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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)

Jerry
2014-Dec-02, 04:34 AM
Nice, concise summary. There are non-elastic, non-linear tidal effects that should defy every attempt to nail down an exacting solution. It is good to see something so close with so few terms.

lpetrich
2014-Dec-02, 08:21 AM
Yes. From Ernest William Brown - Wikipedia (https://en.wikipedia.org/wiki/Ernest_William_Brown),

Observations showed that Brown's tables were indeed superior to those of Hansen, which had been in use since 1857, but there was still a large unexplained fluctuation in the Moon's mean longitude of the order of 10 arcseconds. A 'great empirical term', of magnitude 10.71 arcseconds and period 257 years, was introduced to eliminate this as far as possible. Given the precision of Brown's calculations, it must have come as a great disappointment to have to introduce this arbitrary adjustment.
It turned out to be due to irregularities in the Earth's rotation.

From Lunar theory - Wikipedia (https://en.wikipedia.org/wiki/Lunar_theory), Newton's colleagues suffered from similar headaches for a while:

They also wished to put the inverse-square law of gravitation to the test, and for a time in the 1740s it was seriously doubted, on account of what was then thought to be a large discrepancy between the Newton-theoretical and the observed rates in the motion of the lunar apogee. However Clairaut showed shortly afterwards (1749–50) that at least the major cause of the discrepancy lay not in the lunar theory based on Newton's laws, but in excessive approximations that he and others had relied on to evaluate it.
Alexis Claude Clairaut had to do by hand what I recently did with Mathematica, my favorite computer-algebra-software package.

Returning to the evection, that eccentricity-variation perturbation in longitude, here's what the first few terms look like in it. Expanding its relative value in powers of (ωEarth / ωMoon),
I get: 0.14, 0.05, 0.01 adding up to 0.20 in good agreement with observation.

Gigabyte
2014-Dec-02, 08:21 PM
Now I finally understand why a tide table isn't considered accurate for a location, until they have recorded the tides for 18.5 years.

Hornblower
2014-Dec-03, 03:55 AM
Let us remember that while Newton was a giant in his field and made major advances in mathematics, he was not omnipotent or omniscient. Subsequent mathematicians stood on his shoulders and became giants in their own right, and in doing so made further advances beyond what he was able to do during his life. Thus we have the means to handle multibody gravitational problems in ways far more advanced than Newton's best efforts.

lpetrich
2014-Dec-03, 05:50 AM
The Planet Mars: A History of Observation and Discovery. Chapter 1: Motions of Mars. University of Arizona Press. (http://www.uapress.arizona.edu/onlinebks/MARS/CHAP01.HTM) Something similar happened earlier. Copernicus found that he had to introduce a wobble in Mars's orbit inclination to get Mars's orbit to fit. But when Kepler was working on that planet's motions, he discovered that Copernicus had made a mistake. He had used the "mean Sun", where the Sun would be if its motions were averaged out. In the spirit of heliocentrism, Kepler tried the true Sun's position, and Mars's orbit inclination no longer had that wobble. Kepler exclaimed "Copernicus did not know his own riches!"

Gigabyte
2014-Dec-03, 07:20 PM
The history of astronomy is ripe with these sort of things.

2014-Dec-04, 12:37 AM
I get better solution for the node precession:

p = 2/3 n^2/n-synodic

just 2/3 in the place of 3/4.

for the Moon: n = 1 / year, and n-synodic = 12.4 / y, which gives just the period 18.6 years, exactly.

We can improve this, taking into account an eccentricity and inclination, multiply by cos(i) and something like (1 + 3/2 e^2) probably.

lpetrich
2014-Dec-04, 05:54 PM
The history of astronomy is ripe with these sort of things.
Ripe? Do you mean rife? Also, any favorite examples?

marmolada, that sort of method is more suited for Against the Mainstream. You ought to learn how to do celestial mechanics to get real predictions. That will mean learning Newtonian mechanics and how to solve differential equations.

Back to examples of resolved discrepancies, I'll get back to Johannes Kepler. Tycho Brahe only let JK have TB's Mars data, because TB was afraid that JK might scoop him, and because Mars was the most difficult planet to study. After fixing which Sun to refer Mars's orbit to, JK turned to Mars's motions. He tried fitting Ptolemy's hypothesis of an equant, a point which the planet moves around at constant angular velocity. He found one whose maximum error was only about 8 minutes of arc, but that was rather clearly worse than TB's observations. Also, there wasn't anything at an equant point, which did not seem to make much physical sense. So JK considered some Sun-centered possibilities, and he discovered his equal-area law, his second in the usual count. That is a version of conservation of angular momentum.

The shape of Mars's orbit was more difficult, and he first used Mars to find the Earth's orbit. He considered observations of Mars at different times but with Mars at the same place in its orbit. He found that the Earth's orbit was as eccentric as the orbits of the other planets. He then turned to the problem of Mars's orbit shape, and he discovered that it was an ellipse, a squashed or stretched circle, with the Sun at one focus. This was his first law of planetary motion in the usual count. He published both laws in 1609 in Astronomia Nova (The New Astronomy).

As to his third law, he first proposed that the spacings of the planets were (circumradius)/(inradius) of each of the five Platonic solids. The planets, with which one:

Inner
Outer
Ratio
S Rat
Solid

Mercury
Venus
1.869
1.732
Octahedron

Venus
Earth
1.382
1.258
Icosahedron

Earth
Mars
1.524
1.258
Dodecahedron

Mars
Jupiter
3.415
3
Tetrahedron

Jupiter
Saturn
1.833
1.732
Cube

From Mysterium Cosmographicum (The Cosmographic Mystery), 1596.

S Rat is the Platonic-solid ratio of the radii of the circumscribed and the inscribed circles. Their analytic formula is sqrt(3)*tan(pi/n), where n = 3 for the tetrahedron, 4 for the cube and octahedron, and 5 for the dodecahedron and icosahedron. He had originally thought of regular polygons, but they did not fit very well. The ratio for an n-sided one is sec(pi/n): (3: 2, 4: 1.414, 5: 1.236, 6: 1.154). He later concluded that his regular-solid scheme did not work very well either. He considered some other possibilities, and he hit upon his square-cube law or third law: (period)^2 ~ (distance)^3. He published it in 1619 in Harmonices Mundi (The Harmony of the World).

2014-Dec-04, 06:38 PM
marmolada, that sort of method is more suited for Against the Mainstream. You ought to learn how to do celestial mechanics to get real predictions. That will mean learning Newtonian mechanics and how to solve differential equations.

No, I can use directly the oryginal Newton's style of reasoning to prove the precession of nodes is just:

p = 2/3 n^2/n_s cos(i) x (1 + 3/2 e^2);

The standard solution is like this:

p = 3/4 n^2/n' + higher orders...
and this is incorrect slution, because there is an error about 10% for the spimplest case: n << n'.

it gives just 3/4 instead of 2/3, so the error is equal: 1/12 = 8.3%, so, this is very bad approx.

The same formula is applicable to the planets precession also, thus it's general - universal, not for the Moon only like these numerical perturbative series.

Hornblower
2014-Dec-04, 06:54 PM
No, I can use directly the oryginal Newton's style of reasoning to prove the precession of nodes is just:

p = 2/3 n^2/n_s cos(i) x (1 + 3/2 e^2);

The standard solution is like this:

p = 3/4 n^2/n' + higher orders...
and this is incorrect slution, because there is an error about 10% for the spimplest case: n << n'.

it gives just 3/4 instead of 2/3, so the error is equal: 1/12 = 8.3%, so, this is very bad approx.

The same formula is applicable to the planets precession also, thus it's general - universal, not for the Moon only like these numerical perturbative series.

Your line of thought remains unconvincing to me. All I can see is that you noticed that the number of lunar synodic months in a year is approximately 2/3 of the number of years in the Moon's node precession cycle, and from there somehow made a great leap to inferring that a certain coefficient in the standard solution should be 2/3 instead of 3/4, without showing any rigorous analysis of the gravitational action.

2014-Dec-04, 07:16 PM
Yes. I just notice: 3/2 x 12.4 = 18.6 in years.
This simplicity very intrigued me, so, I tested this for other Moon's distances,
and the relation was ever better for lower orbits - bigger: n_s = synodic, say 20-30 cycles / year!

Then I improved this a little, including on an inclination dependency: cos(i);
And further I take into account the eccentricity:
a^2(1 + 3/2 e^2), it's an average of a square of distance to the moon, and in a real time, not a geometrical.

And now the precession of the Moon's node is exact!
Up to 4 digits only, because the Earth's quadrupole generates about 1/10000 of the node motion. :)

lpetrich
2014-Dec-04, 08:32 PM
Now for some later Solar-System discrepancies.

First, the outer Solar System.

The story starts in 1781 when William Herschel observed what looked like a distant comet. After some months of observing, it became evident that this "comet" was on a nearly circular orbit, and one that orbited outside Saturn's orbit. It was a planet. WH named it Georgium Sidus (Georgian Star or George's Star) after King George III. His colleagues considered some other names before settling on Uranus.

It has an orbit period of about 84 Earth years, but before it completed its first observed period, astronomers had discovered discrepancies in its motion from what was calculated for it from the Sun and the planets inside its orbit. Urbain Leverrier and John Couch Adams both predicted its position, and Johann Gottfried Galle discovered it in 1846. François Arago called it Leverrier, but it was later named Neptune.

Both UL and JCA had used Bode's law in trying to find Neptune, and they thus overestimated Neptune's distance. From Bode's law, its major axis is 38.8 AU, while its actual one is 30.06. Why did UL and JCA succeed? They worked from observations over only part of Uranus's synodic period with Neptune, and rather close parts at that. Neptune's Bode period is 241.68 years and its actual period is 164.79 years. This gives synodic periods 128.77 and 171.38 years, the real one being about (4/3) of the Bode one. Since UL and JCA worked from about 65 years of observations, they say about 0.50 (Bode) and 0.38 (actual) synodic periods of observation. Nowadays, it would be easy to try out a lot of possible orbit elements for a trans-Uranian planet, but UL and JCA had to do everything by hand.

That was not the end of it. There were still some unresolved discrepancies in Uranus's motion, and that led to searches for trans-Neptunian planets. In 1930, Clyde Tombaugh found one: Pluto. But Pluto turned out to be far too small to produce the expected gravitational effects. It has also been recently demoted from planethood, but that's another issue.

A solution to this problem came from a surprising source. An input into the search for orbit discrepancies is the masses of the planets. These could be determined by observing the moons of all the planets that have them, and that includes all the outer planets. So the calculations were done with these masses. But starting in the early 1960's, another source of data became available: spacecraft tracking. This made possible much-improved estimates of the masses of the planets. In particular, Voyager 2 flew by Jupiter in 1979, Saturn in 1981, Uranus in 1986, and Neptune in 1989. Radio tracking of it made possible improved mass values of Uranus and Neptune. These were less than the IAU's 1976 recommended values by 3.6% for Uranus and 1.4% for Neptune. E. Myles Standish plugged them into JPL's ephemeris software and he found that the discrepancies vanished. There was no evidence of trans-Neptunian planets massive enough to produce observable effects on Uranus or Neptune (Source: Planet X - No dynamical evidence in the optical observations (http://adsabs.harvard.edu/cgi-bin/bib_query?1993AJ....105.2000S)).

The mass discrepancies were likely from imprecision in the measurements of the moons' positions. The moons' periods, however, can be measured over a large number of orbits, making them much less of a problem.

2014-Dec-04, 08:40 PM
One nice example of the power of the simplicity:

The precession of the circular obrital plane and with a small inclination is equal:

p = 2/3 GM/d^3 / w_rel;
w_rel is a relative angular speed = 2pi/Synodic period.

Then I modify this in someway to directly compute the arcsecons per year:

1 / w = Ty 86400*365.25/2pi

GM/d^3 (86400*365.25)^2/2pi * 180/pi * 3600'' = 1296000'' / year

And now we can check the Mercury's nodal precession.
Some synodic periods of the Mercury:
Tm-e = 0.31727; earth
Tm-v = 0.3958; venus
Tm-j = 0.24584; jupiter

and average distances to these planets:
Rm = 0.387098 au; Rv = 0.723332; Rj = 5.2; Rs = 9.582;

OK. Now I can compute the precession in arcsecs / y:

pm-v = 2/3 1296000'' * 1/408590 * 1/0.723^3 * 0.3958 = 2.2146'';
pm-e = 2/3 1296000'' * 1/333000 * 1/1 * 0.3173 = 0.8233; --> 0.8335; including the Moon
pm-j = 1.4428;
pm-s = 0.0653; and due to the saturn

sum: pm = 4.556''/year = 455.6'' / century

And according to the observations the precession of Mercury's orbital plane is about: 450'' / cy;

So, it's rather visible the formula works very well.

lpetrich
2014-Dec-05, 05:06 AM
Now the inner Solar System.

In the mid 19th cy., Urbain Leverrier worked on the orbits of the planets, taking into account their mutual perturbations and fitting his solution to observations as far back as 1750. But in 1859, he discovered a discrepancy in the precession of Mercury's perihelion. It was about 38 seconds of arc per century greater than what one would calculate from the other planets' pulls on it (its modern value: 43s/cy). There was, however, a certain problem. Mercury and Venus have no moons and Mars has no moons that were known back then. That made the masses of Mercury, Venus, and Mars adjustable parameters in UL's calculations. He could bump up Venus's mass, but it would cause excess precession of the Earth and Mars. Likewise, bumping up Mars's mass would cause excess precession of Venus and the Earth.

But in 1877, Asaph Hall discovered Mars's two moons. That made it possible to find a good value of Mars's mass independent of its perturbations of the other planets' orbits. It was very close to UL's mass value. So it was hard for there to be discrepancies there.

UL proposed that there was some intra-Mercurian planet, and he proposed the name Vulcan, after the Roman god of the forge, on account of such a planet being very hot from being very near the Sun. In the late 19th cy., various astronomers claimed to observe intra-Mercurian planets, often as objects transiting the Sun, but none of the observations added up to a planet in a well-defined orbit. More recent searches have revealed no evidence of intra-Mercurian "Vulcanoids" larger than a few km, the size of a small asteroid.

So that solution was out.

Late in the 19th cy., various astronomers and physicists proposed various modifications of Newtonian gravity for doing so. But these solutions had various problems. Adjusting the exponent in the inverse-square law would produce the same relative amount of precession in *all* the planets' orbits, and there was no reason to suppose that to be happening. It was also theoretically ugly, since the resulting gravitational potential would violate Poisson's equation.

But in 1907, Albert Einstein turned his attention to gravity. He wanted a theory that (1) has Newtonian gravity in its Newtonian limit, (2) had special relativity in its zero-gravity limit, and (3) had no special space-time coordinate dependence, being like Maxwell's equations. He recognized that space-time has to be curved in it, with time being treated as an extra space dimension, and he studied the formidable mathematics of differential geometry. In 1915, he published his General Theory of Relativity: (space-time curvature) = (gravitational constant) * (energy-momentum density). He calculated Mercury's excess precession with it, and he found agreement. However, he did a weak-field calculation with the excess precession coming from a second-order calculation, with (first-order field)^2 source terms. He despaired of finding strong-field solutions, but Karl Schwarzschild soon found one for a spherically-symmetric, time-independent gravitational field.

So what worked was modifying the law of gravity.

Not surprisingly, tracking spacecraft improved the results, with continued agreement with general relativity: [1306.5569] Use of MESSENGER radioscience data to improve planetary ephemeris and to test general relativity (http://arxiv.org/abs/1306.5569)

lpetrich
2014-Dec-05, 09:50 AM
Let us remember that while Newton was a giant in his field and made major advances in mathematics, he was not omnipotent or omniscient. Subsequent mathematicians stood on his shoulders and became giants in their own right, and in doing so made further advances beyond what he was able to do during his life. Thus we have the means to handle multibody gravitational problems in ways far more advanced than Newton's best efforts.
That's right, and I will demonstrate some alternate formulations of Newtonian mechanics with a simple example. An object with mass m at position q moving in a potential V(q):
\displaystyle{ m \frac{d^2 q}{dt^2} = - \frac{dV}{dq} }

This can be broken down into coupled equations:
\displaystyle{ \frac{dq}{dt} = v ,\ m \frac{dv}{dt} = - \frac{dV}{dq} }

Or with momentum p = m*v,
\displaystyle{ \frac{dq}{dt} = \frac{p}{m} ,\ \frac{dp}{dt} = - \frac{dV}{dq} }

The first big reformulation of Newtonian mechanics was devised by Joseph Louis Lagrange in 1788. In it, a physical system is described by an action I, and its behavior tries to minimize it. The action is given by the integral of the "Lagrangian" L over time:
\displaystyle{ I = \int L dt ,\ \delta I = \delta \int L dt = 0 }

where
\displaystyle{ L = \text{(kinetic energy) - (potential energy)} = \frac12 m v^2 - V }

Doing the minimization gives the Euler-Lagrange equations of motion, after what mathematician Leonhard Euler introduced to solve a similar sort of problem.
\displaystyle{ \frac{d}{dt} \left( \frac{\partial L}{\partial v} \right) - \frac{\partial L}{\partial q} = 0 }

One can easily extend them to multiple coordinates, and one can use them for many-body problems, coordinates other than rectangular, even coordinates like orientation angles. One can do constraints with a method named after JLL himself: "Lagrange multipliers". One can extend this formulation to fields, doing the integration over space as well as time.

In 1833, William Rowan Hamilton invented another big reformulation. He thought of a quantity called the Hamiltonian, a quantity equal to the total energy of a system, and a quantity that one can get equations of motion from:
\displaystyle{ H = \text{(kinetic energy) + (potential energy)} = \frac{p^2}{2m} + V }
\displaystyle{ \frac{dq}{dt} = \frac{\partial H}{\partial p} ,\ \frac{dp}{dt} = - \frac{\partial H}{\partial q} }

with appropriate generalizations to multiple coordinates. The p's are "generalized momenta" that go along with the q's, "generalized coordinates". For an orbit, one can make the q's and p's some simple functions of the orbit elements. Charles-Eugène Delaunay used the Hamiltonian formulation to work on the Moon's motions, using a series of changes of variables to reduce the problem's Hamiltonian into a much simpler form.

Still another big reformulation is named after him and Carl Gustav Jacob Jacobi. It involves Hamilton's principal function or action S, which is a function of coordinates q and constants of the motion a. One can get momentum p and additional constants of the motion b from it as follows:
\displaystyle{ p = \frac{\partial S}{\partial q} ,\ b = \frac{\partial S}{\partial a} }

The overall equation of motion is
\displaystyle{ H + \frac{\partial S}{\partial t} = 0 }

The Hamilton-Jacobi formulation is especially convenient for cases where one can separate variables. With certain coordinate variables and kinetic and potential energy forms, one can express S as a sum of S(coordinate, constants) functions, and get relatively simple solutions.

The Lagrangian, Hamiltonian, and Hamilton-Jacobi formulations are useful not only in Newtonian mechanics, but also in relativity, where one can use a particle's proper time as its time coordinate. They are also related to various formulations of quantum mechanics:

LagrangianFeynman Path-Integral
HamiltonianHeisenberg
Hamilton-JacobiSchrödinger

2014-Dec-06, 07:49 PM
Very impressive maneuvers.
But there is one problem: what is the advantage of this type of formulation, especially where is this better than the old,
and probably naive or even stupid, original methods of Newton?

I guess I should not, but I have to remind the final solution of the equations of this type provide
almost 10% error in the case of the spmple problem of the precession of the orbital plane!

Hornblower
2014-Dec-06, 11:55 PM
Very impressive maneuvers.
But there is one problem: what is the advantage of this type of formulation, especially where is this better than the old,
and probably naive or even stupid, original methods of Newton?

I guess I should not, but I have to remind the final solution of the equations of this type provide
almost 10% error in the case of the spmple problem of the precession of the orbital plane!
As has been pointed out before, Newton was not stupid, but he did not live long enough to develop the mathematical techniques that are needed to make reasonably accurate calculations for three-body problems such as the Moon's motion, which is severely perturbed by the Sun. As pointed out in the OP, Hill and Brown came close. With modern computer techniques a Newtonian calculation works very well for the Moon.

lpetrich
2014-Dec-07, 01:33 AM
Very impressive maneuvers.
But there is one problem: what is the advantage of this type of formulation, especially where is this better than the old,
and probably naive or even stupid, original methods of Newton?
Because one can change to coordinates where the problem becomes MUCH easier to solve. Newton's original formulation was not dumb, just very inconvenient in many cases.

Let's say that you want to solve a central-force problem. In Newton's original formulation, it would be in rectangular coordinates:
\displaystyle{ \frac{d^2 {\mathbf x}}{dt^2} = a(r) \frac{{\mathbf x}}{r} ,\ r = |{\mathbf x}| }

Calculate the angular momentum
\displaystyle{ {\mathbf h} = {\mathbf x} \times {\mathbf v} ,\ {\mathbf v} = \frac{d {\mathbf x}}{dt} }

It is easy to show that h is a constant of the motion. One can show from this that {\mathbf x} \cdot {\mathbf h} = 0 , implying that the motion is planar, with h being perpendicular. Since the motion is around a center, let us try 2D polar coordinates: {\mathbf x} = r \{\cos \theta, \sin \theta, 0\} .

It's easier to change coordinates in the Lagrangian formulation than in the original:
\displaystyle{ L = \frac12 v^2 - w(r) = \frac12 \left[ \left(\frac{dr}{dt}\right)^2 + r^2 \left(\frac{d\theta}{dt}\right)^2 \right] - w(r) }
where a(r) = w'(r).

Using the Euler-Lagrange equations gives
\displaystyle{ \frac{d^2 r}{dt^2} - r \left( \frac{d\theta}{dt} \right)^2 = a(r) ,\ \frac{d}{dt} \left( r^2 \frac{d\theta}{dt} \right) = 0 }

It has a solution:
\displaystyle{ \frac{d\theta}{dt} = \frac{h}{r^2} ,\ \frac{d^2 r}{dt^2} - \frac{h^2}{r^3} = a(r) }

where h is the magnitude of the angular-momentum vector. As you can see, the second equation is a differential equation for one variable, r, and its solution can be plugged into the first equation to give θ.

Now for the Hamilton formulation. For that, we need generalized momenta p corresponding to generalized coordinates q, and the Hamiltonian from the Lagrangian:
\displaystyle{ p = \frac{\partial L}{\partial (dq/dt)} ,\ H = p \frac{dq}{dt} - L }

Thus, in this central-force problem, p_r = dr/dt ,\ p_\theta = r^2 (d\theta/dt) giving
\displaystyle{ H = \frac12 \left( p_r{}^2 + \frac{p_\theta{}^2}{r^2} \right) + w(r) }

In our problem, H is independent of θ, so one can easily show that pθ is constant. It is equal to h, and from the Hamiltonian formulation, one can also get the equations of motion I'd derived earlier. Since H is also independent of time, its value is the total energy E. So one gets
\displaystyle{ \frac{dr}{dt} = \sqrt{ 2(E - w(r)) - \frac{h^2}{r^2} } }

Finally, let's see how the Hamilton-Jacobi solution works. The action function S can be separated: S = - E*t + h*θ + R(r,E,h)

One gets
\displaystyle{ - t_0 = \frac{\partial S}{\partial E} = - t + \frac{\partial R}{\partial E} ,\ \theta_0 = \frac{\partial S}{\partial h} = \theta + \frac{\partial R}{\partial h} }
\displaystyle{ p_r = \frac{dr}{dt} = \frac{\partial S}{\partial r} = \frac{\partial R}{\partial r} ,\ p_\theta = h = \frac{\partial S}{\partial \theta} ,\ H = - \frac{\partial S}{\partial t} = E }

So,
\displaystyle{ R = \sqrt{ 2(E - w(r)) - \frac{h^2}{r^2} } }

giving
\displaystyle{ t = t_0 + \int \frac{dr}{\sqrt{ 2(E - w(r)) - (h^2/r^2) } } ,\ \theta = \theta_0 + \int \frac{h}{r^2} \frac{dr}{\sqrt{ 2(E - w(r)) - (h^2/r^2) } } }

I guess I should not, but I have to remind the final solution of the equations of this type provide
almost 10% error in the case of the spmple problem of the precession of the orbital plane!
One actually gets very good agreement for the Moon. But one has to expand to rather high order in the small parameters, or else do numerical integration.

2014-Dec-07, 06:31 PM
I can't see nothing special in these equations.
These are the typical newtonian equations, but You use the final pre-computed formulas.

You simply omit the whole stage of the investigation, reasoning, which is the most important in this game. :)

lpetrich
2014-Dec-08, 04:53 PM
I don't know what would have satisfied marmolada, because to put in the details of the derivations would have made for much longer posts with little additional informative value. I also don't think that I would have wanted to present an advanced-undergraduate or graduate-level course in classical mechanics. The sort that may use Goldstein's classic textbook on the subject.

But I've found Astronomical, Magnetic and Meteorological Observations Made at the United States Naval Observatory -- United States Naval Observatory -- 1885 -- at books.google.com -- likely also at archive.org

It lists lots of old measurements, so one can get an idea of the state of the art back then. Here's for (Sun's mass) / (Mars's mass)

Leverrier 1858: 2,994,790 ... from its perturbations of the Earth's orbit
Hall 1878: 3,093,500 +- 3,295 ... from Mars's moons
IAU 1994: 3,098,708 +- 9 ... likely from spacecraft tracking

Present-day comparison: Astrodynamic Constants (http://ssd.jpl.nasa.gov/?constants) at JPL -- has masses of the Sun and the planets, using the numbers from a 1994 publication by the IAU.

So Leverrier got Mars right.

lpetrich
2014-Dec-08, 06:39 PM
Is Phobos Hollow?

In 1944, Bevan Sharpless discovered that Mars's moon Phobos was spiraling in to Mars (Secular accelerations in the longitudes of the satellites of Mars (http://adsabs.harvard.edu/abs/1945AJ.....51..185S)). He found for Phobos:

Angular acceleration: 188.2 += 17.1 *10^(-5) deg/yr^2
Inspiral rate: 5.71 +- 0.35 cm/yr

Around 1958, Iosif Shklovsky considered atmospheric drag, and he concluded that Phobos has *very* low density. Something like the density of an iron sphere 16 km across with a thickness of 6 cm, about 1.8*10^(-4) g/cm^3. Some people concluded from this that Phobos was artificial.

But when spacecraft were sent to Mars, their orbits got perturbed by Mars's two moons, giving us their masses and thus their mean densities. Phobos's is 1.876 g/cm^3, making it unlikely that it is hollow.

Shklovsky had ignored tidal drag from Phobos making tides on Mars, and Phobos's "normal" density means that it has enough mass to make enough tidal drag to account for the observed inspiral.

Here is some recent work: Improved estimate of tidal dissipation within Mars from MOLA observations of the shadow of Phobos - Bills - 2005 - Journal of Geophysical Research: Planets (1991–2012) - Wiley Online Library (http://onlinelibrary.wiley.com/doi/10.1029/2004JE002376/full)

Angular acceleration: 136.7 +- 0.6 *10^(-5) deg/yr^2
Inspiral rate: 4.14 +- 0.01 cm/yr
Not much less than Sharpless's results.