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Re: Bulges



There have been some good articles in AJP on the tides over the
years and I'm pretty sure that at least some of them specifically
perform the equipotential type calculations that have been
referred to in this thread. Nevertheless, I'm not at my office
just now and I'd never actually done the calculations myself, so I
fired up Maple and pretty easily got some results that I think
make sense.

I wrote a function that gives the potential at any point in the
orbital plane of the earth-moon system due to the earth, the moon,
and the centrifugal force. The function was written in polar
coordinates with the origin of the reference frame at the center
of the earth and the angle measured from the "to the moon"
direction. I used the function to calculate the potential at (r =
one earth radius, theta = 90 degrees) and then solved for the
radii at different theta's that give the same potential.

In the three tables that follow I show (1) the results of the
calculation, (2) the results if we delete the centrifugal term
and, (3) the results if we delete the moon term. In each case I
give the excess radius (in meters) on the near side and the far
side of the earth along a line cutting the earth-moon line at
angle theta and the overall excess diameter along that line as
functions of theta. (The diagram below may help explain the
terms.)


Near Side
/
/ Theta
---o-----------------------------------------o--
/ Earth Moon
/
Far Side


RESULTS
Excess Excess Excess
Theta Radius Radius Diameter
(Near) (Far)
0 .54281 .53093 1.0737
10 .52607 .51526 1.0413
20 .47797 .47007 .9480
30 .40449 .40064 .8051
40 .31478 .31508 .6299
50 .21992 .22349 .4434
60 .13152 .13672 .2682
70 .06029 .06519 .1255
80 .01469 .01762 .0323
90 0 0 0

WITHOUT CENTRIFUGAL
Excess Excess Excess
Theta Radius Radius Diameter
(Near) (Far)
0 22.11704 -21.04303 1.0740
10 21.77255 -20.73094 1.0416
20 20.75111 -19.80282 .9483
30 19.08831 -18.28297 .8053
40 16.84158 -16.21155 .6300
50 14.08754 -13.64402 .4435
60 10.91860 -10.65029 .2683
70 7.43907 -7.31357 .1255
80 3.76099 -3.72868 .0323
90 0 0 0

WITHOUT MOON
Excess Excess Excess
Theta Radius Radius Diameter
(Near) (Far)
0 -21.57406 21.57432 .0003
10 -21.24630 21.24655 .0003
20 -20.27299 20.27322 .0002
30 -18.68370 18.68389 .0002
40 -16.52672 16.52686 .0001
50 -13.86758 13.86766 .0001
60 -10.78708 10.78711 0
70 -7.37881 7.37881 0
80 -3.74633 3.74631 0
90 0 0 0

(Sorry for all the excess sig figs. It's late and I don't have the
patience to edit the output.)

Notice that it is the moon's gravity--and not the centrifugal force--that
is almost entirely responsible for the tides as shown by the excess
diameters. Notice also that the moon pulls the equipotential surface
toward it while the centrifugal effect is virtually identical in the
opposite direction.

'nuff for now.

John
----------------------------------------------------------------
A. John Mallinckrodt email: mallinckrodt@csupomona.edu
Professor of Physics voice: 909-869-4054
Cal Poly Pomona fax: 909-869-5090
Pomona, CA 91768 office: Building 8, Room 223
web: http://www.sci.csupomona.edu/~mallinckrodt/