Pixel Scale Advice Appreciated

Over-sampling depends on your seeing and any other limitation in your system’s ability to resolve detail.  With my Planewave CDK14 which is kept at a remote site with seeing down to about 1.0”, I found that my 0.73” image scale is quite under-sampled.   When I moved to a smaller image scale of 0.60” the FWHM of my best subs dropped by about 0.20” to around 1.6”.   At 0.40” I got down to FWHM of 1.5” but at 0.30” it was no better.  So I started hitting some limitations beyond seeing when I got below about 1/3 of the seeing.

If the smallest resolvable detail is 1” (that’s what 1” seeing is supposed to mean), you can’t capture that detail with a 1” image scale.   Nyquist’s theory suggests you need two samples to define the crest and trough of a wave.  This was talking about audio waves but applies to light waves as well. And some have suggested that three samples a better for photography.   So assuming your system’s resolving power is limited by seeing and guiding, focus, aperture, etc;  the right image scale is probably your seeing divided by either two or three - somewhere in between.

My experience shooting with my FSQ106 at a scale of 1.46” bears this out.  With good seeing I get subs that are measured at a FWHM of just below 2 pixels at times. But I believe that’s just some sloppy measuring when the number goes below 2 pixels.  This system would be better off with a smaller image scale.  But, with only a 106mm aperture seeing may not always be the limiting factor.   So my guess is that around 1” image scale would be best with the FSQ106.

TL;DR: for a small system in average seeing divide your seeing by 2, for a large system in great seeing divide your seeing by 3 to wrench out the last bit of resolution.

That's a great point about OSC's….the RGGB array….of course!

There are a number of considerations to determine the optimum sampling rate and it helps to understand some of the basic factors that go into the rules of thumb that often get tossed about.

1) Start with the fact that an optical system all on it’s own is a bandwidth limited system.  Spatial frequencies (which look like cosinusoidal waves in the image plane of the form I(1 + B cos(wx)) where B is the optical contrast and w is the spatial frequency) are limited simply by the focal ratio of the optical system.  No spatial frequency above 1/(F*lamda) can get be modulated by the optical system.  At that frequency, there are 4.88 samples across the Airy disk—and that’s a hard limit for all telescopes.  A small complicating factor is that we don’t actually sample the image with point sensors and when you throw in the fact that pixels are generally square, things get modified a bit.   Explaining how that complication works gets a little messy and the changes are mostly minor so for most practical purposes we simply use the Nyquist limit of 4.88 samples across the Airy disk.

2) Unless you are working with a telescope in space, the atmosphere adds an additional limitation to the maximum spatial frequency that you can detect.  On an instantaneous level, the atmosphere simply distorts the wavefront at the entrance pupil.   The time averaged effect of the distortion is what blurs the image increasing the size and form of the point spread function of the whole system (telescope + atmosphere).  The net result is that the typical atmospheric conditions play a major role in determining how small the time average star image is in the image plane.  That’s why it’s hard to come up with a single, clean number for how to best sample the image—and that’s where rules of thumb become useful.  The important thing to understand is that smaller telescopes can actually achieve diffraction limited performance under reasonably good seeing so there’s a dividing point where you compute the size of the PSF using the diffraction limit and where you use the seeing limit.  In general, if the aperture of your scope is less than about 200 mm and you have “pretty good seeing” (<~2 arc-sec) you can generally get away with using the diffraction limited PSF to determine the blur diameter.   For larger apertures, you should estimate the diameter of the PSF using a conservative estimate of the best seeing conditions.  In general, setting the sampling anywhere between 2-3 samples across the blur diameter generally works pretty well.

3) CMOS cameras are now available with really small pixels so why not simply get a camera with the smallest possible pixels?  The problem with that approach is that the signal strength is determined by the area of the pixel multiplied by the responsivity of the sensor (times the incident optical irradiance.)  Making the pixels smaller, decreases the output and that’s the trade off.  In order to maximize the signal, you want the largest possible sensor, but you don’t want to make it so small that you gain nothing in terms of resolving detail in the image.  And this is where the rule of thumb of using 2-3 samples across the blur diameter comes from.  This isn’t a hard number.  The rule of thumb provides a range of “pretty good” configurations—and that’s a good place to start.

John

John Hayes:
And this is where the rule of thumb of using 2-3 samples across the blur diameter comes from.

John,

What's the best way to determine the blur diameter?  Is there a formula?  I'm imagining it is sort of a waterfall of limitations such as the seeing, aperture, focus (hopefully this can be ignored), and Strehl of the optics.  But is it simply the worst of the four on that list or do they interact in some way?

Thanks for your expertise!

Kevin