[Placidus (Mike and Trish)]

Introduction

We continue to see claims that with the arrival of the CMOS chip, the days of the 1 hour sub are over, and that we can start routinely doing subs of a few seconds only.

There can be compelling reason for really short subs, such as poor polar alignment, dodgy guiding, shallow quantum wells, light pollution, wind buffet, airplanes, etc. Let us assume that we have an arbitrarily dark site, an observatory dome not under a flight path, and permanent polar alignment.

We will derive an easy formula for signal to noise ratio as a function of quantum efficiency, source brightness, readout noise, sub duration, and total number of subs.

We will see that increasing the quantum efficiency alone cannot produce a truly huge improvement over say a 16803 chip.

We will see that for a sufficiently faint target, ten 1-hour subs will yield a 600% better signal to noise ratio than the same exposure time in 1-second subs.

We will also see that it is not a good idea to be testing really short subs on ultra-bright easy targets (say M42) if we are later on hoping to photograph truly faint things.

The results will bear out and put in context comments made by Peter Ward, Rick Stevenson, myself and several others in other forums.

Some easy mathematics

Let the quantum efficiency be Q, the photon flux at a given pixel be P photons per second, the duration of a single sub be T seconds, the number of subs be N.

The signal, or total number of photo-electrons captured in a single sub will be QPT. The higher the quantum efficiency, the brighter the object, and the longer the subs, the more signal we will have.

There are two components to the noise that we are interested in here. The first is shot noise. If we have a very faint source of photo-electrons appearing at random intervals, then during any short interval, we may or may not get a photo-electron, so for some subs our count may be zero, in others it may be 1, in others 2, etc, but not always the same.

The actual number captured in a single sub will follow a Poisson distribution, with an expected value (population mean) for any single sub of QPT as mentioned already, a variance also of QPT, and a standard deviation of root(QPT).

The second source of noise that we are interested in here is read noise. Let the read noise in a single sub have a standard deviation of R electrons.

Over a set of N subs, the expected total signal will be NQPT. That makes sense: the more subs, the higher the quantum efficiency, the brighter the target, and the longer each sub, the more photons we will collect. Dim subjects will have poor signal.

At a single pixel, the two sources of noise under discussion (shot noise and read noise) are uncorrelated with each other and uncorrelated from frame to frame. The variances (the squares of the standard deviations) add linearly.

The total variance is therefore N x (shot noise variance + read noise variance) = N ( QPT + RR ), and the total noise is the square root of that.

(Because we can't type superscripts here, I've written the read noise variance R x R as RR.)

The signal to noise ratio is therefore NQPT / root [ N (QPT + RR)].

We will now examine some important and interesting limiting cases.

The first limiting case is for God's Own Camera, where Q = 1, and R = 0. This does not yet exist. The signal to noise ratio is snr = NPT / root [ NPT] = root(NPT). Since P is a constant set by the Almighty, the signal to noise ratio is proportional to root(NT). With the perfect camera, we can choose to do one ten-hour sub, or ten one-hour subs, and get the same result.

The second is an arbitrarily bright source. Think lunar photography. In the limit as P goes to infinity, QPT + RR approximates QPT as closely as we like. The signal to noise ratio becomes root(NQPT). Since Q is set by the manufacturer and P by the choice of target, the signal to noise ratio is proportional to root(NT). Thus again we can choose to do 1 ten second sub, or ten 1 second subs, and the snr will be the same.


The real world of very faint targets

Our third limiting case is the big one. We will imagine an incredibly faint source, where P (the number of photons per second arriving at a given pixel) is as small as we choose. Recall that the signal to noise ratio is NQPT / root [ N (QPT + RR)]. As we choose ever fainter sources, QPT becomes tiny compared with RR, and in the limit, the signal to noise ratio becomes

snr = NQPT / root [NRR] = QPT root(N) / R.

What does this mean?

(1) Traditional CCD chips have a quantum efficiency around 0.3 to 0.7. Even a perfect chip cannot have Q > 1, so if it takes tens of hours to photograph the outer chevrons of the helix using particular scope with a Q of 0.5, no camera in creation can do it in less than half that time. The advantage of a CMOS chip has little to do with efficiency or sensitivity.

(2) The big advantage of a CMOS chip is the greatly reduced read noise. If, all else being equal, we can reduce the read noise R by a factor of ten, then on a sufficiently faint object, we will improve the signal to noise ratio by a factor of ten. That is astonishing and is a good reason for hoping that Santa will produce a 4000x4000 chip with 100Ke quantum wells.

(3) All else being equal, then for the very faintest targets, the snr is proportional to exposure time T, but proportional only to the square root of the number of subs N. That means that for very faint targets, it will be hugely better to do ten 3600 second subs than 36,000 one-second subs. It will be better by a ratio of 3600 root(10) to 1 root(36000), or a ratio of six to one. Long subs rule.

The caveats are "very faint targets" and "all else being equal". Thus if your mount can't do a 1-hour sub, or there are intermittent clouds about, or aeroplanes, or wind buffet, or your quantum wells are too small, or you are overwhelmed by sky glow, or you only want to photograph the moon, shorter subs make sense. If you don't have those limitations, longer subs make sense.

(4) Test your new CMOS camera on very faint targets. Under super-bright conditions, the read noise is, as already discussed, almost irrelevant. It is not so informative to be doing your first test shots on easy targets like M42 or the Lagoon, unless that is all you are ever going to photograph. Have a crack at the outer chevrons of the Helix, or a crack at the faint OIII super-bubble in NGC 602.

(5) We will always want to do enough subs to be able to do good data rejection to get rid of satellite trails and cosmic rays, and to dither so that we can also handle hot pixels and bad columns using statistical data rejection. However, ten 1-hour subs is adequate for good data rejection. 36,000 1 second subs will not add any important protection and is kind of unmanageable.

Someone else might like to write an equation for signal to noise ratio that includes sky glow.