= Solution
For a <singular isothermal sphere>, spherical integration of the <mass density> gives
$$
M(<R)=\int_0^R4\pi r^2\frac{\sigma^2}{2\pi Gr^2}\,dr=\frac{2\sigma^2R}{G}.
$$
The <spherical shell theorem> makes the radial attraction $GM(<R)/R^2$. Balancing it with circular centripetal acceleration yields
$$
\boxed{M(<R)=\frac{2\sigma^2R}{G},\qquad V_c(R)=\sqrt{\frac{GM(<R)}R}=\sqrt2\sigma.}
$$
Thus the model has a <flat galaxy rotation curve>, resembling the outer <galaxy rotation curves> of many <spiral galaxies>. Central rotation curves usually rise, and not all real curves are flat throughout. A pressure-supported <elliptical galaxy> does not itself rotate at this test-particle <circular speed>.
The model has divergent central <mass density>, although finite enclosed mass at each finite radius, and an infinite total mass because $M(<R)\propto R$. Its potential is logarithmic, requiring an outer cutoff for a finite escape energy. A globally constant isotropic <velocity dispersion>, spherical symmetry, and one common profile for all gravitating components are further idealizations. Stellar light commonly has a quite different radial profile from the <dark matter halo>; real disks are flattened, and halos are neither exact isothermal spheres nor infinite. Tidal truncation also changes the distribution near the satellite boundary.
Back to article page
