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A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
Re: Exorcizing the Evil Effects of Parallactic Retardation (Modified by Fred Hebard)
From: Fred Hebard
Date: 2004 Mar 20, 16:10 -0500
From: Fred Hebard
Date: 2004 Mar 20, 16:10 -0500
This is a repost of my previous message, with the table included as text in the body of the file, as well as a pdf, as requested by George Huxtable. (The pdf was made from a jpg, so the text is not editable). My previous version of the table contained a blunder, which, however, did not affect the net result. The blunder was that my ephemerides for Jupiter were advanced by one hour. I guess I should have noticed something was up when the distance between the Moon and Jupiter was reputed to have changed 17 degrees in 5 hours! This time, I double checked the calculations with the U.S. Naval Observatory site's computer (http://aa.usno.navy.mil/). In January, George Huxtable surprised us all by saying that he no longer felt that parallactic retardation affected the accuracy of observation of time by lunar distance. I've finally had time to look at this numerically, and, by golly, George is right, which should come as little surprise to most of us. The table below gives some simulated observed distances between the Moon and Jupiter on 1/14/04 at 1*31.3'S latitude and 0*0.0'W longitude. The altitudes of the Moon and Jupiter, and the Moon's refraction, parallax and semi-diameter plus Jupiter's refraction were calculated using the time in the first column. The horizontal parallax of the Moon was taken from the Nautical almanac and used to calculate the real parallax for the Moon at that altitude and latitude, using Young's Method of Clearing that George Huxtable presented to the list in 2002-2003. Observations near the equator give the greatest "parallactic retardation," combined with the time when the moon is near zenith, which is 5:00:01 hours in the table below. The retardation can be seen by comparing the difference between the "observed sextant reading" of lines 1 and 4, 19'34" of arc, with that of lines 5 & 8, 29'52" of arc. The Moon is only moving through 19'34" of apparent arc between 5:00:01 hours and 5:59:59 hours (lines 1 & 4), when it is near zenith, compared to 29'52" of arc between 0:00:01 hours and 1:59:59 hours (lines 5 & 8), when it is near the horizon. The thinking has been that if the moon is moving through such a smaller swath of apparent arc (19'34"), then any errors in measurement of distance will be magnified. If the error is 30" of arc, one will be measuring to only 1 part in 39 (30"/19'34"), rather than 1 part in 60 (30"/29'52"). This is the supposed "evil effect of parallactic retardation." However, George has been trying to point out to us that there are two components to measuring a lunar distance, one the distance itself, and two the altitude of the bodies. The Moon's parallax is fixed by observation of its altitude, not the distance, and it is the large shift in the _COMPUTED_ parallax between 5:00:01 and 5:59:59 that gives rise to the retardation. Between 5:00:01 and 5:59:59, the computed parallax of the Moon increases from 3.2' to 11.6' of arc, almost a 4-fold difference. In contrast, between 0:00:01 and 0:59:59, the parallax decreases from 56.3 to 51.2, only a 10% difference. But if one holds the parallax constant, in this table by holding the assumed time constant, such as at 5:00:01 between lines 1 and 2, then the observed sextant reading has to increase by 34'7" of arc, from 22*28'12" to 23*2'19", to move the time by lunar up by one hour. You can also see that the observed sextant reading decreases 34'27" between lines 3 and 4 of the table to move the lunar time back one hour from 5:59:59 to 5:00:01. The error has dropped back to 1 part in 60. AND IN A REAL LUNAR, PARALLAX IS HELD CONSTANT BECAUSE IT IS FIXED BY THE ALTITUDE MEASUREMENT. So this is what George has been trying to tell us. Perhaps he can explain these numbers more clearly than I have; but it was the numbers themselves that convinced me. Fred Hebard Moon-Jupiter at 1d31.3'S,0d0.0'W on 1/14/04 time assumed observed sextant time by Hc Moon Refraction Parallax SemiDiameter Hc Jupiter Refraction to calculate reading lunar decimal Moon Moon Moon decimal Jupiter HPM, SD, etc degree min sec degrees min min min degrees min 5:00:01 22 28 12 5:00:01 86.85463 0.1 3.2 16.1 70.77158 0.3 5:00:01 23 2 19 5:59:58 86.85463 0.1 3.2 16.1 70.77158 0.3 5:59:59 22 13 19 5:00:01 78.61850 0.2 11.6 16.1 56.36603 0.7 5:59:59 22 47 46 5:59:58 78.61850 0.2 11.6 16.1 56.36603 0.7 0:00:01 20 28 53 0:00:01 14.15560 4.1 56.3 15.8 32.29832 1.6 0:00:01 20 58 45 0:55:53 14.15560 4.1 56.3 15.8 32.29832 1.6 0:59:59 20 28 53 0:04:27 28.69067 1.9 51.2 15.8 47.15395 0.9 0:59:59 20 58 45 0:59:59 28.69067 1.9 51.2 15.8 47.15395 0.9