Modern analytic celestial mechanics started withIsaac Newton'sPrincipia (1687). The namecelestial mechanics is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later withGottfried Leibniz, and over a century after Newton,Pierre-Simon Laplace introduced the termcelestial mechanics. Prior toKepler, there was little connection between exact, quantitative prediction of planetary positions, usinggeometrical ornumerical techniques, and contemporary discussions of the physical causes of the planets' motion.
Isaac Newton is credited with introducing the idea that the motion of objects in the heavens, such asplanets, theSun, and theMoon, and the motion of objects on the ground, likecannon balls and falling apples, could be described by the same set ofphysical laws. In this sense he unifiedcelestial andterrestrial dynamics. Usinghis law of gravity, Newton confirmedKepler's laws for elliptical orbits by deriving them from the gravitationaltwo-body problem, which Newton included in his epochalPhilosophiæ Naturalis Principia Mathematica in 1687.
After Newton,Joseph-Louis Lagrange attempted to solve the three-body problem in 1772, analyzed the stability of planetary orbits, and discovered the existence of theLagrange points. Lagrange also reformulated the principles ofclassical mechanics, emphasizing energy more than force, and developing amethod 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 andcomets and such (parabolic and hyperbolic orbits areconic section extensions of Kepler'selliptical orbits). More recently, it has also become useful to calculatespacecrafttrajectories.
Henri Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" (1892–1899) and "Lectures on Celestial Mechanics" (1905–1910). In them, he successfully applied the results of their research to the problem of the motion of three bodies and studied in detail the behavior of solutions (frequency, stability, asymptotic, and so on). Poincaré showed that the three-body problem is not integrable. In other words, the general solution of the three-body problem can not be expressed in terms ofalgebraic andtranscendental functions through unambiguous coordinates and velocities of the bodies. His work in this area was the first major achievement in celestial mechanics since Isaac Newton.[1]
These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" (see, in particular, thePoincaré recurrence theorem) and the general theory ofdynamical systems. He introduced the important concept ofbifurcation points and proved the existence of equilibrium figures such as the non-ellipsoids, including ring-shaped and pear-shaped figures, and their stability. For this discovery, Poincaré received the Gold Medal of the Royal Astronomical Society (1900).[2]
Simon Newcomb was a Canadian-American astronomer who revisedPeter Andreas Hansen's table of lunar positions. In 1877, assisted byGeorge 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 inParis, France, in May 1886, the international consensus was that all ephemerides should be based on Newcomb's calculations. A further conference as late as 1950 confirmed Newcomb's constants as the international standard.
Celestial motion, without additional forces such asdrag forces or thethrust of arocket, is governed by the reciprocal gravitational acceleration between masses. A generalization is then-body problem,[3] where a numbern of masses are mutually interacting via the gravitational force. Although analytically notintegrable in the general case,[4] the integration can be well approximated numerically.
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 thepatched conic approximation)
In the case (two-body problem) the configuration is much simpler than for. In this case, the system is fully integrable and exact solutions can be found.[5]
A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, theorbiting body, is much smaller than the other, thecentral body. This is also often approximately valid.
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 innumerical analysis, whichare ancient.) The earliest use of modernperturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics:Newton's solution for the orbit of theMoon, which moves noticeably differently from a simpleKeplerian ellipse because of the competing gravitation of theEarth and theSun.
Perturbation methods start with a simplified form of the original problem, which is carefully chosen to be exactly solvable. In celestial mechanics, this is usually aKeplerian ellipse, which is correct when there are only two gravitating bodies (say, theEarth and theMoon), 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 itstime-rate-of-change equations for the object's position closer to the values from the real problem, such as including the gravitational attraction of a third, more distant body (theSun). The slight changes that result from the terms in the equations – which themselves may have been simplified yet again – are used as corrections to the original solution. Because simplifications are made at every step, 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. In principle, for most problems the recycling and refining of prior solutions to obtain a new generation of better solutions could continue indefinitely, to any desired finite degree of accuracy.
The common difficulty with the method is that the corrections usually 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 theMoon's orbit"It causeth my head to ache."[6]
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" methodused anciently with numbers.
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Problems in celestial mechanics are often posed in simplifying reference frames, such as thesynodic reference frame applied to thethree-body problem, where the origin coincides with thebarycenter of the two larger celestial bodies. Other reference frames for n-body simulations include those that place the origin to follow the center of mass of a body, such as the heliocentric and the geocentric reference frames.[7] The choice of reference frame gives rise to many phenomena, including theretrograde motion ofsuperior planets while on a geocentric reference frame.
General relativity is a more exact theory than Newton's laws for calculating orbits, and it is sometimes necessary to use it for greater accuracy or in high-gravity situations (e.g. orbits near the Sun).
Ephemeris is a compilation of positions of naturally occurring astronomical objects as well as artificial satellites in the sky at a given time or times.
Lunar theory attempts to account for the motions of the Moon.
Numerical analysis is a branch of mathematics, pioneered by celestial mechanicians, for calculating approximate numerical answers (such as the position of aplanet in the sky) which are too difficult to solve down to a general, exact formula.
Creating anumerical model of the solar system was the original goal of celestial mechanics, and has only been imperfectly achieved. It continues to motivate research.
Anorbit 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.
Retrograde motion is orbital motion in a system, such as a planet and its satellites, that is contrary to the direction of rotation of the central body, or more generally contrary in direction to the net angular momentum of the entire system.
Apparent retrograde motion is the periodic, apparently backwards motion of planetary bodies when viewed from the Earth (an accelerated reference frame).
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 anynatural satellite of the other planets.
Tidal force is the combination of out-of-balance forces and accelerations of (mostly) solid bodies that raises tides in bodies of liquid (oceans), atmospheres, and strains planets' and satellites' crusts.
Two solutions, calledVSOP82 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.
^Guerra, André G C; Carvalho, Paulo Simeão (1 August 2016). "Orbital motions of astronomical bodies and their centre of mass from different reference frames: a conceptual step between the geocentric and heliocentric models".Physics Education.51 (5).arXiv:1605.01339.Bibcode:2016PhyEd..51e5012G.doi:10.1088/0031-9120/51/5/055012.
Newtonian Dynamics Undergraduate level course by Richard Fitzpatrick. This includes Lagrangian and Hamiltonian Dynamics and applications to celestial mechanics, gravitational potential theory, the 3-body problem and Lunar motion (an example of the 3-body problem with the Sun, Moon, and the Earth).
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