How Do You Calculate Photon Detection Probabilities for Different Stars?

[Lynx1390]

1. A 2.5m aperture telescope obverses a star through an R filter. Assume that there is no noise associated with the detection system. The CCD has a full well depth of 20,000 counts and a gain correction factor of 1.00000.
a. On average, the telescope detects 3 photons/sec from this star. What is the probability that it will detect less than three photons in an 1 second observation

I'm pretty sure I got this one, use the Poisson distribution as shown below.

P = (3^0e^(-3))/0! + (3^1e^(-3))/1! + (3^2e^(-3))/2! = 0.42


b. Another star is observed. On average, the telescope detects 1,230 photons/sec from this star. What is the probability that will detect more than 1,230 photons in anyone second observation?

It is this part I am a little unsure of, I know I need to use the normal distribution but not sure how to reach from it.

Thanks in advance

 

[BvU]

Hello Lynx, and welcome to PF. Please use the template.

You want to approach the Poisson distribution by a normal distribution in the second case, which is reasonable. Do you know why ? And do you have an expression for the normal distribution ? It has a certain characteristic that makes answering b) a piece of cake...

 

[Lynx1390]

The reason why I want to use the normal distribution is because I know with a larger mean Poisson 'turns' Gaussian. And from a quick look at the bell curve, the answer would be rounded to 1/2. I'm just not sure how to express it mathematically.

 

[Orodruin]

Don't look at the curve, look at the mathematical expression for the probability distribution function of the normal distribution. What is its value in x=μ-ε if its value in x=μ+ε is p₀?

 

[Lynx1390]

Ok, so the probability distribution function is -

P = 1/(σ√2∏)*e^(-1/2((x-μ))/σ)^2

But I'm not sure how to find the standard deviation with the information given? I know its a silly question but I don't understand how to figure it out?

 

[BvU]

Placing brackets in the right place is an art too. The square is inside the exponent, meaning the distribution is symmetrical around x = μ. Symmetrical means half of its area is on the side > μ. That's nice and that is the reason you want to look at the Poisson distribution as an almost Gauss distribution. Quite justifyable for μ = 1230.