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    Re: Star to Star Distances taken on a Second Hand Sextant
    From: Brad Morris
    Date: 2019 Dec 11, 20:39 -0500

    Your concept of a calibrating rotary device is well founded.  That is, in fact, precisely what an Ultradex is.

    The one in my posession has 360 stops, one per each degree.  The total guaranteed error in that Ultradex is 0.5 arcseconds, from any one stop to any other.  There are two plates which lock at the required setting, unlock, rotate to a new position and re-lock.  No airbearings or other exotica, its simply two specially ground plates.  The top plate that moves can easily support a sextant.  The price of a functional Ultradex can be numbing.  

     One can also purchase precisely ground optical polygons, but those are phenomenally expensive. [ http://www.starrett-webber.com/PG1.html  ] They suffer from another, second order problem.  As the number of facets grow, say from 4 to 8, the nominal diameter of the block must also increase OR the facet size gets smaller.  As the size of the facet grows smaller, the ability to use it with the Autocollimator becomes much more difficult.   Note the maximum number of faces at Starrett Weber for an iptical polygon is 12, or every 30°.

    You still need to hold that optical polygon, may I suggest co-axial to the index arm axis of rotation.  Align facet one to sextant index using the autocollimator.  Rotate index arm (which now carries the optical polygon) to facet 2.  Since we know the precise angle between facet 1 and 2, any error shown by the autocollimator is directly attributable to either a poor sextant setting or arc error.  With a bit of practice, we can eliminate sextant setting errors, leaving arc error as the sole culprit.  This can be repeated for the remainder of the arc, and must be at least 6 readings at each facet, to get a firm grasp of the repeatability.   Of course, you need more than an 12-faceted polygon, that's only 30° per facet, yielding 4 calibration points on your arc (0°,30°,60° and 90°)

    The Ultradex is the equivalent of a 360 facet polygon.  With a re-arrangement of the components, the same principles apply.  Using the Ultradex as the reference, attribute any error measured by the autocollimator to arc error.  


    On Wed, Dec 11, 2019, 7:18 PM Peter Monta <NoReply_PeterMonta@fer3.com> wrote:
    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.


    - 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.


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