Question

You are trying to take an image of a particular star with apparent magnitude m=10, and need to figure out how long you will need to expose for with your telescope. Your friend tells you that her telescope of diameter 0.05 metres can detect the star in 119.5 minutes.
a) If your telescope has diameter 0.18 metres, how long do you need to expose for? Answer in minutes.

Answer 1

You haven't told us anything about the detectors being used.  We don't know how the sensitivity of the detector is related to the total number of photons absorbed, and we don't even know whether you and your friend are both using the same type of detector.

All we can do, in desperation, is ASSUME that the minimum time required to just detect a star is inversely proportional to the total number of its photons that strike the detector.  That is, assume . . .

(double the number of photons) ===> (detect the source in half the time) .

-- The intensity of light delivered to the prime focus of a telescope is directly proportional to the AREA of its objective lens or mirror, which in turn is proportional to the square of its radius or diameter.

So your telescope gathers (0.18/0.05)² = 12.96 times as much light as your friends telescope does.

-- So we'd expect your instrument to detect the same star in

(119.5 min) / (12.96) = 9.22 minutes  .

We're simply comparing the performance of two different telescopes as they observe the same object, so the star's magnitude doesn't matter.