I wanted to run the numbers again and realized I had garbled the equations somewhat.
I didn't realize the Nav had slipped in the equation for Earth's gravity into the mix.
Since that's 10,000 times stronger than any of the other forces involved, it isn't
really relevant, unless you're trying to calculate the actual tide heights.
Here's the numbers I'll use. These don't really have enough precision to do this
properly:
Gravitational Constant G 6.67x10^-11
Mass of Moon M 7.35x10^22 kg
Distance to Moon R 3.8x10^8 m
Radius of Earth r 6.3x10^6 m
Earth center to E-M barycenter s 4.641x10^6
The proper formulas for the Moon's effect a
Farside:
GmM/(R+r)^2 - m s omega^2
Near side:
GmM/(R-r)^2 - m s omega^2
This does not count the Earth direct force, which is huge, and use a constant
Centrifugal Force.
I'll work this in terms of the acceleration felt by a body, which is numerically the
same as the force on a one kg body. All numbers are m/sec^2.
The CF comes out to 3.29x10^-5 . This is using an orbital period of 27.3 days (not
the 28 nav mentions)
The gravitation acceleration using GM/r^2 is 3.39x10^5. These numbers are actually
the same, differing only due to the rough approximations used. They must be the same,
because the orbital velocity is determined by the gravitational pull. Thus, we can
use the number computed with GM/r^2.
So, the nearside acceleration becomes:
GM/(R-r)^2 - GM/r^2 = 3.51x10^-5 - 3.39x10^5 = 1.2x10^-6
for the farside:
GM/(R+r)^2 - GM/r^2 = 3.28x10^-5 - 3.39x10^5 = -1.1x10^-6
Thus, the net force is almost equal and opposite, and in agreement with the
traditional values. (The difference between near and farside may even be less, given
roundoff issues.)
If, however, we used Nav's equations, the number are radically different.
The Nearside CF for Nav is (r-s)omega^2, or 1.17x10^-5
The Farside CF is (r+s)omega^2 or 7.74x10^5
Nav said this must be subtracted from the Grav force, which would result in:
Nearside 3.51x10^-5 - 1.17x10^-5 or 2.34x10^5
Farside 3.39x10^-5 - 7.74x10^-5 or -4.46x10^-5
Note that this is a serious imbalance between near and farside, which is certainly not
supported by observation. My hunch is that Nav would prefer to ADD the nearside grav
and CF, which would result in 4.68x10^-5, which is close enough to the farside to
actually make some sense. However, this is roughly 40 times the traditional value
that has been used for many years. Many calculations have been presented that show
the traditional values generate the observed tides; Nav has not presented any credible
explanation for how the experts could be off by a factor of 40.
But, lets take this one more step. What is the contribution from the Sun? First,
running the net gravitational and Centrifugal forces at the Earth's center yields
5.93x10^-3 for grav, and 5.94x10^-3 for CF. Again, these numbers must actually be
equal.
The next step is to calculate the differential gravity. Frankly, since that is so
small compared to the direct gravity that using these approximations would be futile.
I'm content to accept the traditional value: the Sun's differential gravity is a bit
under half of the Moon's, which would be about 5x10^-7.
Finally, to calculate the Centrifugal Force from the Earth-Sun system according to
Nav's formula, we can take advantage of the fact that it is linear with distance. The
delta from the CF at the Earth's center, according to Nav, will be small, 2.5x10-7.
On the near side the tides would be reduced by this amount, on the far side they would
be increased. When added to the Differential Gravity, the net result is the farside
is increased to about 7.5x10^-7, and the nearside is reduced to 2.5x10^-7. This
presents the problem that the night tides have triple the contribution from the Sun as
the day tides.
Using Nav's formulas for the tides pass one test: the near and far side contributions
from the Moon are roughly equal. However, in all other regards they fail miserably.
The are 40 times the accepted and well studied values for tidal forces. But worse,
the Moon's contribution is 100 times the contribution from the Sun. I don't there is
any way to reconcile these discrepancies, and Nav doesn't seem willing to explain it.
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