Derivation of Kepler’s laws from Newton’s law

Kepler derived his three laws in an exceptionally empiric way, on the basis of analysis of observations of the positions of planets. In Kepler’s epoch, the law of gravitation had not yet been discovered. However, Kepler did realize that such a law must exist, and that his three laws should be consequences of a more general law determining the essence of the interaction between celestial bodies. The latter was discovered later by Newton on the basis of careful examination of Kepler’s laws. However, even when Newton’s law of gravitation is known, one needs to make definite efforts to find its consequences for an isolated system of two bodies. Preliminary knowledge of the type of orbit is extremely important for preliminary calculations of a position of a celestial body. The eccentricities of elliptic orbits for almost all planets of the Solar system are very small, and, hence, to a first approximation these orbits may be considered to be circular.