# Celestial Mechanics - Planetary Perturbations, Resonance Phenomena, Tidal Effects, Precession, Non-gravitational Effects, The Three-body Problem

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bodies center law orbits

Modern celestial mechanics began with Isaac New ton's generalization of **Kepler's laws** published in his *Principia* in 1687. Newton used his three **laws of motion** and his law of universal gravitation to do this. The three generalized Kepler's law are:

1) The orbits of two bodies around their center of **mass** (barycenter) are **conic sections** (ellipses, circles, parabolas, or hyperbolas) with the center of mass at a focus of each conic sections; 2) The line joining the center of the two bodies sweeps out equal areas in their orbits in equal **time** intervals. Newton showed that this is a consequence of conservation of angular **momentum** of an isolated two-body system unperturbed by other forces (Newton's third law of motion); 3) From his law of universal gravitation, which states that Bodies l and 2 of masses M_{1} and M_{2} whose centers are separated by a **distance** r experience equal and opposite attractive gravitational forces F_{g} of magnitudes

where G is the Newtonian gravitations factor, and from his second law of motion, Newton derived the following general form of Kepler's third law for these bodies moving around the center of mass along elliptical or circular orbits:

where P is the sidereal period of revolution of the bodies around the center of mass, π is the **ratio** of the circumference of a **circle** to its diameter, X, m_{1} and m_{2 }are the same as in Equation 1 and a is the semi-major axis of the *relative* orbit of the center of the less massive Body 2 around the center of the more massive Body l.

These three generalized Kepler's Law form the basis of the two-body problem of celestial mechanics. **Astrometry** is the branch of celestial mechanics which is concerned with making precise measurements of the positions of celestial bodies, then calculating precise orbits for them based on the observations. In theory, only three observations are needed to define the **orbit** of one celestial body relative to a second one. Actually, many observations are needed to obtain an accurate orbit.

However, for the most precise orbits and predictions, the vast majority of systems investigated are not strictly two-body systems but consist of many bodies (the **solar system**, planetary **satellite** systems, multiple **star** systems, star clusters, and galaxies).

## Additional Topics

To a first approximation, the solar system consists of the Sun and eight major planets, a system much more complicated than a two-body problem. However, use of Equation 2 with reasonable values for the astronomical unit (a convenient unit of length for the solar system) and for G showed that the Sun is far more massive than even the most massive planet Jupiter (whose mass is 0.000955 the Sun…

Ceres, the first asteroid or minor planets, was discovered to orbit the Sun between the orbits of liars and Jupiter in 1801. Thousands of other asteroids have been discovered in that part of interplanetary space, which is now called the Main asteroid Belt. Daniel Kirkwood (1815-1895) noticed in 1866 that the periods of revolution of the asteroids around the Sun did not form a continuous distributi…

Tides raised in the Moon's solid body by Earth have slowed its rotation until it has become tidally locked to Earth (the Moon keeps the same hemisphere turned towards Earth, and its periods of rotation and revolution around Earth are the same, 27.32 mean solar days). Eventually Earth's rotation will be slowed to where Earth will be tidally locked to the Moon, and the durations of the…

Rapidly rotating planets and satellites have appreciable equatorial bulges as a consequence of Newton's First Law of Motion. If the rotation axis of such a body is not perpendicular to its orbit, other bodies in the system will exert stronger gravitational attractions on the near part of the bulge than its far part. The effect of this difference is to tend to turn the body's rotation…

No closed general solution has been found for the problem of systems of three or more bodies whose motions are controlled by their mutual gravitational attractions in a form analogous to the generalized Kepler's Laws for the two-body problem. …

In the last 30 years high performance computers have been used to study the n-body problem (n = 3 to n = 10 or more) by stepwise integration of the orbits of the gravitationally interacting bodies. Earlier computers were incapable of performing such calculations over sufficiently long time intervals. The study of the stability of Pluto's orbits over the last 10,000,000 years mentioned above…

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