[Phtha]

Sir Isaac Newton once mentioned that the only thing that gives him a headache is the moon.

I thought this was an interesting read, ill paste it here:

Newton was a remarkable mathematician; he not only developed calculus, but he also worked out how his laws of motion and his law of gravity produce Kepler's Three Laws.

One of his less well known achievements, however, is his attempt to try to find solutions for more than two bodies, in particular, a solution for the Moon's orbit, which is significantly perturbed by the Sun.

The Moon's orbit around the Earth is approximately elliptical and inclined to the ecliptic, the plane of the Earth's orbit around the Sun. But due to the Sun's perturbations, the Moon's line of nodes (ecliptic - orbit-plane intersection) precesses backwards and its line of apsides (perigee - apogee line) precesses forwards. And also due to those perturbations, the Moon has some periodic wobbles in its motion, some of which have been known for centuries.

Periods:
Nodes: 18.5996 years (backwards)
Apsides: 8.8504 years (forwards)

The next question is why the apsides move twice as fast as the nodes. Newton intended to answer that question by developing perturbation theory, treating the Sun as an add-on to the Earth-Moon system. However, he could not make the numbers work out right, and he was forced to fall back on the more-or-less empirical approach of all his predecessors. He used his theory to tell him what effects to watch out for, and then he determined the sizes of those effects from observations.


But I have recently verified the cause of his headache. Using the computer-algebra package Mathematica, I have verified the most important parts of the Hill-Brown lunar theory (their way of solving the equations of motion), including the source of Newton's headaches.

The lowest-order perturbation theory produces equal precession rates, and predicts a rate of

(3/4)*(ns^2/nm)

where nm is the Moon's rate of orbiting of the Earth relative to the stars, and ns is the Sun's apparent rate. It gives a period of

17.825 years

That is reasonably close to the nodes' rate, but is only half that of the apsides' rate. So I think that Newton had succeeded in finding the lowest-order result, and only that result.

However, the Hill-Brown and similar theories do much better, and taking six extra terms yields:

Nodes: 18.7042
Apsides: 8.72755

The numbers are much better, though about a percent off. The first two extra terms have the answer:

Multiply the nodal rate by 1 - (3/8)*(nm/ns) + (26207/9216)*(nm/ns)^2 + ...

Multiply the apsidal rate by 1 + (75/8)*(nm/ns) + (390815/9216)*(nm/ns)^2 + ...

These terms have sizes:

Nodal: 1, -0.0281, -0.0159, -0.0027, -0.0003, ...
Apsidal: 1, 0.7013, 0.2373, 0.0723, 0.0218, 0.0067, 0.0022, ...

Which has been aspirin for Newton's more recent successors.

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Something like that had happened before him. Johannes Kepler was working on Mars's motions, and he found an error in Copernicus's work. Copernicus made the plane of Mars's orbit pass through the "mean Sun" (where it would be if it followed its averaged-out motion) instead of the true Sun, and got a wobble in Mars's orbit inclination. Kepler corrected that by making the plane of Mars's orbit pass through the true Sun instead, and he found that that pesky wobble was gone.

"Copernicus did not know his own riches!" he said.

And neither did Newton, as my discussion has shown.

From:
http://www.bautforum.com/astronomy/3...-headache.html