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    Re: GMT from Moon and Body Observed Altitudes...??
    From: Frank Reed
    Date: 2019 Feb 21, 09:24 -0800

    Geoffrey, you wrote:

    George G. Bennett, "Longitude from Lunar Altitudes Simplified", Journal of The Institute of Navigation, Vol. 53, No. 2, Summer 2006. It is a viable idea at lower latitudes, where the ecliptic is pretty much on your Prime Vertical. But at higher latitudes - not so.

    Yes. It's an idea that has been re-discovered many times. There are articles about it back in the early 19th century. The argument back then and also today has been that a longitude by lunar altitude is easier for a competent celestial navigator to understand and also easier to implement. Observing altitudes is the basic task of ordinary celestial navigation, so there's some logic to this. Leverage what you know.

    Geometry is the first problem. You want the ecliptic more or less vertical. The "more or less" here can be +/- 30° with no significant loss in accuracy (accuracy is proportional to the cosine of the angle relative to vertical so at +/- 30° we're only down by 13%, which is minor compared to other sources of error in this process). As you note, Geoffrey, this is best in low latitudes. In higher latitudes, even above 50°, it's still possible (given the +/- 30 condition) but this occurs less often. Even in the tropics, it's not valid all day long. You have to think about the orientation of the ecliptic, at the time of the sight, as you're observing the Moon... and that's not obvious. In fact, there is an easier, observational cue that works in every case. If the horns of the Moon are visually horizontal (within that same +/- 30° range), then the motion of the Moon across the celestial sphere, relative to the other celestial bodies, is nearly perpendicular to the horizon. And that's the key: if we're measuring altitudes instead of a proper lunar distance, then we need to observe the Moon when its motion across the sky has a big impact on its altitude. That happens when the ecliptic is vertical relative to the horizon at the time of sight, or, observationally, it happens when the horns of the Moon are horizontal. And for purists, the latter condition is ever-so-slightly more accurate since it better accounts for the 5° tilt of the Moon's orbit relative to the ecliptic. The opportunities to find the right geometry are common in low latitudes and increasingly rare in mid-latitudes.

    A larger problem with longitude by lunar altitudes is that the horizon is uncertain. With proper lunar distances we can measure the distances, especially with averaging, within a tenth of a minute of arc --assuming a good metal sextant with a moderately powerful telescope and an observer with some practiced skill (like Ed Popko's recent excellent results). By contrast, like all altitudes, altitudes of the Moon have a typical uncertainty due to uncertain dip and anomalous refraction near the horizon of about one minute of arc. This can be managed to some extent by shooting the other body's altitude above the same horizon as the Moon and under similar circumstances. So if the Moon is in the east about 45° high, you would look for a bright star that's also in the east and whose altitude is between perhaps 35 and 55 degrees. This approach can reduce the error to perhaps 0.5 minutes of arc. Note that this method can only rarely approach the accuracy of a proper lunar distance sight. Of course things improve rapidly if we can ditch the sea horizon. These uncertainties in the horizon disappear if you're doing this on land with an artificial horizon, assuming it's been carefully leveled.

    In terms of computation and methodology, longitude by lunar altitudes is simple, too. A popular procedure (see, for example, Letcher's "Self-Contained Celestial Navigation") is to plot the lines of position for the Moon and star(s) twice using two different values for UT/GMT -- maybe your best guess for UT minus five minutes and also your best guess plus five minutes. If you shoot two or three stars, their LOPs will cross as normal, but if the UT is wrong, the Moon's LOP will not cross the fix from the stars. You compare the distance away from the star fix for the two trial UTs that you used and with some simple interpolation or visual estimating you select a UT that would make the Moon's LOP pass right through the fix for the stars. Repeat the calculations with your interpolated UT, and there you have it: the three-body fix that includes the Moon will only work if you have used a reasonably close UT. The longitude and latitude of the fix are then correct.

    Those of you that have backyard artificial horizons, carefully leveled, should give this a try. The next time you see the horns of the Moon more-or-less horizontal, grab your sextant and shoot the altitude of the Moon and then an altitude of a bright star on a similar or opposing azimuth relative to the Moon. Also get a couple of other star altitudes on different azimuths if you can. Record all the sights with a clock reading the wrong time (get a friend to set your watch wrong by some minutes and seconds). When you work up your sights and plot the LOPs, the star sights should all cross in a nice fix. The latitude should be right, but the longitude will be wrong. Ignore that for now, or pretend that your longitude is unknown. Then plot the Moon LOP. It will not pass through the fix from the stars. That's the key! Then adjust the UT/GMT, as described above, until the Moon LOP passes right through the star fix. You will then find that your selected UT is about right, within some tens of second of the correct UT, and that your longitude is also now nearly correct (in error at the usual rate of 1' of longitude for every four seconds of time error in UT).

    I mentioned at the top that this idea of lunar altitudes for longitude has been re-discovered many times. In 1854 an article was published in the MNRAS (Monthly Notices of the Royal Astronomical Society) by a Lieutenant Ashe. He noted that he has seen lunars used to check the chronometers only once in over over twenty years of sea experience --which says a lot all by itself. That statement, from a witness alive and kicking in the era, is telling us that in the Royal Navy, lunars were essentially gone by 1830 or so. And this matches other lines of evidence. The editor of the MNRAS in 1854 wrote a reply that nicely covers the issue:

    The practical objection to using altitudes of the moon at sea, for getting the longitude, is, that the horizon is seldom so well defined as to allow of great accuracy, and that, unless the moon's orbit makes a considerable angle with the horizon, her motion in her orbit may not be shown satisfactorily by motion in altitude. The lunar distance observation is capable of much greater accuracy; and by using stars on both side of the moon, a large portion of the necessary errors of observation are diminished; the motion in her orbit is more favourably shown. The calculations are by no means laborious or complicated; but it must be admitted that great nicety is required to make the observations well, and that the instrument must be of the best kind. On land, perhaps, lunar altitudes would be available in lowish latitudes, as the angle is doubled by the mercurial horizon, and the observation is easy and of great exactness.

    That editor 165 years ago makes another good point that I have said many times, but it's worth repeating. The calculations for a proper lunar distance observation "are by no means laborious or complicated". Many modern sources get this terribly wrong. Lunars are and were easy. The advantage gained by shooting lunar altitudes instead of lunar distances is modest (more familiar process of observation and calculation) while the disadvantage is substantial (considerably reduced accuracy unless an excellent artificial horizon can be used and greatly reduced opportunities for the observation especially in higher latitudes).

    A final thought: the expressions "lunars" and "shooting and working lunars" and "longitude by lunars" should be reserved for cases where lunar distances --angles measured across the sky from the Moon to some other celestial body-- are used to find UT/GMT and longitude. The process described above should be carefully distinguished as "longitude by lunar altitudes". They're related techniques, but the difference are significant.

    Frank Reed
    Clockwork Mapping / ReedNavigation.com
    Conanicut Island USA

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