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 $\sqrt{2}\sigma$.

Truncating the singular isothermal gas profile at radius $r_0$ gives $M=2c_s^2{r}_0/G$ and boundary pressure $p_0=c_s^4/(2\pi G{r}_0^2)$. The internal energy of a monatomic gas is $U=3M{c}_s^2/2$, while shell integration gives gravitational self-energy $\Omega =-G{M}^2/r_0=-2M{c}_s^2$. Thus $2U+\Omega =3p_0\mathcal{V}$, with $\mathcal{V}=4\pi r_0^3/3$. The confining pressure supplies the virial surface term. This singular-central-density model differs from a regular pressure-confined isothermal sphere.

For $p=a_T^2\rho$, spherical hydrostatic equilibrium admits $\rho =a_T^2/(2\pi G{r}^2)$ with $M(r)=2a_T^{2}r/G$. Matching an external pressure $p_0$ fixes $r_s=a_T^2/\sqrt{2\pi G{p}_0}$. The central density is singular although the enclosed mass tends to zero; this is one branch, not every finite-centred isothermal cloud.

With potential zero at infinity, a truncated profile $\rho =k/(2\pi G{r}^2)$ has $\Phi (r)=-2k[1+\ln (r_0/r)]$ inside the truncation. Exterior spherical shells have zero force inside but contribute a nonzero constant potential there; omitting them changes escape work.

For $\rho =k/(2\pi G{r}^2)$ truncated at $r_0$, the enclosed mass is $M(r)=2kr/G$. The shell-assembly integral gives $W=-\int GM(r)dM/r=-G{M}_0^2/r_0$. The origin has divergent density but finite total mass and binding energy.

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.