VE - Revision history

Jet53man: /* See Also */

2021-10-28T17:11:46Z

Jet53man: /* See Also */

2021-10-27T16:52:59Z

Jet53man: /* Preface */

2021-10-27T16:39:55Z

Preface

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	'''Why Bother?'''<br />		'''Why Bother?'''<br />

	~~<font size="-1">~~Excerpts drawn from the introductory chapter (p. 3) of ~~[http://ads.harvard.edu/books/1978vtsa.book/ ''The Virial Theorem in Stellar Astrophysics'' (2003)],~~ <br />		Excerpts drawn from the introductory chapter (p. 3) of<br />
	~~by George W. Collins, II</font>~~		{{ Collins78figure }}
	</td>		</td>
	</tr>		</tr>

Jet53man: /* See Also */

2021-10-26T20:39:19Z

Jet53man: /* Whitworth (1981) and Stahler (1983) */

2021-10-08T21:37:55Z

Whitworth (1981) and Stahler (1983)

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	==Whitworth (1981) and Stahler (1983)==		==Whitworth (1981) and Stahler (1983)==
	The above formulation of a [[#Free_Energy_Expression\|Gibbs-like free energy]] has been motivated by the {{ Stahler83 }} analysis of stability of isothermal gas clouds, and it closely parallels the {{ Whitworth81full }} ~~discussion~~ of "global gravitational stability for one-dimensional polytropes." Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"		The above formulation of a [[#Free_Energy_Expression\|Gibbs-like free energy]] has been motivated by the {{ Stahler83 }} analysis of stability of isothermal gas clouds, and it closely parallels the discussion by {{ Whitworth81full }} of "global gravitational stability for one-dimensional polytropes." Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"
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Jet53man: /* Whitworth (1981) and Stahler (1983) */

2021-10-08T21:36:39Z

Whitworth (1981) and Stahler (1983)

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	==Whitworth (1981) and Stahler (1983)==		==Whitworth (1981) and Stahler (1983)==
	The above formulation of a [[#Free_Energy_Expression\|Gibbs-like free energy]] has been motivated by {{ ~~Stahler83full~~ }} analysis of ~~the~~ stability of isothermal gas clouds, and it closely parallels the {{ ~~Whitworth81~~ }} discussion of "global gravitational stability for one-dimensional polytropes." Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"		The above formulation of a [[#Free_Energy_Expression\|Gibbs-like free energy]] has been motivated by the {{ Stahler83 }} analysis of stability of isothermal gas clouds, and it closely parallels the {{ Whitworth81full }} discussion of "global gravitational stability for one-dimensional polytropes." Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"
	<div align="center">		<div align="center">
	<table border="0">		<table border="0">

Jet53man: /* Whitworth (1981) and Stahler (1983) */

2021-10-08T21:34:51Z

Whitworth (1981) and Stahler (1983)

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	==Whitworth (1981) and Stahler (1983)==		==Whitworth (1981) and Stahler (1983)==
	The above formulation of a [[#Free_Energy_Expression\|Gibbs-like free energy]] has been motivated by ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler's (1983, ApJ, 268, 16)]~~ analysis of the stability of isothermal gas clouds, and it closely parallels ~~[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth's (1981, MNRAS, 195, 967)]~~ discussion of ~~the~~ "global gravitational stability for one-dimensional polytropes." Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"		The above formulation of a [[#Free_Energy_Expression\|Gibbs-like free energy]] has been motivated by {{ Stahler83full }} analysis of the stability of isothermal gas clouds, and it closely parallels the {{ Whitworth81 }} discussion of "global gravitational stability for one-dimensional polytropes." Whitworth introduces a "global potential function," <math>\mathfrak{u}</math>, that is the sum of three "internal conserved energy modes,"
	<div align="center">		<div align="center">
	<table border="0">		<table border="0">

Jet53man: /* Illustration */

2021-10-08T21:19:12Z

Illustration

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	'''Isothermal Systems''': If, upon compression or expansion, the configuration remains isothermal~~, [~~see~~, also,~~ Appendix A of ~~[http://adsabs.harvard.edu/abs/1983ApJ...268..165S Stahler] (1983, ApJ, 268, 16)],~~ the reservoir of thermal energy is,		'''Isothermal Systems''': If, upon compression or expansion, the configuration remains isothermal — also see Appendix A of {{ Stahler83full }} — the reservoir of thermal energy is,

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Jet53man: /* Virial Equations (Rotating Frame) */

2021-10-08T20:55:45Z

Virial Equations (Rotating Frame)

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	END OF DELETED COMMENT -->		END OF DELETED COMMENT -->

	In his discussion of the ''Oscillation and Collapse of Interstellar Clouds,'' ~~[http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber (1976, ApJ, 208, 113)]~~ begins with this form of the second-order virial, but adds to it a contribution due to pressure-confinement by an external medium, as [[VE#Generalization\|introduced above in the context of Shu's generalization]].		In his discussion of the ''Oscillation and Collapse of Interstellar Clouds,'' {{ Weber76full }} begins with this form of the second-order virial, but adds to it a contribution due to pressure-confinement by an external medium, as [[VE#Generalization\|introduced above in the context of Shu's generalization]].
	Specifically, Weber opens up his discussion with the following form of the tensor virial equations:		Specifically, Weber opens up his discussion with the following form of the tensor virial equations:

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	~~[http://adsabs.harvard.edu/abs/1976ApJ...208..113W Weber (1976)]~~, Eq. (1)		{{ Weber76 }}, Eq. (1)
	</td>		</td>
	</tr>		</tr>

Jet53man: /* Generalization */

2021-10-08T20:36:54Z

Generalization

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	</div>		</div>

	[It should be noted that ~~[http://adsabs.harvard.edu/abs/1953ApJ...118..116C Chandrasekhar & Fermi] (1953, ApJ, 118, 116)~~ and ~~[http://adsabs.harvard.edu/abs/1956MNRAS.116..503M Mestel & Spitzer] (1956, MNRAS, 116, 503)~~ provide early discussions of virial equilibrium conditions that take into account the energy associated with a magnetic field.]		[It should be noted that {{ CF53full }} and {{ MS56full }} provide early discussions of virial equilibrium conditions that take into account the energy associated with a magnetic field.]

	==Virial Equations (Rotating Frame)==		==Virial Equations (Rotating Frame)==

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	=See Also=		=See Also=
	<ul>		<ul>
	<li>~~Clausius~~, ~~R. J. E. (1870). "~~On a Mechanical Theorem Applicable to Heat~~," [https://www.tandfonline.com/doi/abs/10.1080/14786447008640370 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Ser. 4, vol. 40, issue 265], pp. 122-127.~~</li>		<li>{{ Clausius1870full }}, ''On a Mechanical Theorem Applicable to Heat''</li>
	<ul>		<ul>
	<li>[http://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA122&lpg=PA122&dq=Mechanical+Theorem+Applicable+to+Heat&source=bl&ots=dhToAIe8k9&sig=0TaKVTmMnZ5qWvxLkdzuExMoU-U&hl=en&sa=X&ei=YFd-U5qiGdizyASS74CIAw&ved=0CDYQ6AEwAQ#v=onepage&q=Mechanical%20Theorem%20Applicable%20to%20Heat&f=false Google Book Reference]		<li>[http://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA122&lpg=PA122&dq=Mechanical+Theorem+Applicable+to+Heat&source=bl&ots=dhToAIe8k9&sig=0TaKVTmMnZ5qWvxLkdzuExMoU-U&hl=en&sa=X&ei=YFd-U5qiGdizyASS74CIAw&ved=0CDYQ6AEwAQ#v=onepage&q=Mechanical%20Theorem%20Applicable%20to%20Heat&f=false Google Book Reference]
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	</ul>		</ul>
	<li>[http://en.wikipedia.org/wiki/Virial_theorem Wikipedia discussion of the virial theorem]</li>		<li>[http://en.wikipedia.org/wiki/Virial_theorem Wikipedia discussion of the virial theorem]</li>
	<li>~~[http://ads.harvard.edu/books/1978vtsa.book/ The Virial Theorem in Stellar Astrophysics (1978)] by George W. Collins, II~~</li>		<li>{{ Collins78full }}</li>
	<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>		<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
	</ul>		</ul>

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	=See Also=		=See Also=
	<ul>		<ul>
	<li>Clausius, R. J. E. (1870). "On a Mechanical Theorem Applicable to Heat," [~~http~~://www.tandfonline.com/doi/abs/10.1080/~~.U35XrK1dUys ''~~Philosophical Magazine, Ser. 4, vol. 40, issue 265], pp. 122-127.</li>		<li>Clausius, R. J. E. (1870). "On a Mechanical Theorem Applicable to Heat," [https://www.tandfonline.com/doi/abs/10.1080/14786447008640370 The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, Ser. 4, vol. 40, issue 265], pp. 122-127.</li>
	<ul>		<ul>
	<li>[http://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA122&lpg=PA122&dq=Mechanical+Theorem+Applicable+to+Heat&source=bl&ots=dhToAIe8k9&sig=0TaKVTmMnZ5qWvxLkdzuExMoU-U&hl=en&sa=X&ei=YFd-U5qiGdizyASS74CIAw&ved=0CDYQ6AEwAQ#v=onepage&q=Mechanical%20Theorem%20Applicable%20to%20Heat&f=false Google Book Reference]		<li>[http://books.google.com/books?id=Zk0wAAAAIAAJ&pg=PA122&lpg=PA122&dq=Mechanical+Theorem+Applicable+to+Heat&source=bl&ots=dhToAIe8k9&sig=0TaKVTmMnZ5qWvxLkdzuExMoU-U&hl=en&sa=X&ei=YFd-U5qiGdizyASS74CIAw&ved=0CDYQ6AEwAQ#v=onepage&q=Mechanical%20Theorem%20Applicable%20to%20Heat&f=false Google Book Reference]

← Older revision		Revision as of 16:39, 26 October 2021
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	<li>[http://en.wikipedia.org/wiki/Virial_theorem Wikipedia discussion of the virial theorem]</li>		<li>[http://en.wikipedia.org/wiki/Virial_theorem Wikipedia discussion of the virial theorem]</li>
	<li>[http://ads.harvard.edu/books/1978vtsa.book/ The Virial Theorem in Stellar Astrophysics (1978)] by George W. Collins, II</li>		<li>[http://ads.harvard.edu/books/1978vtsa.book/ The Virial Theorem in Stellar Astrophysics (1978)] by George W. Collins, II</li>
	<li>[[~~User:Tohline/~~SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>		<li>[[SphericallySymmetricConfigurations/IndexFreeEnergy#Index_to_Free-Energy_Analyses\|Index to a Variety of Free-Energy and/or Virial Analyses]]</li>
	</ul>		</ul>


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