1. [5 pts] Use the filter bandwidths for the UBV system and the effective temperature of 9500 K for Vega to determine through which filter Vega would appear brightest to a photometer. Assume S(lambda) = 1 inside the filter bandwidth and S(lambda) = 0 outside the filter bandwidth.
2. [10 pts] (a) The mass density of stars in the solar neighborhood is approximately 0.05 Msun pc-3. Assuming that the mass density is constant and that all of the stars are main sequence M stars, estimate the fraction of the Milky Way's disk volume that is occupied by stars. (b) Suppose that an intruder star (a main-sequence M star) travels perpendicularly through the Galactic disk. What are the odds of the intruder colliding with another star during its passage through the disk?
3. [5 pts] How does the surface brightness, defined as luminosity per unit solid angle, of a transparent, infinitely thin disk galaxy, depend on the disk's inclination angle? (Note: a face-on disk has inclination angle i=0, while an edge-on disk has inclination i=90). Specifically, derive the line-of-sight correction that should be applied to the observed central surface brightness in magnitudes per square arcsec.
4. [5 pts] Consider a set of galaxies with infinitely thin circular disks and random orientations; that is, the normal vectors of these disks are distributed uniformly in all directions. What is the distribution function of apparent axial ratios b/a of these galaxies, where a and b are the projected major and minor axes, respectively?
5. [5 pts] Show that the central surface brightness of 15 mag arcsec-2 in the I band corresponds to 19000 Lsun pc-2.
6. [20 pts]
According to Freeman (1970), the disks of high surface brightness spirals reach IB ~ 21.7 mag arcsec-2.
(a) How many Lsun does the central square arcsecond of
such a galaxy radiate?
(b) If its absolute magnitude, MB = -20.5, how many
Lsun does the galaxy emit in the B-band?
(c) If its distance d=50 Mpc, and we
ignore additional light from a bulge, show that the exponential scale radius is 22", while
80% of the light falls within R25.
(d) For a low-surface brightness galaxy with IB(0) = 24.5 mag arcsec-2, show
that less than 10% of the light comes from R < R25.
(e) Now consider a set of spiral disks, all with MB = -20.5 and d = 50 Mpc, but
with differing exponential scale radii. Plot R25 in arcsec vs the disk
scale length over the range 5" < hR < 120", and plot R25 vs. I(0).
(f) Explain why galaxies with centers much fainter than IB(0) = 21.7 mag arcsec-2
might have been missed from Freeman's 1970 sample.