Let us define two unit vectors, and
. (A unit vector is
simply a vector whose length is unity.) As shown in Fig. 105, the *radial*
unit vector always points from the Sun towards the instantaneous position
of the planet. Moreover, the *tangential* unit vector
is always
normal to , in the direction of increasing .
In Sect. 7.5, we demonstrated that when acceleration is written in terms of polar
coordinates, it takes the form

(562) |

(563) | |||

(564) |

These expressions are more complicated that the corresponding cartesian expressions because the unit vectors and

Now, the planet is subject to a single force: *i.e.*, the force of gravitational attraction exerted
by the Sun. In polar coordinates, this force takes a particularly simple
form (which is why we are using polar coordinates):

(565) |

According to Newton's second law, the planet's equation of motion is written

(566) |

Equation (568) reduces to

(569) |

(570) |

(571) |

(572) |

(573) |

The quantity has another physical interpretation. Consider Fig. 106. Suppose that
our planet moves from to in the short time interval . Here, represents
the position of the Sun. The lines and are both approximately of length .
Moreover, using simple trigonometry, the line is of length
, where
is the small angle through which the line joining the Sun and the planet
rotates in the time interval . The area of the triangle is approximately

(574) |

Clearly, the fact that is a constant of the motion implies that the line joining the planet and the Sun sweeps out area at a

Let

(576) |

The last step follows from the fact that . Differentiating a second time with respect to , we obtain

Equations (567) and (578) can be combined to give

(579) |

(580) |

The above formula can be inverted to give the following simple orbit equation for our planet:

(581) |

where

(583) |

Formula (582) is the standard equation of an *ellipse* (assuming ), with the
origin at a focus. Hence, we have now proved Kepler's first law of planetary motion.
It is clear that is the radial distance at . The radial distance at
is written

(586) |

According to Eq. (575), a line joining the Sun and an orbiting planet sweeps area
at the constant rate . Let be the planet's orbital period. We expect the line to
sweep out the *whole area* of the ellipse enclosed by the planet's orbit in the time
interval . Since the area of an ellipse is , where and are the
*semi-major* and *semi-minor* axes, we can write

Thus, is essentially the planet's mean distance from the Sun. Finally, the relationship between , , and the eccentricity, , is given by the well-known formula

This formula can easily be obtained from Eq. (582).

Equations (584), (585), and (588) can be combined to give

(591) |