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    Re: Luni-Solar Distance
    From: Paul Hirose
    Date: 2010 Oct 30, 15:21 -0700

    I wrote:
    > George's lunar distance is observed from 0°N 0°E at midnight, and also 
    > from 0°N 5°E at 00:15:17 UTC. There are an infinite number of such time 
    > / longitude pairs. Without some additional constraint, there's no way to 
    > know which one is correct.
    But as George has already noted, a time sight resolves the ambiguity. In
    this case, a time sight would have measured GHA Aries = 183.538°. (I'll
    use decimal degrees to make the computation easier.)
    Let's say the chronometer is 5 minutes slow and the longitude 1° off, so
    the lunar distance (36.815º, Regulus to near limb) was observed at
    assumed 2005-03-25 23:55:00 UTC from assumed 0°N 1°E.
    At the assumed time and place, calculated lunar distance = 36.773°. This
    value changes +.0052° per minute of time and -.016 per degree of
    longitude east.
    At the assumed time and place, calculated GHA Aries = 183.286°. This
    value changes +.25° per minute of time and +1.00 per degree of longitude
    To make the calculated lunar distance and GHA Aries equal to the
    observations, the increments are +.042° and +.252°, respectively. The
    unknown increments in time (∆t) and longitude (∆λ) to attain this goal
    may be expressed in a pair of linear equations, with coefficients from
    the paragraphs above.
    .042 = .0052∆t - .016∆λ
    .252 = .25∆t   + ∆λ
    The solution is ∆t = 5.00 minutes and ∆λ = -.999 degrees, which we know
    to be the truth.
    GHA Aries is linear with respect to time and longitude, so the second
    equation is practically perfect. Strictly speaking, the .25 coefficient
    ought to be multiplied by 1.002738 since the rate is sidereal, not
    solar. But I believe it's not worth the bother because lunar distance is
    not really a linear function of time and longitude. In this case it was
    close enough. With larger initial errors this may not be true, and the
    process would have to be repeated.
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