# Betelgeuse problem

I've been given this problem:

(a) A telescope is required to image an area of the sky of angular size 1 arc second onto a 10 mm square CCD. What focal length must the telescope mirror have to fully utilize the CCD area?

With this part, I had no problem. I used the formula $$h= f_0.\alpha$$ with $$\alpha=1$$ arc seconds converted into radians, so around $$4.84×10^{-8}$$ radians, and with $$h=0.01$$ meters. I obtained the value $$2062.65$$ meters for the focal length.

(b) Between October 2019 and January 2020, the star Betelgeuse was observed to get significantly fainter, with its apparent magnitude in the visible band changing from 0.5 to 1.5. Betelgeuse was observed at its brightest and faintest using a telescope with a CCD detector on the focal plane. Both observations were carried out using identical hardware and the same observation time. Find

(i) the factor by which the brightness of Betelgeuse changed between the two observations, From Pogson's equation and the definition of apparent magnitude I was able to determine the factor to be around $$2.512$$.

(ii) the factor by which the signal-to-noise ratio changed,

(iii) how much longer the second observation time would need to be to maintain the same signal to noise ratio as the first. You may assume that Poisson statistics apply and that there are no other significant sources of noise.

However, on parts ii) and iii) I'm completely stuck. Can I assume that the optical band is centered at $$\lambda=500$$ nm? Or how could I approach these problems?

• You are correct that you need to use the definition of "apparent magnitude" which IIRC includes the human scotopic response curve. Then assume there's an IR cutoff filter so the CCD is only collecting in the visible band -- make this explicit when you hand in your homework. As to SNR, the problem says there is no electronics noise, so you only need to calculate the Poisson SNR after you determine the input irradiance (photons/pixel/frame) – Carl Witthoft Mar 27 at 17:25
• How do I determine the number of photons? – space nerd Mar 27 at 17:49
• Convert magnitude to flux, thus power. And use the average wavelength (or a black body curve for the star's temperature and filter window if you want it more elaborate and complicated) to get the number of photons per unit time – planetmaker Mar 28 at 7:19
• Thank you everyone, I think I got it now! – space nerd Mar 28 at 12:41
• By the way it's certainly okay to post an answer to your own question once you've found the answer. This way future readers may also find it helpful. – uhoh Aug 19 at 18:22