The Hernquist model for spherical galaxies states that the mass density at a distance 
to the galactic center is given by
(Here, represents the variable radius to the center of the galaxy.) This model was chosen so that the density has the behavior 
as 
and 
as 
. Let 
represents the total mass contained within a sphere of radius 
, centered at the galactic center. (I.e., is a cumulative mass function–it tells the total mass that lies within a distance to the galaxy’s center.) Determine a formula for . Hint: Use spherical shells.
+33658 In a spherical shell of thickness ds at a radius s, the approximate volume, dV is
dV=4πs2ds
Hence the mass, dM in this shell is given by ρ*dV or:
dM=4πs2s(1+s)3ds
Integrating this from 0 to r we get:
M(r)=4π∫r0s2s(1+s)3ds=2πr2(r+1)2
.
