Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 321 / 2 / a / ii / Solution

Put $K(r)=e^{s(r)}$. The perfect gas relation in the question becomes the polytropic equation of stateAt fixed $r$, vertical hydrostatic balance isAfter one integration,Defining the midplane adiabatic sound speed by $c_{s0}^2=\gamma P_0/{\rho }_0=\gamma K{\rho }_0^{\gamma -1}$ givesThis is a vertically truncated polytropic atmosphere; the physical perfect-gas case has $\gamma >1$.