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From: Robin Stuart
Date: 2020 May 28, 08:29 -0700
My compliments to Dave Walden for posting a number of interesting and thought provoking navigational puzzles recently. With regard to parallax in azimuth; the topic of corrections to parallax in altitude due to the ellipsoidal figure of the Earth was discussed a while ago here and here.
For the particular problem posed
e2 0.00669454
Mean equatorial radius of the Earth, a 6378.14 km
Moon's geocentric distance, R 395191.48 km
Topocentric altitude, h 32.05569°
Latitude, L 39°
Topocentric azimuth, Z 88.83136°
The geocentric horizontal parallax is, sin HP = (a/R)
The result expressed as a series expansion in e2 and (a/R) for the geocentric altitude h' is
h' = h + (a/R) cos h - (a/2R) e2 sin2L cos h + (a/2R) e2 sin 2L cos Z sin h
Assuming a mean HP of 57.7' and plugging in the numbers reproduces the formula found in the Nautical Almanac. At time I didn't derive the equivalent series result for the parallax in azimuth but, having set the problem up as described in the attached document here, it's a simple extension. The result for the geocentric azimuth, Z', is
Z' = Z + (a/2R) e2 sin 2L sin Z / cos h
I haven't found this formula in Chauvenet (although it might be there somewhere) and it seems more convenient to use than the ones he gives.
Plugging the numbers gives Z' = Z + 12.858" in close agreement with Kermit.
Robin Stuart
