= Solution
A <phase-space distribution function> describes the mass in a small cell of positions and velocities: $F(\mathbf x,\mathbf v,t)\,d^3x\,d^3v$ is that mass. Its velocity integral is the spatial <mass density>, and $F/\rho$ at a fixed position is a normalized velocity probability density. A number-normalized <galactic distribution function> differs by the stellar mass convention.
Take $a>0$ and $v_0>0$. For the <cusped logarithmic spherical galaxy model>, differentiate the <gravitational potential> and use the spherical <Poisson equation for Newtonian gravity>:
$$
\Phi'=\frac{v_0^2}{a+r},\qquad 4\pi G\rho=\frac1{r^2}\frac{d}{dr}(r^2\Phi')=v_0^2\frac{2a+r}{r(a+r)^2}.
$$
Thus
$$
\boxed{\rho(r)=\frac{v_0^2}{4\pi G}\frac{2a+r}{r(a+r)^2},\qquad M(r)=\frac{v_0^2r^2}{G(a+r)}.}
$$
A <circular orbit> requires $v_c^2/r=\Phi'$. Its <galaxy rotation curve> is therefore
$$
\boxed{v_c(r)=v_0\sqrt{\frac r{a+r}}.}
$$
It rises as $v_0\sqrt{r/a}$ at small radius and approaches $v_0$ from below at large radius.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/iii/paper-62-logarithmic-rotation.png]
{title=Rotation curve of the cusped logarithmic spherical galaxy model}
The outer <flat galaxy rotation curve> is qualitatively useful for galaxies. However, the density has an inner $1/r$ cusp and an outer $1/r^2$ tail, while $M(r)\sim v_0^2r/G$ diverges. The model therefore requires an outer truncation for a finite galaxy, and does not describe a central constant-density core. Its <gravitational potential> cannot be normalized to zero at infinity.
For a star, the specific energy is $E=v^2/2+\Phi$ and the <specific angular momentum> is $\mathbf L=\mathbf x\times\mathbf v$. Since the <gravitational potential> is stationary and spherical,
$$
\frac{dE}{dt}=\mathbf v\cdot(\dot{\mathbf v}+\nabla\Phi)=0,\qquad\frac{d\mathbf L}{dt}=\mathbf x\times\left(-\Phi'\frac{\mathbf x}{r}\right)=0.
$$
Thus \b[both the specific energy and the entire angular-momentum vector are conserved].
For <Jeans theorem>, follow a stellar trajectory in <phase space>. The <Collisionless Boltzmann equation> is precisely
$$
\frac{dF}{dt}=\partial_tF+\mathbf v\cdot\nabla_xF-\nabla\Phi\cdot\nabla_vF=0.
$$
In a steady system $\partial_tF=0$, so the value of the <galactic distribution function> is constant along every trajectory: it is itself an <integral of motion>. Conversely, if $I_1,\ldots,I_m$ are time-independent <integrals of motion>, then $F=h(I_1,\ldots,I_m)$ satisfies $dF/dt=\sum h_{,a}\dot I_a=0$. Locally, a complete set of invariant orbit labels expresses any steady <galactic distribution function> this way. This is the weak form of <Jeans theorem>; a stronger assertion about a small set of isolating integrals needs additional orbital regularity or mixing assumptions. In particular, the theorem does not say that every steady galaxy must have a <galactic distribution function> depending on only $E$ and $L$.
For the proposed <constant-anisotropy distribution function>, write $v_T=(v_\theta^2+v_\phi^2)^{1/2}$ and $L=rv_T$. Introduce a polar angle $\psi$ in the tangential velocity plane. Then $d^3v=dv_r\,v_Tdv_T\,d\psi$, so the apparent $1/L$ singularity cancels against the velocity Jacobian. This <inverse-angular-momentum velocity integration> makes the density finite for every $r>0$.
Let $t=e^{-\Phi/v_0^2}=a/(a+r)$, $c_1=1$, $c_2=2$, and $\alpha_k=k/(2v_0^2)$. Each exponential component contributes
$$
\rho_k=\frac{c_kt^k}{8\pi^3Gar}\int_{-\infty}^{\infty}e^{-\alpha_kv_r^2}\,dv_r\int_0^\infty e^{-\alpha_kv_T^2}\,dv_T\int_0^{2\pi}d\psi=\frac{v_0^2}{4\pi Gar}\frac{c_k}{k}t^k.
$$
Here the two <Gaussian integrals> and the angular integral have product $\pi^2/\alpha_k$. Consequently,
$$
\rho=\frac{v_0^2}{4\pi Gar}(t+t^2)=\frac{v_0^2}{4\pi G}\frac{2a+r}{r(a+r)^2}.
$$
The <galactic distribution function> is nonnegative, gives the required density, and depends on conserved $E,L$; by <Jeans theorem> it is a self-consistent steady solution.
Differentiating the <Gaussian integral> with respect to $\alpha_k$ shows that each component has $\langle v_r^2\rangle_k=1/(2\alpha_k)=v_0^2/k$, and the same value for $\langle v_T^2\rangle_k$. Angular averaging gives $\langle v_\theta^2\rangle_k=\langle v_\phi^2\rangle_k=v_0^2/(2k)$. Weighting these by $\rho_1,\rho_2$ yields the <velocity dispersions>
$$
\boxed{\sigma_r^2=v_0^2\frac{t+t^2/2}{t+t^2}=\frac{v_0^2}{2}\frac{3a+2r}{2a+r},\qquad\sigma_\theta^2=\sigma_\phi^2=\frac{v_0^2}{4}\frac{3a+2r}{2a+r}.}
$$
The radial numerator includes $2r$, as in the original PDF. All mean velocities vanish by parity. The same parity, or the zero angular average of $\sin\psi\cos\psi$, gives \b[$\langle v_rv_\theta\rangle=\langle v_rv_\phi\rangle=\langle v_\theta v_\phi\rangle=0$].
The <velocity-anisotropy parameter> is
$$
\boxed{\beta=1-\frac{\sigma_\theta^2+\sigma_\phi^2}{2\sigma_r^2}=\frac12.}
$$
The <constant-anisotropy distribution function> therefore favors radial motion at every radius: the radial variance is twice either individual tangential variance. It still has a substantial population with nonzero <angular momentum>, rather than exclusively radial orbits. The $L^{-1}$ factor enhances low-angular-momentum stars. The radial <velocity dispersion> approaches $3v_0^2/4$ at the centre and $v_0^2$ far out; each individual tangential variance is half as large. There is no mean rotation, despite the well-defined <galaxy rotation curve> for hypothetical <circular orbits>.
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