Assume an <ideal-gas atmosphere> of constant mean molecular mass and approximately constant gravity. If temperature decreases linearly with altitude, write $dT/dz=-\Gamma$. Combining the ideal-gas law with <hydrostatic equilibrium> gives
$$
\frac{d\log P}{dz}=-\frac{\mu m_Hg}{k_BT},
\qquad
\frac{d\log P}{d\log T}=\frac{\mu m_Hg}{k_B\Gamma}.
$$
The endpoint conditions determine the power law directly:
$$
\boxed{
T(P)=T_t\left(\frac{P}{P_t}\right)^a
=T_b\left(\frac{P}{P_b}\right)^a,
\qquad
a=\frac{\log(T_b/T_t)}{\log(P_b/P_t)}}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 2 / a / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-315/2/a/i/solution