= Self-gravitating isothermal slab
{title2=$\rho(z)=\rho_0\operatorname{sech}^2(z/z_0)$}
A plane-symmetric, self-gravitating <stellar population> with constant vertical <velocity dispersion> satisfies the <vertical Jeans equation> and the <Poisson equation for Newtonian gravity>. With $\Phi'(0)=0$, their solution is
$$
\rho=\rho_0\operatorname{sech}^2(z/z_0),\quad \Phi-\Phi(0)=2\sigma_z^2\log\cosh(z/z_0),\quad z_0^2=\frac{\sigma_z^2}{2\pi G\rho_0}.
$$
Its <surface density> is $\Sigma=2\rho_0z_0$, so $z_0=\sigma_z^2/(\pi G\Sigma)$. Its outer density tail is exponential with <scale height> $z_0/2$; its central profile is not exponential.
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