Orbital Motion and Kepler's Laws

- 1.
- All planets move in elliptic orbits around the Sun, with the Sun
at one of the focal points of the ellipse. An ellipse is a stretched
circle, as shown below.

It is convenient to define the semi-major axis*a*of the ellipse, which is half the distance between the point of closest approach (the perihelion) and the point of farthest approach (aphelion) of the orbit. The semi-major axis can be thought of as an ``average radius'' for the orbit. The other defining characteristic of an elliptic orbit is its ``eccentricity'' which is simply the ratio of the width of the ellipse to its length. The closer the eccentricity is to one, the more circular the orbit. The shape of an orbit is completely specified if one knows the values of the semi-major axis and the eccentricity. - 2.
- The radius vector drawn from the Sun to a planet sweeps out equal areas in equal time intervals. This implies that the planet must move slower at the furthest points of the orbit than at the closest points. For example if the planet in the Figure below moves from position 1 to 2 in the same time it takes to go from 2 to 3 (the aphelion), the areas swept out by the radius vectors (dotted lines) will only be equal if the distance from 1 to 2 is greater than from 2 to 3. Thus, the planet moves the slowest at aphelion and fastest at the perihelion. This is also consistent with energy conservation, since the planet has greatest kinetic energy when its graviational potential energy is lowest (at perihelion).
- 3.
- The square of the orbital period,
*T*, of any planet is proportional to the cube of the semimajor axis*a*of the orbit:

1999-09-29