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

For early theories of the causes of planetary motion, see Dynamics of the celestial spheres.

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.

Laws of planetary motion

For detailed treatments of how his laws of planetary motion can be used, see Kepler's laws of planetary motion and Kepler problem.

Johannes Kepler as 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 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 Newton developed his law of gravitation in 1686.

Newtonian mechanics and universal gravitation

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 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 Philosophiæ Naturalis Principia Mathematica in 1687.

Three-body problem

After Newton, Joseph-Louis Lagrange attempted to solve the three-body problem in 1772, analyzed the stability of planetary orbits, and discovered the existence of the Lagrange 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 extensions of Kepler's elliptical orbits). More recently, it has also become useful to calculate spacecraft trajectories.

Henri Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms of algebraic and transcendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton.

These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" (see, in particular, the Poincaré recurrence theorem) and the general theory of dynamical systems. He introduced the important concept of bifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).

Standardisation of astronomical tables

Simon Newcomb 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.

Anomalous precession of Mercury

Albert Einstein explained the anomalous precession of Mercury's perihelion in his 1916 paper The Foundation of the General Theory of Relativity. General relativity led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy.

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

Main article: 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.

Orbital mechanics

This section is an excerpt from Orbital mechanics.[edit]

A satellite orbiting Earth has a tangential velocity and an inward acceleration.

Orbital mechanics or astrodynamics is the application of ballistics and celestial mechanics to the practical problems concerning the motion of rockets, satellites, and other spacecraft. The motion of these objects is usually calculated from Newton's laws of motion and the law of universal gravitation. Orbital mechanics is a core discipline within space-mission design and control.

Celestial mechanics treats more broadly the orbital dynamics of systems under the influence of gravity, including both spacecraft and natural astronomical bodies such as star systems, planets, moons, and comets. Orbital mechanics focuses on spacecraft trajectories, including orbital maneuvers, orbital plane changes, and interplanetary transfers, and is used by mission planners to predict the results of propulsive maneuvers.

General relativity is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).

See also

• Astrometry is a part of astronomy that deals with measuring the positions of stars and other celestial bodies, their distances and movements.
• Astrophysics
• Celestial navigation is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.
• Developmental Ephemeris or the Jet Propulsion Laboratory Developmental Ephemeris (JPL DE) is a widely used model of the solar system, which combines celestial mechanics with numerical analysis and astronomical and spacecraft data.
• Dynamics of the celestial spheres concerns pre-Newtonian explanations of the causes of the motions of the stars and planets.
• Dynamical time scale
• Ephemeris is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.
• Gravitation
• Lunar theory attempts to account for the motions of the Moon.
• Numerical analysis is a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of a planet in the sky) which are too difficult to solve down to a general, exact formula.
• Creating a numerical model of the solar system was the original goal of celestial mechanics, and has only been imperfectly achieved. It continues to motivate research.
• An orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.
• Orbital elements are the parameters needed to specify a Newtonian two-body orbit uniquely.
• Osculating orbit is the temporary Keplerian orbit about a central body that an object would continue on, if other perturbations were not present.
• Retrograde motion is orbital motion in a system, such as a planet and its satellites, that is contrary to the direction of rotation of the central body, or more generally contrary in direction to the net angular momentum of the entire system.
• Apparent retrograde motion is the periodic, apparently backwards motion of planetary bodies when viewed from the Earth (an accelerated reference frame).
• 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.
• Tidal force is the combination of out-of-balance forces and accelerations of (mostly) solid bodies that raises tides in bodies of liquid (oceans), atmospheres, and strains planets' and satellites' crusts.
• Two solutions, called VSOP82 and VSOP87 are versions one mathematical theory for the orbits and positions of the major planets, which seeks to provide accurate positions over an extended period of time.

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.
• Doggett, LeRoy E. (1997), "Celestial Mechanics", in Lankford, John (ed.), History of Astronomy: An Encyclopedia, New York: Taylor & Francis, pp. 131–140, ISBN 9780815303220
• J.M.A. Danby, Fundamentals of Celestial Mechanics, 1992, Willmann-Bell
• Alessandra Celletti, Ettore Perozzi, Celestial Mechanics: The Waltz of the Planets, 2007, Springer-Praxis, ISBN 0-387-30777-X.
• Michael Efroimsky. 2005. Gauge Freedom in Orbital Mechanics. Annals of the New York Academy of Sciences, Vol. 1065, pp. 346-374
• Alessandra Celletti, Stability and Chaos in Celestial Mechanics. Springer-Praxis 2010, XVI, 264 p., Hardcover ISBN 978-3-540-85145-5

Further reading

• Encyclopedia:Celestial mechanics Scholarpedia Expert articles

External links

• Calvert, James B. (2003-03-28), Celestial Mechanics, University of Denver, archived from the original on 2006-09-07, retrieved 2006-08-21
• Astronomy of the Earth's Motion in Space, high-school level educational web site by David P. Stern
• Newtonian Dynamics Undergraduate level course by Richard Fitzpatrick. This includes Lagrangian and Hamiltonian Dynamics and applications to celestial mechanics, gravitational potential theory, the 3-body problem and Lunar motion (an example of the 3-body problem with the Sun, Moon, and the Earth).

Research

• Marshall Hampton's research page: Central configurations in the n-body problem Archived 2002-10-01 at the Wayback Machine

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

Associations

• Italian Celestial Mechanics and Astrodynamics Association

Simulations