The Phases of Earth as Seen from the Moon

Author: Congyu
Reviewed by: Yuandao and Shiguang

  What does Earth look like from the Moon? If you know something about Earth, the Moon, and lunar phases, you may readily notice that something is amiss in this image. The screenshot comes from Horizon Lunar Colony, a map in the game Overwatch. Setting its artistry and visual impact aside, we will begin with Earth and the Moon and introduce a few basic ideas about the phases of celestial bodies.

Figure 1: Screenshot from Overwatch. The image greatly exaggerates Earth's apparent size. From the Moon, Earth has an apparent diameter of about 1.9 degrees, only 3.6 times that of the Sun.
  The phase of a celestial body is determined by the relative positions of the Sun, the body, and the observer. It reflects how much of the body's sunlit portion we can see. Take the familiar phases of the Moon. Because the distance from the Sun to Earth is far greater than the distance from Earth to the Moon, we can use the Sun–Earth–Moon angle—that is, the angular separation between the Sun and Moon—as a simple measure of how much of the Moon's illuminated face is visible. As the angular separation between the Moon and Sun changes, the lunar phases follow this sequence: new moon, waxing crescent, first quarter, waxing gibbous, full moon, waning gibbous, last quarter, waning crescent, and new moon. A quarter moon is separated from the Sun by about 90 degrees, while a full moon is about 180 degrees from it.
  A Moon at a given phase occupies a definite position relative to the Sun. You can use the Moon's direction in the sky to infer the Sun's position and then estimate the local time. For example, if we ignore the inclination between the Moon's orbit and the ecliptic, a first-quarter moon must rise or set six hours after the Sun, while a last-quarter moon does so six hours before it.
Figure 2: The relationship between the relative positions of the Sun, Earth, and Moon and the lunar phases. The diagram also shows why the sidereal month and synodic month have different lengths.
  The same rules apply to the “phases of Earth” observed from the Moon. Look at Earth's terminator in the opening game screenshot (Figure 1). Earth appears to be in a waning gibbous phase close to last quarter, yet its angular separation from the Sun at upper left is clearly less than 90 degrees. This means that most of its illuminated side lies on the far side of Earth, out of view, so it should appear as a waning crescent.
  The angle used to describe a celestial body's phase accurately is actually the Sun–Moon–Earth angle, known as the phase angle. As Figure 3 shows, it ranges from 0° to 180°, and the illuminated portion corresponds to an angle of 180°-ψ. Seen from a distance, the terminator separating the bright and dark sides of a celestial body is an elliptical arc—one half of an ellipse divided along its major axis. The extremely curved crescents seen in some animated films and other screen works do not occur as normal phases of celestial bodies. During a lunar eclipse, moreover, the curvature of the edge obscuring the Moon differs from the curvature of a lunar phase. Near the Moon's orbit, the radius of Earth's umbra is 2.6 times the Moon's radius, so the obscuring edge is less strongly curved than the Moon itself.
Figure 3: In the left-hand diagram, ∠SME is the phase angle ψ and is readily seen to equal ∠AMB. The right-hand diagram shows the illuminated and dark sides from the observer's perspective.
  The Moon is not the only Solar System body with phases; every other body has them as well. The inner planets pass through a complete set of phases, but in the reverse order to the Moon: inferior conjunction corresponds to new moon, greatest western elongation to last quarter, superior conjunction to full moon, and greatest eastern elongation to first quarter. Their respective phase angles are 180°, 90°, 0°, and 90°. Because the outer planets orbit beyond Earth, their phase angles never exceed 90°. By analogy with greatest elongation for an inner planet, it is easy to see that when Earth is at greatest elongation relative to an outer planet, that planet's phase angle as observed from Earth reaches its maximum. We therefore cannot see an outer planet in a crescent phase from Earth. Even Mars, whose orbit lies closest to ours, presents at least 84% of its illuminated face.
  Finally, let us admire the haunting view of Earth photographed in 1968 by Apollo 8 from lunar orbit. The astronauts aboard Apollo 8 marveled as Earth appeared to “rise” above the Moon's horizon outside their window. It is worth noting that such earthrises and earthsets would be difficult to see from the lunar surface rather than lunar orbit. The Moon is tidally locked to Earth and always keeps the same side facing us. To someone stationed on the Moon, Earth would therefore remain suspended at a fixed altitude and direction in the sky, moving only slightly because of lunar libration.
Figure 4: A digitally remastered image of Earth. The original was a black-and-white photograph taken by Apollo 8.