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# Determination of Ecliptic Longitude

Figure 29 compares and contrasts heliocentric and geocentric models of the motion of a superior planet (i.e., a planet which is further from the sun than the earth), , as seen from the earth, . The sun is at . In the heliocentric model, we can write the earth-planet displacement vector, , as the sum of the earth-sun displacement vector, , and the sun-planet displacement vector, . The geocentric model, which is entirely equivalent to the heliocentric model as far as the relative motion of the planet with respect to the earth is concerned, and is much more convenient, relies on the simple vector identity
 (155)

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 guide-point, , from the earth. Since is also the displacement of the planet, , from the sun, , it is clear that executes a Keplerian orbit about the earth whose elements are the same as those of the orbit of the planet about the sun. The ellipse traced out by is termed the deferent. The vector gives the displacement of the planet from the guide-point. However, is also the displacement of the sun from the earth. Hence, it is clear that the planet, , executes a Keplerian orbit about the guide-point, , whose elements are the same as the sun's apparent orbit about the earth. The ellipse traced out by about is termed the epicycle.

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)

where and are the mean longitude and equation of center for the deferent, whereas and are the corresponding quantities for the epicycle--see Cha. 5. The epicyclic anomaly is generally written in the range to . The angle is termed the equation of the epicycle, and is usually written in the range to . It is clear from the figure that
 (157)

where and are the radial polar coordinates for the deferent and epicycle, respectively. Moreover, according to Equation (81), , where
 (158)

and
 (159) (160)

are termed radial anomalies. Finally, the ecliptic longitude of the planet is given by (see Fig. 32)
 (161)

Now,

 (162)

is a function of two variables, and . It is impractical to tabulate such a function directly. Fortunately, whilst has a strong dependence on , it only has a fairly weak dependence on . In fact, it is easily seen that varies between and , where
 (163) (164)

Let us define
 (165)

This variable takes the value when , the value when , and the value when . Thus, using quadratic interpolation, we can write
 (166)

where
 (167) (168) (169)

and
 (170) (171)

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:

 (172) (173) (174) (175) (176) (177) (178) (179) (180) (181) (182) (183)

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 .

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Richard Fitzpatrick 2010-07-21