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Re: lat/long from meridian passage
From: George Huxtable
Date: 2011 Jan 24, 00:51 -0000
From: George Huxtable
Date: 2011 Jan 24, 00:51 -0000
John Karl noted that Patrick has neglected to correct for refraction. But he has taken the overall Sun correction, for lower-limb, relevant to the time-of-year, to be 14.8', and that correction includes both semidiameter and refraction. So I doubt if there's any omission on those grounds. The wording was somewhat misleading, in saying- "Because I am using an artificial horizon I made no dip or altitude corrections", because the altitude correction for refraction had been included. Jim Wilson has pointed out that "you definitely need to consider north-south vessel movement and declination change". In Patrick's case he is a stationary observer, so the first point doesn't arise. The second will be considered below. Patrick has told us that we need to make no allowance for index error or watch error. Can we take it, then, that any such errors were actually zero, or otherwise, that they have been allowed for first, before noting the corrected values? Patrick has taken Sun declination from the Almanac, quoting S 19� 38.5, though he doesn't state which Greenwich hour this corresponded to. Not having a 2011 Almanac in front of me, I'll assume (perhaps wrongly) that he took that figure from the nearest, and previous, whole hour GMT, 17:00 h. If that's wrong, I hope he will correct me, as it will affect what's stated in the next paragraph.. Then he states "d=.06", which I presume should really be d=0.6. That indicates that the Sun is travelling North, at just over half-an-arc-minute in each hour, which has to be allowed for as a change in declination, from that at 17:00 GMT. He has worked out that local meridian passage will be around 17:17 GMT, so that extra 17 minutes has to be corrected for. That's easy to estimate, as a correction of 0.2', and it's clear that as the Southerly declination is reducing, at 17:17 the declination will have changed from S 19� 38.5 to S 19� 38.3 over that 17 minute interval. This differs from Patrick's value. If we consider the three middle observations to correspond all to the same altitude, we see a difference between their average, and that of the two outer observations, of 9.9'. But my calculation of the true difference between those altitudes indicates that it should be more like 8.7'. Which indicates to me that the observations are not all that precise, but have a scatter in them in the neighbourhood of +/- 0.5', with a scatter of the reflected double-altitudes being twice that. It looks as if there's room for a bit of improvement there, in observing technique. I'll leave it to Patrick to recalculate his latitude appropriately, but it looks as if it will move a bit further still from his GPS value. This leaves some doubt about his three central altitude observations. Timing and longitude. Patrick places a question mark against the 12:17 observation, perhaps because it shows a slightly smaller altitude then others taken a couple of minutes earlier and later. But over that period, of a couple of minutes of time either side of noon, I would not expect to see any change in altitude greater than about 0.1'. Those three observations closest to noon could just as well be regarded as three attempts to measure effectively the same angle, to be averaged accordingly. The differences between them could well be attributed to the inevitable scatter in making such observations. For the same reason, no real value should be placed in the timing of that fourth observation, noted as 12:19:23. Could Patrick really put his hand on his heart and declare that that was the moment when the Sun descended through the same altitude that it had at 12:15? That would be wishful thinking. So I would put no reliance on the averaged timing of the 12:15 and the 12:19:23 observations. Combining their average, with the outer pair taken at 12:00 and at 12:34:20, just dilutes the precision of those outer observations, the only ones to carry real timing information. So forget the timing of the inner pair, just take the outer pair, giving a mid-value of 12:17:10. However, that is the moment of maximum altitude, which is not quite the same as the moment of meridian passage, because the changing declination shifts the peak of the curve. Over the time-period of the observations, 34 m 20s, the Sun's Southerly declination will have reduced by 0.3 arc-min. That in itself would have increased the Sun's altitude by 0.3'. We have to allow for that when correcting the symmetrical point of the altitude curve to find the moment of meridian passage. One way would be to correct each individual point for the changing declination at the moment of observation, then note the time of the resulting peak. Another approach is to determine the time of the maximum amplitude, then apply a correction, as explained in the following extract from a website I put together years ago about Lewis and Clark's celestial navigation- ================ Max-to-meridian time-shift. As explained above, the mid-time between the am and pm observations would give the moment of maximum altitude, which was near, but not quite at, the moment of apparent noon, when the Sun would be on the meridian. The reason for the time-difference is because the Sun's declination is changing slowly during the day. At summer and winter solstices, the declination is unchanging, and the correction is zero. Near the equinoxes the Sun's declination is changing fastest, at about +24 arc-minutes per day in the spring, and at -24 in the autumn. This rate-of-change can be obtained from the difference in dec. values, in the Almanac, on successive days. The daily rate-of-change (with its sign) in arc-minutes per day, has to be multiplied by 0.637(tan dec - tan lat) :the term in brackets always being negative for Lewis and Clark. Appropriate signs for lat and dec are that North is positive, South negative. The result is the amount (in seconds of time) by which the moment of maximum altitude precedes the moment when the Sun is on the meridian (or apparent noon). This correction can take values up to about �17 seconds, for L&C's circumstances. For them, maximum altitude always preceded apparent noon, from Summer solstice to Winter solstice, and followed it the rest of the year. ===================== In Patrick's case, the daily rate of change of declination, taken from the Almanac, is +13.9', and this results in a time correction of -9.8 seconds. So taking the maximum altitude to be at 12:17:10, that would make the meridian passage to be at 12:17:00. Which would reduce the Westerly longitude by 2.8' to put it at 76�21.8', in fortuitously good agreement with the GPS value of 76� 21.7' I should point out that the name of this thread, "lat/long from meridian passage", is misleading. The lat certainly comes from the moment of meridian passage. The longitude measurement is only possible becaused it's based on two observations taken WELL AWAY from meridian passage, and indeed, the further they are spaced, in time, from meridian passage, the better the result will be. This is a point that I have made many times before, and no doubt will need to do so into the future. George. contact George Huxtable, at george{at}hux.me.uk or at +44 1865 820222 (from UK, 01865 820222) or at 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.