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Mass Upper Limits

Whitworth's (1981) Isothermal Free-Energy Surface
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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 because the degree of compressibility can be adjusted by simply changing the value of the polytropic index, ~n, across the range, 0 \le n \le \infty. (Alternatively, one can vary the effective adiabatic exponent of the gas, γg = 1 + 1 / n.) In particular, n = 0 (\gamma_g = \infty) represents a hard equation of state and describes an incompressible configuration, while n = \infty (γg = 1) represents an isothermal and extremely soft equation of state.

ISOLATED POLYTROPES: Polytropic spheres exhibit three attributes that are especially key in the context of our present discussion:

  1. The equilibrium structure is dynamically stable if n < 3.
  2. The equilibrium structure has a finite radius if n < 5.
  3. The equilibrium structure can be described in terms of closed-form analytic expressions for n = 0, n = 1, and n = 5.

Virial Equilibrium

By examining the virial equilibrium of nonrotating, spherically symmetric configurations that are embedded in an external medium of pressure Pe, one can begin to appreciate that there is a mass above which no equilibrium exists if the effective adiabatic exponent of the gas is γg < 4 / 3. 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,


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/(4-3\gamma_g)} \biggl[ \frac{\bar{c_s}^8}{G^3 P_e} \biggr] \, ,

and,


R_\mathrm{eq}^2 = \frac{1}{5^2} \biggl[ \frac{4}{3\gamma_g} \biggr]^{2/(4-3\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/(4-3\gamma_g)} \biggl( \frac{4}{3\gamma_g} - 1 \biggr) \biggl[ \frac{\bar{c_s}^4 }{GP_e} \biggr] \, .

Polytropes Embedded in an External Medium

Polytropes embedded in an external medium

Summary of Analytic Results

Example:   Specify cs, Pe and n (or η = 1 + 1 / n)

~n

η

Mmax

Req

αvirial

αDHB

scale

αvirial

αDHB

scale

\infty

1

\biggr(\frac{3^4 \cdot 5^3}{2^{10} \pi}\biggr)^{1/2}
 
 \approx 1.77408

--

\biggl[ \frac{c_s^8}{G^3 P_e}\biggr]^{1/2}

\biggl(\frac{3^2 \cdot 5}{2^6\pi} \bigg)^{1/2}
 
 \approx 0.47309

--

\biggl( \frac{c_s^4}{GP_e} \biggr)^{1/2}

5

6/5

\biggl( \frac{3^{19}}{2^{12}\cdot 5^9 \pi} \biggr)^{1/2}
 
 \approx 0.21505

\biggl( \frac{1}{2} \biggr)^{3/10} \biggl( \frac{3^7}{2^8 \pi} \biggr)^{1/2}
 
 \approx 1.33943

\biggl[ \frac{\bar{c_s}^8}{G^3 P_e}\biggr]^{1/2}

\frac{3^{9}}{2^{7}\cdot 5^{6} \pi}
 
 \approx 0.0031326

--

\biggl( \frac{\bar{c_s}^4}{GP_e} \biggr)^{1/2}

Bonnor-Ebert 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] one-dimensional polytropes," Whitworth (1981, MNRAS, 195, 967) normalizes (or "references") various derived mathematical expressions for configuration radii, R, and for the pressure exerted by an external bounding medium, Pex, to quantities he refers to as, respectively, Rrf and Prf. The paragraph from his paper in which these two reference quantities are defined is shown here:

Whitworth (1981, MNRAS, 195, 967)

In order to map Whitworth's terminology to ours, we note, first, that he uses M0 to represent the spherical configuration's total mass, which we refer to simply as M; and his parameter η is related to our ~n via the relation,

\eta = 1 + \frac{1}{n} \, .

Hence, Whitworth writes the polytropic equation of state as,

P = K_\eta \rho^\eta \, ,

whereas, using our standard notation, this same key relation is written as,

~P = K_\mathrm{n} \rho^{1+1/n} ;

and his parameter Kη is identical to our ~K_\mathrm{n}.

According to the second (bottom) expression identified by the red outlined box drawn above,


P_\mathrm{rf} = \frac{3^4 5^3}{2^{10} \pi} \biggl( \frac{K_1^4}{G^3 M^2} \biggr) \, ,

and inverting the expression inside the green outlined box gives,


K_1 = \biggl[ K_n (4 P_\mathrm{rf})^{\eta - 1} \biggr]^{1/\eta} \, .

Hence,


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} \, ,

or, gathering all factors of Prf to the left-hand side,


P_\mathrm{rf}^{(4-3\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] \, .

Analogously, according to the first (top) expression identified inside the red outlined box,


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)} \, .


Related Wikipedia Discussions

Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2021 by Joel E. Tohline
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Recommended citation:   Tohline, Joel E. (2021), The Structure, Stability, & Dynamics of Self-Gravitating Fluids, a (MediaWiki-based) Vistrails.org publication, https://www.vistrails.org/index.php/User:Tohline/citation

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