= Cauchy velocity distribution
{c}
{title2=$k_J=\sqrt{4\pi G\rho_0}/b$}
The isotropic three-dimensional <Cauchy velocity distribution> $f_0=\rho_0b/[\pi^2(v^2+b^2)^2]$, $b>0$, has density $\rho_0$ and one-dimensional marginal $F(u)=\rho_0b/[\pi(u^2+b^2)]$. Its second <velocity> moment diverges, so $b$ is a scale rather than a finite <velocity dispersion>. At zero phase <speed>, $\int F'(u)/u\,du=-\rho_0/b^2$, giving $k_J=\sqrt{4\pi G\rho_0}/b$. In the upper-half frequency <plane>, the marginal response is $\rho_0/(\omega/k+ib)^2$; the growing branch has $\omega=i(\sqrt{4\pi G\rho_0}-kb)$ for $0<k<k_J$.
Back to article page
