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    Re: Friendly challenge : Jupiter Lunar Exercise 09 Feb 2011
    From: Frank Reed
    Date: 2011 Feb 11, 21:34 -0800

    Antoine, you wrote:
    "It thus replicates our Illustrious Forebears Lunars environment as follows : although their Watches indications did differ from UT (by an amount which is what they needed to determine), on the other hand all intervals elapsed between their observations were to be recorded with the utmost care."

    Well, sure, to some extent. Historically, shooting lunars at sea, a navigator, if working alone, would have shot a sequence of sights something like this:

    Before nightfall:
    --Sun altitude (or several) to set the watch to local apparent time (or local mean time if desired, by applying the equation of time).

    Then after dark, recording the local time from the watch for each sight:
    --Moon altitude
    --Jupiter altitude
    --several lunar distances (Jupiter to Moon angle)
    --Jupiter altitude
    --Moon altitude

    Then you could just average the pair of Jupiter altitudes and average the pair of Moon altitudes to make them coincident, or nearly so, with the average of the several lunar distances. The Moon and Jupiter altitudes have no purpose except to provide inputs to help clear the lunar distance. The sights you took for the altitudes of the Moon and Jupiter are rather a long way from the time of the lunar distance sights which makes extrapolation in this fashion tricky and prone to error.

    Instead of going the historical way, we can treat this as a modern navigation problem and focus on crossing lines of position. It's three-dimensional instead of two: we use three observations to get three unknowns, namely latitude, longitude, and GMT. You've got a series of Sun altitudes around 00:03 watch time, a series of Moon altitudes around 01:43 watch time, a series of Jupiter altitudes around 01:49 and finally a set of Jupiter-Moon lunar distances around 02:07 by the watch. All of these together can be "shaken out" to yield a latitude and longitude (and necessarily set the watch to GMT) but it's a process that would have been rather exotic historically and it would have been an unusual navigator who would have known how to get the latitude from all of this, too, at least back before c.1850 when lunars were commonly used.

    Folks who have read Chichester (from what I hear) or Letcher will know how to do this in terms of a Moon altitude. If you have two altitudes of other bodies, like the Sun and Jupiter in your case, and we also have an altitude of the Moon, with an unknown but constant watch error affecting all of the observations, we can adjust the watch error until all three sights coincide. The Sun and Jupiter sights would, of course, yield a standard two-body fix for some estimated value of GMT. And so long as the watch error doesn't amount to more than an hour or a few, imperfect knowledge of GMT would affect that fix only by shifting it east or west. The latitude of the crossing point of the Sun and Jupiter LOPs would only change a small amount for a fairly significant change in GMT. But the Moon is different. Since it's moving more rapidly relative to the stars on the celestial sphere, an LOP for the Moon will move when relative to the other LOPs we clear it with a different GMT. The idea then is to select a GMT such that the Moon LOP crosses the fix from the other two bodies. Unfortunately, this only works when the Moon's motion in the sky is relatively perpendicular to the horizon, which can be judged by looking for situations when the Moon's "horns" (equivalently the line through the Moon's north and south poles) are relatively parallel to the horizon, or at least within 45 degrees or so. But in this case that condition is not met, so we can't use the Moon's altitude to get GMT. We have to turn to the lunar distances...

    Like an altitude sight, every lunar distance makes a line of position for a given instant of GMT, as I've detailed on NavList and other places a number of times. This is not a standard celestial LOP. It's a different beast, and it was apparently quite unknown back in the heyday of lunars. The concept is simple enough: at a given instant of GMT, navigators anywhere along this line of position would measure the same value for the lunar distance. We can even use pairs of lunar distance LOPs to get a position fix without using any horizon at all. The existence of a "lunar distance line of position" means we can do the same trick as above and simply vary the GMT until the lunar distance LOP intersects the fix from the other sights. The rest is just calculation and plotting. When all is said and done, I get a position about two miles west of the Ile-d'Yeu off the western coast of France, Latitude: 46d 42.4'N, Longitude: 2d 26.0'W, GMT: 18:07:30 (watch error 16:00:12). I would put error bars of roughly +/-1' on the latitude and +/-8' on the longitude. Since that error ellipse covers the west coast of the island with an airport right close by, and you said you were on the shore of an island apparently facing west, and you flew there, that all seems consistent.

    How did I do?

    -FER

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