= Solution
A spherically symmetric stellar model has no preferred direction in the tangent plane at a fixed radius. Rotations about the radial axis exchange its two transverse <velocity> directions, so $\langle v_\theta^2\rangle=\langle v_\phi^2\rangle$. This uses <spherical symmetry> of the <velocity> distribution, not just a spherical <mass density>.
The <augmented density> $\rho(r,\psi)$ treats radius and <relative potential> as independent variables, agreeing with the physical density only along $\psi=\psi(r)$. For any regular bound spherical <galactic distribution function> $F(E,L)$, write $P_r=\int v_r^2F\,d^3v$ and $P_t=\int v_\theta^2F\,d^3v$. At fixed $r$, $F_\psi=F_E$ and $\partial_{v_r}F=-v_rF_E$. <Integration by parts> therefore gives
$$
\partial_\psi P_r=\int v_r^2F_E\,d^3v=-\int v_r\partial_{v_r}F\,d^3v=\rho.
$$
For $v_t^2=v_\theta^2+v_\phi^2$, the <velocity> measure is $2\pi v_t\,dv_t\,dv_r$, and $rF_L=\partial_{v_t}F+v_tF_E$. Integrating the transverse derivative gives $-2P_r$, while integrating the radial derivative in $\int v_r^2v_t^2F_E\,d^3v$ gives $2P_t$. Consequently
$$
\partial_rP_r=\frac{2}{r}(P_t-P_r),\qquad\partial_r(r^2P_r)=2rP_t,
$$
where $\psi$ is fixed. These identities show that the augmented representation holds for any such spherical distribution, not only a monomial ansatz. With radial pressure vanishing at the zero-binding boundary, set
$$
A(r,\psi)=P_r(r,\psi)=\int_0^\psi\rho(r,q)\,dq.
$$
Along the physical curve, the chain rule gives $dA/dr=\int_0^\psi\partial_r\rho(r,q)\,dq+\rho(r,\psi)\psi'(r)$. The <Spherical Jeans equation> then requires
$$
B\equiv\rho\langle v_\theta^2\rangle=A+\frac r2\int_0^\psi\partial_r\rho(r,q)\,dq=\frac{1}{2r}\int_0^\psi\partial_r[r^2\rho(r,q)]\,dq.
$$
Thus the <Jeans moments from an augmented density> are
$$
\boxed{\rho\langle v_r^2\rangle=\int_0^\psi\rho(r,q)\,dq,\qquad\rho\langle v_\theta^2\rangle=\frac1{2r}\int_0^\psi\partial_r[r^2\rho(r,q)]\,dq.}
$$
The partial derivative holds $q$ fixed. Substitution cancels the integral of $\partial_r\rho$ and leaves precisely $\rho\psi'$ in the <Spherical Jeans equation>. This is a representation by a chosen <augmented density>, not a unique solution determined by the one-variable density: different extensions off the physical curve encode different anisotropies, and not every formal extension necessarily gives a nonnegative <galactic distribution function>.
For the <hypervirial density-potential family>, differentiate the potential and use the <spherical shell theorem>:
$$
\psi'=-\frac{GM r^{p-1}}{(a^p+r^p)^{1+1/p}},\qquad M(<r)=-\frac{r^2\psi'}{G}=\frac{M r^{p+1}}{(a^p+r^p)^{1+1/p}}.
$$
Differentiating $M(<r)$ and dividing by $4\pi r^2$ gives
$$
\boxed{\rho=\frac{(p+1)M}{4\pi}\frac{a^p r^{p-2}}{(a^p+r^p)^{2+1/p}}.}
$$
The enclosed mass tends to $M$ at infinity and to zero at the centre for every $p>0$. The same result follows from $\nabla^2\psi=-4\pi G\rho$, with the stated sign convention.
From now on set $G=M=a=1$. The separable <augmented density> is $\rho(r,q)=(p+1)r^{p-2}q^{2p+1}/(4\pi)$. Integrating it gives
$$
A=\frac{\rho\psi}{2(p+1)},\qquad B=\frac p2 A,
$$
and hence
$$
\boxed{\langle v_r^2\rangle=\frac{\psi}{2(p+1)},\qquad\langle v_\theta^2\rangle=\langle v_\phi^2\rangle=\frac{p\psi}{4(p+1)}.}
$$
With $\psi=(1+r^p)^{-1/p}$ these are the requested radial expressions. The <velocity-anisotropy parameter> is $\beta=1-p/2$, and the total mean square speed is $\langle v^2\rangle=\psi/2$.
The local kinetic-energy density is $t=\rho\langle v^2\rangle/2=\rho\psi/4$. With the potential zero at infinity, the gravitational energy density is $w=\rho\Phi/2=-\rho\psi/2$, where $\Phi=-\psi$. Therefore
$$
\boxed{2t+w=0\quad\text{at every radius}.}
$$
This is the <local virial relation of the hypervirial model>, a special property stronger than the global <virial theorem>. To compute the global energies as well, set $I=\int\rho\psi\,d^3x$. The substitution $u=r^p$ yields the <global binding integral of the hypervirial model>
$$
I=\frac{p+1}{p}\int_0^\infty u^{1/p}(1+u)^{-2-2/p}\,du=\frac{p+1}{p}\frac{\Gamma(1+1/p)^2}{\Gamma(2+2/p)}.
$$
The <beta function> integral converges for all $p>0$. Thus
$$
\boxed{T=\frac I4,\qquad W=-\frac I2,\qquad2T+W=0.}
$$
Restoring dimensions multiplies $I,T,W$ by $GM^2/a$.
To derive the <hypervirial distribution function>, let $E=\psi-v^2/2$ be the positive <relative energy> and $L=rv\sin\vartheta$ the magnitude of the <specific angular momentum>. Seek $F=C L^{p-2}E^s$ for $E>0$ and zero otherwise. Its density is
$$
\rho=2\pi C r^{p-2}\int_0^\pi\sin^{p-1}\vartheta\,d\vartheta\int_0^{\sqrt{2\psi}}v^p(\psi-v^2/2)^s\,dv.
$$
The angular integral is $\sqrt\pi\Gamma(p/2)/\Gamma((p+1)/2)$. Substitution $z=v^2/(2\psi)$ makes the radial integral
$$
2^{(p-1)/2}\psi^{s+(p+1)/2}B\left(\frac{p+1}{2},s+1\right).
$$
Matching the augmented-density power $\psi^{2p+1}$ gives $s=(3p+1)/2$. Matching its coefficient gives the <normalization of the hypervirial distribution function>
$$
\boxed{F(E,L)=C L^{p-2}E^{(3p+1)/2},\qquad C=\frac{(p+1)\Gamma(2p+2)}{2^{(p+5)/2}\pi^{5/2}\Gamma(p/2)\Gamma((3p+3)/2)}.}
$$
All integrals are finite in <velocity> for $p>0$, and $C>0$. Because $E$ and $L$ are orbital integrals, this is a steady solution by the <Jeans theorem>. As checks, $p=1$ gives $C=3/(4\pi^3)$ for the <Hernquist model>, and $p=2$ gives $C=24\sqrt2/(7\pi^3)$ for the isotropic <Plummer model>.
\b[The density and potential do not uniquely fix the stellar distribution.] The preceding coefficient is fixed within the chosen separable power-law model, but <nonuniqueness of spherical galactic distribution functions> remains without that extra restriction. An explicit positive counterexample uses the same $p=2$ density. Define the <Osipkov-Merritt distribution function> variable $Q=E-L^2/2$ and take
$$
F_{\mathrm{OM}}(E,L)=\frac{3}{\sqrt2\pi^3}Q^{3/2}\quad(Q>0),\qquad F_{\mathrm{OM}}=0\quad(Q\leq0).
$$
Rescale the transverse <velocity> by $\sqrt{1+r^2}$. Then $Q=\psi-[v_r^2+(1+r^2)v_t^2]/2$ and the <velocity> Jacobian contributes $(1+r^2)^{-1}$. Direct integration gives
$$
\rho_{\mathrm{OM}}=\frac{3}{4\pi}\frac{\psi^3}{1+r^2}=\frac{3}{4\pi}\psi^5,
$$
exactly the same <Plummer model> density. This <anisotropic Plummer distribution with unit anisotropy radius> has $\beta=r^2/(1+r^2)$, so it is distinct from the isotropic $E^{7/2}$ distribution while remaining positive and spherical. Its second moments differ, so it does not share the extra augmented-density and local-virial restrictions of the chosen hypervirial model.
Back to article page
