# NavList:

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

**Re: ] Re: Possible limitaion for lunar distance measurement**

**From:**George Huxtable

**Date:**2009 Mar 3, 22:05 -0000

One reason for perceived imprecision in using the Dunthorne formula for clearing lunar distances may be in its implementation by tables rather than in the formula itself. It appears to me that the formula itself is exact, for a spherical Earth, and provided the correction that Herbert refers to is applied (it usually isn't) it remains exact for an oblate Earth. Another small adjustment could be made, for the effect of refraction on the correction for semidiameter(s), to the between-limb spacing, to get to the between-centres spacing. But these corrections apply to any clearing procedure, not just to Dunthorne's, if extreme precision is being sought.. But there's a factor that's used with the Dunthorne procedure, and with Borda's and other clearance procedures also, that is usually obtained from a table, and isn't absolutely precise. It appears in the Dunthorne formula , quoted by Kent, which was- cos D’ = cos Δ’ + cos m’ * cos s’ * ( cos D – cos Δ ) cos m * cos s as the multiplying factor- cos m’ * cos s’ or as its log, to add. cos m * cos s where the dashed quantities are the apparent altitudes of Moon and Sun, and the undashed quantties are the true altitudes, after they have been corrected for parallax and refraction. As a multiplier, it's always slightly less than 1, so as a (nautical-style) log, it's always around 9.99xxxx , that is, to six places. If could be calculated, to the necessary precision, directly from the cosines of those angles, but instead, that is usually short-cut by using an appropriate nautical table. I'll describe this as it appears in Raper's "Practice of Navigation" (my edition is 1864), as Table 73, "The logarithmic difference", which covers 9 pages. You look up the Moon altitude, to the nearest 10', in the range 3º to 90º, against the Moon's horizontal parallax, to the nearest arc-minute in the range 53' to 61', using an auxiliary interpolation table to get the resulting log to the sixth decimal place. By far the most important part of this factor results from Moon parallax, with a bit for refraction, and this table takes no account of non-standard atmospheric conditions, for refraction, stating that it applies to 30 inches of Mercury pressure, and 50ºF. Of course, any allowance for atmospheric differences would be no more than a correction to a correction, and would have little effect except in extreme conditions. But if a navigator found himself working well away from those standard conditions, there would be no way to allow for the fact. Then Raper adds a pair of liitle one-line tables at the foot of each page, giving a small correction to add; either from a table for the Sun (for Sun parallax and refraction), or from another for a star (for refraction alone).. For the Sun this varies between 17 units (in the 6-fig log) at low Sun altitudes, through a minimum of 7 units around 14º, and back up to 18 units at 90º. In "Norie's Navigation" (my tables date from 1914) , the same log factor appears as "Logarithmic difference" in table 39. Dunthorne's original explanation, from 1766, is to be found in Maskelyne's first "Tables Requisite", "A new method of computing the effect of refraction and parallax upon the Moon's distance from the Sun or a fixed star". In that, there is no table provided for correcting this factor for the effects of the Sun's refraction and parallax . Instead, Dunthorne seems to have allowed for a fixed amount, independent of Sun amplitude, and included it in the main table. Although Dunthorne doesnt explain this, the justification for it might be as follows- The effects of Sun parallax (which increases with altitude) and refraction (which decreases with altitude), compensate to some extent, so the sum of the two, varies with altitude by an amount that, if neglected, doesn't cause any great inaccuracy. Dunthorne was simplifying the procedure for his readers, when making Sun observations, but at the expense of precision, reducing that precision to correspond, effectively, to the use of 5-figure tables rather than 6 figures. I just wonder if that may be the source of the imprecision that the German handbook associates with the "Dunthorne method".. George. contact George Huxtable, at george@hux.me.uk or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK. --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To unsubscribe, email NavList-unsubscribe@fer3.com -~----------~----~----~----~------~----~------~--~---