Telescope Optics & Pixel Size

"Why do my stars look square?"

"Why do my exposures have to be so long to get any signal?"

"What is the size of my field of view?"

These questions illustrate the three major issues of matching a telescope to a CCD camera: 1) sampling, 2) pixel sensitivity, and 3) field of view. These issues must be addressed in order to take CCD images with your telescope that are as pleasing to look at as they are scientifically useful.

Sampling
Sampling refers to how many pixels are used to produce details. A CCD image is made up of tiny square-shaped pixels. Each pixel has a brightness value that is assigned a shade of gray color by the display routine. Since the pixels are square, the edges of features in the image will have a stair-step appearance. The more pixels and shades of gray that are used, the smoother the edges will be.

Images that have blocky or square stars suffer from undersampling. That is, there aren't enough pixels being used for each star's image. The number of pixels that make up a star's image is determined by the relationship between the telescope focal length, the physical size of the pixels (usually given in microns, or millionths of a meter), and the size of the star's image (usually given in arcseconds).

The CCD user has some control over focal length and pixel size. Telescopes come in various focal lengths, and focal reducers can be utilized. Similarly, CCD cameras come in a variety of pixel sizes. The KAF-0400, KAF-1600, and KAF-4200 based CCD cameras have 9µ pixels, the KAF-1300 has 16µ pixels. The KAF-1000 and SITe cameras have 24µ pixels. Further, all of these cameras can be binned (see CCD103), giving the user a variety of pixel sizes from the same CCD camera.

Unfortunately, we don't have as much control over the size of the star image, which will vary, depending mostly on the seeing conditions of the observing site. Mountaintop observatories often have 1 arcsecond (or better) seeing, whereas typical backyard observing sites at low elevations in towns or cities might have 3 to 5 arcsecond seeing.

A good rule of thumb to avoid undersampling is to divide your seeing in half and choose a pixel size that provides that amount of sky coverage. For example, if your seeing conditions are generally 4 arcseconds, you should achieve a sky coverage of 2 arcseconds per pixel. If your seeing conditions are often 1 arcsecond, you'll want a pixel size that yields 0.5 arcseconds per pixel. The following formula can be used to determine sky coverage per pixel with any given pixel size and focal length:

Sampling in arcseconds = (206.265 / (focal length in mm) )* (pixel size in microns)

   or

Sampling in arcseconds = (8.12 / (focal length in inches) )* (pixel size in microns)

Pixel Sensitivity
The larger the pixel, the more sensitive the camera will be for any given focal length. This is also a sampling issue. Under excellent seeing conditions, a camera with 24µ pixels on a telescope of 2000 mm focal length will produce images that are very close to being undersampled. For faint deepsky objects, however, these large pixels will outperform (in terms of sensitivity) a camera with 9µ pixels on the same telescope. This is because a camera with 9µ pixels being used on a telescope with 2000 mm focal length will produce images that are nearly oversampled. That is, there are too many pixels making up each star image. The result will be reduced sensitivity, but better resolution.

Recall earlier in this discussion that undersampled images meant having too few pixels for each star image. Consider the case, for a moment, where all the light of a star was contained in a single pixel. If the star were bright enough to produce 10,000 counts, including the background, then the pixel would have a value of 10,000. Now imagine that we cut the pixel size in half; that is, instead of an 18µ x 18µ pixel size, we have a 9µ x 9µ pixel size. The same amount of starlight would fall on the pixels, but since the pixels are half as big, the light would be spread out across four pixels. If the star's image was exactly centered on the pixels, the result would be 2,500 counts per pixel. In other words, the star would appear one fourth as bright as before.


Undersampled
 
Oversampled

We can see, then, that in an effort to match pixel size to focal length, we need to find proper balance between undersampled and oversampled images. For the increased resolution of smaller pixels, we risk oversampled images and reduced sensitivity. If we increase the pixel size to improve sensitivity, we risk reduced resolution and blocky star images. By following the above formula, a good match of pixel size, focal length, and seeing conditions will optimize both sensitivity and resolution.

There are some other considerations for proper image sampling. One is the target object of the CCD image. For bright planets, oversampling will provide better resolution and it will help to cut down the glare that can saturate the CCD pixels. On the other hand, for faint deepsky objects like galaxies or nebulae, moving toward undersampling will give better sensitivity, allowing shorter exposure times. Another consideration is image processing. For instance, if the image is to be sharpened, you will get best results with images that are well sampled.

Field of View
After determining the sky coverage per pixel, the field of view for any sensor and focal length is easily found. From the formula above, multiply P by the number of pixels along each side of the CCD chip, and then divide by 60 to convert arcseconds to arcminutes. The table below gives field of view and sky coverage for 3 popular sensors attached to telescopes of various apertures and focal ratios.

KAF-0402e
Telescope Aperture (inches) 6 6 8 8 10 10 12 12
Focal Ratio f / 5 f / 8 f / 6 f / 10 f / 6 f / 10 f / 6 f / 10
Arcseconds per Pixel 2.4 1.5 1.5 0.9 1.2 0.7 1.0 0.6
Field of View (arcminutes) 31 x 21 20 x 13 20 x 13 12 x 8 16 x 10 9 x 6 13 x 9 8 x 5
KAF-1602e
Telescope Aperture (inches) 6 6 8 8 10 10 12 12
Focal Ratio f / 5 f / 8 f / 6 f / 10 f / 6 f / 10 f / 6 f / 10
Arcseconds per Pixel 2.4 1.5 1.5 0.9 1.2 0.7 1.0 0.6
Field of View (arcminutes) 62 x 42 39 x 26 39 x 26 23 x 16 31 x 21 19 x 13 26 x 17 16 x 10
KAF-1301e
Telescope Aperture (inches) 6 6 8 8 10 10 12 12
Focal Ratio f / 5 f / 8 f / 6 f / 10 f / 6 f / 10 f / 6 f / 10
Arcseconds per Pixel 4.3 2.7 2.7 1.6 2.2 1.3 1.8 1.1
Field of View (arcminutes) 92 x 74 58 x 46 58 x 46 35 x 28 46 x 37 28 x 22 39 x 31 23 x 19

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