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    Re: Lunar altitudes
    From: Jan Kalivoda
    Date: 2003 Apr 14, 10:08 +0200

    George is fully right in my opinion. For neutralizing the errors in both 
    altitudes owing to irregular dip, only their azimuths matter and the nearer 
    they are, the better.
    But for neutralizing the effects of irregular refraction on both altitudes, 
    the altitudes are more important, although the azimuths can be relevant as 
    And of course, the time of both observations should be the same, otherwise a 
    change of atmospheric conditions can wreck the result completely.
    Therefore the advice of old authors seems logical - the same time, 
    neighbouring azimuths, neighbouring altitudes of both bodies; and both should 
    stay near the prime vertical, so that both LHA's obtained might be as 
    reliable as possible. These are the cases 1) and 2) in George's reply.
    Of course, such combination of positions of the Moon and a star or planet in 
    the twighlight period (the Sun we must leave aside) under the clear sky can 
    arise twice in a month at most (or once in two months in higher latitudes?). 
    And even in this moment the altitude observation over the sea horizon is 
    always less accurate than an observation of lunar distances, as the horizon 
    is always a bit blurred, and more in twighlight than during the day.
    But on the other hand, the accuracy of a lunar distance is much more critical 
    for finding the GMT - 1 arc-second of the distance error creates the error of 
    2 time-seconds in GMT on average. This should be considered, too.
    Jan Kalivoda
    ----- Original Message -----
    From: "George Huxtable" 
    Sent: Sunday, April 13, 2003 10:19 PM
    Subject: Re: Lunar altitudes
    I agree with Jan, but only in part.
    He is right, that in some circumstances the possible altitude errors,
    inherent in using the horizon as a reference, can cancel out (for the Moon
    and the other-body) rather than add. But he and I seem to differ somewhat
    in our view of what those circumstances are.
    Let's consider a really simple state of affairs. The observer is on the
    Equator, and observes Sun and Moon on a day when both have a declination of
    zero (or nearly so).
    In that case the Sun and the Moon will follow the same path, rising due
    East, passing exactly overhead, and setting due West. One will be ahead of
    the other, and the angle between them gives a measure of GMT. A
    lunar-distance observer measures that angle in the sky directly, without
    involving the horizon.
    Instead, our observer measures their altitudes up from the horizon. Let's
    imagine that on that day, because of unusual refraction within a few feet
    of the sea-surface, the horizon dip differs from its expected value for the
    observer's height-of-eye. Let's presume that dip is actually 1 arc-minute
    greater than the dip table in the almanac predicts. This would not be an
    uncommon state of affairs.
    When our observer corrects his sextant altitude by subtracting the dip
    taken from the table, he won't subtract quite enough. If the corrected
    altitude ought truly to be 30deg 00', our observer will make it 30deg 01'.
    And exactly that same error would apply to any altitude of any body
    measured at that time, whatever its altitude and whatever its azimuth.
    The observer needs to deduce the angle-in-the-sky between these bodies from
    those two altitudes. Because they are both in a straight East-West line
    through the zenith, the arithmetic is simple and obvious. There are just
    three relevant possibilities-
    1. They are both East of the observer, rising toward the zenith, and at the
    same azimuth of 90deg. In that case the errors in the altitudes of the two
    bodies will both be the same amount and in the same direction. The angle
    between the two bodies will be taken by subtracting their altitudes, so the
    dip error will cancel out, exactly. It doesn't matter a fig what those
    amplitudes are.
    2. They are both West of the observer, setting at an azimuth of 270deg.
    Exactly the same arguments apply.
    3. One is East of the zenith, at azimuth 90deg, and the other is West, at
    270deg. Now the angle between the two bodies is given by taking 180deg -
    (alt1 + alt2). In this case, it's obvious, isn't it, that the dip error
    will now give rise to a 2 arc-min error in the resulting angle, roughly
    corresponding to a 4-minute error in GMT and therefore to a 1deg error in
    the longitude (or 60 miles at low latitudes). This would be true for any
    combination of altitudes of the two bodies.
    [Note. As long as one of the bodies is at an altitude greater that 60deg,
    there's a trick the observer could play here to get around this problem. He
    could measure one of the altitudes by a back-observation, using the fact
    that the sextant can measure altitudes up to 120deg. This involves facing
    away from the body being observed, so that it's somewhere up behind his
    head, and measuring up from the horizon in the opposite direction to the
    azimuth of the body: then subtract the two altitudes.]
    The argument above illustrates the principle in a simple situation, but it
    remains valid in more complex cases. The effect of dip errors will be
    greatest when the azimuths of the two bodies are furthest apart, and will
    tend to zero in cases where the azimuths converge, irrespective of the
    values of the two altitudes.
    So I doubt that there is any virtue at all in waiting for a situation where
    the two altitudes are the same. It's the azimuths that matter.
    Is that argument convincing? If not, please argue back.

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