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    Re: Follow up to comments on March 22 Lunars
    From: George Huxtable
    Date: 2002 Mar 26, 14:23 +0000

    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
    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.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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