lpetrich

2014-Dec-01, 04:11 PM

Isaac Newton once wrote to Edmond Halley that the theory of the motions of the Moon made his head ache and kept him awake so often that he would think of it no more. What gave him such a great headache?

The Moon's motions are more complicated that it might seem. It orbits the Earth in an elliptical orbit that's inclined to the Earth's orbit plane, but its orbit does not stay fixed because the Sun pulls on it also. The Earth's equatorial bulge and the other planets do also, but their effects are much smaller. This makes the Moon's orbit's line of apsides precess forward and its line of nodes precess backward.

Line of apsides -- periapsis and apoapsis -- closest and farthest distance -- period = 8.85 Earth years

Line of nodes -- ascending and descending node -- northward and southward crossing points -- period = 18.60 Earth years

The Moon also has lots of shorter-period variations in its motion. The largest one is called the "evection", and it may be interpreted as a variation of the Moon's eccentricity and a wobble in its line of apsides. The Moon's eccentricity varies by about 20%, going up to 0.66 when the Sun is along its line of apsides, and down to 0.44 when the Sun is perpendicular to that line.

Newton tried to calculate these perturbations of the Moon's orbit using his famous law of gravity. Here is what I think he succeeded in doing. I think that he found the lowest-order part of the precession rates:

ωprec = (3/4) * (ωEarth2 / ωMoon)

This makes both precession periods about 17.8 years. That is reasonably close to the line-of-nodes precession, but it's only half of the line-of-apsides precession.

Also, the lowest-order evection is (15/4) * (ωEarth / ωMoon) * (eccentricity), giving 14%, noticeably less than 20%.

After his triumphs with his law of gravity, it is not surprising that Isaac Newton developed some splitting headaches.

But the problem with his calculations was that he did not go far enough. Remember that he lived long before computer-algebra software was developed. His successors developed alternative formulations that made the calculations simpler, like rotating coordinates and treating orbit elements as a kind of coordinate system. They also used much nicer notation than Newton had. Instead of Euclid-style constructions, they used algebra. Newton might have used a lot of algebra, but if so, preferred to present his results with Euclid-style constructions.

Here is a rather successful approach. George William Hill and Ernest William Brown used rotating coordinates and treating the Sun as very far away and very massive and the Earth as having a circular orbit. They included the Sun's finite distance and the Earth's orbit eccentricity later. Back to how they started, they first found a closed coplanar orbit in rotating coordinates, then treated eccentricity and inclination effects as perturbations of it. The closed coplanar orbit was found as a trigonometric series, as were the eccentricity and inclination effects. These series had a small parameter (ωEarth / (ωMoon - ωEarth)), close to (ωEarth / ωMoon).

Using (ωEarth / ωMoon), here is how the series terms contribute to the precession rates:

Apsides: 1, 0.701, 0.237, 0.072, 0.022, 0.007, 0.002, 0.001, ...

Nodes: 1, -0.028, -0.016, -0.003, ...

Apsides Total: 2.043 -- Period = 8.73 years -- Actual = 8.85 years

Nodes Total: 0.953 -- Period = 18.70 years -- Actual = 18.60 years

Very close, and the discrepancies are likely due to second-order eccentricity and inclination effects. The reason that Newton's apsidal-precession estimate was bad was that the next few terms combined were as big as the first one. Omitting them was more successful for nodal precession, however.

So this and other such techniques were successful analgesics for lunar-theory calculators, even if they required very tiresome calculations before the development of computer-algebra software.

Orbit of the Moon - Wikipedia (https://en.wikipedia.org/wiki/Orbit_of_the_Moon), Lunar theory - Wikipedia (https://en.wikipedia.org/wiki/Lunar_theory), Evection - Wikipedia (https://en.wikipedia.org/wiki/Evection), Variation (astronomy) - Wikipedia (https://en.wikipedia.org/wiki/Variation_(astronomy)), Success and failure in Newton's lunar theory (http://astrogeo.oxfordjournals.org/content/41/6/6.21.full)

Literal Solution for Hill's Lunar Problem, by Dieter S. Schmidt (he used computer-algebra software) Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton - Sir David Brewster - Google Books (http://books.google.com/books?id=Bp8RAAAAYAAJ&pg=PA158&lpg=PA158&dq=Newton+%22made+his+head+ache+and+kept+him+awake +so+often+that+he+would+think+of+it+no+more%22&source=bl&ots=Q9TF5jdtKo&sig=IPWkK7D4UxrUubNPiPRyC7gzS1o&hl=en&sa=X&ei=tJJ8VPHLOdDXoASu3YHIBg&ved=0CCAQ6AEwAA#v=onepage&q=Newton%20%22made%20his%20head%20ache%20and%20kep t%20him%20awake%20so%20often%20that%20he%20would%2 0think%20of%20it%20no%20more%22&f=false) (reference for Newton's lunar-theory headaches)

The Moon's motions are more complicated that it might seem. It orbits the Earth in an elliptical orbit that's inclined to the Earth's orbit plane, but its orbit does not stay fixed because the Sun pulls on it also. The Earth's equatorial bulge and the other planets do also, but their effects are much smaller. This makes the Moon's orbit's line of apsides precess forward and its line of nodes precess backward.

Line of apsides -- periapsis and apoapsis -- closest and farthest distance -- period = 8.85 Earth years

Line of nodes -- ascending and descending node -- northward and southward crossing points -- period = 18.60 Earth years

The Moon also has lots of shorter-period variations in its motion. The largest one is called the "evection", and it may be interpreted as a variation of the Moon's eccentricity and a wobble in its line of apsides. The Moon's eccentricity varies by about 20%, going up to 0.66 when the Sun is along its line of apsides, and down to 0.44 when the Sun is perpendicular to that line.

Newton tried to calculate these perturbations of the Moon's orbit using his famous law of gravity. Here is what I think he succeeded in doing. I think that he found the lowest-order part of the precession rates:

ωprec = (3/4) * (ωEarth2 / ωMoon)

This makes both precession periods about 17.8 years. That is reasonably close to the line-of-nodes precession, but it's only half of the line-of-apsides precession.

Also, the lowest-order evection is (15/4) * (ωEarth / ωMoon) * (eccentricity), giving 14%, noticeably less than 20%.

After his triumphs with his law of gravity, it is not surprising that Isaac Newton developed some splitting headaches.

But the problem with his calculations was that he did not go far enough. Remember that he lived long before computer-algebra software was developed. His successors developed alternative formulations that made the calculations simpler, like rotating coordinates and treating orbit elements as a kind of coordinate system. They also used much nicer notation than Newton had. Instead of Euclid-style constructions, they used algebra. Newton might have used a lot of algebra, but if so, preferred to present his results with Euclid-style constructions.

Here is a rather successful approach. George William Hill and Ernest William Brown used rotating coordinates and treating the Sun as very far away and very massive and the Earth as having a circular orbit. They included the Sun's finite distance and the Earth's orbit eccentricity later. Back to how they started, they first found a closed coplanar orbit in rotating coordinates, then treated eccentricity and inclination effects as perturbations of it. The closed coplanar orbit was found as a trigonometric series, as were the eccentricity and inclination effects. These series had a small parameter (ωEarth / (ωMoon - ωEarth)), close to (ωEarth / ωMoon).

Using (ωEarth / ωMoon), here is how the series terms contribute to the precession rates:

Apsides: 1, 0.701, 0.237, 0.072, 0.022, 0.007, 0.002, 0.001, ...

Nodes: 1, -0.028, -0.016, -0.003, ...

Apsides Total: 2.043 -- Period = 8.73 years -- Actual = 8.85 years

Nodes Total: 0.953 -- Period = 18.70 years -- Actual = 18.60 years

Very close, and the discrepancies are likely due to second-order eccentricity and inclination effects. The reason that Newton's apsidal-precession estimate was bad was that the next few terms combined were as big as the first one. Omitting them was more successful for nodal precession, however.

So this and other such techniques were successful analgesics for lunar-theory calculators, even if they required very tiresome calculations before the development of computer-algebra software.

Orbit of the Moon - Wikipedia (https://en.wikipedia.org/wiki/Orbit_of_the_Moon), Lunar theory - Wikipedia (https://en.wikipedia.org/wiki/Lunar_theory), Evection - Wikipedia (https://en.wikipedia.org/wiki/Evection), Variation (astronomy) - Wikipedia (https://en.wikipedia.org/wiki/Variation_(astronomy)), Success and failure in Newton's lunar theory (http://astrogeo.oxfordjournals.org/content/41/6/6.21.full)

Literal Solution for Hill's Lunar Problem, by Dieter S. Schmidt (he used computer-algebra software) Memoirs of the Life, Writings, and Discoveries of Sir Isaac Newton - Sir David Brewster - Google Books (http://books.google.com/books?id=Bp8RAAAAYAAJ&pg=PA158&lpg=PA158&dq=Newton+%22made+his+head+ache+and+kept+him+awake +so+often+that+he+would+think+of+it+no+more%22&source=bl&ots=Q9TF5jdtKo&sig=IPWkK7D4UxrUubNPiPRyC7gzS1o&hl=en&sa=X&ei=tJJ8VPHLOdDXoASu3YHIBg&ved=0CCAQ6AEwAA#v=onepage&q=Newton%20%22made%20his%20head%20ache%20and%20kep t%20him%20awake%20so%20often%20that%20he%20would%2 0think%20of%20it%20no%20more%22&f=false) (reference for Newton's lunar-theory headaches)