# Phase

## Description

Phase is a technical term used in astronomy to mean the theoretical fraction of a planetary body that is in sunlight. Its computation is always based on the assumption that the body acts like an idealized sphere, and hence should not be expected to correlate precisely with the fraction actually observed (the actual fraction of the surface that is illuminated by sunlight will differ because some areas that would be expected to be in sunlight on a perfect sphere may, in fact, be in shadow and vice versa). A planetary body's phase can be expressed either as a fraction between 0 and 1, or as a percentage (also known as the percent illumination -- equal to the fraction illuminated times 100). A phase of 0 means the object is (theoretically) completely dark. A phase of 1 (100%) means it is completely illuminated. The observed phase depends on the positions of both the Sun and the observer relative to the body, so it has no single value at a particular time. As observed from Earth, the Moon's phase varies from near 0 to near 1 during each month-long lunar cycle, but it seldom attains either limit. Attaining an exact value of 0 or 1 requires a precise alignment of the Sun and Moon, something that seldom happens, even during eclipses. - Jim Mosher

• The phase of a lunar photo can be estimated by noting the fraction of the Moon's equatorial diameter that is in sunlight. If chords are drawn perpendicular to the "vertical" axis of the image (a line through the illumination-defined "poles" of a crescent or gibbous Moon), and the position of the dark limb estimated, then the fraction of the length of each "horizontal" chord (the lines drawn from limb to limb parallel to the "equator") that is in sunlight will be approximately the same, and this number will be equal to the phase. For a perfect sphere, the illuminated fraction along each of these chords would be exactly the same, and hence equal to illuminated fraction for the sphere as a whole, so the relationship would be exact. - Jim Mosher
• The Moon's phase is related to, but should not be confused with, the Moon's Primary Phases (New, First Quarter, and so on: the names used to describe the appearance of the changing patterns of illumination observed from Earth during the lunar cycle). First, the Phases are defined in terms of an imaginary observer at the Earth's center (whereas the observed phase is specific to the observer -- not necessarily on Earth). Second, even for the imaginary geocentric observer, the phase at a particular Primary Phase is not precisely the same each month, nor does it have quite the expected value: for example, the precise phase at First Quarter is a little different each month, and (because of the non-circular shape of the Moon's orbit, and the definition of the Primary Phases as angular -- rather than illumination -- points in the orbit) the average over many months is not quite 0.5 (50%). - Jim Mosher
• Mathematically, the Moon's phase is often in expressed in terms of the phase angle, from which it can be computed. The phase angle is the angle between the Sun and the observer's location on Earth as seen from the Moon's center. Since the phase is defined for a perfect sphere, the following relationship connecting the theoretical phase (P) and phase angle (PA) is exact:

• P = (1 + Cosine(PA))/2

• To obtain the ("theoretical") percent illumination (PI), multiply P by 100.

• PI = 50 x (1 + Cosine(PA))

• These relationships can also be inverted to determine the elongation angle based on the phase or percent illumination:

• PA = ArcCosine(2 x P - 1)
• PA = ArcCosine(PI/50 - 1)

• If the phase or percent illumination is an observational estimate, the resulting phase angle will also be only an estimate, since the input will not precisely match the values expected for a perfect sphere. If the phase or percent illumination is a theoretical value calculated for a sphere, then the resulting phase angle will be exact.

• Since it is much easier to observe the apparent angle (in the sky) between the Sun and Moon (the Moon's elongation angle, or EA), than to visualize the relative positions of the Sun and Earth as seen from the Moon, it is useful to recast these formulae in terms of EA.

• The EA and PA correspond to two corners of the triangle connecting Sun, Moon and observer. As in all triangles, the three angles add to 180°. The third angle is the one between the observer and Moon as seen from the Sun's center. Since the Moon is never more than 406,000 km from Earth, and it is roughly 150,000,000 km from the Sun, this angle can never exceed 0.16°. So we have approximately:

• EA = 180° - PA

• This means the cosine of the EA is roughly the negative of the cosine of the EA; so the exact formulae relating PI to PA can be recast in the following approximate forms:

• PI = 50 x (1 - Cosine(EA)) (approximately)
• EA = ArcCosine(1 - PI/50) (approximately)

• This approximation is especially good when the Moon is nearly New or Full, and the third angle of the Sun-Moon-Observer triangle is extremely small. It is off very slightly when the Sun and Moon appear 90° apart in the sky. The approximate formulae would predict that the illumination at that moment is exactly 50%, but in fact the phase angle between Sun and observer as seen from the Moon (which really determines the illumination) is less, at that moment, by the third angle, which is then in the range 0.14 to 0.16° -- so the illumination has not quite reached 50% at the moment we see the Sun and Moon 90° apart.

• The theoretical difference between the moments of 50% illumination and 90° apparent separation is a classic problem of astronomy related to the relative sizes of the Earth-Moon and Earth-Sun distances. Although various estimates and claims were made, the only real conclusion that could be obtained with the naked eye observations of antiquity was that the difference was extremely small so the Sun must be much farther away than the Moon.