Solar System Geometry, Modern Understandings of

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Solar System Geometry, Modern Understandings of

Aristarchus (c. 310 b.c.e.–230 b.c.e.), an ancient Greek mathematician and astronomer, made the first claim that the planets of the solar system orbit the Sun rather than Earth. However, it was Nicolas Copernicus (1473–1543) who would spur modern investigations that would ultimately overthrow the ancient view of a geocentric universe. Johannes Kepler (1571–1630) and Galileo Galilei (1564–1642) initially carried out these investigations. Through the work of Copernicus, Kepler, and Galileo, the geocentric view of circular orbits with constant velocities was gradually replaced by a heliocentric perspective in which planets travel in elliptical orbits of changing velocities. Kepler and Galileo worked during the beginning of what has come to be known as the "Century of Genius," a remarkable time of mathematical and scientific discovery lasting from the early 1600s through the early 1700s. Isaac Newton (1642–1727), perhaps the greatest mathematician of the modern period, lived his entire life during the Century of Genius, building on the foundation laid by Kepler and Galileo.

Advances in Celestial Mechanics

Isaac Newton was born the year that Galileo died, so there is a sense that the investigations begun by Kepler and Galileo continued in an unbroken chain through the life of Newton. Newton was a mathematician and scientist who was profoundly influenced by the work of his predecessors. Mathematically, he was able to synthesize the work of Kepler and others into perhaps the most useful mathematical tool ever devised, the calculus . Scientifically, among Newton's many accomplishments, his work in celestial mechanics can be considered a generalization and explanation of Kepler's three laws of planetary motion. As early as 1596, Kepler had made reference to a force (he used the word "souls") that seemed to emanate from the Sun, resulting in the planets' orbits. Despite many years of work, Kepler was unable to specifically identify this force. Newton not only named the force, but developed a way of mathematically describing its magnitude in a two-body system:

where m ₁ and m ₂ are the masses of the two bodies (for instance, the Sun and Earth), r is the distance between the bodies, and G is the gravitational constant. This formula is known now as the "Universal Law of Gravity."

As a result of his work on the two-body problem, Newton concluded that such a system may be bound or unbound or in "steady-state," referring to the situation in which the potential energy of the system is greater than, less than, or equal to (respectively) the kinetic energy of the system. If bound, then one body will orbit the central mass on an elliptical path, as Kepler had stated in his second law of motion; however, if unbound, the relative orbit will be hyperbolic ; and if steady-state, the relative orbit will be parabolic . The two-body problem was applicable not only to planets of the solar system, but to the paths of asteroids and comets as well. Of course, the reality of the solar system is that forces beyond just the two bodies in question influence gravitation and orbital paths. Newton noted in his influential book Principia that he believed the solution to a three-body problem would exceed the capacity of the human mind.

From the 1700s onward, mathematical contributions to our understanding of the solar system came from individuals who, unlike before, were not necessarily astronomers. Lagrange, Euler and Laplace—all considered eminent mathematicians—provided new tools for more accurately describing the complexities of the universe. Joseph-Louis Lagrange (1736–1813) studied perturbations of planetary orbits. He lent his name to what has come to be called "Lagrangian points," which are equally spaced locations in front and behind a planet in its orbit, containing bodies (such as asteroids) that obey the principles of the three-body problem. Lagrange and Leonhard Euler (1707–1783) shared the 1772 prize from the Académie des Sciences of Paris for their work on the three-body problem, though it appears that Euler was the first to work on the general problem for bodies outside the solar system. In his masterwork Mécanique Céleste, Pierre-Simon Laplace (1749–1827) showed that the eccentricities and inclinations of planetary orbits to each other always remain small, constant, and self-correcting. He also addressed the question of the stability of the solar system and sowed the seeds for the discovery of Neptune.

The Discoveries of Neptune and Pluto

Despite the seismic nature of the mathematical and scientific discoveries through the seventeenth and eighteenth centuries, mathematicians and astronomers of the time could scarcely have imagined the discoveries that would be made in the following two centuries. Perhaps the greatest surprise of the period following the time of Newton, Lagrange, Euler, and Laplace was the extent to which mathematics would be used not only to confirm astronomical theories but also to predict as yet undetected phenomena.

Until 1781, there were only six known planets in the solar system. That year, an amateur astronomer discovered the planet that would later be named Uranus. By about 1840 enough irregularities had been noted in the orbit of Uranus, including those contained in Laplace's book, that astronomers were actively seeking the cause. One popular theory was that Newton's law of gravitation began to break down over large distances. The other primary theory assumed the opposite—it used Newton's law to predict the existence of an eighth, previously undetected planet that was affecting the orbit of Uranus. In fact, observations of the planet that would come to be known as Neptune had been made by various astronomers as far back as Galileo, but it was assumed to be another star.

It was John Adams (1819–1892) of England and Urbain Le Verrier (1811–1877) of France, working simultaneously using Newton's formulas and the calculus, who accomplished what seemed impossible at the time: making accurate predictions of the position of an unknown planet based solely on the gravitational effects on a known planet. Remarkably, the astronomers were even able to determine the new planet's mass without direct observation. In 1846, an astronomer in Berlin very close to Le Verrier's predicted position observed Neptune. This use of mathematics as a prediction, rather than just a confirmation tool, established it as an essential method for astronomical investigation.

Interestingly, even Neptune's orbit did not behave as expected, and the existence of Neptune did not completely account for the irregularities in the orbits of either Uranus or Jupiter. As a result, in 1905 Percival Lowell (1855–1916) hypothesized the existence of a ninth planet. Lowell would die before finding the planet, but in 1930 the discovery of Pluto was made using the telescope at Lowell's observatory in Flagstaff, Arizona. The planet's orbit was found to be highly eccentric and inclined at a much greater angle to the ecliptic than the other planets. In addition, its orbit actually intersects the interior of the orbit of Neptune, allowing it to approach Uranus more closely than Neptune. Though Pluto's orbit obeys Kepler's and Newton's laws, astronomers consider the orbit to be unpredictable over a long period of time (2 × 10⁷ years).

This unpredictability, or sensitivity to initial conditions, is what mathematicians and astronomers refer to as chaos. In a chaotic region, nearby orbits diverge exponentially with time. Though a recent innovation, chaos theory confirms the work of the mathematician Henri Poincaré (1854–1912) on perturbation series, which he did at the beginning of the last century. Though Poincaré's work called into question Laplace's conclusion about the stability of the solar system, it is now believed that our planetary scheme is an example of bounded chaos, where motion is chaotic but the orbits are relatively stable within a billion year timeframe. The realization that, as a result of chaos, even few-body systems can in fact be very complex in their behavior is arguably the greatest discovery about the geometry of the solar system in the final two decades of the twentieth century.

Relativity and the New Geometries

A major discovery of the first two decades of the twentieth century was Albert Einstein's (1879–1955) theory of relativity. In 1859, Le Verrier had found a small discrepancy between the predicted orbit of Mercury and its actual path. Le Verrier believed the cause of the difference was some undetected planet or ring of material that was too close to the Sun to be observed. However, if this was not the case, Newton's theory of gravitation appeared to be inadequate to explain the discrepancy.

In 1905, Poincaré posited an initial form of the special theory of relativity, and argued that Newton's law was not valid and proposed gravitational waves that moved with the velocity of light. Only weeks later, Einstein published his first paper on special relativity, where he suggested that the velocity of light is the same whether the light is emitted by a body at rest or by a body in uniform motion. Then, in 1907, Einstein began to wonder how Newtonian gravitation would have to be modified to fit in with special relativity. This ultimately resulted in his general theory of relativity, whose litmus test was the discrepancy in the orbit of Mercury. Einstein applied his theory of gravitation and discovered that the difference was not due to unseen planets or a ring of material, but could be accounted for by his theory of relativity. Einstein found Euclidean geometry , the geometry typically taught in high school, to be too limited for describing all gravitational systems that might be encountered in space. As a result, he ascribed a prominent role to the non-Euclidean geometries that had recently been developed during the nineteenth century.

In the early 1800s, Janos Bolyai (1802–1860) and Nicolai Lobachevsky (1793–1856) had boldly published their revolutionary views of one kind of non-Euclidean geometry . Essentially, they independently confirmed that a geometry was possible where there existed not just one but an infinite number of parallel lines to a given line through a point not on the line. In the classical mechanics of Newton, a key assumption was that rays of light travel in parallel lines that are everywhere equidistant from each other. In contrast, Bolyai and Lobachevsky claimed that parallel lines converged and diverged, though still not intersecting. This kind of geometry is known as Lobachevskian or Hyperbolic, whereas Euclidean or plane geometry is referred to as Parabolic. There is also another kind of non-Euclidean geometry, known as Elliptic, which resulted from the later work of Poincaré.

The great mathematician Karl Freidrich Gauss (1777–1855) had early on advanced the idea that space was curved, even suggesting a method for measuring the curvature. However, the connection between Gauss's work with the curvature of space and the new geometries was not immediately apparent. In fact, the immediate effect of the non-Euclidean geometries was to cause a crisis in the mathematical community: Was there any longer a relationship between mathematics and reality? Euclidean geometry had been the basis not only for all of mathematics up to and including the calculus, but also for all scientific observation and measurement, and even the philosophies of Emmanuel Kant (1724–1804) and his followers.

Einstein immediately saw how the various geometries co-existed in space. The genius of his insight was the realization that, whereas light might travel in a straight line in some restricted situation (say, on Earth) when the vast expanse of the universe is considered, light rays are actually traveling in non-Euclidean paths due to gravity from masses such as the Sun. In fact, a modern theoretical model of the universe based on Einstein's general theory of relativity depends on a more complex non-Euclidean geometry than either Hyperbolic or Elliptic. The assumption is that the universe is not Euclidean, and it is the concentration of matter in space (which is not uniform) that determines the extent of the deviation from the Euclidean model.

The modern period of observational science has revolutionized the way we view our universe. The geometry of our solar system is infinitely more complex than was considered possible during the time of Kepler, Galileo or even Newton. With new tools such as the Hubble telescope available to twenty-first century astronomers, it is difficult to imagine the discoveries that await us.

see also Astronomer; Chaos; Solar System Geometry, History of.

Paul G. Shotsberger