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## perturbation theory, precession

Hi

I've been reading that the orbits of the Planets are precessing. i.e. the actual orbit moves round. This is along the line of apsides, which is the widest part of the elliptical orbit. So the line itself is moving anticlockwise.

Anyway Newtons universal Gravitation in no way predicts this but the as the other planets move round the Sun in an anticlockwise direction they pull each others orbits round. Mercury's precession was greater than expected, which was corrected by Einstein.

Anyway, Lagrange proposed that the influence of other planets would cause precession but would not lengthen or shorten an orbit. And the planets were made into rings (to average things out) i.e. the orbital path was seen as a ring. This is known as perturbation theory. Perturbation theory was also used when (the Church) it was thought everything went round the Earth. In other words as stated on wikipedia:

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem.

So my point is, As we don't know how the Planets orbits came about (although we presume the solar system condensed out of a cloud), we cannot be confident about why they are precessing. In other words as we don't know their origin we don't know whether they would be precessing anyway. Just wondering if anyone else feels this presumption (and perturbation theory) may be unwise or erroneous

2. Are you saying that 'we don't know how the Planets orbits came about' is a problem, as the word problem is used in that definition of pertubation theory? In that case I think your presumption fails because they are different classes of problems.

Also, what exactly do you mean by 'we don't know how the Planets orbits came about'? It may be a valid statement but there can be many misunderstandings hidden behind it.

3. Originally Posted by stitt29
[Snip!] I've been reading that the orbits of the Planets are precessing. i.e. the actual orbit moves round. This is along the line of apsides, which is the widest part of the elliptical orbit. So the line itself is moving anticlockwise.

Anyway Newton's universal Gravitation in no way predicts this but the as the other planets move round the Sun in an anticlockwise direction they pull each others orbits round.
Wrong. Newton's gravity does predict this and is used to calculate this to a first approximation. It also predicts that the nodes of the orbits precess in the clockwise direction as well, and again is used as a first approximation in calculating this.
Originally Posted by stitt29
Mercury's precession was greater than expected, which was corrected by Einstein.
True.
Originally Posted by stitt29
Anyway, Lagrange proposed that the influence of other planets would cause precession but would not lengthen or shorten an orbit. And the planets were made into rings (to average things out) i.e. the orbital path was seen as a ring. This is known as perturbation theory.
Lagrange, Laplace, and Euler were among the first to develop perturbation theory in celestial mechanics. Lagrange showed that up to the third order in masses there were no secular terms in the semimajor axis, which (by Kepler's third law) means no secular increase or decrease in the orbital period. The method of replacing a planet with a ring of matter with density proportional to the time spent near the point was developed by Gauss.
Originally Posted by stitt29
Perturbation theory was also used when (the Church) it was thought everything went round the Earth. In other words as stated on wikipedia:

Perturbation theory comprises mathematical methods that are used to find an approximate solution to a problem which cannot be solved exactly, by starting from the exact solution of a related problem.
Wrong as to the time. Perturbation theory in a recognizable form was developed starting after the 1770's by the above-mentioned Lagrange, etc., long after the matter of heliocentricity/geocentricity was settled.

(While Newton did work on such problems as the lunar motion, his work is couched in geometrical language and very hard to read. That is why I referred above to "recognizable form".)
Originally Posted by stitt29
So my point is, as we don't know how the planets' orbits came about (although we presume the solar system condensed out of a cloud), we cannot be confident about why they are precessing. In other words as we don't know their origin we don't know whether they would be precessing anyway. Just wondering if anyone else feels this presumption (and perturbation theory) may be unwise or erroneous.
Again, wrong. We don't have to know exactly how the planets came to be in the presently observed orbits. All we need is a set of observations good enough to infer initial conditions (here meaning at some time close to our own) and we can calculate the motion to a reasonable degree of precision. From this we can infer the precessional rates of the perihelia and nodes, and we can apply small corrections due to general relativity where needed. As far as the motions of major bodies (planets and larger asteroids) are concerned Newtonian gravity with minor corrections from general relativity are more than adequate.

Small asteroids and comets do experience some non-gravitational forces due to heating and out-gassing. It is only when we look at small bodies (space probes) that we see even the hint of small deviations from Newtonian/Einsteinian gravitation. (Pioneer anomaly, flyby anomaly, e.g. That is where some of the action is nowadays.)

In summary, there is no "presumption" about perturbation theory. It works, it works quite well, and indeed worked so well as to show the way to the next refinement of gravity, general relativity. It is entirely possible that the Pioneer and flyby anomalies, arising from the difference between observation and perturbation theory, may do the same.

4. Originally Posted by stitt29
Hi
So my point is, As we don't know how the Planets orbits came about (although we presume the solar system condensed out of a cloud), we cannot be confident about why they are precessing. In other words as we don't know their origin we don't know whether they would be precessing anyway. Just wondering if anyone else feels this presumption (and perturbation theory) may be unwise or erroneous
No, because we can measure the precession of satellite orbits around the Earth (driven by lunar and solar gravity, and for low orbits the oblateness of the Earth itself) and see how accurately perturbation calculations hold for their orbits, for which we know exactly where and how they started their motion. Motion under gravity depends on the location and velocity of the object at a particular time, not on any additional history. (Recognizing the approximations inherent in that statement - it's more complicated when we can't neglect internal structure of the masses, such as the Earth's nonspherical shape). There are satellites whose mission function relies on precession, in fact - sun-synchronous orbits keep them passing over the same pieces of Earth at the same solar time each day, useful for reconnaissance or astronomical surveys.

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Hi Celestial Mechanic you said

"It also predicts that the nodes of the orbits precess in the clockwise direction as well"

I thought precession was anticlockwise. Can you explain. Also what are nodes of the orbits?

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Hi ngc3314 you said

Motion under gravity depends on the location and velocity of the object at a particular time, not on any additional history.

good point. But if the planets are precessing this could have been going on for millions of years and could be an inherent part of their orbit. In other words How can we be sure the planets are precessing because of the influence of the other planets. It just might not be the case.

7. Originally Posted by stitt29
Hi ngc3314 you said

Motion under gravity depends on the location and velocity of the object at a particular time, not on any additional history.

good point. But if the planets are precessing this could have been going on for millions of years and could be an inherent part of their orbit. In other words How can we be sure the planets are precessing because of the influence of the other planets. It just might not be the case.
Can you give us any reason, in appropriate mathematical detail, to doubt the findings of Lagrange, Laplace, Euler, Gauss, et. al.?

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because no one has solved the 3 body problem

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If I'm not mistaken, the correct terminology is prograde and retrograde, as clockwise and counterclockwise are arbitrary.

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Originally Posted by Hornblower
Can you give us any reason, in appropriate mathematical detail, to doubt the findings of Lagrange, Laplace, Euler, Gauss, et. al.?
If you look from above all the planets are orbiting anticlockwise round the Sun. Their orbits are also precessing anticlockwise. Lets say the nearest planet to you (another planet) is on the right hand side of the Sun and you are on the left hand side. Why would this pull your orbit further to the left (or anticlockwise) If both planets are on the LHS I could see how they could pull each other round anticlockwise. BUT if they were on the RHS it is just as likely they could pull the other clockwise

11. Originally Posted by stitt29
If you look from above all the planets are orbiting anticlockwise round the Sun. Their orbits are also precessing anticlockwise. Lets say the nearest planet to you (another planet) is on the right hand side of the Sun and you are on the left hand side. Why would this pull your orbit further to the left (or anticlockwise) If both planets are on the LHS I could see how they could pull each other round anticlockwise. BUT if they were on the RHS it is just as likely they could pull the other clockwise
You have not analyzed the methods of the aforementioned mathematicians, let alone point out what you might think are errors on their part. All you have done is give an oversimplified, misleading attempt at analyzing the gravitational effect of one planet on another one's motion relative to the Sun.

Let's do a simplified thought experiment with the Sun, Mercury and Jupiter. The Sun and Mercury will be gravitating toward Jupiter roughly in unison, since bodies of different masses at any given distance from a third body accelerate at the same rate. We need to look at the gravitational gradient between Mercury and the Sun, not the absolute value of Jupiter's action.

When Mercury and the Sun are in conjunction with Jupiter, the latter will tend to separate them. The effect resembles a weakening of the Sun's gravity as experienced by Mercury. When they are at a right angle to Jupiter, they are accelerating at the same rate, but are converging toward Jupiter. The effect now resembles a strengthening of the Sun's gravity as experienced by Mercury.

Here we have a distortion in the Sun's gravitational field as seen by Mercury, and that distortion is revolving in a prograde direction with a period of about 12 years. Here I am unable to do a rigorous calculation as Euler and others have done, but my hunch is that an elliptical orbit would precess in the same direction. My hunch is in good agreement with the actual prograde precession, whose magnitude is many times that of the small Einsteinian component that distinguishes it from a Newtonian solution.

I don't know which way Mercury's orbit would precess, if at all, if we could magically stop Jupiter's orbital motion and drag it directly away from the Sun and let its gravity drag Mercury and the Sun toward it in a straight line. If you can match mathematical technique with Euler, fire away.

12. Originally Posted by stitt29
Anyway Newtons universal Gravitation in no way predicts this ...
Newtons universal gravitation DOES predict this. It is a result of the pull of Jupiter, and to a lesser degree the other planets. Mercury has an excess beyond what can be accounted for from Newtonian multi-body gravitation (as do the other planets to a much less observable degree).

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Originally Posted by stitt29
...But if the planets are precessing this could have been going on for millions of years and could be an inherent part of their orbit. In other words How can we be sure the planets are precessing because of the influence of the other planets. It just might not be the case.
One way is to do an n-body simulation. And I've done one for you using Gravity Simulator. There's nothing in the code that tells the orbits of the planets to precess. However, they precess anyway. The following graph shows Earth's precession under 3 different conditions:

1. All planets considered
2. All planets except Jupiter considered
3. All planets except Earth are removed

As you can see by the graph, which was generated by the n-body code, the precession is clearly caused by the planets, as it completely goes away in the absence of additional planets. And Jupiter causes the bulk of the precession, as Earth precesses slower in Jupiter's absence.

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Originally Posted by stitt29
because no one has solved the 3 body problem
Three body problems can be solved under certain conditions , as the one you describe , not the general case .

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Originally Posted by Hornblower
You have not analyzed the methods of the aforementioned mathematicians, let alone point out what you might think are errors on their part. All you have done is give an oversimplified, misleading attempt at analyzing the gravitational effect of one planet on another one's motion relative to the Sun.

Let's do a simplified thought experiment with the Sun, Mercury and Jupiter. The Sun and Mercury will be gravitating toward Jupiter roughly in unison, since bodies of different masses at any given distance from a third body accelerate at the same rate. We need to look at the gravitational gradient between Mercury and the Sun, not the absolute value of Jupiter's action.

When Mercury and the Sun are in conjunction with Jupiter, the latter will tend to separate them. The effect resembles a weakening of the Sun's gravity as experienced by Mercury. When they are at a right angle to Jupiter, they are accelerating at the same rate, but are converging toward Jupiter. The effect now resembles a strengthening of the Sun's gravity as experienced by Mercury.
If the effect is to strengthen the Sun's gravity why is Mercury not accelerated into a higher (wider or larger orbit i.e. an increase in the semi major axis). The Earth and moon have a similar increase in their gravitational pull due to the moon pulling on the oceans and this is increasing the moons orbit by 4cm a year. Why doesn't it just precess faster?

This is my point, not well enough made to begin with, If other bodies are adding and subtracting to the Sun's gravitational effect why are the planets orbital lengths rigidly fixed? Why do these additions to the Sun's gravitational pull cause precession and nothing else?

As to addressing the Maths of Euler etc. Please direct me to where I can study this. The confidence that there can be no secular increase seems to me that an assumption has been made resulting in a cancellation of terms. The 3 body problem has yet to be solved but the nine bodies of the solar system interact in such a way that precession occurs but orbital length can only ever remain constant. Why because a brilliant mathematician said so

16. Originally Posted by stitt29
If the effect is to strengthen the Sun's gravity why is Mercury not accelerated into a higher (wider or larger orbit i.e. an increase in the semi major axis). The Earth and moon have a similar increase in their gravitational pull due to the moon pulling on the oceans and this is increasing the moons orbit by 4cm a year. Why doesn't it just precess faster?
You appear to be confusing two different types of perturbation here. The ongoing expansion of the Moon's orbit is caused by a continuous prograde gravitational component from the Earth's tidal bulge. It is transferring energy from Earth's spin to the Moon's orbital motion. It would disappear if the Earth were not spinning. That is separate from the strong perturbations inflicted by the Sun on the Earth/Moon combination. That is the action that causes the rapid prograde precession of the Moon's line of apsides, which is over 40 degrees per year.
This is my point, not well enough made to begin with, If other bodies are adding and subtracting to the Sun's gravitational effect why are the planets orbital lengths rigidly fixed? Why do these additions to the Sun's gravitational pull cause precession and nothing else?
.
The mean distance, orbital period and orbital eccentricity of an orbiting body in the presence of other massive bodies are not rigidly fixed. As a case in point let's go back to our Moon, whose orbit is severely perturbed by the Sun's gravity. The Moon's mean distance from the Earth oscillates over a range of about 1.5% as the Sun's gravity vector revolves with respect to the line of apsides. This is enough to make its orbital period vary by several hours on either side of its mean value, though that mean value remains virtually constant over the long haul. At the same time, the line of apsides, that line connecting the perigee and apogee points, precesses prograde at an average rate around 40 degrees per year, but in a herky-jerky manner. That line does not even go through the barycenter all the time. We could describe the Moon's orbit as wiggling and jiggling like a bowl of jelly. All of this can be found by plotting the monthly movements of the Moon from sources from sources such as Sky and Telescope and the Astronomical Almanac, without doing the calculations.

Jupiter's perturbation of the Sun/Mercury combo are much milder than the Sun's effect on the Earth/Moon combo, but I would expect to see similar patterns on a proportionally much smaller scale, with any oscillation of the orbital distance and period being vanishingly small.
As to addressing the Maths of Euler etc. Please direct me to where I can study this. The confidence that there can be no secular increase seems to me that an assumption has been made resulting in a cancellation of terms. The 3 body problem has yet to be solved but the nine bodies of the solar system interact in such a way that precession occurs but orbital length can only ever remain constant. Why because a brilliant mathematician said so
Make sure you have a solid regimen of college-level calculus under your belt first. I think that was the prerequisite for the celestial mechanics course at the University of Virginia in my college days.

If you still are skeptical about the findings of the aforementioned mathematicians, the burden of proof is on you to make a convincing case against them.

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