Adventure of the Rainbow Moon
by Art WinfreeIt would be easy to notice some things that never happen. For example, I bet you never saw a stone fall upwards, and you would have noticed. Some people say Darwininan natural selection accounts for this: such stones are no longer with us :-) . Other things that never happen are hard to notice. For example, I have been told that no fish ever blinks both eyes at once, though sharks do. I have also heard that certain animals never get any kind of cancer, e.g., sharks. Here's a thing that Ill bet you have never seen happen. How much can you figure out about the reasons why you never saw this, and what questions does your analysis raise that might be settled by what kinds of deliberate observation? What Discoveries are made in the process?
I bet you never saw the Moon in the sky behind the color band of an earthly rainbow. Why not? or have you? Fred Schaaf, contributing editor of Sky and Telescope magazine, drew attention to this in January 2001 on page 112. It is worth dwelling on.
This requires noticing a thing or two about the Moon, irrespective of rainbows, and a thing or two about rainbows, irrespective of moons, then putting them together. In the process, some little surprises emerge, the things I call Adventures in Discovery.
Rainbows first. The rainbow is an arc of a circle centered by the light source, usually the Sun. Sometimes you can see the whole circle, e.g., when flying over clouds with the Sun above you. Usually you just see a part of it, viz., the part where the sky is full of water droplets.
Notice the next rainbow quantitatively. With the Sun shining over your shoulder this circle looks about 90 degrees in diameter around the shadow of your head, centered directly opposite the Sun. More exactly (did you get a sextant? see 7 December column for possible sources) its diameter is 82 degrees. The color band is about 2 degrees wide along a circle about 41 degrees from the anti-solar point.
Is the brightest rainbow's color band wide enough to cover the Moon? The colors span roughly a little fingertip width at arm's length, whereas the Moon falls easily within half that width. So, Yes.
(This is only the brightest rainbow. There are others. I see another 10 degrees outside the first, at 51 degrees, and with inverted colors. And against black cloud background I see a few ripples of repeating color inside the bright one. And there are some effects I see only with the polarizing filter I keep in my wallet. Plenty of small Discoveries to be made here --- don't spoil it for yourself by looking in books --- and more when you try to make sense of such observations in terms of geometrical optics. But lets not be lured off the track now. When you are done, for a lucid and succinct quantitative presentation of the geometry involved, see RJD Tiley.)
Next the Moon. Do you suppose it could ever be where the rainbow's color band decorates the sky? I think you know it is sometimes near the anti-solar point: this is called Full Moon. And you know it runs in roughly a circle around the Earth once a month (whence the similar names) in roughly the same plane as Earth, Sun, and other planets, and in the same direction. So it is sometimes near the Sun rather than 180 degrees away from it. This seems a logical inference, and logic is sometimes valid, but if you are ruthlessly honest with yourself you have to admit never actually seeing it very near the Sun. Why not? For one thing the Moon's visible crescent gets surprisingly thin, not as I naively expected a few months ago in proportion to the Moon-Sun phase angle, but (I think now) more like to its square, specifically, to 1 minus its cosine. A really thin crescent can be visible if the Moon rises an hour or more before dawn, but not at or after dawn. Romantics like to watch the full moon rise, but who watches the new moon rise? In the world of Islam it is very important to determine the exact time of new moon, but the crescent being vanishingly thin at that moment, and this also being exact sunrise time, it proves to be a substantial challenge to observers.
This is worth thinking on and drawing a few diagrams. Doing so allows you to Discover for yourself something about the relative distances of Moon and Sun. Pay attention in your diagram to as much as they allow you to see of the sunlit hemisphere. Just how your diagrams quantify this stuff depends on your guess (pretend to be naive) about the distance ratio. This kind of thinking about a personal everyday observation provides a way to independently at least put a lower bound on that ratio. Can you follow through?
As I write, it is 11 AM on 14 December, about half a cycle after the full moon celebrated in the 30 November column. So it occurs to me to get up and look for the almost-new moon. How close should it be to the Sun? Yikes: just a couple diameters. And in fact I see on the calendar that it will become visible three hours hence by sliding right in front of the Sun. If this really happens, it dramatizes the next point:
that you have occasionally seen the dark new moon cross the Sun during a solar eclipse, such as anticipated this afternoon. So we don't just surmise, but we really know the Moon is sometimes near the Sun (at least we do if we are pretty sure that black disk is actually the familiar Moon, not some alien spacecraft or a demon.) And we remember from two weeks earlier that it is sometimes inside the interesting 82 degree ring around the anti-solar point. In getting from one place to the other in relation to the Sun during the intervening two weeks, mustn't it twice every month cross that ring ? If so then it would seem that the Moon must cross the color band. We might be able to see both superposed if we are at that time on the proper side of the Earth to be seeing the Moon in the sky, and if the Sun is also in the sky to be lighting a natural or artificial rainbow. (If no rainfall, you may have to make your own with a garden hose or a spray bottle.)
Is this a little Discovery, or a mirage woven of words?
We can go ask Mother Nature. Exactly when would these putative crossings occur? Asking this question brings you to notice the Moon falling back against the stars every hour by about one diameter (half a degree of arc). This fits with the idea that it traverses 360 degrees in almost 30 days. So the 41 degrees from Full Moon to rainbow band might be traversed in about 82 hours before or after Full Moon: about three and a half days, thus every month you get two chances a week apart. That's when to look, unless it is night-time or the Moon has already set or not yet risen. You have a window of almost 2 degrees width, or 4 hours of Moon motion, twice a month. That is about one half percent of the time, times half again, considering that it must be day where you live, and then since the Sun is only 41 degrees = 3 hours from exactly opposite the Moon, you have at most one eighth of the day until one or the other sets. One 32nd of a percent of the time.
Another way to put it: If the Sun were right on the horizon, the Moon would be 41 degrees above the opposite horizon, or if the Moon were right on the horizon, the Sun would be 41 degrees above the opposite horizon. Either way, one or the other will drop below the horizon in about 3 hours. So you have to be in the right place on the Earth (within 3 out of 24 time zones) to see them both during the moment when they are the right distance apart ... plus a few hours slop for seeing them not quite the right distance apart but still within the thickness of the color band. Two coincidences are required: a window of maybe 3 hours out of the month's 709 hours between full moons must overlap a window of about 3/24 for the spin position of the Earth carrying your observation platform.
For yet another way to see why Rainbow Moon is a rarity, switch your perspective around to look from the Moon. Download from John Walker (of Bending Space-Time three weeks ago) his wonderful Windows utility "HPLANET.EXE". >From the Display menu choose "View Earth from the Moon" and you will see the Earth with its night half shaded. Set the date to 6 September 2001, UT (not local time) 14:33 (which is Tucson time 07:33, when we know conditions were right.) You will see that at this required phase angle only a sliver of the Earth's rim stands in sunlight. Tucson is in that sliver: it has sunlight and it can see the Moon. Twice each month the Sun-Earth-Moon phase angle is 180-41=139 degrees and this same-size crescent sliver correspondingly appears on one or another edge of the Earth's disk. Everyplace in that sliver, and noplace else, can see a Rainbow Moon (with a garden hose, and if not overcast; I suppose everyplace on the entire Moon-side hemisphere shown below can see a "Rainbow Moon" by also providing an artificial light source where the Sun is expected to rise hours later.)
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From
John Walker's Home Planet, a public download for Windows from www.fourmilab.ch.
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You might note the required size of this sliver (disk diameter times half of 1 - cosine 41 degrees) with a calipers or by cutting a paper disk to fit just inside it, then while leaving that disk pasted over the center of the Earth, use the Edit/ Set Universal Time menu then the Animate menu to search hours near 82 hours before or after nominal full moon (when Earth from Moon shows no sliver of daylight) until the crescent is exactly this thickness. If you can see your home in the lighted crescent, this is when to look for a Rainbow Moon, weather permitting. There must be a rainstorm or fog in progress across your line of sight to the Moon, but not so dense as to obstruct either Moon or Sun from view. How likely is all that? No wonder we never noticed it. A majority of the undergraduate biology seniors in my classes have never even noticed the Moon in the daytime sky, but it is there every day, and as more than an invisible sliver most of that time, patiently awaiting an open eye. What about the Rainbow Moon?
Suppose you wanted to see it, what would you do? You might trust the Home Planet simulator, but you don't know how that works, so in principle you don't know whether it works correctly, and anyway you want to do it yourself. Maybe start by getting a table of full moons, good to the hour, then lie in wait with your spray bottle or garden hose three and a half days before. If circumstances hide from you the Moon's passage into the circle of the rainbow, maybe you can yet see it exit again a week later (twice 82 hours).
Try this and, if you are as much in need of education as I was during all of 2001, you will encounter the punch line for this Adventure in Discovery: this schedule doesn't work at all! (If you don't care to wait weeks or months, try it with the Home Planet simulator.)
First of all, it is hard to find full moon tables that agree to within a few hours. One error I made was to trustingly download moon-oriented software from the web. They don't all report the same numbers. I finally took the US Naval Observatory as my standard. But the problem remains. Refining your observation schedule to the Moon's average period of not 30 days but 29.53 days or 709 hours, i.e., watching 41/360 709 = 80.7 (not 82) hours before and after full moon, accurately determined, you still find that the Moon is never in the right place. Whatever can be going on?
Another error I made (I am really good at this; my first lab at Princeton was called "the Mistake Factory": where I make mistakes) was to get full moon schedules in Universal Time (Zulu, Greenwich Mean Time) and deliberately neglect to convert to my local time zone. True enough, the Moon is 41 degrees from anti-solar at the same moment wherever I might be looking from, independent of time zone, but the point learned here is that my watch is wrong if it is not set to UT. But with that fixed, the Moon still pops up in the most surprising places!
One thing you learn (I learned) this way is that the real interval between full moons varies from month to month around the 709-hour average, within a range at least half a day wide. So measuring 709-80.7 hours from prior full moon, you get a different time than by measuring 80.7 hours before imminent full moon. Surprise: the Moon's orbit is not periodic! I went 59 years without noticing that, iving confidently for at least 47 of those years in a dream world of periodic orbits while the real thing, daily observed, behaved quite otherwise. This leaves me in no position to feel supercilious condescension toward superstitious ancestors.
But straightening that out, it still doesn't work. Must have done some arithmetic wrong. Figure it out over again, apply to the next opportunity several months hence. (This is a clipping from my notebook: this went on from January through August of 2001.) Ooops. Ainít there.
Finally, by accident I happened to notice (with a spray bottle, by now carried habitually in my car, but without camera) a Rainbow Moon in Tucson at 07:33 Sept 6, 2001, fully 89 hours after prior full moon and so 8 hours after I expected it, when I accordingly had no intention of looking.
This provides a great "how to do it" lesson about Discovery: having notions in mind, I sought to confirm my predictions by looking at the appointed time and place, just as in the famous case of Clyde Tombaugh's fortuitous discovery of Pluto. I would have done better to use observation to check the assumptions from which those untested predictions follow. That is really the only sensible way to learn anything unexpected. This different attitude --- of finding out new surprises rather than of smugly confirming prior assurance --- would have entailed both looking at the time Rainbow Moon is predicted to happen and looking at times it is predicted to not happen, to check both.
Well, the "not happen" won. Unless I made another silly arithmetic mistake, this reveals, much to my astonishment, that the angular velocity of the Moon differs from one full moon to the next and/or differs all along its orbit every month.
This is how discoveries are
made: by noticing when our mental visions of the world screw up, admitting
it without excuses, and then checking to find out what was wrong with those
visions. In this case what doesn't work may be our rough mental
model of what we plainly see in the sky:
that the Moon's orbit is nearly a circle traversed at uniform speed around us,and
that we are watching the Moon from a fixed base.
The implicit idea was
that more realistic details can safely be left out from this approximation,
which is plenty good enough for an event that lasts a few hours thanks
to the width of the rainbow's color band. Evidently not so. What
we learn by these frustrating repeated disappointments is that one or both
of the foregoing bullets is seriously goofy. Should we examine this mental
vision more closely? Sure, we can look in books and get as many exact details
as wanted, or just buy TheSky or some such computer program to implement
all details numerically.
But in this column's spirit of personal engagement, I choose instead to examine the Moon by daily observation, using a plastic sextant and a home-made cross-staff, as seen in the 7 December column. Every chance I got for a month after 18 September, I jotted down the angle between the Moon and the Sun or between Moon and anti-solar point (the shadow of my head and sextant) or between Moon and some other star along the path of the Moon, together with date and hour. >From such data we may see whether the Moon does in fact traverse its path at uniform speed as supposed. If so then something unanticipated will be Discovered. If not, maybe the amount of non-uniformity suffices to account for the discrepancies between observed rainbow Moon time and superstitious anticipation based on uniform speed. If that is not sufficient then some other assumption is at fault, and we have something even more surprising to learn.
This is a good thing to actually do, not just read about. Doing it substantially enhanced my awareness of lots of things. In the next column (two weeks hence) I will present the fruit of my month of observing between intervals of cloud cover. By then you might have better data of your own.
Heads Up: Taking advantage of my time machine to read solutions printed in future columns, I think Rainbow Moon will be visible on the Friday of the next column, two weeks hence, 25 January, in mid-afternoon just as the Moon rises. If so, see if you can photograph it. Next chance is two months later.
It is now 2 PM on Friday 14 December,
and peering through my arc welder's helmet I do see a lens of blackness
at the bottom of the Sun. The invisible Moon really is there. 
Copyright 2002 by A.T.Winfree. All rights reserved. Used by permission.