NavList:

Alex Eremenko, you wrote:

As Frank once wrote ...:
"I bet that nobody will be able to correct the numbers in a certificate by ordinary observations."
Corrections that your certificate shows are too small to check them by ordinary methods, like star-to-star distances.

Assuming that the 'Frank' you reference in your comment is me, I'll say that this doesn't sound like something I have said! No worries if you're remembering someone else. I just want to clarify that I don't think that was me. :) In fact, star-to-star distances can be used to test arc error, but lunars are better. And I agree with Modris Fersters that, with some effort and repetition and, of course, care in observation, a fairly good sextant calibration can be re-created using home observations.

Maybe it's just a matter of "degree" (or actually seconds) that we're talking about. With lunars, a reasonable expectation of home certification would be +/-0.2' which of course is +/-12 seconds of arc. That's less quality than can be achieved with a proper tabletop collimation apparatus, but who has one of those now? And I have certainly encountered medium-quality metal sextants with arc errors at some angles as large as two minutes of arc. If we can tabulate those by home calibration, then a sextant like that can have nearly all of its arc error eliminated (converted into a correctable error). A proper calibration like this can provide real value for ordinary celestial navigation. If I can take a sextant with possible error that results in +/-2 nautical mile error in fixes and convert it, by calibration, into an instrument that has only +/-0.2 nautical mile errors, that's a positive outcome --a real, practical improvement!

More generally, regarding star-star distances for sextant calibration, I'll add a few paragraphs... I should start by saying that "star-star distances" do not constitute a navigational method (unlike "Moon-star distances", also known as "lunars"). Star-star distances are fundamentally a method for testing a sextant and secondarily a good training technique for lunars.

When a sextant is set to, let's say, 30°00.0' exactly, we just assume that it is accurately measuring an angle of thirty degrees after applying the usual index correction. But this is not generally true. Sextants have centering errors and arc graduation errors and other issues. These small errors combine to give something that we often call "arc error." A century ago, someone suggested calling it "Kew error" since the certificate from Kew Observatory (and nowhere else at that time in the UK) listed the errors at various angles (and "Kew error" was apparently a common expression for the barometers and thermometers that they tested). The majority of modern sextants come with similar certificates. It's not uncommon to see listed errors of 20 or 30 seconds of arc at various angles. Many sextant certificates state simply that the instrument is "free from error" for practical use, if we believe that. This arc error is like index error, but normally it cannot be tested without special equipment or procedures. Navigators have usually just trusted the certificate (even though it may date from decades earlier) or ignored the matter entirely as negligible for practical navigation. Star-star sights provide one possible means of testing sextant arc error at home or at sea. And by the way, it's something that you can do from an inland observing site --like your backyard. No horizon needed.

The idea behind "star-star" sights is that we can calculate the exact angle between any pair of stars at any time and then compare against that angle measured with a sextant. To do this, you need two things: calculated distances between prominent, easily-identified stars accurate to a fraction of a minute of arc, and also a method for clearing the sights for the effect of refraction. Neither of these is difficult in principle. We can take the positions of the stars from an almanac or any star catalog of sufficient accuracy and then the distance between them is a great circle distance calculation, just like calculating the great circle distance across an ocean between two ports on the surface of the Earth. Some of the stars are moving fast enough across the celestial sphere (proper motion) that they either have to be avoided or the distances need to be recalculated every few years. Also, star positions are shifted by up to 40 seconds of arc by annual aberration so really we need to get the positions for the current date, within a week, if we expect any serious accuracy from these calculations. As for clearing for the effects of refraction, this is just a matter of taking the normal altitude corrections for stars and applying them to an arc that may be at some funny angle across the sky. It's a straight-forward geometry problem. No big deal.

As it turns out, as long as both of the stars in a star-star observation are above 45 degrees, the refraction can be "cleared" from the measured star-star angle by a simple rule: add a tenth of a minute of arc for every five degrees of measured distance (the more general case can be solved easily with a little spherical trigonometry). Example: two stars are observed to be 26°23.4' apart, and they are high in the sky, falling under the scope of the rule that they both be above 45 degrees... 26 divided by 5 is 5 so the correction is about 0.5'. We add that to the observed distance giving 26°23.9'. We have in this way removed or "cleared" the effect of refraction from our observation. Remember that refraction always "squishes" the stars together so the cleared distance, after removing refraction, is always greater than the observed distance.

We compare our "cleared" star-star distance to the pre-calculated distance. If the numbers don't match, any difference is a combination of arc error mixed up with any residual random error in the observation. If we do it four or five times and average and always get the same bias, perhaps the observed distance is on average 1.0 minutes greater than the pre-calculated, then it's a fair bet that we have made a good estimate of the arc error for an angle of 26 degrees. We write that down and move on to another pair of stars. With enough star pairs, we can build up a complete sextant certificate, without paying five shillings to Kew (that was the going rate, and a very good deal, back in 1900).

As for making the observation itself, sometimes it's a little inconvenient and if you've never used a sextant for anything but altitudes (vertical angles), it can be a little weird at first. Generally the sextant has to be held so that the frame of the sextant is in the plane containing the two stars and your eye. It's easy to make the observation if you know the stars fairly well, and if you pre-compute the distance. You pre-set the sextant for the un-refracted distance, aim right at the more convenient of the two stars so that you see it in the horizon glass. Then rotate the instrument slowly about that line of sight (keeping the first star always centered in the horizon glass) until the second star pops into view. It should line up almost exactly with the first, and unless we're really unlucky, a false match is very unlikely. Then we adjust until the images overlap as exactly as possible and read off the numbers.

Some of us have discovered years ago that dark adaptation is a "bad idea" when taking these star-star sights. When the eye is fully dark adapted, stars can look like spikey "blobs," the pointy star shapes of cartoons, rather than small sharp points. In my own experience, I usually get results with star-star sights that are "good but not great". I can detect arc errors larger than 0.5 minutes using these sights and frequently somewhat better than that, but not as fine as a tenth of a minute of arc. Lunars are better ...in my experience. For any such observations, you should use the highest power telescope included with your instrument. The basic resolution of the human eye, properly focused, is 1.0 minutes of arc or slightly better. If you use a 10x telescope, you can resolve a tenth of a minute of arc.

By the way, it's important to eliminate all other sources of error before trying this sort of test. The index mirror has to be as exactly perpendicular as possible. The horizon mirror should be nearly perpendicular (but this is not critical). The telescope should be collimated (its axis parallel to the instrument frame). And the index correction must be found as exactly as possible. This last check is critically important. And as a reminder, I highly recommend zero-ing out index error as best as you possibly can. 

Frank Reed