NavList:

George Brandenburg, you asked:
"Could you give a simple explanation for the asymmetry between the two equations, namely that the first contains a tan(D) term and the second a sin(D) term. To someone who hasn't worked through the equations (me), the problem appears to be completely symmetric with respect to the moon's altitude and the sun's altitude."

The asymmetry results from the parallax of the Moon. The altitude correction for a star is entirely due to refraction and is approximately 1'*tan(z.d.) or 1'/tan(h_star), reasonably accurate for altitudes above 10 or 15 degrees. For the Sun and planets, this is also very nearly correct. For the Moon, there is a much larger parallax correction because the Moon's mean distance is only about 60 times greater than the radius of the Earth. The Moon's altitude correction is about
-57'*cos(h_moon)+1'/tan(h_moon),
or the negative of that, depending on how signs are defined. In fact, for the purposes of this error analysis, you can treat the refraction for both the Moon and star as negligible. So then the Moon's altitude correction is simply proportional to cos(h) and the Sun's altitude correction is zero. The cancellation between the Moon's parallax and the geometry of the lunar clearing process for distances around 90 degrees has no simple explanation that I have been able to think of, no "obvious" reason why it should have to be that way a priori. There's no "symmetry" or "logic" to it --it's a fortuitous cancellation. That's why I have called it the "90 degree miracle".

-FER

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