Lunar mathematics

Summary: Though mankind has always looked up at the moon and even visited, most of the body of lunar knowledge is actually contributed by mathematics, which continues to attempt to model its motion.

The moon is the sole natural satellite of the Earth. Specific astronomical searches have established positively that the Earth has no other satellites larger than a few meters. The lunar body is nearly a sphere with a mean radius of 1738 kilometers (km) or 1000 miles—only 3.7 times less than the Earth. The mean distance of the moon from the Earth is 384,400 km (238,855 miles). The moon is the fifth largest satellite in the solar system and the largest one relative to the size of its planet. The moon is so near and so large in comparison with its “host” that the entire system is often dubbed the “double planet.”

Viewed from above the North Pole of the Earth, the moon travels around it counterclockwise in a slightly elliptical path. The sideric month (one orbit around the Earth with respect to the stars) is 27.3217 days. The synodic month (the cycle of phases visible from the Earth; for example, the time interval between two successive “new moon phases”) is 29.5306 days.

The period of one spin of the moon around its axis (a “lunar day”) is exactly equal to the sideric month because of tidal breaking. This phenomenon is also known as “synchronous rotation,” or tidal coupling.” As a result, from the Earth, people can observe only half of the lunar surface (called the “near,” or “visible,” “side”). The “far” (called “invisible”) hemisphere was photographed for the first time in 1959 by the Soviet robotic spacecraft Luna-3, an episode of the space race between the United States and the Soviet Union. On the moon, the disk of the Earth does not rise and set. It is observable only from the near side in an almost permanent point of the lunar sky (fluctuating a little from a small phenomenon called “libration”).

The face of the moon was influenced by both internal and external factors. On the surface, observers distinguish so-called darker “maria” (flat “seas” without water) and brighter highlands. All of them are covered with numerous craters, the highlands more so than the seas. The far side of the moon has practically no seas. Because of constant bombardment by various small interplanetary particles, the entire surface is enveloped with thin fractured material called “regolith.” There is no atmosphere on the moon. As a result, the difference in temperatures between a lunar day and a lunar night is very high: between –170 degrees Celsius and +130 degrees Celsius (–274 degrees Fahrenheit to +266 degrees Fahrenheit). Water in the form of subsurface ice exists in polar regions. There are no traces of modern tectonics on the surface.

From the Earth, the visible angular diameter of the moon is 0.5 degrees and fairly close to the angular diameter of the sun. This property is essential because sometimes the three bodies, the sun, the Earth and the moon, align along a straight line. In this case, humans observe either a total lunar (if the moon is farther from the sun than the Earth) or a total solar (if the moon is between the sun and the Earth) eclipse. The latter is visible only within narrow strips on the Earth. Such observations are important for solar physics. To see these phenomena, astronomers regularly organize special expeditions. Eclipses often held great religious significance. Scholar Anaxagoras of Clazomenae explained the phenomenon using mathematics. He was imprisoned for asserting that the sun was not a god and that the moon reflected the sun’s light.

The age of the moon is about 4.5 billion years, which is close to the age of the sun and the entire solar system. Of the various concepts of the moon’s origin, the prevailing hypothesis is that the Earth-moon system was formed by a giant impact: a planet-sized body hit the nearly formed proto-Earth, ejecting material into orbit around the proto-Earth, which accreted to form the moon.

The mean density of the moon is just 3.34 grams per centimeter3 and, as a result, the mass of the moon is 81 times less than that of the Earth. The interior of the moon is geochemically differentiated: it has a distinct crust, mantle and core. Surface gravity on the moon is six times less than on the Earth. The general magnetic field of the moon is practically absent.

The moon has always played a significant role in religion, science, art, and culture. Since the Paleolithic, the lunar orb in the sky has been utilized for calendar purposes. That is why the similarity of the terms “moon” and “month” is not coincidental. For the philosopher Aristotle, the moon marked a great border between a mortal and corruptible sublunar (terrestrial) world and an immortal world of ideal heavenly bodies. It became a significant symbol for Islam. For Isaac Newton, the moon was the prime test body to demonstrate mathematically that the fall of an apple and the orbiting of a celestial body are ruled by a single natural law of universal gravity.

Mathematical Modeling

Many mathematicians have developed theoretical models for the motion of the moon. The exact path of the moon around the Earth is affected by many perturbations and is extremely complicated. That is why, after Newton, research of lunar motion (lunar theory) became the central problem of celestial mechanics. Consequently, it appeared among the most critical and difficult tasks for applied mathematics. The moon’s gravitational influence on the Earth produces the ocean tides and the tiny lengthening of the calendar year. Most of what we know about the moon’s size, shape, and other properties has been derived largely through mathematical computations, using mathematical theory and data from Earth-based observations, satellite imagery, and direct measurements made by astronauts.

Human Exploration

Starting at least from Roman times, science fiction authors were the forerunners for delivering terrestrials to the moon. In reality, the first space robots to the moon were launched by the Soviets in 1959. But they failed in the space race with the United States to realize manned expeditions. The first terrestrials to visit the moon were the American astronauts of the Apollo program. After preliminary robotic programs (Ranger, Lunar Orbiter, and Surveyor) and Apollo flybys, American manned landings on the moon occurred in 1969–1972. Among seven planned landings (from Apollo-11 up to Apollo-17), six missions were tremendously successful. Twelve crewmembers stepped down on the near side of the moon, and six more orbited it. Astronauts performed a number of experiments and returned to the labs about 382 kg of lunar matter. Since 2004, Japan, China, India, the United States, and the European Space Agency have each sent successful automatic lunar orbiters.

Among the many thousands of contributors to lunar programs, mathematicians often played outstanding roles. One significant individual was mathematician Richard Arenstorf, who solved a special case of the three-body problem with figure-eight trajectories now called “Arenstorf periodic orbits.” In 1966, he was awarded a NASA medal for exceptional scientific achievement for this work. Another was Evelyn Boyd Granville, who used numerical analysis to aid in the design of missile fuses. She later worked on trajectory and orbit analyses for several space missions, including Apollo. She said, “I can say without a doubt that this was the most interesting job of my lifetime—to be a member of a group responsible for writing computer programs to track the paths of vehicles in space.” In fact, mathematicians occupied many seats in the first row of the Mission Control center. Their work was critical for calculating trajectories and for maneuvers that involved the meeting of two objects in space, including landing on the moon. They also played a significant role in determining a rapid and feasible solution that would safely return the damaged Apollo 13 manned spacecraft to Earth.

Among mathematicians in Russia, the most noticeable contribution to flights to the moon was made by Efraim L. Akim of the Keldysh Institute for Applied Mathematics at the Russian Academy of Sciences in Moscow. He was the principal investigator for special lunar orbiters to create a mathematical model of the lunar gravitational field and the leader of a team to calculate trajectories of the Russian lunar robotic spacecraft.

Several international treaties regulate mutual relations of various states with respect to modern space explorations of the moon. The most important among them are the Outer Space Treaty (1967) and the Agreement Governing the Activities of States on the Moon and Other Celestial Bodies (1979).

Bibliography

Gass, S. I. “Project Mercury’s Man-in-Apace Real-Time Computer System: ‘You Have a Go, at Least Seven Orbits.’”Annals of the History of Computing, IEEE 21, no. 4 (1999).

Eckart, Peter, ed. The Lunar Base Handbook. New York: McGraw-Hill, 1999.

Stroud, Rick. The Book of the Moon. New York: Walker and Co., 2009.

Ulivi, Paolo, and David Harland. Lunar Exploration: Human Pioneers and Robotic Surveyors. Chichester, England: Praxis Publishing, 2004.