OSP-based Programs for Advanced Physics:
Classical Orbit Demos

 

One of the standard topics in classical mechanics, at any level, is that of Kepler's laws of planetary motion. This topic rests on Newton's law of universal gravitation,

F12 = −G m1 m2 r12/r123,

for the gravitational force on mass m1 due to mass m2, given a separation r12 = |r1 r2| = (r12 + r22 − 2r1·r2)1/2, and where G = 6.67 × 10−11Nm2/kg2. The force is attractive and lies along the line separating the two masses.

Kepler's laws can be simply expressed as (1) planets move in ellipses with the Sun at one focus (2) planets sweep out equal areas in equal times as it orbits the Sun, and (3) the square of the period of a planet's orbit is proportional to the cube of the semimajor axis, a, of its orbit: T2 = (4π2/GM) a3.

Introductory treatments of Kepler's laws focus on the special cases of two-body systems where the central object is much more massive than the orbiting body, which itself usually orbits in circular motion.  More advanced treatments includes the general study of elliptical orbits and occasionally cases where the objects are nearly the same mass.  Less frequently, systems with more than two objects are considered, and for good reason.  These systems are, in general, not exactly solvable.  However, as we shall see, there are special cases in which the three-, four-, and five-body problems can be solved.

 

© Mario Belloni and Wolfgang Christian (2006).