User:Tohline/SSC/Structure/BonnorEbert - Revision history

Tohline: /* New Derivation */

2019-03-15T03:34:20Z

New Derivation

← Older revision		Revision as of 03:34, 15 March 2019
Line 993:		Line 993:
	<math>~\frac{\xi_e}{\psi^'}		<math>~\frac{\xi_e}{\psi^'}
	\biggl[e^{-\psi_e}- \frac{1}{2} (\psi^')^2 \biggr]_e		\biggl[e^{-\psi_e}- \frac{1}{2} (\psi^')^2 \biggr]_e
	\biggl[ 1 - \frac{1}{2} \xi_e \psi_e^' \biggr]^{-1} \, .		\biggl[ 1 - \frac{1}{2} \xi_e \psi_e^' \biggr]^{-1}
			</math>
			</td>
			</tr>

			<tr>
			<td align="right">

			</td>
			<td align="center"><math>~=</math></td>
			<td align="left">
			<math>~\frac{\xi_e}{\psi^' e^{\psi_e}}
			\biggl[2 - (\psi^')^2 e^{\psi_e} \biggr]_e
			\biggl[ 2 - \xi_e \psi_e^' \biggr]^{-1} \, .
	</math>		</math>
	</td>		</td>

Tohline: /* New Derivation */

2019-03-15T03:25:23Z

New Derivation

← Older revision		Revision as of 03:25, 15 March 2019
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	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~R</math>		<math>~\frac{dR}{d\xi_e}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 938:		Line 938:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{c_s^~~2}{~~4~~\pi G \rho_c} \biggr)^{1/2} \xi </math>~~		<math>~
	~~</td>~~		\biggl( \frac{c_s^4}{4\pi G P_e} \biggr)^{1 / 2} \biggl[
	~~</tr>~~		1
			- \frac{1}{2} \xi_e \psi_e^'
	~~<tr>~~		\biggr]e^{-\psi_e/2}
	~~<td align="right">~~		\, .</math>
	~~ ~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\biggl( \frac{1~~}{4\pi G P_e} \biggr)^{1/2} \~~xi e^{\psi/2}</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~<math>~\Rightarrow~~~\frac{dR}{d\xi}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~\biggl(~~ \frac{1}{~~4\pi G P_e} \biggr)^{1/~~2} \~~biggl[1 + \frac{~~\~~xi}{2} \cdot \psi~~^' \biggr]e^{\~~psi~~/2} </math>
	</td>		</td>
	</tr>		</tr>
Line 974:		Line 955:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{dM}{dR} = \frac{dM/d\xi}{dR/d\xi}</math>		<math>~\frac{dM}{dR} = \frac{dM/d\xi_e}{dR/d\xi_e}</math>
	</td>		</td>
	<td align="center"><math>~=</math></td>		<td align="center"><math>~=</math></td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{c_s^4}{4\pi G^3 P_e} \biggr)^{1/2} \biggl[		<math>~
	e^{-\~~psi~~} +\frac{1}{2} (\psi^')^2		\biggl( \frac{c_s^8}{4\pi G^3 P_e} \biggr)^{1 / 2}
	\biggr] \xi^2 e^{\~~psi~~/2}		\biggl[e^{-\psi_e}- \frac{1}{2} (\psi^')^2 \biggr]_e \xi_e^2 e^{-\psi_e/2}
	\biggl\{\biggl( \frac{1}{4\pi G P_e} \biggr)^{1/2} \biggl[1 + \frac{~~\xi~~}{2} \~~cdot~~ \~~psi~~^' \biggr]e^{\~~psi~~/2}\biggr\}^{-1}		\biggl\{
			\biggl( \frac{c_s^4}{4\pi G P_e} \biggr)^{1 / 2} \biggl[ 1 - \frac{1}{2} \xi_e \psi_e^'
			\biggr]e^{-\psi_e/2}
			\biggr\}^{-1}
	</math>		</math>
	</td>		</td>
Line 992:		Line 976:
	<td align="center"><math>~=</math></td>		<td align="center"><math>~=</math></td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{c_s^2}{G} \biggr) \biggl[		<math>~
	2e^{-\~~psi~~} + (\psi^')^2		\biggl( \frac{c_s^4}{G^2} \biggr)^{1 / 2}
	\biggr] \xi^2		\biggl[e^{-\psi_e}- \frac{1}{2} (\psi^')^2 \biggr]_e \xi_e^2
	\biggl[2 + \xi \~~psi~~^' \biggr]^{-1}		\biggl[ 1 - \frac{1}{2} \xi_e \psi_e^'
			\biggr]^{-1}
	</math>		</math>
	</td>		</td>
Line 1,006:		Line 991:
	<td align="center"><math>~=</math></td>		<td align="center"><math>~=</math></td>
	<td align="left">		<td align="left">
	<math>~\~~xi e^~~{\psi/2} \biggl[ ~~\xi^2 \psi^'~~ e^{\~~psi/2~~}\~~biggr]^~~{-1}		<math>~\frac{\xi_e}{\psi^'}
	~~\biggl[ 2e^~~{~~-\psi~~} + (\psi^')^2 \biggr] ~~\xi^2~~ \biggl[2 + \xi \~~psi~~^' \biggr]^{-1}		\biggl[e^{-\psi_e}- \frac{1}{2} (\psi^')^2 \biggr]_e
			\biggl[ 1 - \frac{1}{2} \xi_e \psi_e^' \biggr]^{-1} \, .
	</math>		</math>
	</td>		</td>
	</tr>		</tr>
			</table>
			</div>


	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center"><math>~=</math></td>~~
	~~<td align="left">~~
	~~<math>~\xi \biggl[ \psi^' \biggr]^{-1}~~
	~~\biggl[ 2e^{-\psi} + (\psi^')^2 \biggr] \biggl[2 + \xi \psi^' \biggr]^{-1}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~

	~~<tr>~~
	~~<td align="right">~~
	~~ ~~
	~~</td>~~
	~~<td align="center"><math>~=</math></td>~~
	~~<td align="left">~~
	~~<math>~~~
	~~\frac{\xi [2e^{-\psi} + (\psi^')^2 ]}{ \psi^' [2 + \xi \psi^']}~~
	~~</math>~~
	~~</td>~~
	~~</tr>~~
	~~</table>~~
	~~</div>~~
	Now, we can also write,		Now, we can also write,
	<div align="center">		<div align="center">

Tohline: /* New Derivation */

2019-03-14T23:39:52Z

New Derivation

← Older revision		Revision as of 23:39, 14 March 2019
Line 795:		Line 795:
	===New Derivation===		===New Derivation===

	For use in a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Isothermal\|related discussion]], let's examine how the mass varies with radius ~~— at fixed~~ <math>~~~c_s^2~~</math> ~~and~~ fixed <math>~~~P_e~~</math> ~~— in the limit as~~ <math>~~~\xi_e \rightarrow 0~~</math>. From above, we have,		For use in a [[User:Tohline/SSC/Stability/InstabilityOnsetOverview#Isothermal\|related discussion]], let's examine how the mass, <math>~M_R</math>, varies with radius, <math>~R</math>, at fixed <math>~c_s^2</math> and <math>~P_e</math>. From above, we have,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 830:		Line 830:
	</div>		</div>
	and,		and,

	<div align="center">		<div align="center">
			<math>R^2 = \biggl( \frac{c_s^2}{4\pi G \rho_c} \biggr) \xi_e^2 </math> ;<br />
			<p></p>

			<p></p>
			<math>
			M_R^2 = \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr) \biggl[ \xi^2 \frac{d\psi}{d\xi} \biggr]^2_e \, .
			</math>
			</div>

			Given that, <math>P_e = (c_s^2 \rho_c) e^{-\psi_e}</math>, we can also write,
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">

	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~P_e~~</math>		<math>~R</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 841:		Line 852:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(c_s^2 \~~rho_c)~~ e^{-\~~psi~~}</math>		<math>~\biggl( \frac{c_s^4}{4\pi G P_e} \biggr)^{1 / 2} \xi_e e^{-\psi_e/2} \, ,</math>
	</td>		</td>
	</tr>		</tr>
Line 847:		Line 858:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~~~\Rightarrow ~~~ \rho_c~~</math>		<math>~M_R</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 853:		Line 864:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~(c_s^2 P_e) e^{-\~~psi~~}</math>		<math>~\biggl( \frac{c_s^8}{4\pi G^3 P_e} \biggr)^{1 / 2} \biggl[ \xi^2 \psi^' \biggr]_e e^{-\psi_e/2} \, .</math>
	</td>		</td>
	</tr>		</tr>
	</table>		</table>
	~~</div>~~

	Hence,		Hence,
Line 865:		Line 876:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~M</math>		<math>~\frac{dM_R}{d\xi_e}</math>
	</td>		</td>
	<td align="center">		<td align="center">
Line 871:		Line 882:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{c_s^6}{4\pi G^3 ~~\rho_c~~} \biggr)^{1/2} \biggl[ \xi^2 \frac{~~d\psi~~}{d\xi} \biggr]</math>		<math>~\biggl( \frac{c_s^8}{4\pi G^3 P_e} \biggr)^{1 / 2}
			\biggl[
			2\xi \psi^'
			+ \xi^2 \psi^{''}
			- \frac{1}{2} \xi^2 (\psi^')^2
			\biggr]_e e^{-\psi_e/2} </math>
	</td>		</td>
	</tr>		</tr>
Line 883:		Line 899:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{c_s^4}{4\pi G^3 P_e} \biggr)^{1/2} \biggl[ \xi^2 \psi^' \~~biggr]e~~^~~{\psi/~~2~~}</math>~~		<math>~\biggl( \frac{c_s^8}{4\pi G^3 P_e} \biggr)^{1 / 2}
	~~</td>~~		\biggl[
	~~</tr>~~		2\xi \psi^'
			+ \xi^2 \biggl( e^{-\psi_e} - \frac{2}{\xi_e} \psi_e^' \biggr)
	~~<tr>~~		- \frac{1}{2} \xi^2 (\psi^')^2
	~~<td align="right">~~		\biggr]_e e^{-\psi_e/2} </math>
	~~<math>~\Rightarrow~~~\frac{dM}{d\xi}</math>~~
	~~</td>~~
	~~<td align="center">~~
	~~<math>~=</math>~~
	~~</td>~~
	~~<td align="left">~~
	~~<math>~~~\biggl( ~~\frac{c_s~~^4}{~~4\pi G^3 P_e}~~ \~~biggr)^{1/2~~} ~~\biggl[~~
	\frac{2}{\xi} \~~psi~~^' + \~~psi^{''} +~~\frac{1}{2} (\psi^')^2
	\biggr] ~~\xi^2~~ e^{\~~psi~~/2}
	</math>
	</td>		</td>
	</tr>		</tr>
Line 910:		Line 916:
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl( \frac{c_s^4}{4\pi G^3 P_e} \biggr)^{1/2} \biggl[		<math>~\biggl( \frac{c_s^8}{4\pi G^3 P_e} \biggr)^{1 / 2}
	e^{-\~~psi~~} +\frac{1}{2} (\psi^')^2		\biggl[
	\biggr] \xi^2 e^{\~~psi~~/2}		e^{-\psi_e}
	</math>		- \frac{1}{2} (\psi^')^2
			\biggr]_e \xi_e^2 e^{-\psi_e/2} \, .</math>
	</td>		</td>
	</tr>		</tr>

Tohline: /* Mass versus Radius Equilibrium Sequence */

2019-03-14T22:50:19Z

Mass versus Radius Equilibrium Sequence

Show changes

Tohline: /* Following Bonnor's Presentation */

2019-03-14T21:19:39Z

Following Bonnor's Presentation

← Older revision		Revision as of 21:19, 14 March 2019
Line 419:		Line 419:
	4.45 \biggl( \frac{c_s^6}{G^3 \rho_c} \biggr)^{1/2}\, .</math>		4.45 \biggl( \frac{c_s^6}{G^3 \rho_c} \biggr)^{1/2}\, .</math>
	</div>		</div>
	Dividing this expression for <math>M_R</math> by the expression for <math>R</math> gives,		Dividing this expression for <math>~M_R</math> by the expression for <math>~R</math> gives,
	<div align="center">		<div align="center">
	<math>		<math>
Line 432:		Line 432:

	<span id="cychenMaryland">		<span id="cychenMaryland">
	Finally, using this relation to eliminate <math>\rho_c</math> from the expression for <math>~M_R</math> gives,</span>		Finally, using this relation to eliminate <math>~\rho_c</math> from the expression for <math>~M_R</math> gives,</span>
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5">		<table border="0" cellpadding="5">

Tohline: /* P-V Diagram */

2019-03-14T21:15:21Z

P-V Diagram

← Older revision		Revision as of 21:15, 14 March 2019
Line 283:		Line 283:
	In an effort to demonstrate more explicitly how Bonnor's 1956 work and Ebert's 1955 work tie back to the original work on isolated isothermal spheres that was published by Emden in 1907, the right-hand panel of our Figure 1 shows a plot of <math>~P_e/P_0</math> versus <math>~V/V_0</math>, and the right-hand panel of our Figure 2 shows a plot of <math>~\log(P_e/P_0)</math> versus <math>~\log(R/R_0)</math> that we have constructed using data drawn directly from [[User:Tohline/SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].		In an effort to demonstrate more explicitly how Bonnor's 1956 work and Ebert's 1955 work tie back to the original work on isolated isothermal spheres that was published by Emden in 1907, the right-hand panel of our Figure 1 shows a plot of <math>~P_e/P_0</math> versus <math>~V/V_0</math>, and the right-hand panel of our Figure 2 shows a plot of <math>~\log(P_e/P_0)</math> versus <math>~\log(R/R_0)</math> that we have constructed using data drawn directly from [[User:Tohline/SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].


			===Whitworth's Normalization===
			[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) examined the behavior of model sequences that result from embedding polytropes with various effective adiabatic indexes <math>~(\gamma_g = 1/3,~ 2/3,~ 1,~ 4/3,~ 5/3)</math> in an external medium of pressure <math>~P_e</math>. Following Ebert's lead, in his Figure 1b, Whitworth showed how the size of the configuration varies with <math>~P_e</math> along each sequence. A reproduction of Whitworth's Figure 1b has been displayed [[User:Tohline/SSC/Structure/PolytropesEmbedded#WhitworthFig1b\|in our separate discussion of pressure-bounded polytropic configurations that have analytic prescriptions]], but we have again displayed the figure here, in the right-most panel of our Figure 2, because Whitworth included pressure-bounded isothermal configurations <math>~(\gamma_g = 1)</math> among his models. The red arrow identifies the isothermal sequence. Whitworth's Figure 1b effectively serves as an extension of Ebert's Figure 2, shown above in the left-most panel of our Figure 2. In order to assist comparison with Ebert's figure, this time we have rotated and flipped Whitworth's figure relative to its original orientation because Whitworth chose to plot <math>~R</math> versus <math>~P_e</math> whereas Ebert plotted <math>~P_e</math> versus <math>~R</math>.

	<div align="center" id="Fig3">		<div align="center" id="Fig3">
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	</div>		</div>


	~~===Whitworth's Normalization===~~
	[http://adsabs.harvard.edu/abs/1981MNRAS.195..967W Whitworth] (1981, MNRAS, 195, 967) examined the behavior of model sequences that result from embedding polytropes with various effective adiabatic indexes <math>~(\gamma_g = 1/3,~ 2/3,~ 1,~ 4/3,~ 5/3)</math> in an external medium of pressure <math>~P_e</math>. Following Ebert's lead, in his Figure 1b, Whitworth showed how the size of the configuration varies with <math>~P_e</math> along each sequence. A reproduction of Whitworth's Figure 1b has been displayed [[User:Tohline/SSC/Structure/PolytropesEmbedded#WhitworthFig1b\|in our separate discussion of pressure-bounded polytropic configurations that have analytic prescriptions]], but we have again displayed the figure here, in the right-most panel of our Figure 2, because Whitworth included pressure-bounded isothermal configurations <math>~(\gamma_g = 1)</math> among his models. The red arrow identifies the isothermal sequence. Whitworth's Figure 1b effectively serves as an extension of Ebert's Figure 2, shown above in the left-most panel of our Figure 2. In order to assist comparison with Ebert's figure, this time we have rotated and flipped Whitworth's figure relative to its original orientation because Whitworth chose to plot <math>~R</math> versus <math>~P_e</math> whereas Ebert plotted <math>~P_e</math> versus <math>~R</math>.

	It should be noted that the scaling adopted along both axes in Whitworth's Figure 1b is different from the scaling used by Ebert. As is detailed in the [[User:Tohline/SSC/Structure/PolytropesASIDE1\|accompanying ASIDE]], Whitworth "referenced" <math>P_e</math> and <math>R</math> to, respectively,		It should be noted that the scaling adopted along both axes in Whitworth's Figure 1b is different from the scaling used by Ebert. As is detailed in the [[User:Tohline/SSC/Structure/PolytropesASIDE1\|accompanying ASIDE]], Whitworth "referenced" <math>P_e</math> and <math>R</math> to, respectively,

Tohline: /* P-V Diagram */

2019-03-14T21:11:17Z

P-V Diagram

← Older revision		Revision as of 21:11, 14 March 2019
Line 279:		Line 279:
	</table>		</table>
	</div>		</div>


			In an effort to demonstrate more explicitly how Bonnor's 1956 work and Ebert's 1955 work tie back to the original work on isolated isothermal spheres that was published by Emden in 1907, the right-hand panel of our Figure 1 shows a plot of <math>~P_e/P_0</math> versus <math>~V/V_0</math>, and the right-hand panel of our Figure 2 shows a plot of <math>~\log(P_e/P_0)</math> versus <math>~\log(R/R_0)</math> that we have constructed using data drawn directly from [[User:Tohline/SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].


	<div align="center" id="Fig3">		<div align="center" id="Fig3">
Line 335:		Line 339:
	</div>		</div>

	In an effort to demonstrate more explicitly how Bonnor's 1956 work and Ebert's 1955 work tie back to the original work on isolated isothermal spheres that was published by Emden in 1907, the right-hand panel of our Figure 1 shows a plot of <math>~P_e/P_0</math> versus <math>~V/V_0</math>, and the middle panel of our Figure 2 shows a plot of <math>~\log(P_e/P_0)</math> versus <math>~\log(R/R_0)</math> that we have constructed using data drawn directly from [[User:Tohline/SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].

	===Whitworth's Normalization===		===Whitworth's Normalization===

Tohline: /* Pressure-Bounded Isothermal Sphere */

2019-03-14T20:57:02Z

Pressure-Bounded Isothermal Sphere

← Older revision		Revision as of 20:57, 14 March 2019
Line 2:		Line 2:
	=Pressure-Bounded Isothermal Sphere=		=Pressure-Bounded Isothermal Sphere=
	{{LSU_HBook_header}}		{{LSU_HBook_header}}
	As has been [[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere_(structure)\|derived and discussed elsewhere]], an ''isolated'' isothermal sphere has a density profile that extends to infinity and, correspondingly, an unbounded total mass. In an astrophysical context, neither of these properties is desirable. A more realistic isothermal configuration can be constructed by embedding the structure in a low density, but hot external medium whose pressure, <math>P_e</math>, confines the isothermal configuration to a finite size. In a mathematical model, this can be accomplished by ripping off an outer layer of the ''isolated'' isothermal configuration down to the radius — label it <math>\xi_e</math> — at which the configuration's original (internal) pressure equals <math>P_e</math>; the interior of the configuration that remains — containing mass <math>M_{\xi_e}</math> — should be unaltered and in equilibrium. (This will work only for spherically symmetric configurations, as the gravitational acceleration at any location only depends on the mass contained inside that radius.) [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) are credited with constructing the first such models and, most significantly, discovering that, for any specified sound speed and applied external pressure, there is a mass above which no equilibrium configuration exists. We present, here, the salient elements of these (essentially equivalent) derivations.		As has been [[User:Tohline/SSC/Structure/IsothermalSphere#Isothermal_Sphere_(structure)\|derived and discussed elsewhere]], an ''isolated'' isothermal sphere has a density profile that extends to infinity and, correspondingly, an unbounded total mass. In an astrophysical context, neither of these properties is desirable. A more realistic isothermal configuration can be constructed by embedding the structure in a low density, but hot external medium whose pressure, <math>~P_e</math>, confines the isothermal configuration to a finite size. In a mathematical model, this can be accomplished by ripping off an outer layer of the ''isolated'' isothermal configuration down to the radius — label it <math>~\xi_e</math> — at which the configuration's original (internal) pressure equals <math>~P_e</math>; the interior of the configuration that remains — containing mass <math>~M_{\xi_e}</math> — should be unaltered and in equilibrium. (This will work only for spherically symmetric configurations, as the gravitational acceleration at any location only depends on the mass contained inside that radius.) [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) are credited with constructing the first such models and, most significantly, discovering that, for any specified sound speed and applied external pressure, there is a mass above which no equilibrium configuration exists. We present, here, the salient elements of these (essentially equivalent) derivations.

	Prior to studying this discussion of pressure-bounded isothermal spheres, we recommend studying our [[User:Tohline/SSC/Structure/PolytropesEmbedded#n_.3D_5_Polytrope\|related discussion of pressure-bounded {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytropes]]. As with isolated isothermal spheres, isolated {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytropes extend to infinity. But, unlike their isothermal counterparts, the structure of {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytropes is describable analytically. Hence, an analysis of their structure and its extension to pressure-bounded configurations avoids the clutter introduced by a model — such as the isothermal sphere — that can only be described numerically. As it turns out, the pressure-bounded {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytrope exhibits a ''Bonnor-Ebert'' type limiting mass that is analytically prescribable. Its derivation is mathematically quite clean and provides a firm foundation for understanding the better known — but only numerically prescribable — Bonnor-Ebert limiting mass.		Prior to studying this discussion of pressure-bounded isothermal spheres, we recommend studying our [[User:Tohline/SSC/Structure/PolytropesEmbedded#n_.3D_5_Polytrope\|related discussion of pressure-bounded {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytropes]]. As with isolated isothermal spheres, isolated {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytropes extend to infinity. But, unlike their isothermal counterparts, the structure of {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytropes is describable analytically. Hence, an analysis of their structure and its extension to pressure-bounded configurations avoids the clutter introduced by a model — such as the isothermal sphere — that can only be described numerically. As it turns out, the pressure-bounded {{User:Tohline/Math/MP_PolytropicIndex}} = 5 polytrope exhibits a ''Bonnor-Ebert'' type limiting mass that is analytically prescribable. Its derivation is mathematically quite clean and provides a firm foundation for understanding the better known — but only numerically prescribable — Bonnor-Ebert limiting mass.

Tohline: /* P-V Diagram */

2019-03-14T20:46:27Z

P-V Diagram

← Older revision		Revision as of 20:46, 14 March 2019
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	</td>		</td>
	<td align="center" rowspan="2">		<td align="center" rowspan="2">
	[[File:~~WhitworthFig1bFlipped~~.~~jpg~~\|~~250px~~\|center\|Whitworth (1981) Figure 1b]]		[[File:WhitworthFig1bFlippedClipped.png\|300px\|center\|Whitworth (1981) Figure 1b]]
	<!-- [[Image:AAAwaiting01.png\|250px\|center\|Whitworth (1981) Figure 1b]] -->		<!-- [[Image:AAAwaiting01.png\|250px\|center\|Whitworth (1981) Figure 1b]] -->
	</td>		</td>

Tohline: /* P-V Diagram */

2019-03-14T20:31:47Z

P-V Diagram

← Older revision		Revision as of 20:31, 14 March 2019
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	</div>		</div>

	<div align="center" id="~~Fig2b~~">		<div align="center" id="Fig3">
	<table border="1">		<table border="1" width="80%">
	<tr>		<tr>
	<td align="center" colspan="3">		<td align="center" colspan="3">
	'''Figure 2b: Whitworth's P-R Diagrams'''		'''Figure 3: Whitworth's P-R Diagrams'''
	</td>		</td>
	</tr>		</tr>
	<tr>		<tr>
	<td align="center">		<td align="center" width="30%>
	[[File:WhitworthFig1bCopy.jpg\|~~600px~~\|center\|Whitworth (1981) Figure 1b]]		[[File:WhitworthFig1bCopy.jpg\|400px\|center\|Whitworth (1981) Figure 1b]]
	<!-- [[Image:AAAwaiting01.png\|250px\|center\|Ebert (1955) Figure 2]] -->		<!-- [[Image:AAAwaiting01.png\|250px\|center\|Ebert (1955) Figure 2]] -->
	</td>		</td>
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	</td>		</td>
	<td align="center" rowspan="2">		<td align="center" rowspan="2">
	[[File:WhitworthFig1bFlipped.jpg\|~~350px~~\|center\|Whitworth (1981) Figure 1b]]		[[File:WhitworthFig1bFlipped.jpg\|250px\|center\|Whitworth (1981) Figure 1b]]
	<!-- [[Image:AAAwaiting01.png\|250px\|center\|Whitworth (1981) Figure 1b]] -->		<!-- [[Image:AAAwaiting01.png\|250px\|center\|Whitworth (1981) Figure 1b]] -->
	</td>		</td>
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	<tr>		<tr>
	<td align="center">		<td align="center">
	[[File:WhitworthFig1bUsingEmdenData.png\|~~450px~~\|center\|P-R Diagram Using Emden's (1907) data]]		[[File:WhitworthFig1bUsingEmdenData.png\|300px\|center\|P-R Diagram Using Emden's (1907) data]]
	</td>		</td>
	</tr>		</tr>