- Celestial mechanics
Celestial mechanics is the branch of
astrophysicsthat 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 ephemerisdata. 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 reasoningin the 2nd century BC. This may have involved the phenomenon of tides, [ Lucio Russo, "Flussi e riflussi", Feltrinelli, Milano, 2003, ISBN 88-07-10349-4.] which he correctly theorized to be caused by attraction to the Moonand notes that the height of the tides depends on the Moon's position relative to the Sun. [ Bartel Leendert van der Waerden(1987). "The Heliocentric System in Greek, Persian and Hindu Astronomy", "Annals of the New York Academy of Sciences" 500 (1), 525–545  .] 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. [ Bartel Leendert van der Waerden(1987). "The Heliocentric System in Greek, Persian and Hindu Astronomy", "Annals of the New York Academy of Sciences" 500 (1), 525–545 [527-529] .]
Claudius Ptolemywas 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, [B. L. van der Waerden (1970), "Das heliozentrische System in der griechischen,persischen und indischen Astronomie," Naturforschenden Gesellschaft in Zürich, Zürich: Kommissionsverlag Leeman AG. ( cf.Noel Swerdlow (June 1973), "Review: A Lost Monument of Indian Astronomy", "Isis" 64 (2), p. 239-243.)
B. L. van der Waerden (1987), "The heliocentric system in Greek, Persian, and Indian astronomy", in "From deferent to equant: a volume of studies in the history of science in the ancient and medieval near east in honor of E. S. Kennedy", "
New York Academy of Sciences" 500, p. 525-546. ( cf.Dennis Duke (2005), "The Equant in India: The Mathematical Basis of Ancient Indian Planetary Models", "Archive for History of Exact Sciences" 59, p. 563–576.).] [Thurston, Hugh (1994), "Early Astronomy", Springer-Verlag, New York. ISBN 0-387-94107-X, p. 188: quote|"Not only did Aryabhata believe that the earth rotates, but there are glimmerings in his system (and other similar systems) of a possible underlying theory in which the earth (and the planets) orbits the sun, rather than the sun orbiting the earth. The evidence is that the basic planetary periods are relative to the sun."] [ Lucio Russo(2004), "The Forgotten Revolution: How Science Was Born in 300 BC and Why It Had To Be Reborn", Springer, Berlin, ISBN 978-3-540-20396-4. ( cf.Dennis Duke (2005), "The Equant in India: The Mathematical Basis of Ancient Indian Planetary Models", "Archive for History of Exact Sciences" 59, p. 563–576.)] and Albumasar (787-886), a Persian astronomer, to be heliocentric models. [ Bartel Leendert van der Waerden(1987). "The Heliocentric System in Greek, Persian and Hindu Astronomy", "Annals of the New York Academy of Sciences" 500 (1), 525–545 [534-537] .] In the 9th century AD, the Persian physicist and astronomer, Ja'far Muhammad ibn Mūsā ibn Shākir, hypothesized that the heavenly bodies and celestial spheresare 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. [Harvard reference |last=Saliba |first=George |authorlink=George Saliba |year=1994a |title=Early Arabic Critique of Ptolemaic Cosmology: A Ninth-Century Text on the Motion of the Celestial Spheres |journal=Journal for the History of Astronomy |volume=25 |pages=115-141  ] He also proposed that there is a forceof attraction between heavenly bodies, [citation|first=K. A.|last=Waheed|year=1978|title=Islam and The Origins of Modern Science|page=27|publisher=Islamic Publication Ltd., Lahore] vaguely foreshadowing the law of gravity. [Harvard reference |last=Briffault |first=Robert |authorlink=Robert Briffault |year=1938 |title=The Making of Humanity |page=191]
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 physicsand the earliest attempt at applying the experimental method to astronomy and astrophysics. He disproved the universally held opinion that the moonreflects sunlightlike a mirrorand 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 intensityof the light-spot formed by the projection of the moonlightthrough two small apertures onto a screen diminishes constantly as one of the apertures is gradually blocked up."citation|first=G. J.|last=Toomer|title=Review: "Ibn al-Haythams Weg zur Physik" by Matthias Schramm|journal=Isis|volume=55|issue=4|date=December 1964|pages=463–465 [463–4] |doi=10.1086/349914]
He also presented a development of Ptolemy's geocentric epicyclic models in terms of nested
celestial spheres. [Y. Tzvi Langerman (1990), "Ibn al Haytham's On the Configuration of the World", p. 11-25, New York: Garland Publishing.] In chapters 15-16 of his " Book of Optics", he also discovered that the celestial spheres do not consist of solidmatter. [Edward Rosen (1985), "The Dissolution of the Solid Celestial Spheres", "Journal of the History of Ideas" 46 (1), p. 13-31 [19-20, 21] .]
Late Middle Ages
There was much debate on the dynamics of the
celestial spheresduring the late Middle Ages. Averroes(Ibn Rushd), Ibn Bajjah(Avempace) and Thomas Aquinasdeveloped the theory of inertiain the celestial spheres, while Avicenna(Ibn Sina) and Jean Buridandeveloped the theory of impetusin the celestial spheres.
In the 14th century,
Ibn al-Shatirproduced the first model of lunar motion which matched physical observations, and which was later used by Copernicus. George Saliba(2007), [http://youtube.com/watch?v=GfissgPCgfM Lecture at SOAS, London - Part 4/7] and [http://youtube.com/watch?v=0VMBRAd6YBU Lecture at SOAS, London - Part 5/7] ] In the 13th to 15th centuries, Tusi and Ali Kuşçuprovided 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 physicsand 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: [Harvard reference |last=Ragep |first=F. Jamil |year=2001a |title=Tusi and Copernicus: The Earth's Motion in Context |journal=Science in Context |volume=14 |issue=1-2 |pages=145–163 |publisher= Cambridge University Press] [Harvard reference |last=Ragep |first=F. Jamil |year=2001b |title=Freeing Astronomy from Philosophy: An Aspect of Islamic Influence on Science |journal=Osiris, 2nd Series |volume=16 |issue=Science in Theistic Contexts: Cognitive Dimensions |pages=49-64 & 66-71 ]
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 Newtonhad even developed his law of gravitation.
Kepler's laws of planetary motionand the Keplerian problemfor a detailed treatment of how his laws of planetary motion can be used.
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 cannonballs 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.
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 spacecrafttrajectories.
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(March 14, 1879 - April 18, 1955) explained the anomalous precession of Mercury's perihelion, astronomers recognized that Newtonian mechanicsdid not provide the highest accuracy. Today, we have binary pulsarswhose orbits not only require the use of General Relativityfor 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
thrustof 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.
*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)
**Spaceflight to, and stay at a
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.
binary star, e.g. Alpha Centauri(approx. the same mass)
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.
Solar systemorbiting 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.
*The orbit of the
dwarf planet Pluto, ecc. = 0.2488
*The orbit of Mercury, ecc. = 0.2056
Hohmann transfer orbit
Suborbital flightslink to http://dilemna.org.uklink to http://wiki.en.eprocabeed.org.uk
Perturbation theorycomprises 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 theorywas 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 Earthand 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
Earthand 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.
Astrometryis a part of astronomy that deals with measuring the positions of stars and other celestial bodies, their distances and movements.
Astrodynamicsis the study and creation of orbits, especially those of artificial satellites.
Celestial navigationis a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.
Numerical analysisis a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of a planetin 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 elementsare the parameters needed to specify a Newtonian two-body orbit uniquely.
Osculating orbitis the temporary Keplerian orbit about a central body that an object would continue on, if other perturbations were not present.
Satelliteis 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 satelliteof the other planets.
Jet Propulsion Laboratory Developmental Ephemeris(JPL DE) is a widely used model of the solar system, which combines celestial mechanics with numerical analysisand 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.
* [http://www.math.washington.edu/~hampton/research.html Marshall Hampton's research page: Central configurations in the n-body problem]
* [http://www.cmlab.com Celestial Mechanics is a Planetarium Artwork created by D. S. Hessels and G. Dunne]
* [http://orca.phys.uvic.ca/~tatum/celmechs.html Professor Tatum's course notes at the University of Victoria]
* [http://www.mat.uniroma2.it/simca/english.html Italian Celestial Mechanics and Astrodynamics Association]
* Asger Aaboe, "Episodes from the Early History of Astronomy", 2001, Springer-Verlag, ISBN 0-387-95136-9
* 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)
* 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."
* [http://www.annalsnyas.org/cgi/content/abstract/1065/1/346 Annals of the New York Academy of Sciences, Vol. 1065, pp. 346-374]
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