Question

4. a) For background-limited observations show that the signal-to-noise ratio SNR « vt, where t is the exposure time. [5] b) If you reach a limiting magnitude m at given SNR in time t, what magnitude limit will you reach to the same SNR on the same telescope in time t2 = 4t;? (5) c) If you reach a magnitude m = 20 at a SNR of 10 in 5 seconds on a 2-m telescope, how long would it take to reach m = 25 at a SNR of 5 on an 8-m telescope at the same site? [15]

Answer parts a) through c) please.

Answer 1

The Exposure Time Calculator (ETC) is done by using Massey et al. 1992, 2002. The signal-to-noise ratio (SNR) is calculated as follows:

1. First, the Photon Noise coming from the star is given as √(N∗× t), where t is exposure time and N∗ is counting rate (or the magnitude as per the question).
2. Photon Noise coming from the sky is , √ (S × p × t), where S is count per pixel and p is number of pixels.
3. The noise from the measuring device √p × R, where R is noise per pixel.

By using all of these,

SNR =  [ N∗ × t ] / [√ { (N∗ × t) + (S × p × t) + (p × R² ) } ]

Now, to do (c) part of the question we need the formula in which we can calculate exposure time when magnitude and SNR are given. This can be obtained by solving the above quadratic equation of SNR,

Exposure time,  

t = -B+B2 – 4AC 2A

where, A = N∗² ; B =(SNR)² × (N∗ + p × S) and C = (SNR)² × p × R² .

Now, from this above formula of part (c) the answer of part (a) is clear, that means, the exposure time is directly proportional to B which in turn directly proportional to (SNR)².

Therefore,

  

SNR