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The orbit of the Moon around the Earth is completed in approximately 27.3 days. The Earth and Moon orbit about their common center of mass, which lies about 4,700 kilometres from Earth's center (about three quarters of the Earth's radius). On average, the Moon is at a distance of about 385,000 km from the center of the Earth, which corresponds to about 60 Earth radii. With a mean orbital speed of 1.023 km/s, the Moon moves relative to the stars each hour by an amount roughly equal to its angular diameter, or by about 0.5°. The Moon differs from most satellites of other planets in that its orbit is close to the plane of the ecliptic, and not to the Earth's equatorial plane. The lunar orbit plane is inclined to the ecliptic by 5.1°, whereas the Moon's spin axis is inclined by only 1.5°.

Properties of the lunar orbit

Definition of orbital parameters.

Property	Value
Semi-major axis	~384 748 km
Distance at perigee	~364 397 km
Distance at apogee	~406 731 km
Mean eccentricity	0.0549006 (0.044 – 0.067)
Mean inclination of orbit to ecliptic	5.14
Mean obliquity	6.58°
Mean inclination of lunar equator to ecliptic	1.543°
Period of precession of nodes	18.5996 years
Period of recession of line of apsides	8.8504 years

Lunar perigee–apogee size comparison

Because of the relatively large size of the Moon and the small Earth–Moon mass ratio of about 81:1 when compared to other natural satellites in our solar system, some consider the Earth–Moon system to be a double planet. Based on the informal definition that a double planet system must have its barycenter exterior to both bodies, however, others regard the Earth and Moon as an ordinary planet–moon system.

The orbit of the Moon is distinctly elliptical with an average eccentricity of 0.0549. The non-circular form of the lunar orbit causes variations in the Moon's angular speed and apparent size as it moves towards and away from an observer on Earth. The mean angular daily movement relative to an imaginary observer at the barycenter is 13.176358° to the east. The orientation of the orbit is not fixed in space, but precesses over time. One motion is the precession of the line of apsides: the ellipse of the lunar orbit slowly rotates counterclockwise, and completes a full revolution in about 8.850 years (3,233 days). The other motion is associated with the (clockwise) precession of the orbital plane itself about an axis perpendicular to the ecliptic. The points where the lunar orbit intersects the ecliptic (the nodes) precess with time, completing one revolution in about 18.6 years (6,793 days).

The mean inclination of the lunar orbit to the ecliptic plane is 5.145°. The rotation axis of the Moon is also not perpendicular to its orbital plane, so the lunar equator is not in the plane of its orbit, but is inclined to it by a constant value of 6.688° (this is the obliquity). One might be tempted to think that as a result of the precession of the Moon's orbit plane, the angle between the lunar equator and the ecliptic would vary between the sum (11.833°) and difference (1.543°) of these two angles. However, as was discovered by Jacques Cassini in 1721, the rotation axis of the Moon precesses with the same rate as its orbit plane, but is 180° out of phase (see Cassini's Laws). Thus, although the rotation axis of the Moon is not fixed with respect to the stars, the angle between the ecliptic and the lunar equator is always 1.543°.

The Moon orbiting Earth, sizes and distances to scale.

The properties of the orbit described in this way are approximations. The Moon's orbit around the Earth has many irregularities (perturbations), and their study (lunar theory) has occupied astronomers over a long history.

History of observations and measurements

About 3000 years ago, the Babylonians were the first human civilization to keep a consistent record of lunar observations. Clay tablets from that time period found over the territory of present-day Iraq are inscribed with cuneiform recording the times and dates of moonrises and moonsets, the stars that the Moon passed close by, and the time differences between rising and setting of both the Sun and the Moon around the time of the full moon. The Babylonians discovered the three main periods of the Moon's motion and used data analysis to build lunar calendars that extended well into the future. This use of detailed, systematic observations to make predictions based on experimental data may be classified as the first scientific study in human history. However, Babylonians seem to have lacked any geometrical or physical interpretation of their data, and they could not predict future lunar eclipses (although "warnings" were issued before likely eclipse times).

Ancient Greek astronomers were the first to introduce and analyze mathematical models of the motion of objects in the sky. Ptolemy described lunar motion by using a well-defined geometric model of epicycles and evection.

Newton was the first to develop a complete theory of motion, mechanics. The sheer wealth of humanity's observations of the lunar motion was the main testbed of his theory.

Lunar month

The Moon's periods
Name	Value (d)	Definition
sidereal	27.321 661	with respect to the distant stars (13.369 passes per year)
synodic	29.530 589	with respect to the Sun (phases of the Moon, 12.369 cycles per year)
tropical	27.321 582	with respect to the vernal point (precesses in ~26,000 a)
anomalistic	27.554 550	with respect to the perigee (recesses in 3232.6 d = 8.8504 a)
draconic (nodical)	27.212 221	with respect to the ascending node (precesses in 6793.5 d = 18.5996 a)

There are several ways to consider how much time it takes the Moon to complete one orbit. The sidereal month is the time it takes to make one complete orbit with respect to the fixed stars, which is about 27.3 days. In contrast, the synodic month is the time it takes the Moon to reach the same phase, which takes about 29.5 days. The synodic period is longer than the sidereal period because the Earth–Moon system moves a finite distance in its orbit around the Sun during each sidereal month, and a longer time is required to achieve the same relative geometry. Other definitions for the duration of a lunar month include the time it takes to go from perigee to perigee (the anomalistic month), from ascending node to ascending node (the draconic month), and from two successive passes of the same ecliptic longitude (the tropical month). As a result of the slow precession of the lunar orbit, these latter three periods are only slightly different than the sidereal month. The average length of a calendric month (1/12 of a year) is about 30.4 days.

Tidal evolution of the lunar orbit

The gravitational attraction that the Moon exerts on Earth is the cause of tides in the sea. If the Earth possessed a global ocean of uniform depth, the Moon would act to deform both the solid earth (by a small amount) and ocean in the shape of an ellipsoid with high points directly beneath the Moon and on the opposite side of the Earth. However, as a result of the irregular coastline and varying ocean depths, this idealization is only partially realized. While the tidal flow period is generally synchronized to the Moon's orbit around Earth, its phase can vary. In some places on Earth there is only one high tide per day, though this is somewhat rare.

The tidal bulges on Earth are carried ahead of the Earth–Moon axis by a small amount as a result of the Earth's rotation. This is a direct consequence of friction and the dissipation of energy as water moves over the ocean bottom and into or out of bays and estuaries. Each bulge exerts a small amount of gravitational attraction on the Moon, with the bulge closest to the Moon pulling in a direction slightly forward along the Moon's orbit, because the Earth's rotation has carried the bulge forward. The opposing bulge has the opposite effect, but the closer bulge dominates due to its comparative closer distance to the Moon. As a result, some of the Earth's rotational momentum is gradually being transferred to the Moon's orbital momentum, and this causes the Moon to slowly recede from Earth at the rate of approximately 38 millimetres per year. In keeping with the conservation of angular momentum, the Earth's rotation is gradually slowing, and the Earth's day thus lengthens by about 17 microseconds every year. (This would make each Earth day one second longer every 60,000 years or so, and one minute longer every four million years). Looking back, the day was a mere 23 hours in length when the dinosaurs roamed the Earth 65 million years ago.) See tidal acceleration for a more detailed description and references.

So the Moon is gradually receding from the Earth into a higher orbit, and calculations suggest that this will continue for about fifty billion years. By that time, the Earth and Moon will become caught up in what is called a "spin–orbit resonance" in which the Moon will circle the Earth in about 47 days (currently 29 days), and both Moon and Earth will rotate around their axes in the same time, always facing each other with the same side. Beyond this, it is hard to tell what will happen to the Earth–Moon system, considering that the Sun is expected to become a red giant approximately 5 billion years from now.

Libration

Animation of the Moon as it cycles through its phases. The apparent wobbling of the Moon is known as libration.

The Moon is in synchronous rotation, meaning that it keeps the same face turned toward the Earth at all times. This synchronous rotation is only true on average because the Moon's orbit has a definite eccentricity. As a result, the angular velocity of the Moon varies as it moves around the Earth, and is hence not always equal to the Moon's rotational velocity. When the Moon is at its perigee, its rotation is slower than its orbital motion, and this allows us to see up to eight degrees of longitude of its eastern (right) far side. Conversely, when the Moon reaches its apogee, its rotation is faster than its orbital motion and this reveals eight degrees of longitude of its western (left) far side. This is referred to as longitudinal libration.

Because the lunar orbit is also inclined to the Earth's ecliptic plane by 5.1°, the rotation axis of the Moon seems to rotate towards and away from us during one complete orbit. This is referred to as latitudinal libration, which allows one to see almost 7° of latitude beyond the pole on the far side. Finally, because the Moon is only about 60 Earth radii away from the Earth's center of mass, an observer at the equator who observes the Moon throughout the night moves laterally by one Earth diameter. This gives rise to a diurnal libration, which allows one to view an additional one degree's worth of lunar longitude. For the same reason, observers at both geographical poles of the Earth would be able to see one additional degree's worth of libration in latitude.

Path of Earth and Moon around Sun

The Earth and Moon's path around the Sun is always concave to the Sun (far left down)

In representations of the solar system, it is common to draw the trajectory of the Earth from the point of view of the Sun, and the trajectory of the Moon from the point of view of the Earth. This could give the impression that the Moon circles around the Earth in such a way that sometimes it goes backwards when viewed from the Sun's perspective. Since the orbital velocity of the Moon about the Earth (1 km/s) is small compared to the orbital velocity of the Earth about the Sun (30 km/s), this never occurs.

Unlike most other moons in the solar system, the annual trajectory of the Moon is very similar to the one of the Earth. It is always concave towards the Sun, and is nowhere convex or looped. Although if the gravitational attraction of our Sun could be "turned off", the Moon would continue to make one orbit about the Earth with its current sidereal period. And if the Earth–Moon system could be transported to the far reaches of our solar system where the orbital velocity were less than 1 km/s, then the Moon would indeed sometimes move backwards from the vantage point of a non-rotating coordinate system centered at the Sun.

See also

• Ernest William Brown
• Lunar phase
• Lunar standstill
• Lunar theory
• Orbital elements
• Precession
• Lunar Laser Ranging Experiment
• List of orbits
• Axial tilt
• Milankovitch cycles
• Astrological age
• Great year
• Orbital eccentricity
• Equinox
• Perigee