You see the crescent moon all over, not just in the sky. It appears on nation's flags, coffee mugs and boxes of soap powder. Remember the old logo for Proctor and Gamble? The lunar crescent is a common sight in art, from a child's nighttime drawings to Van Gogh paintings.

How would an astronomer, obsessed with accuracy, draw a crescent, a half moon, gibbous or any other phase? Is it just two circular segments joined by their ends? The crescent itself seems part of a larger circle due to a phenomenon known as irradiation. The great naked-eye observer Tycho Brahe estimated the relative diameters as 6:5. Can this be just an illusion? And what is the shape of earthshine, the ashy, purplish light on the disk apart from the crescent moon? While it's surprising how often this is drawn, the crescent cannot be bowed upward in the "night" sky, since the Sun must be nearby and clearly above the horizon.

The bright limb is that illuminated by the Sun. Since the Moon is spherical, a half circle, or a circular segment to be precise, will do for any phase. The border between light and dark, the terminator, requires a bit more thought: it is nearly the Moon's half circle when a thin crescent, but straight at a half moon. It is nearly circular again near the full phase.

In three dimensions, the terminator is the full circle of all sunrises and sets on the Moon. It lies in a plane through the Moon's center, the terminator geometrically being the intersection of this plane and the spherical surface. As the Moon orbits the Earth, increasing its "age" and phase, this plane rotates eastward (on our, visible side), so we see the terminator circle—the half on our side, really—as increasingly foreshortened, reduced to a line, then expanded to a half circle again. The phase angle, the number of degrees from the Sun to the Earth as seen from the Moon, is the important term, but it is more convenient to think in terms of the percentage of the visible face illuminated, from essentially 0% at new to almost 100% at full.

A bit of algebra shows that this terminator curve appears as an ellipse, the elliptical segment we see passing from circle to line back to circle as a function of phase angle. Interestingly, this is the same form of curve Johannes Kepler found for the planets' orbits about the Sun, though surely this is just a coincide of geometry and conservation of angular momentum. For a computer, an ellipse is just another conic curve, a generalization of a circle, so it is a cinch to draw on screen. It's a bit harder to be so accurate with only pencil and paper: I never have been able to get the string and two pushpins to guide my pencil in a respectable ellipse.

So just how full is the Full Moon? Hardly a trick question: the Moon is only truly "full" when at lunar eclipse, directly opposite the Sun, in the shadow of the Earth! Since the Moon's orbit is inclined more than 5° to the ecliptic, the Moon can swing that far above and below the Sun or the point opposite it. Otherwise we would have eclipses twice every month. Full moon technically occurs when the Moon's longitude is 180° from the Sun's whether above or below it. So at Full Moon except at an eclipse, there is a sliver along the top or bottom in the shadow, and the Moon is less than 100% full. Although straightforward enough for the astronomer to calculate, this difference is scarcely noticeable, and the moon in fact appears in the sky full for several days. In contrast, because our vision is far better at detecting straight lines, the moment of the half moon phase can be judged by eye accurately to within hours.

All the planets, even the Earth seen from above, exhibit the phases. Mercury, Venus and Mars all show clearly certain moon-like phases in a telescope. Because of their relatively greater distance from us and the Sun, Jupiter and the rest of the outer planets are never less than 99% full. Jupiter is obviously not round, due to flattening at the poles from its break-neck rotational speed. But if we take its shape to be an ellipsoid, would its terminator, as seen from spacecraft Galileo or one the gas giant's many moons, be an elliptical segment as well? What if we allow for an atmosphere and refraction at the horizon? Is the terminator still a circle appearing as an elliptical segment? What about the twilight lines?

Though the Full Moon lights the nightscape like no other phase, much to the dismay of the backyard observer, it doesn't hold a candle to the crescent in my twilit sky. It's all the same Moon though—joined circle and ellipse—just viewed form a different phase angle.

Moondark is written by Doug Miller and published on the web, in the Delmarva Star Gazers' Star Gazer News and in the Delaware Astronomical Society's FOCUS. Please address comments and suggestions to dmiller@udel.edu. This document was last revised on 10 January '00. All text and images copyright © 2000 Douglas C. Miller, All Rights Reserved. This material may not be reproduced in any form without prior permission.