SSC/Virial/Isothermal - Revision history

Jet53man: /* Bonnor's (1956) Equivalent Relation */

2022-04-30T19:05:47Z

Bonnor's (1956) Equivalent Relation

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	<sup>&dagger;</sup>Y. P. Terletsky (1952, Zh. Eksper. Teor. Fiz., Vol. 22, p. 506) = (~~Proceedings of the~~ Russian ~~Academy of Sciences~~) Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki		<sup>&dagger;</sup>Y. P. Terletsky (1952, Zh. Eksper. Teor. Fiz., Vol. 22, p. 506)<br />
			----
			Notes from J. E. Tohline regarding this referenced article:<br />
			<ul>
			<li>The full title of this (Russian language) journal is, ''Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki'', sometimes abbreviated as, ''ZhETF''.</li>
			<li>English translations of ''ZhETF'' articles dating back to 1967 can be found in the [http://jetp.ras.ru ''Journal of Experimental and Theoretical Physics''] ''(JETP)''; I have been unable to find an English translation (or even the original Russian-language version) of Terletsky's 1952 article.</li>
			<li>A more accessible article by [https://ui.adsabs.harvard.edu/abs/1966AZh....43...96G/abstract I. L. Genkin (1966, Astronomicheskii Zhurnal, Vol. 43, p. 96)] heavily references Terletsky's work.</li>
			</ul>
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Jet53man at 23:06, 29 April 2022

2022-04-29T23:06:35Z

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	and shows the dimensionless external pressure, <math>~\Pi</math>, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius <math>~\chi</math>. The pressure becomes negative at radii <math>~\chi < 1</math>, hence the solution in this regime is unphysical.		and shows the dimensionless external pressure, <math>~\Pi</math>, that is required to construct a nonrotating, self-gravitating, isothermal sphere with an equilibrium radius <math>~\chi</math>. The pressure becomes negative at radii <math>~\chi < 1</math>, hence the solution in this regime is unphysical.

	[[~~User:Tohline/SphericallySymmetricConfigurations~~/Virial#Visual_Representation\|Figure 1]] displays the free energy surface that "lies above" the two-dimensional parameter space <math>~(1.2 < \chi < 1.51</math>; <math>~0.103 < \Pi < 0.104)</math> that is identified here by the thin, red rectangle.		[[SSCpt1/Virial#Visual_Representation\|Figure 1]] displays the free energy surface that "lies above" the two-dimensional parameter space <math>~(1.2 < \chi < 1.51</math>; <math>~0.103 < \Pi < 0.104)</math> that is identified here by the thin, red rectangle.
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Jet53man: /* Bonnor's (1956) Equivalent Relation */

2022-04-29T22:46:02Z

Bonnor's (1956) Equivalent Relation

← Older revision		Revision as of 18:46, 29 April 2022
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	<sup>&dagger;</sup>Y. P. Terletsky (1952, Zh. Eksper. Teor. Fiz., Vol. 22, p. 506)		<sup>&dagger;</sup>Y. P. Terletsky (1952, Zh. Eksper. Teor. Fiz., Vol. 22, p. 506) = (Proceedings of the Russian Academy of Sciences) Zhurnal Eksperimentalnoy i Teoreticheskoy Fiziki
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Jet53man: /* Bonnor's (1956) Equivalent Relation */

2022-04-29T22:33:21Z

Bonnor's (1956) Equivalent Relation

← Older revision		Revision as of 18:33, 29 April 2022
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	<font color="darkgreen">"It has recently been suggested by Terletsky ~~(7)~~ that for a large mass <math>M</math> of gas, of volume <math>V</math> and temperature {{ Math/VAR_Temperature01 }}, containing <math>N</math> molecules under boundary pressure <math>p</math>, the equation of state should be not		<font color="darkgreen">"It has recently been suggested by Terletsky<sup>&dagger;</sup> that for a large mass <math>M</math> of gas, of volume <math>V</math> and temperature {{ Math/VAR_Temperature01 }}, containing <math>N</math> molecules under boundary pressure <math>p</math>, the equation of state should be not

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	where {{ Math/C_BoltzmannConstant }} is Boltzmann's constant, {{ Math/C_GravitationalConstant }} is Newton's constant of gravitation, and <math>\alpha</math> is a constant depending on the shape of the mass. The proposed correction of Boyle's Law arises because, for a large mass, one has to take account of the gravitational interactions between the molecules."</font>		where {{ Math/C_BoltzmannConstant }} is Boltzmann's constant, {{ Math/C_GravitationalConstant }} is Newton's constant of gravitation, and <math>\alpha</math> is a constant depending on the shape of the mass. The proposed correction of Boyle's Law arises because, for a large mass, one has to take account of the gravitational interactions between the molecules."</font>
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			<sup>&dagger;</sup>Y. P. Terletsky (1952, Zh. Eksper. Teor. Fiz., Vol. 22, p. 506)
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Jet53man: /* Bonnor's (1956) Equivalent Relation */

2022-04-29T22:22:32Z

Bonnor's (1956) Equivalent Relation

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	====Bonnor's (1956) Equivalent Relation====		====Bonnor's (1956) Equivalent Relation====
	Inserting the expressions for the coefficients <math>~B_I</math>, <math>~A</math>, and <math>~D</math> gives,		Inserting the expressions for the coefficients <math>B_I</math>, <math>A</math>, and <math>D</math> gives,
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	<math>~		<math>
	3Mc_s^2 ~- \frac{3}{5} \frac{GM^2}{R} = 3 P_e \biggl( \frac{4\pi}{3} R^3\biggr) \, ,		3Mc_s^2 ~- \frac{3}{5} \frac{GM^2}{R} = 3 P_e \biggl( \frac{4\pi}{3} R^3\biggr) \, ,
	</math>		</math>
	</div>		</div>
	or, because the volume <math>~V = (4\pi R^3/3)</math> for a spherical configuration, we can write,		or, because the volume <math>V = (4\pi R^3/3)</math> for a spherical configuration, we can write,
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	<math>		<math>
	~3P_e V = 3Mc_s^2 ~- \frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/3} \frac{GM^2}{V^{1/3}} \, .		3P_e V = 3Mc_s^2 - \frac{3}{5} \biggl( \frac{4\pi}{3} \biggr)^{1/3} \frac{GM^2}{V^{1/3}} \, .
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	~~<!-- [[File:LinMestelShu1965Intro.png\|600px\|Lin, Mestel & Shu (1965) Introductory Paragraph]] -->~~
	<font color="darkgreen">"It has recently been suggested by Terletsky (7) that for a large mass <math>M</math> of gas, of volume <math>V</math> and temperature {{ Math/VAR_Temperature01 }}, containing <math>N</math> molecules under boundary pressure <math>p</math>, the equation of state should be not		<font color="darkgreen">"It has recently been suggested by Terletsky (7) that for a large mass <math>M</math> of gas, of volume <math>V</math> and temperature {{ Math/VAR_Temperature01 }}, containing <math>N</math> molecules under boundary pressure <math>p</math>, the equation of state should be not

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	Once we realize that, for an isothermal configuration, twice the thermal energy content, <math>~2S</math>, can be written as <math>~(3NkT)</math> just as well as via the product, <math>~(3Mc_s^2)</math>, we see that our expression is identical to ~~Bonnor's~~ if we set the prefactor on ~~Bonnor's~~ last term, <math>~\alpha = (4\pi/3)^{1/3}/5</math>. (Indeed, later on the first page of his paper, ~~Bonnor~~ points out that this is the appropriate value for <math>~\alpha</math> when considering a uniform density sphere.)		Once we realize that, for an isothermal configuration, twice the thermal energy content, <math>2S</math>, can be written as <math>(3NkT)</math> just as well as via the product, <math>(3Mc_s^2)</math>, we see that our expression is identical to the one derived by {{ Bonnor56 }} if we set the prefactor on his last term, <math>\alpha = (4\pi/3)^{1/3}/5</math>. (Indeed, later on the first page of his paper, {{ Bonnor56 }} points out that this is the appropriate value for <math>\alpha</math> when considering a uniform-density sphere.)

	====P-V Diagram====		====P-V Diagram====

Jet53man: /* See Also */

2022-04-28T23:41:55Z

Jet53man: /* Quartic Solution */

2022-04-28T23:40:59Z

Quartic Solution

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	In the above <math>~P-V</math> diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give <math>~\Pi</math> for any chosen value of <math>~\chi</math>. Alternatively, the four roots of the quartic equation — <math>~\chi_1</math>, <math>~\chi_2</math>, <math>~\chi_3</math> and <math>~\chi_4</math> in the presentation that follows — will identify the radii at which a spherical configuration will be in equilibrium for any choice of the external pressure, <math>~\Pi</math>, assuming the roots are real.		In the above <math>~P-V</math> diagram discussion, we rearranged the quartic equation governing equilibrium configurations to give <math>~\Pi</math> for any chosen value of <math>~\chi</math>. Alternatively, the four roots of the quartic equation — <math>~\chi_1</math>, <math>~\chi_2</math>, <math>~\chi_3</math> and <math>~\chi_4</math> in the presentation that follows — will identify the radii at which a spherical configuration will be in equilibrium for any choice of the external pressure, <math>~\Pi</math>, assuming the roots are real.
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Jet53man: /* Bonnor's (1956) Equivalent Relation */

2022-04-28T23:39:00Z

Bonnor's (1956) Equivalent Relation

← Older revision		Revision as of 19:39, 28 April 2022
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	It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that ~~was presented in 1956 by [http://adsabs~~.~~harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) as equation~~ (1.2) in ~~a paper titled, "Boyle's Law and Gravitational Instability"~~:		It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that appears as Eq. (1.2) in {{ Bonnor56full }}:

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Jet53man: /* Bonnor's (1956) Equivalent Relation */

2022-04-28T23:34:02Z

Bonnor's (1956) Equivalent Relation

← Older revision		Revision as of 19:34, 28 April 2022
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	It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that was presented in 1956 by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) as equation (1.2) in a paper titled, "Boyle's Law and Gravitational Instability":		It is instructive to compare this expression for a self-gravitating, isothermal equilibrium sphere to the one that was presented in 1956 by [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956, MNRAS, 116, 351) as equation (1.2) in a paper titled, "Boyle's Law and Gravitational Instability":
	~~<div align="center">~~
	~~<table border="2">~~
	~~<tr><td>~~
	~~[[File:Bonnor1951Eq1.2.jpg\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]]~~
	~~<!--[[Image:AAAwaiting01.png\|600px\|center\|Bonnor (1956, MNRAS, 116, 351)]]-->~~
	~~</td></tr>~~
	~~</table>~~
	~~</div>~~


	<table border="1" align="center" cellpadding="8" width="60%">		<table border="1" align="center" cellpadding="8" width="60%">
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	~~<!--~~
	~~[[File:LinMestelShu1965Title.png\|300px\|Lin, Mestel & Shu (1965)]]~~
	~~[http://adsabs.harvard.edu/abs/1965ApJ...142.1431L (1965) ''The Astrophysical Journal'', 142, 143]~~
	~~-->~~
	Reprint of the opening (introductory) paragraph from …<br />		Reprint of the opening (introductory) paragraph from …<br />
	{{ ~~LMS65figure~~ }}		{{ Bonnor56figure }}
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Jet53man: /* Bonnor's (1956) Equivalent Relation */

2022-04-28T23:07:28Z

Bonnor's (1956) Equivalent Relation

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 Once we realize that, for an isothermal configuration, twice the thermal energy content, <math>~2S</math>, can be written as <math>~(3NkT)</math> just as well as via the product, <math>~(3Mc_s^2)</math>, we see that our expression is identical to Bonnor's if we set the prefactor on Bonnor's last term, <math>~\alpha = (4\pi/3)^{1/3}/5</math>.  (Indeed, later on the first page of his paper, Bonnor points out that this is the appropriate value for <math>~\alpha</math> when considering a uniform density sphere.)

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	=See Also=		=See Also=
	<ul>		<ul>
	<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>


	{{ SGFfooter }}		{{ SGFfooter }}