# 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 M1 and M2 whose centers are separated by a distance r experience equal and opposite attractive gravitational forces Fg 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, m1 and m2 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).