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    Re: A Lunar theory question
    From: Frank Reed
    Date: 2010 Apr 7, 11:41 -0700

    Jeremy you asked:
    "As I have been shooting lunars over the past two years or so with different comparing bodies, i often wonder how much does the difference in relative declination matter to lunar accuracy. For example how does the accuracy of the lunar differ if the moon's declination was 25 degrees south, and the sun was 20 degrees north, verses both being of similar declination."

    Declination per se doesn't matter. The difference in *ecliptic* latitude does matter.

    "Is this the reason that we choose bodies close to the ecliptic for lunars?"

    Yes, but you'll notice that Altair, one of the traditional lunars stars from the very first editions of the Nautical Almanac in 1767, is well away from the ecliptic. Let's ignore the five degree inclination of the Moon's orbit to the ecliptic and picture the Moon running right along the ecliptic. It's moving along at an average speed, relative to the celestial sphere, of a bit more than 30 minutes of arc per hour (equivalent to 0.1 minutes of arc in 12 seconds of time --about the best resolution of GMT you can expect from lunars). At perigee it's faster; at apogee slower. Clearly if there's a star aligned dead ahead on the ecliptic, then the rate of change of the angular distance from the Moon to that star will have the maximum value and the ability to resolve GMT from the observation is maximized. Now suppose the star is out of line. Let's call the direction measured from the leading edge of the Moon right along the ecliptic zero degrees. The Moon's poles will be near 90 degrees, and the center of the trailing limb will be near 180 degrees. If a star is 10 degrees out of alignment with the leading or trailing points, the rate of change in the lunar distance would be reduced by a factor of cos(10d) which amounts to less than 2%. This is insignificant. If a star is 45 degrees out of line, then the rate of change is reduced by cos(45) or nearly 30%. That's where we're starting to get a significant reduction in accuracy so that has been the traditional rule. The bearing to the star from the Moon's leading or trailing limb should be no more than 45 degrees out-of-line (see below). This condition can be met for stars that are well away from the ecliptic, like Altair, if the distance is relatively large. Notice that if the Moon happens to be at apogee, the rate of change would be even slower. The safest rule is to look at the actual hourly change in the lunar distance. You want it to be larger than some minimum value. For the distance predictions on my web site, that minimum is set to 20 minutes of arc per hour. If the lunar distance is changing more slowly than that, the distances are considered "not usable". This rate corresponds to a resolution of GMT at a rate of 18 seconds (of time) for a tenth of an arcminute error in the measured distance. Historically, at some point the navigator would have the "P.L. of diff" for the two lunar distances [Note: this is the "proportional log of the difference" in distances for the standard three-hout tabular interval; a "proportional logarithm" of some number x is simply the base ten logarithm of 3/x or equivalently log(3)-log(x)]. In the early period, the navigator had to look up this PL in separate tables. After 1834, the PL Diff was published in the almanacs alongside the distances. The maximum for PL Diff is around 4500 for the same geometric reasons as above. If a navigator was aware of this and had options, it was better to pick an object with a lower PL Diff when deciding when to shoot lunars.

    I said you could picture this condition on the geometry of the Moon and the other body. First, find the poles of the Moon. If the Moon has an appreciable phase, these are very close to the horns of the Moon. Otherwise, you will need to know the Moon well enough to identify its features. Now draw an imaginary X through the Moon' center with the poles in the middle of the open parts at the top and bottom of the X. Stars acceptable for finding GMT by lunars fall in the open parts of the X 90 degrees from the pole. See the attached graphic for a rough idea of this. The yellow X divides the entire celestial sphere into four regions (converging at another X at the point in the heavens opposite the Moon. So long as the star falls in the gray region, it is acceptable for finding GMT by lunars. Naturally results are better closer to the green lines: "dead ahead" and "dead astern" for the Moon's motion among the stars. A star can be a long way from the ecliptic and still work reasonably well. Consider Sirius. As long as the angular distance from the Moon to Sirius is close to 90 degrees, Sirius would fall inside that light gray region. This didn't matter historically since Aldebaran or Spica were always available under those circumstances and they're closer to the ecliptic.


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