Difference between revisions of "User:Tohline/SSC/FreeEnergy/Equilibrium Sequence Instabilities"
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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 | 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 | ||
[[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: | |||
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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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Isothermal spheres | 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. | ||
===Bounded Isothermal Sphere & Bonnor-Ebert Mass=== | ===Bounded Isothermal Sphere & Bonnor-Ebert Mass=== | ||
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 isothermal gas cloud can be | 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. 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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As is [[User:Tohline/SSC/Structure/BonnorEbert#Maximum_Mass|reviewed in a related discussion]] and as is documented in the table accompanying the expression for <math>~M_\mathrm{max}</math>, above, [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) used [[User:Tohline/SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden's (1907) tabulated properties]] of an isothermal sphere to determine that the dimensionless radius of this limiting configuration is <math>\xi_e \approx 6.5</math> and that the leading coefficient, <math>\alpha \approx 1.18</math>. It is worth noting that a [[User:Tohline/SphericallySymmetricConfigurations/Virial#BonnorEbertMass|global virial analysis of the stability of bounded isothermal spheres]] produces the same expression for <math>~M_\mathrm{max}</math> with a leading coefficient that has an exact, analytic prescription, namely, <math>\alpha = (3^4 \cdot 5^3/2^{10}\pi)^{1/2} \approx 1.77408</math>. While it can be advantageous to reference this analytic prescription of <math>\alpha</math>, the virial analysis must be considered more approximate than Bonnor's analysis because it does not require the construction of models that are in detailed force balance. | As is [[User:Tohline/SSC/Structure/BonnorEbert#Maximum_Mass|reviewed in a related discussion]] and as is documented in the table accompanying the expression for <math>~M_\mathrm{max}</math>, above, [http://adsabs.harvard.edu/abs/1956MNRAS.116..351B Bonnor] (1956) used [[User:Tohline/SSC/Structure/IsothermalSphere#Emden.27s_Numerical_Solution|Emden's (1907) tabulated properties]] of an isothermal sphere to determine that the dimensionless radius of this limiting configuration is <math>\xi_e \approx 6.5</math> and that the leading coefficient, <math>\alpha \approx 1.18</math>. It is worth noting that a [[User:Tohline/SphericallySymmetricConfigurations/Virial#BonnorEbertMass|global virial analysis of the stability of bounded isothermal spheres]] produces the same expression for <math>~M_\mathrm{max}</math> with a leading coefficient that has an exact, analytic prescription, namely, <math>\alpha = (3^4 \cdot 5^3/2^{10}\pi)^{1/2} \approx 1.77408</math>. While it can be advantageous to reference this analytic prescription of <math>\alpha</math>, the virial analysis must be considered more approximate than Bonnor's analysis because it does not require the construction of models that are in detailed force balance. | ||
Our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|detailed force-balance analysis of truncated and pressure-bounded, <math>~n=5</math> polytropes]] identifies a physically analogous limiting mass. If the average isothermal sound speed, <math>~\bar{c_s}</math>, [[User:Tohline/SphericallySymmetricConfigurations/Virial#average|as defined elsewhere]], is used in place of <math>c_s</math>, the mathematical expression for <math>~M_\mathrm{max}</math> has exactly the same form as in the isothermal case. But for the <math>~n=5</math> polytrope we know that the limiting configuration has a dimensionless radius given precisely by <math>~\xi_e = 3</math>; and, as a result, the leading coefficient in the definition of <math>~M_\mathrm{max}</math> is prescribable analytically, namely, <math>~\alpha = 2^{-3/10} \cdot (3^7/2^8 \pi)^{1/2} \approx 1.33943</math>. As is documented in the table accompanying the relation for <math>~M_\mathrm{max}</math>, above, in the case of a truncated <math>~n=5</math> polytrope, [[User:Tohline/SphericallySymmetricConfigurations/Virial#Maximum_Mass|the simpler and more approximate virial analysis gives]], <math>~\alpha = (3^{19}/2^{12}\cdot 5^7\pi)^{1/2} \approx 1.07523</math>. | Our [[User:Tohline/SSC/Structure/PolytropesEmbedded#Extension_to_Bounded_Sphere_2|detailed force-balance analysis of truncated and pressure-bounded, <math>~n=5</math> polytropes]] identifies a physically analogous limiting mass. If the average isothermal sound speed, <math>~\bar{c_s}</math>, [[User:Tohline/SphericallySymmetricConfigurations/Virial#average|as defined elsewhere]], is used in place of <math>c_s</math>, the mathematical expression for <math>~M_\mathrm{max}</math> has exactly the same form as in the isothermal case. But for the <math>~n=5</math> polytrope we know that the limiting configuration has a dimensionless radius given precisely by <math>~\xi_e = 3</math>; and, as a result, the leading coefficient in the definition of <math>~M_\mathrm{max}</math> is prescribable analytically, namely, <math>~\alpha = 2^{-3/10} \cdot (3^7/2^8 \pi)^{1/2} \approx 1.33943</math>. As is documented in the table accompanying the relation for <math>~M_\mathrm{max}</math>, above, in the case of a truncated <math>~n=5</math> polytrope, [[User:Tohline/SphericallySymmetricConfigurations/Virial#Maximum_Mass|the simpler and more approximate virial analysis gives]], <math>~\alpha = (3^{19}/2^{12}\cdot 5^7\pi)^{1/2} \approx 1.07523</math>. | ||
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==Bonnor-Ebert Sphere== | ==Bonnor-Ebert Sphere== | ||
Revision as of 21:54, 3 October 2016
Instabilities Associated with Equilibrium Sequence Turning Points
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Introduction
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 polytropic equation of state,
<math>~P = K_\mathrm{n} \rho^{1+1/n}</math>
because the degree of compressibility can be adjusted by simply changing the value of the polytropic index, <math>~n</math>, 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 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
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:
- The equilibrium structure is dynamically stable if <math>~n < 3</math>.
- The equilibrium structure has a finite radius if <math>~n < 5</math>.
- The equilibrium structure can be described in terms of closed-form analytic expressions for <math>~n = 0</math>, <math>~n = 1</math>, and <math>~n = 5</math>.
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 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 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.
Bounded Isothermal Sphere & Bonnor-Ebert Mass
In the mid-1950s, Ebert (1955) and 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. 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.
Bonnor-Ebert Sphere
This information extracted from an accompanying chapter on the Bonnor-Ebert sphere.
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Figures from (left) Ebert (1955) and (center) Bonnor (1956) |
Our Construction |
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Figures from (left) Ebert (1955) and (right) Bonnor (1956) |
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Our Analytic Analysis of n = 5 Polytropic Sequence] |
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Figure from Stahler (1983) |
From our Detailed Analysis of Pressure-Truncated Polytrope Sequences |
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Order-of-Magnitude |
Detailed Force-Balance |
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Schönberg-Chandrasekhar Mass
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Figures from (left) Henrich & Chandraskhar (1941) and (center) Schönberg & Chandrasekhar (1942) |
Bipolytropes with <math>~(n_c,n_e) = (5,1)</math> |
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Figures from (left) Henrich & Chandraskhar (1941) and (center) Schönberg & Chandrasekhar (1942) |
Bipolytropes with <math>~(n_c,n_e) = (5,1)</math> |
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In the early 1940s, Chandrasekhar and his colleagues (see Henrich & Chandraskhar (1941) and 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 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.
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>. 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, 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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Text excerpt from Schönberg & Chandrasekhar (1942) |
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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, 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 Ball, Tout, & Żytkow (2012, MNRAS, 421, 2713)]. Beech's results were not significantly different from those reported by 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 accompanying derivation [see, also, 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:
<math> \biggl(\frac{\pi}{2} + \tan^{-1} \Lambda_i\biggr) (1+\ell_i^2) [ 3 + (1-m_3)^2(2-\ell_i^2)\ell_i^2] - m_3 \ell_i [(1-m_3)\ell_i^4 - (m_3^2 - m_3 +2)\ell_i^2 - 3] = 0 \, , </math>
where,
<math> \ell_i \equiv \frac{\xi_i}{\sqrt{3}} \, ; </math> <math> m_3 \equiv 3 \biggl( \frac{\mu_c}{\mu_e} \biggr)^{-1} \, ; </math> and <math> \Lambda_i \equiv \frac{1}{m_3\ell_i} [ 1 + (1-m_3)\ell_i^2] \, . </math>
Close Binary Stars
- Lai, Rasio, & Shapiro (1993a, ApJL, 406, L63)
- Lai, Rasio, & Shapiro (1993b, ApJS, 88, 205)
- Ostriker & Gunn (1969, ApJ, 157, 1395)
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Figures from (left) Lai, Rasio, & Shapiro (1993) and (right) New & Tohline (1997) |
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See Also
The discussion presented here is supported by detailed reviews and new derivations presented in the following associated chapters:
- A free-energy synopsis.
- Material associated with an overarching PowerPoint presentation.
- Limiting Masses
- An accompanying chapter on the Bonnor-Ebert sphere.
Overlapping discussions of this topic may also be found in the following key references:
- §6.8 of [ST83].
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© 2014 - 2021 by Joel E. Tohline |