# Celestial mechanics

Celestial mechanics is a division of astronomy dealing with the motions and gravitational effects of celestial objects. The field has application of physics, historically Newtonian mechanics, to astronomical objects such as stars and planets. It is distinguished from astrodynamics, which is the study of the creation of artifical satellite orbits.

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## History of celestial mechanics

The Ancient Greeks developed theories regarding celestial mechanics, most of which were geocentric in nature.

### Johannes Kepler

Johannes Kepler was the first to develop laws of orbits, which he did by examining and cataloging the motion of the planets. Johannes Kepler was the first to successfully model planetary orbits to a high degree of accuracy. Years before Isaac Newton had even developed his law of gravitation, Kepler had developed his three laws of planetary motion from empirical observation.

See Kepler's laws of planetary motion and the Keplerian problem for a detailed treatment of how his laws of planetary motion can be used.

### Isaac Newton

Isaac Newton 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 horses and falling apples, could be described by the same set of physical laws. In this sense he unified 'celestial' and 'terrestrial' dynamics.

Using Newton's law of gravitation, proving Kepler's Laws for the case of a circular orbit is simple. Elliptical orbits involve more complex calculations. Using Lagrangian mechanics it is possible to develop a single polar coordinate equation that can be used 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.

### Albert Einstein

After Einstein explained the anomalous precession of Mercury's perihelion, astronomers recognized that Newtonian mechanics did not provide the highest accuracy. Today, we have binary pulsars whose orbits not only require the use of General Relativity for their explanation, but whose evolution proves the existence of gravitational radiation, a discovery that lead to a Nobel prize.

### Open problems

There are a few open problems in celestial mechanics that await solutions. The solution of the n-body problem (which is the problem of finding, given the initial positions, masses, and velocities of n bodies, their subsequent motions as determined by classical mechanics) remains unsolved. The theory of quantum mechanics has not been merged with the theory of general relativity to produce a so-called "theory of everything". Even though Einstein's theory predicts gravitational waves, this radiation has not been directly observed.

## Examples of problems

Celestial motion without additional forces such as thrust of a rocket, is governed by gravitational acceleration of masses due to other masses. A simplification is the n-body problem, where we assume n spherically symmetric masses, and integration of the accelerations reduces to summation.

Examples:

In the case that n=2 (two-body problem), the situation is much simpler than for larger n. Various explicit formulas apply, where in the more general case typically only numerical solutions are possible. It is a useful simplification that is often approximately valid.

Examples:

A further simplification is based on "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.

Examples:

• Solar system orbiting the center of the Milky Way
• a planet orbiting the Sun
• a moon orbiting a planet
• a spacecraft orbiting Earth, a moon, or a planet (in the latter cases the approximation only applies after arrival at that orbit)

Either instead of, or on top of the previous simplification, we may assume circular orbits, making distance and orbital speeds, and potential and kinetic energies constant in time. Notable examples where the eccentricity is high and hence this does not apply are:

Of course, in each example, to obtain more accuracy a less simplified version of the problem can be considered.

### Perturbation theory

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem

## Related topics

• Astrometry is a part of astronomy and deals with the positions of stars and other celestial bodies, their distances and movements.
• Astrodynamics is the study and creation of orbits, especially those of artificial satellites.
• Orbit is the path that an object makes, around another object, whilst under the influence of a source of centripetal force, such as gravity.
• 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 any natural satellite of the other planets.
• Celestial navigation is a position fixing technique that was the first system devised to help sailors locate themselves on a featureless ocean.

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