Rotating, Supermassive Stars

Here we draw upon the work of J. R. Bond, W. D. Arnett, & B. J. Carr (1984, ApJ, 380, 825; hereafter BAC84) who were among the first to seriously address the question of the fate of very massive (stellar) objects.

Equation of State

Our discussion of the equation of state (EOS) that was used by BAC84 draws on the terminology that has already been adopted in our introductory discussion of supplemental relations and closely parallels our review of the properties of the envelope that E. A. Milne (1930, MNRAS, 91, 4) used to construct a bipolytropic sphere.

Expression for Total Pressure

Ignoring the component due to a degenerate electron gas, <math>~P_\mathrm{deg}</math>, the total gas pressure can be expressed as the sum of two separate components: a component of ideal gas pressure, and a component of radiation pressure. That is, in BAC84 the total pressure is given by the expression,

<math>~P_\mathrm{gas} + P_\mathrm{rad} \, ,</math>

where,

Ideal Gas

Radiation

<math>~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T</math>

<math>~P_\mathrm{rad} = \frac{1}{3} a_\mathrm{rad} T^4</math>

Now, BAC84 define the rest-mass density in terms of the mean baryon mass, <math>~m_B</math>, via the expression, <math>~\rho = m_B n</math>, and write (see their equation 1),

<math>~Y_T n k T + \frac{1}{3}a_\mathrm{rad} T^4 \, .</math>

In converting from our notation to theirs we conclude, therefore, that,

<math>~\frac{\Re}{\bar{\mu}} (m_B n) T</math>	<math>~=</math>	<math>~Y_T n k T </math>
<math>~\Rightarrow ~~~~ Y_T </math>	<math>~=</math>	<math>~\frac{\Re}{k} \cdot \frac{m_B}{\bar{\mu}} \, .</math>

Ratio of Radiation Pressure to Gas Pressure

Following Milne (1930), we have defined the parameter, <math>~\beta</math>, as the ratio of gas pressure to total pressure. That is, in the context of BAC84, we have,

<math>\beta \equiv \frac{P_\mathrm{gas} }{P} \, ,</math>

in which case, also,

<math>\frac{P_\mathrm{rad}}{P} = 1-\beta </math> and <math>\frac{P_\mathrm{gas}}{P_\mathrm{rad}} = \frac{\beta}{1-\beta} \, .</math>

Using a different notation, BAC84 (see their equation 5) define <math>~\sigma</math> as the ratio of the radiation pressure to the gas pressure. Therefore, in converting from our notation to theirs we have,

<math>\sigma = \frac{1-\beta}{\beta} ~~~~\Rightarrow ~~~~ \beta = (1 + \sigma)^{-1} \, , </math>

as well as,

<math>\sigma = \frac{P_\mathrm{rad}}{P_\mathrm{gas}} = \frac{a_\mathrm{rad} T^3}{3} \cdot \frac{\bar\mu}{(\Re m_B)n} = \frac{a_\mathrm{rad} T^3}{3Y_T n k} \, ,</math>

which is precisely the definition provided in equation (5) of BAC84.

Expression for Adiabatic Exponent

From equation (2) of Ledoux & Pekeris (1941, ApJ, 94, 124) — see, for example, our brief summary of this work — we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,

<math>\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,</math>

is,

<math>~\Gamma_1</math>	<math>~=</math>	<math>~\beta + \frac{g(4-3\beta)^2}{\beta + 12g (1 - \beta)} </math>
	<math>~=</math>	<math>~\beta + \frac{g}{\beta}\biggl[4-3\beta\biggr]^2\biggl[1 + \frac{12g (1 - \beta)}{\beta}\biggr]^{-1} \, ,</math>

where we are using the temporary shorthand notation, <math>g \equiv (\gamma - 1)</math>. Replacing <math>~\beta</math> with its expression in terms of <math>~\sigma</math> gives us,

<math>~\Gamma_1</math>	<math>~=</math>	<math>~\frac{1}{(1+\sigma)} + g(1+\sigma)\biggl[4-\frac{3}{(1+\sigma)}\biggr]^2\biggl[1 + 12g \sigma\biggr]^{-1} </math>
	<math>~=</math>	<math>~\frac{1}{(1+\sigma)} + \frac{g[4(1+\sigma)-3]^2 }{(1 + 12g \sigma)} </math>

Mass Normalization

Now, according to BAC84 (see their equation 8), when the total pressure is written in polytropic form — specifically, if we set,

— the mass-scaling for relativistic configurations will depend on <math>~G</math>, <math>~c</math>, <math>~K</math>, and <math>~n_p</math> via the expression,

<math>~M_u = K^{n_p/2} G^{-3/2} c^{3-n_p} = \biggl( \frac{K}{G}\biggr)^{3/2} \biggl(\frac{K}{c^2}\biggr)^{(n_p-3)/2} \, .</math>

Polytropic Index Equals 3

Referencing our separate discussion of Milne's (1930) work, when <math>~n_p = 3</math>, the polytropic constant is related to the relevant set of physical parameters via the relation,

<math>~\biggl[ \biggl( \frac{\Re}{\bar\mu}\biggr)^4 \biggl(\frac{1-\beta}{\beta^4}\biggr) \frac{3}{a_\mathrm{rad}} \biggr]^{1/3} \, .</math>

Adopting the BAC84 terminology, this means that,

<math>~\biggl(\frac{K_{3}}{G}\biggr)^3</math>	<math>~=</math>	<math>~\biggl( \frac{\Re}{\bar\mu}\biggr)^4 \biggl[\frac{1-\beta}{\beta^4}\biggr] \frac{3}{G^3 a_\mathrm{rad}} </math>
	<math>~=</math>	<math>~\biggl( \frac{k Y_T}{m_B}\biggr)^4 \biggl[\sigma^4(1+\sigma^{-1})^3\biggr] \frac{3}{G^3 a_\mathrm{rad}} </math>
<math>~\Rightarrow ~~~~ M_\mathrm{norm} \equiv \biggl(\frac{K_{3}}{G}\biggr)^{3/2}</math>	<math>~=</math>	<math>~(1+\sigma^{-1})^{3/2} \biggl( \frac{k Y_T}{m_B}\biggr)^2 \biggl(\frac{3}{G^3 a_\mathrm{rad}}\biggr)^{1/2} \sigma^2 </math>
	<math>~\approx</math>	<math>~\biggl(1+\frac{3}{2\sigma} \biggr) \biggl( \frac{k Y_T}{m_B}\biggr)^2 \biggl(\frac{3}{G^3 a_\mathrm{rad}}\biggr)^{1/2} \sigma^2 \, .</math>

When radiation pressure significantly dominates over gas pressure — that is, in the limit <math>~\sigma \gg 1</math> — the leading factor is approximately unity, in which case we see that this expression for <math>~M_\mathrm{norm}</math> exactly matches the expression for <math>~M_{u,3}</math> given by equation (10) of BAC84.

Polytropic Index Slightly Less Than 3

More generally, equating the two expressions for the total pressure and drawing (twice) on the expression for <math>~\sigma</math> provided above, we have,

<math>~K\rho^{(1 + 1/n_p)}</math>	<math>~=</math>	<math>~Y_T n k T + \frac{a_\mathrm{rad}}{3} T^4 </math>
	<math>~=</math>	<math>~\frac{a_\mathrm{rad}}{3} (1+\sigma^{-1})T^4 </math>
	<math>~=</math>	<math>~\frac{a_\mathrm{rad}}{3} (1+\sigma^{-1})\biggl[ \frac{3Y_T n k \sigma}{a_\mathrm{rad}} \biggr]^{4/3} </math>
	<math>~=</math>	<math>~\biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3}(1+\sigma^{-1})\biggl[ Y_T n k \sigma \biggr]^{4/3} \, .</math>

Now, from above we have,

<math>~1 + \frac{1}{n_p} = \Gamma</math>

<math>~\approx</math>

<math>~\frac{4}{3} + \frac{1}{6\sigma} \, ,</math>

so the lefthand-side of this last expression can be written as,

<math>~\approx</math>

<math>~K\rho^{(4/3+1/6\sigma)} = K(m_B n)^{4/3} \rho^{1/6\sigma} \, .</math>

This means that, for any <math>~\sigma \gg 1</math>,

<math>~\approx</math>

<math>~\biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3} \biggl( \frac{Y_T k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \, .</math>

This matches exactly expression (7) in BAC84. Again from above — and continuing to assume <math>~\sigma \gg 1</math> — we have,

<math>~1 + \frac{1}{n_p} \approx \frac{4}{3} + \frac{1}{6\sigma} </math>	<math>~~~~\Rightarrow ~~~~</math>	<math>~\frac{1}{n_p} \approx \frac{1}{3}\biggl(1 + \frac{1}{2\sigma}\biggr) </math>
	<math>~~~~\Rightarrow ~~~~</math>	<math>~n_p \approx 3\biggl(1 + \frac{1}{2\sigma}\biggr)^{-1} \approx 3\biggl(1 - \frac{1}{2\sigma}\biggr)</math>
	<math>~~~~\Rightarrow ~~~~</math>	<math>~\frac{(n_p-3)}{2} \approx - \frac{3}{4\sigma} \, .</math>

Hence, when the polytropic index is slightly less than 3, the mass normalization is,

<math>~M_u</math>	<math>~\approx</math>	<math>~\biggl( \frac{K}{G}\biggr)^{3/2} \biggl(\frac{K}{c^2}\biggr)^{-3/4\sigma}</math>
	<math>~=</math>	<math>~\biggl[\frac{1}{G} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3} \biggl( \frac{Y_T k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr]^{3/2} \biggl[\frac{1}{c^2} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3} \biggl( \frac{Y_T k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr]^{-3/4\sigma} </math>
	<math>~=</math>	<math>~\biggl[\biggl( \frac{3}{G^3 a_\mathrm{rad}} \biggr)^{1/2} \biggl( \frac{Y_T k }{m_B} \biggr)^{2} (1+\sigma^{-1})^{3/2} \sigma^2 \biggr] \rho^{-1/4\sigma} \biggl[\frac{1}{c^2} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3} \biggl( \frac{Y_T k \sigma}{m_B} \biggr)^{4/3} (1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr]^{-3/4\sigma} </math>
	<math>~=</math>	<math>~\biggl[M_\mathrm{norm} \biggr] \biggl\{\frac{1}{c^2} \biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3} \biggl( \frac{Y_T k \sigma}{m_B} \biggr)^{4/3} (m_B n)^{1/3} \biggl[(1+\sigma^{-1}) \rho^{-1/6\sigma} \biggr] \biggr\}^{-3/4\sigma} </math>

Drawing again from the definition of <math>~\sigma</math> provided above, we have,

<math>~\biggl(\frac{3}{a_\mathrm{rad}}\biggr)^{1/3}</math>

<math>~T (\sigma Y_T n k)^{-1/3} \, ,</math>

so this last relation can be rewritten as,

<math>~\approx</math>

<math>~ M_\mathrm{norm} \cdot f \biggl[\frac{m_B c^2}{T \sigma Y_T k} \biggr]^{3/4\sigma} \approx f \cdot M_{u,3} \biggl(1+\frac{3}{2\sigma} \biggr) \biggl[\frac{m_B c^2}{T \sigma Y_T k} \biggr]^{3/4\sigma} \, ,</math>

where,

<math>f \equiv \biggl[(1+\sigma^{-1})^{-1}\rho^{1/6\sigma} \biggr]^{3/4\sigma} \approx \biggl( 1 - \frac{3}{4\sigma^2}\biggr) (n m_B)^{1/8\sigma^2}\, ,</math>

which certainly is close to unity when <math>~\sigma \gg 1</math>. After setting <math>~f=1</math>, this last expression for <math>~M_u</math> exactly matches the expression presented as equation (9) in BAC84.

User:Tohline/Apps/SMS

Contents

Rotating, Supermassive Stars

Equation of State

Expression for Total Pressure

Ratio of Radiation Pressure to Gas Pressure

Expression for Adiabatic Exponent

Mass Normalization

Polytropic Index Equals 3

Polytropic Index Slightly Less Than 3

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