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    Picturing lunar distances.
    From: George Huxtable
    Date: 2003 May 15, 16:18 +0100

    Picturing Lunar distances.
    Kieran Kelly asks about rigorous and approximate solutions to the clearing
    of lunar distances.
    I am sure that Kieran understands these matters well, but some others seem
    to find difficulty in picturing what goes on when a lunar distance is being
    "cleared". Clearing, in this sense, means being corrected, very precisely,
    for the combined effects of refraction and parallax.
    Explaining such matters for the list is made somewhat harder by its
    discouragement of graphic attachments. In this case, however, flat pictures
    don't help greatly: it's impossible to transmit a solid model. You will
    have to make (or imagine) your own, as follows-
    Take a nice, big, round, orange. Bisect it into two halves so the navel is
    at the centre of one half. You can eat the other half. Indeed, you can eat
    both halves, as long as you leave a half-skin intact.
    Put the half-skin, navel up, on a flat plate. Imagine a tiny observer
    (you), just at the centre point of this orange hemisphere, in the plane of
    the plate, which represents your true horizontal. The orange half-skin
    represents your view of the sky. The navel is at your zenith point Z.
    Between the plate and the navel, measured around the skin, is 90 degrees,
    so that gives you a scale of angle. Somewhere in the sky you can see the
    Moon and the other body you observe: we will presume here that it's the
    Sun, but you should keep in mind that it could just as well be a planet or
    a star, as long as it's somewhere near the Moon's path.
    This orange half-globe is quite a good analogy to the sky seen by an
    observer, as long as one difference is kept in mind. The observer sees the
    bowl of the sky from the inside, looking out. You view the orange from the
    outside, looking in. This causes some differences: if, to an observer, the
    Sun is to the left of the Moon, viewed on the outer surface it will appear
    to be on its right. This doesn't detract from the orange model in our
    present application.
    What a lunar observer does is to measure, at a known time, and as precisely
    as possible, the apparent angular distance, diagonally in the sky, between
    the center of the Moon and the centre of the Sun. This will lie somewhere
    between 0 and 180 degrees. We will call this angle d.
    To arrive at that angle he has had to make corrections for semidiameter and
    index error, and we will presume that's been done.
    At the same time, he measures the apparent altitudes, above the true
    horizontal, of the centres of the Moon (m) and the Sun (s) (having
    corrected where necessary for index error and dip, and semidiameter, but
    not for refraction or parallax).
    We can now mark these in on the outer skin of our orange. Put a dot
    somewhere on the half-orange that's at an angle m degrees up from the
    plate. This represents the centre of the apparent Moon Mark it with an m.
    Now you need to find where to mark the apparent Sun. You know that it's an
    angle s up from the horizontal, so lightly mark in a horizontal line, of
    constant altitude s degrees up from the plate, and the Sun must lie on
    that. Now set a pair of dividers, or knot a piece of thread, to subtend an
    arc equal to the apparent lunar distance d. Swing it about centre m (to
    left or right, it doesn't matter) until it intersects the horizontal line.
    That point is where the apparent Sun lies. Mark it with an s. Join the
    three points to form a triangle Zms.
    All three sides of this triangle are parts of great-circles, so here we
    have a spherical triangle, which can be solved using well-known formulae.
    It's very similar to the standard PZX astronomical triangle, though it
    represents different quantities. What we need to know is the angle at Z,
    the difference in azimuth of Sun and Moon that the observer sees.
    It's not very complicated. We get Z from-
    cos Z = (cos d - sin s sin m) / (cos s cos m)   (equation 1)
    in which Z is angle between azimuths of Moon and Sun,
    d is apparent lunar distance
    m is apparent moon altitude
    s is apparent Sun altitude.
    The "clearing" operation involves calculating what the corrected lunar
    distance D should be, if it was seen by a mythical observer at the centre
    of the Earth, because that's what the predicted lunar distances in the
    almanac are given for. We have to allow for the effects on the apparent
    altitudes of the Moon and Sun, caused by parallax and refraction. The
    correction to the apparent moon altitude always increases it, by an amount
    corr(m), the Moon parallax always greatly exceeding the refraction, which
    works in the opposite direction. So the corrected Moon altitude M = m +
    On the other hand, the Sun's apparent alatitude id always reduced by the
    amount of the correction, because for the Sun refraction always exceeds
    parallax, so the corrected Sun altitude S = s - corr(s).
    Defined in this way, these corrections, which work in opposite directions,
    are always treated as positive quantities. This isn't entirely logical, but
    it's been done that way for so many years it's hard to change it now.
    The effect of these corrections is to jack up the position of the Moon,
    toward Z, by an amount corr(m), and drop the position of the Sun, further
    from Z, by an amount corr(s). Note that parallax and refraction act only to
    raise and lower these positions along vertical lines, and don't give rise
    to any sideways shift, so the azimuth angle between Moon and Sun stays
    exactly as it was before.
    You can now put in a new dot for the corrected Moon, up a bit towards Z,
    and for the corrected Sun, down a (much smaller) bit away from Z. These
    shifts are both small angles, a degree or less for corr(m), and no more
    than a few minutes for corr(s), far too small to show on the skin of an
    orange, so to show the principle you will have to exaggerate them greatly.
    Now draw in a new skewed great-circle line joining these corrected Moon and
    Sun positions, and the exact length of that line is the corrected lunar
    distance D, which we need to find. If you have drawn it right, it should
    cross over the original lunar distance line at just one point between the
    Moon and Sun positions.
    So now there's a different spherical triangle, embracing exactly the same
    zenith angle Z, but with the lengths slightly different from before. A
    similar formula applies, as follows-
    cos Z = (cos D - sin S sin M) / (cos S cos M)    (equation 2)
    where, as stated before, M = m + corr(m), and S = s - corr(s)
    Because the right-hand sides of equations 1 and 2 are both equal to cos Z,
    they are equal to each other, and we can write
    (cos D - sin S sin M) / (cos S cos M) = (cos d - sin s sin m) / (cos s cos
    m), or
    (cos D - sin S sin M) = (cos d - sin s sin m) (cos S cos M) / (cos s cos m), or
    cos D = ((cos d - sin s sin m) (cos S cos M) / (cos s cos m)) + (sin S sin
    M), or
    D = arccos (((cos d - sin s sin m) (cos S cos M) / (cos s cos m)) + (sin S
    sin M)) (equation 3).
    The whole complex business of clearing the lunar distance boils down to
    obtaining a value for D, to high accuracy, from  equation 3. No
    approximations have been made in deriving it: it's completely rigorous.
    Nowadays, it can be computed as it stands to 10 digit accuracy on a pocket
    calculator in a couple of minutes.
    Not so in previous centuries, however. The only aid available was log
    tables and log trig tables, and these were needed to 6-figures or better to
    preserve enough precision. However, logs only help in multiplications and
    divisions, not in additions and subtractions. As Bruce Stark has explained,
    whenever an add or subtract crops up, it's necessary to take antilogs, do
    the addition, then take logs again (though he has used clever dodges to
    bypass that problem).
    Also, cos d comes into the expression, and because it goes negative when d
    exceeds 90 degrees, log cos d becomes meaningless above 90 degrees.
    For those reasons, equation 3 was ill-suited to solution using logs. Many
    methods were invented which minimised such complications, and equation 3
    was bent and twisted to an extraordinary extent to achieve this. In many
    cases, these methods remained rigorous, without approximations: the
    precision was limited only by the precision of the tables available. These
    included Borda, Dunthorne, Delambre, Raper, Young, Kraaft.
    Other methods, however, arrived at D by adding  or subtracting a small
    correction to the measured value d. This couldn't be done by a rigorous
    geometrical process, but by devising small triangles to fit onto the ends
    of the line d which provide amounts to add and subtract to equate it to D.
    Because those correction triangles are always small, no more than 1 degree
    in extent, they can be closely approximated by plane triangles, using
    ordinary plane geometry. If it was necessary, a further level of correction
    could be applied to arrive at even higher precision. The great advantage of
    such non-rigorous methods was this. Because the correction was a small
    quantity to be added to or subtracted from d, it didn't need to be
    calculated to a high accuracy (as a fraction of itself), so even
    four-figure tables would suffice. If the navigator didn't carry those bulky
    and expensive six-figure or seven-figure tables, he would in fact end up
    with a more accurate answer using a non-rigorous method that a rigorous
    Lyons provided one of the earliest non-rigorous methods, followed by
    Merrifield, Airy, Hall.
    C.H.Cotter, "A History of Nautical Astronomy", Hollis and Carter 1968,
    provides the best assessment of all these methods of clearing the distance,
    though as the Nav-L list is by now well aware, he is prone to make careless
    George Huxtable.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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