in the range, <math>~2 > \gamma_g > 4/3</math>, configurations with larger mass have smaller equilibrium radii. (Note that the related result for [[SSC/Virial/Isothermal#Virial_Equilibrium_of_Isothermal_Spheres\|isothermal configurations]] can be obtained by setting <math>~\gamma_g = 1</math> in this adiabatic solution, because <math>~K = c_s^2</math> when <math>~\gamma_g = 1</math>.)

	==Role of Structural Form Factors==		==Role of Structural Form Factors==

Jet53man: /* Virial Equilibrium */

2021-10-12T21:24:47Z

Virial Equilibrium

← Older revision		Revision as of 17:24, 12 October 2021
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	(The [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Structural_Form_Factors\|dimensionless structural form factors]], <math>~\mathfrak{f}_i,</math> that appear in these expressions are defined ''for isolated polytropes'' in our accompanying introductory discussion and are [[~~User:Tohline/~~SSC/Virial/Polytropes#Role_of_Structural_Form_Factors\|discussed further, below]].) Once the pressure exerted by the external medium (<math>~P_e</math>), and the configuration's mass (<math>~M_\mathrm{tot}</math>), angular momentum (<math>~J</math>), and specific entropy (via <math>~K</math>) have been specified, the values of all of the coefficients are known and <math>~\chi_\mathrm{eq}</math> can be determined.		(The [[SSCpt1/Virial#Structural_Form_Factors\|dimensionless structural form factors]], <math>\mathfrak{f}_i,</math> that appear in these expressions are defined ''for isolated polytropes'' in our accompanying introductory discussion and are [[SSC/Virial/Polytropes#Role_of_Structural_Form_Factors\|discussed further, below]].) Once the pressure exerted by the external medium (<math>P_e</math>), and the configuration's mass (<math>M_\mathrm{tot}</math>), angular momentum (<math>J</math>), and specific entropy (via <math>K</math>) have been specified, the values of all of the coefficients are known and <math>\chi_\mathrm{eq}</math> can be determined.

Jet53man: /* Virial Equilibrium of Adiabatic Spheres */

2021-10-12T21:17:47Z

Virial Equilibrium of Adiabatic Spheres

← Older revision		Revision as of 17:17, 12 October 2021
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	=Virial Equilibrium of Adiabatic Spheres=		=Virial Equilibrium of Adiabatic Spheres=
	Highlights of the rather detailed discussion presented below have been [[~~User:Tohline/~~SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29\|summarized in an accompanying chapter]] of this H_Book.		Highlights of the rather detailed discussion presented below have been [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Virial_Equilibrium_of_Adiabatic_Spheres_.28Summary.29\|summarized in an accompanying chapter]] of this H_Book.

	==Review==		==Review==
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	===Virial Equilibrium===		===Virial Equilibrium===
	Also in our introductory discussion — see especially the section titled, [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Energy_Extrema\|''Energy Extrema'']] — we deduced that an adiabatic system's dimensionless equilibrium radius,		Also in our introductory discussion — see especially the section titled, [[SSCpt1/Virial#Energy_Extrema\|''Energy Extrema'']] — we deduced that an adiabatic system's dimensionless equilibrium radius,
	<div align="center">		<div align="center">
	<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, ,</math>		<math>~\chi_\mathrm{eq} \equiv \frac{R_\mathrm{eq}}{R_\mathrm{norm}} \, ,</math>

Jet53man: /* Adopted Normalizations */

2021-10-12T21:16:00Z

Adopted Normalizations

← Older revision		Revision as of 17:16, 12 October 2021
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	==Review==		==Review==
	===Adopted Normalizations===		===Adopted Normalizations===
	In an [[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations\|introductory discussion]] of the virial equilibrium structure of spherically symmetric configurations, we adopted the following physical parameter normalizations for adiabatic systems.		In an [[SSCpt1/Virial#Virial_Equilibrium_of_Spherically_Symmetric_Configurations\|introductory discussion]] of the virial equilibrium structure of spherically symmetric configurations, we adopted the following physical parameter normalizations for adiabatic systems.
	<div align="center">		<div align="center">
	<table border="1" align="center" cellpadding="5">		<table border="1" align="center" cellpadding="5">

Jet53man: Created page with "FORCETOC =Virial Equilibrium of Adiabatic Spheres= Highlights of the rather detailed discussion presented below have been User:To..."

2021-10-12T21:13:58Z

Created page with "__FORCETOC__  =Virial Equilibrium of Adiabatic Spheres= Highlights of the rather detailed discussion presented below have been User:To..."

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