Celestial mechanics is a term for the application of physics, historically Newtonian mechanics, to astronomical objects such as stars and planets. After Einstein explained the anomalous precession of Mercury's perihelion, astronomers recognized that Newtonian mechanics did not provide the highest accuracy. Today, we have binary pulsars whose orbits not only require the use of General Relativity for their explanation, but whose evolution proves the existence of gravitational radiation, a discovery that lead to a Nobel prize: http://www.naic.edu/vscience/astro/nobel.htm

Isaac Newton is credited with introducing the idea that the motion of objects in the heavens, such as planets, the Sun, and the Moon, and the motion of objects on the ground, like horses and falling apples, could be described by the same set of physical laws. In this sense he unified 'celestial' and 'terrestrial' dynamics.

Celestial motion without additional forces such as thrust of a rocket, is governed by gravitational acceleration of masses due to other masses. A simplification is the n-body problem, where we assume n spherically symmetric masses, and integration of the accelerations reduces to summation.

Examples:

• 4-body problem: spaceflight to Mars (for parts of the flight the influence of one or two bodies is very small, so that there we have a 2- or 3-body problem; see also the patched conic approximation)
• 3-body problem:
  • quasi-satellite
  • spaceflight to, and stay at a Lagrangian point

In the case that n=2 (two-body problem), the situation is much simpler than for larger n. Various explicit formulas apply, where in the more general case typically only numerical solutions are possible. It is a useful simplification that is often approximately valid.

Examples:

• a binary star, e.g. Alpha Centauri (approx. the same mass)
• a double planet, e.g. Pluto with its moon Charon (mass ratio 0.147)
• a binary asteroid, e.g. 90 Antiope (approx. the same mass)

A further simplification is based on "standard assumptions in astrodynamics", which include that one body, the orbiting body, is much smaller than the other, the central body. This is also often approximately valid.

Examples:

• Solar system orbiting the center of the Milky Way
• a planet orbiting the Sun
• a moon orbiting a planet
• a spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)

Either instead of, or on top of the previous simplification, we may assume circular orbits, making distance and orbital speeds, and potential and kinetic energies constant in time. Notable examples where the eccentricity is high and hence this does not apply are:

• the orbit of Pluto, ecc. = 0.2488 (largest value among the planets of the Solar System)
• the orbit of Mercury, ecc. = 0.2056
• Hohmann transfer orbit
• Gemini 11 flight
• suborbital flights

Of course, in each example, to obtain more accuracy a less simplified version of the problem can be considered.

Related topics

• Astrometry
• Astrodynamics
• Orbit
• Satellite, Natural satellite
• n-body problem
• Celestial navigation

External link

• Marshall Hampton's research page: Central configurations in the n-body problem