SSCpt2/IntroductorySummary - Revision history

Jet53man: /* Bipolytropic Configurations */

2022-10-17T16:57:25Z

Bipolytropic Configurations

← Older revision		Revision as of 12:57, 17 October 2022
Line 162:		Line 162:
	Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' — also sometimes referred to as ''composite polytropes.'' These are models in which the configuration's "core" is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the configuration's "envelope" is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.		Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' — also sometimes referred to as ''composite polytropes.'' These are models in which the configuration's "core" is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the configuration's "envelope" is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.

	Following the lead of ~~[http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]~~ — see, also, ~~[http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon (1998)]~~ — we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface. As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>. Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>. As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme — specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here — there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>. The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch. It is usually referred to as the Schönberg-Chandrasekhar mass limit.		Following the lead of {{ SC42full }} — see, also, {{ EFC98full }} — we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface. As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>. Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>. As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme — specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here — there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>. The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch. It is usually referred to as the Schönberg-Chandrasekhar mass limit.

	====Details====		====Details====

174.64.8.247: /* Applications */

2021-07-28T16:44:36Z

Applications

← Older revision		Revision as of 12:44, 28 July 2021
Line 6:		Line 6:
	\|-		\|-
	! style="height: 125px; width: 125px; background-color:lightgreen;" \|		! style="height: 125px; width: 125px; background-color:lightgreen;" \|
	<font size="-1">[[H_BookTiledMenu#~~Spherically_Symmetric_Configurations~~\|Hydrostatic<br />Balance<br />Equations]]</font>		<font size="-1">[[H_BookTiledMenu#Equilibrium_Structures\|Hydrostatic<br />Balance<br />Equations]]</font>
	\|}		\|}
	==Spherically Symmetric Configurations==		==Spherically Symmetric Configurations==
Line 162:		Line 162:
	Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' — also sometimes referred to as ''composite polytropes.'' These are models in which the configuration's "core" is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the configuration's "envelope" is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.		Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' — also sometimes referred to as ''composite polytropes.'' These are models in which the configuration's "core" is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the configuration's "envelope" is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.

	Following the lead of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] — see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon (1998)] — we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{~~User:Tohline/~~Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface. As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>. Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>. As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme — specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here — there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>. The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch. It is usually referred to as the Schönberg-Chandrasekhar mass limit.		Following the lead of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] — see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon (1998)] — we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface. As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>. Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>. As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme — specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here — there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>. The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch. It is usually referred to as the Schönberg-Chandrasekhar mass limit.

	====Details====		====Details====

174.64.8.247: /* Applications */

2021-07-28T16:15:33Z

Applications

← Older revision		Revision as of 12:15, 28 July 2021
Line 3:		Line 3:

	=Applications=		=Applications=
			{\| class="PGEclass" style="float:left; margin-right: 20px; border-style: solid; border-width: 3px border-color: black"
			\|-
			! style="height: 125px; width: 125px; background-color:lightgreen;" \|
			<font size="-1">[[H_BookTiledMenu#Spherically_Symmetric_Configurations\|Hydrostatic<br />Balance<br />Equations]]</font>
			\|}
	==Spherically Symmetric Configurations==		==Spherically Symmetric Configurations==

Line 164:		Line 168:


	{{~~LSU_HBook_SSC_Structure~~}}		{{ SSCstructure }}


	{{ SGFfooter }}		{{ SGFfooter }}

Jet53man: /* Pressure-Truncated Configurations */

2021-07-18T23:51:33Z

Pressure-Truncated Configurations

← Older revision		Revision as of 19:51, 18 July 2021
Line 76:		Line 76:

	====Pressure-Truncated Configurations====		====Pressure-Truncated Configurations====
	If the physical system under consideration — such as a protostellar gas cloud — is not isolated but is, instead, surrounded and ''truncated'' by a hot, tenuous medium that exerts on the system a confining external pressure, <math>~P_e</math>, the configuration's equilibrium parameters will be related via the expression,		If the physical system under consideration — such as a protostellar gas cloud — is not isolated but is, instead, surrounded and ''truncated'' by a hot, tenuous medium that exerts on the system a confining external pressure, <math>P_e</math>, the configuration's equilibrium parameters will be related via the expression,

	<div align="center">		<div align="center">
Line 83:		Line 83:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>		<math>\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W} \biggl[ \mathfrak{f}_A - \frac{P_e}{P_c} \biggr]		<math>\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W} \biggl[ \mathfrak{f}_A - \frac{P_e}{P_c} \biggr]
	\, ;</math>		\, ;</math>
	</td>		</td>
Line 95:		Line 95:
	</table>		</table>
	</div>		</div>
	or, fixing {{~~User:Tohline/~~Math/MP_PolytropicConstant}} instead of <math>~P_c</math>, the [[~~User:Tohline/SSC~~/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_2\|relevant virial equilibrium expression]] is,		or, fixing {{ Math/MP_PolytropicConstant }} instead of <math>P_c</math>, the [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_2\|relevant virial equilibrium expression]] is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 101:		Line 101:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~P_e </math>		<math>P_e </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~K_\mathrm{n} \mathfrak{f}_A \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \cdot \frac{1}{\mathfrak{f}_M} \biggr)^{1 + 1/n}		<math>K_\mathrm{n} \mathfrak{f}_A \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \cdot \frac{1}{\mathfrak{f}_M} \biggr)^{1 + 1/n}
	- \frac{\mathfrak{f}_W}{5} \biggl(\frac{3GM^2}{4\pi R_\mathrm{eq}^4} \cdot \frac{1}{\mathfrak{f}_M^2} \biggr) \, .</math>		- \frac{\mathfrak{f}_W}{5} \biggl(\frac{3GM^2}{4\pi R_\mathrm{eq}^4} \cdot \frac{1}{\mathfrak{f}_M^2} \biggr) \, .</math>
	</td>		</td>
Line 113:		Line 113:
	</table>		</table>
	</div>		</div>
	It is a virial expression specifically of this form <math>~(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbow-colored free-energy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, [[~~User:Tohline/SSC~~/Virial/PolytropesEmbeddedOutline#ASIDE:__Isothermal_Configurations\|displayed above]]. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>~P_e</math> is greater than some limiting value, <math>~P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>~P_e = P_\mathrm{crit}</math>, is marked by a red dot on the above-displayed free-energy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[~~User:Tohline/~~SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]].		It is a virial expression specifically of this form <math>(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbow-colored free-energy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, [[SSCpt1/Virial/PolytropesEmbeddedOutline#ASIDE:__Isothermal_Configurations\|displayed above]]. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>P_e</math> is greater than some limiting value, <math>P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>P_e = P_\mathrm{crit}</math>, is marked by a red dot on the above-displayed free-energy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|Bonnor-Ebert spheres]].

	After rearranging terms, for any specified values of the parameters <math>~P_e</math> and {{~~User:Tohline/~~Math/MP_PolytropicConstant}}, this virial equilibrium expression can also be viewed as a [[~~User:Tohline/SSC~~/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_3\|mass-radius relation of the form]],		After rearranging terms, for any specified values of the parameters <math>P_e</math> and {{ Math/MP_PolytropicConstant }}, this virial equilibrium expression can also be viewed as a [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_3\|mass-radius relation of the form]],
	<table border="0" align="right" cellpadding="5">		<table border="0" align="right" cellpadding="5">
	<tr>		<tr>
Line 128:		Line 128:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~a R_\mathrm{eq}^4 - b M^{(n+1)/n} R_\mathrm{eq}^{(n-3)/n} + c M^2</math>		<math>a R_\mathrm{eq}^4 - b M^{(n+1)/n} R_\mathrm{eq}^{(n-3)/n} + c M^2</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~ 0\, ,</math>		<math>0\, ,</math>
	</td>		</td>
	</tr>		</tr>
Line 141:		Line 141:
	where the constants,		where the constants,
	<div align="center">		<div align="center">
	<math>~a \equiv \frac{4\pi}{3} \cdot P_e \, ,</math>      <math>~b \equiv \biggl( \frac{3}{4\pi} \biggr)^{1/n} \cdot K_\mathrm{n} \mathfrak{f}_A \mathfrak{f}_M^{-(n+1)/n} \, ,</math>       and      <math>~c \equiv \frac{G\mathfrak{f}_W}{5\mathfrak{f}_M^2} \, .</math>		<math>a \equiv \frac{4\pi}{3} \cdot P_e \, ,</math>      <math>b \equiv \biggl( \frac{3}{4\pi} \biggr)^{1/n} \cdot K_\mathrm{n} \mathfrak{f}_A \mathfrak{f}_M^{-(n+1)/n} \, ,</math>       and      <math>c \equiv \frac{G\mathfrak{f}_W}{5\mathfrak{f}_M^2} \, .</math>
	</div>		</div>

	Using this virial equilibrium relation (and, for illustration purposes, assuming a = b = c = 1), the [[~~User:Tohline/~~SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation\|curves drawn in the figure that is displayed here, on the right]], show how the equilibrium radius of an embedded, pressure-truncated polytropic sphere varies with mass for seven different adopted polytropic indexes. In direct analogy with the critical pressure that is associated with Bonnor-Ebert spheres, for systems having <math>~n \ge 3</math>, there is a mass, <math>~M_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, when <math>~n > 3</math>, multiple equilibrium configurations having different radii can be constructed for any system having a mass, <math>~M < M_\mathrm{max}</math>.		Using this virial equilibrium relation (and, for illustration purposes, assuming a = b = c = 1), the [[SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation\|curves drawn in the figure that is displayed here, on the right]], show how the equilibrium radius of an embedded, pressure-truncated polytropic sphere varies with mass for seven different adopted polytropic indexes. In direct analogy with the critical pressure that is associated with Bonnor-Ebert spheres, for systems having <math>~n \ge 3</math>, there is a mass, <math>M_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, when <math>n > 3</math>, multiple equilibrium configurations having different radii can be constructed for any system having a mass, <math>M < M_\mathrm{max}</math>.

	A [[~~User:Tohline/~~SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|detailed force-balance analysis of the structure of embedded, pressure-truncated polytropic configurations]] reveals, for each choice of {{~~User:Tohline/~~Math/MP_PolytropicIndex}}, a mass-radius relationship that is qualitatively similar to the one deduced from a virial equilibrium analysis. However, the resulting mass-radius relationship is invariably different in detail and quantitatively more correct than the one prescribed by the virial theorem. At its foundation are models whose internal structural profile — of, for example, the fluid pressure and fluid density — is not uniform but, rather, is precisely that which is required to achieve detailed force balance throughout. Hence, we appreciate that even as <math>~P_e</math> and {{~~User:Tohline/~~Math/MP_PolytropicConstant}} are held fixed, in essence the structural form factors must vary from model to model along the more precise "detailed force-balanced" equilibrium ''sequence.''		A [[SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres\|detailed force-balance analysis of the structure of embedded, pressure-truncated polytropic configurations]] reveals, for each choice of {{ Math/MP_PolytropicIndex }}, a mass-radius relationship that is qualitatively similar to the one deduced from a virial equilibrium analysis. However, the resulting mass-radius relationship is invariably different in detail and quantitatively more correct than the one prescribed by the virial theorem. At its foundation are models whose internal structural profile — of, for example, the fluid pressure and fluid density — is not uniform but, rather, is precisely that which is required to achieve detailed force balance throughout. Hence, we appreciate that even as <math>P_e</math> and {{ Math/MP_PolytropicConstant }} are held fixed, in essence the structural form factors must vary from model to model along the more precise "detailed force-balanced" equilibrium ''sequence.''

	====Bipolytropic Configurations====		====Bipolytropic Configurations====

Jet53man: /* Isolated Configurations */

2021-07-18T23:43:33Z

Isolated Configurations

← Older revision		Revision as of 19:43, 18 July 2021
Line 38:		Line 38:
	If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression\|global free energy]]. An illustrative, undulating free-energy surface is displayed here, on the right; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.		If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression\|global free energy]]. An illustrative, undulating free-energy surface is displayed here, on the right; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.

	In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem\|''scalar virial theorem'']]. Specifically, for [[~~SSCpt1~~/Virial/Polytropes#TwoPointsOfView\|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:		In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem\|''scalar virial theorem'']]. Specifically, for [[SSC/Virial/Polytropes#TwoPointsOfView\|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 56:		Line 56:
	</table>		</table>
	</div>		</div>
	where all three of the dimensionless ''structural form factors'', <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math>, are unity, under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, and are otherwise generically ''of order unity'' for detailed force-balanced models having a wide range of internal structures. Alternatively, if the parameter, {{~~User:Tohline/~~Math/MP_PolytropicConstant}} (which defines the specific entropy of fluid elements throughout the configuration), rather than the central pressure, is held fixed while searching for extrema in the free-energy, the [[~~User:Tohline/~~SSC/Virial/Polytropes#TwoPointsOfView\|virial equilibrium relation for isolated polytropes]] is,		where all three of the dimensionless ''structural form factors'', <math>\mathfrak{f}_M</math>, <math>\mathfrak{f}_W</math>, and <math>\mathfrak{f}_A</math>, are unity, under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, and are otherwise generically ''of order unity'' for detailed force-balanced models having a wide range of internal structures. Alternatively, if the parameter, {{ Math/MP_PolytropicConstant }} (which defines the specific entropy of fluid elements throughout the configuration), rather than the central pressure, is held fixed while searching for extrema in the free-energy, the [[SSC/Virial/Polytropes#TwoPointsOfView\|virial equilibrium relation for isolated polytropes]] is,
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 62:		Line 62:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~M^{(n-1)/n} R_\mathrm{eq}^{(3-n)/n} \biggl( \frac{G}{K_\mathrm{n}} \biggr) </math>		<math>M^{(n-1)/n} R_\mathrm{eq}^{(3-n)/n} \biggl( \frac{G}{K_\mathrm{n}} \biggr) </math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggl(\frac{3}{4\pi \mathfrak{f}_M}\biggr)^{1/n} \, .		<math>\frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggl(\frac{3}{4\pi \mathfrak{f}_M}\biggr)^{1/n} \, .
	</math>		</math>
	</td>		</td>

Jet53man: /* Spherically Symmetric Configurations */

2021-07-18T23:37:18Z

Spherically Symmetric Configurations

← Older revision		Revision as of 19:37, 18 July 2021
Line 20:		Line 20:
	<tr>		<tr>
	<td align="center">		<td align="center">
	[[~~SSCpt2~~/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|Free-Energy Surface]]		[[SSCpt1/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|Free-Energy Surface]]
	</td>		</td>
	</tr>		</tr>
Line 36:		Line 36:
	As our various discussions illustrate (see the table of contents, below), simply varying the power-law index, {{ Math/MP_PolytropicIndex }}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.		As our various discussions illustrate (see the table of contents, below), simply varying the power-law index, {{ Math/MP_PolytropicIndex }}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.

	If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression\|global free energy]]. An illustrative, undulating free-energy surface is ~~[[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres\|~~displayed here, on the right]]; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.		If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression\|global free energy]]. An illustrative, undulating free-energy surface is displayed here, on the right; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.

	In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem\|''scalar virial theorem'']]. Specifically, for [[SSCpt1/Virial/Polytropes#TwoPointsOfView\|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:		In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem\|''scalar virial theorem'']]. Specifically, for [[SSCpt1/Virial/Polytropes#TwoPointsOfView\|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:

Jet53man: /* Structure: */

2021-07-18T22:42:12Z

Structure:

← Older revision		Revision as of 18:42, 18 July 2021
Line 8:		Line 8:
	===Structure:===		===Structure:===

	Here we show how the set of [[~~User:Tohline/~~PGE#Principal_Governing_Equations\|principal governing equations]] (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds.		Here we show how the set of [[PGE#Principal_Governing_Equations\|principal governing equations]] (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds.

	====Isolated Configurations====		====Isolated Configurations====
	After supplementing the PGEs by specifying an [[~~User:Tohline/~~SR#Supplemental_Relations\|equation of state of the fluid]], the system of equations is usually solved by employing one of [[~~User:Tohline/SphericallySymmetricConfigurations~~/SolutionStrategies#Solution_Strategies\|three techniques]] to obtain a "detailed force-balanced" model that provides the radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration — given its mass, <math>~M</math>, and central pressure, <math>~P_c</math>, for example — as well as details regarding the internal radial profiles of the fluid density and fluid pressure. If a ''polytropic'' equation of state,		After supplementing the PGEs by specifying an [[SR#Supplemental_Relations\|equation of state of the fluid]], the system of equations is usually solved by employing one of [[SSCpt2/SolutionStrategies#Solution_Strategies\|three techniques]] to obtain a "detailed force-balanced" model that provides the radius, <math>R_\mathrm{eq}</math>, of the equilibrium configuration — given its mass, <math>M</math>, and central pressure, <math>P_c</math>, for example — as well as details regarding the internal radial profiles of the fluid density and fluid pressure. If a ''polytropic'' equation of state,
	<table border="0" align="right" cellpadding="5">		<table border="0" align="right" cellpadding="5">
	<tr>		<tr>
Line 20:		Line 20:
	<tr>		<tr>
	<td align="center">		<td align="center">
	[[~~User:Tohline/SSC~~/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|Free-Energy Surface]]		[[SSCpt2/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface\|Free-Energy Surface]]
	</td>		</td>
	</tr>		</tr>
Line 26:		Line 26:

	<div align="center">		<div align="center">
	{{~~User:Tohline/~~Math/EQ_Polytrope01}}		{{ Math/EQ_Polytrope01 }}
	</div>		</div>
	is adopted, for example, a detailed force-balanced model is fully described by the radially dependent function, <math>~\Theta_H(\xi) = [\rho(\xi)/\rho_c]^{1/n}</math>, which is obtained by solving the		is adopted, for example, a detailed force-balanced model is fully described by the radially dependent function, <math>\Theta_H(\xi) = [\rho(\xi)/\rho_c]^{1/n}</math>, which is obtained by solving the
	<div align="center">		<div align="center">
	Lane-Emden Equation<br />		Lane-Emden Equation<br />

	{{ ~~User:Tohline/~~Math/EQ_SSLaneEmden01 }}		{{ Math/EQ_SSLaneEmden01 }}
	</div>		</div>
	As our various discussions illustrate (see the table of contents, below), simply varying the power-law index, {{~~User:Tohline/~~Math/MP_PolytropicIndex}}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.		As our various discussions illustrate (see the table of contents, below), simply varying the power-law index, {{ Math/MP_PolytropicIndex }}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.

	If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[~~User:Tohline/~~VE#Free_Energy_Expression\|global free energy]]. An illustrative, undulating free-energy surface is [[~~User:Tohline/SSC~~/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres\|displayed here, on the right]]; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.		If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[VE#Free_Energy_Expression\|global free energy]]. An illustrative, undulating free-energy surface is [[SSCpt1/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres\|displayed here, on the right]]; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.

	In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[~~User:Tohline/~~VE#Scalar_Virial_Theorem\|''scalar virial theorem'']]. Specifically, for [[~~User:Tohline/SSC~~/Virial/Polytropes#TwoPointsOfView\|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:		In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[VE#Scalar_Virial_Theorem\|''scalar virial theorem'']]. Specifically, for [[SSCpt1/Virial/Polytropes#TwoPointsOfView\|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:
	<div align="center">		<div align="center">
	<table border="0" cellpadding="5" align="center">		<table border="0" cellpadding="5" align="center">
Line 44:		Line 44:
	<tr>		<tr>
	<td align="right">		<td align="right">
	<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>		<math>\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
	</td>		</td>
	<td align="center">		<td align="center">
	<math>~=</math>		<math>=</math>
	</td>		</td>
	<td align="left">		<td align="left">
	<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W}		<math>\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W}
	\, ,</math>		\, ,</math>
	</td>		</td>

Jet53man: Created page with "FORCETOC =Applications= ==Spherically Symmetric Configurations== ===Structure:=== Here we show how the set of User:Tohline/PGE..."

2021-07-18T22:33:07Z

Created page with "__FORCETOC__  =Applications= ==Spherically Symmetric Configurations== ===Structure:=== Here we show how the set of User:Tohline/PGE..."

New page

__FORCETOC__


=Applications=

==Spherically Symmetric Configurations==

===Structure:===

Here we show how the set of [[User:Tohline/PGE#Principal_Governing_Equations|principal governing equations]] (PGEs) can be solved to determine the equilibrium structure of spherically symmetric fluid configurations — such as individual, nonrotating stars or protostellar gas clouds.

====Isolated Configurations====
After supplementing the PGEs by specifying an [[User:Tohline/SR#Supplemental_Relations|equation of state of the fluid]], the system of equations is usually solved by employing one of [[User:Tohline/SphericallySymmetricConfigurations/SolutionStrategies#Solution_Strategies|three techniques]] to obtain a "detailed force-balanced" model that provides the radius, <math>~R_\mathrm{eq}</math>, of the equilibrium configuration — given its mass, <math>~M</math>, and central pressure, <math>~P_c</math>, for example — as well as details regarding the internal radial profiles of the fluid density and fluid pressure. If a ''polytropic'' equation of state,
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:FreeNRGpressureRadiusIsothermal.png|250px|right|border|Whitworth's (1981) Isothermal Free-Energy Surface]]
</td>
</tr>
<tr>
<td align="center">
[[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#3DIsothermalSurface|Free-Energy Surface]]
</td>
</tr>
</table>

<div align="center">
{{User:Tohline/Math/EQ_Polytrope01}}
</div>
is adopted, for example, a detailed force-balanced model is fully described by the radially dependent function, <math>~\Theta_H(\xi) = [\rho(\xi)/\rho_c]^{1/n}</math>, which is obtained by solving the
<div align="center">
Lane-Emden Equation<br />

{{ User:Tohline/Math/EQ_SSLaneEmden01 }}
</div>
As our various discussions illustrate (see the table of contents, below), simply varying the power-law index, {{User:Tohline/Math/MP_PolytropicIndex}}, gives rise to equilibrium configurations that have a wide variety of internal structural profiles.

If one is not particularly concerned about details regarding the distribution of matter ''within'' the equilibrium configuration, a good estimate of the size of the equilibrium system can be determined by assuming a uniform-density structure then identifying local extrema in the system's [[User:Tohline/VE#Free_Energy_Expression|global free energy]]. An illustrative, undulating free-energy surface is [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_of_Embedded_Polytropic_Spheres|displayed here, on the right]]; blue dots identify equilibria associated with a "valley" of the free-energy surface while white dots identify equilibria that lie along a "ridge" in the free-energy surface.

In the astrophysics community, the mathematical relation that serves to define the properties of configurations that are associated with such free-energy extrema is often referred to as the [[User:Tohline/VE#Scalar_Virial_Theorem|''scalar virial theorem'']]. Specifically, for [[User:Tohline/SSC/Virial/Polytropes#TwoPointsOfView|''isolated'' systems in virial equilibrium]], the following relation between configuration parameters holds:
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_A \cdot \mathfrak{f}_M^2}{\mathfrak{f}_W}
\, ,</math>
</td>
</tr>
</table>
</div>
where all three of the dimensionless ''structural form factors'', <math>~\mathfrak{f}_M</math>, <math>~\mathfrak{f}_W</math>, and <math>~\mathfrak{f}_A</math>, are unity, under the assumption that the equilibrium configuration has uniform density and uniform pressure throughout, and are otherwise generically ''of order unity'' for detailed force-balanced models having a wide range of internal structures. Alternatively, if the parameter, {{User:Tohline/Math/MP_PolytropicConstant}} (which defines the specific entropy of fluid elements throughout the configuration), rather than the central pressure, is held fixed while searching for extrema in the free-energy, the [[User:Tohline/SSC/Virial/Polytropes#TwoPointsOfView|virial equilibrium relation for isolated polytropes]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~M^{(n-1)/n} R_\mathrm{eq}^{(3-n)/n} \biggl( \frac{G}{K_\mathrm{n}} \biggr) </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\frac{5\mathfrak{f}_A \mathfrak{f}_M}{\mathfrak{f}_W} \biggl(\frac{3}{4\pi \mathfrak{f}_M}\biggr)^{1/n} \, .
</math>
</td>
</tr>
</table>
</div>

====Pressure-Truncated Configurations====
If the physical system under consideration — such as a protostellar gas cloud — is not isolated but is, instead, surrounded and ''truncated'' by a hot, tenuous medium that exerts on the system a confining external pressure, <math>~P_e</math>, the configuration's equilibrium parameters will be related via the expression,

<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~\frac{GM^2}{P_c R_\mathrm{eq}^4}</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~\biggl(\frac{2^2\cdot 5 \pi}{3} \biggr) \frac{\mathfrak{f}_M^2}{\mathfrak{f}_W} \biggl[ \mathfrak{f}_A - \frac{P_e}{P_c} \biggr]
\, ;</math>
</td>
</tr>
</table>
</div>
or, fixing {{User:Tohline/Math/MP_PolytropicConstant}} instead of <math>~P_c</math>, the [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_2|relevant virial equilibrium expression]] is,
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~P_e </math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~K_\mathrm{n} \mathfrak{f}_A \biggl( \frac{3M}{4\pi R_\mathrm{eq}^3} \cdot \frac{1}{\mathfrak{f}_M} \biggr)^{1 + 1/n}
- \frac{\mathfrak{f}_W}{5} \biggl(\frac{3GM^2}{4\pi R_\mathrm{eq}^4} \cdot \frac{1}{\mathfrak{f}_M^2} \biggr) \, .</math>
</td>
</tr>
</table>
</div>
It is a virial expression specifically of this form <math>~(</math>with <math>n = \infty</math> and <math>\mathfrak{f}_M = \mathfrak{f}_W = \mathfrak{f}_A = 1)</math> that identifies extrema (e.g., valleys or ridges) in the rainbow-colored free-energy surface, <math>\mathfrak{G}^*(R_\mathrm{eq}, P_e)</math>, [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#ASIDE:__Isothermal_Configurations|displayed above]]. As can be determined from this algebraic expression and as the figure illustrates, for any specified mass no equilibrium states exist if <math>~P_e</math> is greater than some limiting value, <math>~P_\mathrm{crit}</math>; the equilibrium configuration associated with the limiting condition, <math>~P_e = P_\mathrm{crit}</math>, is marked by a red dot on the above-displayed free-energy surface. The astrophysical significance of this critical state was first discussed in the mid 1950s in the context of star formation and, specifically, [[User:Tohline/SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere|Bonnor-Ebert spheres]].

After rearranging terms, for any specified values of the parameters <math>~P_e</math> and {{User:Tohline/Math/MP_PolytropicConstant}}, this virial equilibrium expression can also be viewed as a [[User:Tohline/SSC/Virial/PolytropesEmbeddedOutline#Virial_Equilibrium_3|mass-radius relation of the form]],
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:MassRadiusVirialLabeled.png|250px|right|border|Virial Mass-Radius Relation]]
</td>
</tr>
</table>
<div align="center">
<table border="0" cellpadding="5" align="center">

<tr>
<td align="right">
<math>~a R_\mathrm{eq}^4 - b M^{(n+1)/n} R_\mathrm{eq}^{(n-3)/n} + c M^2</math>
</td>
<td align="center">
<math>~=</math>
</td>
<td align="left">
<math>~ 0\, ,</math>
</td>
</tr>
</table>
</div>
where the constants,
<div align="center">
<math>~a \equiv \frac{4\pi}{3} \cdot P_e \, ,</math>      <math>~b \equiv \biggl( \frac{3}{4\pi} \biggr)^{1/n} \cdot K_\mathrm{n} \mathfrak{f}_A \mathfrak{f}_M^{-(n+1)/n} \, ,</math>       and      <math>~c \equiv \frac{G\mathfrak{f}_W}{5\mathfrak{f}_M^2} \, .</math>
</div>

Using this virial equilibrium relation (and, for illustration purposes, assuming a = b = c = 1), the [[User:Tohline/SSC/Virial/PolytropesEmbedded/SecondEffortAgain#Plotting_Concise_Mass-Radius_Relation|curves drawn in the figure that is displayed here, on the right]], show how the equilibrium radius of an embedded, pressure-truncated polytropic sphere varies with mass for seven different adopted polytropic indexes. In direct analogy with the critical pressure that is associated with Bonnor-Ebert spheres, for systems having <math>~n \ge 3</math>, there is a mass, <math>~M_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, when <math>~n > 3</math>, multiple equilibrium configurations having different radii can be constructed for any system having a mass, <math>~M < M_\mathrm{max}</math>.

A [[User:Tohline/SSC/Structure/PolytropesEmbedded#Embedded_Polytropic_Spheres|detailed force-balance analysis of the structure of embedded, pressure-truncated polytropic configurations]] reveals, for each choice of {{User:Tohline/Math/MP_PolytropicIndex}}, a mass-radius relationship that is qualitatively similar to the one deduced from a virial equilibrium analysis. However, the resulting mass-radius relationship is invariably different in detail and quantitatively more correct than the one prescribed by the virial theorem. At its foundation are models whose internal structural profile — of, for example, the fluid pressure and fluid density — is not uniform but, rather, is precisely that which is required to achieve detailed force balance throughout. Hence, we appreciate that even as <math>~P_e</math> and {{User:Tohline/Math/MP_PolytropicConstant}} are held fixed, in essence the structural form factors must vary from model to model along the more precise "detailed force-balanced" equilibrium ''sequence.''

====Bipolytropic Configurations====
<table border="0" align="right" cellpadding="5">
<tr>
<td align="center" bgcolor="white">
[[File:PlotSequencesBest02.png|300px|right|border|Virial Mass-Radius Relation]]
</td>
</tr>
</table>
Additional insight into the structural properties and evolution of stars can be gained by studying ''bipolytropes'' — also sometimes referred to as ''composite polytropes.'' These are models in which the configuration's "core" is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the configuration's "envelope" is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. We have found it particularly instructive to examine bipolytropes having <math>~(n_c, n_e) = (5, 1)\, ,</math> in part, because equilibrium models of such systems can be completely described analytically whether they are constructed via a detailed force-balance analysis or by identifying extrema in the free-energy function, that is, via a virial theorem analysis.

Following the lead of [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] — see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon (1998)] — we have pieced these bipolytropic configurations together mathematically in such a way that the mean molecular weight, {{User:Tohline/Math/MP_MeanMolecularWeight}}, of the fluid is allowed to change in a discontinuous fashion at the core-envelope interface. As is illustrated in the figure, shown here on the right, a physically interesting equilibrium model "sequence" can be constructed by monotonically shifting the location of the core-envelope interface from the center of the configuration to its surface while holding fixed the value of the envelope-to-core mean molecular weight ratio, <math>~\mu_e/\mu_c</math>. Each curve shows how the relative mass of the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, correlates with the relative ''size'' of the core, as measured by the ratio of the radial position of the core-envelope interface to the equilibrium radius of the composite polytropic configuration, <math>~q \equiv r_i/R_\mathrm{eq}</math>. As the figure illustrates, if the jump in the mean molecular weight is sufficiently extreme — specifically, if <math>~\mu_e/\mu_c < 1/3</math> for the bipolytropic configurations being considered here — there is a core mass, <math>~\nu_\mathrm{max}</math>, above which equilibrium configurations do not exist; and, two equilibrium configurations having different core ''sizes'', <math>~q</math>, can be constructed for any system having a core mass, <math>~\nu < \nu_\mathrm{max}</math>. The astrophysical significance of <math>~\nu_\mathrm{max}</math> was first identified in the early 1940s in bipolytropic configurations having <math>~(n_c, n_e) = (\infty, 3/2)</math>, and has been discussed extensively in the context of the evolutionary transition of stars from the main sequence to the giant branch. It is usually referred to as the Schönberg-Chandrasekhar mass limit.

====Details====
In the following table, each green check mark identifies and provides a link to an H_Book chapter that presents a detailed discussion of the topic that is identified on the left — for example, the equilibirum structure of "isolated polytropes" or an "isothermal sphere embedded in an external medium." Mathematical models that provide full solutions to the PGEs, including details regarding the internal structural profiles of equilibrium configurations, are derived in chapters whose check marks fall under the column labeled "Detailed Force-Balance." Insight into the properties of equilibrium systems that is revealed via an analysis of a system's free-energy and the corresponding scalar virial theorem is presented in chapters whose check marks fall under the column labeled "Virial Equilibrium." Motivated by a striking similarity between the sets of model sequences displayed in the two "mass-radius" diagrams shown, above, a link also is provided to a chapter that discusses the relationship between the limiting mass associated with Bonnor-Ebert spheres and the Schönberg-Chandrasekhar mass limit.

{{LSU_HBook_SSC_Structure}}

{{ SGFfooter }}

SSCpt2/IntroductorySummary - Revision history

Jet53man: /* Bipolytropic Configurations */

174.64.8.247: /* Applications */

174.64.8.247: /* Applications */

Jet53man: /* Pressure-Truncated Configurations */

Jet53man: /* Isolated Configurations */

Jet53man: /* Spherically Symmetric Configurations */

Jet53man: /* Structure: */

Jet53man: Created page with "__FORCETOC__ =Applications= ==Spherically Symmetric Configurations== ===Structure:=== Here we show how the set of User:Tohline/PGE..."

Jet53man: Created page with "FORCETOC =Applications= ==Spherically Symmetric Configurations== ===Structure:=== Here we show how the set of User:Tohline/PGE..."