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The Hernquist model for spherical galaxies states that the mass density at a distance to the galactic center is given by

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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.

Guest Dec 14, 2014

#1
+27223
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In a spherical shell of thickness ds at a radius s, the approximate volume, dV is

$$dV=4\pi s^2ds$$

Hence the mass, dM in this shell is given by ρ*dV or:

$$dM=4\pi \frac{s^2}{s(1+s)^3}ds$$

Integrating this from 0 to r we get:

$$M(r)=4\pi \int_0^r \frac{s^2}{s(1+s)^3}ds=2\pi \frac{r^2}{(r+1)^2}$$

Alan  Dec 14, 2014
#1
+27223
+10

In a spherical shell of thickness ds at a radius s, the approximate volume, dV is

$$dV=4\pi s^2ds$$

Hence the mass, dM in this shell is given by ρ*dV or:

$$dM=4\pi \frac{s^2}{s(1+s)^3}ds$$

Integrating this from 0 to r we get:

$$M(r)=4\pi \int_0^r \frac{s^2}{s(1+s)^3}ds=2\pi \frac{r^2}{(r+1)^2}$$

Alan  Dec 14, 2014
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Thanks Alan,

I think I have got my head around that.  :)

How would you do that integration by hand?

Melody  Dec 15, 2014
#3
+27223
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.

Alan  Dec 15, 2014
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Thank you Alan, this was a great question and answer :))

Melody  Dec 15, 2014