4N: Notes to Chapter 4
- Page ID
- 141467
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The last integral can be integrated by parts so that
\[\begin{aligned}
& \mathrm{\int_v \frac{4 \pi G \operatorname{Pr}^2 \rho}{c^2} d V=\left.\frac{4 \pi G \operatorname{Pr}^2 m(r)}{c^2}\right|_0 ^R-\int_0^R \frac{4 \pi r^2 G m(r)}{c^2} \frac{d P}{d r} d r-\int_0^R \frac{8 \pi G m(r) P}{c} r d r} \\[4pt]
& \mathrm{=\int_v \frac{G^2 m^2(r) \rho}{r^2 c^2} d V-\left.\frac{8 \pi G}{c^2} m(r) P r\right|_0 ^R+\frac{8 \pi G}{c^2} \int_0^R m(r)\left[P+\frac{d P}{d r}\right] d r} \\[4pt]
& \mathrm{=\int_v \frac{G^2 m^2(r) \rho}{r^2 c^2} d V+\left.\frac{8 \pi G}{c^2} m(r) P(r)\right|_0 ^R-\frac{8 \pi G}{c^2} \int_0^R m(r) \frac{d P}{d r} d r+\frac{8 \pi G}{c^2} \int_0^R m(r) \frac{d P}{d r} d r}
\end{aligned}\tag{N4.1.1}\]
The third term vanishes since \(\mathrm{m(0)=0}\) and \(\mathrm{P(R)=0}\), the last two integrals cancel so that
\[\mathrm{\int_v \frac{4 \pi G \operatorname{Pr}^2 \rho}{c^2} d V=\int_v \frac{G^2 m^2(r) \rho}{r^2 c^2} d V}.\tag{N4.1.2}\]
4.2
Start1ng w1th the polytropic Equation of state
\[\mathrm{p=K \rho^\gamma}.\tag{N4.2.1}\]
It is not hard to convince yourself that
\[\gamma=\frac{\mathrm{d} \ell \mathrm{nP}}{\mathrm{d} \ell \mathrm{n} \rho}=\frac{\rho}{\mathrm{P}} \frac{\mathrm{dP}}{\mathrm{d} \rho}.\tag{N4.2.2}\]
This can be reduced to a sing1e parameter by considering Chandrasekhar's parametric Equation of state for a nearly re1ativistic degenerate gas6
\[\mathrm{P}=\mathrm{Af}(\mathrm{x}), \quad \rho=\mathrm{Bx}^3,\tag{N4.2.3}\]
where \(\mathrm{f(x)=x\left(2 x^2-3\right)\left(x^2+1\right)+3 \operatorname{Sinh}^{-1}(x)}\). The limit of the hyperbolic sine is:
\[\mathrm{\frac{\operatorname{Lim}}{x \rightarrow \infty}\left[\sinh ^{-1}(x)\right]=\ln2x.}\tag{N4.2.4}\]
Now consider the behavior of \(\mathrm{f}(\mathrm{x})\) as \(\mathrm{x} \rightarrow \infty\).
\[\left.\begin{array}{l}
\mathrm{f}(\mathrm{x}) \cong \mathrm{x}\left(2 \mathrm{x}^2-3\right)\left(\mathrm{x}^2-3\right)\left(\mathrm{x}+\frac{1}{2} \mathrm{x}^{-1}+\cdots+\right)=2 \mathrm{x}^4+\mathrm{x}^2-3 \mathrm{x}^2-\frac{3}{2} \mathrm{x}^{-1}+\cdots+ \\[4pt]
\text {or } \\[4pt]
\mathrm{f}(\mathrm{x}) \cong 2\left(\mathrm{x}^4-\mathrm{x}^2\right)
\end{array}\right\}.\tag{N4.2.5}\]
Simi1ar1y
\[\mathrm{\frac{d P}{d \rho}=\frac{d P}{d x} \frac{d x}{d \rho}=\frac{A}{B} \frac{\left(8 x^3-4 x\right)}{3 x^2}.}\tag{N4.2.6}\]
Thus
\[\left.\begin{array}{l}
\varepsilon=\gamma-\frac{4}{3} \cong \frac{\mathrm{Bx}^3}{2 \mathrm{A}\left(\mathrm{x}^4-\mathrm{x}^2\right)}\left[\frac{\mathrm{a}}{\mathrm{b}} \frac{\left(8 \mathrm{x}^3-4 \mathrm{x}\right)}{3 \mathrm{x}^2}\right]^{-\frac{4}{3}}=\frac{2}{3}\left[\frac{\left(2 \mathrm{x}^2-1\right)}{\mathrm{x}^2-1}\right]-\frac{4}{3} \\[4pt]
\text {or } \\[4pt]
\varepsilon=\frac{2}{3}\left(1+\frac{\mathrm{x}^2}{\mathrm{x}^2-1}\right)-\frac{4}{3}=\frac{2}{3}\left(2+\mathrm{x}^{-2}+\mathrm{x}^{-4}+\cdots+\right)-\frac{4}{3} \cong \frac{2 \mathrm{x}^{-2}}{3}
\end{array}\right\}.\tag{N4.2.7}\]