 
Summary: ASTR 3830: Problem Set 2
(due in class Friday February 13th)
1. The area inside the Einstein ring is given by
A = pdL
2
qE
2
, where dL is the distance to the
lens and qE is the angular size of the Einstein ring. Assume that the distance to the
sources, dS, is fixed, and write
dL = xdS :
Sketch how A varies with x for 0 < x < 1 (don't forget that qE depends on both dL and dLS).
By differentiating and setting the resultant expression to zero, find the value of x that
maximizes the physical area inside the Einstein ring.
(this proves one of the most important results for the study of gravitational lensing,
which we'll return to later in the semester)
2. Assume that early in the history of the Milky Way, a short burst of star formation
formed a population of stars in the halo with a Salpeter mass function,
