F

where ¿/pixel is the size of the pixel and F is the focal length of the optical system. Be sure to convert all measurements to the same units. To convert, recall that one micrometer (or micron) equals 1 /1000 of a millimeter, and 1,000 micrometers (microns) equal one millimeter.

What is the smallest angular feature that a sensor with 7.5-micron pixels can distinguish on a telescope with a focal length of 1,000 millimeters? Substituting a pixel size of 7.5 microns (= 0.0075 millimeters) and a focal length of 1,000 millimeters into Equation 4.3 gives the angular pixel size as 1.55 arcseconds. During exposures lasting more than a few seconds, telescope tremor, small tracking errors, and slight residual defocusing can enlarge star images from a tiny diffraction disk to 2 or 3 arcseconds. If the seeing is around 3 arcseconds (that is, rather poor), the match is excellent—but if the seeing settles down to 1.5 arcseconds, a longer focal length would be a better match for 7.5-micron pixels.

However, it is important to remember that in deep-sky imaging especially, resolution is not everything; many times, the field of view you need to cover an object will dictate the focal length of the optical system. This is a judgment call which you will need to make when you select equipment.

For planetary imaging, resolution becomes the primary consideration. If the seeing is good, the size of the diffraction disk is the limiting factor. To capture all the detail present in the telescopic image, the sensor must be able to fully resolve the diffraction disk.

Given a sensor with 7.5-micron pixels, what focal length is necessary to capture all of the detail on the planet Jupiter visible with a 10-inch telescope? The first step is to compute the angular diameter of the bright central region of the diffraction disk, $HWHM (half-width at half-maximum), from:

where X is the wavelength of the light forming the image and A is the aperture of the telescope. (For more detail, see Section 1.2.3.) The telescope aperture is 250 mm, and the effective wavelength for a typical sensor is 600 nanometers (600x10-6 mm), yielding a diameter for the core of the diffraction disk of 0.51 arc-seconds. To capture diffraction-limited detail, the Nyquist sampling theorem states that it takes two pixels to sample the core of the diffraction disk. One pixel should therefore fill an angle of 0.25 arcseconds at the focal plane.

Knowing that essential fact, you solve Equation 4.3 for the focal length;

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