Consider Figure 17. is half of an elliptical planetary orbit. Furthermore, is the
geometric center of the orbit, the focus at which the sun is located,
the instantaneous position of the planet, the perihelion point (*i.e.*, the planet's point of closest approach to the sun),
and the aphelion point (*i.e.*, the point of furthest distance from the sun). The ellipse is symmetric about
, which is termed the *major axis*, and about ,
which is termed the *minor axis*.
The length is called the orbital
*major radius*. The length represents the displacement of the sun from
the geometric center of the orbit, and is generally written , where is termed the
orbital *eccentricity*, where . The length
is called
the orbital *minor radius*.
The length represents the radial distance of the planet from the sun.
Finally, the angle is the angular bearing of the planet from the sun,
relative to the major axis of the orbit, and is termed the *true anomaly*.

is half of a circle whose geometric center is , and whose
radius is . Hence, the circle passes through the perihelion and aphelion
points. is the point at which the perpendicular from
meets the major axis . The point where produced
meets circle is denoted . Finally,
the angle is called the *elliptic anomaly*.

Now, the equation of the ellipse is

(60) |

(61) |

(62) |

Now, . Furthermore, it is easily demonstrated that
, , , and . Consequently, Eq. (63) yields

Taking the square root of the sum of the squares of the previous two equations, we obtain

which can be combined with Eq. (65) to give

(67) |

Now, according to Kepler's second law,

(68) |

(69) |

But,

(70) |

(71) |

Hence, we can write

(72) |

(73) |

(74) |

In summary, the radial and angular polar coordinates, and , respectively,
of a planet in a Keplerian orbit about the sun are specified as *implicit* functions
of the mean anomaly, which is a *linear* function of time, by the
following three equations:

(75) | |||

(76) | |||

(77) |

It turns out that the earth and the five visible planets all possess

(78) | |||

(79) | |||

(80) |

Finally, these equations can be combined to give and as