Past exam of the astrophysics course of the University of Oxford / 2022 / 2 / d

For a regular spherical star, let ${\rho }_c=\rho (0)$ and $P_c=P(0)$. The stellar mass conservation equation and stellar hydrostatic equation areAssuming the smoothness needed for the requested Taylor series, write $M=a_0+a_{1}r+a_2{r}^2+a_3{r}^3+\cdots$. There is no point mass at the centre, so $a_0=0$, and matching the stellar mass conservation equation gives $a_1=a_2=0$, $a_3=4\pi {\rho }_c/3$. Substitution in the stellar hydrostatic equation gives $P^{\prime }(r)=-(4\pi G/3){\rho }_c^{2}r+O(r^3)$, henceThe remainder orders use a smooth spherically symmetric scalar density, for which $\rho (r)={\rho }_c+O(r^2)$. Merely finite continuous density gives the same leading coefficients with $o(r^3)$ and $o(r^2)$ remainders. In particular $P^{\prime }(0)=0$, despite the apparent $r^{-2}$ in the stellar hydrostatic equation.