 Solution

ID: past-exam-of-the-astrophysics-course-of-the-university-of-oxford/2022/5/a/solution

Let

n (x) = \int f d^{3} v

and

\overline{A} = n^{- 1} \int A f d^{3} v

. Assume the one-particle distribution function and its weighted tails decay enough that all required velocity integrals converge and boundary terms at infinite velocity vanish. Multiply the steady Collisionless Boltzmann equation by

v_{j}

and integrate over velocity. In Cartesian coordinates, spatial differentiation commutes with this integration, while integration by parts gives

\int v_{j} \frac{\partial f}{\partial v _{i}} d^{3} v = - δ_{ij} n .

Consequently the three force-balance equations are

i = 1 \sum 3 \frac{\partial}{\partial x _{i}} (n \overline{v_{i} v_{j}}) = - n \frac{\partial Φ}{\partial x _{j}}, j = 1, 2, 3.

Explicitly the

x

component is

\partial_{x} (n \overline{v_{x}^{2}}) + \partial_{y} (n \overline{v_{x} v_{y}}) + \partial_{z} (n \overline{v_{x} v_{z}}) = - n Φ_{x}

, with cyclic counterparts for

y, z

. For equal stellar masses, multiplying by the common mass replaces

n

ρ

; for multiple stellar masses, use a mass-weighted one-particle distribution function or derive a separate equation for each tracer population.

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