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Re: GMT from Moon and Body Observed Altitudes...??
From: Wendel Brunner
Date: 2019 Feb 20, 09:27 -0800
The question is raised: Can you determine longitude and GMT just from observations of the altitudes of the moon and a body near the ecliptic (like the sun) without measuring the lunar distance between the two bodies? The answer is: Yes, those altitudes geometrically and mathematically determine the longitude and time of the observer, but it is a very inaccurate and impractical way to do it, and sextant observations are not precise enough for this approach. Of course, you need an almanac that gives very precise values for the GHA and declination of the bodies. The Nautical Almanac, giving values only to the nearest 0.1' - 0.2' or so, is probably not precise enough.

To determine longitude on a steadily spinning planet requires a clock. In the absence of a timepiece on the vessel, the heavens provide a number of clocks which can be read from astronomic observation. The most practical celestial clock - really, the only practical one - is the moon. The moon passes through its cycle of phases moving 360 degrees with respect to the sun in 28 days, or about 30 minutes of arc per hour. If you can measure the distance between the sun and moon to within 0.5 minutes of arc, you can determine time to within one minute, or longitude to within plus or minus 15 minutes.  But it is really hard to make sextant measurements to much better that 0.5', no matter how good you think you are. And you can see that the faster the celestial clock moves, the more accurately one can determine the time using a sextant. The moon is about as slow a clock as one can use.
Determining the lunar distance and the altitudes of both bodies simultaneously simplifies the calculations, but is observationally awkward and not necessary. Turns out, any two of the three values of Lunar Distance and altitudes of the moon and the other body, and the time between the observations, are enough to determine the GMT and Longitude of the observer, and that solution can be calculated using an iterative approach (   https://www.starpath.com/resources2/brunner-lunars.pdf ).
The article in the link shows how to calculate GMT and Longitude using the LD and the altitude of one body. To see how the two altitudes determine GMT, consider the following thought observations: The altitudes of two fixed stars do not determine longitude. The altitudes of both stars for one observation are the same as the altitudes of those bodies one (sidereal) hour later and 15 degrees to the West, or at any linked combination of Longitude and time. Two star observations without a chronometer cannot determine Longitude - as is well understood. However, the respective altitudes of the moon and say a star near the ecliptic or the sun do change over time and longitude. A (sidereal) hour later and 15 degrees to the west, the star will have the same altitude, but the moon's altitude will have changed as the moon moves in its orbit with respect to the star. This change in altitude can be calculated from almanac data in a straightforward manner, and used to calculate GMT and Longitude by the iterative methods described in the article.
However, this method won't be practical to find Longitude with a sextant. The change in Lunar altitude with time is too slow to get a useful time with a sextant measurement. The rate of change of altitude will be the rate of change of LD, multiplied by something like the sine of the angle the moon's orbit makes wth the horizon - a number considerably less than one - and the Longitude error will be correspondingly magnified.
BTW, trying to measure the bodies' Zn's with a pylorus will not work at all. Those Zn's have to be calculated to 0.1' or so from an assumed position and time.
So the conclusion is: To determine Longitude by the Method of Lunar Distance, you actually have to measure the LD, and one other identified celestial body. There is no other practical way to do it.
Wendel Brunner - wendel.brunner on gmail.
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