To write an equation for SNR we need to know the various noise sources that contribute to an astronomical measurement with a CCD. These are:

• Random fluctuations in the detected photons from the source, i.e. the shot noise on the source signal, (Sobj)0.5.
• Random fluctuations in the detected photons from the sky background, i.e. the shot noise on the sky signal, (Ssky)0.5.
• Random fluctuations in the thermally-generated electrons produced in the CCD, ie. the shot noise on the dark current, (Sdark)0.5.
• Time-independent detector noise, i.e. readout noise, R. Note that no square root is required here as there is no signal associated with this noise source and hence Poissonian statistics do not apply.
Assuming that all of the above noise sources are independent, the total noise, N, is given by the square root of the sum of the squares of the individual errors:

N = (Sobj + Ssky + Sdark + R2)0.5.

Hence the SNR is given by the equation:

SNR = S / N = Sobj / (Sobj + Ssky + Sdark + R2)0.5.

It is important to realise that the above equation applies even after subtracting the mean sky background and dark current levels from each pixel, since the shot noise from these sources will still be present.

To use the above equation correctly, one must be careful with the units used for each of the terms. Typically, when predicting the SNR of an observation, we have:

• Sobj in units of photons per second.
• Ssky in units of photons per second per pixel.
• Sdark in units of electrons per second per pixel.
• R in units of electrons per pixel.
A few things are worthy of note in the above list. First, Sobj is the total number of photons from the object, which will probably be spread out over a number of pixels, whereas Ssky is the number of sky photons per pixel. Second, Sobj and Ssky are in photon units not electrons. Third, Sobj, Ssky and Sdark will increase with exposure time, but R will not.

So, to clarify the SNR equation given above, we need to account for the exposure time of the CCD image, t, the number of pixels that the object is spread over, npix, and the conversion efficiency of photons to electrons, which is given by the quantum efficiency, QE, of the CCD expressed as a number between 0 and 1. The latter conversion from photons to electrons is essential, as otherwise one would predict a higher signal-to-noise than measured, i.e. the signal in the SNR equation must be the detected signal, not the signal emitted by the source. The resulting equation is sometimes referred to as the CCD equation:

SNR = (Sobj . t . QE) / [ (Sobj . t . QE) + (Ssky . t . QE . npix) + (Sdark . t . npix) + (R2 . npix) ]0.5,

which can be simplified to:

SNR = [Sobj . (t . QE)0.5] / [ Sobj + npix (Ssky + (Sdark / QE ) + (R2 / QE . t)) ]0.5.

Sometimes, Sobj and Ssky are given in counts. In this case, we must also convert them into electron units. This is because Poisson statistics are only applicable when counting independent, random events, which means that the noise is only equal to the square root of the signal if the signal is in units of the detected quantity, i.e. electrons. The conversion from counts to electrons can be performed by replacing QE in the above equation by the CCD gain, g, in units of e-/ADU:

SNR = (Sobj . t . g) / [ (Sobj . t . g) + (Ssky . t . g . npix) + (Sdark . t . npix) + (R2 . npix) ]0.5,

which can be simplified to:

SNR = [Sobj . (t . g)0.5] / [ Sobj + npix (Ssky + (Sdark / g ) + (R2 / g . t)) ]0.5.

Similarly, Sobj and Ssky are sometimes given in flux units, and these must be converted into electrons before the CCD equation can be used. In this case, one must first divide the flux by the energy of a single photon to give the number of photons, and then multiply by the QE to give the number of electrons.

We can use the CCD equation to define three limiting cases: the object-limited case, the background-limited case and the readout-noise limited case.
• Object limited: In this case, the object signal per pixel is much larger than the sky signal, dark current or readout noise per pixel. Hence,

SNR = Sobj / (Sobj + Ssky + Sdark + R2)0.5 ~ (Sobj)0.5.

Hence the SNR increases as the square root of the object signal. Since the object signal is proportional to the exposure time, this means that the SNR is proportional to the square root of the exposure time. If the exposure time is doubled, the SNR will increase by 20.5 ~ 1.4. The object signal is also proportional to the area of the telescope aperture, which is proportional to the square of the diameter. Hence the SNR is proportional to the diameter of the telescope aperture - if the diameter is doubled, the SNR will double.

• Background limited: In this case, the sky signal per pixel is much larger than the object signal, dark current or readout noise per pixel. Hence,

SNR = Sobj / (Sobj + Ssky + Sdark + R2)0.5 ~ Sobj / (Ssky)0.5.

This is similar to the object-limited case. Both the object and sky signal increase linearly with exposure time, hence the SNR is proportional to the square root of the exposure time. Both the object and sky signal also increase linearly with telescope area, hence the SNR is proportional to the diameter of the telescope aperture.

For a given sky signal, the SNR will increase linearly with the object signal. For a given object signal, however, the SNR decreases as the square root of the increasing background level. This is why it is so important to minimize light pollution and observe faint objects when the Moon is new.

• Readout-noise limited: In this case, the readout noise per pixel is much larger than the object signal, sky signal or dark current per pixel. Hence,

SNR = Sobj / (Sobj + Ssky + Sdark + R2)0.5 ~ Sobj / 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.

To maximise SNR, one should always try to expose for long enough to obtain object- or background-limited data, as otherwise one pays a significant penalty for reading out the CCD. However, it isn't always possible to avoid the readout-noise limited regime, particularly when exposure times must be kept short in order to sample short time-scale variability.

Some calculations illustrating how to use the CCD equation are given in the example problems.