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z-earth-moonsyst - RE LE rot Earth Moon System 2 = IEE = IE...

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I m 2 2 2 ( ) ( ) tot E M E M orb L R T T π π - = + 27 days R E R L tot =L E-rot +L M-orbit +L M-rot very small (neglect) Δ E E = Δ K E-rot = -W tidal-friction Earth –Moon System solid Earth – rotates under ocean tidal budge - frictional energy loss Earth’s rotation slows with time decreases increases 2 2 3 E (2 ) T R GM π = -2 E 2 3 GM T (2 ) R π = E 3/2 GM 1 1 = T (2 ) R π I 1/ 2 2 ( ) tot E M E E L m GM R T π = + 3 2 m G skg = moon moves away with time L transferred to moon constant I I 2 ( ) E rot E E E E L T π ω - = = I m 2 2 ( ) M orb M orb M orb M M orb L R T π ω - - - - = = 1 day I m 2 2 2 ( ) ( ) tot E M E M orb L R T T π π - = + 2 E E E 2 I = M R 5 constant 1 1 I I 2 2 2 ( ) 2 2 E rot E E E E K T π ω - = = I 2 2 2 (2 ) 1 1 [ ]( ) 2 ( ) ( ) tidal E rot E E E W K T later T earlier π - - = - ( ) ( ) E E T later T earlier >
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R E R=rR E 2 2 3 E (2 ) T R GM π = R=rR E M-orbit E-rot = T ( ) T now τ M-orbit E-rot27 = 27 days = T ( ) T now 2 3 2 3 E 2 E E (2 ) R GM T r π τ = [ ] 2 3 0.003584 r τ = [ ] 2 3 0.0033375 r τ = moon-orbit at 91 R E distant future day’=month’=51 days (current) distant past moon- much closer day shorter
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L tot =L E-rot +L M-orbit +L M-rot very small (neglect) R E R=rR E 2 E E E 2 I = M R 5 2 E M-orbit E E-rot M 2 L = R ( ) 80 ( ) r T π τ totM-orbit E-rot M-orbit L = L + L M-orbit E-rot = T ( ) T now τ 2 2 E E totM-orbit E M R 2 L = [ ] 80 2 { } 5 r t T π τ + 2 M-orbit M M-orbit 2 L = m R ( ) T π
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