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# Solution of lunar equations of motion

The lunar equations of motion, Equations (11.86)-(11.93), take the general form

 (11.106) (11.107) (11.108) and (11.109)

where

 (11.110) (11.111) and (11.112)

Here, use has been made of the definitions (11.99)-(11.102). Furthermore, it follows from Equations (11.94)-(11.96), (11.103)-(11.105), and (11.106) that

 (11.113) (11.114) and (11.115)

As before, we have neglected terms that are third order, or greater, in the small parameters , , , , , and .

Let us write

 (11.116) (11.117) (11.118) (11.119) (11.120) (11.121) (11.122) (11.123) and (11.124)

Here, the , , , , et cetera, are constants.

Equations (11.110)-(11.124) can be combined to give

 (11.125) (11.126) (11.127) (11.128) (11.129) (11.130) (11.131) (11.132) (11.133) (11.134) (11.135) (11.136) (11.137) (11.138) and (11.139)

as well as

 (11.140) (11.141) (11.142) (11.143) (11.144) (11.145) (11.146) (11.147) (11.148) (11.149) (11.150) and (11.151)

as well as

 (11.152) (11.153) (11.154) (11.155) and (11.156)

as well as

 (11.157) (11.158) (11.159) (11.160) (11.161) (11.162) (11.163) (11.164) (11.165) (11.166) (11.167) (11.168) (11.169) (11.170) and (11.171)

as well as

 (11.172) (11.173) (11.174) (11.175) (11.176) (11.177) (11.178) (11.179) (11.180) (11.181) (11.182) and (11.183)

as well as

 (11.184) (11.185) (11.186) (11.187) and (11.188)

Here, and in the following, we have retained and corrections to the parameters , , , , , , , , , , , and , and corrections to the parameters , , , , , , , , , , , , , , , and , while neglecting similar corrections for all of the other parameters appearing in Equations (11.116)-(11.124).

Substitution of Equations (11.116), (11.117), (11.119), and (11.120) into Equations (11.107) and (11.108) yields

 (11.189)

for , as well as

 (11.190) and (11.191)

as well as

 (11.192) and (11.193)

for . In the previous two equations, for , and otherwise. Moreover,

 (11.194) (11.195) (11.196) (11.197) (11.198) (11.199) (11.200) (11.201) (11.202) (11.203) (11.204) and (11.205)

In calculating the previous frequencies, we have assumed that the ecliptic longitude of the lunar perigee takes the form

 (11.206)

where is an arbitrary constant, and is an, as yet, unknown constant that parameterizes the precession of the lunar perigee. Here, use has been made of the facts that and . Likewise, we have assumed that the ecliptic longitude of the lunar ascending mode takes the form

 (11.207)

where is an arbitrary constant, and is an, as yet, unknown constant that parameterizes the regression of the lunar ascending node.

Substituting Equations (11.118) and (11.121) into Equation (11.109), we obtain

 (11.208)

and

 (11.209)

for . In the previous equation, for , and otherwise. Moreover,

 (11.210) (11.211) (11.212) (11.213) and (11.214)

In the following few sections, we shall develop our solution of the lunar equations of motion in a systematic fashion by considering groups of similar terms separately.

Next: Variation Up: Lunar motion Previous: Useful definitions
Richard Fitzpatrick 2016-03-31