Microlensing as discussed in class magnifies the source, instead of producing multiple images because they

cannot be resolved.

a$)(2$ pts$)$ Show that when the source is at Einstein radius from the lens $($angularly$),$ the total magnification

is $1\mathrm{.}34.$

b$)$ Microlensing is detectable because the source and lens move relative to each other, and thus the magnifi$-$

cation changes with time. Let the source be stationary, while the lens moves in a straight line across the sky

$($proper motion$)$ with angular speed $.$ Let t$=0$ be the instant the source and lens are the closest in the sky$.$

We will look at the example of a $0\mathrm{.}1$ solar mass $($dark$)$ lens located $48$ kpc from us$,$ and $2$ kpc from the

source. During the microlensing event, the source is observed to have a peak flux increase of a factor of $6\mathrm{.}16,$

and is brighter by a factor of $1\mathrm{.}34$ for a total of $115$ days. Calculate the following. Note that since these are

small distances, they can be added or subtracted directly.

b$-$i$)(4$ pts$)$ What is the closest angular distance between the source and lens? You need to specify units.

b$-$ii$)(2$ pts$)$ What is the average angular speed of the lens across the sky$?$ Again, specify units.

b$-$iii$)(1$ pt$)$ What is the average linear speed?

$2$