Kepler's Laws

The fact remained, however, that, in spite of Kepler's successful modeling of the solar system with his three laws of planetary motion, he had discovered them by trial and error without any basis in physical law. More than 60 years after Kepler published his third law, Isaac Newton published his Principia, in which he developed his three laws of motion and his theory and law of universal gravitation. By using these laws, Newton was able to derive each of Kepler's laws in a more general form than Kepler had stated them, and, moreover, they were now based on physical theory. Kepler's laws were derived by Newton from the basis of the two-body problem of celestial mechanics. They are:

1. The orbits of two bodies around their center of mass (barycenter) are conic sections (circles, ellipses, parabolas, or hyperbolae) with the center of mass at one focus of each conic section orbit. Parabolas and hyperbolas are open-ended orbits, and the period of revolution (P) is undefined for them. One may consider a circular orbit to be a special case of the ellipse where the two foci of the ellipse, F₁ and F₂, coincide with the ellipse's center (C), and the ellipse becomes a circle of radius (a).
2. The line joining the bodies sweeps out equal areas in their orbits in equal intervals of time. Newton showed that this generalized law is a consequence of the conservation of angular momentum (from Newton's third law of motion) of an isolated system of two bodies unperturbed by other forces.
3. From his law of universal gravitation, which states that two bodies of masses, M₁ and M₂, whose centers are separated by the distance "r" experience equal and opposite attractive gravitational forces (F_g) with the magnitude

Where G is the Newtonian gravitational factor, and from his Second Law of Motion, Newton derived the following generalized form of Kepler's third law for two bodies moving in elliptical orbits around their center of mass where π is the ratio of the circumference of a circle to its diameter, "a" is the semi-major axis of the relative orbit of the body of smaller mass, M₂, around the center of the more massive body of M₁.

Some of the applications of these generalized Kepler's laws are briefly discussed below.