 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

Differentiating the <semi-grey irradiated atmosphere> profile gives
$$
4T^3\frac{dT}{d\tau}
=a\left[T_{\rm int}^4
+fT_{\rm irr}^4(1-\gamma^2)e^{-\gamma c\tau}\right].
$$
Hydrostatic balance gives $d\tau/dP=\kappa_{\rm IR}/g$ when the thermal mean opacity is locally constant. Hence
$$
\boxed{
\frac{dT}{dP}
=\frac{a\kappa_{\rm IR}}{4gT^3}
\left[T_{\rm int}^4
+fT_{\rm irr}^4(1-\gamma^2)e^{-\gamma c\tau}\right]}.
$$
For variable opacity, replace $\kappa_{\rm IR}$ by its local value and integrate $d\tau=\kappa_{\rm IR}(P,T)dP/g$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 1 / c / Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 315 / 1 / c / Solution

 Ancestors (11)

 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 / 1 / c / Solution

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)