 
Summary: ASTR 3830: Problem Set 3
(due in class Friday February 27th)
1) A spiral galaxy has a rotation curve which rises linearly from zero at the center to 200
km/s at 5 kpc from the center. The rotation curve then remains flat at a constant value of
200 km/s out to 15 kpc, beyond which it can't be measured.
(a) Make a plot of M(r), the total mass enclosed within radius r, as a function of radius.
Assume that the mass distribution in the galaxy is spherically symmetric. Note: 1 kpc =
3.086 x 1021
cm. Use grams for the yaxis on the plot.
(b) Assume that the form of the rotation curve is dominated by dark matter. Plot the
density of the dark matter in g cm3
versus radius.
2) (a) Show that if a single star in a galaxy has apparent magnitude m*, then the apparent
magnitude of a cluster of N such stars mN is given by:
mN  m* = 2.5log10 N .
(b) The surface brightness of a galaxy at distance d is I0 magnitudes per square arcsecond
in some waveband. If the galaxy is made up of stars of absolute magnitude M (measured
in the same waveband), use the above result, together with the definition of the absolute
magnitude, to show that the number of stars contained within a one arcsecond square
