# Mass Upper Limits

## 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,

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

## 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:

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

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