Reading: Chapter 5 of Sparke & Gallagher
The rules: Collaborative work is encouraged. This homework can be done in consultation with your fellow classmates, the AI, or the professor. However, everyone must submit their own solutions to get credit, and all help should be acknowledged (a single sentence mentioning the others in your study group is sufficient). Show your work.
1. [11 pts] (a) Show that a central surface brightness of 15 mag arcsec-2 in the I band corresponds to 18000 Lsun pc-2. (b) Explain why this does not depend on the galaxy's distance (in the low redshift regime). (c) About how far does IB = 25 mag arcsec-2 fall below the brightness of the night sky, given in Table 1.9?
2. [24 pts] (a) For NGC 7331, the radial velocity is 820 km/s; if Ho = 60 km/s Mpc-1, find its distance by using Hubble's law. (b) The corrected total V magnitude of NGC 7331 is 8.75, what is its V-band luminosity? Is NGC 7331 more or less luminous than the Milky Way and M31? (c) The observed scale length is 55"; what is the disk scale length in physical units (kpc)? The observed R25 = 315"; at what radius (in kpc) does the surface brightness equal 25 mag arcsec-2? How does it compare with the size of the Milky Way's disk? (d) Use equation 5.1 to show that the disk luminosity, LD = 2 pi Io hR2. If the central surface brightness is 350 Lsun pc-2, what is LD? What is the surface density of starlight at R = 8 kpc?
3. [15 pts] (a) In a galaxy where the potential follows the Plummer model, find the rotation curve V(R). (b) Show that Vmax2 = 2 G M/(33/2a). (c) Sketch V(R) for R < 4 a.