 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

2026

Paper 315

Assume full heat redistribution, so
$$
T_{\rm eq}=T_*\sqrt{\frac{R_*}{2a_{\rm orb}}}
(1-A_B)^{1/4}.
$$
For $T_*=5778\,\mathrm K$, $T_{\rm eq}=600\,\mathrm K$, and <Bond albedo> $A_B=0.5$,
$$
\frac{a_{\rm orb}}{R_*}
=\frac12\left(\frac{T_*}{T_{\rm eq}}\right)^2
\sqrt{1-A_B}\simeq32.8.
$$
Taking $R_p=R_{\rm Nep}=0.0354R_*$ gives
$$
\left(\frac{R_p}{a_{\rm orb}}\right)^2
\simeq1.17\times10^{-6}.
$$
The reported $50\,\mathrm{ppm}$ visible eclipse would therefore imply
$$
\boxed{A_g\simeq\frac{50\times10^{-6}}{1.17\times10^{-6}}simeq43},
$$
which is impossible. At least one assumption, the measurement, or the stated system parameters must fail; even $A_g=1$ gives only about $1.2\,\mathrm{ppm}$.

At $20\,\mu\mathrm m$, reflected light is negligible. The <thermal eclipse depth> is
$$
\left(\frac{R_p}{R_*}\right)^2
\frac{e^{hc/(\lambda k_BT_*)}-1}
{e^{hc/(\lambda k_BT_p)}-1}.
$$
With $T_p=600\,\mathrm K$ this is
$$
\boxed{F_p/F_*\simeq7.2\times10^{-5}\simeq72\,\mathrm{ppm}}.
$$

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 2 / a / ii / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 2 / a / ii / Solution

 Ancestors (12)

 Discussion (0)

 Articles by others on the same topic (0)

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 2 / a / ii / Solution

 Ancestors (12)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)