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Rotating, Supermassive Stars

Whitworth's (1981) Isothermal Free-Energy Surface
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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, ~P_\mathrm{deg}, 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,

~P

~=

~P_\mathrm{gas} + P_\mathrm{rad} \, ,

where,

Ideal Gas Radiation

~P_\mathrm{gas} = \frac{\Re}{\bar{\mu}} \rho T

~P_\mathrm{rad} = \frac{1}{3} a_\mathrm{rad} T^4

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

~P

~=

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

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

~\frac{\Re}{\bar{\mu}} (m_B n) T

~=

~Y_T n k T

~\Rightarrow ~~~~ Y_T

~=

~\frac{\Re}{k} \cdot \frac{m_B}{\bar{\mu}}  \, .

Ratio of Radiation Pressure to Gas Pressure

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

\beta \equiv \frac{P_\mathrm{gas} }{P}  \, ,

in which case, also,

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

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

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

as well as,

\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} \, ,

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

Expression for Adiabatic Exponent

Three Equivalent Expressions

Equation (24) from A. S. Eddington (1918, MNRAS, 79, 2)

~\Gamma_1 - \frac{4}{3}

~=

~\frac{(\gamma-\tfrac{4}{3})(4-3\beta)}{1 + 12(\gamma-1) (1 - \beta)/\beta}


Equation (131) from Chapter II of [C67]

~\Gamma_1

~=

~\beta + \frac{(\gamma-1)(4-3\beta)^2}{\beta + 12(\gamma-1) (1 - \beta)}


Derived below … noting that ~\beta = (1+\sigma)^{-1}

~\Gamma_1 - \frac{4}{3}

~=

~\frac{(3\gamma -4)(1+4\sigma)}{3(1+\sigma)[1 + 12(\gamma-1)\sigma]}


For any value of the ratio of specific heats

From equation (2) of Ledoux & Pekeris (1941, ApJ, 94, 124) — see, for example, our brief summary of this work — or, equally well, from equation (131) in Chapter II of [C67], we see that when the total pressure is of the form being considered here, a general expression for the adiabatic exponent,

\Gamma_1 \equiv \frac{d\ln P}{d\ln\rho} \, ,

is,

~\Gamma_1

~=

~\beta + \frac{(\gamma-1)(4-3\beta)^2}{\beta + 12(\gamma-1) (1 - \beta)} \, ,

where, ~\gamma is the ratio of specific heats associated with the ideal-gas component of the equation of state. Notice that ~\beta = 1 represents a situation where there is no radiation pressure. In this limit the expression simplifies to,

~\Gamma_1\biggr|_{\beta=1}

~=

~\gamma \, ,

which makes sense. On the other hand, setting ~\beta = 0 represents the other extreme, where there is no ideal-gas contribution to the pressure. In this case, we have,

~\Gamma_1\biggr|_{\beta=0}

~=

~\frac{16(\gamma-1)}{12(\gamma-1) } = \frac{4}{3} \, .

Used by BAC84

On the other hand, without derivation BAC84 state (see their equation 4) that the adiabatic exponent is,

~\Gamma_1

~=

~\frac{4}{3} + \frac{4\sigma + 1}{3(\sigma+1)(8\sigma + 1)} \, .

They also point out that, in the case where a system is dominated by radiation pressure ~(\sigma \gg 1), this expression becomes,

~\Gamma_1\biggr|_{\sigma \gg 1}

~\approx

~\frac{4}{3} + \frac{1}{6\sigma} \, .

Clearly, in the limit ~\sigma \rightarrow \infty, this gives ~\Gamma_1 \rightarrow 4/3, which, as it should, matches the limiting value obtained from the Ledoux & Pekeris (1941) expression when ~\beta = 0.

BAC84 do not explicitly state what value they used for the ratio of specific heats when deriving their expression for the adiabatic exponent. But this can be deduced by examining how their expression behaves in the limit of no radiation pressure, that is, for ~\sigma = 0. In this limit, the BAC84 expression gives,

~\Gamma_1\biggr|_{\sigma = 0}

~\approx

~\frac{4}{3} + \frac{1}{3} = \frac{5}{3} \, .

The general BAC84 expression should therefore match the (even more) general Ledoux & Pekeris (1941) expression if we set ~\gamma = \tfrac{5}{3}. Let's check this out. Inserting this specific value of ~\gamma, and remembering (from above) that,

\sigma = \frac{1-\beta }{\beta} \, ,

the Ledoux & Pekeris (1941) expression for the adiabatic exponent becomes,

~\Gamma_1

~=

~\beta + \frac{\tfrac{2}{3}(4-3\beta)^2}{\beta + 8(1 - \beta)}

 

~=

~\frac{1}{3\beta} \biggl[ \frac{2(4-3\beta)^2}{1 + 8\sigma} \biggr] + \beta

 

~=

~\frac{1}{3\beta(1 + 8\sigma)} \biggl[ 2(4-3\beta)^2  + 3\beta^2(1 + 8\sigma) \biggr]

 

~=

~\frac{1}{3\beta(1 + 8\sigma)} \biggl[ 32 -48\beta + 18\beta^2  + 3\beta^2 + 24\beta^2\sigma \biggr]

 

~=

~\frac{1}{3\beta(1 + 8\sigma)} \biggl[ 32 -48\beta + \beta^2(  21 + 24\sigma) \biggr]

~\Rightarrow ~~~~\Gamma_1 - \frac{4}{3}

~=

~\frac{1}{3\beta(1 + 8\sigma)} \biggl[ 32 -48\beta + \beta^2(  21 + 24\sigma) -4\beta(1+8\sigma)\biggr]

 

~=

~\frac{1}{3\beta(1 + 8\sigma)} \biggl[ 32 -4\beta(13+8\sigma) + 3\beta^2(  7 + 8\sigma) \biggr]

 

~=

~\frac{1}{3\beta(1 + 8\sigma)} \biggl[ 32 -4\beta(8+8\sigma) - 20\beta + 3\beta^2(  8 + 8\sigma) -3\beta^2\biggr]

 

~=

~\frac{1}{3\beta(1 + 8\sigma)} \biggl[ 32 -32 - 20\beta + 24\beta -3\beta^2\biggr]

 

~=

~\frac{(4-3\beta)}{3(1 + 8\sigma)} = \frac{\tfrac{4}{\beta}-3}{\tfrac{3}{\beta}(1 + 8\sigma)}

 

~=

~\frac{4(1+\sigma) - 3}{3(1+\sigma)(1 + 8\sigma)}

 

~=

~\frac{1+4\sigma}{3(1+\sigma)(1 + 8\sigma)} \, .

Or, even more generally, we can show that,

~\Gamma_1 - \frac{4}{3}

~=

~\frac{(3\gamma -4)(1+4\sigma)}{3(1+\sigma)[1 + 12(\gamma-1)\sigma]} \, .



Mass Normalization

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

P = K\rho^{(1+1/n_p)}

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

~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} \, .


Polytropic Index Equals 3

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

~K_{3}

~=

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

Adopting the BAC84 terminology, this means that,

~\biggl(\frac{K_{3}}{G}\biggr)^3

~=

~\biggl( \frac{\Re}{\bar\mu}\biggr)^4 \biggl[\frac{1-\beta}{\beta^4}\biggr] \frac{3}{G^3 a_\mathrm{rad}}

 

~=

~\biggl( \frac{k Y_T}{m_B}\biggr)^4 \biggl[\sigma^4(1+\sigma^{-1})^3\biggr] \frac{3}{G^3 a_\mathrm{rad}}

~\Rightarrow ~~~~ M_\mathrm{norm} \equiv \biggl(\frac{K_{3}}{G}\biggr)^{3/2}

~=

~(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

 

~\approx

~\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 \, .

When radiation pressure significantly dominates over gas pressure — that is, in the limit ~\sigma \gg 1 — the leading factor is approximately unity, in which case we see that this expression for ~M_\mathrm{norm} exactly matches the expression for ~M_{u,3} 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 ~\sigma provided above, we have,

~K\rho^{(1 + 1/n_p)}

~=

~Y_T n k T + \frac{a_\mathrm{rad}}{3} T^4

 

~=

~\frac{a_\mathrm{rad}}{3} (1+\sigma^{-1})T^4

 

~=

~\frac{a_\mathrm{rad}}{3} (1+\sigma^{-1})\biggl[ \frac{3Y_T n k \sigma}{a_\mathrm{rad}} \biggr]^{4/3}

 

~=

~\biggl( \frac{3}{a_\mathrm{rad}} \biggr)^{1/3}(1+\sigma^{-1})\biggl[ Y_T n k \sigma \biggr]^{4/3} \, .

Now, from above we have,

~1 + \frac{1}{n_p} = \Gamma

~\approx

~\frac{4}{3} + \frac{1}{6\sigma} \, ,

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

~K\rho^{(1+1/n_p)}

~\approx

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

This means that, for any ~\sigma \gg 1,

~K

~\approx

~\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} \, .

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

~1 + \frac{1}{n_p} \approx \frac{4}{3} + \frac{1}{6\sigma}

      ~~~~\Rightarrow ~~~~     

~\frac{1}{n_p} \approx \frac{1}{3}\biggl(1 + \frac{1}{2\sigma}\biggr)

 

      ~~~~\Rightarrow ~~~~     

~n_p \approx 3\biggl(1 + \frac{1}{2\sigma}\biggr)^{-1} \approx 3\biggl(1 - \frac{1}{2\sigma}\biggr)

 

      ~~~~\Rightarrow ~~~~     

~\frac{(n_p-3)}{2} \approx - \frac{3}{4\sigma} \, .

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

~M_u

~\approx

~\biggl( \frac{K}{G}\biggr)^{3/2} \biggl(\frac{K}{c^2}\biggr)^{-3/4\sigma}

 

~=

~\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}

 

~=

~\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}

 

~=

~\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}

Drawing again from the definition of ~\sigma provided above, we have,

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

~=

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

so this last relation can be rewritten as,

~M_u

~\approx

~
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} 
\, ,

where,

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}\, ,

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


Whitworth's (1981) Isothermal Free-Energy Surface

© 2014 - 2019 by Joel E. Tohline
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