Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 315 / 1 / b / Solution

In a thin atmosphere in hydrostatic equilibrium, take gravity $g$ and mean molecular mass $\mu m_H$ as constant. The ideal gas equation of state and hydrostatic balance giveFor $P>1\mathrm{mbar}$,so the atmospheric lapse rate follows from the chain rule:It vanishes in the assumed isothermal region above $1\mathrm{mbar}$. If natural logarithms are used instead, the factor $\ln 10$ is absent and the fitted numerical value of $A$ changes accordingly.