        # Elliptical Orbits Revisited

From Unit 2, here are Kepler's three laws of planetary motion.

1.The orbits are not circles: The orbits of the planets are ellipses with the Sun at one focus. The above diagram is representative of an elliptical orbit.
The planet would travel along the line of the ellipse and the
Sun is at one "focus" inside the ellipse. Notice that the other
focus in the ellipse would be empty space.

2. The speed is not constant: The line joining the planet to the Sun sweeps out equal areas in equal intervals of time. Thus, the planet moves faster when it is closer to the Sun.

3. We can relate the periods of the planets' orbits to the sizes of their orbits: The square of the sidereal period, in years, equals the cube of the semi-major axis, in AU.

In Unit 5 you've already seen how the first two laws are important in spaceflight. Kepler's third law will be an important part of our analysis of traveling from Earth to another planet.

As you recall, an ellipse is an elongated circle. Actually, a circle is a form of an ellipse with an eccentricity equal to zero (e = 0). a in the diagram above is called the semi-major axis and c is the distance from the focus to the circle of the ellipse. In the case of orbit transfers from one planet to another, the focus will be the Sun. 1) When the eccentricity is zero, an ellipse is commonly called a:

Square
Circle
Parabola

Keep in mind that e, the eccentricity is defined as c divided by a (e = c/a). We call it eccentricity because it describes how far from center the focus is (just as my eccentric aunt is far from being like the average person). The eccentricity determines how "flat" an eclipse is. The orbits of the planets are actually not very eccentric - e is small for nearly all the planets. To illustrate the principles of space flight we will simplify the problem by ignoring this small eccentricity. So in what follows we assume that the orbits of the planets are, for all practical purposes, perfect circles even though we know this is not exactly true. 