Matching a CCD Camera to a Telescope
Every kind of astronomical optical system serves the primary purpose of collecting light over a larger area and concentrating it into a smaller area. In the process, the relative positions of light sources in the field are faithfully reproduced. Thus an image is formed. 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.
What the Airy disk and diffraction rings are is the Fourier transform of the circular telescope aperture convolved with the incoming point light source. Differently shaped apertures produce differently shaped Airy disks and rings. The familiar diffraction spikes seen on images taken with a Newtonian telescope are an example. In this case, the vanes of the secondary spider cause the diffraction spikes. Any object placed in the optical path will have its own effects. If you are curious some evening (when the seeing or tranparency is lousy, or the moon is up) try making an aperture mask for your telescope with different shapes cut in it and see how it affects the Airy disks and diffraction rings of your stars.
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. The reason why a star image is important here is that every object we image is made up of an infinite number of points, all overlapping at the focal plane. 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.
The first question that comes to mind is: how large is the FWHM?
The short form of that answer is:
FWHM = 1.02 * (wavelength) * (Focal Ratio)
(For those wanting a more thorough description of the math, I would refer you to chapter 1 in The Handbook of Astronomical Image Processing by Berry and Burnell).
The equation for the FWHM is kind of interesting in itself, as you can see it depends on only 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 f/10 telescope. Our FWHM would be:
1.02 * 656nm * 10 = 6691.2nm = ~ 6.7 microns
(656nm is the wavelength of the dominant line of H-alpha).
This, of course, is assuming perfect seeing, more about that later.
Now let’s take a picture of our star with a CCD camera. With the CCD camera, what we are attempting to do is to sample the focal plane using a device that records the image as in a 2-dimensional array of tiny points. Each point, or pixel, contains a measurement of all the light that falls on it. In order to later view this image we will convert it into a series of pixels that we will display on the computer screen or print on a printer. The idea here 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.
To our rescue comes Harry Nyquist. In 1933, Nyquist determined that in order to extract all of the information contained in a signal it must be sampled at twice the highest frequency that it contains. It is for this reason that your CD player samples music at 44kHz so that you can hear (if you’re young enough), music with frequency content up to 22kHz. This same 1-dimensional sampling theory applies directly in our 2-dimensional images as well. Since a star image represents the smallest piece of detail we have present, the Nyquist Theorem tells us that we have to sample it twice in each dimension (2 pixels across x 2 pixels up and down).
So what this all boils down to is that you need your CCD camera’s pixels to be half 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 is largely out of our control. On the average night, it tends to settle down as the land cools and comes into equilibrium with the air. But a lot of the seeing is pretty constant, and is due to geography and weather patterns. Here in the NY area, the wind generally comes from the west. As it passes across the Pocono Mountains in PA, the ridges deflect the air upwards and it gets turbulent. This results in a lot of high-altitude turbulence that pretty much limits our best local seeing 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. Since the stars are no longer point sources, we cannot expect out telescopes to treat them as such. Hence, we end up with a larger FWHM at the image plane.
So lets apply all this to our example. First we need to figure out how small our CCD camera’s pixels really need to be when seeing is taken into account. We can determine the linear size of the FWHM of the star image at the focal plane with this formula with the formula:
Star Size = (Star Angular Size * Focal Length)/206265
where the star angular size is in arc seconds And The focal length is in mm.
Using the f/10 telescope example used earlier, let’s assume the scope is a C8,
With a 2000mm focal length. If we assume our seeing is 2.5 arc seconds, then
Star Size = (2.5 * 2000)/206265 = 0.0242mm = 24.2 microns
Sampling this star image twice, we would need to have pixels half this size, or 12 microns.
If we cut the focal length using an f/6.3 focal reducer, the focal length become 1260mm and the star size at focus becomes 15.2 microns, requiring pixels of ~7.5 microns to adequately sample it. So as our focal length goes down, so does our pixel size. If we decide to use a camera lens, like a 500mm telephoto, our required pixel size drops even more, until it gets to the point where no CCD camera made has pixels small enough to precisely meet the Nyquist criterion.
So what happens when we use pixels that are too big?
This situation is called undersampling, and at its worst 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. (Plus, you can always count on the seeing to smear the stars around, making small pixels much less necessary ;-) ).
What about when your pixels are too small?
In this case, 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. But that is the subject of another article. The biggest problem here 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.
So how do we figure out what the pixel size is for a given CCD/telescope combination?
The formula used to determine the angular size of a pixel is the same formula used to determine the linear size of a star FWHM at the focal plane, but solving for the angular size:
Pixel Angular Size = 206265 * (Pixel Size/1000)/(Focal Length)
Where the Pixel Angular size is given in arc seconds, the Pixel Size is given in microns, and the Focal Length is given in mm. For your convenience, I have posted a spreadsheet to my website at http://www.jburnell.com/Pixels.xls where I have solved the equation for a number of CCD camera/telescope combinations. Feel free to download it and plug in your own numbers. Here are a few combinations using our C8 example from above coupled to a Starlight Xpress SXV-H9 CCD camera using 1x1, 2x2 and 3x3 binning, along with an even shorter focal length scope, a Tele Vue NP-101:
Knowing all this now, we can discuss matching the CCD camera to the scope. The main concern is to make sure that the CCD camera you choose can capture all the resolution that your scope can deliver, taking into account your local seeing conditions. If you have a long focal length scope, you will benefit from larger pixels, because their larger surface area makes them more sensitive, and their greater full-well capacity means you will be able to capture more photons before the onset of blooming. If you have a shorter focal length telescope, you want smaller pixels so as to better sample the image. If you have a variety of instruments, or use a focal reducer on a longer focus instrument, a camera with smaller pixels can be binned when shooting at longer focal lengths and operated unbinned for the shorter focal lengths. Using the formula presented here, or the spreadsheet, you can determine for yourself whether the CCD camera you want to use is capable of adequately sampling the image formed by your telescope.
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