User:Tohline/SSC/Structure/LimitingMasses
Mass Upper Limits
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Spherically symmetric, selfgravitating, 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 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_g = 1 + 1/n</math>.) In particular, <math>n = 0</math> (<math>\gamma_g = \infty</math>) represents a hard equation of state and describes an incompressible configuration, while <math>n = \infty</math> (<math>\gamma_g = 1</math>) represents an isothermal and extremely soft equation of state.
Isolated Polytropes
Isolated polytropic spheres 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 closedform analytic expressions for <math>n = 0</math>, <math>n = 1</math>, and <math>n = 5</math>.
Isothermal Spheres
Isothermal spheres (polytropes with index <math>n=\infty</math>) 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 mathematical models of 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 infinite in extent and is dynamically unstable, but it is less cumbersome to analyze because its structure can be described by closedform analytic expressions.
Bounded Isothermal Sphere & BonnorEbert Mass
In the mid1950s, Ebert (1955) and Bonnor (1956) independently realized that an isothermal gas cloud can be stabilized by embedding it in a hot, tenuous external medium. The relevant mathematical model is constructed by chopping off the 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. But for a given mass and temperature, there is a value of <math>P_e</math> below which the truncated isothermal sphere is dynamically unstable, like its isolated and untruncated counterpart. Viewed another way, given the value of <math>P_e</math> and the isothermal sound speed, <math>c_s</math>, a bounded isothermal sphere will be dynamically stable only if its mass is below a critical value,
BonnorEbert Mass


where <math>\alpha</math> is a dimensionless coefficient of order unity. This limiting mass is often referred to as the BonnorEbert mass. It appears most frequently in the astrophysics literature in discussions of star formation because that is the arena in which both Bonnor and Ebert were conducting research when they made their discoveries.
As is reviewed in a related discussion and as is documented in the table accompanying the expression for <math>M_\mathrm{max}</math>, above, Bonnor (1956) used 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 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 detailed forcebalance analysis of truncated and pressurebounded, <math>n=5</math> polytropes identifies a physically analogous limiting mass. If the average isothermal sound speed, <math>\bar{c_s}</math>, 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, the simpler and more approximate virial analysis gives, <math>\alpha = (3^{19}/2^{12}\cdot 5^7\pi)^{1/2} \approx 1.07523</math>.
SchönbergChandrasekhar Mass
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 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 coreenvelope 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 SchönbergChandrasekhar mass limit, although it was Henrich & Chandrasekhar who were the first to identify the instability.
Text excerpt from Schönberg & Chandrasekhar (1942) 

In an effort to develop a more complete appreciation of the onset of the instability associated with the SchönbergChandrasekhar 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. His 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 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 massfraction, <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 + (1m_3)^2(2\ell_i^2)\ell_i^2]  m_3 \ell_i [(1m_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 + (1m_3)\ell_i^2] \, . </math>
Work in Progress
Virial Equilibrium
By examining the virial equilibrium of nonrotating, spherically symmetric configurations that are embedded in an external medium of pressure <math>P_e</math>, one can begin to appreciate that there is a mass above which no equilibrium exists if the effective adiabatic exponent of the gas is <math>\gamma_g < 4/3</math>. Assuming uniform density and uniform specific entropy configurations for simplicity, there is an analytic expression for this limiting mass and for the equilibrium radius of that limiting configuration. Specifically,
<math> M_\mathrm{max}^2 = \frac{3\cdot 5}{2^2 \pi} \biggl(\frac{4}{3\gamma_g}  1 \biggr) \biggl[ \frac{3 \gamma_g}{4} \biggr] ^{4/(43\gamma_g)} \biggl[ \frac{\bar{c_s}^8}{G^3 P_e} \biggr] \, , </math>
and,
<math> R_\mathrm{eq}^2 = \frac{1}{5^2} \biggl[ \frac{4}{3\gamma_g} \biggr]^{2/(43\gamma_g)} \biggl[ \frac{GM_\mathrm{max}}{\bar{c_s}^2 } \biggr]^2 = \frac{3}{2^2 \cdot 5\pi} \biggl[ \frac{3\gamma_g}{4} \biggr]^{2/(43\gamma_g)} \biggl( \frac{4}{3\gamma_g}  1 \biggr) \biggl[ \frac{\bar{c_s}^4 }{GP_e} \biggr] \, . </math>
Polytropes Embedded in an External Medium
Polytropes embedded in an external medium
Summary of Analytic Results
Example: Specify <math>c_s</math>, <math>P_e</math> and <math>n</math> (or <math>\eta = 1+1/n)</math>  
<math>~n</math> 
<math>\eta</math> 
<math>M_\mathrm{max}</math> 
<math>R_\mathrm{eq}</math> 

<math>\alpha_\mathrm{virial}</math> 
<math>\alpha_\mathrm{DHB}</math> 
<math>\mathrm{scale}</math> 
<math>\alpha_\mathrm{virial}</math> 
<math>\alpha_\mathrm{DHB}</math> 
<math>\mathrm{scale}</math> 

<math>\infty</math> 
1 
<math>\biggr(\frac{3^4 \cdot 5^3}{2^{10} \pi}\biggr)^{1/2} </math> 
 
<math>\biggl[ \frac{c_s^8}{G^3 P_e}\biggr]^{1/2}</math> 
<math>\biggl(\frac{3^2 \cdot 5}{2^6\pi} \bigg)^{1/2}</math> 
 
<math>\biggl( \frac{c_s^4}{GP_e} \biggr)^{1/2}</math> 
5 
6/5 
<math>\biggl( \frac{3^{19}}{2^{12}\cdot 5^9 \pi} \biggr)^{1/2} </math> 
<math>\biggl( \frac{1}{2} \biggr)^{3/10} \biggl( \frac{3^7}{2^8 \pi} \biggr)^{1/2} </math> 
<math>\biggl[ \frac{\bar{c_s}^8}{G^3 P_e}\biggr]^{1/2}</math> 
<math>\frac{3^{9}}{2^{7}\cdot 5^{6} \pi} </math> 
 
<math>\biggl( \frac{\bar{c_s}^4}{GP_e} \biggr)^{1/2}</math> 
BonnorEbert Limiting Mass
Here we will refer to the original works of Ebert (1955, ZA, 37, 217) and Bonnor (1956, MNRAS, 116, 351) .
In his study of the "global gravitational stability [of] onedimensional polytropes," Whitworth (1981, MNRAS, 195, 967) normalizes (or "references") various derived mathematical expressions for configuration radii, <math>R</math>, and for the pressure exerted by an external bounding medium, <math>P_\mathrm{ex}</math>, to quantities he refers to as, respectively, <math>R_\mathrm{rf}</math> and <math>P_\mathrm{rf}</math>. The paragraph from his paper in which these two reference quantities are defined is shown here:
In order to map Whitworth's terminology to ours, we note, first, that he uses <math>M_0</math> to represent the spherical configuration's total mass, which we refer to simply as <math>M</math>; and his parameter <math>\eta</math> is related to our <math>~n</math> via the relation,
<math>\eta = 1 + \frac{1}{n} \, .</math>
Hence, Whitworth writes the polytropic equation of state as,
<math>P = K_\eta \rho^\eta \, ,</math>
whereas, using our standard notation, this same key relation is written as,
<math>~P = K_\mathrm{n} \rho^{1+1/n}</math> ;
and his parameter <math>K_\eta</math> is identical to our <math>~K_\mathrm{n}</math>.
According to the second (bottom) expression identified by the red outlined box drawn above,
<math> P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{K_1^4}{G^3 M^2} \biggr) \, , </math>
and inverting the expression inside the green outlined box gives,
<math> K_1 = \biggl[ K_n (4 P_\mathrm{rf})^{\eta  1} \biggr]^{1/\eta} \, . </math>
Hence,
<math> P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{1}{G^3 M^2} \biggr)\biggl[ K_n (4 P_\mathrm{rf})^{\eta  1} \biggr]^{4/\eta} \, , </math>
or, gathering all factors of <math>P_\mathrm{rf}</math> to the lefthand side,
<math> P_\mathrm{rf}^{(43\eta)} = 2^{2(4+\eta)} \biggl( \frac{3^4 5^3}{\pi} \biggr)^\eta \biggl[ \frac{K_n^4}{G^{3\eta} M^{2\eta}} \biggr] \, . </math>
Analogously, according to the first (top) expression identified inside the red outlined box,
<math> R_\mathrm{rf} = \frac{2^2 GM}{3\cdot 5 K_1} = 2^{2/\eta} \biggl( \frac{GM}{3\cdot 5}\biggr) K_n^{1/\eta} P_\mathrm{rf}^{(1\eta)/\eta} ~~~~\Rightarrow~~~~ R_\mathrm{rf}^\eta = \frac{2^{2}}{K_n} \biggl( \frac{GM}{3\cdot 5}\biggr)^\eta P_\mathrm{rf}^{(1\eta)} \, . </math>
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