A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2012 Sep 18, 12:53 -0700
Greg, your diagram is fine, but there was no need to place the zenith right between the objects. Indeed the WHOLE POINT of the neat little approximation that I have been talking about is that it does not depend on the locations of the stars relative to the zenith either in location or in orientation, just so long as both stars are above 45 degrees in altitude. The correction is 0.1' per 5 degrees of distance. And since refraction shrinks the constellations, what we measure is slightly smaller than what would be seen by that fictitious geocentric observer for whom the stars are charted.
So WHY does this work? FIRST, let's recall that refraction for altitudes not too close to the horizon is approximately given by 1'*tan(z). That is, one minute of arc multipled by the tangent of the zenith distance. This is a good approximation down to 20 or 30 degrees altitude depending on the level of accuracy you want. NEXT, remember that the tangent of small angles is nearly equal to the angle (when the angle is expressed as a pure number or "in radians" as they say). And in fact, even up to 45 degrees, the tangent is a rather slowly varying function. If you remember your high school trig, the graph of the tangent function takes off as you approach 90 degrees, but below 45 degrees, it's nearly a straight line. This means that we can get a fairly good approximation for the refraction of a single star's altitude by dropping the tangent altogether and using 1'*z (z still in radians). The KEY POINT here is that the refraction is directly proportional to the zenith distance. The amount is near enough to 0.1' for every five degrees. So far this gives us one special case: a single star at some arbitrary altitude above 45 degrees and another star at the zenith. Refraction compresses that distance by 0.1' for every five degrees of distance.
TO SOLVE THE GENERAL CASE, we need to remember a little something about similar triangles. If two triangles are similar, meaning that the corner angles are the same, then the corresponding sides have the same proportion to each other. In other words, if I draw one triangle abc, and then draw another triangle ABC with the same shape but twice as big, then the ratio a/A and b/B and c/C will all be the same (and the ratios will all be equal to 2 if the second triangle is, in fact, twice as big, of course). DRAW A DIAGRAM of two stars near the zenith. Make it flat (don't worry about any spherical trig here). Label the sides from each star to the zenith z1 and z2. Label the distance between them d. Now draw another diagram where the sides are each contracted by 25% (much bigger than refraction but nice for visualization): z1' is 25% shorter than z1 and z2' is 25% shorter than z2. The angle at the top of the triangle which is the difference in azimuth between the stars has not changed. The other corner angles are unchanged in the case of the flat triangle and insignificantly different in the spherical triangle case. THEREFORE d' is shorter than d by 25%. And that's it. If the zenith distances are compressed in direct proportion to each zenith distance, THEN the star-to-star distance is compressed in the SAME proportion relative to the original distance (regardless of the specific zenith distances and regardless of the orientation). Incidentally, you can work this out using the spherical trig law of sines, and you'll get the same result.
There are two points of approximation in all of this. The most important is the assumption that the tangent of the zenith distance is approximately equal to the zenith distance (in radians). This approximation eventually breaks down, but it's not bad up to zenith distances of about 45 degrees. There is also an approximation that comes into play when we assume that the corner angles don't change. They do, but by a very small amount. How good is this approximate rule? Plenty good enough for most of the "practical" cases we've discussed. It's rarely off by more than a tenth of a minute of arc for all cases where the altitudes are higher than 45 degrees. Naturally the most extreme cases give worse results. So if you have two stars 45 degrees high on opposite sides of the sky, this approximate rule predicts that refraction compresses the angular distance by 1.8' while the actual number should be just about 2.0'.
So once again, for any pair of stars above 45 degrees, refraction compresses the distance by 0.1' for every five degrees of distance between them regardless of their orientation relative to each other and regardless of their distance from the zenith. Any case in geometry where distances are changed in direct proportion to the distances is effectively "center free" (the expansion of the universe, e.g.). The location of the zenith doesn't matter so you might as well pretend that one star is at the zenith. The refraction is the same. And if you prefer using the tables in the almanac, that suggests another way to do it: enter the refraction table with the observed distance subtracted from 90 degrees. The refraction in the distance is the corresponding refraction in altitude except that you add it to a star-star distance rather than subtracting it as you would for an altitude (if you do this, then you also should not use this method for distances greater than 45 degrees).
It's come up peripherally in some of the messages here that you have to include some other corrections. This is true only if you're using "stale" almanac data. If you have star positions (in SHA, Dec or equivalent) for the current date or at least the current week, you're fine. Aberration is the most significant issue here. During the year, the finite speed of light slightly shifts the constellations, pushing them towards the direction in which the Earth is travelling. For stars 90 degrees apart, this amounts to more than 0.3' back and forth every six months (+/-20"). But if you have current almanac data (assuming it's calculated correctly), this is already included. Certain other phenomena like precession and nutation are not relevant since these are really coordinate rotations. They don't change the distances between stars.
I previously suggested using "Stellarium" for altitudes, both true and refracted. Unfortunately, like many "free" open source products, it's buggy and it appears that the refraction is incorrect, at least in the version that I have.
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