OPTICS FOR TASS: run the numbers

[hjohnson@pluto.njcc.com (Herbert R Johnson)]

Tom has said he's ordered the Mak optic, so for the short term the
discussion of optics is moot. However, for some reason we never got
down to the "numbers", the objective specifications, in the discussion,
except for some comments. I decided to do the arithmetic after Chris's
comments:

On Wed, 07 Jan 1998 14:24:42 -0800, Chris Albertson <chris@topdog.pas1.logicon.com> wrote:

*>Is the math correct below?
*>
*>With only 2k pixels across 40mm we only require 25 lines/mm resolution.
*>as we are sampling at 50pixels/mm.  If the lens projected an image with
*>highr spacial frequency it would be a waste. This is in the range of what 
*>a single use disposable camera can do.
*>
*>--Chris Albertson

Thanks for the reality check.  Let's see how much trouble I can get into
with my cross-check and a little review of the Mark III.

My recollection is that "film" provides about 200 lines/mm, an order of
magnitude more resolution. I'm not sure the eye is that good
across a *large* field. So, an optical system of photographic quality
is OK, and maybe one of visual quality, provided either has good characteristics
(coma, focus) across the field of view. As the center of most optic systems
is usually pretty good, oversized lens or mirror systems with added stops
(or maybe one field stop in the front) may do the job if the resulting
image size is acceptable. What numbers do we need to determine these things?

If that 40mm holds true we'd need an *inch and a half* film
image to test the CCD's "minimum" field, and root two times that (2 inches)
to look corner-to-corner of the CCD.  2 inches is a standard eyepiece size
so the size at the focal plane may be "about right" for a visual telescope
(if it is good to the edge, maybe not). My email archives says the goal
is a 5 degree or 4 degree field of view. For 2000 pixels
across, that's 4 X 3600 / 2000 = 7.2 arc seconds per pixel minimal (more
for the 5 degree field).

The resolutions defined above are no great test of optics. BUT, we are
using a 12-bit A/D on a comparable CCD device. Let's say we end up
with 10 bits of "usable" dynamic range. One part in one thousand (1024)
is 60 db dynamic range, for images resolved to 7 arc seconds.

Now, I'll throw it to the TASS observers: what is the best PSF's for Mark
III images (say in the center of the field) in terms of size and dynamic
range? May as well learn from our own experience. So, I search my archives
again to answer my question.

*To: tass@wwa.com
*From: Glenn Gombert <ggombert@infinet.com>
*Date: Wed, 05 Mar 1997 18:17:49 -0500
*Subject: "Average" FWHM PSF In Tom's Images
*
*Hello,
*        While I still had the data on the hard disk I went through and
*calculated the "average" FWHM of the detected objects in each of Tom's
*images. Here are the numbers that I came up with:
*
*        File:           SkyBackground(ADU)     RMS Sky        FWHM(Average)
*        G0483967.FTS    11167.6                55.4055        3.61
*        G0493669.FTS    11698.1                97.5382        3.88
*        G0483977.FTS    10926.8                49.9414        4.02
*        G0493678.FTS    11753.2                95.8544        3.81
*        G1493995.FTS    11439.9                52.8192        3.62
*        G1493669.FTS    13155.8                63.7128        2.66
*        G1493678.FTS    13154.2                62.4993        2.91
*        G1493995.FTS    11705.8                52.7644        3.44
*        G2483931.FTS    99972.7                45.9182        4.23
*        G2493623.FTS    12642.9                96.1175        5.49
*        G2493900.FTS    10704.7                49.7170        4.08
*        G2493909.FTS    10772.3                50.5891        4.10
*
***** [end quote by Glen]

...so for some range from 100 units to (my recollection of about) 32000 units -
about 300:1 range incidently, close enough to 1000:1 - we detect "several"
pixels in diameter (4 to 6) for our stellar point images. (FIve is often
mentioned otherwise as a PSF window.) Let's guess that the center of the
image is twice as good, so we get two or three pixel diameter PSF's. The Mark
III pixel scale is 13.75 arcsec/pixel I believe: my math says this
is about twice what the Mark IV design goal.

Conclusion:

These would seem to be the numbers to test against: 1.5 inches to 2 inches
of image size, 4 to 5 degrees of field of view with minimal distortion,
distortion *theoretically* defined as about 25 lines/mm at the image plane
or about 7 arc seconds per pixel of theoretical resolution overall. But, in
the context of about 30 to 60 db of dynamic range (300:1 to 1000:1),
less-than-perfect skies, and issues of focus and optical variation, we may
expect PRACTICAL PSF's of "a few" pixels (2-3-4) in diameter.

My guess: if we can get an optic with coma-free field (visually or
photographically) and full illumination across the desired image size,
AND with the desired field of view, the rest will follow.

Herb Johnson

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