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

**Re: Correction to measuring Octant instrument error**

**From:**George Huxtable

**Date:**2000 Sep 21, 2:27 PM

Clive Sutherland has suggested using the angle-in-the sky between pairs of stars to correct the scale on Bill Murdoch's octant. I've never tried measuring angles between stars, but wouldn't doubt that a more experienced observer than I am could accomplish such a feat, especially from a firm platform on land. All uncertainties connected with observing the horizon would then, as Clive says, be removed. I'm glad that Clive has corrected an initial error and agrees that spherical geometry has to be used to work out the angles between stars. However, another error needs to be corrected, when Clive states- >Several angles on each pair would be measured and averaged,(just before to just >after the calculated Meridian Pass) to ensure that the two stars of the >pair will >have the same Altitude. >No correction will then be needed for Refraction. It's true that the AMOUNT of the refraction correction would be the same for two stars, both at the same altitude. However, that doesn't imply that the angle between those stars is unaffected by the refraction. That would be true only for two stars which were very nearly at the same azimuth. For two stars at different azimuths, the trouble is that the refractions, though the same in amount, are in quite different planes. This becomes obvious when we take an extreme example. Consider two stars which both, for a certain observer at a particular moment, have a calculated altitude of 20 degrees (a zenith angle of 70 degrees). However, refraction would cause both stars to appear to be a bit higher in the sky, giving each (with a refraction taken from Norie at 2.6 minutes) a corrected zenith angle of 69 degrees 57.4 minutes. As yet, we haven't considered how far apart in azimuth these two stars happen to be: now suppose it differs by 180 degrees, so the stars happen to be in opposite directions in the sky (say, one to the East, one to the West). The angle in the sky that the observer sees between these stars is the arc between them that passes through his zenith, so it is 139 degrees 54.8 minutes (the two zenith angles added). So the refraction correction to the angle between these two stars is 5.2 minutes, DOUBLE the refraction correction for each star. If the difference in azimuth was less than 180 degrees, the refraction correction required would be less. The conclusion has to be that for ANY two bodies, in comparing the measured angle-in-the-sky with the tabulated value, the refraction for both bodies has to be taken into account. And to do so the altitude of each body has to be known, either by measurement or from tables. There's no advantage in choosing two bodies with the same altitude. No great accuracy in the altitude is required, because at any altitude, refraction is only a small correction. This is the problem that faced navigators of the late-18th and 19th century when obtaining longitude from a Moon-Star or a Moon-Sun distance; that of "clearing" the Lunar distance, or correcting it for the effects of refraction of the two bodies and (for the Moon) the greater effect of parallax. After that, the semidiameters had to be allowed-for. The problem was that the Lunar Distance tables had to be suitable for use from anywhere on the Earth, so they were compiled on the basis that the angles between the centres of the bodies was shown, as if they had been viewed from the centre of the Earth, and as if there were no atmosphere to refract. Hundreds of papers were written about methods of making these corrections, and dozens of different calculation strategies were employed, using many different sets of lookup tables. It tested to the limit the mathematical ability of the navigators of the day, many of whom would go through the routine motions without having the faintest idea what they were doing. Really, the only calculation tool available to help was that of logarithms, and as these can not deal with negative quanities, the trig. formulae had to be twisted and deformed to fit in with that limitation. This only made the process more abstruse to the ordinary navigator. So a problem that we can solve in a moment with a pocket calculator became an arcane mystery. Getting back to Clive's proposal, I think he is overstating things a bit when he says- >Of course the final proof of the instrument calibration will be the accuracy of >sights taken of the Sun etc as George Huxtable suggests. However this is >the end >point not the starting point of the exercise. All the corrections that we know >and love, necessary for a Sun sight e.g. Height of eye, Dip, Refraction >Semi-diam, Irradiation, all dilute the confidence in the calibration so >obtained Let's take these corrections one by one- The total dip correction IS the height-of-eye correction; they are not separate items. The height of eye should be easy to estimate to a foot or so, and as the major component to dip is entirely geometrical (only the Earth's diameter is involved) it can be calculated precvisely. There remains a refraction-from-the-horizon contribution to the dip which can vary markedly, but seldom exceeds two minutes, and this provides the greates uncertaincy in sextant observation in otherwise-good conditions. The purpose of a well-calibrated second sextant, alongside the octant under test, would be to check whether abnormal refraction at the horizon is occurring. Provided observations are confined to angles over say 20 degrees altitude then the refraction correction will be no greater than 2.6 minutes, and any variation in the refraction from its standard value is unlikely to exceed 0.1 minutes. To calibrate smaller angles than 20 degrees, Clive's proposal to measure angles-in the sky between two stars might well be appropriate. Semi-diameter is a geometrical quantity that is known very precisely and its accuracy is not in question. Irradiation is a supposed effect within the human eye that makes bright objects appear slightly larger than they actually are. It was not universally accepted as real, though a suggested amount (for the Sun's lower limb) was once included in tabulated values for its semidiameter. As a result, an irradiation correction used to be required if measuring to the Sun's upper limb. But in recent years, after the reality of the effect had been questioned, all reference to irradiation has been dropped, as I understand it. So really, no strong reason remains to dismiss the use of Sun observations in the calibration of Bill's octant. I do hope that Bill appreciates all the attention that his proposal is receiving. Whether or not he thinks the advice he is getting is appropriate to his plans, he has raised some intriguing general questions. 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. ------------------------------