It's finally time for me to report on the results of my
Interactive Physics model of the Moon synchronization process.
This has been a fairly long, but very interesting side
investigation for me, one that has--I think--greatly increased my
intuition for the process. I hope that a few readers will find
this similarly long post, at least, somewhat similarly
interesting.
Initially I modeled the Earth and Moon, as did Ludwik, with each
composed of two point masses (PM's). I connected them with
springs and added dashpots to provide internal dissipation.
Eventually, however, I came to the conclusion that this model was
inadequate because it did not allow for induced tidal bulges to
"push or pull to the side" on anything and, thus, create any
accelerating or decelerating torques.
Accordingly, I improved the simulation so that both the Earth and
the Moon were modeled as systems of *four* PM's attached to each
other by springs and dashpots with adjustable force and damping
constants. My Earth is 10 times as massive as my Moon. The two
bodies are given adjustable initial angular velocities and the
system is given an adjustable orbital velocity. Unfortunately, no
amount of twiddling with force and damping constants seemed
capable of overcoming an inherent instability of both objects to
"gravitational collapse" due primarily to the unbounded
gravitational attraction between the PM's but also partially to
numerical artifacts.
My wife and I then went skiing for a week and celebrated our tenth
anniversary over quarter break at Timberline Lodge, Mt. Hood.
(Highly recommended and awe-inspiring testimony to the WPA, BTW.)
I thought about the problem in a strictly-enforced computer-
deprived state and, when I returned home, I added a short range
repulsive force between PM's to reduce the tendency for
gravitational collapse and rigid separators for good measure.
With these additions, the simulation now seems quite stable if
somewhat more artificial.
I monitor both the orbital radius of the Moon and its tidal
distortion. The tidal distortion is calculated simply as the
difference of the two diagonal distances between the four PM's
that make up the Moon.
The simulation is not as polished as others that I have placed on my
IP web page (www.intranet.csupomona.edu/~ajm/myweb/index.ip.html),
but I have put it there anyway for anyone who may be interested.
There is also a screenshot which might help with the following.
Now (finally!) for some results:
1. The initial "rapid" rotation of the Moon decreases quickly as a
result, I think, of the phase lag of the tidal *bulges* with
respect to the tidal *stresses* that produce them and the
resulting average decelerating torque.
2. This initial phase is accompanied by an *increasing* orbital
eccentricity as the apogee increases while the perigee remains
relatively constant. This seems to be the result of a coupling of
both bodies' relatively large rotational angular momenta into the
orbital angular momentum (via the tidal deformations), a coupling
that is most effective at perigee. My sense watching the
simulation is that the relatively rapid rotation of the tidal
bulges past each other essentially acts to give a small forward
swat to the Moon concentrated near each perigee and, thus,
increasing the next apogee.
3. Eventually the Moon's rotation locks into some specific phase
relationship with its orbital motion around the Earth. I will
call this an "overrotating phase lock" (OPL) because it is *not*
the kind of synchronous 1:1 rotation observed in the case of the
*real* Moon. The first time I got the simulation to run this far,
the OPL involved the Moon turning through one quarter of an extra
rotation with each revolution. The second time, using different
damping constants, the OPL involved an extra *half* rotation per
revolution. Clearly, an OPL can only occur with an elliptical
orbit in which the OPL is maintained by tidal tugs which pulsate
in magnitude between perigee and apogee. Equally clearly, the
specific OPL conditions that *I* obtained were artifacts of the
quadrilateral symmetry of my moon. If I had constructed my moon
from a pentagon of PM's I would have seen OPL's involving an extra
fifth, two-fifths, etc. of a rotation per revolution. In a
stiffened prolate body one could *only* have an integer number of
additional *half* rotations per revolution as exhibited by the
planet Mercury, I believe.
Keep in mind that my moon and earth have symmetrical (square)
equilibrium forms in the absence of tidal stresses, are relatively
easily deformed by tidal stresses, and that such deformations
relax with only a small--but, I think, critical--time lag as the
tidal stresses vary. This is quite different from the case of a
body that is essentially frozen into a prolate form and that
suffers minor tidal deformations from that form. To improve my
model I would probably need to build in a time-dependent hardening
factor to model the transition from an early, more plastic moon to
the current hardened form as discussed by David Bowman.
4. Once the Moon is phase locked, orbit circularization begins.
The perigee increases AND the apogee decreases. This seems to be
due to the fact that, once the rotation is phase locked, there is
a consistent tendency to overaccelerate the Moon as it approaches
perigee and overdecelerate it as it leaves, the result of a lunar
tidal bulge which somewhat lags the Moon's CM on the way in and
somewhat leads on the way out due to the higher than average
orbital speed near perigee. (O.K. It's hard to explain; draw some
pictures and/or run the simulation yourself and perhaps you'll see
what I mean.)
5. Eventually the OPL is broken and the Moon's rotation begins,
once again, to slow. This is clearly the result of orbital
circularization and the attendant reduction in the magnitude of
the tidal pulsations that are responsible for maintaining the OPL.
(The Moon should be able to lock again into a slower OPL for a
while although I have not seen this behavior in my simulation.)
6. The Moon finally slows enough that one of its tidal bulges is
permanently locked by the tidal stress and it begins approaching
synchronous rotation (SR). It is interesting, however, to observe
the damped oscillations about the average SR rate that occur as
the now permanent tidal bulge is alternately tugged forward and
backward relative to the instantaneous direction to the Earth.
These oscillations--having an amplitude of nearly 45 degrees (the
angle of unstable equilibrium for these oscillations) at first--
are gradually reduced to a much smaller value that depends only on
the small residual eccentricity of the orbit. This is the
condition of the *real* Moon and the reason for its observed
librations today. Furthermore, since the restoring torque is of
the "soft spring" variety--with relatively smaller torques at
large angular displacements than would be observed in the linear
case--these oscillations increase in frequency as they decrease in
amplitude (much like an oscillating spring door stop.)
7. After the Moon settles into SR, its orbital radius continues to
increase slowly as the Earth transfers additional angular momentum
to it via the usual "dragged ahead tidal bulge model" that most
people (except Jim Green, perhaps) accept as the explanation for
the slow increase in the duration of the day and in the Moon's
orbital radius.
John
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A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:ajmallinckro@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
Pomona, CA 91768-4031 office:Building 8, Room 223