Re: Bulges

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

David took on the daunting task of summarizing. Leigh Palmer's last two
posts also summarize the results, with which I am in agreement. 

On Sun, 2 Mar 1997, David Dockstader wrote:

> Have I got this right?
> Don Simanek doesn't think the fact that the earth & moon orbit about their
> center of mass needs to be included in an explaination of tidal bulges.

Well, ultimately we need to include everything which affects the result. 
To the specific question of "Why are there two tidal bulges rather than
only one," the field gradient is the primary reason, somewhat modified by
the smaller rotational effect resulting from rotation about the center of
mass of the earth-moon system. But, if there were no rotation, there would
still be the *two* bulges on opposite sides of earth. 

> Mark Sylvester explains that without the orbiting there would be only one
> bulge.

I disagree. Leigh seems to say that without the rotation the problem would
be unrealistic (perhaps because earth and moon would crash into each
other?). I say, imagine a laboratory test in the spirit of Henry
Cavendish, with two elastic initially spherical bodies near each other,
but secured by spikes into a table. Now, with incredibly sensitive
instruments, measure their shape. You'd have the elongation along the line
joining them, two bulges in each, *no mater where the spikes were
located*. 

> Don and Lee disagree with Mark?
> Certainly we all agree that an elastic body, or a fluid body (a more realistic
> model for the earth in this case) placed in a gravitational gradient will
> deform to an oblate spheroid, don't we?

It seems that we do, hearing no dissent. Elongated, approximately along
(but lagging behind due to friction) the line joining earth and moon. 

> I think the difference is what one uses as a reference for measuring a bulge.
> Mark wants to nail the earth and moon in space.

That's a very interesting point I don't think has been fully addressed. If
you are talking about *ocean* tides, with the complications of water
sloshing in ocean basins and crashing into shorelines, you've got a
resonance phenomena, with something like standing waves set up in these
large bodies of water. The details are *very* messy, and at some locations
on earth there are no tidal variations, and at some other locations
there's only one tide per day, etc. But *these* tides are measured with
respect to the solid shoreline, or to an averaged sea level, *which
already includes the tidal bulges of earth* as well as the equatorial
bulge all round the earth. That is, I think, what was confusing our
discussion. Some people wanted to concentrate on these relatively small
perturbations in shoreline water level (which seem large if you are
intending to spend the day at the beach, or in a boat or ship). Some of us
wanted to concentrate on the reason for the *two* tidal bulges. 

Incidentally, the moon has tidal bulges also.

> If we nail up a spherical earth and then bring in the moon, certainly all the
> distortion of the earth is the result of earth material being pulled toward
> the moon.  

I really don't like the phrase 'pulled toward the moon'. Pulled with
respect to what? I don't like 'accelerated toward' either. They conjure up
wrong conclusions in student minds.

> Mark calls this one bulge.  No part of the earth is repelled by the
> moon!  I think his reference is the nails holding the earth and moon "fixed".
> Do you disagree with his conclusion?
> Lee wants to throw water on the nailed down earth and moon.  I claim, and I
> think Mark would agree, that if you throw water on the earth all the water will
> run over to the moon side of the earth, one bulge.

Well, I certainly don't agree. Even if the entire earth were water all the
way to the center, you'd still get two bulges on opposite sides. This is,
remember, not simply due to the *force* the moon exerts on the earth, but
on the gradient of that force, the fact that the force on the lunar side
is less than the force at the center and poles of the earth, and least at
the far side of the earth. This, combined with the earth's own field,
results in the net force gradient which distorts the shape of the earth. 

> To get water on the other
> side of the earth one needs to spin the earth moon system about the center of
> the mass.

I strongly disagree. That's one of the false notions which elementary
textbook explanations promote, and which was the reason for my entry into
this fray. Lawrence S. Lerner's problem 14.41 in his _Physics For
Scientists And Engineers_ has decent (though small) diagrams, and asks the
students to calculate the departure from sphericity of an earth covered
completely with water, and to explain why there are *two* bulges. His
result for the departure from sphericity at is given

                             3
    delta(r) = (m/M)[r/(R+r)] r

r is the radius of the earth, R is the distance to the external body, m is
the mass of the external body and M is the mass of the earth. He also
asks students to show that the height difference between high and low
tides at mid-ocean is (3/2) delta(r). Remember, this is a water-covered
earth, so no ocean sloshing or standing waves need be considered. And,
yes, Lerner's diagram shows the *two* bulges, and never once mentions
rotational (centrifugal) effects, since he's ignoring rotation. He also
asks students to use the fact that the mass of the moon is 1/81 that of
earth, and its distance from earth is 60 times the earth's radius, to find
the height of midocean tides. He remarks that "the result is quite close
to what is observed at such midocean locations as Hawaii."

> The obiting of the center of mass is not needed to get oblate shapes
> it is needed to get water on both sides of the earth!

I disagree, as noted above.

> The Kowalski experiment is great.  However, note that his earth is
> falling, it is not nailed down.  As Mark said, this is what is needed to
> get water on both sides of the earth.  If Kowalski nails down the earth
> & moon he gets one bulge. 

I very much doubt that. I'll bet he gets two.

This whole thing got a complete airing on the Skeptic discussion group
about a year ago. Vic Stenger, physicist at the University of Hawaii, was
kind enough to e-mail me the whole thread today. It arose from a
discussion of the tidal forces on a *person*. Some astrologers claim that
astrological influences at the time of birth are due to tidal forces on
the baby from moon, sun and planets. The whole notion is, of course,
absurd, but it made an interesting calculational problem. The question was
whether the tidal effect of the doctor delivering the baby was greater
than that of moon, sun or planets. Vic worked out all the math, and a neat
way to estimate the tidal forces on any body. But as the thread is untidy,
he hasn't given me permission to repost it until he cleans it up. I may,
in a couple of weeks (spring break--whee!), have time to clean it up for
posting. 

	-- Donald

.......................................................................
Dr. Donald E. Simanek                            Office: 717-893-2079
Prof. of Physics                    Internet: dsimanek@eagle.lhup.edu
Lock Haven University, Lock Haven, PA. 17745          CIS: 73147,2166 
Home page: http://www.lhup.edu/~dsimanek            FAX: 717-893-2047
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