UNDER SAMPLING vs OVER SAMPLING
The limit for the smallest object that can be observed by a telescope is called telescope resolution. When two objects are closer together than the the telescope resolution, then those two objects will start to appear in the telescope as just one object because the telescope can not resolve (discern) one object from the other. Telescope resolution is measured in angular units called arc-seconds. It takes 60 arc-seconds to make 1 arc-minute and 60 arc-minutes to make 1 degree. For example, a full moon is roughly 1/2 degree or 30 arc-minutes, or 1,800 arc-seconds.
Dawes’ Limit and Rayleight Limit formulas are theoretical values for estimating telescope resolution. Dawes’ limit is always lower than Rayleight limit. Which one is correct? All light traveling through a telescope is diffracted (scattered) around the object that creates one or more diffraction rings. Rayleight Limit defines the resolution where the 1st diffraction ring occurs. The Dawes’ limit ignores all diffraction rings estimates resolution of the airy disk (what’s left of the star without the diffraction rings). Telescope aperture (where light enters the telescope) directly limits telescope resolution. A larger aperture telescope will be able to resolve smaller objects. For example, a 116 mm aperture telescope has a Dawes’ Limit of 1.00 arc-seconds. By doubling the aperture to 232 mm, the Dawes’ Limit is now 0.50 arc-seconds.
CCD or CMOS digital technologies are used to capture the telescope images. The best photograph with fine detail occurs with a camera that can accurately sample the smallest objects produced by the telescope. Harry Nyquist is attributed for what is called the Nyquist (Sampling) Rate that simply states 2 samples are required to represent the highest frequency (smallest object) in a bandwidth limited system (telescope). A telescope with a Dawes’ Limit of 1.00 arc-seconds requires a camera that can sample that object 2 times (ie. 0.50 arc-seconds). Under ideal sampling conditions that means each pixel in the digital camera needs to capture 0.50 arc-seconds of telescope light.
Without tweaking the imaging system, the camera will either under sample (less than 2 samples for the smallest object) or over sample (more than 2 samples per smallest object). Under sampling means a loss in fine detail that the telescope is capable of producing. Over sampling means more than 2 samples are taken per the smallest object. Some over sampling is always better than under sampling because you will capture the most detail the telescope can produce. Over sampling can cause a reduction in camera sensitivity, increase in noise levels and tracking becomes more critical. The rule is, some is better none.
There are three parameters that define optimum pixel size. They are: light wavelength, scope aperture, and scope focal length. For a given telescope, calculate the optimum pixel size and use a camera that has a pixel that is close to, but greater than the optimum value. Next step is to reduce the aperture of the scope to match the optimum pixel size to the camera pixel size. That’s right, less aperture is better in this case. You will need to prepare a shield to place over your scope to limit the amount of light entering the scope. Of course, you could adjust the focal length of the scope, but that is very difficult to do on the fly.
Simple stated, for a sharp, detailed photograph, the telescope and camera must be matched to sample the smallest object that the telescope can resolve at the Nyquist Rate (2 camera pixels). Under sampling occurs when fewer than 2 camera pixels are available to sample the smallest object that the telescope can resolve. This results in a loses of fine detail and causes image distortion. Over sampling occurs when more than 2 pixels are used to define the smallest object that the telescope can resolve. This results in lower pixel sensitivity, increase in noise, and increases tracking difficulty.
Tuning these four parameters will produce the most detail in your images: aperture, focal length, pixel size and color of light (wavelength). Obviously you will pick a telescope of some aperture and focal length. You will pick a camera with a specific pixel size. Finally, you will point that system at some color in the sky. Telescope aperture and focal extenders (barlows) are the only parameters that are easily adjustable on the fly during an imaging session. Optimal Nyquist sampling can be achieved by tweaking those two parameters.
Before you try these techniques, be aware sky conditions can and do impact the resolving power of any telescope. On a poor night, your equipment will over sample, every time. Just try tracking a star when seeing is lousy – it’s very diffcult.
EXAMPLE 1
Consider a 120 x 840 mm refractor with a camera with the KAF8300 chip. You would need at least a 2,500 mm focal length scope. Very expensive for a refractor. Or you could change the camera, but that can also be expensive. You could reduce the aperture to 40 mm or use a 2x barlow and reduce the aperture to 80 mm. Those simple on the fly tweaks will produce the sharpest image possible under ideal skies. Use above calculator to reduce telescope aperture until the optimum pixel size @ 502 nm matches the camera pixel size.
EXAMPLE 2
Same telescope but change the camera to a new CMOS with 3.8 um pixels. Now you would need a 1,750 mm minimum focal length scope. Now, you use that 2x barlow and reduce the aperture to 110 mm to sample the best detail. Just by changing the camera you picked up 30 mm of aperture.
EXAMPLE 3
Consider an 8″ f/10 SCT that with a 200 mm aperture and the CMOS camera. The scope focal length should be 2,900 mm. Insert a 1.5x barlow and you keep all that precious aperture. No barlow, no problem, just reduce the aperture to 135 mm and you are good to go.
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