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22 February 2002
Moon and Sun Violate Reason? Nope: "Reason" Takes a Lesson From Them
by Art Winfree

Last time, we saw that the position of the Moon relative to the Sun in the sky smoothly increases through a full cycle once a month and carries on into the next cycle and month. Yet every ephemeris calculator says the "phase angle" between the two smoothly increases and decreases, never becoming even as much as 180 degrees, not crossing 180, and in most months turning around while still falling short of 180 by a few degrees! This seems to require some revision of concept in one place or the other. Paradox can only be resolved by realizing something fundamental, but what, and how to find it?
The "phase angle" is the angle between Sun and Earth as seen from the Moon, thus 0 at perfect full moon, which would accordingly also be an eclipse. This measurement "as seen from the Moon" being inconvenient for an Earth-bound observer, I use instead the angle between the Moon and the Sun's antipodes, which is the same thing but for a slight parallax correction that never exceeds 0.15 degree. So the "phase angle" is 180 degrees more or less than the angle we see between Moon and Sun. When Sun and Moon looks to us 180 degrees apart, one rising while the other sets, phase angle=0.

(In this connection it is helpful to be aware of jargon in this area that puzzled me for a long time: the "phase" of the Moon is quite a different thing, in fact not an angle at all, but the illuminated area fraction of the ostensible disk, 0.5 [1+ cosine(phase angle)]. And the "phase position" is yet something else, to do with the crescent's orientation. )
"Paradox can only be resolved by realizing something fundamental, but what, and how to find it?" Soldiering on, in this Adventure I found "it" in two ways. If you found others, please inform me.
Way 1: Longitude and Latitude on the Celestial Sphere
As we found on 7 December, the manifold of possible directions around an observer is like a sphere. Let's decorate it with longitude and latitude coordinates, and let the Sun's annual path across the background of stars be its equator, at latitude 0. One way to think about the angle between Sun and Moon along the ecliptic is to find the ever-increasing ecliptic longitude of the Moon as a function of time (something like 360 degrees/ 30 days * time in days) and that of the Sun as a function of time (something like 360 degrees/ 365 days * time in days) and subtract Sun's from the Moon's. Just as expected on 8 February in Figure 1, this does give a perpetually rising, nearly-straight line that goes through 360n at full moons (not +180, because the phase angle is defined as above, from the lunar perspective):

But this subtraction of longitudes does something else too: it implicitly draws our attention to the ecliptic latitude, and to the fact that we have been ignoring it. Notice it in whatever tabulation you acquired for the position of the Moon. Ecliptic latitude is 0 by definition for the Sun, and amounts to only a few degrees for the Moon, so presumably it doesn't matter for such dramatic qualitative issues as seem to be at stake here, right? Well, let's keep this in mind, and carry on.
Things to notice in Way 1:
At full moon, when the Sun's and Moon's longitudes are opposite, so the difference between them is 180, is the "phase angle" 0 as assumed? The Sun and Moon are at opposite longitudes but not quite at opposite points on the sphere because the Moon does not exactly ride on the Sun's path, the ecliptic. In other words they may differ a little in latitude. If the latitudes differ by a couple degrees, then the phase angle can never be less than those couple degrees. This is just what seemed so perplexing in JPL's HORIZONS ephemeris:

Similarly, at new moon when longitudes are equal, so their difference is 0, the computed ephemeris nevertheless says the phase angle falls a couple degrees (different in different months) short of 180 degrees. The directions are not quite the same when the longitudes are equal unless also the latitudes are equal. So the phase angle can never be as big as 180 degrees, again just as the ephemeris reported:

Noticing these things leads us to Way 2:
Measuring angles at the center of the Celestial Sphere
to be continued next time.

Just a short note on the supplementary observation, at the end of the prior column, that the Sun Violates Reason, Too:
Theorizing really didn't help me a bit on this one. What helped was trying to think what I might be missing in the observations. They were about sunrise, while I was getting together breakfast in each of those dark days. Why no attention to sunset? Just because I was busy at work in a lab with no windows. Not a good scientific reason. So why not try looking at the riddle from the other end, by noting times of sunset, too? Too late to do it personally, but they can be looked up retrospectively. Turns out sunsets are also getting later, but that is only what was expected after solstice. Qualitatively. Are they getting later at the expected rate, according to my vision of a ball spinning on a tilted axis as it revolves about 1 degree/day about the Sun? Or an easier question: what about the interval from rise to set? That is supposed to minimize on the solstice, then increase from the bottom of a sine function. And it did. So my conceptual model seems right on target .... except for this: that both rise and set times, while getting farther apart exactly as expected, are drifting together later and later. That is what was not expected. Later relative to what? Well, to the kitchen clock, or to noon, or to anything based on 24-hour periods. Maybe the problem is not with sunrise or sunset so much as with "Noon".
This shift of focus, brought on by looking at the data from the ignored other side, provides the missing key. The actual Sun, in other words, might be getting ahead of the average schedule of 24 hours: its motion across the background stars is not uniform! This is the same sort of exception to Aristotelian astronomy as we encountered in trying to time the Moon's appearances for catching a Rainbow Moon. I was eventually able to infer from the data that the Sun progresses fastest somewhere between 2 and 14 January (when sunrise is as late as it ever gets and does not noticeably change.) Why? A possibility familiar to most of us since the time of Keeper and Newton is that the Earth's orbit around the Sun has some (though very much less than the Moon's) eccentricity, and Earth moves faster when nearer the Sun. Rummaged web next: http://www.analemma.com/ confirmed this, with estimated perihelion date 2 January, blowing away my childhood misconception that summer is when the Sun is closer (which, ironically because the reasoning it nutty in any case, is not a mistake for kids in the southern hemisphere!) I think with a little more effort, i.e., quantifying the excess angular velocity as a percentage of the mean and conserving angular momentum by requiring a constant product of distance² * angular velocity, you might figure out how much closer and then estimate the percentage increase of solar irradiation: it should be same as the angular velocity increment, both being inversely proportional to distance². How much is this compared to the effect of changed tilt of the land to noontime sunlight (twice 23.5 degrees) ?

Next time: Conclusion to Reason Takes a Lesson, and wrap-up of trysts with the Moon.

Copyright 2002 by A.T.Winfree. All rights reserved. Used by permission.