To a first approximation, the solar system consists of the Sun and eight major planets, a system much more complicated than a two-body problem. However, use of Equation 2 with reasonable values for the astronomical unit (a convenient unit of length for the solar system) and for G showed that the Sun is far more massive than even the most massive planet Jupiter (whose mass is 0.000955 the Sun's mass). This showed that the gravitational forces of the planets on each other are much weaker than the gravitational forces between the Sun and each of the planets, which enabled astronomers to consider the gravitational interactions of the planets as producing small changes with time perturbations) in the elliptical orbit of each planet around the center of mass of the solar system (which is always in or near the Sun). If the Sun and a planet (say the Earth or Jupiter) were alone in empty space, we would have an ideal two-body problem and we would expect the two-body problem as defined by the generalized Kepler's law to exactly describe their orbits around the systems center of mass. Then the seven orbital elements (of which a and y are two) of a planet's orbit should remain constant forever.
However, the gravitational forces of the other planets on a planet cause its orbit to change slightly over time; these changes can be accurately allowed for over limited time intervals by calculating the perturbations of its orbital elements over time that are caused by the gravitational forces of the other planets.
Historically, perturbation theory has been more useful than merely providing accurate predictions of future planetary positions. Only six major planets were known when Newton published his Principia. William Herschel (1738-1822) fortuitously discovered Uranus, the seventh major planet from the Sun, in March 1781. The initial orbital elements calculated for Uranus did not accurately allow prediction of its future position even after inclusion of the perturbations caused by the six other major planets. Before 1821, Uranus was consistently observed to be ahead of its predicted position in its orbit; afterwards, it lagged behind its predicted positions.
John Couch Adams (1819-1892) in England and Urbain Leverier (1811-1877) in France, hypothesized that Uranus had passed an undiscovered massive planet further than it was from the Sun in the year 1821. They both made detailed calculations to locate the position of the undiscovered planet perturbing the motion of Uranus. Johann Galle (1812-1910) in Berlin, Germany used Leverier's calculations to discover the unknown planet in September 1846, which was then named Neptune.
Further unexplained perturbations of the orbits of Uranus and Neptune led Percival Lowell (1855-1916) and several other astronomers to use them to calculate predicted positions for another undiscovered (trans-Neptunian) planet beyond Neptune's orbit. Lowell searched for the trans-Neptunian planets he predicted from 1906 until his death in November 1916 without finding it. The search for a trans-Neptunian planet was resumed in 1929 at Lowell Observatory, where Clyde Tombaugh (1905–1997) who discovered Pluto in February 1930.
Lowell had predicted that a planet more massive than Earth produced the unexplained perturbations. During the years following Pluto's discovery, however, detailed studies of its perturbations of the orbits of Uranus and Neptune showed that Pluto is considerably less massive than Earth. The discovery of Pluto's satellite, Charon, in 1978 allowed the determination of the total mass of Pluto and Charon from Equation 2 which is about 0.00237 Earth's mass (about 0.2 the mass of Earth's moon). There are two consequences of this discovery; Tombaugh's discovery of Pluto may have been fortuitous, and one may make the case that Pluto is not a major planet.
The discrepancy in mass between the masses predicted by Lowell and others for the trans-Neptunian planet and the mass of the Pluto-Charon double planet has led to a renewed search for one or more additional trans-Neptunian planet(s) that still continues. The opinion also exists that the unexplained perturbations of the orbits of Uranus and Neptune are caused by systematic errors in some early measurements of their positions and that no trans-Neptunian planets with masses on the order of Earth's mass exist.
Science EncyclopediaScience & Philosophy: Categorical judgement to ChimaeraCelestial Mechanics - Planetary Perturbations, Resonance Phenomena, Tidal Effects, Precession, Non-gravitational Effects, The Three-body Problem