Using a ccd detector installed on a 1-m diameter telescope to record the image of a star, the light of the star is spread over a circular area of radius r pixels. the number of photoelectrons accumulated over the area during the exposure due to the light of the star is nstar, while light from the sky produces nsky photoelectrons per pixel.

calculate the signal-to-noise (s/n) ratio of the photometric measurement of the star, assuming poisson statistics (the noise is the square root of the total number of counts from all sources). then, assuming constant rate of photon arrival, find the functional dependence of s/n on exposure time t, in two limiting cases: counts from stars > > > counts from sky and vice-versa.

if we observe the same star with a 10-m diameter telescope, the light is spread over an area of r/3 due to larger light-gathering area. by what factor does the exposure time with the 1-m telescope need to be increased to reach the s/n obtained with the 10-m telescope, for each of the two limiting cases?