Question

1 Photon noise and more When counting photons with a detector, we have an inherent uncertainty in the time interval between arriving photons and therefore also the total number of photons counted during an exposure. The Poisson distribution describes measurements made by counting uncorrelated events such as the arrival of photons. The mean and variance for Poisson distribution is given by: $E(X) = Var(X) = \mu$ for $X \sim Poisson(\mu)$ with $\mu$ being the mean number of events. (1) a) Maaike wants to measure a star's brightness with a precision of 5 percent. She has a perfect imaging system and a noiseless detector. How many photons does she need to count? How many photons does she need to count if she wants a relative precision of 0.5 percent? b) Maaike realizes that when she points her telescope to an empty part of the sky near the star, she measures a background count of 50 percent that of what she measures for the star. Now how many measurement photons would Maaike need to achieve a relative precision of 5 percent on the star' brightness taking into account the sky background? What about for 0.5 percent relative precision? c) Suddenly Maaike realizes that her noiseless camera has a readout noise of 4 electrons. How many photons does she now need to count to reach a relative precision of 5 percent? What about a relative precision of 0.5 percent?

          1 Photon noise and more
When counting photons with a detector, we have an inherent uncertainty in the time interval between arriving
photons and therefore also the total number of photons counted during an exposure. The Poisson distribution
describes measurements made by counting uncorrelated events such as the arrival of photons. The mean and
variance for Poisson distribution is given by:
$E(X) = Var(X) = \mu$ for $X \sim Poisson(\mu)$
with $\mu$ being the mean number of events.
(1)
a) Maaike wants to measure a star's brightness with a precision of 5 percent. She has a perfect imaging system
and a noiseless detector. How many photons does she need to count? How many photons does she need to
count if she wants a relative precision of 0.5 percent?
b) Maaike realizes that when she points her telescope to an empty part of the sky near the star, she measures
a background count of 50 percent that of what she measures for the star. Now how many measurement
photons would Maaike need to achieve a relative precision of 5 percent on the star' brightness taking into
account the sky background? What about for 0.5 percent relative precision?
c) Suddenly Maaike realizes that her noiseless camera has a readout noise of 4 electrons. How many photons
does she now need to count to reach a relative precision of 5 percent? What about a relative precision of 0.5
percent?

1 Photon noise and more
When counting photons with a detector, we have an inherent uncertainty in the time interval between arriving
photons and therefore also the total number of photons counted during an exposure. The Poisson distribution
describes measurements made by counting uncorrelated events such as the arrival of photons. The mean and
variance for Poisson distribution is given by:
E(X) = Var(X) = μ for X ∼ Poisson(μ)
with μ being the mean number of events.
(1)
a) Maaike wants to measure a star's brightness with a precision of 5 percent. She has a perfect imaging system
and a noiseless detector. How many photons does she need to count? How many photons does she need to
count if she wants a relative precision of 0.5 percent?
b) Maaike realizes that when she points her telescope to an empty part of the sky near the star, she measures
a background count of 50 percent that of what she measures for the star. Now how many measurement
photons would Maaike need to achieve a relative precision of 5 percent on the star' brightness taking into
account the sky background? What about for 0.5 percent relative precision?
c) Suddenly Maaike realizes that her noiseless camera has a readout noise of 4 electrons. How many photons
does she now need to count to reach a relative precision of 5 percent? What about a relative precision of 0.5
percent?

Added by Denise R.

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