Re: Moon's Spin

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

Jim Green wrote:

> I think David Bowman was saying that (David help us out here) if the Moon
> was molten 5^9 years ago and in the same orbit it has now, then the then
> Earth's gravity gradients would have forced the solidifying Moon into a mass
> quatdrupole configuration and that configuration would have prevailed intact
> to the present.  Thus if the current Earth had the same gravity gradients,
> THEN the Earth would now continue to hold the solid Moon in the same
> orientation it had then.

That's pretty close to what I was trying to say.  Only a few nit picky points
of clarification here.  1) I never gave any time number for when the moon
solidified.  I think when you say 5^9 above you mean 5 X 10^9 or possibly
5E9.  Is this correct?  I personally don't particularly care when the moon
solidified, but I think (off the top of my head without checking any
references) those that are paid to think about such things give for their
best estimate for the time of the formation of the moon as closer to
something like 4, or 3.9 X 10^9 years ago based on the ages of the oldest
lunar rocks brought back by the Apollo astronauts.  2)  I didn't mean to
imply that the moon's current shape was precisely unchanged from the time
that it solidified.  Rather, that (since the moon is quite stiff) it doesn't
change its shape on a very short time scale, such as during the stress and
torque fluctuations that it is subjected to during the monthly lunar cycle.
The moon keeps its shape long enough so that any temporary rotation of it
will cause a restoring torque to return it to its equilbrium orientation
quickly enough so the moon doesn't have time enough to deform into a new
shape as it is rotated in a nonequilibrium orientation.  Over the eons of
time I'm perfectly willing to suppose that the moon's shape probably relaxed
from a more prolate shape in the past to a lesser prolate shape now as moon
receded from the earth as the tidal stress acting on it due to the earth's
gravitational field gradient weakened with distance.  I certainly did not do
any calculation to check to see if the moon is stiff enough to not change its
shape at all as it receded from the earth in response to the weakened tidal
stresses at greater distances from the earth.

> I expressed concern that assumptions that can't be verified tend to lead us
> into stories which are convoluted to fit current data.  Further the same
> people who talk about the above also conclude that the Moon started out MUCH
> closer to the Earth than now AND if the Moon were molten so likely was the
> Earth THUS the gravity gradients would have been much different at the Moon
> then than they are now.  Hence there is no reason for the Earth to be able
> to hold the Moon in the same orientation now as the Earth did then. 

I'm no planetary scientist with a pet theoretical ax to grind, but from what
I have read most such people (i.e. planetary scientists) have formed a
consensus about what the senario for the formation of the moon was.  The
standard story is that relatively soon after the planets had finished taking 
shape some 4.5 X 10^9 yrs ago there were many left over planetessimals, and
partially accreted planets left over from the planet forming materials.  Most
of this material eventually either crashed as meteorites onto the planets and
moons that had already had been formed (making lots of craters of all sizes),
or it was deflected out of the (inner part of) the solar system by numerous
near misses with the planets.  This left the remaining planets well separated
from each other with relatively stable orbits with essentially nothing between
them.  During the early time of still intense meteoritic bombardment an
anomalously large (Mars-sized) object struck the young earth at an oblique
angle.  This ejected a huge chunk of material from the earth made of matter
from both the earth and the foreign object.  Regardless of whether or not the
earth was molten before the collision, its surface and the ejected material
certainly was (at least) molten (if not mostly vaporized as well) just
afterward.  The earth's gravity recaptured a large amount of the volitile
stuff (water, atmospheric gases. etc.)  A large mass of the ejected material
made of mostly refractory rocks and stuff coalesed into the moon.  The new  
moon's gravity was too weak to hold onto the high temperature volitile
materials so it became barren of an atmosphere and its rocks were completely
desicated of any water.  The newly-formed moon then orbited the earth much
more closely than now.  Over the eons the moon has since been gradually
receding from the earth as it's orbit has acquired some of the earth's lost
spin angular momentum due to the moon's tidal breaking of the earth's spin.
Since the moon was formed from the loose ejecta of a major collision it's
inital motion was primarily translational and not rotational.  I think the
moon was never supposed to have had a spin out of sync with its orbit since
the time of its formation.  This collision-ejection senario does explain many
previously loose ends about the current state of the earth-moon system.  It
explains the systematic chemical similarities *and* differences between the
moon and the earth.  It explain's the earth's rapid spin and its tilted spin
axis.  (Before the collision it was thought that the earth's spin was very
slow like that of Mercury and Venus because the accretion process that built
up the inner planets from planetesimals resulted in the growing body being
pummelled from all sides which tended to cancel out any net accumulated spin
angular momentum.)  It explains the relative orientation's of the moon's
orbital plane and the earth's spin axis.  It explains the moon's age
~3.9 X 10^9 yr (as well as the age of the oldest surface rocks on the earth).
It explains how the moon could have gotten to its current orbital position in
the time allowed.  (If the moon had formed with the earth at 4.5 X 10^9 yr
ago then it should have receded further out from the earth by now than its
current orbital radius.)  Because this senario ties so many separate loose
ends together, planetary scientists tend to prefer it over other conceivable
lunar formation senarios.

> BUT someone privately has given me to think that it might be possible for
> this to happen if the mass configuration frozen into the Moon were DIPOLE --
> just like the "tidal bulges" this list likes to talk about.  IE surely the
> Moon is oblate with respect to the orbital axis, but suppose the Moon were
> frozen oblate with respect to the Moon/Earth axis as well.

The moon (as well as the earth) does not have an intrinsic mass dipole
moment.  This is simply because there is no such thing as negative mass.  You
need both positive and negative mass to make a mass dipole.  It is a
straightforward calculation to show that the mass dipole moment always
cancels out to zero for *any* mass distribution if the origin of the
coordinate system for the calculation is at the distribution's center of
mass.  The moon (and any tidal bulges on the earth) is a prolate (i.e.
football-like) shape *not* an oblate shape.  The earth's overall shape is
oblate, but that is due to the effects of the earth's spin--not anything due
to tides.

>                                                             Then if the Moon       
> were to rotate slightly (as it does during the year), the gravitational
> attraction to the front lobe would be greater than to the rear lobe and
> there would be an impressed torque to bring it back.

This is true but the restoring torque is on a quadrupole moment not a dipole
moment.  An electric dipole in an electric field has one stable equilibrium
orientation of minimum potential energy.  If it is rotated by 180 degrees 
from this orientation the orientational potential energy is a maximum.  The
dipole experiences a maximum restoring torque if it is rotated by 90 degrees.
In contrast, OTOH a prolate shaped mass distribution in a uniform
gravitational field *gradient* has a minimum of its potential energy in *two*
opposite orientations.  (If you turn a prolate spheroid around by 180 degrees
it is equivalent to no rotation at all by symmetry.) The orientations of
maximum potential energy is at a 90 degree rotation from stable equilibrium.
The angle of maximum torque is at 45 degrees.  (The different rotational
properties are due the fact that a dipole involves projections onto the l=1
spherical harmonics and a quadrupole involves a projection onto the l=2
spherical harmonics.)

>                                                      This would be the case
> if the gravitational field had spherical symetry and tht symetry would
> continue to the present.

This would be the case even if the gravitational field of the earth was more
complicated than spherically symmetric as long as its gravitational field
gradient was relatively uniform in the vicinity of the moon and it was
changing slowly in time.  But in any event the earth's gravitational field in
the vicinity of the moon *is* quite well approximated as spherically
symmetric (even if the earth was substantially out of round) because the
field strength pattern for the higher multipole moments falls off with
distance much faster than the spherically symmetric monopole term.

> It is just the converse of this that the planetary geophysicists assume
> happens between the Earth's "tidal bulges" and the Moon IE the phase lag of
> the "bulges" causes a torque on the Moon to slow it down.  But of course
> there are no such things as "tidal bulges" so their arguments are vacuous.
> (Sorry, Folks, I couldn't resist -- write privately if you want to do a side
> bar in that! -- Or maybe David or Dan want to take the heat -- I do not.)  

I don't care about any heat, and I'll go on record here on this issue as a
firm fence straddler on the issue of the earth's tidal bulges.  I agree with
Jim that a picture of the tides on the earth in terms just two tidal bulges
synchronized with (toward and away from the moon) is a very misleading and
*incorrect* view of the situation which is *dominated* by a host of
complicating factors which influence the actual behavior of the ocean tides.
But I think that underlying all of these other tidal motions is a background
level which has the tidal bulge picture present.  Its just that the signal
of these bulges (of a small fraction of a meter of height) is swamped by the
noise of the effects of the complicating factors which influence the actual
oceanic tides in a given place of multiple meters in height.  If there were
truly no such thing as tidal bulges (synchronized with the moon) then there
would be no handle for the moon to brake the earth's spin and to fling the
moon further out into space.  Since the orbital recession of the moon and the
secular angular deceleration of the earth's spin are measured effects, it
seems clear to me that the tidal bulges must exist even if they are too small
to show up and notice under all the other complicated asynchronous tidal
motions.

> In any case my arguments satisfy me -- and my rampant speculation is as good
> as anybody's rampant speculation. (:-)

Same here.

David Bowman
dbowman@gtc.georgetown.ky.us