Matching image sensor size to telescope resolution

Good images of planets push the bounds of most telescopes' resolution. Planetary imagers want detailed images of bright objects. Deep sky imagers are more concerned with capturing photons from dim objects; other issues come into play that favor larger pixels. Under sampling planetary images will loose details that cannot be recovered in post processing. Understanding how your telescope's resolution relates to your image sensor pixel size will guide you in camera and auxiliary lens selection. Two numbers are important to know:

  • the pixel size of your sensor - more accurately the pixel pitch
  • the size of your telescope's diffraction resolution limit at prime focus

These are not hard to calculate. First you need to know the diffraction limited resolution of your telescope. There are two standard ways to quantify this: the Rayleigh Criterion and Dawes Limit. Let's use Dawes Limit:

R = 116 / D
where Resolution is in arc sec & Diameter of the objective is in mm

For the Questar's 89 mm objective this is:
116/89 = 1.3 arc sec

We will be going digital and sampling the image with a photo sensor. We must use the Nyquist–Shannon sampling theorem. Basically this says: you can't separate two lines without a space between them. In our case it means that we need to sample the image a half the diffraction limited optical resolution. For the Questar this is
0.65 arc sec (= 1.3 / 2).

Next we need to know how large our resolution is at prime focus. This is simply the focal length times the angle in radians. For the Questar with a camera the focal length is about 1350mm.

h = F * a
where image hight and Focal length are the same units, the object angle is in radians
(1 radian = 60 * 60 * 57.3 arc sec).

for the Questar
h = 1350mm * ( 0.65 arc sec / ( 60 * 60 * 57.3) = 0.00425 mm or 4.25 microns

For the Questar 89mm telescope, the sensor element pitch (~ pixel size) must be less than 4.25 microns or we are throwing away resolution in our sensor.

We can arrive at a simple formula, by substituting the formula for Dawes Limit into our expressing for h, the sensor pitch at prime focus. Here is an approximate simple formula that will give you your maximum diffraction limited resolution in microns from your objective size and focal length in mm:

h = F * a

now substituting Dawes Limit/2 as the angle in Radians

h = F * (116 /2 D) * 1 / (60 * 60 * 57.3)

h = F / (D * 3566) in mm

h = F / (D * 3.6) in microns

however F/D is just the focal ratio of the telescope f. So using the small angle approximation, Dawes Limit, and the sampling theorem we can approximate the sensor pitch required to capture the diffraction limited resolution of any telescope (telephoto lens) as a very simple function of the focal ratio:

h = f / 3.6 in microns

Calculating your sensor pixel pitch is even easier. Just divide the sensor physical dimension by the number of pixels in the same direction. The Sony a6300 sensor for example is 23.5mm and 6000 pixels wide giving a pixel pitch of 0.00425 mm or 4.25 microns! How does the prime focus diffraction limited resolution of your telescope compare to the pixel pitch of your camera?

If you are under-sampling your telescope's resolution you can add auxiliary lenses (e.g. focal expanders or Barlow lenses) and trade field of view for resolution. If your processing includes deconvolution, you may want to over-sample your telescope's resolution.

Content created: 2015-05-18 and last modified: 2017-09-25




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