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    Errors in earlier posting on corrections to calculated altitudes..
    From: George Huxtable
    Date: 2002 Oct 10, 10:08 +0100

    May I please take the list back six months or so, to a posting from me on
    26 March 02 entitled-
    
    "Re:  [NAV-L] Follow up to comments on March 22 lunars"
    
    I will copy its contents in full. Here goes-
    
    +++++++++++++++++(start of copied March 26 posting)+++++++++++++++++
    
    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.
    
    +++++++++++++++++(end of copied March 26 posting)++++++++++++++++
    
    (Now back to today, 10 Oct 02)
    
    Arthur Pearson is implementing lunars on a computer spreadsheet, and has
    discovered that the expressions quoted above gives some small discrepancies
    when compared with the "wrong-way" correction tables in Bruce Stark's
    useful "Tables for clearing the lunar distance"
    
    I have checked his findings and confirm that these differences are the
    result of two errors in my formula for "parallax correction P in minutes",
    which I gave as-
    
    P = 60*(1-.0032*(sin L)^2)*atn(cosC/(3478/HP) - sinC) IN ARC-MINUTES
    
    whereas it should have been-
    
    P = 60*(1-.0032*(sin L)^2)*atn(cosC/((3438/HP) - sinC)) IN ARC MINUTES
    
    You will see that in the incorrect version I got the bracketing wrong, and
    also put in a slightly wrong value (3478 where it should have been 3438)
    for the constant term, which should be the number of minutes of arc in a
    radian. Sorry about that. Thanks to Arthur for his painstaking diligence.
    
    The above expression expects L (Latitude) and C (Calculated altitude) to be
    provided in degrees, and HP (Horizontal Parallax) in arc-minutes, and will
    provide a "reverse" parallax correction in arc-minutes. Its results should
    now agree with Bruce's "wrong-way" tables within 0.1 arc-minutes or so, and
    Arthor confirms that where he has checked, this is true..
    
    Users of the above formula for P will need to avoid a pithole when HP is
    given as zero, as it always is for stars and (effectively) for Saturn and
    Jupiter, and is often approximated by zero for the Sun (though .15 arc-min
    is better). On reaching the term (3438/HP) when HP is zero, many machines
    (though not all) will throw up on arriving at an infinity. It's sensible to
    trap out that possibility by testing in advance whether HP=0, and if it is,
    setting P to zero without going through any calculation. Otherwise the
    formula for P can easily be rearranged to avoid putting HP into a
    denominator.
    
    Thanks to Arthur for his really useful input. It would be too much to hope
    that there are no further such errors lurking in my past stuff on lunars
    and awaiting discovery. I would welcome notice of any other discrepancies
    or doubts. The errors noted above affect only that email posting on 26
    March and do not affect my postings "About Lunars", parts 1 to 4, of which
    I am acutely aware that part 5 is long overdue.
    
    My own corrections, by programmable pocket-calculator, of calculated
    altitudes, remain unaffected by the errors referred to above, which were
    made in transcribing for that posting to the list.
    
    George Huxtable.
    
    
    ------------------------------
    
    george---.u-net.com
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.
    ------------------------------
    
    
    

       
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