# two-body problem

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## two-body problem

A special case of the n-body problem in which a general solution can be found for the orbits of two bodies under the influence of their mutual gravitational attraction. The motion of a planet around the Sun is an example as long as the attractive forces of other planets are assumed to be negligible. A solution of the two-body problem is often acceptably realistic. Most theories of celestial motion thus use functions and principles, such as orbital elements and Kepler's laws, that have been derived by consideration of a two-body problem.

## Two-Body Problem

(in astronomy), the problem of the motion of two bodies that are mutually attracting in accordance with Newton’s law of gravitation. In the two-body problem attracting bodies are taken to be material points, which is valid if they are spherical in structure or if the distances between them are very great compared to their size. This requirement is largely met for the sun and each of the planets. In solving the two-body problem the motion of one body in relation to the other is usually considered. The motion in this problem occurs in conic sections—a circle, ellipse, parabola, hyperbola, or straight line—in accordance with Kepler’s laws. The two-body law, describing so-called unperturbed motion, is the first approximation in studying the true motion of celestial bodies.

N. P. ERPLYLEV

## two-body problem

[′tü ¦bäd·ē ′präb·ləm]
(mechanics)
The problem of predicting the motions of two objects obeying Newton's laws of motion and exerting forces on each other according to some specified law such as Newton's law of gravitation, given their masses and their positions and velocities at some initial time.
References in periodicals archive ?
In case of two interacting particles a general solution to the problem is known and it enables to specify main types of motion in the two-body problem (see [17]).
Caption: Figure 6: The global error at each integration point when solving the two-body problem (Problem 4) with [DELTA]t = 1/2.
A wide range of numerical integrators have been developed for performing such simulations; see, for example,[1, 2].The primary objective of this paper is to analyze and compare the error growth and efficiency of different ODE solvers applied to the Kepler's two-body problem. Throughout this paper, the error growth is examined in terms of the global error in the position and velocity, and the relative error in terms of total energy and angular momentum of the system.
To construct an analytical or semianalytical planetary theory, it is necessary to obtain a set of developments for the two-body problem, and from that, we can evaluate the inverse of the distance between the two planets and so to develop the second member of the differential equations of motion.
This controller is applied to some astrodynamics to achieve some interesting conclusions, including stable lissajous orbits in solar sail's three-body problem and degenerated two-body problem, quasiperiodic formation flying on a [J.sub.2]-perturbed mean circular orbit, controlled frozen orbits for a spacecraft with high area-to-mass ratio.
At points A and B, the particle or spacecraft is assumed to be far from the Earth and its motion can be modeled by a two-body problem with the Sun.
Shchepetilov, Reduction of the two-body problem with central interaction on simply connected spaces of constant sectional curvature, J.
Laplace supposed such a speed as a result of his soltion of the gravitational two-body problem, which concerns the motion of two point particles that interact only with each other, due to gravity.
Each of these interactions is considered a two-body problem because it asks how two objects--such as Earth and a spacecraft--behave if the only forces acting on them are each other's gravity.
The students cannot simply apply elementary calculus functions to the two-body problem to get an idea of how the two bodies move with time.
Physics has a long history of reducing many-body problems to one-or two-body problems in order to find more powerful solutions, and Alder and his colleagues have high hopes of doing it for this one.

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