Moon Violates Reason
by Art WinfreeThis column recently presented several Adventures involving the Moon. This was not intended, but you know how Adventures go: one thing gets you into another. Because all of these traverse absolute terra incognita so far as I am concerned, they present exactly the needed opportunity for Discovery in the special sense of this column: upper case to distinguish "personally, first time for this amateur" from the usual "first time in our society." So it seemed reasonable to contiue engaging the Moon, and I even proffer one more encounter now, this one intended to be the last in this area lest terminal boredom afflict readers (if any) allergic to the sky.
This last one really boggled me, so I am hopeful that it might also give you pause to practice examining un-articulated beliefs. I present it in the form of a paradox to unravel. You know what is involved in paradoxes: fundamentally changing the way you think about something, often by Discovering an implicit assumption that you might otherwise never have noticed, let alone challenged.
Here it is.
"Obvious facts"
From our backyard point of view the Moon, Sun, and planets all traverse a circular band across the celestial sphere, called the ecliptic. The ecliptic runs through the 12 conventional constellations of the zodiac. It is a circle in the sky, like an equator. More exactly, it is defined as the Sun's annual path across the background of stars, even though we can't see the stars near the Sun. The Moon also scoots around the ecliptic every month, or more exactly, every 27.32 days. The Moon is constantly racing endlessly farther ahead of the Sun along this track. Starting from any full moon, every 29.53 days the Moon gets fully 360 degrees further beyond the Sun and we see another full moon. We might depict it so:
This idealizes. In the 11 January column we failed to anticipate the time of Rainbow Moon on account of just such idealization. With the help of a sextant, by 25 January we Discovered that the Moon traverses its orbit not at constant angular velocity but varying its pace as much as 1 part in 4. However, replotting the above with such more accurate data does not visibly alter its appearance so we can leave out that refinement here.
High-precision computational models
Next look at computer software intended to inform you of the phases of the Moon, for example the HORIZONS ephemeris generator used last time at http://ssd.jpl.nasa.gov/cgi-bin/eph, or $20 LunarPhase, borrowable for a month at no cost. At whatever time you prescribe, any of these will tell you the "Phase Angle" of the Moon, which means the angle subtended in the Moon's sky between Earth and Sun (0 at full moon), almost exactly the angle between the Moon and the Sun's antipodes from our perspective as observer at the center of the ecliptic.
(A caution: any two software packages are apt to differ in this result. Results may also differ within one package. TheSky draws the Moon on screen quite accurately from the perspective of a surface observed in Tucson, for example, but its information box numerically presents what seems to be the Phase Angle for an observer at the center of the Earth. The Gold Standard in all such matters is Myles Standish's HORIZONS ephemeris. All the commercial software try to match it with a minimum of mistakes.)
Strangely, in the most conspicuous possible contrast to the expectation above, no phase angle is ever reported by any software outside the range 0-180 degrees. For example, here is the plot from HORIZONS for the next two months:
Well, you say, that is doubtless merely a matter of some archaic convention, like so many things in astronomy inherited ultimately from the Babylonians: probably the program reports the absolute value of the positive or negative angle measured from 0, or chooses either the angle modulo 360 or its complement to 360, whichever is smaller. Replotted with any such convention, the "Phase Angles Expected" figure above would look like this Febr-Apr sample from HORIZONS.
Almost. But there is a surprising difference, still.
Look closer at your tabulated computations or the sample plotted above for Febr-Apr: they never get very near to either 0 or 180 degrees. The increasing angle doesn't proceed at some smoothly time-varying but nearly uniform pace to 180 then abruptly start to decline again at the identical, but inverted, pace. We have not just switched from reporting an angle to reporting its complement to 360. Rather, the reported phase angle gradually slows down and ceases to increase while still a few degrees below 180, then starts backing up, and similarly as it nears 0. Here, magnified, are the first minimum (full moon):
:
and the first maximum (new moon):
Why does this curious evasion of 0 and 180 come to our attention first from computer models? Shouldn't we have seen it in the sextant data collected during September and October (see 25 January column)? Well, we didn't see it because overcast happened to obscure an interval around full moon (phase angle 0) and the Sun's glare always precludes observation around new moon (phase angle 180 degrees).
But let's believe that JPL,
NASA, and the Naval Observatory have realistic impressions of the Moon's
orbit, and so let's accept their computer model where we cannot conveniently
observe.
The paradox
How can this be? No doubt there are small details of astronomical modeling left out from our world-view. For example, the Moon's path along the nominal ecliptic tilts a few degrees from the Sun's, and the the Moon's elliptical orbit precesses fully around in a couple of decades, so these diagrams do not repeat exactly every month. But can such trivia affect the need expressed in Figure 1 that the Moon-Sun angle keeps increasing through 0 and 180 and 360 degrees over and over? What monstrous departure from our simple-minded view of astronomy lurks behind the radically different impression given by calculations from every reliable ephemeris program?
I present you this riddle for an Adventure in Discovery. Two weeks hence you might be interested to return here to compare your Adventures to mine.
Bonus trouble:
Sun Violates Reason, too!
If you like being mystified and finding your own way out of it, here is another mystery that gave me a good workout recently. I won't much elaborate on it later, since you can learn all the answers easily enough on the web. I resisted mightily against that temptation for two weeks, even while despairing that I could ever figure this out by myself and find testable implications to go check. Then it was all the more satisfying finally to do so (like the thrill when picking a lock, when it suddenly clicks open), and only then to rummage google with "earliest sunset" solstice "latest sunrise" and find out how much I had missed. Your own discipline might give you the same satisfaction.
Here is the riddle. On 21 December we got the shortest day, and that was encouraging. But I was surprised to notice for weeks after that the sunrise did not get any earlier. In fact it kept getting later! How the devil can that be? Skewering ping pong balls on knitting needles, marking latitude rings on their surfaces, tilting the needle 23 degrees from vertical, rotating before a flashlight beam, and so on seemed like the right moves to make. But all such and corresponding diagrams and trig only re-confirmed my prejudice that sunrise should start getting earlier and sunset later, from the 21st. Have I observed carelessly during the rush to get breakfast and beat the traffic to work?
A "Skywatcher's Almanac" graphically displays those times throughout the year and robustly confirms my impression that the Sun is not doing like any of my models, and that the sunset (which I hadn't opportunity to observe in those weeks) is likewise misbehaving, though with a different asymmetry. Finally I got the HORIZONS ephemeris (see prior column) to provide me numerical "observations" spanning a year, and plotted them this way and that, using a spreadsheet program. They confirm that the almanac was reliably made and printed, and my observations are compatible with both. For another surprise, they additionally testify that the effect is more conspicuous at lower latitudes.
If you like good puzzles, this is one that will open your eyes, iff you can stay your hand from just looking up the answer in an introductory astronomy text.
What happened about Rainbow Moon on 25 January?
The Moon rose at 14:40 in Tucson, but from my perspective in the Catalina Foothills, until 15:12 it remained behind a mountain ridge. At 15:46 the calculated phase angle of its center was 39.6 deg, marginal for observation on the primary rainbow at 41+-1 degrees from the solar antipodes. My sextant confirmed 39 degrees (less precise because I must guess the position of my eye in the shadow of my head for the direction of solar antipodes) and you can see from the photo below that is about right: the Moon is already maybe 4 diameters inside the rainbow and falling further inside toward full moon at a rate of 1 diameter per hour while the whole picture cartwheels overhead at 15 degrees per hour.
An error turned up in SoftwareBisque's TheSky: it apparently reports Moon phase angle numerically without taking account of parallax, which, for us observers up here on the surface, at times of rise and set is as much as 1 degree = 2 Moon diameters = 2 hours for Rainbow Moon purposes. For geocentric observers (if such could exist) the HORIZONS ephemeris gives exactly TheSky's numbers.
It was a clear day and I made my own rainbow with a garden hose, and verified with sextant that it stood 41 degrees from the shadow of my head. An uncommonly strong wind was blowing spray in my face and onto the lenses. In occasional moments of clarity the secondary rainbow (39 degrees from antipodes, inside the primary, inadequately captured to CCD here) perfectly skewered the Moon.
Two hours earlier would have been
perfect, when the Moon was still way below the horizon in Tucson. Observers
further East in USA had their Moonrise 2-3 hours before mine and could have
got a better photo. None were reported. Maybe next
time. 
Copyright 2002 by A.T.Winfree. All rights reserved. Used by permission.