RESEARCH STARTER

Earth-Moon Relations

Earth-Moon Relations refer to the complex gravitational interaction between Earth and its natural satellite, the Moon. The Moon orbits Earth at an average distance of about 384,000 kilometers and revolves around a common center of mass known as the barycenter, which is located about 4,680 kilometers from the center of Earth. This relationship affects various phenomena, such as ocean tides, which occur due to the Moon's gravitational pull creating bulges in Earth's waters. Over time, this tidal interaction has led to a gradual increase in the length of Earth's day and the Moon's orbital radius.

The dynamics of the Earth-Moon system are intricate, as both bodies influence each other's motion through gravitational forces. The Moon's rotation is synchronized with its orbit, resulting in the same side always facing Earth. This phenomenon, combined with the elliptical shape of their orbits, leads to additional movements known as libration, where the Moon appears to rock slightly. Historically, the Earth-Moon relationship has been the subject of significant scientific inquiry, contributing to our understanding of celestial mechanics and the origins of the Moon itself, which is believed to have formed from debris after a collision between Earth and a Mars-sized body. As a result, exploring these relations continues to reveal insights into both our planet's history and the mechanics of our solar system.

Full Article

The Moon is the closest astronomical body to 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 Earth, orbiting at an average center-to-center separation of 384,000 kilometers (238,606.5 miles). The Moon has a polar radius of 1,736 kilometers (1,081 miles); at this distance, it appears to be 0.5 degrees in angular width. The mass of the Moon is 7.34 × 1022 kilograms (1.62 × 1023 pounds), and the density of the Moon is 3,344 kilograms per cubic meter (208.76 pounds per cubic foot). Earth, by contrast, has a mass of 5.97 × 1024 kilograms (1.32 × 1025 pounds) and a polar radius of 6,356.8 kilometers (3949.9 miles), giving it a density of 5,514 kilograms per cubic meter (344.22 pounds per cubic foot), 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, unlike Earth, which has one.

Earth is close enough for material thrown off the Moon by a 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, Earth oscillates above and below the ecliptic (the plane of the 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 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 the gravitational tidal force. Neither 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 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 Earth's crust. The rotation of 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 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. The Moon is responsible for the long-term slowing of Earth's rotation through tidal friction. An article published in 2026 shows that human-caused climate change is now also lengthening the day. Melting polar ice and rising sea levels are redistributing mass away from Earth's poles, slowing the planet's spin at a rate of about 1.33 milliseconds per century in the early twenty-first century — a pace not seen in the past 3.6 million years.

Growth-ring counts in fossil corals 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 and has continued to lengthen gradually at a rate of roughly 2.3 milliseconds per century.

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, Earth will always present the same face to the Moon, just as the Moon always presents the same face to Earth today.

The rotation of the Earth gives it an oblate shape that is thicker at the equator than at the poles. The gravitational pull of the Sun and Moon on this equatorial bulge acts as a torque that causes Earth to precess like a top. The spin axis of 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 the Earth-Moon interactions are gravitational and are known from Earth-bound 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 the 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 the Moon's gravitational influence on 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 the 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, Earth and a body approximately the size of Mars collided some 4.5 billion years ago. The collision threw a substantial amount of 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 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. Many potential solutions have been proposed 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, various 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." In 2023, an international team of mathematicians found 12,000 new solutions to the infamous problem. Because of the problem's unsolvable nature, scientists will continue to hypothesize and study the three-body problem.

Earth and the Moon are also too close for either to be regarded as point masses. Further, neither is purely spherical: 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 Earth. The rotational period of the Moon equals its orbital period, so that one face perpetually faces 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 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 extent of the libration waxes and wanes. This variation in eccentricity is called evection.


Bibliography

Cartwright, Jon. "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem." Science, 8 Mar. 2013, www.science.org/content/article/physicists-discover-whopping-13-new-solutions-three-body-problem. Accessed 6 Apr. 2026.

Comins, Neil F. What if the Moon Didn't Exist? Voyages to Earths That Might Have Been. HarperCollins, 1993.

Cowing, Keith. "The Length of a Day on Earth and Its Impact on the Evolution of Life." Astrobiology, 19 Aug. 2024, astrobiology.com/2024/08/the-length-of-a-day-on-earth-and-its-impact-on-the-evolution-of-life.html. Accessed 6 Apr. 2026.

ETH Zurich. "Climate Change Is Slowing Earth's Spin at Unprecedented Rate Compared to Past 3.6 Million Years." Phys.org, 12 Mar. 2026, https://phys.org/news/2026-03-climate-earth-unprecedented-million-years.html. Accessed 6 Apr. 2026.

Ferguson, Kitty. Tycho and Kepler: The Unlikely Partnership That Forever Changed Our Understanding of the Heavens. Walker, 2002.

Hamer, Ashley. "Physicists Crack Unsolvable Three-Body Problem Using Drunkard's Walk." LiveScience. 4 Jan. 2022, www.livescience.com/three-body-problem-solution. Accessed 6 Apr. 2026.

Kiani Shahvandi, Mostafa, and Benedikt Soja. "Climate-Induced Length of Day Variations Since the Late Pliocene." Journal of Geophysical Research: Solid Earth, vol. 131, no. 3, 2026, e2025JB032161, doi:10.1029/2025JB032161. 

Kolerstrom, Nicholas. Newton's Forgotten Lunar Theory: His Contribution to the Quest for Longitude. Green Lion, 2000.

Lewis, Briley. "Mathematicians Find 12,000 New Solutions to 'Unsolvable' 3-Body Problem." Space, 24 Sept. 2023, www.space.com/mathematicians-unsolvable-3-body-problem-12000-solutions. Accessed 6 Apr. 2026.

"Moon Fact Sheet." NASA Goddard Space Flight Center, 10 Dec. 2025, nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html. Accessed 7 Apr. 2026.

Moore, Patrick. On the Moon. Cassell, 2001.

Full Article

The Moon is the closest astronomical body to 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 Earth, orbiting at an average center-to-center separation of 384,000 kilometers (238,606.5 miles). The Moon has a polar radius of 1,736 kilometers (1,081 miles); at this distance, it appears to be 0.5 degrees in angular width. The mass of the Moon is 7.34 × 1022 kilograms (1.62 × 1023 pounds), and the density of the Moon is 3,344 kilograms per cubic meter (208.76 pounds per cubic foot). Earth, by contrast, has a mass of 5.97 × 1024 kilograms (1.32 × 1025 pounds) and a polar radius of 6,356.8 kilometers (3949.9 miles), giving it a density of 5,514 kilograms per cubic meter (344.22 pounds per cubic foot), 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, unlike Earth, which has one.

Earth is close enough for material thrown off the Moon by a 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, Earth oscillates above and below the ecliptic (the plane of the 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 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 the gravitational tidal force. Neither 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 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 Earth's crust. The rotation of 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 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. The Moon is responsible for the long-term slowing of Earth's rotation through tidal friction. An article published in 2026 shows that human-caused climate change is now also lengthening the day. Melting polar ice and rising sea levels are redistributing mass away from Earth's poles, slowing the planet's spin at a rate of about 1.33 milliseconds per century in the early twenty-first century — a pace not seen in the past 3.6 million years.

Growth-ring counts in fossil corals 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 and has continued to lengthen gradually at a rate of roughly 2.3 milliseconds per century.

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, Earth will always present the same face to the Moon, just as the Moon always presents the same face to Earth today.

The rotation of the Earth gives it an oblate shape that is thicker at the equator than at the poles. The gravitational pull of the Sun and Moon on this equatorial bulge acts as a torque that causes Earth to precess like a top. The spin axis of 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 the Earth-Moon interactions are gravitational and are known from Earth-bound 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 the 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 the Moon's gravitational influence on 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 the 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, Earth and a body approximately the size of Mars collided some 4.5 billion years ago. The collision threw a substantial amount of 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 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. Many potential solutions have been proposed 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, various 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." In 2023, an international team of mathematicians found 12,000 new solutions to the infamous problem. Because of the problem's unsolvable nature, scientists will continue to hypothesize and study the three-body problem.

Earth and the Moon are also too close for either to be regarded as point masses. Further, neither is purely spherical: 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 Earth. The rotational period of the Moon equals its orbital period, so that one face perpetually faces 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 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 extent of the libration waxes and wanes. This variation in eccentricity is called evection.


Bibliography

Cartwright, Jon. "Physicists Discover a Whopping 13 New Solutions to Three-Body Problem." Science, 8 Mar. 2013, www.science.org/content/article/physicists-discover-whopping-13-new-solutions-three-body-problem. Accessed 6 Apr. 2026.

Comins, Neil F. What if the Moon Didn't Exist? Voyages to Earths That Might Have Been. HarperCollins, 1993.

Cowing, Keith. "The Length of a Day on Earth and Its Impact on the Evolution of Life." Astrobiology, 19 Aug. 2024, astrobiology.com/2024/08/the-length-of-a-day-on-earth-and-its-impact-on-the-evolution-of-life.html. Accessed 6 Apr. 2026.

ETH Zurich. "Climate Change Is Slowing Earth's Spin at Unprecedented Rate Compared to Past 3.6 Million Years." Phys.org, 12 Mar. 2026, https://phys.org/news/2026-03-climate-earth-unprecedented-million-years.html. Accessed 6 Apr. 2026.

Ferguson, Kitty. Tycho and Kepler: The Unlikely Partnership That Forever Changed Our Understanding of the Heavens. Walker, 2002.

Hamer, Ashley. "Physicists Crack Unsolvable Three-Body Problem Using Drunkard's Walk." LiveScience. 4 Jan. 2022, www.livescience.com/three-body-problem-solution. Accessed 6 Apr. 2026.

Kiani Shahvandi, Mostafa, and Benedikt Soja. "Climate-Induced Length of Day Variations Since the Late Pliocene." Journal of Geophysical Research: Solid Earth, vol. 131, no. 3, 2026, e2025JB032161, doi:10.1029/2025JB032161. 

Kolerstrom, Nicholas. Newton's Forgotten Lunar Theory: His Contribution to the Quest for Longitude. Green Lion, 2000.

Lewis, Briley. "Mathematicians Find 12,000 New Solutions to 'Unsolvable' 3-Body Problem." Space, 24 Sept. 2023, www.space.com/mathematicians-unsolvable-3-body-problem-12000-solutions. Accessed 6 Apr. 2026.

"Moon Fact Sheet." NASA Goddard Space Flight Center, 10 Dec. 2025, nssdc.gsfc.nasa.gov/planetary/factsheet/moonfact.html. Accessed 7 Apr. 2026.

Moore, Patrick. On the Moon. Cassell, 2001.

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