Adventure of the Rainbow Moon: crucial experiment this afternoon in Tucson
by Art WinfreeLast time we entertained a puzzle, basically "How might my awareness be expanded by physically engaging the absence of an experience that "ought" to be remembered, but isn't, viz., the apparition of the Moon behind the color band of a rainbow?"Engagement consisted of predicting times when that phenomenon "should" be visible, then finding that it isn't. So now it becomes a personal argument with the Moon, to be settled in person by accosting the Moon. I entered into a spreadsheet a month of observation times and sextant angles (called the "phase angle") between Moon and the antipodes to the Sun or (since it is awkward to measure angles exceeding about 90 degrees) between Moon and more convenient other markers. That done, the next step is to apply appropriate offsets to blend together observations made by different methods in four 90-degree ranges of the 360 degree monthly orbit of the Moon: at night in one week against a star, and in another week against a different star, by day in one week, against the Sun, and in the other week against the anti-Solar shadow. This makes a new column in the spreadsheet, and the next subtracts the linear average angular progress at 360 degrees/ 709 hours in order to see how much the observed phase angles differ from what was expected at uniform speed. Presumably not much, since an ignorant biologist such as myself, still living in an Aristotelian world, expects the Moon's orbit is pretty nearly circular. Presumably random errors of measurement will be conspicuous, though, along with gaps during overcast.
Well, without apologies, here are my actual results as first-time observer with the plastic sextant shown in the 7 December column.
28 observations as described, interrupted by an interval of overcast in the middle. data re missing at the edges because the near-new moon is hard to see so close to the Sun. The slope here is the phase angle velocity. Relative to 0.5 degree/hour linear average here subtracted, this ranged about 1 part in 4. An unexpected incidental observation: the Moon looked bigger at left and right edges of this plot, when phase angle was also accumulating faster, and looked smaller near full moon in the middle of this interval when phase angle was also falling behind fastest. Diameter ranged about 1 part in 8.
Of course many data are missing for many reasons. But what remains does show that the Moon's progress is far from uniform, getting ahead of mean schedule by 7-8 degrees at some times and as far behind at other times. That means its appearance at a particular place can be early or late by as much as 15 hours, ranging over a full day! No wonder predictions for Rainbow Moon proved so unreliable. The 6 September observation was off by 8 hours, well within this range of now-expected error.
Why so uneven? And another question: as I write this three months later , the range seems less now, more like 1 part in 10 or 12. Did I make some mistake in earlier observations?
But let's keep our eyes on the ball. The aim was to anticipate a Rainbow Moon. Can we do better by working into our predictions this sinusoidal departure from from the naively constant schedule? In other words, if the angular velocity varies on a regular schedule, we might anticipate Rainbow Moon times before and after full moon by using 41 degrees / the scheduled velocity rather than / the mean velocity. This attempt also leads to unexpected Discoveries. One is that the sinusoid does not repeat on a simple schedule! The Moon's angular velocity looks almost a function of its position on the background of stars, but its period does not finish one cycle and start the next quite on the same schedule as full moons. It drifts noticeably from one full moon to the next, enough to be no good for predictions a couple years hence, so it is hardly worth the trouble to formalize this correction.
Back to the data plot above: it seems that my sextant/cross-staff observations wobble by as much as a degree (= 2 hours of Moon phase) to either side of a smooth curve threading them. Is this just because I haven't yet learned to use a sextant with consummate skill? Repeatedly measuring the angles between mountain peaks, I find I can reliably do an order of magnitude better than +- a degree. So there is something worth checking here. What else might be involved? How about the second assumption of the 11 January column, that we observe from a fixed base, effectively motionless relative to the Moon?
Insight! We are not watching from the center of the Earth, but from its surface, from the rim of a rotating wheel whose radius might not be negligibly small compared to the Moon's distance. It turns out to be about 1/60 the distance to the Moon: so we get parallax effects as large as +- 1 degree = 2 Moon diameters = 2 hours. If we didn't already know that the Earth is a rotating ball and the Moon is only 30 diameters distant, we might have been able to infer it from enough sextant observations of the Moon. (Jump ahead as you read to see the figure below: the fine wiggles are this daily parallax.)
The complexity of the Moon's orbit is such a puzzle that Isaac Newton at age 60, while Director of the Mint and busy hanging counterfeiters, and involved in alchemy experiments after swearing off physics, nevertheless felt he had to take time to figure it out because it threatened to subvert his published Law of Gravity. See http://www.ucl.ac.uk/sts/nk/newton.htm. He did not succeed, so things things looked bad for theory, and the matter was set aside. Others finally worked through most of the details in mid-18th century. You and I have only Discovered the existence of these subtleties, far from clarifying them. But without taking the trouble to ask a simple question and look, we would still remain in the pre-Keplerian, even pre-Ptolemaic, even pre-Hipparchian dream world of circular orbits supported by concentric crystalline spheres.
Cheating
I think it makes sense to engage Nature yourself with a sextant, for a month at least, but if your skies are always cloudy, here is a half-way house of interest, the one I finally used for cheating a few paragraphs above. Go to http://ssd.jpl.nasa.gov/cgi-bin/eph and you will find input forms for the Horizons ephemeris, from which you can download whatever geometrical data you might have wanted to observe for yourself, about whatever solar system objects. I learned about this only last month, by looking up "phase angle" on the web then politely emailing to people who know about such matters to ask where I might find a graphing calculator. Called upon about 400 times a day, 5-year-old Horizons is a very wonderful service, now containing exact updated orbital elements for 16 spacecraft, 75 artificial satellites, 315 comets, 63,000 asteroids, and the planets and their abundant natural satellites. Its Help pages are very complete and clear. Specify the range of dates and the interval between observations, the Moon as Target, your home location (or code “500” for the center of the Earth to get rid of the +-1 degree daily wobble), and list the data wanted, such as item 24, the phase angle Sun-Target-Observer. Ask to have the data decimal degrees (rather than hours, minutes, seconds) and in csv format suitable to cut/paste into a spreadsheet for calculation and plotting. Then an algorithm will spare you the ravages of cloud cover, frostbite, or mosquitoes. It is fun to study such "data" like raw observations free from instrumental error, and see what you can infer from them as though you did not know they come from a formula in which all the principles you seek are already encoded (just as they are already encoded in the physical Moon.)
For example, in just a few minutes I took a whole year of Moon observations at hourly intervals. Here is a plot of the segment overlapping the sextant observations above (after subtracting the mean rate of phase advance, as above):
JPL / NASA ephemeris calculations to compare to observations with sextant, above
Extraordinary care with sextant would not be needed to notice the +- 1 degree (2 diameter) fluctuation of the Moon's apparent rate of travel on the background of stars during consecutive 12 hour intervals. If you didn't know the Earth is a spinning ball that carries your observation platform back and forth on a circle with diameter several percent of the distance to the Moon, here is where you might Discover it for yourself. This simple observation could have helped the defenders of Copernicus and Galileo, when confronted by indignant incredulousness that the Earth could be moving.
Continuing through 2002 at Tucson, the HORIZONS ephemeris, with daily wobble of +-1 degree = +-2 hours included, anticipates a dozen opportunities to observe the Moon at phase angle 41 degrees. Striking out the ones that in Tucson occur before moonrise or after moonset and likewise for the Sun, remarkably, eight still remain. The first is anticipated today, i.e., on the Friday of this column's scheduled appearance: did it happen?
25 January, 2002 mid-afternoon moonrise 25 March, 2002 mid-afternoon moonrise 31 March, 2002 early morning moonset 20 July, 2002 after late afternoon moonrise 26 August, 2002 morning moonrise 17 September, 2002 late afternoon moonrise 23 November, 2002 before moonset about 10 AM 15 December, 2002 sunset
You are not at Tucson so half of these might not work for you (or all of them, if I figured wrong again.)
Unexpectedly ballooning and shrinking disk
There remains a troublesome business to clarify. Why does the Moon's speed vary so? Maybe because the Moon's orbit not so symmetric as fondly imagined before looking? By having to look sharply in order to record angles, you may have also noticed as I did for the first time in my life, that the apparent size of the Moon in the sky also varies (see caption under the first figure above). This proves at least that the orbit is not a circle concentric to the Earth.
How can you quantify this? You might start by laser printing a nice scale into transparency (or just cut a piece of transparent ruler) and mount it inside the eyepiece of your telescope, at the focal plane. Calibrating the scale is not essential for quantifying % changes. Would it be still simpler to just time the 120+-several seconds it takes for the Moon's diameter to move west and exit your binoculars' or telescope's field of view? Yes, if you are willing to observe at every hour of night and there is always enough earthshine to distinguish the dark edge of the Moon (there isn't tonite, for example). But if you prefer the convenience of taking observations by day then north-south measurement seems your only consistent option. Best, of course, would be photos for concentric comparison at leisure.
The diameter ballooned by about 1 part in 8 during the month I watched, then shrunk back. Does this fit with the observed variation in phase velocities? Well, if you believe Kepler's empirical second law or if you believe in the conservation of angular momentum in a central field, then you expect the product of radial distance from Earth times (distance times angular velocity) to remain constant. "Angular velocity" means "relative to the fixed stars", i.e., the phase angle velocity relative to the Sun (the slope of the sinusoid in the first figure above plus the removed mean velocity of 360 degrees/ 29.53 days) plus the relatively small 1.02 to 0.95 degree/day angular velocity of the Sun. Since any sphere's apparent visual diameter is inversely proportional to distance, we then expect phase velocity/diameter2 to be constant. This does seem loosely compatible with our unexpected observations, the phase velocity varying 1 part in 4 and the diameter 1 part in 8. Here is a branch point for another potential Adventure: checking this out by more careful observations.
Cheating, an exact check of angular velocity/diameter2 from the HORIZONS ephemeris does show a fixed value, at least to 2-3 decimal places. Curiously, though, it is not quite exact but shows little wiggles in bursts half a year apart. An exact look at the distance to the Moon also shows excessive monthly range at the same times as the wiggles, half a year apart. This reminds of another loose screw encountered above: that the "1 part in 8" variation of disk size doesn't seem to fit the facts a few months later. Last Sept-Oct 2001, I measured 1 part in 8, whereas now, in Dec-Jan 2002, it seems more like 1 part in 12. This is really mysterious to an Adventurer conceptually blinkered by implicit adherence to 2-body, 2-dimensional visions. The virtue of actually looking at the Moon to notice how much its apparent diameter varies is that it may cause these blinkers to fall off.
According to HORIZONS this was not a mistaken observation, but just unforeseen plain fact about an orbit complicated by something that seems most severe twice a year and least severe twice a year. Whatever could that be?
Think about it. We ignore the Sun only on the presumption that the Earth-Moon system is in free fall around the Sun, both parts (Moon and Earth) falling at the same rate, i.e., in the same orbit. But are they? At most times of the month the Moon is nearer or farther from the Sun than is Earth. So there is a differential acceleration called the "tidal" influence of the Sun. The observed irregularities, in other words, have to do with the 3rd body in this problem, the Sun, which pulls with varying strength and from seasonally different directions relative to the long axis of the ellipse. If it pulled Earth and Moon equally there would be no net effect, but their distances from the Sun are not quite the same so there is a tidal residue with monthly and semi-annual components affecting the idealized ellipse.
How big an effect can that be? We can estimate the acceleration of Moon and Earth together toward the Sun as SunMass = 300,000 EarthMasses/ 100 million miles distance2. This should affect nothing, since the Earth-Moon package is in free fall around the Sun ... except that Earth and Moon are not really at quite the same distance: the Moon is sometimes a quarter million miles farther or closer. So it accelerates less or more rapidly than the Earth. And how great is that monthly discrepancy compared to the Moon's acceleration toward Earth that determines its elliptic orbit? That is 1 EarthMass/ 1/4 million miles distance2. And how much might this effect vary semi-annually on account of the "circle" really being somewhat elliptical and keeping almost same orientation to fixed stars from month to month, while the Sun moves 12 degrees per month?
You might be able to compare these rough quantities to see if this tidal effect has plausible magnitude to be a candidate for interpreting the last two pictures quantitatively. If not too ridiculously small, then we might go on to think of the Moon in its orbit as analogous to a gyroscope like a bicycle wheel, torqued by this tidal leverage and therefore precessing: its plane of rotation would not be expected to keep fixed orientation. Maybe this is connected to the 18-year cycle of eclipses familiar to the builders of Stonehenge 4000 years ago.
Here is another good branch point for a new Adventure in Discovery perfectly accessible to anyone with sextant, with a transparent scale inserted into her binoculars or telescope eyepiece (or a stopwatch), and with patience and curiosity to Discover things for herself whether or not someone's great great grandfather did so before her. We adopt such vigorous attitude about other things, e.g., about discovering sex, carpentry, mountain tops and coral reefs, all of which were well explored by others before us and their adventures published in novels and biographies: so why not also about those parts of the natural world that have been explored in academic science journals?
At this point it seems clear that anticipating the moment of a Rainbow Moon is a job best executed in terms of algorithms meticulously summing a long series of "effects" or directly simulating the gravitational interactions of three bodies. If you care to go well beyond this Adventure in Discovery, e.g., if you care to import/tinker algorithms that suffice to various levels of accuracy, see the definitive compilations by Meeus and by Chapront. If you would only like to observe in 5 minutes that the Moon's position in the sky is sufficiently puzzling that only a naif or someone as desperate as the aging Isaac Newton would engage it, here is all you need to do:
Open a software package like TheSky, find and center the Moon, set up a stepwise time skip by the Moon's mean sidereal period of 27.32 days, and let the picture start iterating. The stars stay put, but the Moon dances erratically throughout a patch about 15 degrees across. That is a 30 hour range of predicted Moon position relative to the rainbow. At the outset of this exercise I fancied it would look motionless at such stroboscopic intervals, and accordingly never saw Rainbow Moon when anticipated.
Then are we finished with this engagement? Not unless you want to be. Here is maybe the germ of another little Discovery: did you think of this one? In addition to a Rainbow Moon there may also be a Moon Rainbow, a rainbow in water droplets 139 degrees from the Moon rather than from the Sun. With garden hose in hand against a dark sky, full Moon over your shoulder, this might be easy to check on 8 February. If true then the moon rainbow must also be there in daytime, and the Sun will be in the Moon Rainbow's color band at the same time the Moon is in the color band that we studied so single-mindedly that we never even thought of this other one. Of course it will be too faint to see in daytime. But in your mind's eye can you visualize them both on the celestial sphere? Do the two rainbows cross? Would they if we used the 51-degree secondary rainbow?
And another little Discovery, about refractive indices. Long before accidentally witnessing the 6 September 2001 Rainbow Moon, I thought I had seen one about 4PM in Tucson on 6 March 2001. And in a way, I had. But such calculations as finally arrived at for this column now deny the possibility. They say the phase angle then was only 39 degrees, even taking into account parallax and atmospheric refraction. So how could I have seen it? Well, thinking back, I realize I was then not too fussy about what I used for a spray bottle. The one ready to hand at the lab that afternoon was mostly alcohol, a distinction I'd have considered to be of no consequence, had I thought of it at all. But alcohol refracts red light (the outside rim of the primary rainbow) more than water refracts violet light (inside rim), so the alcohol-pickled rainbow arc I guess was shrunken enough to catch the Moon at 39 degrees! So the little Discovery is that witnessing (or just missing!) Rainbow Moon in a real rainfall also delicately assays the refractive index of those distant droplets. Is an Acid-Rain-Bow noticeably different from pristine?
And, back to the original pursuit: the currently best photo of something like a Rainbow Moon is in Fred Schaaf's column on page 112 of the January 2001 issue of Sky and Telescope. Has all our wrestling with the Moon equipped us to do better, e.g., this afternoon?.
Copyright 2002 by A.T.Winfree. All rights reserved. Used by permission.