The density exponent in $\epsilon=\epsilon_0\rho^aT^b$ reflects the number of reacting pairs per unit mass; for ordinary two-body hydrogen burning, $a\simeq1$. The temperature exponent is the local logarithmic sensitivity of the thermally averaged nuclear cross-section. Near $2\times10^7\,\mathrm K$,
$$
\begin{array}{c|cc}
&a&b\\ \hline
\text{<proton–proton chain>} &1&\simeq4\\
\text{<CNO cycle>} &1&\simeq17
\end{array}
$$
are standard approximations.

Spherical regularity makes central scalar profiles even in $r$. Since density and temperature decrease outward in a normal stellar core, they have expansions
$$
T=T_c(1-\alpha r^2+O(r^4)),
\qquad
\rho=\rho_c(1-\beta r^2+O(r^4)),
$$
with $\alpha,\beta>0$. The high power $T^b$ then suppresses burning rapidly away from the centre, especially for the CNO cycle.

The luminosity equation is
$$
\frac{dL}{dr}=4\pi r^2\rho\epsilon
\propto r^2\rho^{a+1}T^b.
$$
Expanding and integrating gives
$$
L(r)\propto r^3\left(1-\gamma r^2+\cdots\right),
\qquad
\boxed{\gamma=\frac35[(a+1)\beta+b\alpha]}.
$$
The luminosity generated by a thin shell scales locally as $r^2\exp[-((a+1)\beta+b\alpha)r^2]$. Its maximum therefore has
$$
r_{\rm burn}\simeq[(a+1)\beta+b\alpha]^{-1/2}
\propto b^{-1/2}
$$
when the temperature term dominates. Comparable central structures then give
$$
\boxed{\frac{r_{\rm CNO}}{r_{pp}}\simeq
\sqrt{\frac{b_{pp}}{b_{\rm CNO}}}\simeq
\sqrt{\frac4{17}}\simeq0.49}.
$$

Near the centre, mass conservation gives $m(r)=4\pi\rho_cr^3/3+O(r^5)$. Substitution into hydrostatic equilibrium and integration yields
$$
\boxed{P(r)=P_c-\frac{2\pi}{3}G\rho_c^2r^2+O(r^4)}.
$$
For the perfect-gas law $P=\rho\mathcal RT/\mu$,
$$
\alpha+\beta=\frac{2\pi G\rho_c\mu}{3\mathcal RT_c}.
$$
Taking the two-body value $a=1$ in the earlier expression for $\gamma$ and eliminating $\alpha$ gives
$$
\boxed{\gamma=\frac35\left[(2-b)\beta+
\frac{2\pi b}{3}\frac{G\rho_c\mu}{\mathcal RT_c}\right]}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 317 / 2 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-317/2/solution