# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Altitudes for lunars. was Re: Lunars - Finding Bermuda in 1807**

**From:**George Huxtable

**Date:**2007 May 21, 00:15 +0100

Ken Muldrew made some interesting observations in his posting a week ago, commenting on Frank's account about a passage to Bermuda in 1807 Frank had quoted- | > "As the night approached, the sky became beautifully clear, and shortly | > after sunset I got my sextant to work. Before the twilight was ended, and | > the horizon too faint to admit of the altitudes being taken with accuracy, | > I had observed four or five sets of lunars. and Ken asked- | What do you suppose he means by "four or five sets of lunars"? The almanac | would only have 2 stars for a given date. I don't suppose he would | calculate the distances to other stars without mentioning the fact. Would | a "set" consist of an altitude, a lunar distance, and the other altitude | in succession (possibly with only a single observation of each)? and I wonder whether that was really the case, in 1807. It's true that in the earliest Nautical Almanacs, only one object to the East of the Moon, and one to the West, would be offered on most days, just as Ken says. I have an Almanac for 1864, and by then the Almanac was giving (where available) the Sun, a planet or two, and 5 or 6 stars, some to East of the Moon, some to the West, for the navigator to choose between, or make multiple observations. I don't know from what date that greater choice was offered. Has anyone access to an Almanac from around 1807, to see how many lunar distances were tabulated then? ===================== Ken went on to say- | Frank also mentioned, in reply to George, that he agreed that the | altitudes for the later lunars would not have been calculated without | noting the fact. This brings up the curious difference between those who | navigated on land, and those who navigated at sea during this period, | despite their training (and their textbooks) being essentially the same. | The land navigators always calculated their altitudes. Even when taking | lunars using the sun, where the time sight is from a solar altitude, they | calculate the sun's altitude at the moment of the lunar. They take a time | sight with every lunar, so there is no question of there being any | difficulty associated with measuring the altitudes. They often take | meridian altitudes of stars for latitude, so they seem comfortable enough | sitting about in the middle of the night looking into their artificial | horizon. The fur trade navigators were always far enough North to allow | altitudes with an ordinary sextant. It is something of a mystery. My comment- Perhaps it ought not to be a great surprise. After all, mariners usually had a useful horizon to measure up from, so measured altitudes came rather easily (though perhaps not so easy, at night). Harder for the inland traveller, who needed an artificial horizon to do that job. To see stars in an artificial horizon, a mercury reflector was called for, though a water-surface would do for Sun or Moon. Depending on the observer's latitude, it could well happen that at the convenient time for taking a lunar, one or both of the bodies involved might be above 60 degrees altitude, so that its doubled, reflected, value was out of range of a sextant. So the inland traveller had to be able to cope with the situation where the altitudes couldn't be directly measured, and had to be calculated from the Almanac. However, that is in itself quite a complex matter, which I will discuss after a digression about lunar corrections. First, let's review why the altitudes of the Moon, and the other-body, are needed when a lunar distance is measured. Because the makers of the almanac didn't know where on Earth an observer would be when taking a lunar, they tabulated the lunar distances as though they had been taken by a fictitious observer at the centre of a transparent Earth. The lunar distance measured by a real observer can differ significantly from that tabulated value, mainly because of the altered direction of the Moon, due to parallax. Because the Moon is so near, that parallax can give rise to a correction as great as 1 degree, either way. To a lesser extent, refraction in the atmosphere changes the observed altitude of the Moon and other-body. All these corrections change with the altitude, and can be calculated if the altitudes of the two bodies are known. They combine in a complex way and can be precisely allowed for in the operation famously known as "clearing the lunar distance". We can estimate that in the worst-case, an error of 1 degree in the altitude will give rise to an error of no more than an arc-minute or so in the corrected lunar distance, so that altitudes are not needed to be known to better than about a quarter-degree or so. It's not a precise measurement that is called for, then, but it is vital that such a correction is properly made, because of the high precision that's called for in the lunar distance itself. The necessary altitudes can be measured, up from a real or artificial horizon, or instead they can be calculated, in a similar manner to the way that you get a calculated altitude, when working a position line. It doesn't matter at all how those altitudes are obtained, as long as they are accurate enough. ========================== A lunar observation tells you nothing about your latitude, only longitude, so to obtain an overall position some sort of altitude observation, most easily (if possible) a noon Sun, is called for. So the inland observer has to be competent to use an artificial horizon, if only with the Sun (the easiest body). And we will see that he has also has to take some sort of time-sight, and that again can be an altitude of the Sun, when it's well away from the meridian. We will assume that he carries some sort of watch, the error of which is known to within some fraction of a minute each day. That will allow him to measure time intervals with reasonable accuracy, but isn't good enough to show accurate elapsed time since his departure. Not a chronometer, in other words, but at least a useful watch. We will assume that our observer has had clear enough skies to keep track of his latitude at each noon, and can interpolate latitude at in-between times using dead reckoning. At some time, noted on his watch, our observer took a lunar distance, but was unable to observe the altitudes of Moon and other-body at the time. He will have to calculate them, then. How is he to go about it? The problem is, he doesn't know his longitude (that's what he is trying to find). Nor does he know his true GMT, or the error of his watch on GMT. If he did, it would be a simple matter to find longitude. Instead, he has to derive these things from a lunar distance and a time-sight. At some time, as close as possible to the moment of taking the lunar (within a few hours either way), the observer should take a time-sight, noting the time on his watch. This won't necessarily employ either of the bodies involved in the lunar distance. It will involve getting out the artificial horizon, which can be used with a water-surface if it's the Sun being measured, and taking the altitude of a body when it's well toward the East or West. From that, knowing its declination and the latitude at that moment, it's easy to get the Local Hour Angle (LHA) of the Sun or other body being observed, as has been discussed recently on this list. How did we get the declination, which changes with time, when we didn't know the GMT? Well, for the Sun, it changes rather slowly with time, so you could guess, well enough, at the GMT, to take dec from the Almanac, for a first-try. For stars, of course, dec doesn't change at all. Next, we want to get the Greenwich Hour Angle of the body at that moment of observation. That amounts to the observer's Westerly longitude plus (if the body is somewhat to his West) or minus (if to his East) the LHA of the body, in degrees. But we don't know that longitude. No matter; just guess a trial value that's roughly in the right ballpark. It will all work out in the wash. That will result in a trial value for the GHA of the body. There's no reason to think that it's the true GHA; that would be the case only if the guessed longitude happened to be precise. But that trial GHA will do, for the present purpose. Next go to the Almanac (for the right day - be careful!), look up the column showing GHA for that body, and search for two hourly values that bracket the trial GHA. By reverse interpolation, find a trial GMT that corresponds to it. Again, there's no reason for that to be the true GMT for the timesight, but it will do. Check whether the declination of the body, at that time, corresponds with the initial dec that was chosen before. If it's significantly different you may have to change the value for dec and recalculate. Now, we have a (trial) GMT for the timesight. From the watch, we know the time-gap between the moment of the timesight and the moment of the lunar, which may be before or after. So now, we can get a (trial) GMT for the moment of the lunar. Now we are on familiar ground. From the Almanac, we can now get the GHA and dec of the two bodies involved in the lunar. From the DR, we have the observer's lat at the moment of the lunar. The trial value to take for his long at that moment is the trial long used for the timesight, adjusted according to the shift in long in the interval, if any, taken from the DR. From that lat and long, and dec and RA, we can now easily deduce the altitude of each body, using tables or computer program or calculator, using the familiar formula. At long last, we have calculated the necessary altitudes. The rest of the calculation then follows as for a traditional lunar. We can and should now dismiss those trial values for long and GMT, which will not be used again. Using those calculated altitudes, clear the measured lunar distance, and then compare it with the tabulated lunar distance from the Almanac or some computer ephemeris, interpolating to find out what was the actual true GMT at the moment of taking the lunar. That then gives the error in the watch. From that known error, we can now get the true GMT at which the timesight was taken. The Almanac will provide the true GHA (not the trial value, as before) of the body used for the timesight. Adding or subtracting the LHA of that body, as appropriate, provides the longitude. Looking back on that operation, you can see that calculating the altitudes has not in any way eliminated the need to measure altitudes. What it has done is to allow measurement of altitude of a more convenient body at a more convenient moment, and transferring that information to deduce those altitudes needed for the correction. I have gone through all the steps rather long-windedly to show what's involved, and have little doubt that there are short-cuts to be found by cleverer readers. What I have described are the steps involved using a modern Almanac, which works with GHA. In the heyday of lunars, everything was given in terms of Right Ascension (RA) and the operations were rather different, and somewhat simpler. George. contact George Huxtable at george---.u-net.com or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. --~--~---------~--~----~------------~-------~--~----~ To post to this group, send email to NavList@fer3.com To unsubscribe, send email to NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---