# Spreadsheet for lunar recession calculation

Whorton, Mark (mark.whorton@msfc.nasa.gov)
Wed, 11 Mar 1998 07:26:53 -0600

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With respect to the recent mention of the Walter Brown Jr. Video, a
friend (Hill Roberts) sent the attached Xcel 4.0 spreadsheet and a
discussion of the lunar recession rate that might be of interest to the
group. I am forwarding his message and spreadsheet:

Here's some additional thoughts relative to earth-moon recession, since
they
didn't really respond to that. Current recession rate is about 1 cm/yr
or
less. Current distance is ~3.8E05km. In 4E9 yrs a linear regression
places the
moon at ~3.40E05km. This is no major problem of "Brownian" proportion.
Of
course the phenomenon is certainly not linear since its fundamentally a
deceleration effect.

Since
L = I w (where w = omega, ang vel)

let w > w' (where w' is slowed ang vel)

conservation of ang mom gives

L = I w = I' w' where I = M R*R, I' = M R' * R'

so R' * R' ~ 1 / w' or R' ~ sqrt (1 / w' )

Assuming that since the earth moon system is slowing due to tidal
friction
which should be a roughly constant retarding force, then the slowdown
rate
should be roughly constant.

Since the day loses about 0.0016 sec/century this corresponds to a
slowdown
rate of about 5.5% per billion years -- not slowing very fast.

Using these parameters shows that a linear regression overestimates the
distance closed over time. The nonlinear regression based on
conservation of
ang mom gives an earth-moon distance 4.5 billion years ago as ~3.44E05
km,
compared to 3.8E05km today.

In either case the change is less than 10% from today! No big deal.

Hill

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