Past exam of the astrophysical fluid dynamics course of the University of Cambridge / 2017 / 3 / c / Solution

Let $a_T^2=k_{B}T/(\mu m_u)$ be the constant isothermal sound speed squared. The stationary galactic gas obeys hydrostatic equilibrium in the specified gravitational potential:Integrating from the pressure-matching radius $r_0$ gives $\rho (r)=\rho (r_0)\exp [-(\psi (r)-\psi (r_0))/a_T^2]$. Since $\rho (r_0)=p_0/a_T^2$ and $\psi =-A{r}^{-1/2}$,The exponent is dimensionless: the quantity in parentheses times $A$ is a difference of gravitational potential, of units ${\mathrm{m}}^2{\mathrm{s}}^{-2}$. Density increases inward, as expected. The idealized potential is singular at the centre, so this formal profile should not be extrapolated into a region where the prescribed potential or the neglect of gas self-gravity fails.