# NavList:

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

**Re: Lunars.**

**From:**George Huxtable

**Date:**2001 Jul 04, 11:36 PM

Nigel Gardner asked- >Anyone out there with a good algorithm and process for longitude by lunars? > >NG ============== I am aware that what follows is not a full answer to Nigel's question. It proposes a successive-approximation procedure for correcting the chronometer error: not finding the longitude directly, but a first step toward it. It's a procedure which could be implemented on a computer, but mould be quite inappropriate to other situations such as hand-calculation using tables. The produre described below has not been written or tested by me, so it is rather a tentative suggestion. Some details have been omitted, such as automatic decision of which limb of the moon is the appropriate one to measure to. It's quite possible that there errors or ambiguities, and I would be grateful if any list member would bring them to my attention. The proposal has been stood up to be shot down, if necessary. ============== I would treat the problem in its simplest form to start with, not attempting to establish longitude immediately, but simply to find the chronometer error. Once this error is known, longitude can then be deduced by measurement of an altitude of a body that's well away from the meridian, as a separate problem. To find the chronometer error, the observer only needs to take the lunar distance and needs to measure no altitudes at all. The necessary altitudes can be computed rather than measured, so no horizon is needed. Star-moon distances can then be measured in pitch-darkness, when stars are bright. I would presume a rough initial position (lat and long) and an assumed time of the measurement: probably the chronometer reading with its unknown error. I know of no way of expressing time as an explicit funtion of lunar distance; only how to calculate lunar distance in terms of time. So the method will be one of successive approximation. The accurate position in the sky of the moon and the other body must be calculated, to a precision of 10 sec of arc or so, for the assumed time. For this I would use the methods given by Meeus, in his Astronomical Algorithms, using all the many coefficients that he provides. These sky positions should be converted to altitude and azimuth, as seen from the assumed lat and long. These are the positions the bodies would be seen at if viewed from the centre of the earth, and if our atmosphere did not refract. Correct these two altitudes for the effect of refraction and also parallax of moon and sun (augmented parallax in the case of the moon). Note that these corrections are to be made in the opposite sense to what we are used to, because we are aiming to calculate the direction in the sky the observer would see the body if he were to measure its altitude, by correcting the geocentic position. Not the other way round, as we usually do it. It may be worthwhile to adapt those corrections slightly because we are using a slightly different angle as the argument, but this is a minute adjustment. Now we calculate the angle-in-the-sky between these two apparent bodies by one of the standard spherical-geometry formulae. Exactly the same as calculating the great-circle distance between A and B on a spherical earth. This is the calculated lunar distance. Note that in this process we have not needed to "clear" the lunar distance of parallax and refraction in the complex way that is usually called for; it was simple addition and subtraction. Now we take our measured lunar distance, corrected for index error. We allow for the semidiameter of the moon (remembering which was the appropriate limb) and if called for, the Sun. These semidiameters and the parallaxes used previously are taken from the calculated earth distance, which was a by-product of the Meeus calculations. After all this, we have a measured lunar distance and a calculated lunar distance to compare. The difference between these quantities, if we divide by the angular velocity of the moon in its orbit, is roughly the error in the chronometer. Adjust the assumed time by this quantity, and if necessary repeat the whole calculation again, using the new assumed time. At each iteration the time-error will shrink and you choose when to stop the process. George Huxtable ------------------------------ george@huxtable.u-net.com George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. Tel. 01865 820222 or (int.) +44 1865 820222. ------------------------------