NavList:

David, you wrote:
"I’m familiar with the expression ‘neat, plausible, and wrong’ but I’m struggling a bit in this case.  I can see Bauer’s explanation is neat, I can see that it’s plausible, but why exactly is it wrong.  Is it meteorological?  I.e. the atmosphere might not be the same in every direction.  Or is it geometrical?"

It's basic great circle geometry (with meteorological concerns only for the lowest altitudes). We know that refraction is a vertical correction (and Bauer explains the other simple case, one body above the other, correctly on that same page). The mistake is thinking that a horizontal celestial angle is actually measured along an arc parallel to the horizon or a "circle of altitude". It's not. It's a great circle. The most blatant failure of this old idea that the refraction is zero when the two stars are at the same altitude can be seen when the stars are on opposite azimuths. Suppose I have a star 45° high due east and another star 45° high due west. By Bauer's advice, there's no refraction correction. In fact, the refraction is substantial for this sort of test: about 1.9' of arc. One might reply, "oh but that's a special case". No, it isn't. Try it out for two stars separated by 90° both at 45° altitude. The rule given by Bauer is simply wrong. And I'm fairly certain he didn't invent it. It seems to have been "common knowledge" (common ignorance) among many USN navigators in the late twentieth century.

Does this mean that there are no simple rules for removing the effect of refraction in star-star angles? No, in fact the proper, exact calculation is quite straight-forward and identical to the corrections for clearing lunars (see my two-part essay from 15 years ago here). There's also a nice nomogram method, which you can find, for example, in Letcher's "Self-Contained Celestial Navigation". 

And there's an even more delightful correction when both stars are higher than 45°. In that case, the refraction is 0.1' per 5° of angular separation. That's all there is to it! This works no matter how high the individual stars are and no matter how they are oriented, so long as both are higher than 45°. The refracted (observed) distances are shorter than the true (calculated) distances by 0.1' for every 5° distance between the stars. I take 100% of the blame for bringing this neat trick to the attention of NavList. I worked it up some years ago, and you won't find it published anywhere else. It works because refraction above 15° is almost exactly proportional to tan(ZD) and for angles less than 45°, the tangent is nearly proportional to the angle. In addition the refraction values are all less than 1.0' for altitudes higher than 45°. Thus the effect of refraction is to "shrink" the sky for that fraction of it higher than 45° altitude. Every constellation you see above that altitude is reduced in size by refraction at the rate of 0.1' per 5° of size or separation. This nothing by normal visual observation, but it's easily measured with a well-adjusted metal sextant.

There's another problem with Bauer's tables which he does not address sufficiently. He mentioned on that same page that the angle between Aldebaran and Alnilam had changed by only a tenth of a minute of arc between 1895 and 1985 (or something like that). In fact, long-term changes are rare. Only stars with high proper motion (Arcturus, Rigil Kentaurus, and very few others) will show any measurable differences over decades. Over a century, like his example here, a few more stars (usually nearby stars) show some modest change due to proper motion. Though the coordinates of the stars are changed dramatically by precession and nutation over decades, this is a rotation of the entire celestial sphere which does not affect star-star angles (for an analogy, if we used different poles on the Earth for coordinates, lats and lons could be very different, but the distances between cities would remain the same). Unfortunately, there can be substantial changes in star-star angles in just six months. This is "stellar aberration". Aberration can change a distance by 0.6-0.7' over the course of a year. So if anyone is contemplating publishing similar star-star tables, you should make monthly or bi-monthly tables. Or you use an app and call it a day.

Frank Reed

PS: To visualize aberration, find the direction on the celestial sphere towards which the Earth is travelling in its orbit around the Sun. This is easiest just before sunrise. Point to the Sun just below the horizon. Look 90° from there along the ecliptic. This will be a point near the meridian (and in the northern hemisphere, that point will be high in the South in early Fall and low in the South in early Spring). That's where we're headed. Aberration is directly proportional to v/c where v is our speed relative to the Sun and c is the speed of light. The Earth's orbital speed is 30km/sec while c is nearly 300,000km/sec so the ratio is 1/10,000. That's the angular deflection caused by the Earth's motion. This angle is a "pure ratio" or equivalently an angle "in radians". As always, to convert to minutes of arc, multiply by 3438 (=180·60/pi). You get 0.34' change in position. And as the Earth travels around the Sun, the angle changes by +/-0.34' or nearly 0.7' net. The aberration is zero in the direction towards which the Earth is travelling on any given day, and also zero in exactly the opposite direction. It is maximized on a belt halfway between those directions (which is centered on the great circle 90° from the direction of travel and therefore includes the Sun, the anti-Sun point, and the ecliptic poles (in Draco and Dorado, always). The stars are pushed forward in that belt towards the direction of the Earth's motion. Six months later, they're pushed in the opposite direction. NOTE: this is entirely distinct from the refraction correction which I discussed above, but you can visualize stellar aberration and even compensate for it by thinking about this pattern.