# Celestial Mechanics

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

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

- Celestial Sphere: The Apparent Motions of the Sun, Moon, Planets, and Stars
- Celestial Coordinates - Horizon Coordinates, Celestial Latitude B, Galactic Longitude - Equatorial coordinates, Right ascension a, Declination d, Hour angle, Ecliptic coordinate, Celestial longitude l
- Celestial Mechanics - Planetary Perturbations
- Celestial Mechanics - Resonance Phenomena
- Celestial Mechanics - Tidal Effects
- Celestial Mechanics - Precession
- Celestial Mechanics - Non-gravitational Effects
- Celestial Mechanics - The Three-body Problem
- Celestial Mechanics - The N-body Problem
- Celestial Mechanics - Recent Developments
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