ASTR 3830: Problem Set 3 (due in class Friday February 27th) 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 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 Collections: Physics