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    Star-star sights (part 2)
    From: Frank Reed CT
    Date: 2004 Apr 6, 20:03 EDT
    So you've measured some distances between stars with your sextant. Now it's time to "clear" those distances and compare them with the expected distances that we calculated earlier. Clearing distances between stars is very similar to clearing a lunar distance, and this is part of the reason that star-star sights are worth doing.

    First let's compare the clearing process with the clearing process for an ordinary celestial altitude sight. When you measure the altitude of a star, you correct for index correction, dip, and refraction (and while we're here, if you feel like using current almanac data and want to shoot Mars or Venus, you'll also have to throw in a small parallax correction). All of these corrections are direct additions or subtractions from the measured altitude.

    Star-star distance corrections are similar to altitude corrections. You don't need to correct a star-star distance for dip since we're not measuring an angle from the horizon. And this is a good thing; dip depends very strongly on observer height and can be influenced by unusual refraction effects at the horizon. Altitude sights will almost always have an inaccuracy of +/-1 minute of arc because of variable dip. Star-star sights don't have this problem so they can potentially be as much as ten times more accurate. With a star-star sight, you should still correct for index correction (unless you want to think of your star-star sight as a measurement of IC itself). And of course, the IC correction works just the same way as in any other sextant observation: if it's 1.2 minutes on the arc, you subtract 1.2 minutes from the measured angle.

    Correcting star-star sights for refraction (and parallax for Mars and Venus) is the tricky part. Unlike an altitude sight, the corrections are not simple additions and subtractions. But before I get into the tricky math, let's consider an exception where the math is still simple.

    Suppose I observe the angle between one star and another that is directly above it. That is, they have the same azimuth (within +/-5 degrees or so). I can measure the angular distance between those stars by holding the sextant in a "normal" vertical orientation, and whenever this is the case, the correction to the star-star distance is trivial since we're just dealing with a difference in altitudes. The shift in altitude caused by refraction is directed 100% along the line between the stars in this case. The correction to the distance then is just the difference in the refractions for the two stars:
        Correction to Distance = dH1*100% - dH2*100%
                          or  = dH1 - dH2.
    where dH1 is the altitude correction (the "d" is for difference) for star 1 and dH2 is the altitude correction for star 2. I've spelled out the factors of "100%" planning ahead for the next step. Here's an example of this sort of special case:
      It's early June, and I notice that Arcturus is directly above Spica in the wouthwestern sky (they have the same azimuth). I measure the angular distance between them. Carefully bringing the images of the stars together, I find that the angle between them (corrected for IC) is
        Dist. Arcturus-Spica = 32d 46.7'.
    I also measure very roughly the altitude of Spica and find that it's 30 degrees give or take 10 minutes (which in this case implies that Arcturus is about 63 degrees high). So I turn to the altitude correction tables for stars on the inside front cover of the Nautical Almanac and I read out the altitude correction for Spica. At 30 degrees altitude, the correction is -1.7 minutes of arc. For Arcturus, the correction is -0.5 minutes. Notice that we do NOT need to know the individual altitudes of the stars very precisely. We only need them well enough to enter the altitude correction table at the correct line. In the case of a star 65 degrees high, you could be off by 2 degrees in your altitude and you would still have a correction accurate to a tenth of a minute of arc.

    If we want all those tiny tenths of a minute in the cleared distance, we have to be very careful to pick up all the sources of altitude corrections. Let's add a little fun to the puzzle and suppose that it's a balmy June evening and the air temperature is 76 degrees Fahrenheit with an air pressure of 29.95 inches of Hg. Turning the page in the almanac, there's an additional correction for Spica. That warm air is a little thin... it doesn't refract as much so the total altitude correction for Spica is reduced to -1.6 minutes of arc.

    We now have the altitude corrections dH1 and dH2 and we can apply them to correct the measured star-star distance between Arcturus and Spica. But which way? The correction is just dH1-dH2 and I know that the two corrections are 1.6 minutes and 0.5 minutes. The difference between these is either 1.1 minutes or -1.1 minutes depending on which star I pick as star 1. To decide the case, just remember that refraction is "a tide that lifts all stars". Refraction pushes stars up away from the horizon. It squeezes the constellations towards the zenith and so it reduces distances between stars. The distance we measured is a distance that has been reduced by refraction so the correction of 1.1 minutes has to be added onto the measured distance. The measured distance was 32d 46.7 so the final cleared distance is 32d 47.8'. It turns out that this is EXACTLY the predicted distance between those stars proving that our sextant and our skill at handling the instrument is as good as it can get... <g> These numbers, of course, were set up to work out correctly. In general, you might expect that your cleared distance will differ by a few tenths of a minute of arc with a good sextant properly adjusted or maybe a few minutes with a lower quality sextant.

    If you would like to try a star-star sight this month, the distance from Spica to Arcturus is still 32d 47.8'. But there are lots of stars to choose from! It would be best to pick a few pairs of stars with a wide range of distances between them.
    [As a reminder, you can readily calculate the "predicted" distance between the stars using the standard cosine rule:
      cos(DIST) = sin(DEC1)*sin(DEC2)+cos(DEC1)*cos(DEC2)*cos(SHA2-SHA1).
    Remember to get the SHAs and Declinations of the stars you've observed for the month of your observations (or the exact date and time if you decide to use planets instead of stars).]

    There's another "easy case" for clearing a star-star distance that is much less relevant in practice but useful conceptually. Imagine two bright stars both very close to the horizon, widely separated in azimuth. You could even imagine them as real lighthouses. Suppose you measure the angle between those two low-lying stars (or lighthouse beacons). You would hold your sextant horizontally and bring the two stars together. After you've made the measurement, do you have to clear the distance to remove the effects of refraction? Refaction is quite significant close to the horizon. If both stars are 1d 00' high, the refraction in altitude for each is over 24 minutes of arc with significant variations from temperature and pressure. But in this case, the refraction has almost no impact on the angular distance between the two stars because the star-star distance is measured almost exactly parallel to the horizon while refraction lifts stars vertically towards the zenith. So in this case, the equation for the correction of the distance is
        Correction to Distance = dH1*0% + dH2*0%
                          or  = 0.
    There is no correction required to clear a star-star distance when the angle is measured horizontally. Note that only applies very close to the horizon, and so it doesn't have much practical significance. and note especially that if you measure the distance between two stars at the same altitude but high in the sky (e.g. both 45 degrees high) you are not measuring a horizontal angle so there is a non-zero correction in this case (it falls under the general case below).

    How do we clear star-star distances generally? If one star is halfway up the sky in the east and the other star is a third of the way up the sky in the south, the effects of refraction do not act directly along the arc between the two stars. Refraction lifts all stars towards the zenith, but the star-star distance in the general case is measured at some skewed angle across the sky. You can probably guess by now that the general case can be written in a way that resembles the percentage formulas I've used above. In general:
        Correction to Distance = dH1*factor1 + dH2*factor2
    where factor1 and factor2 are "corner cosines". They are percentages between 0 and 100% (and positive or negative depending on the direction to the other star) that tell us how much of the refraction acts along the arc between the two stars. These two factors depend in a simple way on the altitudes of the two stars and the distance between them. Again, we do not need very high accuracy in the altitudes of the stars to calculate these factors. If we can get the factors accurate to about 1% (or in some cases 0.1%), we're doing just fine.

    Once upon a time, coming up with clever methods of calculating the "corner cosines", factor1 and factor2 above, occupied the time and energy of some of the world's most talented mathematical navigators. Most of the numerous so-called "approximative methods" of calculating lunar distances are, at heart, just different ways of calculating these two factors for the more demanding case of lunars. And there's plenty of relatively interesting mathematical history in these methods, but today it's not worth the bother. Calculation is extremely cheap, so let's just jump to the equations in one simple form:
        factor1 = [sin(H2)-cos(Dist.)*sin(H1)]/[cos(H1)*sin(Dist.)]
        factor2 = [sin(H1)-cos(Dist.)*sin(H2)]/[cos(H2)*sin(Dist.)]
    where H1 and H2 are the altitudes of star 1 and star 2 and "Dist." is the distance between them (this can be the cleared or uncleared distance or the pre-calculated "predicted" distance). These factors, if calculated correctly, will always range between -1 and +1, and they tell us what fraction of each altitude correction acts along the line between the two stars. If the sextant is held more or less horizontally when the stars are observed, the factors will be nearer zero. If the sextant is more or less vertical, then the factors will be nearer to +1 and -1.

    We've got all the pieces of the puzzle now. I can measure the distance between two stars wherever they are in the sky. If I record the approximate altitudes of those stars at about the time that I've made the star-star measurement, I can look up their altitude corrections dH1 and dH2 in the Nautical Almanac and then with a few keystrokes on a calculator I can calculate the "corner cosines" factor1 and factor2. I can completely clear the observed distance by multiplying each dH by its corresponding factor and adding the products to my measured distance. By comparing the cleared distance with a calculated "predicted" distance, I have a fairly accurate test of my ability to measure sextant angles in that range.

    In my own experience, I've used measurements like this to test sextants and also to practice the skills necessary for shooting lunars. I have a Davis plastic sextant which I like very much within its limitations. It's great for low to medium altitude sights, but by shooting some star-star sights, I was able to confirm that it has a significant eccentricity for angles above 60 degrees. By mapping out these corrections for large angles, I've made the cheap little thing even more useful. Eccentricity is a correctable sextant error. Star-star sights are a practical way of finding those corrections. And of course, learning how to do these sights and how to clear a star-star sight is an excellent approach to learning lunars.

    I started "part 1" of this topic by mentioning Letcher's approach to star-star sights. As I noted, his tabulation of distances is a little off because he didn't appreciate the annual effect of stellar aberration. But if you're interested in neat mathematical tricks, you might like to see a simplification in the clearing calculation that occurs when Letcher assumes a specific formula for the law of atmospheric refraction. I haven't gone into that here because it obscures the connection with other types of calculations and it doesn't decrease the calculational effort significantly (cheap calculation again). But it's cool and worth reading: see Letcher, "Self-Contained Celestial Navigation with H.O. 208".

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

    [A note to the mathematically inclined (mathematically dis-inclined keep out!): draw the spherical triangle with corners at Z, S1, and S2 (zenith, star 1, star 2). Consider small changes in the lengths of the sides ZS1 and ZS2. These are equivalent to generalized altitude changes dH1 and dH2 (they can be from any cause and have any value, assumed to be small for now). If you hold the zenith angle constant, then you can directly solve for the change in distance by solving for Z "before" the changes in altitude and then using that Z to solve for the new distance "after" the changes in altitude. The trick in these "approximative" methods is to do a multivariate Taylor series expansion treating both H1 and H2 as variable and Z as fixed. Then the "linear order" part of the expansion is D = D0 + dH1*factor1 + dH2*factor2 and straight-forward calculation will yield the factors as above. Assuming that the altitude corrections are just a couple of minutes of arc, as they are for star-star sights, linear order is sufficient for practical purposes. Note to the mathematically dis-inclined: how did you get down here?<g>].

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