In other words, we can get from the earth to the planet by one of two different routes. The first route corresponds to the heliocentric model, and the second to the geocentric model. In the latter model, gives the displacement of the so-called

Figure 30 illustrates in more detail how the deferent-epicycle model
is used to determine the ecliptic longitude of a superior planet.
The planet orbits (counterclockwise) on a small Keplerian orbit
about guide-point , which, in turn, orbits the earth, , (counterclockwise) on a large
Keplerian orbit . As has already been mentioned, the small orbit is termed the epicycle, and the large orbit the deferent. Both orbits are assumed to lie in the plane of the ecliptic. This approximation does not
introduce a large error into our calculations because the orbital inclinations of the visible planets to
the ecliptic plane are all fairly small.
Let , , , , , ,
and denote the geometric center, apocenter (*i.e.*, the point of
furthest distance from the central object), pericenter (*i.e.*, the
point of closest approach to the central object), major radius, eccentricity, longitude
of the pericenter, and true anomaly of the deferent, respectively. Let , , , , , , and denote the corresponding
quantities for the epicycle.

Let the line be produced, and let the perpendicular be
dropped to it from , as shown in Fig. 31. The angle
is termed the *epicyclic anomaly* (see Fig. 32), and takes the form

(156) |

(157) |

and

(159) | |||

(160) |

are termed

(161) |

Now,

(162) |

(163) | |||

(164) |

Let us define

(165) |

(166) |

(167) | |||

(168) | |||

(169) |

and

This scheme allows us to avoid having to tabulate a two-dimensional function, whilst ensuring that the exact value of is obtained when , , or . The above interpolation scheme is very similar to that adopted by Ptolemy in the Almagest.

Our procedure for determining the ecliptic longitude of a superior planet is described below. It is assumed that the ecliptic longitude, , and the
radial anomaly, , of the sun have already been calculated. The latter quantity is tabulated as a function of the solar mean anomaly
in Table 33. In the following, , , , ,
, and represent elements of the orbit of the planet in question
about the sun, and represents the eccentricity of the sun's apparent orbit
about the earth. (In general, the subscript denotes the sun.) In particular, is the major radius of the planetary orbit in
units in which the major radius of the sun's apparent orbit about the
earth is *unity*. The requisite elements for all of the superior planets at the J2000 epoch (
JD)
are listed in Table 30. The ecliptic longitude of a superior
planet is specified by the following formulae:

Here, , , , and . The constants , , , and for each of the superior planets are listed in Table 44. Finally, the functions are tabulated in Table 45.

For the case of Mars, the above formulae are capable of matching NASA ephemeris data during the years 1995-2006 CE with a mean error of and a maximum error of . For the case of Jupiter, the mean error is and the maximum error . Finally, for the case of Saturn, the mean error is and the maximum error .