11

Today, sky coordinates are measured as "Right Ascension" (RA) and declination. These are similar to the angular coordinates we use for the Earth's surface but are measured on the celestial sphere relative to the celestial equator and pole. By using the current sidereal time, it is possible to map the local sky coordinates (ie. a horizontal bearing relative ...


8

There are several major ideas that went into general relativity: finite speed of gravity propagation, not necessarily Euclidean geometry of space, identification of inertia and gravity, mechanics as geometry, and uniformity of physical laws in all coordinate frames, even accelerated ones (general covariance). The last two ideas are specifically Einstein's, ...


8

At the time of Brahe and Kepler they did not use the right ascention and declination to record the movement of planets. These coordinates are related to the Earth, and it is known since the times of Hipparchus and Ptolemy that one has to use the ecliptic coordinates, that is a system related to the Sun motion. (Ecliptic is the large circle in the sky on ...


7

The right hand side picture shows qualitatively what you obtain from Ptolemy's model. The actual curve is a cycloid, in the first approximation. It uses rigorous mathematics. Moreover it describes quite well the actual motion of planets, including Mars. Just look at the actual motion of Mars in the sky (or in the Internet:-) To answer your specific ...


7

Goldstine, A History of Numerical Analysis from the 16th through the 19th Century (1977), describes Kepler's approach (p. 47), which may be found in Kepler's Epitome Astronomiae Copernicanae (1618), Ch. 4, Bk. V., pp. 665f. It is an iterative numerical algorithm Kepler called regula positionum. Goldstine describes the steps of an example, which begins on p....


7

Alexis Clairaut designed and led the detailed calculations of planetary perturbations to estimate the return-date in 1759 of what then became known as Halley's comet. The most detailed account of the work (in which Clairaut was assisted in the calculations by the young Jerome de Lalande and by Mme Lepaute) seems to be that given by Curtis Wilson in the ...


5

It was a theoretical invention. A special case, optimal elliptic transfer between two circular orbits, a.k.a. the Hohmann transfer, was described by Hohmann in The Attainability of Heavenly Bodies (1925), see Washington: NASA Technical Translation F-44, 1960 on Internet Archive. It is accomplished by two burns, one prograde one retrograde. Hohmann, ...


5

The movement of planets is not "chaotic". Otherwise prediction would be impossible. The planets move according to Kepler Laws, with very good accuracy. Strictly speaking the description of the motion as regular or chaotic depends on the time scale. On the scale of few thousand years, it is very regular, obeying the Kepler laws. Deviations from this regular ...


5

Kepler's Proofs will get you started on your quest. This article mentions 987 folio pages of arithmetic; you should also look at the tables and methods of Copernicus.


4

General references on the subject are Whittaker's History of the Theories of Aether and Electricity and Timeline Of History Of Electricity. Newton's persuasive evidence for the inverse square law of gravity was a defining achievent of new science, so of course it invited imitation wherever possible. Newton himself already approaches magnetism with gravity as ...


4

All these words are from astronomy, used to describe the motion of the Sun, Moon and planets. The simplest model is this. Suppose that Earth is the center of a circle. This circle is called deferent. A point moves on the deferent with constant angular velocity, this point is called the center of the epicycle, and epycycle itself is a circle centered at this ...


4

Kepler's third law was discovered on the basis of comparison of periods and distances of the planets. This was in 1619. Only in 1621 Kepler noticed that Galileo moons of Jupiter also satisfy this law. This fact was later used by Galileo as an argument in favor of Copernican system. By the way, Kepler was one of the first astronomers who used logarithms. (...


4

The Wikipedia article on Harmonices Mundi states that Kepler gave only the conclusion. Since he had taken all of Brahe's observations, the presumption is that he used this data, for he was very familiar with it, and it was more than adequate for the task. His published result describes the relationship in terms of the sun and planets, but not planets and ...


3

At the time of Kepler, not only Runge-Kutta, but the very notion of the differential equation was not avalailable:-) If you really would like to see how Kepler calculated the orbits, why don't you look at his own work, Astronomia Nova, which is available in English translation?


3

Poincare was studying the circular restricted three-body problem, where two bodies in a binary system revolve around their barycenter and a third test mass moves in some orbit around them. Birkhoff acknowledges this in his proof of the theorem, Proof of Poincare's Geometric Theorem. Specifically, Poincare was looking to prove that this third body may have ...


3

Actually, I've found some sources that suggest that Apollonius applied epicycles to planetary motion: From here: Apollonius also was an important person in founding Greek mathematical astronomy. He used geometrical models to explain planetary theory. He introduced systems of eccentric and epicyclic motion to explain the motion of the planets. From here: ...


3

I believe the question is asking about the following tables, one for the period 1800 AD to 2050 AD, the other two for 3000 BC to 3000 AD. Note that in these tables, "EM Bary" refers to the Earth-Moon barycenter, the center of mass of the Earth-Moon system. Table 1. Keplerian elements and their rates, with respect to the mean ecliptic and equinox of J2000, ...


3

Let me try to summarize this long discussion to an "answer". All that we know about epicycle theory comes from Ptolemy. Ptolemy credits Apollonius (262-190 BC) with one mathematical theorem (which says that excentric motion is equivalent to epicycle), and Hipparchus (190-120) with using epicycles to describe the motions of Sun, Moon and possibly planets. (...


2

In An IX du calenedrier republicain (1801) Jean-Baptiste Biot published his Analyse du traité de mécanique céleste de P.S. Laplace To develop the relationships existing between movements and forces that produce them, and from here to deduce the nature of the force that would animate celestial bodies in such a way that their movements are the ones ...


2

You can look in Wikipedia for the list of astronomers between Hipparchus and Ptolemy, or for more detail consult Neugebauer. (Or Ptolemy himself:-) The problem is that 90% of our knowledge about astronomy before Ptolemy (except Babylonian for which we have an independent source) is based on Ptolemy books, and this is not an exaggeration. An evidence of this ...


2

You are on to something here. Section X of Book 1 of Principia discusses the motion of particles constrained to curves or surfaces under the action of central forces (that is forces directed towards a fixed point, the center, with the magnitude depending only on the distance to it). This is an example of constrained dynamics, now associated with "D'Alembert'...


1

As it almost always happens it is impossible to tell exactly "who was the first". But Hellenistic scientists certainly knew this. The moon has visible parallax, so that they could prove that Moon is much closer than the Sun and planets. According to Ptolemy, Hipparchus knew about Moon's parallax. They supposed (correctly but without proof) that stars are ...


Only top voted, non community-wiki answers of a minimum length are eligible