08 March 2002"Reason" Takes a Lesson, concluded
by Art Winfree
Last time, we came up with Way 1 to resolve the paradox that the lunar phase angle does not continue cycling through 0-180-360=0-180... degrees month after month as expected. One solution to a paradox is never enough for conviction, so we thought a little more and found another approach, hopefully corroborating Way 1:Way 2:
Measuring angles at the center of the Celestial SphereThis way starts by drawing a picture
of the celestial sphere of directions around the observer, and painting the ecliptic on it as an equator (black circle). By definition, the Sun runs exactly along this circle. Paint the Moon's dashed circular path around the sky not quite identically: at least allow the two to differ, however slightly. Here I exaggerate the tilt of the dashed circle carrying the Moon (little foreground dot). In fact the two paths are almost the same, the Moon's circle tilting only 5 degrees: no big deal for such dramatic qualitative issues as seem to be at stake here, right? Well, let's keep this in mind, and carry on.
Draw a line from the observer at the center of the sphere to the Sun's antipodes and another to the Moon. The angle between them is called the phase angle. (And it doesn't make any important difference of the observer is not exactly at center.)
(Note that on this sphere of directions it would make no sense to draw a line from Sun to Moon, because those are not really Sun and Moon but directions to Sun and Moon.)
Things to notice about Way 2:
If the Moon runs exactly along the ecliptic (i.e., at ecliptic latitude 0), its phase angle is always exactly the difference of ecliptic longitudes plus/minus 180 degrees. It starts from 0 when the two longitudes are opposite at full moon, progresses smoothly through 180 degrees to 360 degrees (which we re-name 0 now) and smoothly continues beyond. The only way we could qualitatively alter this report is at some point to switch how we measure the angle. For example, at the moment when the two radial lines from the observer at the center of the sphere meet as one straight line, if we were measuring the angle on the widening side we might e.g., flip over to measure the complementary angle on the shrinking side. This keeps the angles always <= 180 degrees, with no discontinuity because the switch is made when both ways of measuring report180 degrees. In that case the plot would become a zigzag like our first impression of any ephemeris report:
except that its corners would be perfectly sharp, not rounded as the ephemeris reveals when looked at closely:
The key Discovery for clarifying this muddle is that the two radial lines never do make one straight line if the Moon departs even slightly from the ecliptic. Though this seemed to me a merely trivially different situation, it has a big implication: the difference angle approaches 180 degrees but never gets quite there before it starts to decrease again. With Moon (exaggeratedly) as far off the track of the Sun (the ecliptic) as in the red ball diagram above, there is no way the phase angle (complement of the Sun-observer-Moon angle) could ever become even as small as 7 degrees (circle drawn around the antipodes to the Sun). Similarly (not shown explicitly, when Moon is near Sun) it could never get even as big as 173 degrees.
This picture, together with memory of the partial solar eclipse during the 14 December writing of the 11 January column, reminds us of something that didn't happen. Our vision of the pertinent geometry at the top of the previous column, with Moon circulating exactly along the Sun's ecliptic circle, implies that every full moon should be a lunar eclipse (Sun exactly 180 degrees opposite through the Earth, which thus casts its shadow on the Moon, phase angle 0; or from lunar perspective, phase angle 0 means no angle between Sun and Earth in the sky). And every new moon should be a solar eclipse (Moon-Sun angle 0 so phase angle 180, Moon passing right in front of the Sun.) And so it reminds us that this does not happen: here is a small quantitative detail with a dramatic qualitative consequence.
At the end of the prior column we asked, rhetorically, as though the answer were obviously "No way!", "Can such trivia affect the perceived need that the Moon-Sun angle keeps increasing through 0 and 180 and 360 degrees over and over?" Answer Discovered: "Yes, indeed it can, surprisingly enough! "
And we asked "What monstrous departure from our simple-minded view of astronomy lurks behind the radically different impression (turning back before 0 or 180 degrees) given by calculations from every reliable ephemeris program?" Answer Discovered: 180 degrees from Sun direction to Moon direction (phase angle 0) is not a point on a 1d ring of periodic time, and so not a point you have to pass through recurrently in order to keep cycling. But rather it is a point on a 2d sphere of directions, and you need not go near it in order to keep cycling around the sphere ... not only need not go through it, but cannot unless you can invoke infinitely precise navigational skills.
Without ever explicitly articulating this assumption, I fancied that the Moon's angular position as projected for simplifying convenience onto the ecliptic plane could not differ qualitatively from its exact angle, because the Moon's circular track tilts only 5 degrees off the ecliptic. Mistake. Angle measurement spuriously restricted to the plane (or to a 1d ring in the plane) can produce results entirely different from proper measurement in 3d (or on a 2d sphere in 3d). This change of dimension is the sought-for conceptual earthquake needed to bridge the topological paradox encountered last time. I count this as a Discovery because is changed my worldview. The fact that everybody else had already made this transition detracts nothing from my Discovery in the sense of this column.
End of paradox and Adventure? I don't think so. Not until we make some effort to check. I don't know about you, but most of my "understandings" are self-deceptions or at least incomplete. I am never done before the outcome is tested.
To wrap up an understanding of this paradox, we need to see how (or whether!) last week's intuitively expected situation does in fact transition smoothly to the actual situation when the Moon's path diverges from latitude 0. The answer is that there is no smooth transition, nor any transition at all! If Moon and Sun paths differ, or even if they are identical but not restricted to a plane, the difference angle called the phase angle necessarily falls short of 180 degrees and so it stays in a restricted range of less than a half cycle. The only way to obtain the "intuitively obvious" conceptual figure naively expected:
is to make Moon and Sun traverse identical circular planar paths, and keep angle measurement confined to the plane and so keep it one-sided. This kind of measurement bears scant resemblance to what is required in every other (3d) context. It is not an approximation that should be acceptable if millisecond precision is not required, but rather it is simply un-generic, a misleading fantasy.
Yet another way to say it:
The expected figure (our pre-Adventure expectation, that we thought qualitatively unassailable) is indeed true of ecliptic longitude angles, i.e., of projections of 3d reality onto the 2d plane of Earth's orbit around the Sun (or of the Moon's around the Earth). The surprise Discovered by attention to a tiny "detail" is that this vision not only slightly misleads in quantitative terms, but also utterly fails qualitatively the moment the least bit of three-dimensional reality is acknowledged in the form of non-zero latitude.
And an analogy:
Since summer 1981 when Jupiter and Saturn were pretty close together on the ecliptic, I've been intermittently watching Jupiter race ahead along the ecliptic to catch up to Saturn a cycle later in summer 2000. This is my lifetime-scale "wall clock". In mid-1999 Jupiter was -15 degrees or so behind, and in mid-2001 was +15 degrees or so ahead. So a confusable person like myself might think it would have had to go through 0 degrees in-between. This is where mere imagining foundered. 0 degrees "ahead or behind" implicitly became in imagination "0 degrees away". Looking made a difference. Much to my astonishment Jupiter never went from minus to positive angle from Saturn. It never got closer than a degree from Saturn: it got around 0 without passing through 0. Same stunt the Moon did, same reason: "ahead or behind" in projection onto the nearby ecliptic plane is not the same as angular distance in the sky, on the sphere of directions. The Discovery? That the orbital planes of these two planets differ by 1.2 degrees, a detail I had formerly thought utterly negligible if I ever considered the possibility at all.This Adventure in Discovery reminds early 20th century arctic explorer Vilhjalmur Stefanson, who commented: "An adventure is a sign of incompetence; everything you add to an explorer's heroism you have to subtract from his competence."
Now are we done? Could be, but for full value we ought to first look around to see what else can be seen from this newly acquired perspective. Maybe this little "Discovery" illuminates the 30 November and 14 December "Trouble at Full Moon" columns. There we saw that the moon brightens up sharply when close to full, giving a flash, then we interpreted this flash in terms of reflections visible only at phase angle close to 0, and tested possible mechanisms of such reflection from the moon dust. Now we recognize that full moon is only rarely (only during lunar eclipse!) very near to phase angle 0, so tabulating brightness against hours before or after full moon as a proxy for phase angle (as in the graph used there) may dramatically underestimate the actual angle and so report a spuriously low brightness at each small angle. Different full moons must be expected to have different phase angles and brightness' according to lunar ecliptic latitude at that time: the Moon can miss the ecliptic by as much as 5 degrees, roughly as much as half a day's travel. So the finer-grained data I asked for (at intervals less than a day) would not better reveal the curve's features unless we also factor in this perpendicular offset, which we never would have thought to do had we not gone on to explore this frivolous paradox and so "Discovered" the importance of latitude in the phase angle.
Incidentally, a person watching intelligently during approach to a lunar eclipse could be in danger of making a very alarming (and fortunately false) "Discovery" on account of this very surprising retro-reflection from the Moon only as the phase angle becomes less than a few degrees during several hours of a single night. The Moon is dramatically brightening up! What could that possibly mean? Moonlight is reflected sunlight (we know by noticing the same Fraunhofer lines in its spectrum.) So it must be that the Sun is dramatically brightening up!! Uh, oh! Nova or supernova? I do not look forward to dawn.... :o (
Overview of the past five Adventures, as we take leave of them forever:
11 January: Noticing that we never see the Moon in a rainbow's color band led to predicting when we should, then finding that the Moon never keeps its appointments, and this led to stubbornly engaging it with a sextant.
25 January: Sextant observations revealed that the Moon does not traverse a circular orbit, but moves faster across the stars when nearer and moves slower when more distant; and that professional ephemeris calculators do a pretty good job of matching those observations and going on to still finer resolution. One unexpected aspect of that refinement is +-2 hour variation in Moon timing due to parallax, as our observing station far from the center of the Earth traverses loops several thousand miles wide.
And on the 25th we caught our first deliberately foreseen Rainbow Moon (sort of: rising a bit later in Tucson, the Moon was already in the dimmer secondary rainbow and didn't photograph
well). Note that another chance is coming up on 25 March.
08 February: Astonishingly, the necessarily (it seemed) irreversible advance of the Moon's phase angle is not reflected in those calculations: instead it exhibits sawtooth oscillations well inside the range 0..180 degrees, turning back from both extremes and smoothly reversing every 2 weeks. What is this paradox telling us?
22 February: It tells us, very simply, that thinking in terms of 1-dimensional rings in the sky provides nothing like a good approximation to the observably real angles involved, even though the Moon's 1-dimensional ring path differs from the Sun's by less than 6 degrees.
And today, 08 March, we found other approaches and analogies to the same problem. They corroborate the answer deduced, and so give confidence that maybe our concepts are today less askew than some weeks ago.
Many surprises. Notice where they came from: from merely thinking in a perfectly unsophisticated way about what we see, and from looking to check simple inferences quantitatively. This is how Discoveries are made. And here is one more:
I learned today (Ward and Brownlee: Rare Earth) that Jacques Laskar (Nature 18 February 1993) discovered computationally that without this tricky Moon, we might not be here to observe its absence! The Moon's peculiar orbit apparently stabilizes the tilt of Earth's spin axis (23.5 degrees) against faint but geologically cumulative influences from Jupiter, from Saturn, and from the Sun (which do scramble Venus's tilt and seasonality), and so allows land-based multicellular animals to evolve unchallenged by indecently variable and severe seasonality.
Postscript on an item mentioned 11 January:
Bending SpaceTime in the Basement.You might haved looked up John Walker's wonderful website, including one of the clearest discussions I have ever seen of Henry Cavendish's famous lead-ball experiment to measure the universal constant in Newton's theory of gravitation. Maybe you also tried it. I would love to hear reports if so. I tried it and encountered difficulties.
Next time:
Still more atrocious surprises as chunks of iron fall out of the sky. In December I invited you to get such a chunk from Ebay. For a next Adventure in Discovery we will have a look to see what's inside: surprises, naturally, at least to such innocents as myself.
Copyright 2002 by A.T.Winfree. All rights reserved. Used by permission.