Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: Bulges



I first sent this out last night on my other account. The listproc didn't
post it, so I'm sending it again on the correct account. Mark Sylvester
and Donald have each posted clarifications this morning (we westerners are
a bunch of Sunday lieabeds) and the standard explanation of the far bulge
(the earth is pulled away from the material in the bulge) is invoked by
some third party named Plait who is not here to defend himself.

The most important conceptual improvement in my explanation is that the
Earth and Moon don't have to be nailed down at particular points; they are
nailed down at *all* points in the rotating frame. I then throw an ocean's
worth of water on Earth and let it seek its equilibrium configuration. The
water will flow to seek a level of lower potential in this frame until all
water has reached the lowest potential attainable. At that configuration
the water's surface will be an equipotential, perpendicular to the local
gradient of the potential (what we would call the "local acceleration of
gravity"). My alternative follows:

Donald's contribution leads me to offer one of my own. I find explaining
tides in terms of forces conceptually strenuous. It is a vector problem.
I much prefer to work in terms of a scalar quantity, the potential function,
and to work in what I have touted here before as the natural coordinate
system for such a problem, one in which the Earth and Moon are nailed
down, the rotating coordinate system. If one now chooses the center of
mass of this system as the origin, one finds three terms contribute to the
potential function: the gravitational potentials of the two bodies (which
we will approximate as being spherical and, hence, contributing point mass
potential terms) and the centrifugal potential (horrors, I hear some of
you crying out). If this function is now set equal to a constant (-GMe/Re)
the resulting surface will be found to be a prolate spheroid. Two bulges
fall out of the calculation very naturally.

Once one becomes used to the rotating frame it is easy to "feel" why this
should be, by the way. A physicist who is aculturated to thinking this way
would never say flatly that rotation does not give rise to the tidal
bulges because he invokes the concept of centrifugal force. Those who
choose to look at the world as though centrifugal forces do not exist may
well say that rotation does not cause the bulges. I claim that if the
bulges can be derived exactly by a model which invokes centrifugal force,
then it is logically defensible to say that rotation causes the tidal
bulges in part. Just what fraction of the tidal bulge is due to rotation
can readily be calculated from this model by redoing the calculation and
leaving out the centrifugal term. The bulges remain, but they are smaller
if only the gravitational terms are invoked.

I think it is a shame that this point of view is not taught in our schools.
The zealous censorship of physics textbooks that has removed the entirely
accessible concept of centrifugal force from our problem solving tool bag
has done a great disservice to our students.

Leigh