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Accurate Parallax Computation
From: Antoine Couëtte
Date: 2011 Feb 16, 15:35 -0800

ACCURATE PARALLAX COMPUTATION

Hello to all,

With 2 good NavList Friends, we recently stumbled upon a somewhat "imprecise" definition from Jean Meeus in his very well known "ASTRONOMICAL ALGORITMS" book. Certainly it is not a "false" presentation, but certainly too this one does not quite belong either to the class of his most celebrated "crystal clear" definitions.

It all deals with the way to accurately compute the Moon Parallax in a 2 dimension plan. When one needs to accurately compute Topocentric Height from Geocentric height, generally such computation is performed through iterations, since from Topocentric into Geocentric it is quite quite "straight and forward".

While I first presented them with one such "direct" 2 dimension method, namely with Rectangular-Polar/Polar-Rectangular coordinates transformation, one of these Friends also pointed out that it is possible to use our more familiar Tangent/Arctangent functions to fulfill this aim.

You will find in the enclosed "2 Dimension Parallax Computation.pdf" Annex one detailed way to perform such Tangent/Arctangent computation which does not require successive iterations.

As explained in the bottom of this document :

- If you know H topocentric, in order to compute SD topocentric (i.e. augmented SD) the recommended step is to "travel" through the Geocentric height to get back to Topocentric height and (this time at last) Distance from Body, which - only then - enables to accurately compute Augmented Semi-Diameter without special precaution, and
- If utmost accuracy is required, e.g. for satellites work such GPS, and given that the Earth Center is most generally not on the downwards extension of the Observer's Zenith (i.e. not on the Observer's "Nadir"), then it is necessary to compute in 3 dimension space.

As a conclusion, nothing really amazingly new here, and the idea to perform this computation this way is not even mine, but it is the one of a good and nice Friend. Nonetheless and as far as I am concerned, until now I have not found any detailed explanations anywhere (including in the outstanding "ASTRONOMIE GENERALE" by André Danjon) on the way to accurately treat parallax in Horizontal Coordinates.

Hope it may help some fellow NavList Members, just in case …

Best Regards to all,

Kermit

Antoine M. "Kermit" Couëtte

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