A telescope focuses a star as a round point of light. Assuming high quality optics, the diameter of the point of light is determined by the telescope’s focal length (longer focal lengths result in larger star diameters) and the sky's ‘seeing’ conditions (atmospheric dispersion spreads the point of light, making it larger). Short focal length telescopes and ideal seeing conditions provide the smallest stars, longer focal lengths and less favourable skies produce larger stars.

The challenge

For a star to retain it's round shape when viewed on your screen or photograph it’s diameter must cover a sufficient number of pixels. Too few and the image will be 'under-sampled’, the stars will appear blocky and angular'. For a smoother more natural look more pixels are required, but not too many because if you use more pixels than are necessary to achieve round stars the image is 'over-sampled’. Over-sampled images look rather nice because the stars are round with smooth edges but if you have more pixels than are necessary why not use a reducer to reduce the telescope’s effective focal length, which makes the image brighter and enables you to fit more sky on your sensor. In affect, over-sampling reduces field of view.

The theory

In the 1920s Harold Nyquist developed a theorem for digital sampling of analog signals. Nyquist’s formula suggests the sampling rate should be double the frequency of the analog signal. So, if OK seeing is between 2-4” FWHM then the sampling rate, according to Nyquist, should be 1-2”.

There is some debate around using this for modern CCD sensors because they use square pixels, and we want to image round stars. Using typical seeing at 4” FWHM, Nyquist’s formula would suggest each pixel has 2” resolution which would mean a star could fall on just one pixel, or it might illuminate a 2x2 array, so be captured as a square.

The solution

It is better then to image with a resolution 1/3 of the analog signal, doing this will ensure a star will always fall on multiple pixels so remain circular.

Our calculator, at typical seeing of 2-4”, uses the Nyquist formula of 1/2 and the 1/3 to stop stars becoming square so the optimal range is between 0.67” and 2”. (0.67 = 2 / 3, 2 = 4 / 2).

In summary, we are using Nyquist as a starting point, with a slight tweak, because we are typically sampling very small, circular, stars.

Making it easy

When using our calculator you you don’t need to understand the theory or the maths. Simply enter the telescope's focal length, the camera's pixel size and your sky's seeing conditions to determine if they are a good match :-)

A few notes:

  1. We are assuming OK seeing is between 2-4” FWHM and a resolution between 0.67” and 2” per pixel is the sweet spot. Over 2” is under-sampling and under 0.67" is over-sampling. The scale changes according to the ‘seeing’ conditions entered.
  2. The calculator uses maths to determine its result, it is not influenced by brand or sales spiel. If the result suggests your existing telescope / camera combination is less than ideal please don’t shoot the messenger. Pleasing images can be made using less than optimum equipment.
  3. Depending on the result, a message will suggest how the situation might be improved, i.e. pixel binning or using a barlow.
  4. Under-sampling is worse than over-sampling. When over-sampling you do lose field-of-view and, probably, some sensitivity but the star shapes are pleasing.
  5. Unbinnable small pixel One Shot Colour cameras (mostly DSLRs) are not a good choice for use with with long focal length telescopes, like SCTs.
  6. Pixel-scale (the subject of this calculator) is arguably the most important factor to consider when choosing a camera for your telescope, but there are others: sensor size, sensitivity, dark current, read noise, anti blooming, etc. They all play a part. Please don’t base your purchase decision on pixel-scale alone. Most astronomy retailers understand the significance of pixel-scale so will consider it, together with other factors, before recommending a camera / telescope combination.
  7. If your telescope or camera is not listed please add it.

At Astronomy Tools we want to make useful information available to all. If you can see a way we can improve any of our calculators, or would like us to build a new one, please contact us.

CCD Suitability Calculator

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