 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

2022

Paper 317

The <stellar structure equations> are
$$
\frac{dM_r}{dr}=4\pi r^2\rho,\qquad
\frac{dP}{dr}=-\frac{GM_r\rho}{r^2},
$$
$$
\frac{dL_r}{dr}=4\pi r^2\rho\epsilon,\qquad
\frac{dT}{dr}=-\frac{3\kappa\rho L_r}
{16\pi a_{\rm rad}cr^2T^3},
$$
with
$$
P=P_g+P_{\rm rad},\qquad
P_g=\frac{\rho k_BT}{\mu H},\qquad
P_{\rm rad}=\frac{a_{\rm rad}T^4}{3},
\qquad
\kappa=\kappa_0\rho T^{-7/2}.
$$
The last relation is the <Kramers opacity law>. Uniform energy release per unit mass means $\epsilon$ is constant, so integration of the mass and luminosity equations gives $L_r=\epsilon M_r$. At the surface,
$$
\boxed{\epsilon=\frac LM=M^{-1}L}.
$$
Thus $\epsilon=M^aL^b$ has
$$
\boxed{a=-1,\qquad b=1}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 1 / i / Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 1 / i / Solution

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 317 / 1 / i / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)