Earth-Moon Relations

The moon is the closest astronomical body to the Earth, with a mass approximately 1.2 percent that of Earth. This unusually large fraction gives the moon significant influence over the orbital and rotational motion of Earth, creating tides strong enough to have important geologic and oceanographic effects, among them variations in the length of the day. The moon, along with the sun, causes Earth’s spin axis to precess with a period of 26,000 years.

Overview

The moon is the most prominent astronomical body after the sun. It is the closest astronomical body to the Earth, orbiting at an average center-to-center separation of 384,000 kilometers. The moon has a radius of 1,740 kilometers; at this distance, it appears to be 0.5° in angular width. The mass of the moon is 3.74 10²² kilograms and the density of the moon is 3.3 grams per cubic centimeter. Earth, by contrast, has a mass of 5.97 10²⁴ kilograms and a radius of 6,380 kilometers, giving it a density of 5.5 grams per cubic centimeter, substantially more than that of the moon. The lower density of the moon, along with its lack of a magnetic field, argues that the moon lacks a molten metallic core such as the Earth has.

Earth is close enough for material thrown off of the moon by meteorite impact (called “ejecta”) to fall onto it. A small number of meteorites discovered in desert areas or in Antarctica closely resemble lunar rocks collected by the Apollo astronauts and have been verified as of lunar origin.

The moon and Earth are gravitationally bound. They orbit around a common point, called the barycenter, with a period of 27.3 days. This period is called the sidereal month and represents the time for the Earth-moon system to complete one rotation with respect to the stars. The synodic month, by contrast, is 29.5 days, the time between successive full moons.

The Earth-moon system is gravitationally bound to the sun. Hence, the barycenter orbits the sun in obedience to Johannes Kepler’s three laws of planetary motion: the orbit of the barycenter is an ellipse with the sun at one focus; the line from the center of the sun to the barycenter sweeps out equal areas in equal times; and, the cube of the radius of the barycenter orbit is proportional to the square of the period. The barycenter lies on a line joining the center of Earth to the center of the moon, at a point 4,680 kilometers from the center of Earth. This distance is 73 percent of the radius of Earth. An observer on Mars would see Earth displaced from the ideal elliptical orbit by as much as 3/8 of its diameter.

The motion of Earth about the barycenter is superimposed on the elliptical motion of the barycenter about the sun in a complicated manner. Earth oscillates back and forth across the barycenter ellipse, spending half of a synodic month inside the ellipse (toward the sun) and the other half outside the ellipse. Simultaneously, the Earth oscillates above and below the ecliptic (the plane of Earth’s orbit) spending half of a sidereal month above the plane and the other half below it. These back-and-forth and up-and-down oscillations are not necessarily synchronous. When the Earth is inside the ellipse, the moon is outside it, and vice versa. A similar arrangement holds for the up-and-down displacements. Absent the moon, the center of the Earth would coincide with the barycenter and the planetary motion of the Earth would be close to the elliptical ideal. The Earth-moon system, on the other hand, has one of the most complicated motions in the solar system.

The Earth-moon system is like an unbalanced dumbbell tumbling end over end about the barycenter. The gravitational pull is the bar holding the dumbbell together. The sides of the Earth and moon facing each other can be referred to as the inner sides, and the opposite sides of each can be referred to as the “outer” sides. The gravitational force falls off with distance, making the gravitational pull on the inner side of each body stronger than the gravitational pull on the outer side. This inequality of forces is referred to as gravitational tidal force. Neither the Earth nor the moon is rigid. Each is plastic enough to change shape under the influence of the tidal force. The gravitational pull of the moon raises a bulge in the Earth more or less directly under the moon; the bulge is matched by a similar one at a location more or less directly opposite the moon. The bulge in the ocean presents itself as the familiar tides. Similar but less familiar tides exist in the atmosphere and in the Earth’s crust. The rotation of the Earth attempts to carry these bulges away from the point directly under the moon, resulting in a slight sideways component to the mutual gravitational pull. This sideways pull acts as a brake on the rotational motion of the Earth, slowing it down and increasing the length of the day. The increase is approximately one-thousandth of a second per century, but it has been accumulating since the creation of the moon billions of years ago. Growth-ring counts in fossil coral from 400 million years ago seem to indicate that the year (whose length should not change) consisted of about 400 days back then; now a year consists of about 365 days. In other words, the length of the day has increased by about 10 percent in the past 400 million years.

The sideways pull on the moon is in the direction of its orbital motion around Earth. Extra energy imparted by the pull increases the radius of the moon’s orbit and also increases the length of the sidereal month. Since the length of the day is increasing faster than the length of the sidereal month, eventually the two will become equal and the day and month will be the same. At that time, the Earth will always present the same face to the moon, just as the moon always presents the same face to the Earth today.

The rotation of the Earth gives it an oblate shape that is thicker at the equator than through the poles. The gravitational pull of the sun and moon on this equatorial bulge acts as a torque that causes the Earth to precess like a top. The spin axis of the Earth currently points toward Polaris, the pole star, but this is only an accident of history. In 13,000 years, Vega (in the constellation Lyra) will be the pole star.

Knowledge Gained

The bulk of Earth-moon interactions are gravitational and are known from earthbound observations. The apparent location of the sun in the zodiac on the first day of spring (recognized as the day that the sun rose due east and set due west) held great cultural and religious significance to ancient civilizations and was monitored closely. Over the centuries, it became clear that this location, originally in the constellation Taurus, had moved to the constellation Aries. The Greek astronomer Hipparchus discovered this fact about 130 BCE. and from it deduced the 26,700-year circular motion of the north celestial pole. In 1530, Nicolaus Copernicus recognized this as due to drift of the Earth’s rotational axis with respect to the fixed stars, and Sir Isaac Newton in 1687 showed the phenomenon to be an effect of moon’s gravitational influence on the Earth.

Edmond Halley in 1693 and Immanuel Kant in 1754 used Newtonian gravitational theory to calculate the locations, dates, and times of total solar eclipses discussed in ancient Greek and Roman documents. Their calculations argued that the eclipses could not have taken place at the dates and places recorded. The discrepancies were eventually traced to changes in the length of the day due to tidal braking.

Starting with Apollo 11, each subsequent lunar landing mission (except the ill-fated Apollo 13) brought back significant amounts of lunar rock for scientific study. Oxygen derived from the lunar material proved to have the same ratio of isotopes as oxygen found on Earth. In contrast, oxygen retrieved from meteorites believed to be of Martian origin had substantially different isotopic ratios.

This discovery, in conjunction with the observation that the moon lacks an iron core, led to the impact theory of lunar origin. In this theory, the young Earth and a body approximately the size of Mars collided some 4.5 billion years ago. The collision threw a substantial amount of the Earth’s crust into space, where some of the material coalesced into the moon, with the remainder falling back to Earth. Since this happened after the bulk of the iron in the proto-Earth had sunk into the core, the material that formed the moon was relatively iron-free.

Context

The combined motion of the Earth and the moon around their common barycenter is one of the most complicated problems in celestial mechanics. Newton once referred to it as the only problem that ever gave him a headache. Several factors complicate the solution. The influence of the sun makes the problem a three-body gravitational interaction rather than the simpler two-body problem conquered by Kepler. Unlike the two-body problem, the three-body problem cannot be solved in closed analytic form; particular approximate solutions exist for special configurations, but the sun-Earth-moon trio does not conform to any of them. Several potential solutions have been found for the three-body problem; one of the most important of which is that the related motion of the three bodies is nonrepeating. Since Newton first attempted to tackle the problem, a total of sixteen families of solutions to the problem have been proposed: the Lagrange-Euler family, the Broucke-Hénon family, the figure-eight family, and thirteen additional families discovered by Milovan Šuvakov and Veljko Dmitrašinović from the Institute of Physics Belgrade in 2013. In 2021, a pair of physicists at Technion-Israel Institute of Technology claimed to have solved the three-body problem by calculating the probability of the third body’s movement. They relied on a series of random movements known as the “drunkard’s walk.”

The Earth and moon are also too close for either to be regarded as point masses. Further, neither is purely spherical: the Earth is ellipsoidal, with an equatorial bulge as a product of its rotation; the moon is oval as a result of a permanent tidal bulge on the side facing the Earth. The rotational period of the moon equals its orbital period, so that one face perpetually faces the Earth, but the orbit is not circular, so that the moon moves along the orbit at a varying rate. This causes the side of the moon facing the Earth to rock back and forth, a motion known as libration. The deviation from circularity (called the eccentricity) is itself variable, driven by the gravitational pull of the sun, so that the the extent of the libration waxes and wanes. This variation in eccentricity is called evection.

Bibliography

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