Calculation of Earth's precession



Earth is a gyroscope.
Earth precesses at a rate of 2.45*10^-12 rad/s measured
by astronomers (25,800 Years per turn).

Mach Principle says that without a force / torque a gyro
won't precess. Precession is due to inertia caused by the
centrifugal force of the spinning mass.

Centrifugal force requires a rigid point in space.
Hence, without a rigid fixed point in space gyroscopic
effects cannot occur.

Nevertheless, Earth precesses and many other celestial
bodies precess too, inclusive relativistic pulsars.
As far as I was taught by the experts, relativity pretends
to have field equations that can explain pulsars precession.
Experts also claim that relativity reduces to Newton
Theory for small relative speed, like the Earth for instance.
That's amazing, because if that were true, all we had to do is
to use the obvious simplification of relativity field equations
and get the gyroscope equations.
Since physics has no good equations for the gyroscope,
how am I supposed to believe on relativity field equations?

Earth is precessing and the only cause one can figure
to cause a torque is the gravity force over Earth bulge.
One can assume that Earth's orbit is a fixed line in
space, made of points where one can pick up an
instantaneous fixed point.
A little trigonometry and math is enough to show that
such torque is a sine square function over one Year,
so that to cause Earth's precession on average.
To understand what I'm talking about, it will be good
to read the following explanation by the middle of
the link below:
http://mb-soft.com/public/precess.html
The chapter on "Precession of the Earth's Axis"
explains it very well, starting at 6th paragraph,
and also adds a 19-Years nutation due to the Moon.
So, due to the Earth's equatorial bulge, one must have
nutation changes with 1-Year period and 19-Years
period, respectively (see article).

The problem is very well explained and everybody can
understand it (just read the article above).

The article was written by a physicist that claims:
1 - To have calculated a value of 25,700 Years period of
Earth precession, due to Sun-Moon only, to compare
with the actual measured value 25,800 Years, which
obviously is a very good approximation.
2 - That torque caused by Sun-Moon over the Earth's
bulge is a sine squared function, like this:
- Zero at 21 March.
- Maximum at 21 June.
- Zero at 21 September.
- Maximum at 21 December.
Therefore a net torque on average was calculated.

The claims made by the physicist caused obvious
predictions - A continuous motion due to nutation
over the Year, which doesn't cancel out on average,
since it was that same average term the heart of the
explanation.

By the end of the article, the physicist complains
about the following:
QUOTE:
"Assorted comments have been received since this
page was first placed on the Internet in 1998. I have
been somewhat amazed to find that there are many
people who claim that there is no such motion of the
Earth, that it is too slow of a process to have ever
been confirmed! Wow! I have been sometimes
tempted to ask if those people also believe in a
"flat Earth"!
END QUOTE.

The Earth's precess and one should be able to
calculate the torque and the precession rate to
be close to that of the actual value measured.
The Earth looks like a gyroscope that have found
a fixed orbit/point in space to cause precession
due to applied external torques on the equatorial
bulge.
And of course that Earth's equatorial bulge is
due to centrifugal force - a REAL force that
had REAL consequences at the equator.

What's an interesting point is how the physicist
gets the right result with the wrong equations.

In the chapter "Deeper Math regarding Precession
Beginning" the equation used was Euler's equation:
T3 = I3 dw3/dt + (I2 - I1) w2 w1
being I1 w1 the gyroscopic moment.

The right equation must be:
T3 = I3 dw3/dt + I1 w2 w1 sin(theta) +
+ (I1 - I2) w2^2 sin(theta) cos(theta)

Or else, since I3 dw3/dt and the last term
(I1 - I2) w2^2 sin(theta) cos(theta) are both
much smaller then T3 = T3max sin(theta)
and I1 w2 w1 sin(theta), we can use instead
an approximate solution:
T3average sin(theta) = I1 w2 w1 sin(theta)
T3average = I1 w1 w2

The whole math explanation on the article above
is a mess, since it considers T3=0 (no external
torques) and all the article explanation was about
the calculation of T3 non-zero on average.
(A sine squared term that changes over time
was explained during 90% of the article, but
now it simply is gone (T3=0).
The physicist calculated a torque due to
the Sun T3 = 1.44*10^22 Nm, and for
the Moon the twice of that value, addictive.
By the end of the article he puts T3=0.
Then it mixes the axis 3 and 2 (typo?).
The formula used (see the numbers used
and what they mean) was:
I3 w3 = (I2 - I1) w1 theta2
(the integral of the above formula,
assuming T3=0).
It's a nice cooking of Euler's equation.
Actually a very good cooking in general.
The final part of the article is pure crap and
a sort of magic cooking of values.

In fact, the physicist of the article claims:
- Conservation of Angular Momentum appears
to be violated, where it is always true otherwise.
As the precessional motion begins, angular momentum
"appears" where it had not existed before. This is unique
in the field of Physics!
- And of course, it is up to take energy out of
that violation: http://mb-soft.com/public2/earthrot.html
Basically the physicist looks like a crank.
Even so, that crank is the best that Physics have, so far,
(the article was pointed to me on this thread by
someone that looks like a non-crank physicist).

Now, since dw3/dt is very small and negligible,
if the calculations of I1 and I2=I3 are right, and
T3average = 4.55*10^22 Nm (Sun+Moon) I should
be able to calculate Earth's precession w2.
w2 = T3average / I1 w1 =
= 4.55*10^22 / (8.7*10^37 kg m^2 * 7.292*10^-5 rad/s) =
= 7.17*10^-12 rad/s
The actual value measured is 2.45*10^-12 rad/s

Yep, one must forget the Moon because on average the
torque is zero (one needs to look careful on the error made by
the physicist on Moon's analysis - it's due to the Moon's
eccentric orbit and it cancels on average), and one gets the
correct value:
w2 = T3average / I1 w1 =
= 1.44*10^22 / (8.7*10^37 kg m^2 * 7.292*10^-5 rad/s) =
= 2.27*10^-12 rad/s
(To be compared with 2.45*10^-12 rad/s measured).

Voilá, Earth's precession explained with NON-ZERO
torque as it should.

T3 was calculated by the physicist of the article for
the Sun (Moon is out because on average is zero,
according to my analysis of the problem).
The Earth total inertia moment (of the whole sphere)
is 8.7*10^37 kg m^2, as said in the article.
Earth's daily (24 hours) angular velocity is 7.292*10^-5 rad/s,
as said in the article.
And I got the right value without any cooking or crap.




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