# 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:

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.