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    Lunars - Oblateness Correction
    From: Frank Reed
    Date: 2008 Jul 12, 21:02 -0400

    Since this has come up in a several recent messages, I thought I would spell
    out the way it's handled in my online lunars calculator.
    
    The approach is straight out of Chauvenet. What we call "oblateness" is the
    slight flattening of the Earth at the poles relative to the equator
    converting it into an "oblate spheroid" rather than a perfect sphere. The
    degree of flattening is approximately one part in 300 (1/297 is closer but
    not significantly so for these purposes). Chauvenet usually refers to this
    property as the "compression" of the Earth's spherical shape.
    
    There are two pieces to the correction. First, you correct the HP taken from
    the almanac (this is Chauvenet's "Table XIII" correction):
     HP=HP0*(1+(sin(Lat))^2/300).
    This is a small correction. On average, the error from ignoring it is
    roughly 0.05 minutes of arc in the clearing process which corresponds to an
    error in longitude of about 1.5' or about 1 nautical mile on average. The
    correction is larger at higher latitudes, but of course this has a smaller
    impact on the position fix since the longitude lines converge.
    
    Second, you correct the lunar distance (after clearing it, but it doesn't
    really matter whether you do it at the beginning instead) with a small
    increment:
     inc_LD=sin(Lat)*[(HP/150)*(sin(Dec2)/sin(LD)-sin(Dec1)/tan(LD))]
    where Dec1 is the declination of the Moon and Dec2 that of the Sun or other
    body. The error in longitude, converted to nautical miles, that would result
    from ignoring this second small correction would be, on average, about equal
    to
     (1.35 n.m.)*sin(2*Lat).
    By the way, Chauvenet has a factor of "A" in his equation and tells the
    reader to look it up in a small table giving log(A) as a function of
    latitude. If you're trying to figure it out from that table, bear in mind
    that the table actually shows log(A)+10 which was normal back then. The
    small variation of A with latitude is not at all important. Chauvenet had
    the clever idea that most of this small correction could be included in the
    lunar distance tables in the almanac. Then the correction would have been
    simply
     inc_LD=sin(Lat)*x
    where x is equal to everything in the square brackets above. I have
    considered adding this little addition to the predicted lunar distance
    tables on my web site. Anybody want it?
    
    PLEASE NOTE: these corrections have been included in the calculations of the
    lunar distance clearing tool on my web site for over three years. You can
    directly assess the significance of the total oblateness correction by
    selecting "Ignore Oblateness" in the "Options" section. If you're trying to
    get an exact assessment of your skill or your sextant's arc error, there's
    no reason to turn off the oblateness calculation. But for historical
    NAVIGATIONAL calculations or just for general understanding, there may be
    times when you want to turn off the oblateness correction.
    
    Incidentally, there's no real reason to bother with the remaining details of
    Chauvenet's own, rather idiosyncratic method of clearing lunars. It was
    rarely used, and it offers no really significant advantage. But the general
    discussion in his book is definitely worth reading.
    
     -FER
    www.HistoricalAtlas.com/lunars
    
    
    
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