In a binary system of objects interacting through gravity,Newtonian mechanics can used to produce a set oforbital elements that will predict with reasonable accuracy the future position of the two bodies. This method demonstrates the correctness ofKepler's laws of planetary motion. Where one of the bodies is sufficiently massive,general relativity must be included to predictapsidal precession. The problem becomes more complicated when another body is added, creating athree-body problem that can not be solved exactly.Perturbation theory is used to find an approximate solution to this problem.
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".[1] The term "dynamics" came in a little later withGottfried Leibniz,[2] and over a century after Newton,Pierre-Simon Laplace introduced the termcelestial mechanics.[3] 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.
Following Newton, mathematicians attempted to solve the more complex problem of predicting the future motion of three bodies interacting through gravity: thethree-body problem. The first to provide a periodic solution was the Swiss mathematicianLeonhard Euler, who in 1762 demonstrated three equilibrium points lie along a straight line passing through the two primary masses. If a body of infinitesimal mass occupied one of these points, it would remain there in a stable orbit. French mathematicianJoseph-Louis Lagrange attempted to solve this restricted three-body problem in 1772, and discovered two more stable orbits at the vertices ofequilateral triangles with the two primary masses. Collectively, these solutions became known as theLagrange points.[6]
Lagrange reformulated the principles ofclassical mechanics, emphasizing energy more than force,[7] 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).[8][9] More recently, it has also become useful to calculatespacecrafttrajectories.[10]
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'sPrincipia.[11][12]
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).[13]
Simon Newcomb was a Canadian-American astronomer who revisedPeter Andreas Hansen's table of lunar positions.[14] In 1877, assisted byGeorge William Hill, he recalculated all the major astronomical constants. After 1884 he conceived, withA. 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.[15]
Apsidal precession of Mercury's orbit around the Sun (not to scale)
In 1849,Urbain Le Verrier reported that Mercury's closest approach the Sun, itsperihelion, was observed to advance at the rate of43″ per century. Thisprecession of Mercury's perihelion could not be accounted for by known gravitational perturbations using Newton's law. Instead, Le Verrier later attributed the effect to a proposed planet orbiting inside the orbit of Mercury. DubbedVulcan, subsequent searches failed to locate any such body. The cause remained a mystery untilAlbert Einstein explained theapsidal precession in his 1916 paperThe Foundation of the General Theory of Relativity.General relativity led astronomers to recognize thatNewtonian mechanics did not provide the highest accuracy in proximity to massive bodies.[16] This led to attempts to solve thetwo-body problem in general relativity and the discovery ofgravitational radiation.[17][18]
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,[19] where a numbern of masses are mutually interacting via the gravitational force. Although analytically notintegrable in the general case,[20] 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.[21]
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.[22]
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 the"guess, check, and adjust" method used innumerical analysis, whichis 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.[23] Additional sources of orbital perturbation includeatmospheric drag,solar radiation pressure, and non-uniform gravitational fields.[24]
Perturbation methods start with a simplified form of the original problem, which is chosen to be exactly solvable. In celestial mechanics, this is usually aKeplerian ellipse, which is correct when there are only two gravitating bodies, 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. The changes that result from the terms in the equations are used as corrections to the original solution.[25] Because simplifications are made at every step, the corrections are never perfect, but even one cycle of corrections often provides a better approximation.
A partially corrected solution can be re-used as the new starting point for another cycle of perturbations and corrections. In principle, the recycling of prior solutions to obtain a better solution could continue indefinitely. The difficulty is that the corrections usually progressively make the new solutions more complicated.Newton is reported to have said, regarding the problem of theMoon's orbit"It causeth my head to ache.".[26]
A reference frame is an arbitrary definedcoordinate system, whose origin, orientation, and scale are specified inphysical space. The frame is aligned via a set of reference points, such as distant galaxies.[27] 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.[28] The choice of reference frame gives rise to phenomena such as theretrograde motion ofsuperior planets in a geocentric reference frame.[29]
AnInertial frame of reference is employed for bodies with mass. Thus a Lunar Reference System defines an Earth Inertial frame with Earth as the origin, the Lunar Inertial frame having an origin of the Moon, and an Earth-Moon Barycentric Rotating frame anchored to the rotatingEarth-Moon barycenter.[30]Positioning systems such asGPS orGLONASS use a reference frame based on the Earth. However, these are unsuitable for navigation in space.[31] For interplanetary trajectories, a heliocentric (Sun-centered) coordinate system is used, with the XY plane aligned with theecliptic as defined for a particularepoch.[32]
Thelocal standard of rest (LSR) is a reference frame based on the mean motion of stellar objects in the neighborhood of the Sun. Thepeculiar velocity of the Sun relative to this framework is 13.4 km/s in the direction of thesolar apex.[33] There are two possible definitions for the LSR: the first is based on the kinetic motion of nearby stars, and the second is a dynamical standard that follows the Sun in its orbit around the galaxy. These two drift apart with the passage of time as the stars follow the gravitational potential around the galaxy. Perturbations in a star's galactic orbit result inepicycle motions.[34]
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.
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).
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.
^For example:Gangestad, Joseph W.; et al. (October 2010). "Lagrange's planetary equations for the motion of electrostatically charged spacecraft".Celestial Mechanics and Dynamical Astronomy.108 (2):125–145.Bibcode:2010CeMDA.108..125G.doi:10.1007/s10569-010-9297-z.
^Guerra, André G. C.; Carvalho, Paulo Simeão (August 1, 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).
