= Solution
A <galactic distribution function> $f(\mathbf x,\mathbf v,t)$ is the stellar mass or number per six-dimensional <phase space> volume,
$$
dM=f(\mathbf x,\mathbf v,t)\,d^3x\,d^3v.
$$
Its velocity moments give the spatial density, mean velocity, and <velocity dispersion>; integrating those quantities along the line of sight and weighting by luminosity produces surface-brightness and line-of-sight-velocity observables. A model is compared with data only after the same projection, selection function, and instrumental convolution have been applied.
<Jeans theorem> states that every steady solution of the <Collisionless Boltzmann equation> depends on phase-space coordinates only through <integrals of motion>. Conversely, every nonnegative function of isolating integrals is a steady collisionless distribution function on the region where those integrals are defined.
For the stated <power law> in <relative energy>, isotropy gives
$$
\rho(\Psi)=4\pi F\int_0^{\sqrt{2\Psi}}
\left(\Psi-\frac{v^2}{2}\right)^{n-3/2}v^2\,dv.
$$
With the requested <change of variables> $v=\sqrt{2\Psi}\cos\theta$, the density becomes
$$
\rho=8\sqrt2\pi F\Psi^n
\int_0^{\pi/2}\sin^{2n-2}\theta\cos^2\theta\,d\theta
=2\sqrt2\pi^{3/2}F
\frac{\Gamma(n-1/2)}{\Gamma(n+1)}\Psi^n,
$$
where the last equality uses the <Beta function> and <Gamma function>. Thus
$$
\boxed{\rho\propto\Psi^n}
$$
for $n>1/2$.
The normalized second velocity moment is
$$
\overline{v^2}
=\frac{\int_0^{\sqrt{2\Psi}}v^4(\Psi-v^2/2)^{n-3/2}\,dv}
{\int_0^{\sqrt{2\Psi}}v^2(\Psi-v^2/2)^{n-3/2}\,dv}.
$$
The same substitution and the <Beta-function recurrence> give
$$
\overline{v^2}
=2\Psi\frac{B(5/2,n-1/2)}{B(3/2,n-1/2)}
=\boxed{\frac{3\Psi}{n+1}}.
$$
Consequently the one-dimensional isotropic <velocity dispersion> is $\sigma^2=\overline{v^2}/3=\Psi/(n+1)$, proving the required linear dependence on the <relative potential>.
Back to article page
