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# The Keplerian Orbit

Under Newton's theory of gravity, two gravitating objects in orbit around each other follow Keplerian orbits. The physics of the Keplerian orbit generally provides a excellent approximation to the true orbit of planets and binary stars.

Keplerian orbits were empirically characterized for the planets in our Solar System by Johannes Kepler at the beginning of the 17th century. Three characteristics are expressed in the following three laws:

• Each planet moves on a closed elliptical orbit in a plane containing the Sun, with the Sun at one focus of the ellipse.
• The area swept out per unit time by a line drawn between a planet and the Sun is a constant throughout the planet's orbit.
• The ratio of the cube of the semimajor axis—half the length along the long axis of the ellipse—of a planet's orbit to the square of the orbital period is the same for all planets in the Solar System.

These three empirical laws can be directly derived from the Newtonian theory of gravity under the assumption that the masses of the planets are negligible compared to the mass of the Sun. The second of these laws is a consequence of the conservation of angular momentum, and would hold for any time-independent radial gravitational attraction between two objects. The first of these laws is a consequence of the inverse-square rule of gravitational attraction; this law fails when the gravitational field is modified by the presence of other gravitating bodies. The final law is a consequence of the inverse-square rule and of the mass of the planets being negligible compared to the mass of the Sun.

## The Breakdown of Keplerian Orbits

The solution of the two-body problem in Newtonian mechanics breaks down in several ways when applied to real orbiting bodies. For instance, in deriving the Keplerian orbit, one assumes that the bodies are spherically symmetric, which is equivalent to assuming that the mass of each is concentrated in a point at the center of each. If a body is extended, as Earth and every other planets is, the non-spherical density distribution within that planet causes an orbiting satellite to deviate slightly from a Keplerian orbit. This, in fact, is how one probes the interior of Jupiter and Saturn: by watching how an orbiting spacecraft deviates from the expected Keplerian orbit.

A second cause of error in the Keplerian treatment of real orbits is the neglect of the gravitational fields of other bodies. This is of particular concern for planetary orbits. Over a single orbit, the effect the gravitational attraction of other planets on a given planet is tiny, but over many orbits, the effect grows to significance. The gravitational attraction of Jupiter causes the perihelion of Earth's orbit to slowly drift. Through their gravitational fields, Jupiter shepherds a swarm of asteroids around the Sun, and Neptune locks many Kuiper Belt objects in harmonious orbits.

Strictly speaking, Newtonian gravity is an incorrect theory of gravity; the currently accepted theory of gravity is general relativity. For a weak gravitational field, Newtonian gravity is generally sufficiently accurate for describing orbital motion, but in several notable cases, the effects of general relativity are important. Within the solar system the effects of general relativity can be seen in the drift of Mercury's perihelion. The effect of general relativity is also seen in compact binary systems of neutron stars.