Singular isothermal sphere (SIS)

A singular isothermal sphere with one-dimensional velocity dispersion $\sigma$ has $\rho (r)={\sigma }^2/(2\pi G{r}^2)$, enclosed mass $M(<r)=2{\sigma }^{2}r/G$, and constant circular speed $\sqrt{2}\sigma$.

The projected density $\Sigma (R)={\sigma }^2/(2GR)$ gives cylindrical mass $\pi {\sigma }^{2}R/G$, so the reduced deflection magnitude is the constant displayed angular Einstein radius. The signed thin gravitational lens equation is $\beta =\theta -{\theta }_E\mathrm{sgn}\theta$. A source with $0<\beta <{\theta }_E$ has two images; one with $\beta >{\theta }_E$ has one. The lensing convergence and lensing shear magnitude are both ${\theta }_E/(2|\theta |)$.

For the $r^{-2}$ singular isothermal sphere, the isotropic Spherical Jeans equation gives ${\sigma }_r^2={\sigma }_{\theta }^2={\sigma }_{\varphi }^2=v_0^2/2$, so $⟨v^2⟩=3v_0^2/2$. Randomly oriented circular orbits instead have ${\sigma }_r^2=0$, each transverse second moment $v_0^2/2$, and total mean square speed $v_0^2$.

A finite spherical subvolume has radial-pressure virial surface term $4\pi R^3\rho (R){\sigma }_r^2$. The infinite singular isothermal sphere has divergent total mass and energies, so a finite isolated virial theorem cannot omit this term. Different orbital distributions with the same density can have different radial boundary pressures.