Solar System Stability

If I explain this properly, my readers will be able to understand perturbation (a deviation of a system, moving object, or process from its regular or normal state of path, caused by an outside influence), without having to be a physics geek, however I do expect to lose many readers.  All planets move around the Sun according to specific physical laws that determine their motion.  Using these laws, scientists can predict how an object should move through space even if it’s billions of miles away.  Copernicus put forth the theory that the planets moved around the Sun, with the Earth being just another one of those planets, and his work was followed by Galileo, Kepler, and finally when Isaac Newton stepped into the picture, the heliocentric model won out.

Johannes Kepler tried to make sense out of all the data that was collected by his predecessor Tycho Brahe, and he ended up developing three laws of planetary motion, where the first law stated that “All planets move in elliptical orbits, with the Sun at one focus.”  This differed from what Aristotle taught, saying that the planets were in circular orbits, but an ellipse looks a lot like a circle, so Kepler’s statement did not bother many people.  However it bothered Isaac Newton and it bothered him so much that he had to develop a new form of math to determine who was right, Aristotle or Kepler and Calculus decided in Kepler’s favor.

Most people are familiar with the story of Isaac Newton sitting under an apple tree, but not everyone knows that this incident made Newton think about the Moon.  Supposedly when the apple landed on his head Newton said, “I wonder if this mysterious force called gravity is what makes the Moon orbit the Earth.”  Newton thought that to keep the Moon moving around the Earth, rather than wandering off, that the Earth must exert a pull on the Moon.  Newton concluded that gravity was that the same force that pulled all falling objects downward toward earth.  In 1666, Sir Isaac Newton developed the theories of gravitation and then 20 years later, in 1686, he presented the three laws of motion.

Newton set out to describe the behavior of both prograde (when a planet moves west-to-east relative to the stars) and retrograde (when a planet moves east-to-west relative to the stars) motions of the planets and how they moved about the Sun in ellipses, in his law of universal gravitation.  This law accounted for gravitation not only on Earth, but of all the heavenly bodies and it explained why moons orbit their parent planet, why comets recurred and were often perturbed by the other planets, why our world experiences tides, and why the planets don’t disturb one another and cause frequent ejections.

Newton derived his three laws of motion, after reading about the experiments of Galileo and what Descartes had to say, he then investigated the effects of impact, and analyzed the consequences of a collision, before he espoused the concept of inertia.  The l8^th century astronomers and theorists wondered if Newton’s laws were adequate enough to explain the motions of the heavenly bodies.  Scientists began to wonder if the mutual attraction of the planets would produce perturbations that would alter their regular courses, and some of this already became evident through the increasing accuracy of observations.  A group of the most brilliant theorists came forward, mostly in France, including Clairaut, Euler, d’Alembert, and later Lagrange and Laplace, all set themselves to resolving this task.  For a long time these astute mathematicians were baffled, but later Laplace succeeded in deriving a solution.

Perturbation theory is based on the fact that it is possible to give an approximate description of the system under study using some specially selected ‘ideal’ system which can be correctly and completely studied. Perturbation theory was first proposed for the solution of problems in celestial mechanics, in the context of the motions of planets in the solar system.  Since the planets are very remote from each other, and since their mass is small as compared to the mass of the Sun, being one million times more massive than Earth and Jupiter being three orders of magnitude smaller than the Sun and three orders larger than Earth.  In a logarithmic sense, it is mid-way between Earth and Sun, thus the gravitational forces between the planets can be neglected, and the planetary motion is considered, to a first approximation, as taking place along Kepler’s orbits, which are defined by the equations of the two-body problem, the two bodies being the planet and the Sun.

The strength of the gravitational force between two objects depends on two factors, mass and distance and if the distance increases, the force of gravity decreases and vice a versa, thus when two planets are closer in their orbits to each other, there will be a stronger force of gravity exerted between them.  To some degree, each planet is attracted by all the others in addition to the Sun’s attraction.  The problem of two massive bodies, like the Sun and Earth, gravitationally attracting each other was solved completely by Newton, but with three or more bodies, the problem becomes vastly more complex.  The theory of perturbations, as applied to the Lunar Theory, was developed from the geometrical standpoint by Newton.  Newton assumed the deviations in the orbits of Jupiter and Saturn were due to the mutual attraction of the two planets.

In March of 1746, the prize commission of the Paris Academy of Sciences, met to select a prize problem for the Academy’s contest of 1748.  They chose the mutual perturbations of Jupiter and Saturn.  Since Kepler’s time, Jupiter had been accelerating and Saturn slowing down, and in other ways deviating from the Keplerian rules.  The contest of 1748 was the first academic contest of the eighteenth century in which a case of the three-body problem was posed for solution.  From the moment that the prize problem was set, two members of the prize commission, Alexis-Claude Clairaut and Jean le Rond d’Alembert, each unbeknownst to the other, launched their own assaults on the three-body problem.

The memoirs of Clairaut and d’Alembert contained important advances, making the solutions depend upon the integration of the differential equations in series.  Clairaut was able to apply his process of integration to the perturbations of Halley’s comet by the planets Jupiter and Saturn.  Clairaut computed the perturbations due to the attractions of Jupiter and Saturn, and predicted that the perihelion passage would be April 13, 1759.  He remarked that the time was uncertain to the extent of a month because of the uncertainties in the masses of Jupiter and Saturn and the possibility of perturbations from unknown planets beyond these two.  The comet passed the perihelion on March 13, giving a striking proof of the value of Clairaut’s methods.

In 1785, the French mathematician Pierre Simon de Laplace established a theory of motion for Jupiter and Saturn that agreed with 18th century observations.  He was able to account for Ptolemy’s observations to within one minute of arc, and he showed that Newton’s law was in itself sufficient to explain the movement of the planets throughout known history.  Laplace was confident in his abilities to understand celestial mechanics, so he decided to approach the problem of the stability of the Solar System and his pioneering investigations represented a milestone in the subsequent development of perturbation techniques.  Laplace used perturbation theory to show that the solar system is stable.  He was able to treat the solar system as a many-body problem in which not only the action of the Sun on the planets, but also the effects that the planets have on each other were considered.  He wanted to show that any changes in the motion of the planets due to these perturbations did not lead to ejections of any planets out of our solar system.

Understanding how perturbation effects planets led to the discovery of Uranus in 1846 and because of supposed perturbations of the orbits of Uranus and Neptune, Pluto was discovered in 1930, however the discovery of Pluto was actually coincidental, as Pluto’s mass was lower than predicted and the perturbation to the motions of Uranus and Neptune did not account for a planet (or planetoid) as small as Pluto.  It would be great if science were able to detect a flying rock or icy mud ball that may eventually slam into Earth and change our planet irreversibly and such an impact actually happened 65 million years ago that is widely believed to have killed off the dinosaurs.  Understanding perturbation is important as there is always the possibility that the Kuiper Belt could be holding a large body that could eventually become dislodged from its orbit by a passing star or some type of collision and begin its trek to the Inner Solar System where it might spell the end of all life on Earth, but for now we are safe.

I am not sure how good of a job I did in trying to explain perturbation, but what cannot be defined may still be learned.  I used to cringe anytime I heard the word Laplace when I was studying calculus.  However, he did play a crucial role in many areas of mathematics, physics and engineering, but learning about his methods is not always easy.  In 1867, William Thomson Baron Kelvin and ‎Peter Guthrie Tait collaborated on a book titled ‘Treatise on Natural Philosophy’ where they stated, “There can be but one opinion as to the beauty and utility of this analysis of Laplace; but the manner in which it has been hitherto presented has seemed repulsive to the ablest mathematicians, and difficult to ordinary mathematical students.”

In summary, astronomical perturbation involves a disturbance in the orbit or motion of a heavenly body. In the solar system perturbations affect the planets, their satellites, and comets.  Over long periods of time, perturbations may affect the size, shape, or position of the orbit of a heavenly body.  Over shorter periods, perturbations may affect the position of a body in its orbit, causing it, for example, to be sometimes ahead and sometimes behind where it would otherwise be.  Without understanding perturbation, you will never be able to fully grasp Einstein’s general theory of relativity.