Astronomical Laboratory ASTR:4850,
Fall 2015
by Philip Kaaret
Sections 4.3-4.5, 5.1, and 5.4 in Handbook of CCD Astronomy, second edition, by Steve B. Howell.
In this lab, you will calculate the signal to noise ratio of a
stellar brightness measurement using the images of M39 that you
obtained using the Van Allen Observatory (VAO).
A ideal imaging detector for astronomy would record the number of
photons striking each pixel and nothing else - no counts due to
read noise or dark current. In that case, and if the sky
were perfectly dark, we could measure the brightness of a star by
drawing a circle around the star on a CCD image and then summing
up all the counts inside the circle.
However, as you found in the previous lab, the measurements
produced by a CCD will also have contributions due to noise
associated with the electronics that amplifies and digitizes the
charge signal in the CCD readout (read noise) and due to electrons
released by thermal fluctuations in the silicon (dark
current). In order to measure the brightness of a star these
contributions must be measured and subtracted off. Since the
read noise and dark current arise due to random process, they will
fluctuate and it will never be possible to subtract them off
exactly. If know the magnitude of the read noise and dark
current, we can estimate how much they affect the measurement of
the stellar brightness, i.e. how much noise they add to the
measurement.
In addition, the sky isn't perfectly dark, so we need to subtract
off the background from the sky. To do this, we estimate the
brightness of the sky near the star, being careful to exclude the
star and any other stars, and then subtract that off.
Usually, one does the dark current subtraction on the whole
image. Below, we will handle each image individually, so
that you can see all of the various contributions to the noise.
The signal, S, from the star is the total number of net
electrons, N*, recorded by the CCD from the
star after subtraction of the bias, dark current, and sky
background. The generation of each electron is a random
process and if we take repeated measurements of the star the value
of NT will fluctuate according to a Poisson
distribution (note that we use NT instead of N*
as in the textbook). If you've never heard of the Poisson
distribution, look it up on wikipedia, read this http://www.umass.edu/wsp/resources/poisson/,
or ask an instructor. Also read/ask about the Gaussian or
normal distribution.
In the limit of large NT (actually N*
> 25 is enough), the Poisson distribution is well approximated
by a Gaussian distribution and we can describe the fluctuations in
NT in terms of a standard deviation equal
to sqrt( NT). This is true even with
an ideal CCD and telescope and with zero sky background.
These intrinsic fluctuations contribute a limiting noise term σ*
= sqrt( N*). A 'bright' source is defined
as one where the intrinsic noise dominates and, thus, the signal
to noise ratio, S/N, for a measurement of the source is
S/N = N*/sqrt( N*) = 1/sqrt( N*)
The number of electrons recorded by the CCD from the dark current
and the sky background are also Poisson distributed and fluctuate
in the same way. Let's look at each pixel individually, we
write the number of dark current electrons per pixel as ND.
Even though we subtract off an estimate of this number, the
fluctuations in the number of dark current electrons remains and
adds noise per pixel, σD = sqrt(ND).
The same is true for electrons produced by the sky
background. The number of electrons per pixel from sky
background is NS and each adds noise, σS
= sqrt(NS).
The read noise, σR, is the noise associated with the
electronics that amplifies and digitizes the charge signal in the
CCD readout. Read noise is present even for zero signal and
is usually assumed to be independent of the magnitude of the
signal. Read noise will arise for each pixel in the star
image, so one needs to sum up the read noise for all the pixels in
the source extraction region.
We are then led to the question of how to add these various noise contributions. The key point is that the noise distributions are well described by Gaussian probability distributions. To combine two probability distributions, one performs a convolution. The convolution of f and g is written f∗g. It is defined as the integral of the product of the two functions after one is reversed and shifted:
The wikipedia page on convolution, http://en.wikipedia.org/wiki/Convolution,
has a good explanation with some nice graphics to help you get
some intuition about what convolution means and does to functions.
The convolution of two Gaussians is another Gaussian. If
the width (standard deviation) of the initial Gaussians are σ1
and σ2, then their convolution has a width σ2
= σ12 + σ2 2.
This sort of addition is called "root of sum of
squares" or "addition in quadrature". Noting that we need to sum up the noise
contributions from dark current, sky background, and read noise
for each pixel in the circle containing the star, i.e. for npix
pixels, the total noise is then:
σ2 = σT2
+ npix × (σS 2 + σD
2 + σR 2) = sqrt[NT
+ npix × (NS + ND
+ σR2)]
Contrary to what is stated in the textbook, the reason that the
read noise term appears with a square, unlike the other terms in
the expression on the right, is simply because it is a directly
measured noise, while the other terms are estimates of noise
derived from a number of electrons, e.g. σS = sqrt(NS);
note that the noise terms are uniformly treated in the center
expression. (Also, note that 'shot noise' is another name
for 'Poisson noise', so the footnote on page 74 actually makes no
sense.) The signal to noise equation for CCDs or the "CCD
equation" is then:
S/N = N*/sqrt[NT + npix × (NS + ND + σR2)]
In the previous lab, you measured the read noise of the SBIG
ST-402 and how the dark current depends on temperature and
exposure time. In this lab, you will use the images that you
obtained of M39 using the VAO to estimate the sky background and
the signal level from stars and you will use the dark frames to
estimate the dark current. Using this information, you will
then plot how the S/N depends on various parameters.
Now let's calculate the S/N of your detection of your selected
star. You just calculated values for NT ,
NS, and ND in counts.
Recall that you need to convert these to electrons. Look
through the documentation for the CCD camera on the VAO to find
the gain of the CCD and the read noise. In choosing the
radius of the circle that you used to extract counts for the star,
you picked npix . Record your
calculations and your results in your lab notebook. Which
component of the noise dominates?
Now inspect your sky image and find a much dimmer star, one of
the dimmest that you can see. Repeat the process above,
ending up with a pair of plots (as above) for the dimmer
star.
Put printouts of your four plots in your lab notebook and write
some discussion comparing them with figure 5.6 and the upper panel
of figure 5.7 in the textbook. What extraction radius gives
the best S/N? Does this change depending on the brightness
of the star?