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# Lunar Equations of Motion

It is convenient to solve the lunar equation of motion, (1123), in a geocentric frame of reference, (say), which rotates with respect to at the fixed angular velocity . Thus, if the lunar orbit were a circle, centered on the Earth, and lying in the ecliptic plane, then the Moon would appear stationary in . In fact, the small eccentricity of the lunar orbit, , combined with its slight inclination to the ecliptic plane, , causes the Moon to execute a small periodic orbit about the stationary point.

Let , , and , , be the Cartesian coordinates of the Moon in and , respectively. It is easily demonstrated that (see Section A.16)

 (1128) (1129) (1130)

Moreover, if , , are the Cartesian components of the Sun in then (see Section A.5)
 (1131) (1132) (1133)

giving
 (1134) (1135) (1136)

where use has been made of Equations (1125)-(1127).

Now, in the rotating frame , the lunar equation of motion (1123) transforms to (see Chapter 7)

 (1137)

where . Furthermore, expanding the final term on the right-hand side of (1137) to lowest order in the small parameter , we obtain
 (1138)

When written in terms of Cartesian coordinates, the above equation yields
 (1139) (1140) (1141)

where , and use has been made of Equations (1134)-(1136).

It is convenient, at this stage, to normalize all lengths to , and all times to . Accordingly, let

 (1142) (1143) (1144)

and , and . In normalized form, Equations (1139)-(1141) become
 (1145) (1146) (1147)

respectively, where is a measure of the perturbing influence of the Sun on the lunar orbit. Here, and .

Finally, let us write

 (1148) (1149) (1150)

where , and , , . Thus, if the lunar orbit were a circle, centered on the Earth, and lying in the ecliptic plane, then, in the rotating frame , the Moon would appear stationary at the point , . Expanding Equations (1145)-(1147) to second-order in , , , and neglecting terms of order and , etc., we obtain
 (1151) (1152) (1153)

Now, once the above three equations have been solved for , , and , the Cartesian coordinates, , , , of the Moon in the non-rotating geocentric frame are obtained from Equations (1128)-(1130), (1142)-(1144), and (1148)-(1150). However, it is more convenient to write , , and , where is the radial distance between the Earth and Moon, and and are termed the Moon's ecliptic longitude and ecliptic latitude, respectively. Moreover, it is easily seen that, to second-order in , , , and neglecting terms of order ,

 (1154) (1155) (1156)

Next: Unperturbed Lunar Motion Up: Lunar Motion Previous: Preliminary Analysis
Richard Fitzpatrick 2011-03-31