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.
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.
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.
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.
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:
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.
Calculate the resoution in arc seconds per pixel of a CCD with a particular telescope. Resolution Formula: ( Pixel Size / Telescope Focal Length ) X 206.265