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    Re: Lunar Distance Puzzle
    From: Frank Reed
    Date: 2011 Aug 14, 19:20 -0700

    Dave Fleming, you wrote:
    "My first reaction was that time is easily found from this data but position is not."

    Or the other way around! At a given instant of time (some known time), a pair of lunar distance measurements gives a position fix. It's perhaps easiest to see why this would work by imagining measuring angles between stars and some artificial satellite in low Earth orbit. Since an artificial satellite moves very quickly, we could do that photographically. You take a one-second exposure of some Sputnik with a fast camera recording the exact time. You examine the resulting image, correct as necessary for field distortion, then count pixels and convert to angles to get the "lunar distances" between this Sputnik and two stars, e.g., Vega and Altair. Given a known, up-to-date ephemeris for the satellite, there is only one place (well, two, but easily distinguished) on the Earth where the satellite could have exactly those angular distances from this Sputnik. And if we can get angles accurate to a tenth of a minute of arc, for a satellite 300 miles up, the resulting position should be accurate to about 50 feet. We can do exactly the same thing with the Moon and lunar distances. The Moon advances in its orbit much more slowly so we can use a good sextant instead of a camera. With some effort making sure that the instrument is properly adjusted and by averaging four distances for each star, we can expect angles accurate to a tenth of a minute of arc here, too. But the Moon is 238,000 miles away so the expected accuracy (one s.d.) is 6 nautical miles. Not very accurate but potentially useful in some circumstances and significantly this can be done without any horizon whatsoever. No altitudes are measured. This was not done historically --more on this below.

    You added:
    "Each lunar distance measurement is a determination of the time. The Vega-Lunar is most important as the moon is moving essentially straight toward Vega."

    I'm guessing you just mis-spoke here. The Moon never moves anything like straight towards Vega since Vega is well off the ecliptic. Of the three stars given in DW's example, Antares is closest to the ecliptic, which is why it was included as a standard lunars star historically. The Moon moves nearly straight towards/away from Antares unless it is very close to it in the sky. Altair is also near enough to the ecliptic that it can sometimes be used for the determination of time by lunars but only when the Moon is well away from that part of the ecliptic that's closest to Altair. In the example case that Dave Walden described, both Vega and Altair were nearly perpendicular to the path of the Moon's motion across the celestial sphere, so for determining GMT, they would be nearly useless.

    I should add here that historically lunars were used only for determining Greenwich Time. Anything else is a modern variant. The idea of using lunars to determine position turns up very briefly, marginally even, in the published literature in the twentieth century, but it's something that we're discussing here primarily because I myself introduced it on NavList back in 2008 (originally). On the first pass, I saw it as a means of determining the Moon's altitude when no horizon was visible, since the altitude of the Moon compared to a star varies by nearly a full degree from horizon to zenith thanks to its large parallax: observe the parallax and you can figure out the altitude using the Moon altitude correction table in reverse. In fact, though, these lunar distance measurements at KNOWN time yield a complete position fix. Essentially with a pair of lunar distance observations, we're determining the exact apparent (topocentric) location of the Moon's center on the celestial sphere. It will only have that position from certain places on the Earth. In effect, this pair of angles generates a line of position which emanates from the Moon's center and touches the Earth's surface at one single point. That's the observer's position subject to an error of 6 nautical miles for every tenth of a minute of arc error in the observed angles. Notice that this line from the Moon's center will strike the Earth at a shallow angle when the Moon is low in the sky. If the Moon is at a low altitude, the accuracy of the position is reduced to about (6 n.m.)/cos(altitude) in the direction along the Moon's azimuth from the observer's position, but it remains accurate to 6 n.m. (with no adjustment) in the direction perpendicular to the Moon's azimuth. So if you have a case where the Moon is low in the sky all night long, you're better off doing a running fix with observations separated by a few hours.

    -FER

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