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Re: Moon's synchronism



On Tue, 17 Mar 1998, A. John Mallinckrodt wrote:

Of the moon's rotational velocity perturbations due to the periodic tidal
torque. I just did a "back of the envelope" (actually more like "two
sheet of paper") derivation of the period for small oscillations of a
nearly spherical but slightly prolate ellipsoid in a gravitational field
gradient and got the sensible result that

T = 2 pi/sqrt(f g')

where f is the fractional difference between the maximum and minimum
rotational inertias about axes through the CM and g' is the local field
gradient. For the moon g' is about 1.4 x 10^-11 s^-2. I don't know the
value of f, but if it were as large as 1% (which I sincerely doubt), the
period would still be over 6 months.

Last night I tried my hand at doing this calculation and wish to report
that I didn't quite get exactly the same result that you (John) do.
According to my calculations it looks like the natural resonant period
for the moon's rocking motion is about 14 years. Since it is being
driven with a period of 27.3 days this means that the driving frequency
seems to be around 190 times higher than the natural resonant frequency.

The formula I get for the resonant period T is similar to the one John
gave above but I get an extra factor of (3/2) in the argument of the
sqrt: T = 2*[pi]/sqrt((3/2)*f*g') where f is the relative difference in
the moon's moments of inertia about its principal axes, i.e.
f = (I_yy - I_xx)/I_zz, and g' = 2*(M_e)*G/(R_em)^3 is the magnitude of
the earth's gravitational field gradient at the location of the moon.
Above (in the formula for f) I_ii is the moment of inertia of the moon
about its i-th principle body axis through its center of mass, and the
definitions of the x, y & z axes for the moments of inertia are as
follows: the x-axis goes through the moon's prolate 'long' dimension;
the z-axis is the spin/rocking axis of the moon; the y-axis is the lunar
equatorial axis which is mutually perpendicular to the x & z directions.
In the formula for g' M_e is the earth's mass, G is Newton's
gravitational constant and R_em is the mean earth-moon distance.

If we assume that the moon's shape is that of a *slightly* prolate
spheroid (i.e. a very low eccentricity prolate ellipsoid of revolution)
then f can be expressed as the product of 3 different dimensionless
factors: f = p*q*s. Here p is the 'prolateness', i.e. the difference in
the relative lengths of the lunar ellipsoid's axes (prolate axis minus
transverse axis over mean value, c/a - 1), q = [rho]_s/[rho]_bar
([rho]_s being the mass density near the lunar surface, and [rho]_bar
being the mean overall lunar mass density), and s = (k_e/k)^2 = square of
the ratio of what the moon's effective radius of gyration k_e about its
spin axis would be if the moon was a uniform sphere (with the correct
mass and radius) divided by the moon's *actual* radius of gyration k
about its spin axis. (This means we can also write s = (2/5)*(R_m/k)^2
where R_m is the moon's mean radius.)

If we assume that the prolate shape of the moon is the *equilibrium* shape
such that the lunar surface is everywhere an equi-potential surface
(equi-potential, that is, if the moon wasn't rocking but was in
equilibrium in the earth's gravitational field gradient, this being a
relaxed soft-moon approximation), is probably not too bad of an
approximation considering how much time the moon has had to relax to
its current shape. With this assumption the value of p becomes:
p = (3/2)*(M_e/M_m)*((R_m/R_em)^3) where M_m and R_m are the lunar mass
and mean radius respectively. This gives a value of p = 1.13 x 10^(-5).
If we estimate (by guessing that the moon's surface mass density is
typical of terrestrial crustal rocks) the lunar surface density-to-mean
density ratio as q = 0.86, we can use this estimate for q to find an
estimate for s as follows: We first (very) crudely model the radial
density profile of the moon as as a function of radial distance r as:
[rho](r) = [rho]_0 - ([rho]_0 - [rho]_s)*(r/R_m)^2 where [rho]_0 and
[rho]_s are the central core and surface densities respectively. Using
this crude model and evaluating the moment of inertia of the moon we get
that the value of our needed s-factor parameter is s = (5/7) + (2/7)*q.
Thus, if we take q = 0.86 we get s = 0.96. Putting these factors
together gives f = p*q*s = (1.13 x 10^(-5))*(0.86)*(0.96) = 9.3 x 10^(-6)
and putting this f value into the expression of T gives a prediction for
T = 2*[pi]/sqrt((3/2)*(9.3 x 10^(-6))*(1.41 x 10^(-11)) s^(-2)) =
= 4.5 x 10^8 s = 14 yr.

It seems that the ratio of the driving frequency for the rocking motion
to its natural resonant frequency is higher than I would have guessed.

David Bowman
dbowman@gtc.georgetown.ky.us