Celestial mechanics

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As a branch of astronomy, celestial mechanics describes the movement of astronomical objects based on physical theories with the help of mathematical modeling . The description of the planetary motion by Kepler's laws is a mathematical modeling, which was subsequently theoretically founded by Newtonian mechanics . The term astrodynamics is sometimes used synonymously, but specifically describes the movement of artificial bodies in the gravitational field . A separate sub-area, both due to its special interest and its complexity, is the lunar theory , which deals with the movement of the moon . The creation of tabular overviews of the movement of astronomical objects is known as ephemeris calculation.

Celestial mechanics is essentially based on the law of gravitation and a precise definition of coordinate and time systems . As a subject, it is closely related to astrometry and theoretical astronomy .


Antiquity and the Middle Ages

At the beginning of celestial mechanics there is the problem of predicting the movement of the planets, which originally included not the earth, but also the sun and moon. The first to deduce regularities from already very precise observations of these movements were from the 3rd millennium BC onwards. The inhabitants of Mesopotamia . This is passed down in later cuneiform texts of the Babylonians and Assyrians , for example the Venus tablets of Ammi-saduqa . Her findings also include the discovery of the regularity in the occurrence of solar or lunar eclipses , which is now known as the Saros cycle . The Egyptians also succeeded in the 3rd millennium BC. By observing the heliacal rising of Sirius, the length of the year was determined to be 365.25 days, which existed in Europe until the introduction of the Gregorian calendar in modern times.

The Greeks took the next big step by developing mathematical methods and models. Using geometric methods, Eratosthenes determined in the 3rd century BC The circumference of the earth with 252,000 stadia or 50 times the distance from Alexandria and Aswan , i.e. 41,750 km, which was very close to the actual value (40,075 km at the equator). Hipparchus in the 2nd century BC Chr. Calculated the distance of the moon with 30 diameters of the earth (= 382,260 km), which also almost corresponds to the mean distance measured today of 385,000 km. In addition, Hipparchus discovered the precession of the vernal equinox on the basis of a comparison with older measurements , a phenomenon that occurs when the earth's axis wobbles over a period of over 25,000 years.

In the middle of the 2nd century AD, the astronomical knowledge of antiquity was worked out by Claudius Ptolemaeus into a detailed geocentric worldview (→  Ptolemaic worldview ). His work Almagest remained decisive for all practical calculations of movements in the sky for around 1400 years. The model is based on a stationary earth and assigns the sun, moon and planets movements that are composed exclusively of uniform circular movements, because according to the Aristotelian philosophy these are the only possible form of movement without permanent drive. Ptolomew achieved approximate agreement with the observations of the individual planets by assuming complex orbits consisting of a larger circle ( deferent ) on which one (or more) smaller circles revolve ( epicyclic ). In addition, he had to start that the earth is not in the center of the deferents, but rather eccentrically , and that the circular movements on the deferents only run at a constant angular speed if they are related to different center points ( equants ). Despite the complicated construction, the observed positions of the planets deviated from the calculated positions in an irregular manner, often by up to 10 ′ (this corresponds to 1/3 the diameter of the moon).

Copernican turn

The turn to the heliocentric worldview , also known as the Copernican turn , was prepared by Nicolaus Copernicus at the beginning of the 16th century through his work Commentariolus and underpinned in 1543 by his main work De revolutionibus orbium coelestium . The model is based on the same (partly incorrect) observational data as Ptolomew, but classifies the earth under the planets, whose orbits now all lead around the sun.

With this, Copernicus achieved, above all, a strong conceptual simplification, because the uneven movements of the planets, as far as they are caused by observation from Earth, no longer have to be modeled individually for each planet. In addition, the distance between the planets and the sun could be determined in his system (in units of the radius of the earth's orbit, which thus became an astronomical unit ), and thus also their orbital speed. Only then did z. B. that with the distance the cycle time increases and the path speed decreases. Copernicus stuck to the Aristotelian basic idea that the heavenly bodies would only move on predetermined orbits. A noticeable improvement in accuracy could therefore not result from the Copernican model, so that the tables based on the Ptolemaic model continued to be used for the calculation of ephemeris and horoscopes .

In the Copernican system, the earth is downgraded from the center of the solar system to one of several planets, which is considered to be one of the triggers of the upheaval from the Middle Ages to the modern age. But the earth still played a special role. The earth's orbit is the only one that has an exact circular orbit, in the center of which the central sun rests and the orbital planes and apsidal lines of all other planets intersect.

The basic Aristotelian idea of ​​the uniform circular motions of the planets was only given up at the beginning of the 17th century by Johannes Kepler . With the help of Tycho Brahe's long-term observations , which were much more accurate than before and, above all, extended over the entire visible part of the planetary orbits, he was able to determine the shape of the orbits and the variation in orbital speed. He worked out a model in which the planets move on an ellipse with the (true) sun at one focal point ( 1st Kepler's law ), whereby the orbital speed varies according to a certain law depending on the distance to the sun ( 2nd Kepler's law ) . Kepler's Law ). The planetary positions calculated afterwards only deviated from the observations by up to 1 ′.

Mechanics of planetary movements

Kepler also made detailed considerations that these movements are determined by a constant influence from the sun. The leap to physical theory, in which the orbital movements could have been mathematically derived from simple statements about the forces acting between bodies, was not yet complete. This was only achieved by Isaac Newton , who in his work Philosophiae Naturalis Principia Mathematica ("Mathematical Principles of Natural Philosophy") not only formulated the mechanism of action of gravity , but also provided the tools through the development of infinitesimal calculus (which he called fluxion calculus ) which the movements resulting from its law of gravitation could be calculated. According to these calculations, Kepler's laws are only exactly valid if the consideration is limited to only two celestial bodies, e.g. B. Sun and a planet. Even for the irregularities of the moon's movement, he had to take into account the forces of the earth and the sun. The Principia Mathematica remained the authoritative standard work on celestial mechanics and mechanics in general until the end of the 18th century.

Newton's law of gravitation made it possible to calculate the positions of the planets much more precisely than before. The deviations from Kepler's orbits , known as orbital disturbances , were successfully traced back to the attraction of the other planets. It became famous later in the 19th century that the existence of another unknown planet could be deduced from the orbital disturbances of Uranus and its approximate position calculated (see below. Discovery of Neptune ).

Following Newton, his theory was applied, developed and refined. At the beginning of the 18th century Edmond Halley was able to come to the conclusion through the investigation of comet paths that several comets observed so far were not individual phenomena, but the periodic appearance of one and the same comet, namely Halley's comet named after him , its new one He successfully predicted the turn of the year 1758/1759. In the further development and refinement of celestial mechanical instruments, which went hand in hand with advances in mathematics, the mathematicians Euler , Clairaut and d'Alembert made significant contributions through their work on the three-body problem , perturbation theory and the lunar theory. The findings of this period were summarized in the monumental work Traité de mécanique céleste by Pierre-Simon Laplace .

The next big step came in connection with the discovery of the dwarf planet Ceres . The object was discovered by Giuseppe Piazzi on January 1st, 1801 and followed for a few weeks, then disappeared behind the sun and could not be found again despite great efforts. From September onwards, Carl Friedrich Gauß devoted himself to the problem, pursuing a completely new approach to orbit calculation, namely to find the Keplerellipse that best corresponded to the present observations without making any assumptions about the shape and position of the orbit. This extreme value task of minimizing errors is known today as the method of least squares and has countless applications outside of celestial mechanics. Based on Gauss' calculations, Ceres was then found again in December 1801 by Franz Xaver von Zach .

A further advance of celestial mechanical methods resulted from initially inexplicable deviations in the position of the planet Uranus, discovered in 1781, from the previously determined orbit (as already mentioned above). After first questioning the quality of older observations, considering deviations from Newton's law of gravitation and investigating possible disturbances from a hypothetical moon of Uranus, the view prevailed from 1840 that only disturbances from a previously undiscovered planet would make the observations in a satisfactory manner can explain. A complex problem of the “inverse” perturbation theory arose, in which the position of the interfering body had to be deduced from the observed perturbations. Almost at the same time, Urbain Le Verrier and John Couch Adams worked on its solution and in 1845 they came to the first results, which, however, have not yet received any attention. Only when George Biddell Airy , then Astronomer Royal in Greenwich , noticed that the results of Le Verrier and Adams were approximately identical, did he initiate a search. In the meantime, however, Le Verrier had asked the German astronomer Johann Gottfried Galle to search for the presumed planet at the calculated position. As a result, on September 23, 1846, Galle was able to find an undisclosed star at a distance of only one degree of arc from the forecast, which soon turned out to be a planet through its movement, the newly discovered planet Neptune .

general theory of relativity

The next big step arose at the beginning of the 20th century again from inexplicable deviations, this time in the orbit of the planet Mercury . It was found that the perihelion of Mercury changed slightly more (43 ″ per century) than could be explained by the gravitation of the sun and the known planets. The attempt to infer an unknown planet in the usual way, which was temporarily called " volcano " and which should have moved in the immediate vicinity of the sun, failed. Only through Albert Einstein's general theory of relativity could the perihelion of Mercury be fully explained by the curvature of space caused by the sun . In the decades that followed, the observation accuracy was improved to such an extent that relativistic corrections are now also included in the movements of all other bodies in the solar system .

New questions

Finally, the celestial mechanics of the present is characterized by new possibilities as well as new problems. On the one hand, new possibilities arose through the use of computers and thus a tremendous increase in the available computing power. Problems that would have required years of calculation in the past can now be solved with great accuracy within minutes. The performance of modern telescopes , which has increased by orders of magnitude, and the availability of instruments in space make completely new celestial mechanical phenomena visible today, for example exoplanets and their orbits. Problems that previously could only be dealt with in the beginning, such as the question of the stability of the solar system, the dynamics of the development of planetary systems or the formation and collisions of entire galaxies , can now be simulated using appropriately powerful computers.

Classic texts


  • Hans Bucerius: Lectures on celestial mechanics (2 volumes). Bibliographisches Institut, Mannheim 1966f.
  • Andreas Guthmann: Introduction to celestial mechanics and ephemeris calculation - theory, algorithms, numerics. Spectrum, Heidelberg 2000, ISBN 3-8274-0574-2 .
  • Jean Meeus : Astronomical Algorithms. Barth, Leipzig 1992, ISBN 3-335-00318-7 .
  • Franz Pichler: From planetary theories to celestial mechanics. Trauner, Linz 2004, ISBN 3-85487-780-3 .
  • Manfred Schneider : Celestial Mechanics (4 volumes). Spectrum, Heidelberg 1992ff.
  • Karl Stumpff : Celestial Mechanics (3 volumes). VEB Deutscher Verlag der Wissenschaften, Berlin
    • Vol. 1. The two-body problem and the methods of determining the orbit of planets and comets. 2nd edition, 1973.
    • Vol. 2. The three-body problem. 1965.
    • Vol. 3. General disorders. 1974.
  • Alessandra Celletti et al .: Modern celestial mechanics - from theory to applications. Kluwer, Dordrecht 2002, ISBN 1-4020-0762-0 .
  • Norriss S. Hetherington: Planetary motions - a historical perspective. Greenwood Press, Westport 2006, ISBN 0-313-33241-X .
  • Archie E. Roy: Orbital motion. Inst. Of Physics, Bristol 2005, ISBN 0-7503-1015-4 .

Web links

Commons : Celestial Mechanics  - Collection of images, videos and audio files
Wiktionary: celestial mechanics  - explanations of meanings, word origins, synonyms, translations

Individual evidence

  1. Duden article Astrodynamics
  2. celestial mechanics or astrodynamics? (Blog post by Florian Freistetter)
  3. Guthmann: Introduction 2000, p. 17
  4. a b C.A. Gearhart: Epicycles, eccentrics, and ellipses: The predictive capabilities of Copernican planetary models . In: Archive for History of Exact Sciences . tape 32 , no. 3 , 1985, pp. 207-222 , doi : 10.1007 / BF00348449 .
  5. Guthmann: Introduction 2000, p. 20
  6. Thomas Bührke: Great moments of astronomy: from Copernicus to Oppenheimer. Munich 2001, p. 150.
  7. Guthmann: Introduction 2000, pp. 24-26
  8. James Lequeux: Le Verrier - Magnificent and Detestable Astronomer. Springer Verlag, 2013, p. 23.