# Signal to noise ratio

Summary:: An image was taken with a ##60## second exposure time of a 6th magnitude star and the signal to noise ratio was detected to be ##S/N = 20##.

a. What should the exposure time be if you wanted a ##S/N = 100##?
b. Now calculate the ##S/N## if it were a 2nd magnitude star for a ##10## second exposure.

For part a I got the following:

Let ##S = \mu t##, where ##\mu## is the count and ##t## is time, therefore we have $$S/N = 20$$ $$\frac{\mu t}{\sqrt(\mu t)} = 20$$ $$(\frac{\mu t}{\sqrt(\mu t)})^2= 20^2$$ $$\frac{\mu^2 t^2}{\mu t} = 400$$ therefore the count ##\mu## is ##400##. Therefore in order to get ##S/N = 100## we have $$S/N = 100$$ $$\frac{\mu t}{\sqrt(\mu t)} = 100$$ $$(\frac{400t}{\sqrt(400t)})^2= 100^2$$ $$\frac{400^2 t^2}{400t} = 10000$$ now solving for ##t## I got ##1500## seconds.

But for part b of the question, do I apply the same logic even though the magnitude of the star is different? Or will the logic be different?

• Delta2

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mfb
Mentor
The count rate ##\mu## will be different and you can calculate how much it will differ based on the magnitude.

You only consider Poisson noise here - which might be fine (and might be expected within the scope of the problem), but in general there can be other noise sources, too.

The count rate ##\mu## will be different and you can calculate how much it will differ based on the magnitude.

You only consider Poisson noise here - which might be fine (and might be expected within the scope of the problem), but in general there can be other noise sources, too.
Yes, the scope is only Poisson. So the count rate for part b will be different due to the change of magnitude? So since the magnitude difference is ##4## will I get the following: $$2.5^4 * \sqrt\frac{400}{\frac{60sec}{10sec}}?$$

mfb
Mentor
Why did you put the magnitude difference outside the brackets?

Why did you put the magnitude difference outside the brackets?
Hmm which brackets?

mfb
Mentor
Eh, I meant the square root.

Eh, I meant the square root.
I thought the equation was magnitude times the count? Or does the magnitude need to be inside the square root?

mfb
Mentor
2.54 is the ratio of photons per time. Just like all the other terms that scale with the signal it should be in the square root if you calculate signal to noise ratios.

• learningastronomy