The Nyquist Theorem was developed by Harry Nyquist and gives a criteria for the minimum sampling of an analog signal waveform to preserve all of the information in the original. This theorem, developed for acoustic audio waves, says that the sampling rate must be equal to, or greater than, twice the highest frequency of the analog signal. If this criteria is met, the original waveform can be reconstructed without error from samples taken at equal time intervals.

This theorem has been applied to the spatial domain where analog stars in the form of Airy Disks are sampled by pixels to convert the information into digital form.

For astrophotography, this would mean that the pixel size should theoretically be 1/2 the size of the Airy Disk to sample it so that no information is lost. Practically, it means that the pixel should be 1/2 the size of the seeing disk since even a perfect telescope is limited by the seeing. A scope that could produce a 1 arc second in diameter Airy Disk might grow to 5 arcseconds on a night of poor seeing.

If a star is sampled with a pixel that is larger than 1/2 the Airy Disk, it is said to be undersampled. If it is sampled with a pixel that is smaller than 1/2 the Airy Disk, it is said to be oversampled.

However, the Nyquist Theorem is insufficient in many ways for astrophotography, both for stars recorded in deep-sky work, and particularly for high-resolution planetary work.

The Airy Disk and the Nyquist Sample Size

Examples of an Airy Disk that is progressively sampled by larger pixels.

Consider a scope that can form a 12.8 micron Airy Disk. According to the Nyquist theorem, it can theoretically be sampled sufficiently by pixels that are 6.4 microns square. The problem is that after this round star is sampled at this size, the result is 4 pixels that are square. Hardly our idea of a normal round star with a gradual brightness falloff from the bright center of the star.

For astrophotography, to preserve the stellar profile that we are used to, stars must be oversampled. According to Brad Wallis, the Nyquist theorem's criteria for sampling size needs to be a minimum of 1/3x to 1/4x the size of the Airy Disk.

The Nyquist Theorem is also insufficient for high-resolution planetary work because it can only sufficiently sample detail that is in the same direction as the columns and rows of pixels. For planetary detail, such as fine lines on Mars, or Enke's Gap in Saturn's ring system, the finest detail may be at an angle to the sensor's grid of pixels. This requires a higher sampling standard.

For planetary photography, the focal length of the imaging telescope can be increased by adding Barlow lenses or using eyepiece projection to increase the image size and the Airy Disk so that fine details can be sampled at 3x to 4x the normal Nyquist sampling criteria.

For deep-sky astrophotography, we are generally stuck with the size of the pixels we have in the camera, and while we can change focal length, we usually do so to change the field of view that we intend to photograph, not to satisfy the Nyquist theorem. In the real world, we are stuck with the pixel size we have, and we just go ahead and use it, and it turns out that it's not such a big deal after all. Pictures still come out looking great.

Consider our 5 inch f/8 scope example with a Canon 20Da DSLR camera. That scope can form an Airy Disk that is 2.18 arcseconds, or 10.7 microns. The pixel size in the 20Da is 6.4 microns. This undersamples the stars. And, in fact, in pictures, when they are enlarged so much that you can see individual pixels, the smallest stars in the image are square. But it usually doesn't matter because a deep-sky image is usually never enlarged to a size where this can be seen.

The situation gets even worse with the 20Da with faster optical systems, such as a 400mm f/2.8 telephoto lens. Theoretically such a system can form a star that is only 3.7 microns in diameter, but the pixels are much bigger at 6.4 microns. Again, it doesn't matter that much. Yes, the smallest stars are just a single pixel. However you are hardly ever going to see this unless the image is enlarged to gigantic proportions.

For long-exposure astrophotos at long focal lengths, accumulated errors in focus, seeing, and guiding will all contribute to make the Airy Disk grow in size. For short focal length images with wide angle lenses, the small aperture will be the limiting factor in how small the Airy Disk can be, with lenses under 50mm forming an Airy Disk larger than 2x the pixel size in most DSLR cameras.

The only time the Nyquist theorem is really critical is for high-resolution planetary or double star work. Then a criteria making the pixel size at least 1/3 to 1/4 the size of the Airy Disk should be used.

Assuming a night of perfect seeing, the limiting factor will be the resolution of the telescope. A 5 inch f/8 scope can form an Airy disk that has a FWHM diameter of 4.5 microns, so the pixel size should be around 1 micron. There are no digital cameras that have pixels this small. The only solution to correctly sample the image with this telescope is to increase the focal ratio and magnify the image. Making the image larger makes the Airy Disk larger. The 6.4 micron pixels in a Canon 20D would need an Airy Disk that is about 24 microns in diameter, so the image needs to be magnified by a factor of about 5x. This can be accomplished with either a 5x Barlow, or with eyepiece projection. A 5x Barlow in an f/8 optical system will increase the focal ratio to f/40.

The problem with making the image larger by magnifying it and increasing the focal ratio is that you also make the image dimmer and it will require a longer exposure. For high resolution planetary work longer exposures mean more degradation of the image due to seeing. So a compromise must be reached between the image magnification needed to achieve correct sampling and the length of the exposure in relation to the quality of the seeing.