SSC/FreeEnergy/EquilibriumSequenceInstabilities - Revision history

Joel2: /* Schönberg-Chandrasekhar Mass */

2024-07-05T22:49:48Z

Schönberg-Chandrasekhar Mass

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	In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[~~User:Tohline/~~SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.		In the early 1940s, Chandrasekhar and his colleagues (see [http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] and [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]) discovered that a star with an isothermal core will become unstable if the fractional mass of the core is above some limiting value. They discovered this by constructing models that are now commonly referred to as ''composite polytropes'' or ''bipolytropes'', that is, models in which the star's core is described by a polytropic equation of state having one index — say, <math>~n_c</math> — and the star's envelope is described by a polytropic equation of state of a different index — say, <math>~n_e</math>. In [[SSC/Structure/BiPolytropes#BiPolytropes\|an accompanying discussion]] we explain in detail how the two structural components with different polytropic indexes are pieced together mathematically to build equilibrium bipolytropes. For a given choice of the two indexes, <math>~n_c</math> and <math>~n_e</math>, a sequence of models can be generated by varying the radial location at which the interface between the core and envelope occurs. As the interface location is varied, the relative amount of mass enclosed inside the core, <math>~\nu \equiv M_\mathrm{core}/M_\mathrm{tot}</math>, quite naturally varies as well.

	[http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] built structures of uniform composition having an isothermal core (<math>~n_c = \infty</math>) and an <math>~n_e = 3/2</math> polytropic envelope and found that equilibrium models exist only for values of <math>~\nu \le \nu_\mathrm{max} \approx 0.35</math>. [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] extended this analysis to include structures in which the mean molecular weight of the gas changes discontinuously across the interface. Specifically, they used the same values of <math>~n_c</math> and <math>~n_e</math> as Henrich & Chandrasekhar, but they constructed models in which the ratio of the molecular weight in the core to the molecular weight in the envelope is <math>~\mu_c/\mu_e = 2</math>. This was done to more realistically represent stars as they evolve off the main sequence; they have inert, isothermal helium cores and envelopes that are rich in hydrogen. Note that introducing a discontinuous drop in the mean molecular weight at the core-envelope interface also introduces a discontinuous drop in the gas density across the interface. As the following excerpt from p. 168 of their article summarizes, in these models, [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] found that <math>~\nu_\mathrm{max} \approx 0.101</math>. This is commonly referred to as the Schönberg-Chandrasekhar mass limit, although it was Henrich & Chandrasekhar who were the first to identify the instability.		[http://adsabs.harvard.edu/abs/1941ApJ....94..525H Henrich & Chandraskhar (1941)] built structures of uniform composition having an isothermal core (<math>~n_c = \infty</math>) and an <math>~n_e = 3/2</math> polytropic envelope and found that equilibrium models exist only for values of <math>~\nu \le \nu_\mathrm{max} \approx 0.35</math>. [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] extended this analysis to include structures in which the mean molecular weight of the gas changes discontinuously across the interface. Specifically, they used the same values of <math>~n_c</math> and <math>~n_e</math> as Henrich & Chandrasekhar, but they constructed models in which the ratio of the molecular weight in the core to the molecular weight in the envelope is <math>~\mu_c/\mu_e = 2</math>. This was done to more realistically represent stars as they evolve off the main sequence; they have inert, isothermal helium cores and envelopes that are rich in hydrogen. Note that introducing a discontinuous drop in the mean molecular weight at the core-envelope interface also introduces a discontinuous drop in the gas density across the interface. As the following excerpt from p. 168 of their article summarizes, in these models, [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)] found that <math>~\nu_\mathrm{max} \approx 0.101</math>. This is commonly referred to as the Schönberg-Chandrasekhar mass limit, although it was Henrich & Chandrasekhar who were the first to identify the instability.
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	In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech (1988)] matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see [http://adsabs.harvard.edu/abs/2012MNRAS.421.2713B Ball, Tout, & Żytkow] (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]; in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.		In an effort to develop a more complete appreciation of the onset of the instability associated with the Schönberg-Chandrasekhar mass limit, [http://adsabs.harvard.edu/abs/1988Ap%26SS.147..219B Beech (1988)] matched an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an isothermal core and, like Schönberg & Chandrasekhar, allowed for a discontinuous change in the molecular weight at the interface. [For an even more comprehensive generalization and discussion, see [http://adsabs.harvard.edu/abs/2012MNRAS.421.2713B Ball, Tout, & Żytkow] (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by [http://adsabs.harvard.edu/abs/1942ApJ....96..161S Schönberg & Chandrasekhar (1942)]; in particular, the value of <math>~\nu_\mathrm{max}</math> was still only definable numerically because an isothermal core cannot be described in terms of analytic functions.

	[[~~User:Tohline/~~SSC/Structure/BiPolytropes/~~Analytic5_1~~#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:		[[SSC/Structure/BiPolytropes/Analytic51#BiPolytrope_with_nc_.3D_5_and_ne_.3D_1\|In an accompanying derivation]] [see, also, [http://adsabs.harvard.edu/abs/1998MNRAS.298..831E Eggleton, Faulkner, and Cannon] (1998, MNRAS, 298, 831)] we have gone one step farther, matching an analytically prescribable, <math>~n_e = 1</math> polytropic envelope to an analytically prescribable, <math>~n_c = 5</math> polytropic core. For this bipolytrope, we show that there is a limiting mass-fraction, <math>~\nu_\mathrm{max}</math>, for any choice of the molecular weight ratio <math>~\mu_c/\mu_e > 3</math> and that the interface location, <math>~\xi_i</math>, associated with this critical configuration is given by the positive, real root of the following relation:

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Joel2: /* Bonnor-Ebert Sphere */

2024-07-05T22:42:17Z

Bonnor-Ebert Sphere

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	==Bonnor-Ebert Sphere==		==Bonnor-Ebert Sphere==
	In the mid-1950s, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) independently realized that an equilibrium isothermal gas cloud can be constructed with a finite radius by embedding it in a hot, tenuous external medium. As has been described in an [[~~User:Tohline/~~SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|accompanying chapter]], the relevant mathematical model is constructed by chopping off the ''isolated'' isothermal sphere at some finite radius — call it, <math>~\xi_e</math> — and imposing an externally applied pressure, <math>~P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere. By varying <math>~\xi_e</math>, a ''sequence'' of equilibrium models can be constructed, as illustrated in Figure 1.		In the mid-1950s, [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert] (1955) and [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) independently realized that an equilibrium isothermal gas cloud can be constructed with a finite radius by embedding it in a hot, tenuous external medium. As has been described in an [[SSC/Structure/BonnorEbert#Pressure-Bounded_Isothermal_Sphere\|accompanying chapter]], the relevant mathematical model is constructed by chopping off the ''isolated'' isothermal sphere at some finite radius — call it, <math>\xi_e</math> — and imposing an externally applied pressure, <math>P_e</math>, that is equal to the pressure of the isothermal gas at the specified edge of the truncated sphere. By varying <math>\xi_e</math>, a ''sequence'' of equilibrium models can be constructed, as illustrated in Figure 1.

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	[[File:EbertFig2Title.~~jpg~~\|200px\|center\|Ebert (1955) Figure 2]]		[[File:EbertFig2Title.png\|200px\|center\|Ebert (1955) Figure 2]]
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	<table border="1">		<table border="1">
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	~~Figures from (left) [http://adsabs.harvard.edu/abs/1955ZA.....37..217E Ebert (1955)] and (right) [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor (1956)]~~
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	~~[[File:EbertFig2Title.jpg\|250px\|center\|Ebert (1955) Figure 2]]~~
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	~~[[File:Bonnor1956Fig1reproduced.jpg\|400px\|center\|Bonnor (1956, MNRAS, 116, 351)]]~~
	~~<!-- [[Image:AAAwaiting01.png\|400px\|center\|Bonnor (1956, MNRAS, 116, 351)]] -->~~
	~~</td>~~
	~~</tr>~~

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Joel2: /* Equation of State Considerations */

2024-07-05T22:27:59Z

Equation of State Considerations

Joel2: /* Instabilities Associated with Equilibrium Sequence Turning Points */

2024-07-05T22:24:39Z

Instabilities Associated with Equilibrium Sequence Turning Points

← Older revision		Revision as of 18:24, 5 July 2024
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	=Instabilities Associated with Equilibrium Sequence Turning Points=		=Instabilities Associated with Equilibrium Sequence Turning Points=

	Detailed derivations and discussions that underpin this chapter can be found in [[~~User:Tohline/~~Appendix/Ramblings/~~Turning_Points~~#Turning_Points\|a chapter of our ''Ramblings Appendix'']].		Detailed derivations and discussions that underpin this chapter can be found in [[Appendix/Ramblings/TurningPoints#Turning_Points\|a chapter of our ''Ramblings Appendix'']].

	==Statement of the Problem==		==Statement of the Problem==

Joel2: Created page with "FORCETOC =Instabilities Associated with Equilibrium Sequence Turning Points= Detailed derivations and discussions that underpin this chapter can be found in a chapter of our ''Ramblings Appendix''. ==Statement of the Problem== We can construct detailed force-balance (DFB) models of spherically symmetric, ''isolated'' polytropes over the entire range of polytropic indexes: $~\infty \ge n \ge 0<..."$

2024-07-05T22:23:08Z

Created page with "__FORCETOC__ =Instabilities Associated with Equilibrium Sequence Turning Points= Detailed derivations and discussions that underpin this chapter can be found in a chapter of our ''Ramblings Appendix''. ==Statement of the Problem== We can construct detailed force-balance (DFB) models of spherically symmetric, ''isolated'' polytropes over the entire range of polytropic indexes: <math>~\infty \ge n \ge 0<..."

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← Older revision		Revision as of 18:27, 5 July 2024
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	==Equation of State Considerations==		==Equation of State Considerations==

	Spherically symmetric, self-gravitating, equilibrium configurations can be constructed from gases exhibiting a wide variety of degrees of compressibility. When examining how the internal structure of such configurations varies with compressibility, or when examining the relative stability of such structures, it can be instructive to construct models using a [[~~User:Tohline/~~SR#Time-Independent_Problems\|polytropic equation of state]],		Spherically symmetric, self-gravitating, equilibrium configurations can be constructed from gases exhibiting a wide variety of degrees of compressibility. When examining how the internal structure of such configurations varies with compressibility, or when examining the relative stability of such structures, it can be instructive to construct models using a [[SR#Time-Independent_Problems\|polytropic equation of state]],

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	{{ ~~User:Tohline/~~Math/EQ_Polytrope01 }}		{{ Math/EQ_Polytrope01 }}
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	because the degree of compressibility can be adjusted by simply changing the value of the polytropic index, {{~~User:Tohline/~~Math/MP_PolytropicIndex}}, across the range, <math>~0 \le n \le \infty</math>. (Alternatively, one can vary the effective adiabatic exponent of the gas, <math>~\Gamma = 1 + 1/n</math>.) In particular, <math>~n = 0 ~~ (\Gamma = \infty)</math> represents a ''hard'' equation of state and describes an incompressible configuration, while <math>~n = \infty ~~(\Gamma = 1)</math> represents an isothermal and extremely ''soft'' equation of state.		because the degree of compressibility can be adjusted by simply changing the value of the polytropic index, {{Math/MP_PolytropicIndex}}, across the range, <math>0 \le n \le \infty</math>. (Alternatively, one can vary the effective adiabatic exponent of the gas, <math>\Gamma = 1 + 1/n</math>.) In particular, <math>n = 0 ~~ (\Gamma = \infty)</math> represents a ''hard'' equation of state and describes an incompressible configuration, while <math>n = \infty ~~(\Gamma = 1)</math> represents an isothermal and extremely ''soft'' equation of state.


	As has been detailed in an [[~~User:Tohline/~~SSC/Structure/Polytropes#Polytropic_Spheres\|accompanying discussion]], the structural properties of spherical polytropes can be described entirely in terms of a dimensionless radial coordinate, <math>~\xi</math>, and by the radial dependence of the dimensionless enthalpy function, <math>~\theta_n(\xi)</math>, and its first radial derivative, <math>~\theta^'_n(\xi)</math>. At the center of each configuration <math>~(\xi=0)</math>, <math>~\theta_n = 1</math> and <math>~\theta^'_n = 0</math>. The surface of each		As has been detailed in an [[SSC/Structure/Polytropes#Polytropic_Spheres\|accompanying discussion]], the structural properties of spherical polytropes can be described entirely in terms of a dimensionless radial coordinate, <math>\xi</math>, and by the radial dependence of the dimensionless enthalpy function, <math>\theta_n(\xi)</math>, and its first radial derivative, <math>\theta^'_n(\xi)</math>. At the center of each configuration <math>~(\xi=0)</math>, <math>\theta_n = 1</math> and <math>\theta^'_n = 0</math>. The surface of each
	[[~~User:Tohline/~~SSC/Structure/Polytropes#Polytropic_Spheres\|''isolated'' polytrope]] is identified by the radial coordinate, <math>~\xi_1</math>, at which <math>~\theta_n</math> first drops to zero. As a class, isolated polytropes exhibit three attributes that are especially key in the context of our present discussion:		[[SSC/Structure/Polytropes#Polytropic_Spheres\|''isolated'' polytrope]] is identified by the radial coordinate, <math>\xi_1</math>, at which <math>~\theta_n</math> first drops to zero. As a class, isolated polytropes exhibit three attributes that are especially key in the context of our present discussion:
	<!--		<!--
	For example, the radial mass-density distribution and the Lagrangian mass coordinate are given, respectively, but the expressions,		For example, the radial mass-density distribution and the Lagrangian mass coordinate are given, respectively, but the expressions,
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	<li>The equilibrium structure is dynamically stable if <math>~n < 3</math>.		<li>The equilibrium structure is dynamically stable if <math>n < 3</math>.
	<li>The equilibrium structure has a finite radius if <math>~n < 5</math>.		<li>The equilibrium structure has a finite radius if <math>n < 5</math>.
	<li>The equilibrium structure can be described in terms of closed-form analytic expressions for [[~~User:Tohline/~~SSC/Structure/Polytropes#n_.3D_0_Polytrope\|<math>~n = 0</math>]],		<li>The equilibrium structure can be described in terms of closed-form analytic expressions for [[SSC/Structure/Polytropes#n_.3D_0_Polytrope\|<math>n = 0</math>]],
	[[~~User:Tohline/~~SSC/Structure/Polytropes#n_.3D_1_Polytrope\|<math>~n = 1</math>]], and		[[SSC/Structure/Polytropes#n_.3D_1_Polytrope\|<math>~n = 1</math>]], and
	[[~~User:Tohline/~~SSC/Structure/Polytropes#n_.3D_5_Polytrope\|<math>~n = 5</math>]].		[[SSC/Structure/Polytropes#n_.3D_5_Polytrope\|<math>~n = 5</math>]].
	</ol>		</ol>


	Isothermal spheres are discussed in a wide variety of astrophysical contexts because it is not uncommon for physical conditions to conspire to create an extended volume throughout which a configuration exhibits uniform temperature. But, as can be surmised from our list of three key polytrope attributes and recognition that equilibrium isothermal configurations are polytropes with index <math>~n=\infty</math>, mathematical models of [[~~User:Tohline/~~SSC/Structure/IsothermalSphere#Isothermal_Sphere\|isolated isothermal spheres]] are relatively cumbersome to analyze because they extend to infinity, they are dynamically unstable, and they are not describable in terms of analytic functions. In such astrophysical contexts, we have sometimes found it advantageous to employ an <math>~n=5</math> polytrope instead of an isothermal sphere. An [[~~User:Tohline/~~SSC/Structure/Polytropes#n_.3D_5_Polytrope\|isolated <math>~n=5</math> polytrope]] can serve as an effective surrogate for an isothermal sphere because it is both infinite in extent and dynamically unstable, but it is less cumbersome to analyze because its structure can be described by closed-form analytic expressions.		Isothermal spheres are discussed in a wide variety of astrophysical contexts because it is not uncommon for physical conditions to conspire to create an extended volume throughout which a configuration exhibits uniform temperature. But, as can be surmised from our list of three key polytrope attributes and recognition that equilibrium isothermal configurations are polytropes with index <math>n=\infty</math>, mathematical models of [[SSC/Structure/IsothermalSphere#Isothermal_Sphere\|isolated isothermal spheres]] are relatively cumbersome to analyze because they extend to infinity, they are dynamically unstable, and they are not describable in terms of analytic functions. In such astrophysical contexts, we have sometimes found it advantageous to employ an <math>n=5</math> polytrope instead of an isothermal sphere. An [[SSC/Structure/Polytropes#n_.3D_5_Polytrope\|isolated <math>~n=5</math> polytrope]] can serve as an effective surrogate for an isothermal sphere because it is both infinite in extent and dynamically unstable, but it is less cumbersome to analyze because its structure can be described by closed-form analytic expressions.


	==Bonnor-Ebert Sphere==		==Bonnor-Ebert Sphere==