Bessel Functions and Kepler's Equation

Abstract

2. KEPLER'S EQUATION. After years of work, Johannes Kepler announced three laws of planetary motion early in the seventeenth century. Kepler's three laws state that the planets move in elliptical orbits in a common plane with the sun at one focus, that for each planet the line connecting the sun with the planet sweeps out equal areas in equal times and that the ratio of the square of the period of revolution of each planet to the cube of the semimajor axis of its orbit is the same for all planets. Kepler stated the first two laws in 1609 in the Astronomia Nova and the third in 1619 in The Harmony of the World. As we know now, these laws are only approximations, but for the six planets known at the time and to the limits of observation then they were essentially exact. Kepler's Equation is a consequence of the first two laws only. Suppose a planet moves in the counterclockwise direction in an elliptical orbit with the sun at one focus which has eccentricity e, 0 < e < 1, has semimajor axis a, and is traveled once in time T. In the figure, A denotes perihelion, C center of the orbit, and S the position of the sun. If, having passed through A, the planet after elapsed time t is at position P, we wish to express the polar coordinates of P, (r, v), relative to S at time t. The quantity v = angle PSA is called the true anomaly of the planet at time t. The circle centered at C with radius a is called the eccentric circle. If we draw the line P perpendicular to radius CA and mark R, its intersection with CA, and Q, its intersection with the eccentric circle, the quantity E = angle QCA is called the eccentric anomaly of the planet at time t.