NavList:

I've always wanted some way to assess sextant arc error, not because I expect the errors to be large, but just to have the assurance that nothing is grossly (or subtly) wrong.  (A reflecting circle has what it needs to be self-calibrating with circle-closure techniques, but a sextant doesn't without external help.)

So what is the easiest, least expensive way to get there?  Here are some half-baked thoughts.

Goals:

- modest accuracy, ~1 arcsecond standard error (state of the art rotary tables are near 1 mas)

- inexpensive

- reasonably automatic

Star distances are one way to go, but they would be difficult and awkward to automate.  Autocollimator hopscotch has been mentioned in the past on NavList and in Bill Morris's book.  A calibration suite consisting of a reflecting circle (or equivalent optical-bench setup) and a couple of collimators would also work, by comparing the sextant with the circle when measuring the angle formed by the collimators.  Again, though, that's quite tedious.

So I'm thinking rotary table.  The old-school, non-electronic rotary table would be a graduated plate with several (4 or 5) reading microscopes.  That would not be automatic enough---each reading would take minutes.  But what if we use small cameras, such as cellphone cameras or webcams, with inexpensive lenses / objectives?  That would be around $20 per camera, and three cameras might be enough.  As for the plate, I'm thinking it could be ground glass.  That is inexpensive and easy to make, with just the slight problem that the "graduations" are a bit random and hard to read.  But the rotary-table firmware would just memorize every small field around the entire circle.  Given an image of the ground glass, the firmware would correlate it with all the known fields to recover its position.

A plate of ground glass has some merit as a calibration artifact I think.  The material, silica, is uniform, robust, reasonably stiff, and does not creep.  (Soda-lime glass would be fine, but a plate of fused silica might be a little better, and it has a lower CTE.  As Brad mentions, Zerodur might be better still, but much more expensive.)  The small pits of the ground surface aren't going anywhere, and there's a large amount of redundancy.  It seems easiest to use the surface of the plate, but using the edge (after being milled circular) would be possible.  A circular plate has a simple geometry and might be expected to respond well under thermal stress:  a uniform expansion or contraction or cupping would have no effect on accuracy; only the low-order transverse thermal gradients would be a problem.

So all this is under $100:  plate, cameras, Raspberry Pi computer.  What about the accuracy?  Well, let's assume the webcams with their front-end lenses can achieve something like a 0.2 numerical aperture over the center of their fields.  Image only in the green because of chromatic aberrations, and use only a tiny portion of the center of the field because of the massive field curvature of these simple lenses (but we need a field of only 100 microns or so).

That gives a nominal resolution of around 1 micron FWHM, which, for a plate radius of 100 mm, is 2 arcseconds.  But the averaging effect of the image correlation should give a precision quite a bit better than this.  One could hope that calibration errors would be a small fraction of the inherent precision; just let the thing self-calibrate all the time with small motors turning the plate.  Speaking of turning the plate, commercial rotary tables have fancy air bearings with submicron runout.  I don't see what that's necessary for this application.  With more than one measuring camera, the eccentricity is jointly estimated with the angle, and it drops out.

Would be a fun project.  Or just buy a commercial rotary table at fabulously high cost.

Cheers,

Peter