Exposure Duration

From TzecMaun

Exposure Duration

One of the most common questions we get from students is: "How long should my exposures be?"

It turns out to be an interesting and complex question.

The controlling factor in any exposure is a concept: Signal to Noise Ratio. You would probably assume that signal is the incoming light - but that is not quite right! Noise is the uncertainty in the data. It comes from two sources:

• The incoming light itself. Light is a quantum phenomenon, and as such it contains its own uncertainty (noise). The noise in the incoming light is equal to the square root of the signal. So if we measure an incoming signal of 100 photons, then the uncertainty is 10 photons. (We might have actually received from 95 to 105 photons).

• Readout from the camera. Incoming photons strike a pixel and most of them are converted into electrons. The electrons are then counted. The read noise is the uncertainty in the readout. If 100 photons strike a pixel, and 90% get converted to electrons abbreviated e-), and the read noise is 4 e-, then we count 90 photons, but the actual total may have been anywhere between 83 and 97, inclusive.

The thing to take note of here is that the two types of noise are very different:

• The shot noise varies with the signal. If the signal increases, the shot noise increases as well. Fortunately, the signal increases faster than the uncertainty. For a signal of 100, the signal to noise ratio is 100:10, or 10:1. For a signal of 200, the S/N ratio is 200:14, or 14:1. (Note that doubling (2x) the exposure improves the S/N by the square root of 2. An exposure 4x as long will have S/N 2x better. The improvement in S/N is the square root of the increase.)

• The read noise is constant. It applies once for each exposure. If you take a very short exposure, with a small signal, then the read noise will be large relative to the signal. For a signal of 100, and a read noise of 13, the S/N ratio is 100:13 or 7.6:1. For a signal of 200, the read noise remains 13, so the S/N ratio would be 15.2:1 (double).

Note that the total noise actually must include shot noise and read noise in the calculation. So far, we have been showing calculations that only include one type of noise or the other.

Since shot noise increases, but read noise stays the same, as we take longer exposures, shot noise gets bigger and bigger and read noise stays the same. At some point, the shot noise becomes overwhelmingly large compared to read noise. In effect, the read noise gets lost in the shot noise and is no longer a significant factor in the signal to noise ratio.

There is one more complication. Shot noise actually comes from two sources: the incoming signal from the object we are imaging, and from sky brightness. For locations with a bright sky, this must be taken into account. But both the New Mexico and Australian locations that we use have very dark skies, so this complication can be ignored.

Finally, we can address the question of how long an exposure to take: the ideal single exposure duration is long enough for the shot noise to overwhelm the read noise. Since read noise is now too small to matter, we can combine these ideal exposures using software like MaxIm DL or CDDSoft to "stack" images and improve our S/N even further. Adding ten of these ideal exposures is very nearly like taking one super-long exposure - both the stacked exposures and the super-long exposure will have similar signal to noise ratios.

You can use the following web resource to calculate the ideal sub-exposure duration for any camera/telescope combination:

http://www.ccdware.com/resources/

Ron Wodaski Observatory Director