SSC/Structure/BonnorEbert - Revision history

Jet53man: /* Related Discussions */

2021-09-20T17:00:36Z

Related Discussions

← Older revision		Revision as of 13:00, 20 September 2021
Line 1,089:		Line 1,089:
	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes embedded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes embedded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]		* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia		** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia

Jet53man: /* Pressure-Bounded Isothermal Sphere */

2021-07-29T15:55:33Z

Pressure-Bounded Isothermal Sphere

← Older revision		Revision as of 11:55, 29 July 2021
Line 4:		Line 4:
	\|-		\|-
	! style="height: 125px; width: 125px; background-color:#ffeeee;" \|		! style="height: 125px; width: 125px; background-color:#ffeeee;" \|
	<font size="-1">[[H_BookTiledMenu#~~Equilibrium_Structures~~\|<b>Bonnor-Ebert<br />(Isothermal)<br />Spheres</b>]]<br />(1955-56)</font>		<font size="-1">[[H_BookTiledMenu#MoreModels\|<b>Bonnor-Ebert<br />(Isothermal)<br />Spheres</b>]]<br />(1955-56)</font>
	\|}		\|}

Jet53man: /* Pressure-Bounded Isothermal Sphere */

2021-07-29T15:46:43Z

Pressure-Bounded Isothermal Sphere

← Older revision		Revision as of 11:46, 29 July 2021
Line 1:		Line 1:
	__FORCETOC__		__FORCETOC__
	=Pressure-Bounded Isothermal Sphere=		=Pressure-Bounded Isothermal Sphere=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:#ffeeee;" \|
			<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|<b>Bonnor-Ebert<br />(Isothermal)<br />Spheres</b>]]<br />(1955-56)</font>
			\|}

	As has been [[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 [[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.

Jet53man: /* Related Discussions */

2021-07-27T17:40:41Z

Related Discussions

← Older revision		Revision as of 13:40, 27 July 2021
Line 1,082:		Line 1,082:

	=Related Discussions=		=Related Discussions=
	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes ~~emdeded~~ in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes embedded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic5_1#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]

Jet53man: /* Mass versus Radius Equilibrium Sequence */

2021-07-27T17:39:21Z

Mass versus Radius Equilibrium Sequence

← Older revision		Revision as of 13:39, 27 July 2021
Line 492:		Line 492:
	==Mass versus Radius Equilibrium Sequence==		==Mass versus Radius Equilibrium Sequence==
	===Old Derivation===		===Old 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 [[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,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
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, <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,		For use in a [[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">

Jet53man: /* Limiting Pressure and Maximum Mass */

2021-07-27T17:37:34Z

Limiting Pressure and Maximum Mass

← Older revision		Revision as of 13:37, 27 July 2021
Line 395:		Line 395:
	<math>~1-\frac{1}{2} e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2 = 0 \, .</math>		<math>~1-\frac{1}{2} e^{\psi} \biggl( \frac{d\psi}{d\xi}\biggr)^2 = 0 \, .</math>
	</div>		</div>
	In the following figure, the function defined by the left-hand-side of this expression is plotted versus <math>~\ln\xi</math> using [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) tabulated data]]. As Bonnor noted, the function first crosses zero when <math>~\xi \approx 6.5</math> <math>~(\ln\xi = 1.87)</math>.		In the following figure, the function defined by the left-hand-side of this expression is plotted versus <math>~\ln\xi</math> using [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) tabulated data]]. As Bonnor noted, the function first crosses zero when <math>~\xi \approx 6.5</math> <math>~(\ln\xi = 1.87)</math>.

	<div align="center">		<div align="center">
Line 413:		Line 413:
	</math>		</math>
	</div>		</div>
	which matches equation (3.3) of [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)]; and, drawing on function values provided in [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) Table 14]], it will have a total mass,		which matches equation (3.3) of [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)]; and, drawing on function values provided in [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) Table 14]], it will have a total mass,
	<div align="center">		<div align="center">
	<math>~M_R = \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[\xi^2 \frac{d\psi}{d\xi} \biggr]_e \approx		<math>~M_R = \biggl( \frac{c_s^6}{4\pi G^3 \rho_c} \biggr)^{1/2} \biggl[\xi^2 \frac{d\psi}{d\xi} \biggr]_e \approx
Line 429:		Line 429:
	<math>P_e = c_s^2 \rho_c e^{-\psi} \approx c_s^2 \rho_c (0.08493 + 0.05833)/2 = 0.0716~ c_s^2 \rho_c \, ,</math>		<math>P_e = c_s^2 \rho_c e^{-\psi} \approx c_s^2 \rho_c (0.08493 + 0.05833)/2 = 0.0716~ c_s^2 \rho_c \, ,</math>
	</div>		</div>
	where, again, numerical values have been drawn from [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) Table 14]].		where, again, numerical values have been drawn from [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Emden's (1907) Table 14]].

	<span id="cychenMaryland">		<span id="cychenMaryland">
Line 473:		Line 473:

	This is the mass upper limit for a stable, pressure-bounded isothermal sphere — the so-called Bonnor-Ebert mass; see, for example, equation (1) of [http://adsabs.harvard.edu/abs/1977ApJ...214..488S Shu (1977)].		This is the mass upper limit for a stable, pressure-bounded isothermal sphere — the so-called Bonnor-Ebert mass; see, for example, equation (1) of [http://adsabs.harvard.edu/abs/1977ApJ...214..488S Shu (1977)].
	[[~~User:Tohline/~~SSC/Structure/LimitingMasses#Bounded_Isothermal_Sphere_.26_Bonnor-Ebert_Mass\|In a separate discussion]], we compare this result to the determinations of other related mass upper-limits.		[[SSC/Structure/LimitingMasses#Bounded_Isothermal_Sphere_.26_Bonnor-Ebert_Mass\|In a separate discussion]], we compare this result to the determinations of other related mass upper-limits.

	===Ebert's Corresponding Presentation===		===Ebert's Corresponding Presentation===

Jet53man: /* P-V Diagram */

2021-07-27T17:35:31Z

P-V Diagram

← Older revision		Revision as of 13:35, 27 July 2021
Line 269:		Line 269:
	<tr><td align="right">Figure No.:</td><td align="left">2 (p. 222)</td></tr>		<tr><td align="right">Figure No.:</td><td align="left">2 (p. 222)</td></tr>
	<tr><td align="right">Copyright:</td><td align="left">[https://www.springer.com/gp/ Springer-Verlag] —</td></tr>		<tr><td align="right">Copyright:</td><td align="left">[https://www.springer.com/gp/ Springer-Verlag] —</td></tr>
	<tr><td align="right"> </td><td align="left">Required reuse permissions [[~~User:Tohline/~~Appendix/Images#MNRAS\|summarized here]] and [https://www.springer.com/gp/authors-editors/book-authors-editors/resources-guidelines/rights-permissions-licensing detailed by the publisher here].</td></tr>		<tr><td align="right"> </td><td align="left">Required reuse permissions [[Appendix/Images#MNRAS\|summarized here]] and [https://www.springer.com/gp/authors-editors/book-authors-editors/resources-guidelines/rights-permissions-licensing detailed by the publisher here].</td></tr>
	</table>		</table>
	</td>		</td>
	<td align="left">		<td align="left">
	<div align="center">'''Pertaining to Our Original Figure''' (immediately above)</div>		<div align="center">'''Pertaining to Our Original Figure''' (immediately above)</div>
	Caption: In an effort to demonstrate more explicitly how Ebert's 1955 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve in the above panel displays 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)]].		Caption: In an effort to demonstrate more explicitly how Ebert's 1955 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve in the above panel displays 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 [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].
	</td>		</td>
	</tr>		</tr>
Line 281:		Line 281:


	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 [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].


	===Whitworth's Normalization===		===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>.		[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 [[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">
Line 308:		Line 308:
	<tr><td align="right">Figure No.:</td><td align="left">1b (p. 971)</td></tr>		<tr><td align="right">Figure No.:</td><td align="left">1b (p. 971)</td></tr>
	<tr><td align="right">Copyright:</td><td align="left">[https://ras.ac.uk Royal Astronomical Society] —</td></tr>		<tr><td align="right">Copyright:</td><td align="left">[https://ras.ac.uk Royal Astronomical Society] —</td></tr>
	<tr><td align="right"> </td><td align="left">Permission to reuse this figure has been granted implicitly as [[~~User:Tohline/~~Appendix/Images#MNRAS\|summarized here]] and as [https://academic.oup.com/mnras/pages/rights_and_new_business_development detailed by the publisher here].</td></tr>		<tr><td align="right"> </td><td align="left">Permission to reuse this figure has been granted implicitly as [[Appendix/Images#MNRAS\|summarized here]] and as [https://academic.oup.com/mnras/pages/rights_and_new_business_development detailed by the publisher here].</td></tr>
	</table>		</table>
	<p></p> <p></p> <p></p>		<p></p> <p></p> <p></p>
Line 325:		Line 325:
	<table border="0" cellpadding="2" align="center" width="90%"><tr><td align="center">'''Pertaining to Our Original Figure''' (on the bottom-left)</td></tr>		<table border="0" cellpadding="2" align="center" width="90%"><tr><td align="center">'''Pertaining to Our Original Figure''' (on the bottom-left)</td></tr>
	<tr><td align="left">		<tr><td align="left">
	Caption: In an effort to demonstrate more explicitly how Whitworth's 1981 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve displays a plot of <math>~P_e/P_\mathrm{rf}</math> versus <math>~V/V_\mathrm{rf}</math> that we have constructed using data drawn directly from [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].		Caption: In an effort to demonstrate more explicitly how Whitworth's 1981 work ties back to the original work on isolated isothermal spheres that was published by Emden in 1907, the blue curve displays a plot of <math>~P_e/P_\mathrm{rf}</math> versus <math>~V/V_\mathrm{rf}</math> that we have constructed using data drawn directly from [[SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution\|Table 14 of Emden (1907)]].
	</td></tr>		</td></tr>
	</table>		</table>
Line 343:		Line 343:


	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 [[SSC/Structure/PolytropesASIDE1\|accompanying ASIDE]], Whitworth "referenced" <math>P_e</math> and <math>R</math> to, respectively,
	<div align="center">		<div align="center">
	<math>~P_\mathrm{rf} = \biggl( \frac{3^4\cdot 5^3}{2^8} \biggr) P_0 \, ,</math>		<math>~P_\mathrm{rf} = \biggl( \frac{3^4\cdot 5^3}{2^8} \biggr) P_0 \, ,</math>

Jet53man: Created page with "FORCETOC =Pressure-Bounded Isothermal Sphere= As has been derived and discussed elsewhere, an ''isolated'..."

2021-07-27T17:31:26Z

Created page with "__FORCETOC__ =Pressure-Bounded Isothermal Sphere= As has been derived and discussed elsewhere, an ''isolated'..."

Show changes

← Older revision		Revision as of 13:00, 20 September 2021
Line 1,089:		Line 1,089:
	* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes embedded in an external medium]]		* [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|Polytropes embedded in an external medium]]
	* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]		* [[SSC/Structure/BiPolytropes#BiPolytropes\|Constructing BiPolytropes]]
	* [[SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]		* [[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|Analytic description of BiPolytrope with <math>(n_c, n_e) = (5,1)</math>]]
	* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]		* [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]]
	** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia		** [http://en.wikipedia.org/wiki/Bonnor-Ebert_mass Bonnor-Ebert Mass] according to Wikipedia

SSC/Structure/BonnorEbert - Revision history

Jet53man: /* Related Discussions */

Jet53man: /* Pressure-Bounded Isothermal Sphere */

Jet53man: /* Pressure-Bounded Isothermal Sphere */

Jet53man: /* Related Discussions */

Jet53man: /* Mass versus Radius Equilibrium Sequence */

Jet53man: /* Limiting Pressure and Maximum Mass */

Jet53man: /* P-V Diagram */

Jet53man: Created page with "__FORCETOC__ =Pressure-Bounded Isothermal Sphere= As has been derived and discussed elsewhere, an ''isolated'..."

Jet53man: Created page with "FORCETOC =Pressure-Bounded Isothermal Sphere= As has been derived and discussed elsewhere, an ''isolated'..."