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# Perturbed Lunar Motion

The perturbed nonlinear lunar equations of motion, (1151)-(1153), take the general form
 (1169) (1170) (1171)

where
 (1172) (1173) (1174)

Let us search for solutions of the general form
 (1175) (1176) (1177)

Substituting expressions (1172)-(1177) into Equations (1169)-(1171), it is easily demonstrated that
 (1178) (1179) (1180) (1181)

where .

Table 3: Angular frequencies and phase-shifts associated with the principal periodic driving terms appearing in the perturbed nonlinear lunar equations of motion.

The angular frequencies, , , and phase shifts, , , of the principal periodic driving terms that appear on the right-hand sides of the perturbed nonlinear lunar equations of motion, (1169)-(1171), are specified in Table 3. Here, and are, as yet, unspecified constants associated with the precession of the lunar perigee, and the regression of the ascending node, respectively. Note that and are the frequencies of the Moon's unforced motion in ecliptic longitude and latitude, respectively. Moreover, is the forcing frequency associated with the perturbing influence of the Sun. All other frequencies appearing in Table 3 are combinations of these three fundamental frequencies. In fact, , , , , , and . Note that there is no .

Now, a comparison of Equations (1151)-(1153), (1169)-(1171), and Table 1119 reveals that

 (1182) (1183) (1184)

Substitution of the solutions (1175)-(1177) into the above equations, followed by a comparison with expressions (1172)-(1174), yields the amplitudes , , and specified in Table 4. Note that, in calculating these amplitudes, we have neglected all contributions to the periodic driving terms, appearing in Equations (1169)-(1171) which involve cubic, or higher order, combinations of , , , , , and , since we only expanded Equations (1151)-(1153) to second-order in , , and .

Table 4: Amplitudes of the periodic driving terms appearing in the perturbed nonlinear lunar equations of motion.
 0

For , it follows from Equation (1178) and Table 4 that

 (1185)

For , making the approximation (see Table 3), it follows from Equations (1179), (1180) and Table 4 that

 (1186) (1187)

Likewise, making the approximation (see Table 3), it follows from Equation (1181) and Table 4 that
 (1188)

For , making the approximation (see Table 3), it follows from Equations (1179), (1180) and Table 4 that

 (1189) (1190)

Likewise, making the approximation (see Table 3), it follows from Equation (1181) and Table 4 that
 (1191)

For , making the approximation (see Table 3), it follows from Equations (1179), (1180) and Table 4 that

 (1192) (1193)

Thus, according to Table 4,
 (1194) (1195) (1196)

For , making the approximation (see Table 3), it follows from Equations (1179), (1180), (1194), and (1195) that

 (1197) (1198)

Likewise, making the approximation (see Table 3), it follows from Equations (1181) and (1196) that
 (1199)

Thus, according to Table 4,
 (1200) (1201) (1202)

Finally, for , by analogy with Equations (1160)-(1162), we expect

 (1203) (1204) (1205)

Thus, since (see Table 3), it follows from Equations (1179), (1200), (1201), and (1203) that
 (1206)

which yields
 (1207)

Likewise, since (see Table 3), it follows from Equations (1181), (1202), and (1205) that
 (1208)

which yields
 (1209)

According to the above analysis, our final expressions for , , and are

 (1210) (1211) (1212)

Thus, making use of Equations (1154)-(1156), we find that
 (1213) (1214) (1215)

The above expressions are accurate up to second-order in the small parameters , , and .

Next: Description of Lunar Motion Up: Lunar Motion Previous: Unperturbed Lunar Motion
Richard Fitzpatrick 2011-03-31