Advances in Understanding Celestial Mechanics

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Advances in Understanding Celestial Mechanics

Overview

Modern celestial mechanics began with the application of Isaac Newton's laws of motion to the observations of astronomy. Mathematicians of the eighteenth and nineteenth centuries worked to understand the workings of the solar system in terms of all its gravitational forces. While the gravitational pull of the Sun keeps the planets in their orbits, each planet's path is slightly disturbed by the presence of the others.

Background

Early astronomy was primarily concerned with celestial mechanics in a broad sense; that is, understanding the apparent motion of the stars and planets. Without telescopes, the Moon was the only celestial body upon which details could be observed. Everything else was too distant, except for the Sun, which was too bright. So theories could be devised about the nature of the other bodies, but the only property that could actually be measured was their motion.

The word planet comes from the Greek word for wanderer. As seen from Earth, the stars all march across the sky at the same steady pace, but the planets move at a variable rate, sometimes even changing direction. We now know that we see the planets in this way because they, like Earth, are moving in their separate orbits around the Sun. The apparent regular motion of the distant stars arises from our changing point of view as Earth rotates on its axis and revolves around the Sun.

Of course, none of this was obvious to the ancients, who assumed that the entire cosmos revolved around Earth. The most influential ancient astronomer, Ptolemy, lived in Alexandria around 150 A.D.. He explained the variable motion of the planets by assuming that they revolved in small orbits called epicycles, on a larger circle called a deferent that orbited Earth. Ptolemy adjusted the speeds and distances in his system until he was able to make accurate predictions of planetary positions. His system, preserved in a work his successors called the Almagest, or the "Greatest," was used until the Renaissance.

The Polish astronomer Nicolaus Copernicus (1473-1543) is generally considered the father of modern-day astronomy. He realized that Earth orbits the Sun like the other planets and that our view of the sky is affected by Earth's motion. He put forth his heliocentric, or sun-centered, view of the cosmos in his great work Concerning the Revolutions of the Celestial Sphere (1543). Since he still assumed, incorrectly, that the orbits of the planets were circular, he had to retain some of Ptolemy's epicycles to accurately match their positions.

Johannes Kepler (1571-1630), using twenty years of precise measurements made by his mentor Tycho Brahe (1546-1601), developed his famous laws of planetary motion, the first of which states that the orbits are elliptical. Kepler's laws were accurate, but empirical; that is, they accounted for the observed data, but had no explanation for why the orbits should be shaped as they are. It was Isaac Newton (1642-1727) who provided that explanation. Newton derived three general laws of motion, and concluded that the gravitational force between two bodies was proportional to the product of their masses and the inverse square of the distance between them.

With the new understanding of gravity and the laws of motion, it was now possible to consider the planets as an n-body problem; that is, a theoretical problem of predicting the behavior of a fixed number of masses interacting by means of their gravitational fields. Such problems are complex because every object affects every other. So, for example, in our Solar System, the main determinant of the planetary orbits is the gravitational field of the Sun. However, each planet experiences perturbations in its own orbit because of all the others. Once the basic mechanism of orbits was understood, the discipline of celestial mechanics began to concentrate on understanding the perturbations.

Impact

Planetary motion is described in terms of differential equations. Differential equations involve derivatives, mathematical expressions for the rate of change of one quantity with respect to another. For example, the derivative, or rate of change, of distance with respect to time is velocity. The rate of change of velocity with respect to time is acceleration, and acceleration is determined by the gravitational or other force on an object. The equations describing position, velocity, and force necessarily involve derivatives.

Differential equations are themselves an important field of study in mathematics. Many are extremely difficult to solve. While some have exact solutions, the solutions to others must be approximated, and many techniques have been developed to do so. Today differential equations are sometimes solved using numerical computing techniques.

In celestial mechanics, a "two-body problem" such as the rotation of a single planet around the Sun without taking any other masses into account is described by a relatively simple differential equation. It can be solved exactly, and the solution reproduces Kepler's laws. Add in the masses of the other planets, however, and the situation becomes much more complicated. Even the great Newton essentially threw up his hands and attributed the stability of the Solar System to occasional divine intervention in which everything was nudged back into place.

The stability problem was taken up by the French astronomer Pierre-Simon Laplace (1749-1827). Observations seemed to indicate that the orbit of Jupiter was continuously shrinking while that of Saturn was expanding. In 1786 Laplace showed that the eccentricities of the planetary orbits and the angles at which they are inclined with respect to one another will remain small and self-correcting. Because the perturbations in the motion are periodic, they do not accumulate and disrupt the stability of the solar system. In the case of Jupiter and Saturn, the effect being observed had a period of 929 years.

Between 1798 and 1827 Laplace's five-volume work on planetary motion, Traité de mecanique céleste ("Treatise on Celestial Mechanics"), was published. In it he provided a complete method for calculating the movements of the planets and their moons, including gravitational perturbations and the tidal disturbances of the bodies' shapes. The book quickly became a classic. It was updated and enlarged by both the American mathematician Nathaniel Bowditch (1773-1838) and the French astronomer Félix Tisserand (1845-1896). Tisserand's version, four volumes published between 1889 and 1896, remains an important reference in the field.

For any given planet, it is generally convenient to consider the sum of the forces from all the others as a net perturbation of the orbital ellipse. The French-Italian mathematician Joseph-Louis Lagrange (1736-1813) expressed this in a set of differential equations sometimes called the Lagrange planetary equations. They must be solved numerically or by means of successive approximations using a series of mathematical terms to increase the range over which a solution is a good fit to the actual motion. Lagrange used his equations to explain the libration of the Moon; that is, the oscillations seen as a slight change in the position of the visible lunar features. The techniques also resulted in the discovery of the planet Neptune in 1846, after its position was predicted from perturbations of the orbit of Uranus.

In 1889 the French savant Henri Poincaré (1854-1912) won a prize offered by King Oscar II of Sweden for a contribution to the n-body problem. Poincaré applied to the problem several of the advances in mathematical analysis that had been developed since the time of Laplace and Lagrange and added a few important techniques of his own invention. His work was published between 1892 and 1899 as Les méthodes nouvelles de la méchanique céleste ("The New Methods of Celestial Mechanics").

Poincaré was more concerned with the mathematics of celestial mechanics than with obtaining precisely accurate predictions of planetary motion. He studied the series approximations used with the differential equations of motion, to understand when they would converge to a useful solution. In cases where they did not converge to a general solution, he showed under what conditions they could be used to approximate the motion for a significant period of time.

The advances made by nineteenth-century mathematicians and astronomers with respect to celestial mechanics set the stage for further developments in the twentieth century. Poincaré's work, for instance, anticipated the modern idea of chaotic motion—or chaos theory—in which for some initial conditions the future state of a system becomes unpredictable within an allowable range.

SHERRI CHASIN CALVO