Mathematics in Space

By James Case

Calculating the Cosmos: How Mathematics Unveils the Universe. By Ian Stewart. Basic Books, New York, NY, October 2016. 360 pages, $27.99.

From the beginning of time, writes Ian Stewart in his latest popular mathenscience¹ book, men have looked at the night sky and questioned, “What’s going on up there?” In fact, virtually all proposed answers to this question have been abandoned in favor of better ones, which are then abandoned in term. Indeed, with the advance of the physical and mathematical sciences, the ability to discriminate between good and bad guesses has grown markedly, rendering not-yet-discredited guesses increasingly hard to dismiss. In Calculating the Cosmos: How Mathematics Unveils the Universe, Stewart offers an extensive catalogue of noteworthy conjectures concerning the nature and extent of the cosmos, beginning with some of the earliest and culminating in current “best guesses.”

SIAM members are likely familiar with many of the earlier notions, such as Ptolemaic cosmology, which placed Earth at the center of the universe—surrounded by an invisible “celestial sphere,” to which fixed stars were attached—while the sun, the moon, and the planets travelled without collision on separate concentric spheres. The Copernican Revolution challenged medieval orthodoxy with evidence that the sun lies at the center of the universe, while Earth and other planets rotate around it in orbits consistent with Isaac Newton’s inverse-square law. Relativity and Edwin Hubble’s discovery of an expanding universe spawned yet another round of cosmological guesswork. More recently still, the succession of manned and unmanned missions that followed Sputnik I’s trip to space in October 1957 has given birth to a phalanx of more modern guesses, no few of which were quickly debunked. Stewart follows space exploration closely, and offers an up-to-date summary of what scientists have learned.

Chapter 1 introduces gravity, conic sections, \(N\)-body problems, general relativity, and the historic realization that nature obeys mathematical laws. After pointing out that Newton solved the two-body problem, and that the three-body problem appears insoluble, Stewart reveals that current investigators continue to discover new and unexpected consequences of Newton’s laws. He mentions particularly a family of planar orbits—the simplest being shaped like a figure eight—around which three equal point masses can pursue one another indefinitely, along with a corkscrew-shaped orbit that spirals around the line segment joining the centers of a binary star. The spirals are loose near the middle of the segment but crowd together by the stars at the ends, somewhat resembling a slinky toy stretched only in the middle. There is some evidence that an exoplanet named Kepler-b may be trapped in such an orbit.

Chapter 2 concerns the origins of the solar system. The current best proposition attributes its formation to the collapse of a giant gas cloud, in which non-uniformities in the initial distribution of matter, together with gravitational attraction, caused gaseous clumps to form and then congeal into solid bodies. These bodies frequently collided and grew in size as smaller bodies were drawn to larger ones. The vast number of craters on the moon, Mercury, and Mars attests to the frequency of such collisions in the early universe.

The growth of computing power in the 1980s, along with the development of accurate computational techniques, allowed scientists to model the collapse of giant gas clouds as N-body problems. A realistic application of this method requires a few hundred billion bodies, rendering the calculations infeasible. Hence, smaller numbers are used. Crude integration techniques cannot be trusted here, since they neglect such physical realities as the conservation of energy and angular momentum. If such oversight were to decrease overall energy, for instance, rather than conserve it, the effect would resemble friction – closed planetary orbits would be replaced by decaying ones that spiral into the sun.

The development of symplectic integrators, numerical methods specifically designed for the integration of ordinary differential equations in Hamiltonian form: \(\dot{p}=H_q(p,q),\)  \(\dot{q}=-H_p(p,q)\), has largely surmounted such difficulties. Since these methods preserve the symplectic \(2\)-form \(dp\wedge dq\), they also preserve linear and angular momentum exactly, along with total energy. As a result, symplectic integrators permit the accurate simulation of systems of mutually-gravitating bodies in free space over extremely long periods of time.

Stewart writes extensively about the rings of Saturn, the study of which has a long and surprisingly complex history. Originally observed by Galileo in 1610, they seemed at first to look like separate moons, and later like ears on a face. But Christiaan Huygens, armed with a better telescope, was able to report in 1655 that Saturn “is surrounded by a thin flat ring, nowhere touching, inclined to the ecliptic.” By 1666, Robert Hooke could observe shadows, both of the globe upon the ring and the ring upon the globe, showing what’s in front of what. Then in 1787, Pierre-Simon Laplace pointed out that a single wide flat ring would break apart since, by Johannes Kepler’s third law, the outer portions must rotate more slowly than the inner ones. He thus concluded that the wide flat ring must be composed of several concentric ringlets, each rotating at a different speed. Next in 1859, James Clerk Maxwell showed that even a narrow flat ringlet is unstable, since the slightest disturbance causes such a surface to buckle, ripple, and bend, immediately snapping like a dry piece of spaghetti with the application of distortive forces. Could the rings be composed of fluid? No, because as Sophie Kovalevsky showed in 1874, fluid rings would also be unstable. It was not until around 1895 that telescopes improved to the point where observers could declare Saturn’s rings to be composed of a truly vast number of small (presumably solid) orbiting bodies.

According to Stewart, no military plan survives contact with the enemy, and no astronomical theory survives contact with better observations. Man’s knowledge of Saturn changed forever in 1980, when Voyager I started sending back pictures of the rings. The images soon revealed, for instance, that one of the rings is not circular, and that dark fuzzy “spokes” seem to emanate from the center of the planet and rotate within the “wheel” formed by the rings. Nothing previously noted concerning the rings had lacked circular symmetry. Voyager II, which had launched before Voyager I but was moving more slowly, confirmed both observations some nine months later. The Voyager missions also revealed that some of the rings appear to be braided, some exhibit strange kinks, and some are incomplete, consisting of discrete, roughly-circular arcs separated by gaps. Before the Voyager encounters, Earth-bound astronomers had observed that Saturn possessed nine moons; Voyager increased the number to 30. Today it’s 62, 53 of which now have official names. The Cassini probe, currently orbiting Saturn, provides a stream of data on the planet, its rings, and its moons.

Later chapters discuss—in something like layman’s terms—the location of asteroids in the solar system; the rings of Saturn; the overall curvature of space; the Big Bang theory; the whereabouts of dark matter and dark energy; and a great deal more. For anyone who hasn’t kept up with space exploration, Calculating the Cosmos is an exceedingly pleasant and highly informative read.

¹ Term coined by the elder President Bush to describe a subject in which few American students excel.

James Case writes from Baltimore, Maryland.