Remember in our earlier lesson on universal gravitation, that the speed of a satellite in circular orbit is constant. In this lesson we will examine the properties of satellites in elliptical orbits satellites which have variable speeds since their distance from the earth is constantly changing. These three empirical laws were published by Johannes Kepler (15711630). His research was based on the extensive data compiled by Tycho Brahe (15461601) on Mars.

Kepler's 1st Law: The Law of Elliptical Orbits
Each planet travels in an elliptical orbit with the sun at one focus.
Here are some properties and vocabulary about ellipses that you should remember.

When the planet is located at point P it is at the perihelion position.

R_{P} = distance from Sun to P and is called the perihelion radius.

When the planet is located at point A it is at the aphelion position.

R_{A} = distance from Sun to A and is called the aphelion radius.

The distance PA = R_{P} + R_{A} is called the major axis which is represented mathematically in formulas as 2a.

The perpendicular bisector of the major axis is called the minor axis and its length is represented mathematically in formulas as 2b.

The center of the ellipse is where the major and minor axes cross each other.
 The average orbital radius is called R_{AV} = a = ½ (R_{A} + R_{P})

Kepler’s 2nd Law: The Law of Equal Areas
A line from the planet to the sun sweeps out equal areas of space in equal intervals of time.
 At the perihelion, the position closest to the sun along the planet’s orbital path, the planet’s speed is maximal.
 At the aphelion, the position farthest from the sun along the planet’s orbital path, the planet’s speed is minimal.
Thus, the satellite’s speed is inversely proportional to its average distance from the sun. v_{A}R_{A} = v_{P}R_{P}
_{
}

Kepler’s 3rd Law: The Law of Periods
The square of a planet’s orbital period is directly proportional to the cube if its average distance from the sun. T^{2} ~ R_{av}^{3} T^{2} / R_{av}^{3} = constant Since the orbits of the planets in our solar system are EXTREMELY close to being circular in shape (the Earth's eccentricity is 0.0167), we can set the centripetal force equal to the force of universal gravitation and, F_{C} = F_{G} m_{planet}v^{2} / R = GM_{Sun}m_{planet} / R^{2} (2 π R / T)^{2} = GM_{Sun}/ R 4 π^{2 }R^{2} / T^{2} = GM_{Sun} / R 4 π^{2 }R^{3} = GM_{Sun} T^{2} R^{3} / T^{2} = GM_{Sun} / 4 π^{2} Thus Kepler’s constant for our solar system equals GM_{Sun} / 4 π^{2} In general, for any system of satellites, the ratio of T^{2} / R_{av}^{3} equals a constant for that system. Thus, for any group of satellites, the ratio of T^{2} / R_{av}^{3} will be the same. This is one way to determine if two satellites are in orbit about the same central body. 