Download 12 - BYU Physics and Astronomy

Physics 427
Introduction to Astrophysics
Problem Assignment #12
Due: Wednesday, November 19, 2014
PHOTON COUNT STATISTICS
Suppose that a process or combination of processes producing random pulses at total rate λ is
maintained for a time interval t so that N = λt pulses are detected. If the observation is repeated several
times there should be a standard deviation in the measurements of σ1 = (λt)½ = N ½, i.e., for a particular
count N has a probability of about 2/3 of being with within σ1 of λt, or the uncertainty in N is ΔN σ1.
Suppose a second set of processes produces pulses at rate η so that in time interval t a count M = ηt is
accumulated with σ2 = M ½. It is clear that if the two counts are added that N + M would have a standard
deviation and associated uncertainty of Δ(N + M) = σ12 = (N +M)½ = (σ12 + σ22)½. Likewise, since a
deviation of a particular magnitude δN (or δM) is equally likely (almost) to occur with either a positive or
a negative sign, the uncertainty in the difference N - M is virtually the same as that in the sum N + M, i.e.,
Δ(N - M) = σ12 = (N +M)½ = (σ12 + σ22)½.
Apply this result to a pixel which is exposed to a star image for an interval of time t wherein it
accumulates counts from the star (at rate λ) plus sky and dark counts (at combined rate η) and then is
exposed for time t only to the sky, accumulating counts at rate η. The star count rate, λ, is then
determined by taking the difference in the two integrated signals, S1 = (λ + η)t and S2 = ηt, and dividing
by t, i.e.,λ = (S1 - S2 )/t.
1. Derive an expression giving the percentage uncertainty in λ, as determined in this manner, as a
function of λ, η, and t.
2. Now suppose that on a dark, moonless night with good seeing the sky plus dark count rate is η =
1/s whereas on a bright moonlit night η = 4/s. Calculate how long a star with λ = 200/s must be
observed on a dark night, t1, and how long it must be observed on a brightly lit night, t2, to
determine λ with an accuracy of 1 per cent. What is the ratio t2/t1?
3. Repeat the calculations of part 2 above for a second star 5 magnitudes fainter than the first.
4. Answer the questions which follow: (a) How many times brighter than the faint star is the
brighter star? (b) How many times longer than the bright star must the faint star be observed on
the dark night? (c) How many times longer than the bright star must the faint star be observed on
the the moonlit night?
(If your results are correct it should be readily apparent why large observatories usually give
higher priority to observers of faint objects than to observers of bright objects when granting
"dark" [near new moon] observing time.)