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    Star-star sights (part 1)
    From: Frank Reed CT
    Date: 2004 Apr 6, 20:01 EDT
    I got very busy on a project a couple of months ago. So first, a re-post:

    In Letcher's "Self-Contained Celestial Navigation with H.O. 208", he cleverly includes a chapter on star-to-star sights just before his chapter on lunar distances. Star-star sights are an excellent test of a sextant and a navigator's skill in using it. You can't estimate how good your lunars or other celestial sights will be unless you can test your sextant over a wide range of angles. In addition, the process of "clearing" a star-star sight is very similar to the steps required to clear a lunar distance sight. Many of the practical tricks that apply to lunar distances also work for distances measured between stars. So these sights have practical value and pedagogic value, too.

    The main sextant correction which every celestial navigator knows about is "index correction". It's almost never zero (and there's no reason to prefer a sextant with zero IC). Every navigator who has practiced celestial has learned to check the IC on a regular basis. With plastic sextants, it's best to check the IC before every sight. The usual test is to line up the horizon so that there is no "step" between the direct image and the reflected image. Then you look at the reading on the arc which might amount to a few minutes. That reading is the index correction. And of course, "if it's on, it's off... if it's off, it's on". For standard line of position celestial navigation, this technique is sufficient; it will yield an index correction accurate to maybe 0.5 minutes of arc or a little smaller. But can we do better? And can we include corrections for "eccentricity"? That is, should there be a different "index correction" for a measured angle of 60 degrees than you would use for a measured angle of 30 degrees. It will always be a very small difference in any modern sextant, but if you want good corrections of your sextant observations for lunars or any other purpose, you need those little differences.

    The distances between stars in the night sky change very slowly and in a completely predictable way due to proper motion and aberration. Measured star-star distances only have to be corrected for refraction, and in most cases we can confidently predict that refraction effect. Comparing the cleared measured star-star distance with the predicted true star-star distance gives us the correction we need to apply to future sextant sights. Unlike the usual index correction, a comparison between a measured star-star distance and a predicted star-star distance can give us a sextant correction that applies directly to large
    measured angles. And when you've made repeated star-star sextant sights and cleared them correctly, if there remains a random difference that just won't go away... sometimes a few tenths of a minute of arc above the predicted distance, sometimes a few tenths below... then you've discovered the real limits of your instrument in your hands. By shooting a few dozen star-star distances, you'll be able to estimate independently the potential quality of other sights you might want to shoot; if you can't get star-star sights better than 0.5 minutes of arc, you probably won't be able to get lunars better than that limit either.

    In "Self-Contained...", Letcher presents a table of selected stars and the calculated distances between them to the nearest tenth of a minute of arc. He has carefully chosen a set of stars which have very low proper motions. As a reminder, proper motion is the angular change in a star's position due to the actual motion of the star through space. Distant stars usually have low proper motions, and very few stars have proper motions large enough to measure with a sextant even after years of observations. Even so, the stars are slowly gliding across the sky. Since Letcher's book was written in the mid-1970s, his table is now slightly out-of-date. Perhaps someone would like to volunteer to update the table (and for every month of the year -- see below)??

    Unfortunately, Letcher made one significant error in his chapter on star-star sights. He correctly notes that precession has no effect on the distances between stars. Precession is just a rotation of the coordinate system (this is similar to noticing that moving the north pole around on the globe would not change the distance between New York and London). But he also says that "aberration" does not affect the relative positions of stars and the distances between them, and that's wrong. Aberration is caused by the finite speed of light, c, and the motion of the Earth in its orbit. It is proportional to the ratio of the Earth's orbital speed which is about 30km/sec to c which is about 300,000km/sec. To convert that angle, 1/10000, from a pure ratio to seconds of arc, we multiply it by 206265. So aberration changes the position of a star by as much as +/-20 seconds or arc or +/-0.34 minutes of arc in a cyclic manner during the course of a year. That's measurable with a sextant -- we can't leave it out. Since aberration depends on the motion of the Earth around the Sun, Letcher should have included tables for star-star distances for every month of the year.

    Today, since computing power is so cheap, we can skip all these tables and directly calculate the true angular distances between stars on-the-fly as we need them:
      cos(DIST) = sin(DEC1)*sin(DEC2)+cos(DEC1)*cos(DEC2)*cos(SHA2-SHA1)
    It's not even necessary to know what proper motion and precession and aberration mean. As long as you're using the SHA's and DEC's for your stars from this week's page of the Nautical Almanac, you can directly calculate the "true" angular distance between any pair of navigational stars using a spreadsheet or a handheld calculator. You might want to look at your evening sky and find a few pairs with apparent distances covering the full arc of your sextant. Look for pairs with separations of 10 degrees, 30, 60, 90, and maybe 120. Pre-calculate the expected true distances for those pairs of stars, and get ready to observe.

    You'll have to hold your sextant at sometimes awkward angles to make measurements of the distances between stars. You'll discover on your own some of the problems involved in bringing together images of stars. Pre-setting the sextant to the calculated true distance will help a great deal, but they should not line up exactly since we have not yet corrected for refraction. Bring the two star images together until they overlap as perfectly as possible. If there is a side-to-side gap when they're as close together as possible, you may need to adjust your sextant.

    In order to clear the distance, you'll need one or two other pieces of information. Either record the time of your sight to the nearest 15 seconds or so, or measure the altitudes of the two stars. Refraction depends on altitude so we'll need these altitudes (or we'll calculate them from the time) in order to determine the amount of refraction.

    OK, so make some observations and report back (or maybe just contemplate doing so if you're experiencing Northern Hemisphere January <g>). Some possibilities: measure from Rigel to Betelgeuse, Procyon, Regulus, and Denebola.

    Next step (coming soon): clearing the sight and comparing it with the prediction.

    Frank E. Reed
    [ ] Mystic, Connecticut
    [X] Chicago, Illinois
       
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