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