For an integrable function $U$ and $\phi \in C^1(\mathbb R;\,\mathbb R_+)$, where $\mathrm{supp}(\phi)\subset (-\infty, E_0]$ for some $0\leq E_0 < \infty$, we define $$\begin{aligned}
\rho_U(r) := \int_{\mathbb R_v^3} \phi\left(\frac{|v|^2}{2} + U(r)\right) \,\mathrm dv^1 \mathrm dv^2 \mathrm dv^3,  \label{def_rho_u} \\
\alpha_U(r) :=  \int_{\mathbb R_v^3} \phi'\left(\frac{|v|^2}{2} + U(r)\right) \,\mathrm dv^1 \mathrm dv^2 \mathrm dv^3. \label{def_alpha_u}\end{aligned}$$ We assume that the functions $\phi$ and $\psi$ in (\[ansatz\_f\]) have the following properties.

1.  $\phi\in C^2(\mathbb R)$ and there exists $E_0 > 0$ such that $\phi(E)=0$ for $E\geq  E_0$ and $\phi(E) > 0$ for $E < E_0$. \[cond\_phi\]

2.  The ansatz $f(x,v)=\phi\left(\frac 12 |v|^2 + U_N(x)\right), x, v \in\mathbb R^3$, leads to a compactly supported, spherically symmetric steady state $(f_N, U_N)$ of the Vlasov-Poisson system for particles with mass $1-q^2$, i.e., there exists a solution $U_N \in C^2(\mathbb R^3)$, of the equation $\Delta U_N = 4\pi (1-q^2) \rho_{N}(x)$, $U_N(0)=0$, where we used the shorthand $\rho_N := \rho_{U_N}$. This solution is spherically symmetric, $U_N(x) = U_N(|x|)$, and the support of $\rho_N \in C_c^2(\mathbb R^3)$ is the closed ball $\overline B_{R_N}(0)$ where $U_N(R_N)=E_0$ and $U_N(r) < E_0$ for $0\leq r < R_N < \infty$, and $U_N(r) > E_0$ for $r> R_N$.

3.  We have $6+ 4\pi (1-q^2) r^2 