Astronomy 341 Fall 2012 | Observational Astronomy Haverford College
CCD Terminology
Read noise
An unavoidable pixel-to-pixel fluctuation in the number of electrons per pixel that
occurs during chip readout. Typical values for read noise are ~ 10 or fewer electrons
per pixel, and it is independent of exposure time. If there are many counts per
pixel, then this noise will have a small impact on the total noise. But for faint
sources, or when adding together a number of exposures, read noise can have a
notable impact.
Dark Current
Over time, electrons are freed from the CCD material itself and collect inside
individual pixels. These electrons are thermally generated; the cooler the chip, the
lower the dark current. The signal and noise introduced by the dark current are
dependent on exposure time. This “dark current” pools together with the electrons
released by photons hitting the chip. The dark current of a chip (at a given
temperature) can be measured by taking dark frames - exposures with the shutter
closed. Dark current is typically negligible in modern instruments which at run at
very cold temperatures.
Gain
The factor which relates the amount of charge collected in a pixel to the number
output data units. Gain is quantified by electrons per Analog-to-Digital unit (e-/
ADU). If the gain of a CCD is 1, then the number of counts that is assigned to a
pixel (and that you read when you hover over that pixel with your mouse) is equal
to the number of photons absorbed by that pixel. Values of gain are commonly ~ a
few.
Quantum Efficiency (QE)
The efficiency with which photons hitting a pixel actually release an electron in that
pixel. This is a function of wavelength. The quantum efficiency of all pixels on a
CCD are not identical. This is a reason why it is necessary to obtain flat fields.
Bias
CCDs are designed so that they never read out a negative value of signal. To do
this, they offset the actual value of signal by some amount - the “bias”. The bias of a
CCD can be measured by taking bias frames - zero length exposures - the shutter
remains closed.
Flat field
The gain and QE are slightly different from pixel to pixel in the CCD. A flat field is
a way to normalize science exposures to divide out the effect of these variations. To
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obtain a perfect measurement of the shape of a chip’s response, a uniform
illumination of the chip is needed. This is difficult to do; astronomers typically take
many exposures of the sky during twilight or sunrise and median the exposures
together to measure a flat. When processing flats, you need to first subtract off the
bias and then normalize the flat field to have a median value of 1. Normalizing the
flats is needed so that when you divide your science image by the flat, the number of
counts is preserved.
Reduced image
Science images need to be corrected for systematic variations in gain and QE across
the field-of-view and corrected for the bias. This is called “reducing the data”. A
typical method of reducing data simply includes: Object = (Raw object - Bias)/(Flat
field). This output object frame is the “reduced image”.
CCD Statistics
To understand CCD statistics, is to understand is the idea of Poisson statistics.
The arrival of photons to a CCD pixel is a Poisson process. The conditions that
must be satisfied for something to be considered a Poisson process are well
articulated by WolframMathworld to be:
1. The numbers of changes in nonoverlapping intervals are independent for all
intervals.
2. The probability of exactly one change in a sufficiently small interval h ≡ /n is
P =ν/n , where ν is the probability of one change and n is the number of trials.
3. The probability of two or more changes in a sufficiently small interval h is
essentially 0.
The numbers of events that occur from a Poisson process follow a Poisson
distribution, with the probability of measuring x events given an expected average
of µ :
P(x; µ) =
e−µ µx
x!
The variance of a Poisson distribution with an average number of events, µ , is also
equal to µ .
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Because they are zero-length exposures, the pixel-to-pixel fluctuations in bias
frames result from readout noise. The bias frame is in units of ADU, and read
noise is in units of electrons, so
σbias = RN/g
Flat frames contain a substantial number of counts (i.e. have a high signal-to-noise
but a number of counts below the non-linear regime of the chip) because they need
to accurately characterize the shape of the flat field. The pixel-to-pixel fluctuations
of the counts are thus dominated by the Poisson noise of the photons hitting each
pixel. The number of electrons accumulated in the pixels thus varies like N0.5. If
gain is large, then a greater amount of charge (more photons) are required to yield a
set number of ADU. There will then be a larger pixel-to-pixel fluctuation in signal,
because the random fluctuations in signal are ~ N0.5. Read noise does not
substantially add to this. If the average value of the flat field in ADU (after
subtracting the bias) is F̄ , then
σf lat =
√
F̄ g
g
The quality of astronomical observations is a function of the signal-to-noise of the
image. The signal-to-noise (S/N) is a function of the number of photons gathered for
a specific measurement:
S
Nγ
=!
N
Nγ + Npix (Nsky + Ndark + RN 2 )
In this expression, the capital N’s are all total numbers of photons (electrons). If
using this expression to compute the signal to noise of your observation, be sure to
convert ADU to electrons using the gain. Nγ is the number of photons (electrons) in
a single pixel or from a set of pixels. For stellar photometry within an aperture, it
would be the total number of photons (electrons) within the aperture. Npix is then
number of pixels within the aperture. The other values of N are all per pixel. Nsky
is the expected number of photons per pixel from sky background, Ndark is the
expected number of photons per pixel from dark current and RN is the read noise
per pixel. Because RN is the actual shot noise, instead of a Poisson variate, it is
added in as RN2
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!
S/N
∼
Nγ . For faint sources the other sources of noise in the
For bright sources,
denominator become as or more important. The sky brightness is typically the most
important source of noise, after the Poisson fluctuations of photons arriving from
the source itself.
Lets get the expression above for S/N in terms of slightly more observable
quantities. Typically, we know the flux (energy per area per second) of a star at the
location of Earth from its apparent magnitude. The total number of photons from
that source will be linearly proportional to time, as will the total number of sky
photons. Using n to represent photon arrival rate (for the object it is over all
pixels), and dropping the dark current (because its usually negligible):
S
nγ t
=!
N
nγ t + Npix (nsky t + RN 2 )
√
We can now easily see that S/N ∝ texp in regimes where read noise isn’t the
dominant source of noise (most regimes). We can also solve for the integration time
needed to obtain a measurement of some signal-to-noise, given the rates of sky
photon arrival per pixel, object photon arrival per aperture, the number of pixels in
the aperture, and the read noise:
S 2
) = n2γ t2
N
S 2
S
2 2
nγ t − (nγ + Npix nsky )( ) t − Npix RN 2 ( )2 = 0
N
N
(nγ t + Npix (nsky t + RN 2 )) × (
This can be solved with the quadratic formula:
t=
−B +
√
B 2 − 4AC
2A
Object and sky brightness
We can use the apparent magnitude of the sky and of our objects to estimate the
number of sky and object photons that should reach a CCD pixel given the collecting
area of the telescope, the pixel area of the CCD, the photometric filter, and the
throughput of the system.
The website:
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http://www.astro.utoronto.ca/~patton/astro/mags.html
summarizes many quantities needed for S/N or exposure time calculations. This
includes a worked out example of how to calculate the number of photons per second
incident on a 1 m2 aperture per second (at the top of the atmosphere) for a star with
V = 23.9:
This website also includes the approximate night sky brightnesses in magnitudes
per square arcsecond in several filters, as a function of the phase of the Moon. This
doesn’t include light from man-made sources.