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 y-axis 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 cm-3
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