Celestial mechanics

Celestial mechanics is the branch of astrophysics that deals with the motions of celestial objects. The field applies principles of physics, historically classical mechanics, to astronomical objects such as stars and planets to produce ephemeris data. Orbital mechanics (astrodynamics) is a subfield which focuses on the orbits of artificial satellites.

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 or more years, as early as the Babylonian astronomers.

Classical Greek writers speculated widely regarding celestial motions, and presented many geometrical mechanisms to model the motions of the planets. Their models employed combinations of uniform circular motion and were centered on the earth. An independent philosophical tradition was concerned with the physical causes of such circular motions. 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.

The only known supporter of Aristarchus was Seleucus of Seleucia, a Babylonian astronomer who is said to have proved heliocentrism through reasoning in the 2nd century BC. This may have involved the phenomenon of tides, which he correctly theorized to be caused by attraction to the Moon and notes that the height of the tides depends on the Moon's position relative to the Sun. Alternatively, he may have determined the constants of a geometric model for the heliocentric theory and developed methods to compute planetary positions using this model, possibly using early trigonometric methods that were available in his time, much like Copernicus.

Claudius Ptolemy

Claudius Ptolemy was an ancient astronomer and astrologer in early Imperial Roman times who wrote several books on astronomy. The most significant of these was the Almagest, which remained the most important book on predictive geometrical astronomy for some 1400 years. 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 data and parameters obtained 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. Although his model was extremely accurate, it relied solely on geometrical constructions rather than on physical causes; Ptolemy did not use celestial mechanics.

Early Middle Ages

Some have interpreted the planetary models developed by Aryabhata (476-550), an Indian astronomer, and Albumasar (787-886), a Persian astronomer, to be heliocentric models. In the 9th century AD, the Persian physicist and astronomer, Ja'far Muhammad ibn Mūsā ibn Shākir, hypothesized that the heavenly bodies and celestial spheres are subject to the same laws of physics as Earth, unlike the ancients who believed that the celestial spheres followed their own set of physical laws different from that of Earth. He also proposed that there is a force of attraction between heavenly bodies, vaguely foreshadowing the law of gravity.

Ibn al-Haytham

In the early 11th century, Ibn al-Haytham (Alhazen) wrote the Maqala fi daw al-qamar (On the Light of the Moon) some time before 1021. This was the first attempt successful at combining mathematical astronomy with physics and the earliest attempt at applying the experimental method to astronomy and astrophysics. He disproved the universally held opinion that the moon reflects sunlight like a mirror and correctly concluded that it "emits light from those portions of its surface which the sun's light strikes." In order to prove that "light is emitted from every point of the moon's illuminated surface," he built an "ingenious experimental device." Ibn al-Haytham had "formulated a clear conception of the relationship between an ideal mathematical model and the complex of observable phenomena; in particular, he was the first to make a systematic use of the method of varying the experimental conditions in a constant and uniform manner, in an experiment showing that the intensity of the light-spot formed by the projection of the moonlight through two small apertures onto a screen diminishes constantly as one of the apertures is gradually blocked up."

He also presented a development of Ptolemy's geocentric epicyclic models in terms of nested celestial spheres. In chapters 15-16 of his Book of Optics, he also discovered that the celestial spheres do not consist of solid matter.

Late Middle Ages

There was much debate on the dynamics of the celestial spheres during the late Middle Ages. Averroes (Ibn Rushd), Ibn Bajjah (Avempace) and Thomas Aquinas developed the theory of inertia in the celestial spheres, while Avicenna (Ibn Sina) and Jean Buridan developed the theory of impetus in the celestial spheres.

In the 14th century, Ibn al-Shatir produced the first model of lunar motion which matched physical observations, and which was later used by Copernicus. In the 13th to 15th centuries, Tusi and Ali Kuşçu provided the earliest empirical evidence for the Earth's rotation, using the phenomena of comets to refute Ptolemy's claim that a stationary Earth can be determined through observation. Kuşçu further rejected Aristotelian physics and natural philosophy, allowing astronomy and physics to become empirical and mathematical instead of philosophical. In the early 16th century, the debate on the Earth's motion was continued by Al-Birjandi (d. 1528), who in his analysis of what might occur if the Earth were rotating, develops a hypothesis similar to Galileo Galilei's notion of "circular inertia", which he described in the following observational test:

Johannes Kepler

Johannes Kepler (December 27, 1571 - November 15, 1630) 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 using his physical principles and 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 (January 4, 1643 – March 31, 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 Newton's law of gravitation, proving Kepler's Laws for the case of a circular orbit is simple. Elliptical orbits involve more complex calculations, which Newton included in his Principia.

Joseph-Louis Lagrange

After Newton, Lagrange (January 25, 1736 - April 10, 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. More recently, it has also become useful to calculate spacecraft trajectories.

Simon Newcomb

Simon Newcomb (March 12, 1835 – July 11, 1909) was a Canadian-American astronomer 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 1896-May, the international consensus was 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

After Albert Einstein (March 14, 1879 - April 18, 1955) 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 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:

• 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. This assumption sacrifices accuracy for simplicity, especially for high eccentricity orbits which are by definition non-circular.

Examples:

• 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

link to http://dilemna.org.uk link to http://wiki.en.eprocabeed.org.uk

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 perturbation theory was to deal with the otherwise unsolveable 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 simple enough to be solved exactly. 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 starting conditions closer to the real problem, such as including the gravitational attraction of a third body (the Sun). The slight changes that result, which themselves may have been simplifed yet again, are used as corrections. Because of simplifications introduced along every step of the way, 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. The common difficulty with the method is that usually the corrections 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.

See also

• Astrometry is a part of astronomy that deals with measuring 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.
• Celestial navigation is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.
• 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.
• 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.
• 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.
• 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.

External links

• Calvert, James B. Celestial Mechanics. University of Denver. Retrieved on 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

Associations

• Italian Celestial Mechanics and Astrodynamics Association

References