From: aephraim@physics3 (Aephraim M. Steinberg)
Newsgroups: sci.physics
Subject: Re: Why does the moon keep the same face to the Earth?
Date: 25 Jan 1993 21:44:39 GMT
Organization: /etc/organization
NNTP-Posting-Host: physics3.berkeley.edu

In article <1993Jan23.154116.13409@aifh.ed.ac.uk> tw@aifh.ed.ac.uk (Toby) writes:
>
>Some friends and I were trying to work out why the moon maintains
>the same face towards the earth. Our favourite theory is that the
>moon is asymmetric, and its orientation is a minimum of gravitional
>energy - this would explain why it maintains this orientation under
>disturbances like large asteroid strikes, gravitational influences
>of the sun .... A rival theory was that its angular velocity is
>a direct consequence of it being spun off the Earth (the "where did the
>Pacific Ocean go?" theory). However, I am more dubious that this
>would be so stable? Can anyone help answer this question? Thanks 
>in advance,

If you allow angular momentum to be "traded" back and forth between the
moon's orbit and its spin, but conserve the total, you find the state
of minimal rotational kinetic energy to be the one in which the orbital
angular velocity is equal to the rotational angular velocity-- i.e., 
the one in which the moon spins around (relative to an inertial frame)
once a month, thus keeping one face towards the Earth at all times.

In order for this to have happened, there must be some dissipative
coupling between the two degrees of freedom.  The standard line is that
this is due to tidal effects, i.e., friction between sand masses sliding
back and forth against one another.  I actually once ran across an 
article that made me guffaw in the middle of the physics library, so much
did it remind me of the "suppose we have a spherical cow" joke, only with
a perverse twist.  A perfectly rigid, symmetric moon would not exhibit
the desired effect.  In order to present a not overly complicated model
calculation, some theorists decided to make the simplest asymmetric
assumption.  They assumed that the moon was a cube, and calculated how
the dissipation would progress!

Anyway, this should happen for all bodies.  The Earth will eventually
face the same side to the sun constantly if nothing else happens to the
solar system first (which it will), and if it's fair to ignore effects
of other gravitational effects (I don't know to what extent it is).  But
it would take a damn long time.  If I remember correctly, Mercury ALMOST
does this-- but on the other hand, that may be an old, incorrect theory
which is now known to be false!  Embarrassingly, I don't remember, so don't
quote me on it.  I know there was some sci-fi (Niven?) story about such
a planet, where the only habitable region was right along the strip of
perpetual sunrise (set).

A related model is one of two masses sliding along on a frictionless
table, connected by a weak, dissipative spring.  Such a coupling does
not affect the total momentum of the two masses, but can lower the
total energy of the system.  If the total momentum
m_1 v_1  + m_2 v_2 = P is a constant, let us try to minimize the energy
(m_1 v_1^2 + m_2 v_2^2)/2.  To do this, we will differentiate with respect
to v_1, after setting v_2 = (P - m_1 v_1)/m_2.

m_1 v_1 + m_2 v_2 (-m_1/m_2) = 0 by the chain rule

m_1 v_1 - m_1 v_2 = 0

or v_1 = v_2 !

It's no harder to do this for the angular case (v -> \omega, m -> I, etc.),
but I find this version instructive because you can actually picture the
two masses and find it believable that the lowest energy state is the one
in which they simply slide along together.


-- 
Aephraim M. Steinberg                    | "WHY must I treat the measuring
UCB Physics                              | device classically??  What will
aephraim@physics.berkeley.edu            | happen to me if I don't??"
                                         |       -- Eugene Wigner
Newsgroups: sci.physics