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We now have a firm understanding of the two main types of noise we have to deal with in an astrophoto, shot noise and read noise. Before we begin doing math we have to consider the units for the values we are going to discuss. How much signal we capture can be represented by multiplying the average rate of the signal with the amount of time we are exposing. The average rate should be, in our case, a measurement of flux. Flux is how much of something passes through a certain area in a certain time. We have to use area, because we want to consider the amount of signal reaching each pixel, and each one takes up a certain area. Since pixels have a size measured in micrometers (μm), we could use units of photons per minute per square micrometer. However, we should remember that the photodiodes do not record every single photon that strike them, but only a certain percentage based on wavelength. When they do record one, an electron of charge is stored. So we can use electrons instead of photons. And, instead of using square micrometers, we should use a unit of area we'll just call the pixel. Since different cameras have different sized pixels, how big this unit really is depends on the camera being used, but it makes things easier to work with. So, the "brightness" of a particular part of an object is measured in electrons per minute per pixel. We can also use this to measure the rate of dark current. This also means we'll be using the minute as our time unit, though using seconds or some other unit is not incorrect. Ultimately the flux value will depend on the telescope (mostly focal ratio) and the camera (sensitivity, pixel size, etc) so no target can be said to have a specific flux universal to all equipment setups.

We know that the two primary sources of noise in our images will be shot noise from signal, and read noise from the camera. Let us define some variables:
r = read noise of the camera
t = signal flux of our target (in electrons per pixel per minute)
d = dark current flux of camera
s = skyglow flux
L = subexposure length
X = number of frames in a stack

We know that two different noise sources, when combined, sum in quadrature. So, if S is the total amount of signal acumulated in an image, the magnitude of the combined shot and read noise from a single frame would be:

When we capture an image, we have three different signal inputs. There is of course the target, then there is the skyglow, and finally there is the dark current. So the SNR of a single subexposure would look like:

The first thing we do with our subs typically is subtract a dark frame to remove the dark current contribution to the overall signal. To do that, we will capture dark frames. Typically, multiple dark frames are captured and stacked to reduce the shot noise component from each. The SNR of a signal dark frame looks like:

We will then combine multiple dark frames, and then divide by the number of frames X_dark (essentially averaging). This leaves us with a final SNR of the dark frame as:

When we subtract the dark frame from the image, we will remove the dark current component from the signal, but we must add (since noise is never negative) the noise component in quadrature to the noise segment. That gives us:

We will create a value called the dark noise factor in order to simplify the equation and substitute, giving us:

Since the number of stacked dark frames will always be a positive number, the dark noise factor will always be greater than one. More dark frames makes this value smaller. Because of this, subtracting a dark actually increases the total noise in the image. But, since the dark noise factor should be fairly close to 1, and since the dark current noise and read noise should be relatively insignificant noise contributors, this increase should be very small.

Typically at this point we would divide the image by a flat frame. However, we are not going to consider this step here. Since the flat frame will have it's own noise, this division will have some effect on the overall SNR of the image, but the division of two normally distributed random variables is an extremely complex mathematical problem and trying to incorporate that would make the math pretty much unintelligible. The effect on SNR from this division, assuming a good flat, should not be significant in any way so it is safe to ignore it for now.

The next step after flats would be stacking the subexposures. We'll assume just a straight summation for now to make the math simpler. We will also subtract a constant value that equals the sky glow (we assume it is relatively uniform across the frame) When we do this, we get an SNR of:

Let us consider the total integration time T = X * L and substitute that in, giving us

So we see that our overall SNR is proportional to the square root of the total integration time. We also see that the longer subexposures will reduce the effect of read noise on the final image, regardless of the total integration time. We also see how skyglow is an important contributor to noise. Not only do darker skies make it easier to image faint objects, darker skies allow you to image more detail with less total signal.

Now that we have this equation, we can use it to help us understand how to get the data we want in the most efficient way possible. We will explore this in the upcoming sections.