I. Perair 16.07 Dynamics Fall 2004 Lecture D28-Central Force Motion: Keplers Laws When the only force acting on a particle is always directed to- wards a fixed point, the motion is called central force motion. This type of motion is particularly relevant when studying the orbital movement of planets and satellites. The laws which gov- ern this motion were first postulated by Kepler and deduced from observation. In this lecture, we will see that these laws are a con- equence of Newtons second law. An understanding of central force motion is necessary for the design of satellites and space vehicles Kepler’ s Problen We consider the motion of a particle of mass m, in an inertial reference frame, under the influence of a force F, directed towards the origin. m k O We will be particularly interested in the case when the force is inversely proportional to the square of the distance between the particle and the origin, such as the gravitational force. In this case, F=-Izmer where u is the gravitational r. r is the lus of the position vector, r, and er=r/ can be shown that, in general, Kepler's problem is equivalent to the two-body problem, in which two masses, M and m, move solely due to the influence of their mutual gravitational attraction. This equivalence is obvious when M>>m, since, in this case, the center of mass of the system can be taken to be at A However, even in the more general case when the two masses are of similar size, the problem can be reduced to a Kepler problem( see note below)J. Peraire 16.07 Dynamics Fall 2004 Version 1.1 Lecture D28 - Central Force Motion: Kepler’s Laws When the only force acting on a particle is always directed to￾wards a fixed point, the motion is called central force motion. This type of motion is particularly relevant when studying the orbital movement of planets and satellites. The laws which gov￾ern this motion were first postulated by Kepler and deduced from observation. In this lecture, we will see that these laws are a con￾sequence of Newton’s second law. An understanding of central force motion is necessary for the design of satellites and space vehicles. Kepler’s Problem We consider the motion of a particle of mass m, in an inertial reference frame, under the influence of a force, F, directed towards the origin. We will be particularly interested in the case when the force is inversely proportional to the square of the distance between the particle and the origin, such as the gravitational force. In this case, F = − µ r 2 mer, where µ is the gravitational parameter, r is the modulus of the position vector, r, and er = r/r. It can be shown that, in general, Kepler’s problem is equivalent to the two-body problem, in which two masses, M and m, move solely due to the influence of their mutual gravitational attraction. This equivalence is obvious when M >> m, since, in this case, the center of mass of the system can be taken to be at M. However, even in the more general case when the two masses are of similar size, the problem can be reduced to a Kepler problem (see note below). 1