Matching a CCD Camera to a Telescope

Every kind of astronomical optical system serves the primary purpose of collecting light which reproduce an image in the focal plane. When one of these light sources is a point source, such as a star, its image at the point of focus consists of a bright central region, surrounded by a series of ever fainter diffraction rings. The central region is called the Airy Disk, named after its discoverer, George Airy. In an unobstructed telescope with perfect optics, the Airy disk contains 84% of the light from the star, the remaining 16% is spread over the diffraction rings. Since the majority of the light is contained in the Airy disk, this is the image of the star that we want to accurately record with our CCD camera. Of this disk, half of the light is confined to the small central core, inside a region called the Full Width Half Maximum or FWHM. A star image represents the smallest unit of meaningful detail we can record. The FWHM is the part of the stellar image that contains the most light.

FWHM = 1.02 * (wavelength) * (Focal Ratio)

From the equation you can see that FWHM only depends on two things, the wavelength of incoming light, and the focal ratio of the system. Let's look at an example. Suppose we are imaging a star using a hydrogen-alpha filter using an 200mm f/8 Newton telescope.

The FWHM would be 1.02 * 656nm * 8 = ~ 5.4μm (656nm is the wavelength of the dominant line of H-alpha).

This, of course, is assuming perfect seeing.

With the CCD camera we record a image as in a 2-dimensional array of pixels. Each pixel contains a measurement of all the light that falls on it. In order to view this image we will convert the image into a series of pixels that can be displayed on a computer screen or can be printed. The idea is that we want to reproduce the image as it was recorded at the focal plane. The problem is to determine how many pixels we need to accurately represent the original image. In 1933, Nyquist determined that in order to extract all of the information contained in an audio sine wave it must be sampled at twice the highest frequency that it contains.

For a 2 dimensional Gaussian PSF, like our astro images, the Nyquist criterion is FWHM = 3.3 pixels (See here for more info).

Deepsky imaging

From this you can see that you need your CCD camera's pixels to be one third the FWHM produced by your telescope but that doesn't take the seeing into account. Seeing, as we all know, is due to atmospheric turbulence, and causes the Airy disks of our star images to be much larger than our optics are capable of producing if they were true point sources.
Seeing is usually specified in arc seconds, and it refers to the size that the FWHM of the point image of a star is smeared into by the turbulent atmosphere and therefore we end up with a larger FWHM at the image plane.
Taking the seeing into account the star size on CCD chip will become :

Star Size = (Seeing * Focal Length)/206.3

where the star size is in μm, the seeing is in arc seconds and the focal length is in mm.

Lets assume a seeing of 3" and a focal length of 1600mm, then the star size on the CCD chip will be :

Star Size = (3 * 1600)/206.3 = 23.3μm

Sampling this star image according the Nyquist criteria we would need pixels of 7μm (23.3 / 3.3).

If we cut the focal length to 1200mm (with a focal reducer) the star size at focus becomes 17.5μm requiring pixels of
~6μm to adequately sample it. So the pixel size decreases with an decreasing focal length.

When we use a CCD camera where the pixels are too big compared to the above is called undersampling, and at its worst it can lead to square, blocky stars, or even stars that are missing completely. Minor undersampling is not too bad, though and many great astrophotos have been taken with undersampled star images. (Seeing smears the stars around, making small pixels much less necessary).

When we use a CCD camera where the pixels are too small compared to te above is called oversampling, you really don't lose anything, but you are not capturing any additional information either. Most CCD cameras will allow you to bin the pixels, making big ones out of little ones, which can really help the signal-to-noise ratio. The biggest problem, using binning, is that you are not taking full advantage of all the image capturing ability of your CCD camera. Using a shorter focal length telescope in this case would give you a wider field of view.

For my imaging scope (12" F/5 Newton) and a general local seeing of about 3" this gives an optimum pixel size of 6.6μm.

As it is a F/5 scope it will generate an Airy disk of 6.7μm at Lambda=550nm (~0.9") under ideal circumstances. With a seeing of 3" the Airy disk will grow to ~20μm. For astrophotography, to presere the stellar profile that we are used to, stars must be oversampled. According the Nyquist theorem's criteria for sampling the pixel size needs to be a minimum of 1/3x to 1/4x the size of the Airy Disk. This would give a pixel size of 5.0μm to 6.7μm. This corresponds very well with calculated result for the optimum pixel size for the telescope.

Most popular CCD cameras fitting this scope have pixels of 5.4μm (Kodak KAF-8300 sensor), 6,45μm (Sony ICX285) or 6.8μm (Kodak KAF-3200ME) where de Kodak KAF-8300 sensor gives the biggest FOV. Another advantage of the KAF-8300 for my set-up is that when we have a better seeing (occasionally) than 3" the pixel size still meets the criteria.

In the DSLR arena there are interesting camera's from Canon for example one with 5.2μm pixels (450D) and one with 5.7μm pixels (1000D). Of course these camera's have to be modded (filter removement and cooling).

Currently I use a QHY10 OSC CCD camera with 6.05μm pixels, fitting perfectly in te calculated range of 5.0μm to 6.7μm.

Planetary imaging

Planetary imaging is normally done with a long focal length (focal ratio of 20 to 30) and at high frame rates (30 to 100 fps) in order to freeze the seeing. In this case the resolving power is more important than the smearing due to seeing.
See here for a more in depth discussion.
© Copyright Rob Kantelberg
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