Mass Upper Limits

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/(4-3\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/(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] \, . </math>

Polytropes Embedded in an External Medium

Polytropes embedded in an external medium

Summary of Analytic Results

Example: Specify <math>c_s</math> and <math>P_e</math>
<math>~n</math>	<math>\eta = 1+1/n</math>	<math>M_\mathrm{max}</math>			<math>R_\mathrm{eq}</math>
<math>~n</math>	<math>\eta = 1+1/n</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{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>\biggl( \frac{\bar{c_s}^4}{GP_e} \biggr)^{1/2}</math>

Example: Specify <math>M</math> and <math>c_s</math>
<math>~n</math>	<math>\eta = 1+1/n</math>	<math>(P_e)_\mathrm{max}</math>			<math>R_\mathrm{eq}</math>
<math>~n</math>	<math>\eta = 1+1/n</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)</math>	--	<math>\biggl[ \frac{c_s^8}{M^2 G^3}\biggr]</math>	<math>\frac{2^2}{3\cdot 5} </math>	--	<math>\biggl( \frac{GM}{c_s^2} \biggr) </math>
5	6/5	--	<math>\biggl( \frac{1}{2} \biggr)^{3/5} \biggl( \frac{3^7}{2^8 \pi} \biggr)</math>	<math>\biggl[ \frac{\bar{c_s}^8}{M^2 G^3}\biggr]</math>	--	<math>\frac{2^{2}}{3^2} </math>	<math>\biggl( \frac{GM}{\bar{c_s}^2} \biggr) </math>

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, <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,

Hence, Whitworth writes the polytropic equation of state as,

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 left-hand side,

<math> 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] \, . </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>

Related Wikipedia Discussions

Bonnor-Ebert Mass

User:Tohline/SSC/Structure/BonnorEbert

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