Wikipedia

User:Besmeraldo/Deferent and epicycle

< User:Besmeraldo

Bad Science

edit

"Adding epicycles" has come to be used as a derogatory comment in modern scientific discussion. The term might be used, for example, to describe continuing to try to adjust a theory to make its predictions match the facts. There is a generally accepted idea that extra epicycles were invented to alleviate the growing errors that the Ptolemaic system noted as measurements became more accurate, particularly for Mars. According to this notion, epicycles are regarded by some as the paradigmatic example of bad science.[1]

Copernicus added an extra epicycle to his planets, but that was only in an effort to eliminate Ptolemy's equant, which he considered a philosophical break away from Aristotle's perfection of the heavens. Mathematically, the second epicycle and the equant produce the same results, and many Copernican astronomers before Kepler continued using the equant, as the mathematic calculations were easier.It is important to keep in mind the importance of Galileo's trial on science at the time. Galileo was a strong proponent for the heliocentric model associated with Kepler, Copernicus', and his observations of the cosmos. More importantly, his model did not include any epicycles or deferents, and this was a feat that he was proud of achieving[2].For as much critics that Ptolomy receives for its epicycle and defferent model, with the excessive use of epicycles as a mean to correct the flaws of his system and make it confirm to his observations, Copernicus was the thinker that applied extra epicycles to correct for defects on his theory, not Ptolomy.[3]

Being a system that is for the most part used to justify the geocentric model, with the exception of Copernicus cosmos, the different and epicycle model was favored over the heliocentric ideas that Kepler and Galileo proposed, this fact is made abundantly clear when the Catholic church is analyzed, as it endorsed this model as it favored its central dogma[4]. Later adopters of the epicyclic model such as Tycho Brahe who considered the Church's scriptures when creating his model[5], were even more favored by it. The Tychonic model was a hybrid model that blended the geocentric and heliocentric characteristics, with a still Earth that has the sun and moon surrounding it, and the planets orbiting the sun. To Brahe, the idea of a revolving and moving Earth was impossible, and the scripture should be always paramount and respected[6]. When Galileo tried to challenge Tycho Brahe's System, the church was dissatisfied with their views being challenged, Galileo's publication did not aid his case on the Galileo's Trial.

History

edit

When ancient astronomers viewed the sky, they saw the Sun, Moon, and stars moving overhead in a regular fashion. Babylonians did celestial observations, mainly of the sun and moon as a means of recalibrating and preserving timekeeping for religious ceremonies[7]. Other early civilizations such as the greeks had thinkers like Thales of Miletus, the first to document and predict a solar eclipse[8] , or Heraclides, the earliest recorded astronomer to propose a geocentric model for the cosmos[9]. They also saw the "wanderers" or "planetai" (our planets). The regularity in the motions of the wandering bodies suggested that their positions might be predictable.

 
The complexity to be described by the geocentric model

The most obvious approach to the problem of predicting the motions of the heavenly bodies was simply to map their positions against the star field and then to fit mathematical functions to the changing positions.[10]The introduction of better celestial measurement instruments, such as the introduction of the Gnomon by Anaximander[11] allowed the greeks to have a better understanding of the passage of time, such as the number of days in a year and the length of seasons[12], indispensable for astronomic measurements.

The ancients worked from a geocentric perspective for the simple reason that the Earth was where they stood and observed the sky, and it is the sky which appears to move while the ground seems still and steady underfoot. Some Greek astronomers (e.g., Aristarchus of Samos) speculated that the planets (Earth included) orbited the Sun, but the optics (and the specific mathematics – Isaac Newton's Law of Gravitation for example) necessary to provide data that would convincingly support the heliocentric model did not exist in Ptolemy's time and would not come around for over fifteen hundred years after his time. Furthermore, Aristotelian physics was not designed with these sorts of calculations in mind, and Aristotle's philosophy regarding the heavens was entirely at odds with the concept of heliocentrism. It was not until Galileo Galilei observed the moons of Jupiter on 7 January 1610, and the phases of Venus in September 1610 that the heliocentric model began to receive broad support among astronomers, who also came to accept the notion that the planets are individual worlds orbiting the Sun (that is, that the Earth is a planet and is one among several). Johannes Kepler was able to formulate his three laws of planetary motion, which described the orbits of the planets in our solar system to a remarkable degree of accuracy utilizing a system that employs ellipsis rather than circular orbits; Kepler's three laws are still taught today in university physics and astronomy classes, and the wording of these laws has not changed since Kepler first formulated them four hundred years ago.

The apparent motion of the heavenly bodies with respect to time is cyclical in nature. Apollonius of Perga realized that this cyclical variation could be represented visually by small circular orbits, or epicycles, revolving on larger circular orbits, or deferents. Hipparchus calculated the required orbits. Deferents and epicycles in the ancient models did not represent orbits in the modern sense, rather, a complex set of circular paths whose centers are separated by a specific distance in order to approximate the observed movement of the celestial bodies.

Claudius Ptolemy refined the deferent-and-epicycle concept and introduced the equant as a mechanism for accounting for velocity variations in the motions of the planets. The empirical methodology he developed proved to be extraordinarily accurate for its day and was still in use at the time of Copernicus and Kepler.

 
The basic simplicity of the Copernican universe, from Thomas Digges' book

Owen Gingerich[13] describes a planetary conjunction that occurred in 1504 that was apparently observed by Copernicus. In notes bound with his copy of the Alfonsine Tables, Copernicus commented that "Mars surpasses the numbers by more than two degrees. Saturn is surpassed by the numbers by one and a half degrees." Using modern computer programs, Gingerich discovered that, at the time of the conjunction, Saturn indeed lagged behind the tables by a degree and a half and Mars led the predictions by nearly two degrees. Moreover, he found that Ptolemy's predictions for Jupiter at the same time were quite accurate. Copernicus and his contemporaries were therefore using Ptolemy's methods and finding them trustworthy well over a thousand years after Ptolemy's original work was published.

When Copernicus transformed Earth-based observations to heliocentric coordinates,[14] he was confronted with an entirely new problem. The Sun-centered positions displayed a cyclical motion with respect to time but without retrograde loops in the case of the outer planets. In principle, the heliocentric motion was simpler but with new subtleties due to the yet-to-be-discovered elliptical shape of the orbits. Another complication was caused by a problem that Copernicus never solved: correctly accounting for the motion of the Earth in the coordinate transformation.[15] In keeping with past practice, Copernicus used the deferent/epicycle model in his theory but his epicycles were small and were called "epicyclets".

In the Ptolemaic system the models for each of the planets were different and so it was with Copernicus' initial models. As he worked through the mathematics, however, Copernicus discovered that his models could be combined in a unified system. Furthermore, if they were scaled so that the Earth's orbit was the same in all of them, the ordering of the planets we recognize today easily followed from the math. Mercury orbited closest to the Sun and the rest of the planets fell into place in order outward, arranged in distance by their periods of revolution.[16]

Although Copernicus' models reduced the magnitude of the epicycles considerably, whether they were simpler than Ptolemy's is moot. Copernicus eliminated Ptolemy's somewhat-maligned equant but at a cost of additional epicycles. Various 16th-century books based on Ptolemy and Copernicus use about equal numbers of epicycles.[17][18][19] The idea that Copernicus used only 34 circles in his system comes from his own statement in a preliminary unpublished sketch called the Commentariolus. By the time he published De revolutionibus orbium coelestium, he had added more circles. Counting the total number is difficult, but estimates are that he created a system just as complicated, or even more so.[20] Koestler, in his history of man's vision of the universe, equates the number of epicycles used by Copernicus at 48.[21] The popular total of about 80 circles for the Ptolemaic system seems to have appeared in 1898. It may have been inspired by the non-Ptolemaic system of Girolamo Fracastoro, who used either 77 or 79 orbs in his system inspired by Eudoxus of Cnidus.[22] Copernicus in his works exaggerated the number of epicycles used in the Ptolemaic system; although original counts ranged to 80 circles, by Copernicus's time the Ptolemaic system had been updated by Peurbach towards the similar number of 40; hence Copernicus effectively replaced the problem of retrograde with further epicycles.[23]

Copernicus' theory was at least as accurate as Ptolemy's but never achieved the stature and recognition of Ptolemy's theory. What was needed was Kepler's elliptical theory, not published until 1609 and 1619. Copernicus' work provided explanations for phenomena like retrograde motion, but really did not prove that the planets actually orbited the Sun.

 
The deferent (O) is offset from the Earth (T). P is the centre of the epicycle of the Sun S.

Ptolemy's and Copernicus' theories proved the durability and adaptability of the deferent/epicycle device for representing planetary motion. The deferent/epicycle models worked as well as they did because of the extraordinary orbital stability of the solar system. Either theory could be used today had Gottfried Wilhelm Leibniz and Isaac Newton not invented calculus.[24]

The first planetary model without any epicycles was that of Ibn Bajjah (Avempace) in 12th century Andalusian Spain,[25] but epicycles were not eliminated in Europe until the 17th century, when Johannes Kepler's model of elliptical orbits gradually replaced Copernicus' model based on perfect circles.

Newtonian or classical mechanics eliminated the need for deferent/epicycle methods altogether and produced more accurate theories. By treating the Sun and planets as point masses and using Newton's law of universal gravitation, equations of motion were derived that could be solved by various means to compute predictions of planetary orbital velocities and positions. Simple two-body problems, for example, can be solved analytically. More-complex n-body problems require numerical methods for solution.

The power of Newtonian mechanics to solve problems in orbital mechanics is illustrated by the discovery of Neptune. Analysis of observed perturbations in the orbit of Uranus produced estimates of the suspected planet's position within a degree of where it was found. This could not have been accomplished with deferent/epicycle methods. Still, Newton in 1702 published Theory of the Moon's Motion which employed an epicycle and remained in use in China into the nineteenth century. Subsequent tables based on Newton's Theory could have approached arcminute accuracy.[26]


References

edit
  1. ^ See e.g., Kolb, Rocky, Blind Watchers of the Sky, Addison–Wesley, 1996. P. 299 (ISBN 0-201-48992-9)
  2. ^ "Whose Revolution? Copernicus, Brahe & Kepler | Modeling the Cosmos | Articles and Essays | Finding Our Place in the Cosmos: From Galileo to Sagan and Beyond | Digital Collections | Library of Congress". Library of Congress, Washington, D.C. 20540 USA. Retrieved 2021-12-06.
  3. ^ Sherwood., Schlafly, Roger (2011). How Einstein ruined physics : motion, symmetry, and revolution in science. [publisher not identified]. ISBN 978-1-4611-2019-3. OCLC 784402136.{{cite book}}: CS1 maint: multiple names: authors list (link)
  4. ^ Gurnett, Donald A. (2009). "The Search for Life in the Solar System". Transactions of the American Clinical and Climatological Association. 120: 299–325. ISSN 0065-7778. PMC 2744519. PMID 19768185.
  5. ^ Håkansson, Håkan (2007), "Tycho the Prophet: History, Astrology and the Apocalypse in Early Modern Science", The Word and the World, London: Palgrave Macmillan UK, pp. 137–156, retrieved 2021-12-06
  6. ^ Repcheck, Jack (2008). Copernicus' secret : how the scientific revolution began (1st Simon & Schuster trade pbk. ed ed.). New York: Simon & Schuster Paperbacks. ISBN 978-0-7432-8952-8. OCLC 209693599. {{cite book}}: |edition= has extra text (help)
  7. ^ Olmstead, A. T. (1938). "Babylonian Astronomy: Historical Sketch". The American Journal of Semitic Languages and Literatures. 55 (2): 113–129. ISSN 1062-0516.
  8. ^ Mosshammer, Alden A. (1981). "Thales' Eclipse". Transactions of the American Philological Association (1974-). 111: 145–155. doi:10.2307/284125. ISSN 0360-5949.
  9. ^ Splinter, Robert (2017). Illustrated encyclopedia of applied and engineering physics. Boca Raton: CRC Press. ISBN 978-1-351-62563-0. OCLC 983744463.
  10. ^ For an example of the complexity of the problem, see Owen Gingerich, The Book Nobody Read, Walker, 2004, p. 50
  11. ^ author., Diogenes Laertius,. The lives and opinions of eminent philosophers. ISBN 978-1-230-21699-7. OCLC 881385989. {{cite book}}: |last= has generic name (help)CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link)
  12. ^ Pedersen, Olaf (1993). Early physics and astronomy : a historical introduction (Rev. ed ed.). Cambridge [England]: Cambridge University Press. ISBN 0-521-40340-5. OCLC 24173447. {{cite book}}: |edition= has extra text (help)
  13. ^ Gingerich, Chapter 4
  14. ^ One volume of De Revolutionibus was devoted to a description of the trigonometry used to make the transformation between geocentric and heliocentric coordinates.
  15. ^ Gingerich, p. 267
  16. ^ Gingerich, p. 54
  17. ^ Palter, Robert (1970). "Approach to the History of Astronomy". Studies in the History and Philosophy of Science. 1: 94.
  18. ^ Owen Gingerich, "Alfonso X as a Patron of Astronomy", in The Eye of Heaven: Ptolemy, Copernicus, Kepler (New York: American Institute of Physics, 1993), p. 125.
  19. ^ Gingerich, "Crisis versus Aesthetic in the Copernican Revolution", in Eye of Heaven, pp. 193–204.
  20. ^ "The popular belief that Copernicus's heliocentric system constitutes a significant simplification of the Ptolemaic system is obviously wrong ... [T]he Copernican models themselves require about twice as many circles as the Ptolemaic models and are far less elegant and adaptable." Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. ISBN 978-0-486-22332-2., p. 204. This is an extreme estimate in favor of Ptolemy.
  21. ^ Koestler, Arthur (1989) [1959]. The Sleepwalkers. Arkana, Penguin Books., p. 195
  22. ^ Palter, Approach to the History of Astronomy, pp. 113–114.
  23. ^ Koestler, Arthur (1989) [1959]. The Sleepwalkers. Arkana, Penguin Books., pp. 194–195
  24. ^ A deferent/epicycle model is in fact used to compute Lunar positions needed to define modern Hindu calendars. See Nachum Dershovitz and Edward M. Reingold: Calendrical Calculations, Cambridge University Press, 1997, Chapter 14. (ISBN 0-521-56474-3)
  25. ^ Goldstein, Bernard R. (1972). "Theory and Observation in Medieval Astronomy". Isis. 63 (1): 39–47 [40–41]. doi:10.1086/350839.
  26. ^ Kollerstrom, Nicholas (2000). Newton's Forgotten Lunar Theory. Green Lion Press. ISBN 1-888009-08-X.
Retrieved from "https://en.wikipedia.org/w/index.php?title=User:Besmeraldo/Deferent_and_epicycle&oldid=1059049301"