Celestial mechanics

Celestial mechanics is the branch of astronomy that deals with the motions of objects in outer space. Historically, celestial mechanics applies principles of physics (classical mechanics) to astronomical objects, such as stars and planets, to produce ephemeris data.

History
Modern analytic celestial mechanics started with Isaac Newton's Principia (1687). The name celestial mechanics is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term celestial mechanics. Prior to Kepler there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical techniques, and contemporary discussions of the physical causes of the planets' motion.

Johannes Kepler
Johannes Kepler (1571–1630) was the first to closely integrate the predictive geometrical astronomy, which had been dominant from Ptolemy in the 2nd century to Copernicus, with physical concepts to produce a New Astronomy, Based upon Causes, or Celestial Physics in 1609. His work led to the modern laws of planetary orbits, which he developed using his physical principles and the planetary observations made by Tycho Brahe. Kepler's elliptical model greatly improved the accuracy of predictions of planetary motion, years before Isaac Newton developed his law of gravitation in 1686.

Isaac Newton
Isaac Newton (25 December 1642 – 31 March 1727) 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 his law of gravity, Newton confirmed Kepler's Laws for elliptical orbits by deriving them from the gravitational two-body problem, which Newton included in his epochal Principia.

Joseph-Louis Lagrange
After Newton, Lagrange (25 January 1736 – 10 April 1813) attempted to solve the three-body problem, analyzed the stability of planetary orbits, and discovered the existence of the Lagrangian points. Lagrange also reformulated the principles of classical mechanics, emphasizing energy more than force, and developing a method to use a single polar coordinate equation to describe any orbit, even those that are parabolic and hyperbolic. This is useful for calculating the behaviour of planets and comets and such (parabolic and hyperbolic orbits are conic section extentions of Kepler's elliptical orbits). More recently, it has also become useful to calculate spacecraft trajectories.

Simon Newcomb
Simon Newcomb (12 March 1835 – 11 July 1909) was a Canadian-American astronomer who revised Peter Andreas Hansen's table of lunar positions. In 1877, assisted by George William Hill, he recalculated all the major astronomical constants. After 1884 he conceived, with A.M.W. Downing, a plan to resolve much international confusion on the subject. By the time he attended a standardisation conference in Paris, France, in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard.

Albert Einstein
Albert Einstein (14 March 1879 – 18 April 1955) explained the anomalous precession of Mercury's perihelion in his 1916 paper The Foundation of the General Theory of Relativity. This led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy. Observations of binary pulsars – the first in 1974 – whose orbits not only require the use of General Relativity for their explanation, but whose evolution proves the existence of gravitational radiation, was a discovery that led to the 1993 Nobel Physics Prize.

Examples of problems
Celestial motion, without additional forces such as drag forces or the thrust of a rocket, is governed by the reciprocal gravitational acceleration between masses. A generalization is the n-body problem, where a number n of masses are mutually interacting via the gravitational force. Although analytically not integrable in the general case, the integration can be well approximated numerically.


 * 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 $$n=2$$ case (two-body problem) the configuration is much simpler than for $$n>2$$. In this case, the system is fully integrable and exact solutions can be found.


 * 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 the "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:
 * The 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)

Perturbation theory
Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly. (It is closely related to methods used in numerical analysis, which are ancient.) The earliest use of modern perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: Newton's solution for the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun.

Perturbation methods start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually a Keplerian ellipse, which is correct when there are only two gravitating bodies (say, the Earth and the Moon), or a circular orbit, which is only correct in special cases of two-body motion, but is often close enough for practical use.

The solved, but simplified problem is then "perturbed" to make its time-rate-of-change equations for the object's position closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (the Sun). The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a remarkably better approximate solution to the real problem.

There is no requirement to stop at only one cycle of corrections. A partially corrected solution can be re-used as the new starting point for yet another cycle of perturbations and corrections. In principle, for most problems the recycling and refining of prior solutions to obtain a new generation of better solutions could continue indefinitely, to any desired finite degree of accuracy.

The common difficulty with the method is that the corrections usually progressively make the new solutions very much more complicated, so each cycle is much more difficult to manage than the previous cycle of corrections. Newton is reported to have said, regarding the problem of the Moon's orbit "It causeth my head to ache."

This general procedure – starting with a simplified problem and gradually adding corrections that make the starting point of the corrected problem closer to the real situation – is a widely used mathematical tool in advanced sciences and engineering. It is the natural extension of the "guess, check, and fix" method used anciently with numbers.

Reference frame
Problems in celestial mechanics are often posed in simplifying reference frames, such as the synodic reference frame applied to the three-body problem, where the origin coincides with the barycenter of the two larger celestial bodies. Other reference frames for n-body simulations include those that place the origin to follow the center of mass of a body, such as the heliocentric and the geocentric reference frames. The choice of reference frame gives rise to many phenomena, including the retrograde motion of superior planets while on a geocentric reference frame.