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waskeyc
journeyman
Reged: 06/27/08
Posts: 15
Loc: Virginia
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Is there an optimum focal length for a given pixel spacing, or is more power always better?
Background
There is much talk about Dawe's limit, and how much magnification per aperture for visual use, but not much about the limit for high power (planetary) photography. We know that for a given focal length, a given angular separation will produce a distance between two objects at the focal plane. To be specific,
(focal plane separation) = tan(angular separation) * (focal length).
Example
For a CCD sensor (camera, webcam, etc.) with pixels 10 microns apart (in the horizontal direction) mounted to a telescope with a Dawe's limit of 1 arc-second, would we get optimal detail by using barlow(s) to achieve a focal length of 2063 mm? Or is more always better, and use the maximum magnification available, because the final image can always be downsampled?
Or is there another focal length that is optimal? Possibilities include 2x the above to be sure that a pixel from each color is included in the limiting angular separation, 1/2x the above due to some sampling theorem, and sqrt(2)x the above due to the diagonal distance between pixels.
Also note that this applies to afocal imaging, where camera lens (actual) focal length / eyepiece focal length = multiplier on telescope's focal length.
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Chris
Celestron C4R
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walt r
Carpal Tunnel
Reged: 02/13/07
Posts: 2423
Loc: Doylestown, PA
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Yes, by calculation there is an optimum fl. I know this is covered in the book "AIP4WIN" in some detail but I don't have the book at hand this moment. I'll try to remember to look it up tonight.
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Walt
Obsession 18" f/4.45 #1370 AN/SC
MK67 Deluxe 6" f/12 Mak-Cass, Super Polaris GEM, JMI MicroMax DSC
DIY 60mm f/6 Achromat
Cookbook 245 CCD
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yock1960
super member
Reged: 06/22/08
Posts: 176
Loc: SW Ohio, USA
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I'm pretty sure I read that it's in the F20 to F25 range. I think I picked that up from "Introduction To Webcam Astrophotography", Hardcover Book by Robert Reeves. Have to look it up when I get home.
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Discovery 6" Dobsonian
ETX80 Backpack Edition + Meade #884 Tripod
Astronomiser ASTSC1C
Meade DSI II
Televue Powermate 2.5x
Gallery
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sang33ta
professor emeritus
Reged: 07/08/08
Posts: 539
Loc: UK
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I don't know about all the maths but i find 2000mm focal length the most ideal for a "do-all" scope. What we need is higher dinsity CCD chips with more pixels per inch.
More focal length gives better planet views but you have problems getting the full moon and many DSOs in an eyepiece, although focal reducer can help. I think there is still a market for <f10 large cats that noone is supplying because 0.33focal reducers won't reach focus on alot of equipment.
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Hioptic 152mm f12.5 Maksutov
Celestron Advanced CG5-GT Mount (Mr Noisy!)
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Casio QV-2900UX
Got fed up of waiting for Meade ETX-150 so put this together for £500/$1000 
Edited by sang33ta (08/15/08 11:10 AM)
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walt r
Carpal Tunnel
Reged: 02/13/07
Posts: 2423
Loc: Doylestown, PA
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OK, I remembered to look this up in AIP by Richard Berry & James Burnell, Willmann-Bell,2000.
It is the focal ratio that determines is size of the Airy disk and the Nyquist sampling theorem the determines the minimum number of pixels to resolve the Airy disk.
So, for critically sampling the equation is:
F <= (A * d) / (0.51 * lambda)
where:
F = focal length
A = aperture
d = pixel size
lambda = wavelength of light (550nm for yellow)
All are in meters.
With
pixels = 10 microns ( .01mm = .00001 meter)
Celestron C4R = 4inch aperture or 100mm (0.1 meter)
yellow light = 550e-9 meter
F <= 3.56 meters for perfect seeing (f/ = 35).
If seeing, and other other errors tripled the effective diameter of the Airy disk then F could be 1188mm or an f/ = 11.8.
Edited by walt r (08/16/08 01:27 PM)
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waskeyc
journeyman
Reged: 06/27/08
Posts: 15
Loc: Virginia
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Thanks walt. 
Solving for pixel size then, d = F Ratio * lambda / 0.15, correct? So Nyquist says it takes roughly 7 pixels (1/.15) to resolve an Airy disk?
In practice, since I don't have that many Barlow's, the answer to my original question is: "More is better until the atmospheric conditions no longer permit." 
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Chris
Celestron C4R
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imjeffp
Senior Space Cadet
Reged: 09/30/03
Posts: 4435
Loc: Cedar Park, Texas
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Use Dawe's Limit to solve the resolving power of your telescope. For example, an 11" telescope can resolve 0.41 arc-seconds.
According to Sampling Theorem, you want to sample at twice the frequency of the signal being sampled. In other words, if 0.41 arc-seconds is the frequency, you want to use two pixels to sample that amount. Thus, the ideal resolution for our 11" telescope would be to sample at about 0.2 arc-seconds/pixel.
The NexImage has 5.6µ pixels.
If you use the formula  , you'll come up with 0.41"/pixel. (There's a handy calculator here.)
A 2x barlow makes our f/10 11" scope's FL 5600 for 0.21"/pixel, and a 2.5x Powermate yields a 7000mm FL, and 0.17"/pixel.
This is a long answer to get around to a general rule of thumb. Most webcams are properly sampled around f/20 to f/25. Too little, and you're undersampling the image--you're not recording all of the detail your optics can deliver. Too much, and while the image may be bigger, you're not actually capturing any more information--there's just not any more to be captured.
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Blog
ST80 • AT80EDT/LXD650
ETX-90/DS-2000 • 10" LX200 Classic ("The Quarter-Meter Telescope at the Heritage Park Observatory")
SPC900NC • DMK21AF04 • Digital Rebel XT
  
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walt r
Carpal Tunnel
Reged: 02/13/07
Posts: 2423
Loc: Doylestown, PA
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Big apologies.....I swapped two numbers in the equation I posted.
0.15 should have been 0.51 (I corrected my post above to fix this error and the calculated results).
The equation is:
F <= (A * d) / (0.51 * lambda)
As imJeff stated "you want to use two pixels to sample" the Airy disk.
The 0.51 comes from the Airy disk equation of 1.02 * Lambda. Since you want two sample of the disk 1.02 is divided by two to get 0.51 * Lambda.
The focal lengths and f/'s I got in the original post seemed a bit too big. I recheck and found my error.
Again sorry and hope the correction is much more useful.
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Walt
Obsession 18" f/4.45 #1370 AN/SC
MK67 Deluxe 6" f/12 Mak-Cass, Super Polaris GEM, JMI MicroMax DSC
DIY 60mm f/6 Achromat
Cookbook 245 CCD
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