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Moon Parallax, Math Trivia (was Re: venus)
From: Frank Reed CT
Date: 2004 Oct 20, 01:30 EDT
From: Frank Reed CT
Date: 2004 Oct 20, 01:30 EDT
Earlier we were talking about the calculation of the Moon's parallax in altitude when you're given the true geocentric altitude instead of the apparent altitude.
George H wrote:
"To increase accuracy, we need to find a better approximation, to bring us
closer to-
OA = TA - HP cos OA, when we don't yet know what OA is.
My own routine, for reverse correction for parallax of the Moon's TA, takes
the Moon's distance D from Earth's centre in Astronomical Units (which has
been computed elsewhere) to determine HP.
The angle to be subtracted from TA to obtain OA is, in this routine-
Arc-tan ( cos TA / ((23455 * D) - sin TA))
The amount of this correction now corresponds very closely (with a change
of sign) to the correction that's added to OA to obtain TA. "
I mentioned that you can also do this calculation by iteration. Now for a bit of trivia: the Arctan formula, which is plenty accurate enough for practical puposes, is actually an approximation. There is no closed form solution for OA above. There cannot be since the equation is identical in form to Kepler's Equation. It makes little difference in practical terms since HP is always a small angle, but it's a bit of mathematical entertainment that the historical method of entering the paper tables twice (thrice is better) is more accurate than a seemingly rigorous equation. It makes not a bit of difference when you can throw the whole calculation on a computer --merely trivia.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
George H wrote:
"To increase accuracy, we need to find a better approximation, to bring us
closer to-
OA = TA - HP cos OA, when we don't yet know what OA is.
My own routine, for reverse correction for parallax of the Moon's TA, takes
the Moon's distance D from Earth's centre in Astronomical Units (which has
been computed elsewhere) to determine HP.
The angle to be subtracted from TA to obtain OA is, in this routine-
Arc-tan ( cos TA / ((23455 * D) - sin TA))
The amount of this correction now corresponds very closely (with a change
of sign) to the correction that's added to OA to obtain TA. "
I mentioned that you can also do this calculation by iteration. Now for a bit of trivia: the Arctan formula, which is plenty accurate enough for practical puposes, is actually an approximation. There is no closed form solution for OA above. There cannot be since the equation is identical in form to Kepler's Equation. It makes little difference in practical terms since HP is always a small angle, but it's a bit of mathematical entertainment that the historical method of entering the paper tables twice (thrice is better) is more accurate than a seemingly rigorous equation. It makes not a bit of difference when you can throw the whole calculation on a computer --merely trivia.
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
