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    Re: Real accuracy of the method of lunar distances
    From: George Huxtable
    Date: 2004 Jan 7, 20:40 +0000

    Frank Reed wrote-
    >Regarding rates of change in apparent lunar distances, George Huxtable wrote:
    >"Yes, that can happen, particularly when the Moon is passing nearly overhead
    >in the tropics."
    >...passing nearly overhead ...and well away from the meridian. If the Moon is
    >close to the meridian, the effect on a lunar from rate of change of parallax
    >would not be serious even if the altitude is high.
    I hope we are not talking at cross-purposes here. I stand by what I said.
    The rate of change in the parallax correction to lunar distance is at its
    MAXIMUM when the moon passes through the meridian, even though the parallax
    correction itself may then be small at high altitudes (or zero, even, when
    the Moon passes directly overhead).
    The parallax correction to the lunar distance is due to the component of
    the Moon's parallax taken in the direction of travel of the Moon. One of
    the effects of the clearing process is to resolve this component. It's this
    component which is changing most quickly when the Moon passes through the
    meridian. Not because the AMOUNT of parallax is changing quickly then: it
    isn't, it's passing through a stationary value, a minimum. But because the
    DIRECTION of parallax is changing quickly then. It's always a vector
    pointing toward the zenith, and the closer the Moon passes the zenith, the
    faster the direction of that vector will change. And it's the effect of
    that rapidly-changing vector, resolved along the Moon's path, that causes
    the effect of parallax on the lunar distance to be changing most quickly as
    the Moon passes meridian.
    To get a simple picture of how it all works, imagine an observer on the
    equator, and the Sun and the Moon both with zero declination. The Moon
    rises exactly in the East and passes right overhead to set in the West. The
    Sun follows exactly the same path, a few hours later. We neglect the Sun's
    parallax, but the Moon is depressed by parallax by about 1 degree on
    rising, and the parallax then follows (cos alt), so there's zero parallax
    when alt = 90. It then changes sign (from being displaced Easterly to
    Westerly) and continues to follow that cos alt curve, increasing Westerly
    until it sets in the West, depressed again by about 1 deg. Because
    everything's in a straight line, the lunar distance is found by simple
    subtraction rather than by a complex "clearing" process. And it's clear,
    isn't it, that the effect of parallax on that distance is changing fastest
    when (cos alt) changes fastest, as the Moon passes overhead, at the point
    where (cos alt), and parallax are zero.
    >And wrote:
    >"Measuring lunar distance to the Sun or a planet, rather than a star, slows
    >the change of
    >lunar distance a bit further still."
    >But not always. It increases the rate of change if the Moon happens to be on
    >the other side. The Sun chases a waxing moon.
    Yes, always. The Moon moves with respect to the starry background at a rate
    of 360 deg per (sidereal) month. The Sun moves with respect to the starry
    background at a rate of 360 deg per year, IN THE SAME DIRECTION as the Moon
    does. So with respect to the Sun, the Moon ALWAYS moves more slowly, by
    about 1 part in 12, than it does with respect to the stars. No matter which
    side of the Sun it's on. On one side, the Moon-Sun angle is decreasing with
    time. On the other side the Moon-Sun angle is increasing with time. But
    however and whenever you measure it, the Moon-Sun angle will change about
    8% more slowly than does a Moon-Star angle. And of course Moon-Planet
    angles will sometimes change more slowly still.
    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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