Celestial mechanics
Celestial mechanics is the branch of astronomy that deals with the motions and gravitational interactions of objects in outer space. Historically, celestial mechanics applies principles of physics to astronomical objects, such as stars and planets, to produce ephemeris data. The computation of the motion of the bodies through orbital mechanics can be simplified by using an appropriate inertial frame of reference. This leads to the use of various different coordinate systems, such as the Heliocentric coordinate system.
In a binary system of objects interacting through gravity, Newtonian mechanics can used to produce a set of orbital elements that will predict with reasonable accuracy the future position of the two bodies. This method demonstrates the correctness of Kepler's laws of planetary motion. Where one of the bodies is sufficiently massive, general relativity must be included to predict apsidal precession. The problem becomes more complicated when another body is added, creating a three-body problem that can not be solved exactly. Perturbation theory is used to find an approximate solution to this problem.
History
Modern analytic celestial mechanics started with Isaac Newton's Principia. The name celestial mechanics is more recent than that. Newton wrote that the field should be called "rational mechanics". The term "dynamics" came in a little later with Gottfried Leibniz, and over a century after Newton, Pierre-Simon Laplace introduced the term celestial mechanics. Prior to Kepler, there was little connection between exact, quantitative prediction of planetary positions, using geometrical or numerical techniques, and contemporary discussions of the physical causes of the planets' motion.Laws of planetary motion
was the first to closely integrate the predictive geometrical astronomy, which had been dominant from Ptolemy in the 2nd century to Copernicus, with physical concepts to produce a New Astronomy, Based upon Causes, or Celestial Physics in 1609. His work led to the laws of planetary orbits, which he developed using his physical principles and the planetary observations made by Tycho Brahe. Kepler's elliptical model greatly improved the accuracy of predictions of planetary motion, years before Newton developed his law of gravitation in 1686.Newtonian mechanics and universal gravitation
is credited with introducing the idea that the motion of objects in the heavens, such as planets, the Sun, and the Moon, and the motion of objects on the ground, like cannon balls and falling apples, could be described by the same set of physical laws. In this sense he unified celestial and terrestrial dynamics. Using his law of gravity, Newton confirmed Kepler's laws for elliptical orbits by deriving them from the gravitational two-body problem, which Newton included in his epochal Philosophiæ Naturalis Principia Mathematica in 1687.Three-body problem
Following Newton, mathematicians attempted to solve the more complex problem of predicting the future motion of three bodies interacting through gravity: the three-body problem. The first to provide a periodic solution was the Swiss mathematician Leonhard 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 mathematician Joseph-Louis Lagrange attempted to solve this restricted three-body problem in 1772, and discovered two more stable orbits at the vertices of equilateral triangles with the two primary masses. Collectively, these solutions became known as the Lagrange points.Lagrange reformulated the principles of classical mechanics, emphasizing energy more than force, and developing a method 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 and comets and such. More recently, it has also become useful to calculate spacecraft trajectories.
Henri Poincaré published two now classical monographs, "New Methods of Celestial Mechanics" and "Lectures on Celestial Mechanics". 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. 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 of algebraic and transcendental 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's Principia.
These monographs include an idea of Poincaré, which later became the basis for mathematical "chaos theory" and the general theory of dynamical systems. He introduced the important concept of bifurcation 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.
Standardisation of astronomical tables
was a Canadian-American astronomer who revised Peter Andreas Hansen's table of lunar positions. In 1877, assisted by George 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 in Paris, 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.Anomalous precession of Mercury
In 1849, Urbain Le Verrier reported that Mercury's closest approach the Sun, its perihelion, was observed to advance at the rate of per century. This precession 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. Dubbed Vulcan, subsequent searches failed to locate any such body. The cause remained a mystery until Albert Einstein explained the apsidal precession in his 1916 paper The Foundation of the General Theory of Relativity. General relativity led astronomers to recognize that Newtonian mechanics did not provide the highest accuracy in proximity to massive bodies. This led to attempts to solve the two-body problem in general relativity and the discovery of gravitational radiation.Examples of problems
Celestial motion, without additional forces such as drag forces or the thrust of a rocket, is governed by the reciprocal gravitational acceleration between masses. A generalization is the n-body problem, where a number n of masses are mutually interacting via the gravitational force. Although analytically not integrable in the general case, the integration can be well approximated numerically.In the case the configuration is much simpler than for. In this case, the system is fully integrable and exact solutions can be found.
A further simplification is based on the "standard assumptions in astrodynamics", which include that one body, the orbiting body, is much smaller than the other, the central 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. The earliest use of modern perturbation theory was to deal with the otherwise unsolvable mathematical problems of celestial mechanics: Newton's solution for the orbit of the Moon, which moves noticeably differently from a simple Keplerian ellipse because of the competing gravitation of the Earth and the Sun. Additional sources of orbital perturbation include atmospheric drag, solar radiation pressure, and non-uniform gravitational fields.Perturbation methods start with a simplified form of the original problem, which is chosen to be exactly solvable. In celestial mechanics, this is usually a Keplerian 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 its time-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. 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 the Moon's orbit "It causeth my head to ache.".