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Re: Simulating Parallactic Retardation in Lunar Distances
From: George Huxtable
Date: 2002 Nov 3, 22:41 +0000
From: George Huxtable
Date: 2002 Nov 3, 22:41 +0000
I'm grateful to Arthur Pearson for making a useful simulation of the effect of changing parallax of the Moon on the rate at Which the Moon appears to move against its starry background. It has given me confidence in the rather tentative notion that I put forward in "About Lunars, part 4" to this list earlier this year, and dubbed "parallactic retardation". I am now fairly sure that this is a serious effect that the observer of lunars ought to be aware of (and which would have been useful knowledge in the heyday of lunars). The main problem with lunar distances is the slowness of the motion of the Moon against its sky background, of the order of 30 arc-minutes (one Moon diameter) in an hour. The Moon's (geocentric) angular speed varies significantly between apogee (when it's furthest from the Earth) and perigee (nearest), but 33 arc-minutes per hour is a rough mid-value. The faster motion of 19 minutes per half-hour, shown up by Arthur's simulations, is because they coincide, nearly, with perigee of the Moon. Another simulation, made a fortnight later near apogee, would (I predict) show a much slower geocentric motion for the Moon, and though the retardation would then be less in real terms, it would become a greater fraction of the Moon's motion. Though the Moon moves through the stars much faster than any other object in the sky, this rate-of-motion limits the accuracy of a measurement of Greenwich time to something like 2 minutes of time for every arc-minute of error in the lunar distance, and therefore that error of 1 arc-minute in distance results in an error in longitude of about 30 minutes. It is this magnification of errors by a factor of about 30 that is the bugbear of the lunar distance method, and why such extreme precision of measurement is called for. What parallactic retardation shows is that the rapidly-changing parallax in the view of the Moon by an observer on Earth can further slow the apparent speed of the Moon through the stars, and by a very significant amount. There is no such daily-parallax for stars, and very little for the Sun and planets: it affects only the Moon, because it is so close to us. If the apparent motion of the Moon is halved (and in extreme cases that can occur) then the magnification factor of longitude error resulting from lunar-distance error can increase from about 30 to about 60. As Arthur's data shows, the effect of parallactic retardation is at its worst when the Moon is near overhead, and is halved for Moon altitudes of 30�. So, as Arthur points out, a useful piece of advice for navigators by lunars would be to avoid (if possible) those high lunar altitudes, but (I would add) also avoiding low altitudes of 10� or less where refraction errors become significant. It's worth pointing out that these parallax effects doe not introduce any actual ERROR, in themselves. The almanac gives the geocentric Moon position, that is, as seen from an imaginary observer at the centre of the Earth. That is the best the almanac can do, as it has no idea where the observer will be. The corrections made in reducing the lunar distance allow for the effects of refraction and parallax, for an observer somewhere on the Earth's surface, and are capable of doing so very accurately. What the parallactic retardation effect does is reduces the SENSITIVITY of the resulting longitude to the measured lunar distance. HERE'S A MIND-PICTURE I keep a picture in my mind of the concept of lunar parallax, which might help others. I imagine an observer, sitting at the Moon's centre (and you can reduce the Moon to a point for this purpose), looking at the Earth, which is enormous in his sky (subtending about 2�). Around the Earth, the star background will always be visible agains the blackness of space. The Earth's centre will appear to be moving against this star background, making a circuit in a month. The line from the Moon to the Earth is exactly the same as from the Earth to the Moon, but pointing in the opposite direction. 180� away, so these two lines, though facing opposite directions, change in exactly the same way. Now imagine our Moon-man looking, not at the Earth's centre, but at a tiny man on the Earth's surface, who is riding round with the Earth as it rotates. The Earth is a ball which is rolling on its axis as it appears to move around against the stars. But it is rolling "backwards", rather like the wagon-wheels seemed to do on old western films. This will give the Earthman, when on the side of the earth which can be seen by the Moonman, a component of velocity, in the direction of apparent motion of the Earth, which is opposed to that motion, and subtracts from it. The biggest such effect would occur at the point directly below the Moon, from which the Moon would be overhead. And someone on the other side of the Earth, invisible from the Moon except if the Earth was transparent, would have a velocity which added to that of the Earth. So you can see that the man on Earth, as seen by the Moonman, would appear to have a daily "rocking" motion, of up to 1� each way, which is superimposed on the motion of the Earth's centre, with respect to the starry background, of about 33 arc-minutes per hour.. And the motion of the Moonman seen by the Earthman would vary in exactly the same way, but would be in the opposite direction, differing by 180�. That is the motion we are trying to deduce, the parallax-affected motion of the Moon's centre against the stars.. I have over-simplified matters greatly by ignoring the tilt of the Earth's axis with respect to the plane of the Moon's orbit, which can reach 28� or so at times. ================== You will see from Arthur's graphs that although the equations in his posting include the correction for parallax, in his simulation he has decreed that the Earth has no atmosphere, and therefore no refraction, to isolate the parallax contribution, for simplicity. What Arthur has simulated is the effect of the expression for correcting a geocentric altitude to give an apparent altitude as seen by an observer on the Earth's surface. He quotes that formula as- P = (1-0.0032*(Sin(Lat))^2) * (ATan(Cos(Mc)/((3438/HP)-Sin(Mc)))) which gives a parallax correction (in degrees) to subtract from the geocentric altitude of the Moon (Mc) in degrees, calculated from the Almanac. HP, the Horizontal Parallax of the Moon, is expected to be given in minutes. Only if the above expression is correct will the simulation be valid. We have seen how easy it is to make errors in these matters, as in first posting that expression to the list several months ago, I happened to make two separate trancription errors, which Arthur Pearson managed to uncover, and which I corrected in a posting last month. What gives me great confidence, however, is that Arthur tells me he has compared predictions from the above formula with Bruce Stark's volume of Lunar Tables, in which Bruce has provided what he calls a "wrong-way" parallax correction. Arthur confirms that he sees good agreement wherever he has checked. As Bruce's table, and my formula, were derived quite independently, then it seems more than likely that both of us are correct. Thanks to Arthur Pearson for much painstaking work to confirm the reality of parallactic retardation. ================ I have accessed Arthur's web page at http://members.verizon.net/~vze3nfrm/files.html but have not been able to download the files therein to my old Mac. Instead, Arthur has kindly sent them to me directly as an attachment to an off-list email, and should you find the same problem, no doubt he will do the same for you, if you ask him nicely onThe text on Arthur's recent mailing to the list, as read by my emailer program, shows a few punctuation oddities, perhaps as a result of being put together in a word-processor. If it's the same for you, you will find that the meaning comes through perfectly well, except perhaps in the equations, where (I take it) the character "�" (which is distinct from"^", "to the power of"} is substituted in some places where there should be a minus sign. Also it appears that "Mc," is printed where "Mc'" was intended. No harm done, though. 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. ------------------------------