Astronomical Laboratory ASTR:4850,
Spring 2018
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 M36 that you
obtained with the Van Allen Observatory 17 inch
telescope and the SBIG STXL-6303E camera. Before you
start the lab, you need a sky image in the B filter, a dark frame
with the same exposure time, and a bias frame. All of the frames
should be taken at the same temperature.
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. However, if
we can determine 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 uncertainty or 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. Sky
background, of course, also adds to the noise in any ground-based
measurement of stellar brightness.
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, NT, recorded by the CCD from the
star (or other astronomical target) 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 because N*
is not a valid Python variable name). If you've never heard of the
Poisson distribution, look it up on wikipedia or read this http://www.umass.edu/wsp/resources/poisson/.
Also read about the Gaussian or normal distribution.
In the limit of large NT, 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 σT = sqrt( NT)
where sqrt is short for square root. 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 = NT/sqrt( NT) = 1/sqrt( NT)
For dimmer sources, we need to worry about noise from the sky background, the dark current, and the read noise. 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. Looking 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 remain and add 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. Noise distributions are typically 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" (NASA people like to call this RSS) 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 makes no sense.) The
signal to noise equation for CCDs or the "CCD equation" is then:
S/N = NT/sqrt[NT + npix × (NS + ND + σR2)]
In the previous lab, you measured the read noise of the StarShoot
camera and how the dark current depends on temperature and
exposure time. In this lab, you will use the images that you
obtained of M36 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 look at how
the S/N depends on various parameters.
Before calculating the S/N, we need to know the gain and read
noise of the camera. Look through the documentation for the CCD
camera on the VAO to find the gain of the CCD. You can also find
the read noise this way, but since it is easy to measure the read
noise if you have two or more bias images (you should have 5),
pull up your python code from the last lab and measure the read
noise of the CCD. Does it agree with the value in the
documentation?
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. Note that you need to convert
these to electrons. 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?