- The orbit of each planet is an ellipse with the sun at one of its foci.
- A circle is a special case of an ellipse with both foci at the center.

- The radius vector of each planet describes equal areas in equal time.
- For an ellipse the radius vector would be the line from the Sun (at one focus) to the planet.

- The squares of the periods of the planets are proportional to the cubes of their mean distances from the Sun.
- This is sometimes called the
*harmonic law*.

- This is sometimes called the

Then, from the third law, Newton was able to prove that the forces acting on the planets are inversely proportional to the squares of their distances and directly proportional to their masses. This fact is easy to demonstrate for circular orbits. We begin with the equation for the force on a body in circular motion:

*f* = mV^{2} / r

and substituting V = 2 r / t (circumference /time)

*f* = 4 ^{2} m r / t^{2}

Then, using the subscripts 1 and 2 for planets 1 and 2,

*f*_{1} / *f*_{2} =
m_{1} r_{1} t_{2}^{2} / m_{2} r_{2} t_{1}^{2}

But from Kepler's third law,

t_{2}^{2} / t_{1}^{2} =
r_{2}^{3} / r_{1}^{3}

Substituting this relationship into the previous expression:

*f*_{1} / *f*_{2} =
m_{1} r_{2}^{2} / m_{2} r_{1}^{2}

describing the propertirs of the gravitational force.

For the case of elliptical orbits, a, the semi-major axis of the orbit, is substituted for r, but the derivation of this relationship is much less simple.

In the figure above we see the inner 4 planets in our solar system and our moon. The physical parameters of these planets are given in the table below.

Planet | Mass | Orbit Radius | Period | Eccentricity |
---|---|---|---|---|

(kg) | (km) | (days) | ||

Mercury | 3.31 × 10^{23} | 5.79 × 10^{7} | 87.97 | 0.206 |

Venus | 4.87 × 10^{24} | 1.08 × 10^{8} | 224.70 | 0.007 |

Earth | 5.98 × 10^{24} | 1.50 × 10^{8} | 365.26 | 0.017 |

Mars | 6.43 × 10^{23} | 2.28 × 10^{8} | 686.98 | 0.093 |

kstrom@hanksville.phast.umass.edu