The CCD SignaltoNoise Equation
For CCD observations there are several sources of noise. Suppose the number of photoelectrons
detected from the object, sky background and dark current are \(S_o, S_s\) and \(S_d\),
respectively. Then the various noise sources, and their amplitudes (standard deviations) are:
 shotnoise in the detected photoelectrons from the source, \(\sqrt{S_o}\);
 shotnoise in the detected photoelectrons from the sky background, \(\sqrt{S_s}\);
 shotnoise in the thermally excited electrons, i.e the dark current, \(\sqrt{S_d}\);
 timeindependent readout noise, \(R\). Note there is no square root here. The readout noise
is the standard deviation in the number of electrons measure  it is not a Poissonian
counting process.
It is extremely important to realise that these noise sources are present even after the mean sky
background or dark current has been subtracted. The measured value of any CCD pixel differs from
it's true value by some
unknown amount due to noise. The values above tell us
the typical size of this unknown noise, but not it's exact value. Subtracting the sky during
photometry, or
subtracting dark frames
does not remove the shot noise from the CCD pixels!
We can assume all the noise sources are independent. When we add Gaussian random variables, the
variance \(\sigma_T^2\) of the result is equal to the sum of the
variances of each Gaussian, \(\sigma^2_T = \Sigma_i^n \sigma^2_i\) . The total noise,
\(N\), is equal to the standard deviation \(\sigma_T\), and is therefore given by:
\[N = \sqrt{S_o + S_s + S_d + R^2}.\]
Hence, the SNR is given by
\[{\rm SNR} = S_o/N = \frac{S_o}{\sqrt{S_o + S_s + S_d + R^2}}.\]
CCD equation in units of counts (ADU)
Typically, when predicting the SNR of a CCD observation we have the following quantities to hand:
 \(S_o\), in units of photons per second;
 \(S_s\), in units of photons per second per pixel;
 \(S_d\), in units of electrons per second per pixel;
 \(R\), in units of electrons per pixel.
A few things are worthy of comment in the above list. First, \(S_o\) is the total number of
photons from the object, which will probably be spread out over a number of pixels, whereas
\(S_s\) is the number of sky photons per pixel. Second, \(S_o\) and \(S_s\) are in photon units
not electrons. Third, \(S_o\), \(S_s\) and \(S_d\) will increase with exposure time, but \(R\)
will not.
We need to use electrons in the SNR equation above, not the number of photons emitted by the
source. Poissonian statistics apply to whatever we are counting, and in this case we are
counting electrons! The total number of electrons detected from the source is given by
\[S_o Qt,\]
where \(t\) is the exposure time in seconds, and \(Q\) is the Quantum efficiency of the CCD,
expressed as a number between 0 and 1. Similarly, the total number of electrons detected from
the sky is
\[S_s Qt n_p,\]
where \(n_p\) is the number of pixels that the object is spread over. Using similar arguments
for the other terms, the CCD SNR equation is written as
\[{\rm SNR} = \frac{S_o Qt}{ \sqrt{ S_o Qt + S_s Qt n_p + S_d t n_p + R^2 n_p } }.\]
Sometimes \(S_o\) and \(S_s\) will be in different units  astronomers do like to be awkward,
after all. We always need to convert them to units of electrons, as electrons are the thing
that's actually being counted, and thus obeys Poissonian statistics. With \(S_o\) and \(S_s\) in
photons, we converted to electrons using the QE. If \(S_o\) and \(S_s\) are given in counts, we
convert from counts to electrons by using the CCD gain \(G\) in place of the QE \(Q\). The SNR
CCD equation becomes
\[{\rm SNR} = \frac{S_o Gt}{ \sqrt{ S_o Gt + S_s Gt n_p + S_d t n_p + R^2 n_p } }.\]
Finally, one might also find \(S_o\) and \(S_s\) given in flux units. In that case we divide the
flux by the average energy of a single photon in this bandpass to find the number of photons
arriving per second, and then use the QE \(Q\) to convert to electrons per second.
Limiting Cases
There are three limiting cases for the CCD SNR equation. Using the simplified form \({\rm SNR} =
S_o / \sqrt{S_o + S_d + S_s + R^2}\), we find that the these limiting cases are:

Object Limited  \(S_o\) is much greater than \(S_s\), \(S_d\) and \(R^2\).
In this case \[{\rm SNR} = S_o / \sqrt{S_o} = \sqrt{S_o}.\] Since \(S_o \propto t\), this
means that \({\rm SNR} \propto \sqrt t\). In addition, \(S_o \propto D^2\), where \(D\) is
the telescope aperture. Therefore \({\rm SNR} \propto D\).

Background Limited  \(S_s\) is much greater than \(S_o\), \(S_d\) and
\(R^2\). In this case \[{\rm SNR} = S_o / \sqrt{S_s}.\] In this case, since both \(S_s\) and
\(S_o\) scale the same way with exposure time and telescope aperture, the SNR scales with
respect to these values in the same way as the object limited case. For a fixed \(S_o\), the
SNR scales with the square root of the sky signal \(S_s\). It is therefore very important to
observe faint sources when the sky is faint. Scattered light from the from the Moon is a
major source of sky background, so faint sources should be observed near new Moon.

Read Noise Limited  \(R^2\) is much greater than \(S_s\), \(S_o\) and
\(S_d\). Since all these quantities scale with exposure time, but \(R\) does not, short
exposures are often read noise limited. In this case \[{\rm SNR} = S_o / R.\] Since the
readout noise is independent of integration time or telescope aperture, the SNR will now
increase linearly with exposure time and as the square of the telescope aperture diameter.