Density profile for the general isothermal sphere, together with its projected form (left). Notice the density law flattens into a core inside r0 ~ 1.
The gradient of the density law in a log-log plot. Notice that the singular isothermal sphere would be -2 at all radii. For the general isothermal sphere, the gradient is near zero at the origin, and only becomes -2 at about 10 r0.