Astro 506

Problem 4: To be handed in on Feb. 14

Define I(r) to be the radial surface-brightness profile of an
astronomical source (azimuthally averaged, and in counts/arcsec^2)
such that the curve of growth, g(r), of the light enclosed by a
circular aperture of radius r is given by:

	d(g(r)) = 2 pi I(r) r dr.

Define eta(r) to be the ratio of the surface brightness at
radius r to the average surface brightness within radius r, namely:

	eta(r) = I(r) / <I(r)>.

Prove the following theorem: 

If an image is taken in the background limit (i.e. the dominate source
of noise is contributed by a uniform background source, e.g. sky), and
an object light profile is a monotonically decreasing function of r,
then the the signal-to-noise of the intergrated light (g(r)) has a
maximum when eta = 1/2.