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    Equal Altitudes at Local Apparent Noon
    From: Mal Misuraca
    Date: 1997 Feb 12, 10:23 EST

    the programs for equal altitudes at local apparent noon, which appeared years
    ago in Rogoff's book on calculator navigation.
            Jan doubts that equal altitudes at noon will work well, because of the
    ambiguity in the sun's altitude near local apparent noon.  I probably did not
    explain myself well, because it is to cure that problem that the technique of
    equal altitudes is used.
            The sun near local apparent noon appears not to change altitude, sometimes
    for three minutes or more.  The length of time of ambiguity is related to the
    zenith distance between the observer and the sun---the greater the distance,
    the longer the episode of ambiguity.  It is exactly true, as Jan says, that
    during this period it is not possible to know the exact moment of LAN, which
    means that it is not possible to know the longitude of the boat, it being
    dependent on accurate GMT.
            If, however, you take a sextant shot well before noon, when the sun is
    clearly increasing in altitude, then after noon reset your sextant to
    precisely the same altitude and wait for the sun to come to the horizon in
    your sextant, the time of local apparent noon will be precisely half the time
    between the two sights of equal altitutude, added to the time of the first
    sight.  If several sets of equal altitudes are taken, an average of the
    estimates of local apparent noon will be obtained.
            Jan correctly points out that movement of the vessel between sights can
    affect the result, and he says that he doubts that this technique, as a result
    of those anomalies, will work.  The declination of the sun will change, thus
    affecting sextant altitude; and he questions whether either east-west or
    north-south movement of the boat would have an affect as well.
            As it happens, these issues have long since been worked out, because the
    technique of equal altitudes around the meridian is so valuable that it is
    worth working them out.  (I teach equal altitudes to my one-day emergency
    celestial class.)
            The Admiralty Manual of Navigation (mine is the 1954 edition) devotes a
    chapter to equal altitudes and provides a table for correcting for the
    problems Jan points out.  Easier to use is Hewitt Schlereth's table in his
    book "Latitude & Longitude by the Noon Sight," published by Seven Seas (and
    maybe out of print, but in used bookstores.  Try Armchair Sailor, in
    Sausalito, California.)
            Schlereth writes that moving directly at the sun during this process
    increases the sun's altitude (because it shortens the zenith distance) and
    thus means that it will take longer for the sun to drop to the equal altitude
    after noon.  This means that the interval between the sights will increase,
    and the time of LAN will be later than it actually occurred, thus moving the
    longitude calculation to the west.  If the boat is sailing away from the sun,
    this error is reversed.
            Fortunately, this error can be corrected; and Schlereth provides a table for
    the small boat owner, which takes into account (1) declination of the sun and
    (2) latitude of the vessel---thus yielding approximate zenith distance, for
    (3) a vessel moving at 6 knots directly toward or away from the sun.  If the
    boat's movement has only a component of north-south direction, he describes
    how to correct for it.
            How accurate are equal altitudes before and after local apparent noon?  The
    observer essentially has control of that question.  If zenith distance is
    great, and the sun's altitude is therefore relatively low at noon, the
    navigator must begin much earlier to take sights before noon, and therefore
    stay later to catch equal altitudes after noon, because the sun's rate of
    increase/decrease in altitude is less than if zenith distance is short, and
    the sun rises and falls quickly.  A good rule of thumb is to begin taking
    altitudes two minutes before the estimated time for LAN for each one degree of
    zenith distance.
            I have used this technique over the years in long zenith distances (46
    degrees) and short (12 degrees), with excellent results.  Its greatest value
    is the ability to teach it to students who will simply not take the time to
    learn celestial except as backup to GPS, hence, my one-day celestial class.
    It requires only the N.A., because there is no spherical triangle to solve,
    and it works for latitude and longitude without a universal plotting sheet,
    assumed positions, and the other adjustments that intefere with intuitive
    understanding.  Latitude is determined the usual way, longitude by the method
    I have described.
            If once each day, the amateur navigator has a decent fix from the sun, the
    boat will not be lost.  (If weather is a factor, with clouds obscuring the sun
    from moment to moment, the navigator would begin much earlier than LAN, in
    order to get a series of spaced shots for equal altitudes.)
            What began this long explanation, with thanks to Jan for prompting it, was my
    search for someone who had taken Rogoff's old HP-67 program and rewritten it
    for the 48GX or another calculator.  I would still like to leave that question
    out there, although there have been some suggestions from readers.
                                            Mal Misuraca
                                            "Celestial in a Day"
                                            595 Market Street, Suite 2450
                                            San Francisco 94105
                                            KO6KR, "Passage East,"

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