Celestial mechanics

Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. The field applies principles of physics, historically Newtonian mechanics, to astronomical objects such as stars and planets to produce ephemeris data. It is distinguished from astrodynamics, which is the study of the creation of artificial satellite orbits.

History of celestial mechanics

Although modern analytic celestial mechanics starts 400 years ago with Isaac Newton, prior studies addressing the problem of planetary positions are known going back perhaps 3,000 years.

Ancient civilizations

The Ancient Babylonians had no mechanistic theories regarding celestial motions, but recognized repeating patterns in the motion of the sun, moon, and planets. They used tabulated positions during similar past celestial alignments to accurately predict future planetary motions.

Imperial Chinese astrologers also observed and tabulated positions of planets and guest stars which can refer to either a comet or a nova. Although their records are a very useful historical source for modern astronomy, there is no known record of them having predicted celestial motions.

The Classical Greek writers speculated widely regarding celestial motions, and presented many mechanisms for the motions of the planets. Their ideas mostly involved uniform circular motion, and were centered on the earth. An extraordinary figure among the ancient Greek astronomers is Aristarchus of Samos (310 BC - c.230 BC), who suggested a heliocentric model of the universe and attempted to measure Earth's distance from the Sun.

Claudius Ptolemy

Claudius Ptolemy was an ancient astronomer and astrologer in early Imperial Roman times who wrote a book on astronomy now called the Almagest. The Almagest was the most influential secular book of classical antiquity. Ptolemy selected the best of the astronomical principles of his Greek predecessors, especially Hipparchus, and appears to have combined them either directly or indirectly with tabulations from the Babylonians. Although Ptolemy relied mainly on the work of Hipparchus, he introduced at least one idea, the equant, which appears to be his own, and which greatly improved the accuracy of the predicted positions of the planets. His model solar system fails to correctly predict the apparent change in the size of the moon (libration), but otherwise is accurate to within the naked-eye observations available to him.

Johannes Kepler

Johannes Kepler was the first to closely integrate the predictive geometrical astronomy, which had been dominant from Ptolemy to Copernicus, with physical concepts to produce a New Astronomy, Based upon Causes, or Celestial Physics.... His work led to the modern laws of planetary orbits, which he developed by carefully analyzing the planetary observations made by Tycho Brahe. Kepler's model greatly improved the accuracy of predictions of planetary motion, years before Isaac Newton had even developed his law of gravitation.

See Kepler's laws of planetary motion and the Keplerian problem for a detailed treatment of how his laws of planetary motion can be used.

Isaac Newton

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 cannon balls and falling apples, could be described by the same set of physical laws. In this sense he unified celestial and terrestrial dynamics.

Using Newton's law of gravitation, proving Kepler's Laws for the case of a circular orbit is simple. Elliptical orbits involve more complex calculations. Using Lagrangian mechanics it is possible to develop a single polar coordinate equation that can be used to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and comets and such. More recently, it has also become useful to calculate spacecraft trajectories.

Albert Einstein

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 led to a Nobel prize.

Examples of problems

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 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 the dwarf planet Pluto, ecc. = 0.2488
• 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.

Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem.

See also

• Astrometry is a part of astronomy that deals with the positions of stars and other celestial bodies, their distances and movements.
• Astrodynamics is the study and creation of orbits, especially those of artificial satellites.
• Orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.
• Satellite is an object that orbits another object (known as its primary). The term is often used to describe an artificial satellite (as opposed to natural satellites, or moons). The common noun moon (not capitalized) is used to mean any natural satellite of the other planets.
• Celestial navigation is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.
• VSOP82 and VSOP87 are two versions of a mathematical theory for the orbits and positions of the major planets to the highest possible accuracy.

External links

• Calvert, James B. (2003-03-28). "Celestial Mechanics". University of Denver. Retrieved 2006-08-21.

Research

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

Artwork

• Celestial Mechanics is a Planetarium Artwork created by D. S. Hessels and G. Dunne

Course notes

• Professor Tatum's course notes at the University of Victoria

References

• Forest R. Moulton, Introduction to Celestial Mechanics, 1984, Dover, ISBN 0-486-64687-4

• John E.Prussing, Bruce A.Conway, Orbital Mechanics, 1993, Oxford Univ.Press

• William M. Smart, Celestial Mechanics, 1961, John Wiley. (Hard to find, but a classic)