托尔曼－奥本海默－沃尔科夫方程

方程形式

${\displaystyle {\frac {dP(r)}{dr}}=-{\frac {G(\rho (r)+P(r)/c^{2})(M(r)+4\pi P(r)r^{3}/c^{2})}{r^{2}(1-2GM(r)/rc^{2})}}.}$

${\displaystyle {\frac {dM(r)}{dr}}=4\pi \rho (r)r^{2}.}$

${\displaystyle ds^{2}=e^{\nu (r)}c^{2}dt^{2}-(1-2GM(r)/rc^{2})^{-1}dr^{2}-r^{2}(d\theta ^{2}+sin^{2}\theta d\phi ^{2}),}$

${\displaystyle {\frac {d\nu (r)}{dr}}=-{\frac {2}{P(r)+\rho (r)c^{2}}}{\frac {dP(r)}{dr}}.}$

${\displaystyle ds^{2}=(1-2GM_{0}/rc^{2})c^{2}dt^{2}-(1-2GM_{0}/rc^{2})^{-1}dr^{2}-r^{2}(d\theta ^{2}+sin^{2}\theta d\phi ^{2})\,}$

${\displaystyle M_{0}=M(r_{B})=\int _{0}^{r_{B}}4\pi \rho (r)r^{2}\,dr.}$

${\displaystyle M_{1}=\int _{0}^{r_{B}}{\frac {4\pi \rho (r)r^{2}}{\sqrt {1-2GM(r)/rc^{2}}}}\,dr.}$

${\displaystyle \delta M=\int _{0}^{r_{B}}4\pi \rho (r)r^{2}((1-2GM(r)/rc^{2})^{-1/2}-1)\,dr,}$

参考资料

1. On Massive Neutron Cores, J. R. Oppenheimer and G. M. Volkoff, Physical Review 55, 374 (February 15, 1939), pp. 374–381.
2. ^ Effect of Inhomogeneity on Cosmological Models, Richard C. Tolman, Proceedings of the National Academy of Sciences 20, 3 (March 15, 1934), pp. 169–176.
3. ^ Static Solutions of Einstein's Field Equations for Spheres of Fluid, Richard C. Tolman, Physical Review 55, 374 (February 15, 1939), pp. 364–373.
4. ^ The maximum mass of a neutron star, I. Bombaci, Astronomy and Astrophysics 305 (January 1996), pp. 871–877.
5. ^ Bombaci, I. The maximum mass of a neutron star. Astronomy and Astrophysics. 1996, 305: 871–877.
6. ^ The evolution and explosion of massive stars, S. E. Woosley, A. Heger, and T. A. Weaver, Reviews of Modern Physics 74, 4 (October 2002), pp. 1015–1071.
7. ^ Black Hole Binaries, Jeffrey E. McClintock and Ronald A. Remillard, arXiv:astro-ph/0306213v4.
8. ^ Observational evidence for stellar-mass black holes, Jorge Casares, arXiv:astro-ph/0612312v1.