Past exam of the mathematics course of the University of Cambridge / 2001 / iii / Paper 41 / 4 / iv / Solution

Let ${\rho }_g(r)$ be the gas density and $M(<r)$ the total gravitating mass. For a spherically symmetric gas in hydrostatic equilibrium, radial force balance isUsing the ideal gas law $P={\rho }_g{k}_{B}T/(\mu m_p)$, with constant mean molecular weight $\mu$, differentiate the product rather than setting $T$ constant:Substitution and cancellation of ${\rho }_g$ give the cluster hydrostatic mass estimatorHere $k_B$ is the Boltzmann constant, denoted $K$ in the question. The prefactor has units of mass; a pressure decreasing outward makes the bracket negative and the inferred mass positive. For an isothermal gas with ${\rho }_g\propto r^{-\beta }$, this reduces to $M(<r)=\beta k_{B}Tr/(\mu m_{p}G)$. The density in the logarithmic derivative is the gas density, not the total density. A varying $\mu$ or nonthermal pressure would require the corresponding extra terms.