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Subject: Re: Follow up to comments on March 22 Lunars
From: George Huxtable (george@XXX.XXX)
Date: Tue Mar 26 2002 - 09:23:15 EST
Arthur Pearson said-
>2) Bruce's "wrong way" tables convert from computed altitude to apparent
>altitude to allow the use of calculated altitudes in clearing lunars.
>Unless I am missing something, they don't allow one to "unclear" a
>calculated lunar distance. If he were willing to share the formulas for
>the wrong way tables, it would be possible to create a spreadsheet to
>derive periodic apparent altitudes from calculated altitudes, and from
>those periodic apparent altitudes, to calculate the apparent lunar
>distance, and then sextant distance as it would appear to an observer.
>This would allow examination of George's "parallactic retardation" over
>an extended period using hourly almanac data. I would be willing to take
>a run at this and share the data with the group.
=======================
Reply from George-
I am keen to support that offer from Arthur.
Here are the "backwards" parallax and refraction corrections that I use in
my calculator program, when converting a calculated "true" altitude C to an
"apparent" altitude A.
From C (in degrees) subtract the parallax correction P as shown below to
give an angle C'. To this, add the refraction correction R as shown below.
L is the observer's latitude in degrees. The corrections are shown in
arc-minutes, and the horizontal parallax HP is in arc-minutes, but for
those using a calculator or computer it will usually be more convenient to
adapt the formulae to keep everything in decimal degrees. How I hate these
damned sexagesimal angles!
C' = C - P C is the altitude calculated from an almanac or from a
computer-prediction, P is the parallax correction in minutes.
where P = 60*(1-.0032*(sin L)^2)*atn(cosC/(3478/HP) - sinC) IN ARC-MINUTES
The quantity in the brackets, (1-.0032*(sinL)^2) is a small correction,
always close to 1, for the reduction in horizontal parallax caused by the
elliptical shape of the Earth. If that bracket is omitted the resulting
error (for the Moon) is less than .2 arc-minutes, and the agreement with
Stark's "wrong-way" parallax table is very precise, as he omits that part
of the correction.
Then apparent altitude A = C' + R R is the refraction in minutes
where R = 1.02* tan (90 - .998797*C' - 10.3 / (C' + 5.11)) IN ARC-MINUTES.
The refraction correction is taken from Meeus equation 16.4 but is slightly
tinkered-with to avoid a possible infinity arising during the evaluation,
and to ensure that the refraction at 90 degrees is zero, as it must be by
symmetry.
Refraction correction formulae differ slightly, and are all an attempt to
obtain an empirical fit to observations, particularly at small altitudes.
The formula above gives results that differ only by 0.1 arc-minutes with
Stark. Which of the two is "right-er" is rather irrelevant.
George Huxtable.
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george@XXX.XXX
George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
Tel. 01865 820222 or (int.) +44 1865 820222.
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