Re: moon's synchronism

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Concerning Ludwik Kowalski's recent comments about the pendulum analogy
for the lunar libration mode:

> In a one-line reply to John M. (or was it to David B. ?)  Brian W.
> compared our Moon to a pendulum. Actually it is a good model for
> describing the Interactive Physics simulation I was watching.

I think Brian's pendulum analogy in response to John Mallinckrodt's
calculation of the moon's natural resonant frequency of libration.

Ludwik's comments then elaborated on the usefulness of the pendulum
analogy in the lunar context, especially regarding his interpretation
of his IP simulations.  I hate to (partially) burst Ludwik's balloon
here, but the pendulum analogy, although having a couple of legitimate
points of contact with the lunar case, is I believe quite misleading
on a number of levels.

First the legitimate analogous points:

1. Both a pendulum and the lunar libration mode involve a rotational
oscillation mode which approaches simple harmonic motion for small
angular oscillation amplitudes.

2. Both a pendulum and the lunar libration mode involve oscillations
about a minimum in a *gravitational* potential energy.  IOW, in both
cases the restoring torque is fundamentally supplied by gravity in some
way.

Now for the important differences:

1. A pendulum oscillation involves an oscillation mode of a mass
distribution which couples to its mass *dipole* moment which pivots about
a point *necessarily off of* the center of mass of the mass distribution.
For the lunar rocking frequency problem, OTOH, the oscillation involves
an oscillation mode coupling to the *quadupole* moment on the moon's
mass distribution *about* its center of mass.  In fact, if we tried to
take an ordinary physical pendulum and shift its pivot point to its
center of mass, it would not work as a pendulum at all (in a uniform
gravitational field) because the restoring torque would vanish in this
field for all orientations of the pendulum.  In this case the pendulum
would be balanced.  This is because all mass distributions have an
exactly zero dipole moment about their own center of mass.  Once the
dipole moment vanishes then so does the restoring torque.  The reason
that a mass distribution cannot have a nonzero dipole moment about its
own center of mass is that there is no such thing as (far as we know as) a
negative mass which is required for the nonzero value.  This is different
for electric and magnetic dipoles.  For instance, a magnetic compass
needle (or dip meter) responds to a magnetic field in a fundamentally
pendulum-like way because it pivots about the CM and has a nonzero
magnetic dipole moment about that point.  Electric dipoles can also exist
about their center of mass because of the existence of both positive and
negative charges on opposite sides of the central point.  For masses,
OTOH, we have a problem because the location of the CM is determined by
the condition that the (mass) dipole moment vanish about that point.  In
this case both sides of the CM have a balanced amount of mass and the
distribution cannot be unbalanced about its balance point (by
definition).

2. The coupling of an ordinary pendulum's (necessarily eccentric) dipole 
moment to an external gravitational field configuration is via the
gravitational field itself.  A uniform gravitational field works just
fine.  OTOH, the gravitational coupling of the moon's quadrupole moment
(in the lunar libration mode problem) is via the *gradient* of the
earth's external gravitational field in the vicinity of the moon.  In
this case a uniform field will *not* work, but a uniform gravitational
field *gradient* would work just fine in providing the restoring torque
needed.  In the lunar case the earth does not actually have a uniform
gravitational field gradient in the vicinity of the moon, but this is
only a higher order complication irrelevant for the main effect discussed
here.  The moon's quadrupole moment then couples to the average (earth-
produced) field gradient at the moon's extended location.  The higher
order derivatives of the earth's gravitational field in the vicinity of
the moon give rise to higher order multipole interactions with the moon's
mass distribution which are significantly smaller than the main
quadrupole interaction.  BTW, such couplings between gravitational
field gradients and (their induction of, and their resulting torques on)
mass quadupole moments are called *tidal* interactions.

3. For an ordinary pendulum (simple or physical) the restoring torque is
proportional to the sine of the angle of rotation away from the
equilibrium orientation.  For the kind of tidal interaction-induced lunar
libration mode discussed in this thread the restoring torque is
proportional to the sine of *twice* the angle of rotation away from the
equilibrium orientation.  Thus for a pendulum the restoring torque is
maximal for a 90 deg rotation away from equilibrium and the pendulum is
in a state of unstable equilibrium of maximal potential energy for a
180 deg (upside down) rotation.  For the tidal-quadrupole problem, OTOH,
the maximal torque occurs at 45 deg of rotation and an unstable
equilibrium with maximal potential energy occurs at 90 deg.  In fact, at
180 deg rotation the system has a second stable equilbrium potential
energy minimum.  For a pure tidal-quadrupole problem both the 0 and
180 deg potential energy minima and orientations are *equivalent*.  Since
the moon, presumably, has a nonzero octupole moment that couples to the
second derivative of the earth's gravitational field we expect that this
higher-order interaction will break this 0-180 deg degenerate symmetry so
that (presumably) the 0 deg orientation has a slightly lower potential 
energy than the 180 degree orientation (which would then be demoted to
a locally stable but globally metastable equilibrium angle).

4. The lunar libratory motions, as we have already discussed, are driven
at the rate of the lunar orbital frequency which is more than 2 orders of
magnitude higher than the natural resonant oscillatory frequency of the
moon's orientation.  Thus, as John M. so elequently emphasized, the
actually observed libratory motion of the moon is almost entirely due to
the moon's rotational inertia, and the tidal restoring torque exerted on
the moon by the earth is effectively irrelevant for understanding this
motion.  So any pendulum analogy would be like driving the a steel
pendulum bob on a long string whose natural period is 3 seconds at 60 Hz
about its equilibrium orientation with an electromagnet.  In such a
system the fact that we have a pendulum at all is quite secondary when
describing the bob's high frequency vibrations.

In conclusion, I don't necessarily recommend against using a pendulum
analogy for this problem, but I do advise that it only be used with
caution and with awareness of the significant limitations of the analogy.

David Bowman
dbowman@gtc.georgetown.ky.us