Past exam of the mathematics course of the University of Cambridge / 2009 / iii / Paper 62 / 2 / a / Solution

Write $u_i=⟨v_i⟩$ and use summation over repeated Cartesian indices. A mass-normalized galactic distribution function obeys the Collisionless Boltzmann equationAssume $f$ decays in velocity sufficiently fast that the necessary boundary terms vanish. Its zeroth velocity moment gives the continuity equationFor the first velocity moment, multiply by $v_j$ and integrate. In the gravitational term, integration by parts gives $\int v_j{\partial }_{v_i}f{d}^{3}v=-{\delta }_{ij}\rho$. ThereforeThe velocity-dispersion tensor is the centered second moment, so $⟨v_i{v}_j⟩=u_i{u}_j+{\sigma }_{ij}^2$. Expanding the streaming terms, the factor multiplying $u_j$ cancels by the continuity equation:Hence the Jeans equation isIts fluid analogue is the momentum equation with pressure tensor $P_{ij}=\rho {\sigma }_{ij}^2$. For isotropic random motion, $P_{ij}=P{\delta }_{ij}$, and this becomes the Euler equations for an inviscid fluid:The tensor form allows anisotropic stellar stresses. In a fluid an equation of state can provide a pressure closure; the Jeans equation alone does not determine the velocity-dispersion tensor of a collisionless stellar system.