 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2019

Paper 317

For the <nonrelativistic white dwarf>, combine the <hydrostatic pressure support equation> and <enclosed mass> equation:
$$
\frac{dP}{dr}=-\frac{Gm\rho}{r^2},\qquad
\frac{dm}{dr}=4\pi r^2\rho.
$$
Eliminating $m$ yields
$$
\boxed{\frac1{r^2}\frac d{dr}\left(\frac{r^2}{\rho}\frac{dP}{dr}\right)=-4\pi G\rho,\qquad P=K_{\rm NR}\rho^{5/3}.}
$$
Introduce the <Lane-Emden variables for a stellar polytrope>,
$$
\rho=\rho_c\theta^{3/2},\qquad r=a\xi,\qquad
 a^2=\frac{5K_{\rm NR}}{8\pi G}\rho_c^{-1/3}.
$$
The resulting <Lane-Emden equation> is
$$
\boxed{\frac1{\xi^2}\frac d{d\xi}(\xi^2\theta')=-\theta^{3/2},\qquad
\theta(0)=1,\quad\theta'(0)=0.}
$$
The <white dwarf> surface is at the first zero $\xi_1$ of $\theta$, with $R=a\xi_1$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 3 / ii / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 3 / ii / a / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 3 / ii / a / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)