[BRrollin]

How do "perturbations...make an underlying model match observation."

As previously discussed, pertubations are calculated on basis of epicycles and are used to make a theory fit observations.

The dynamics of galaxies are also based on epicycles to make a theory fit data:

https://arxiv.org/ftp/arxiv/papers/0911/0911.1594.pdf

Lindblad’s epicycles – valid method or bad science?

In popular culture, epicycles have become almost synonymous with bad science; “adding epicycles” refers to a process of introducing fudges to make a theory fit data, when actually the theory needs to be replaced in its entirety. It is generally believed that epicycles were banished from science when Newton solved his equations of motion and showed that it follows from the inverse square law of gravity that planetary orbits are ellipses. So, it comes as something of a surprise to those unfamiliar with galactic dynamics that the galactic orbits of stars are treated in textbooks using a theory of epicycles revitalized by Bertil Lindblad in the 1920s, and used to introduce density wave theory, which, as reinforced by Lin & Shu (1964), by Lin, Yuan and Shu (1969) and by Kalnajs (1973), has been the leading model of spiral structure for nearly 40 years.

~

Conclusion

The implication to astrophysics is severe. The motions of stars are governed by known mathematical laws. Astrophysics is, or at least it should be, a mathematical science. One should therefore expect that theories in astrophysics are subjected to rigorous mathematical scrutiny. Regrettably, the degree of scrutiny applied to Lindblad’s epicycles and to density wave theory has been seriously lacking. Students should be made aware that these ideas can no longer be considered as science, and authors of textbooks should consider whether they merit anything more than a historical note.

If so, how might FET have done so in this circumstance? How might the underlying model of FET be applied? Do tell.

FE doesn't propose a planetary theory.

FET fails to match heliocentricity.

Interesting that you think that a model with epicycles is valid. However, epicycles do not make a system valid. It is the reason Ptolmy's model was rejected

Interesting find, Tom. Strangely, I found another article by the same author who fits data to current elliptical orbits without appeal to epicycles.

https://arxiv.org/pdf/0901.3503.pdf

It appears reasonable, and empirical.

What do we conclude here, from finding two articles by the same author who seems to simultaneously confirm elliptical Newtonian orbits yet also questions spiral construction by density wave models?

It is interesting.

[GreatATuin]

FE doesn't propose a planetary theory.

I think that's all I needed to know. FE doesn't even try to take into account phenomena that are described, understood and predicted in the heliocentric model.

FET fails to match heliocentricity.

Interesting that you think that a model with epicycles is valid. However, epicycles do not make a system valid. It is the reason Ptolmy's model was rejected

Interesting that you think that repeating the word "epicycle" enough times, especially out of context, will make people think that a model that does work does not actually work. Anyway so far, you've brought absolutely no evidence of epicycles being used to calculate the comet's orbit. It's just your interpretation from completely unrelated papers.

[Tom Bishop]

Interesting find, Tom. Strangely, I found another article by the same author who fits data to current elliptical orbits without appeal to epicycles.

https://arxiv.org/pdf/0901.3503.pdf

It appears reasonable, and empirical.

The author is discussing his finding of a symmetrical Newtonian solution that does not require epicycle theory to exist.

We will describe an alternative mechanism, which does not depend on epicycles, and which also results in spiral structure. We will show that this structure is dynamically stable, and that the observed stream motions are precisely those which the structure predicts.

~

The results of our investigation have lead us to re-examine the hydrogen maps of the Milky Way, from which we identify the possibility of a symmetric two-armed spiral with half the conventionally accepted pitch angle.

~

Appendix C Spiral Galaxy Simulation
http://rqgravity.net/images/spiralmotions/gss.avi.

In this animation using 4 500 stars, each star follows a rosette. This is the form of orbits predicted under Newtonian gravity for mass distributed symmetrically in the galactic plane and in the halo. Rosettes are aligned by mutual gravity between stars. The gravity of the arm causes stars to follow the arm during the ingoing part of their orbit. The simulation uses orbits with random eccentricities between 0.10 and 0.18, corresponding to observations of local stars in the Milky Way. The pattern created is a grand-design two-armed spiral.

The accepted view of galactic dynamics is not this, and requires epicycles, which the author criticizes in the other paper. The author is correct that epicycles are used to make data fit observations and are invalid methods of describing celestial mechanics.

A perfectly symmetrical spiral galaxy might work with Newtonian gravity. As you will recall from the three and n body problems, only symmetrical solutions may exist:
https://wiki.tfes.org/Three_Body_Problem

Ask a Mathematician says: "The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits)."

The sun-earth-moon system is not perfectly symmetric and does not work. Not does the solar system. Moons orbiting around planets which orbit a Sun do not work. Due to the n-body problems dynamical prediction is unavailable. The addition of epicycles are required to fit theory to observation.

Interesting that you think that repeating the word "epicycle" enough times, especially out of context, will make people think that a model that does work does not actually work. Anyway so far, you've brought absolutely no evidence of epicycles being used to calculate the comet's orbit. It's just your interpretation from completely unrelated papers.

https://archive.org/download/the-foundations-of-astrodynamics/The%20Foundations%20of%20Astrodynamics.pdf

Dr. Samuel Herrick (1911-1977) says the same as the previous authors, explaining that epicycles are still used in modern astrodynamics:

Physical celestial mechanics may be said to have begun with Galileo Galilei, Isaac Newton, and the laws of force and gravitation. Astrodynamics and mathematical celestial mechanics, on the other hand, date back at least to Heracleides of Pontus in the fourth century B.C. The Greek invention of epicycles and eccentrics was developed into a system by Apollonius of Perga in the third century and Hipparchus of Alexandria in the second century B.C. It was refined and published by Ptolemy of Alexandria in the second century A.D., and came to be known as the Ptolemaic system. It is generally assumed that the epicycle was discredited by Johannes Kepler some 1500 years later, but in point of fact epicycles have persisted in astrodynamics down to the present day, and have extended their domain into other areas of science under the guise of Fourier series! ”

~

  “ In modern perturbation theory we actually take account of the original epicyclic concept by combining several Fourier series that have arguments based upon different angular variables. ”

Again, another source explaining to us that Modern Perturbation Theory = Epicycles

They never stopped using them. They only called them something else once the term became discredited.

I think that's all I needed to know. FE doesn't even try to take into account phenomena that are described, understood and predicted in the heliocentric model.

Except for retrograde motion FE has yet to explore or discuss the dynamics of the planets. Your presentation of a model based upon epicycles is hardly proof of it. Epicycles are a byword for bad science and a workaround for a lousy theory.

[GreatATuin]

You can go on forever cherrypicking out of context citations in an attempt to associate the ancient concept of epicycles with current astronomy and call it a pseudoscience, but you'll never convince anyone.

Whether you like it or not, astronomical predictions work. The model they're based on works.

I think that's all I needed to know. FE doesn't even try to take into account phenomena that are described, understood and predicted in the heliocentric model.

Except for retrograde motion FE has yet to explore or discuss the dynamics of the planets. Your presentation of a model based upon epicycles is hardly proof of it. Epicycles are a byword for bad science and a workaround for a non-working theory.

FE understandably has very little interest exploring things that don't fit any flat Earth model and are easily explained in the heliocentric model. And you can repeat the word "epicycles" as many times as you want, it won't make your point any more valid. The heliocentric model is not based on epicycles.

Anyway, as long as you refuse to acknowledge that Newton's equations can be used to build a working simulation with approximations through numerical methods, even if the n-body problem doesn't have a formal closed-form solution, I don't even know if there's any point in answering anything you say.

[Tom Bishop]

Newton used epicycles for the Moon:

History of the Inductive Sciences (1846)
By William Whewell, Historian of Science

3.— The Epicyclical Hypothesis was found capable of accommodating itself to such new discoveries. These new inequalities could be represented by new combinations of eccentrics and epicycles: all the realand imaginary discoveries by astronomers, up to Copernicus, were actually embodied in these hypotheses; Copernicus, as we have said, did not reject such hypotheses; the lunar inequalities which Tycho etected might have boen similarly exhibited; and even Newton³⁶ represents the motion of the moon’s apogee by means of an epicycle. As a mode of expressing the law of the irregularity, and of calculating its results in particular cases, the epicyclical theory was capable of continuing to render great service to astronomy, however extensive the progress of the science might be. It was, in fact, as we have already said, the modern process of representing the motion by means of a series of circular functions.

Epicycles were all the rage in the 1800's, hundreds of years after Copernicus, Kepler, and Newton made their contributions to astronomy. Professor De Morgan is cited to explain the state of astronomy in University of Toronto Quarterly (1895):

Of the modern employment of the Ptolemaic epicycles, De Morgan, secretary of the Royal Astronomical Society of London, wrote in 1844: “ The common notion is that the theory of epicycles was a cumbrous and useless apparatus, thrown away by the moderns and originating in the Ptolemaic or rather Platonic notion that all celestial motions must either be circular and uniform motions or compounded of them. But, on the contrary, it was an elegant and most efficient mathematical instrument which enabled Hipparchus and Ptolemy to represent and predict much better than their predecessors had done; and it was probably at least as good a theory as their instruments and capabilities of observation required or deserved. And many readers will be surprised to hear that the modern astronomer to this day resolves the same motions into epicyelic ones. When the latter expresses a result by series of sines and cosines (especially when the angle is a mean motion or a multiple of it) he uses epicycles; and for one which Ptolemy scribbled on the, heavens, to use Milton’s phrase, he scribbles twenty. The difference is that the ancient believed in the necessity of these instruments, the modern only in their convenience; the former used those which do not sufficiently represent actual phenomena, the latter knows how to choose better; the former taking the instruments to be the actual contrivances of nature was obliged to make one set explain everything, the latter will adapt one set to latitude, another to longitude, another to distance. Difference enough no doubt, but not the sort of difference which the common notion supposes.”

Such was the state of affairs fifty years ago; today epicycles may be said to possess the heavens above and the earth beneath and the waters and the air between, nor has the all-pervading ether escaped them. In analytic guise they dominate the mathematics of hydrokinetics and sound, of heat, light and electricity; in fact, wherever there is either periodic or irregular motion, there the mathematician “ scribbles ” his epicycles, and not content like Ptolemy to wheel them on simple circles he rolls epicycle on epicycle to the third, the fourth or the fifth degree. Nor does their influence end here. Machines have been made to record for a sufficient length of time any motions for the character of which a working theory has to be found; other machines analyse the records into epicyclic movements, smoothing out or rejecting accidental irregularities, and still other machines recombine the epicycles to predict the motions as they will occur at a future time or under given changes of condition. Thus we have mechanical tide-predictors, harmonic analyzers of meteorological phenomena, epicyclic tracers of deviation curves for the compasses in iron ships, and a fast increasing array of other such machines.

Some perturbation terms are today still named after the ancient epicycles. Here a reference from modern day, on computing the planets:

How to compute planetary positions
By Paul Schlyter, Stockholm, Sweden

All perturbation terms that are smaller than 0.01 degrees in longitude or latitude and smaller than 0.1 Earth radii in distance have been omitted here. A few of the largest perturbation terms even have their own names! The Evection (the largest perturbation) was discovered already by Ptolemy a few thousand years ago (the Evection was one of Ptolemy's epicycles). The Variation and the Yearly Equation were both discovered by Tycho Brahe in the 16'th century.

To calculate the celestial bodies we are told to use perturbation terms with the names of Ptolmy's ancient epicycles!

[BRrollin]

Interesting find, Tom. Strangely, I found another article by the same author who fits data to current elliptical orbits without appeal to epicycles.

https://arxiv.org/pdf/0901.3503.pdf

It appears reasonable, and empirical.

The author is discussing his finding of a symmetrical Newtonian solution that does not require epicycle theory to exist.

We will describe an alternative mechanism, which does not depend on epicycles, and which also results in spiral structure. We will show that this structure is dynamically stable, and that the observed stream motions are precisely those which the structure predicts.

~

The results of our investigation have lead us to re-examine the hydrogen maps of the Milky Way, from which we identify the possibility of a symmetric two-armed spiral with half the conventionally accepted pitch angle.

~

Appendix C Spiral Galaxy Simulation
http://rqgravity.net/images/spiralmotions/gss.avi.

In this animation using 4 500 stars, each star follows a rosette. This is the form of orbits predicted under Newtonian gravity for mass distributed symmetrically in the galactic plane and in the halo. Rosettes are aligned by mutual gravity between stars. The gravity of the arm causes stars to follow the arm during the ingoing part of their orbit. The simulation uses orbits with random eccentricities between 0.10 and 0.18, corresponding to observations of local stars in the Milky Way. The pattern created is a grand-design two-armed spiral.

The accepted view of galactic dynamics is not this, and requires epicycles, which the author criticizes in the other paper. The author is correct that epicycles are used to make data fit observations and are invalid methods of describing celestial mechanics.

A perfectly symmetrical spiral galaxy might work with Newtonian gravity. As you will recall from the three and n body problems, only symmetrical solutions may exist:
https://wiki.tfes.org/Three_Body_Problem

Ask a Mathematician says: "The problem with the 3-body problem is that it can’t be done, except in a very small set of frankly goofy scenarios (like identical planets following identical orbits)."

The sun-earth-moon system is not perfectly symmetric and does not work. Not does the solar system. Moons orbiting around planets which orbit a Sun do not work. Due to the n-body problems dynamical prediction is unavailable. The addition of epicycles are required to fit theory to observation.

Interesting that you think that repeating the word "epicycle" enough times, especially out of context, will make people think that a model that does work does not actually work. Anyway so far, you've brought absolutely no evidence of epicycles being used to calculate the comet's orbit. It's just your interpretation from completely unrelated papers.

https://archive.org/download/the-foundations-of-astrodynamics/The%20Foundations%20of%20Astrodynamics.pdf

Dr. Samuel Herrick (1911-1977) says the same as the previous authors, explaining that epicycles are still used in modern astrodynamics:

Physical celestial mechanics may be said to have begun with Galileo Galilei, Isaac Newton, and the laws of force and gravitation. Astrodynamics and mathematical celestial mechanics, on the other hand, date back at least to Heracleides of Pontus in the fourth century B.C. The Greek invention of epicycles and eccentrics was developed into a system by Apollonius of Perga in the third century and Hipparchus of Alexandria in the second century B.C. It was refined and published by Ptolemy of Alexandria in the second century A.D., and came to be known as the Ptolemaic system. It is generally assumed that the epicycle was discredited by Johannes Kepler some 1500 years later, but in point of fact epicycles have persisted in astrodynamics down to the present day, and have extended their domain into other areas of science under the guise of Fourier series! ”

~

  “ In modern perturbation theory we actually take account of the original epicyclic concept by combining several Fourier series that have arguments based upon different angular variables. ”

Again, another source explaining to us that Modern Perturbation Theory = Epicycles

They never stopped using them. They only called them something else once the term became discredited.

I think that's all I needed to know. FE doesn't even try to take into account phenomena that are described, understood and predicted in the heliocentric model.

Except for retrograde motion FE has yet to explore or discuss the dynamics of the planets. Your presentation of a model based upon epicycles is hardly proof of it. Epicycles are a byword for bad science and a workaround for a lousy theory.

I agree with some of that. But a symmetric solution permits symmetric solutions using perturbation analysis. So if a symmetric solution is found, and the perturbative expansions are identified, the the asymmetric extensions are defined.

[GreatATuin]

Epicycles were all the rage, hundreds of years after Copernicus, Kepler, and Newton made their contributions to astronomy. From University of Toronto Quarterly (1895):

Of the modern employment of the Ptolemaic epicycles, De Morgan, secretary of the Royal Astronomical Society of London, wrote in 1844: “ The common notion is that the theory of epicycles was a cumbrous and useless apparatus, thrown away by the moderns and originating in the Ptolemaic or rather Platonic notion that all celestial motions must either be circular and uniform motions or compounded of them. But, on the contrary, it was an elegant and most efficient mathematical instrument which enabled Hipparchus and Ptolemy to represent and predict much better than their predecessors had done; and it was probably at least as good a theory as their instruments and capabilities of observation required or deserved. And many readers will be surprised to hear that the modern astronomer to this day resolves the same motions into epicyelic ones. When the latter expresses a result by series of sines and cosines (especially when the angle is a mean motion or a multiple of it) he uses epicycles; and for one which Ptolemy scribbled on the, heavens, to use Milton’s phrase, he scribbles twenty. The difference is that the ancient believed in the necessity of these instruments, the modern only in their convenience; the former used those which do not sufficiently represent actual phenomena, the latter knows how to choose better; the former taking the instruments to be the actual contrivances of nature was obliged to make one set explain everything, the latter will adapt one set to latitude, another to longitude, another to distance. Difference enough no doubt, but not the sort of difference which the common notion supposes.”

Such was the state of affairs fifty years ago; today epicycles may be said to possess the heavens above and the earth beneath and the waters and the air between, nor has the all-pervading ether escaped them. In analytic guise they dominate the mathematics of hydrokinetics and sound, of heat, light and electricity; in fact, wherever there is either periodic or irregular motion, there the mathematician “ scribbles ” his epicycles, and not content like Ptolemy to wheel them on simple circles he rolls epicycle on epicycle to the third, the fourth or the fifth degree. Nor does their influence end here. Machines have been made to record for a sufficient length of time any motions for the character of which a working theory has to be found; other machines analyse the records into epicyclic movements, smoothing out or rejecting accidental irregularities, and still other machines recombine the epicycles to predict the motions as they will occur at a future time or under given changes of condition. Thus we have mechanical tide-predictors, harmonic analyzers of meteorological phenomena, epicyclic tracers of deviation curves for the compasses in iron ships, and a fast increasing array of other such machines.

It looks like you've missed an important part, I tried to help you figure it out.

For Ptolemy and Hipparchus, epicycles were the theory.

After Kepler and Newton, epicycles were, at best, mathematical tools used to find solutions that fit the new theory, especially before the computer was invented and better tools were available. It strikes me that some of your references are not from the last century, but from the previous one.

Some perturbation terms are today still named after the ancient epicycles. Here a reference from modern day, on computing the planets:

All perturbation terms that are smaller than 0.01 degrees in longitude or latitude and smaller than 0.1 Earth radii in distance have been omitted here. A few of the largest perturbation terms even have their own names! The Evection (the largest perturbation) was discovered already by Ptolemy a few thousand years ago (the Evection was one of Ptolemy's epicycles). The Variation and the Yearly Equation were both discovered by Tycho Brahe in the 16'th century.

To calculate the celestial bodies we are told to use perturbation terms with the names of Ptolmy's ancient epicycles!

The Lunar orbit has variations, or perturbations. Some of them were known to Ptolemy, who accounted for them with the only tool he had, epicycles, but any model will have to do it one way or another to be accurate. Giving the perturbation a new name, or an old one, or no name at all doesn't change anything. By the way, according to https://en.wikipedia.org/wiki/Evection, the name was coined by Bullialdus in the 17th century, which makes an already weak point totally moot.

[BRrollin]

How do "perturbations...make an underlying model match observation."

As previously discussed, pertubations are calculated on basis of epicycles and are used to make a theory fit observations.

The dynamics of galaxies are also based on epicycles to make a theory fit data:

https://arxiv.org/ftp/arxiv/papers/0911/0911.1594.pdf

Lindblad’s epicycles – valid method or bad science?

In popular culture, epicycles have become almost synonymous with bad science; “adding epicycles” refers to a process of introducing fudges to make a theory fit data, when actually the theory needs to be replaced in its entirety. It is generally believed that epicycles were banished from science when Newton solved his equations of motion and showed that it follows from the inverse square law of gravity that planetary orbits are ellipses. So, it comes as something of a surprise to those unfamiliar with galactic dynamics that the galactic orbits of stars are treated in textbooks using a theory of epicycles revitalized by Bertil Lindblad in the 1920s, and used to introduce density wave theory, which, as reinforced by Lin & Shu (1964), by Lin, Yuan and Shu (1969) and by Kalnajs (1973), has been the leading model of spiral structure for nearly 40 years.

~

Conclusion

The implication to astrophysics is severe. The motions of stars are governed by known mathematical laws. Astrophysics is, or at least it should be, a mathematical science. One should therefore expect that theories in astrophysics are subjected to rigorous mathematical scrutiny. Regrettably, the degree of scrutiny applied to Lindblad’s epicycles and to density wave theory has been seriously lacking. Students should be made aware that these ideas can no longer be considered as science, and authors of textbooks should consider whether they merit anything more than a historical note.

If so, how might FET have done so in this circumstance? How might the underlying model of FET be applied? Do tell.

FE doesn't propose a planetary theory.

FET fails to match heliocentricity.

Interesting that you think that a model with epicycles is valid. However, epicycles do not make a system valid. It is the reason Ptolmy's model was rejected

Ya know, it just occurred to me that Francis fails to incorporate the spiral arm magnetic fields in his discussions. This seems severely problematic, as it has long been known that B fields are integral to studying these dynamical systems. For example:

https://www.aanda.org/articles/aa/full_html/2018/01/aa29988-16/aa29988-16.html

But there are many, many more examples, which I can provide upon request, but omit until then so as not to spam readables.

[Tom Bishop]

The difference is that the ancient believed in the necessity of these instruments, the modern only in their convenience; the former used those which do not sufficiently represent actual phenomena, the latter knows how to choose better; the former taking the instruments to be the actual contrivances of nature was obliged to make one set explain everything, the latter will adapt one set to latitude, another to longitude, another to distance. Difference enough no doubt, but not the sort of difference which the common notion supposes.

It looks like you've missed an important part, I tried to help you figure it out.

So you think that astronomers are using epicycles, but that's not a problem, because they are only using them "it's convenient"?

That's just an excuse. "We only do it because it's convenient" and "We already know that Newton's system is true (despite that he used epicycles in practice)" are  fairytales. It is admitted that there is no dynamical model of the solar system.

Perturbation Theory for Restricted Three-Body Orbits
1991 Thesis by David A. Ross

I. Introduction

Before the astrodynamics of man-made objects in space can be fully understood, one must first comprehend basic planetary motion. Thanks to Sir Isaac Newton and his three laws of motion, and to Johann Kepler for his three laws of orbital motion, it can be shown that nearly all astrodynamical systems are dominated by a single conservative force known as gravity. In fact, the most general description of the motion of a collection of objects in space is defined by the n-body problem.

In an n-body system, the nth body is acted upon by the other n-1 gravitational masses present. In this way, the motion of any mass in a system affects and is affected by every other mass in the system. The overwhelming task of representing each body is well illustrated by Wiesel.

"Our own solar system consists of one star, nine planets, over fifty moons, tens of thousands of asteroids, and millions of comets. The description of the motion of this system is clearly important, but an exact solution to this problem has not been found in over three hundred years of study." (8:33)

Therefore, the use of the exact n-body description of a dynamical system is not simply a nuisance, it is virtually impossible to implement.

The paper goes on to talk about perturbation theory and fourier methods.

There you have it. Ross quotes Dr. William Weisel who admits that there is no dynamical gravity model. The dynamical way is not simply a nuisance—it is impossible and not even attempted. They are using a workaround.

[BRrollin]

The difference is that the ancient believed in the necessity of these instruments, the modern only in their convenience; the former used those which do not sufficiently represent actual phenomena, the latter knows how to choose better; the former taking the instruments to be the actual contrivances of nature was obliged to make one set explain everything, the latter will adapt one set to latitude, another to longitude, another to distance. Difference enough no doubt, but not the sort of difference which the common notion supposes.

It looks like you've missed an important part, I tried to help you figure it out.

So you think that astronomers are using epicycles, but that's not a problem, because they are only using them "it's convenient"?

That's just an excuse. "We only do it because it's convenient" and "We already know that Newton's system is true (despite that he used epicycles in practice)" are  fairytales. It is admitted that there is no dynamical model of the solar system.

Perturbation Theory for Restricted Three-Body Orbits
1991 Thesis by David A. Ross

I. Introduction

Before the astrodynamics of man-made objects in space can be fully understood, one must first comprehend basic planetary motion. Thanks to Sir Isaac Newton and his three laws of motion, and to Johann Kepler for his three laws of orbital motion, it can be shown that nearly all astrodynamical systems are dominated by a single conservative force known as gravity. In fact, the most general description of the motion of a collection of objects in space is defined by the n-body problem.

In an n-body system, the nth body is acted upon by the other n-1 gravitational masses present. In this way, the motion of any mass in a system affects and is affected by every other mass in the system. The overwhelming task of representing each body is well illustrated by Wiesel.

"Our own solar system consists of one star, nine planets, over fifty moons, tens of thousands of asteroids, and millions of comets. The description of the motion of this system is clearly important, but an exact solution to this problem has not been found in over three hundred years of study." (8:33)

Therefore, the use of the exact n-body description of a dynamical system is not simply a nuisance, it is virtually impossible to implement.

The paper goes on to talk about perturbation theory and fourier methods.

There you have it. Ross quotes Dr. William Weisel who admits that there is no dynamical gravity model. The dynamical way is not simply a nuisance—it is impossible and not even attempted. They are using a workaround.

My concern, Tom, is that you tend to link such esoteric works that lack follow-up by the scientific community. If published, they are never referenced much nor duplicated. It’s...odd.

Here’s a recent paper that removes any doubt that celestial mechanics must appeal to epicycles.

https://www.sciencedirect.com/science/article/abs/pii/S1384107618303336?via%3Dihub.

[Tom Bishop]

You posted a paper with the title "Central-body square configuration of restricted six-body problem".

Have you seen what this problem actually looks like? The available solutions to the n-body problems are inherently symmetrical, unlike real astronomical systems like the supposed Sun-Earth-Moon system.

N Bodies
Caltech Physicist Sean Carroll

This will be familiar to anyone who reads John Baez’s This Week’s Finds in Mathematical Physics, but I can’t help but show these lovely exact solutions to the gravitational N-body problem. This one is beautiful in its simplicity: twenty-one point masses moving around in a figure-8.



The N-body problem is one of the most famous, and easily stated, problems in mathematical physics: find exact solutions to point masses moving under their mutual Newtonian gravitational forces (i.e. the inverse-square law). For N=2 the complete set of solutions is straightforward and has been known for a long time — each body moves in a conic section (circle, ellipse, parabola or hyperbola) around the center of mass. In fact, Kepler found the solution even before Newton came up with the problem!

But let N=3 and chaos breaks loose, quite literally. For a long time people recognized that the motion of three gravitating bodies would be a difficult problem, but there were hopes to at least characterize the kinds of solutions that might exist (even if we couldn’t write down the solutions explicitly). It became a celebrated goal for mathematical physicists, and the very amusing story behind how it was resolved is related in Peter Galison’s book Einstein’s Clocks and Poincare’s Maps. In 1885, a mathematical competition was announced in honor of the 60th birthday of King Oscar II of Sweden, and the three-body problem was one of the questions. (Feel free to muse about the likelihood of the birthday of any contemporary world leader being celebrated by mathematical competitions.) Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. Gosta Mittag-Leffler, chief editor, forwarded Phragmen’s questions to Poincare, asking him to fix up these nagging issues before the prize essay appeared in print. Poincare went to work, but discovered to his consternation that one of the tiny problems was in fact a profoundly devastating possibility that he hadn’t really taken seriously. What he ended up proving was the opposite of his original claim — three-body orbits were not stable at all. Not only were orbits not periodic, they didn’t even approach some sort of asymptotic fixed points. Now that we have computers to run simulations, this kind of behavior is less surprising (example here from Steve McMillan — note how the final “binary” is not made of the same “stars” as the original one), but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincare ended up inventing chaos theory.

But the story doesn’t quite end there. Mittag-Leffler, convinced that Poincare would be able to tie up the loose threads in his prize essay, went ahead and printed it. By the time he heard from Poincare that no such tying-up would be forthcoming, the journal had already been mailed to mathematicians throughout Europe. Mittag-Leffler swung into action, telegraphing Berlin and Paris in an attempt to have all copies of the journal destroyed. He basically succeeded, but not without creating a minor scandal in elite mathematical circles across the Continent. (The Wikipedia entry on Poincare tells a much less interesting, and less accurate, version of the story.)

However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing. The image at the top really is an exact solution to twenty-one equal-mass objects moving in a figure-eight under their mutual gravitational attraction. They’re moving in a plane, of course, but that’s not strictly necessary; here’s a close relative of the figure-8, perturbed outside the plane.



From there you can just go nuts; here’s an example with twelve objects orbiting with cubic symmetry — four distinct periodic paths with three particles each.



Knowledge of this exact solution, plus $3.50, will get you a grande latte at Starbucks. Mathematicians have all the fun.

[BRrollin]

You posted a paper with the title "Central-body square configuration of restricted six-body problem".

Have you seen what this problem actually looks like? The available solutions to the n-body problems are inherently symmetrical, unlike real astronomical systems like the supposed Sun-Earth-Moon system.

N Bodies
Caltech Physicist Sean Carroll

This will be familiar to anyone who reads John Baez’s This Week’s Finds in Mathematical Physics, but I can’t help but show these lovely exact solutions to the gravitational N-body problem. This one is beautiful in its simplicity: twenty-one point masses moving around in a figure-8.



The N-body problem is one of the most famous, and easily stated, problems in mathematical physics: find exact solutions to point masses moving under their mutual Newtonian gravitational forces (i.e. the inverse-square law). For N=2 the complete set of solutions is straightforward and has been known for a long time — each body moves in a conic section (circle, ellipse, parabola or hyperbola) around the center of mass. In fact, Kepler found the solution even before Newton came up with the problem!

But let N=3 and chaos breaks loose, quite literally. For a long time people recognized that the motion of three gravitating bodies would be a difficult problem, but there were hopes to at least characterize the kinds of solutions that might exist (even if we couldn’t write down the solutions explicitly). It became a celebrated goal for mathematical physicists, and the very amusing story behind how it was resolved is related in Peter Galison’s book Einstein’s Clocks and Poincare’s Maps. In 1885, a mathematical competition was announced in honor of the 60th birthday of King Oscar II of Sweden, and the three-body problem was one of the questions. (Feel free to muse about the likelihood of the birthday of any contemporary world leader being celebrated by mathematical competitions.) Henri Poincare was a favorite to win the prize, and he submitted an essay that demonstrated the stability of planetary motions in the three-body problem (actually the “restricted” problem, in which one test body moves in the gravitational field generated by two others). In other words, without knowing the exact solutions, we could at least be confident that the orbits wouldn’t go crazy; more technically, solutions starting with very similar initial conditions would give very similar orbits. Poincare’s work was hailed as brilliant, and he was awarded the prize.

But as his essay was being prepared for publication in Acta Mathematica, a couple of tiny problems were pointed out by Edvard Phragmen, a Swedish mathematician who was an assistant editor at the journal. Gosta Mittag-Leffler, chief editor, forwarded Phragmen’s questions to Poincare, asking him to fix up these nagging issues before the prize essay appeared in print. Poincare went to work, but discovered to his consternation that one of the tiny problems was in fact a profoundly devastating possibility that he hadn’t really taken seriously. What he ended up proving was the opposite of his original claim — three-body orbits were not stable at all. Not only were orbits not periodic, they didn’t even approach some sort of asymptotic fixed points. Now that we have computers to run simulations, this kind of behavior is less surprising (example here from Steve McMillan — note how the final “binary” is not made of the same “stars” as the original one), but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincare ended up inventing chaos theory.

But the story doesn’t quite end there. Mittag-Leffler, convinced that Poincare would be able to tie up the loose threads in his prize essay, went ahead and printed it. By the time he heard from Poincare that no such tying-up would be forthcoming, the journal had already been mailed to mathematicians throughout Europe. Mittag-Leffler swung into action, telegraphing Berlin and Paris in an attempt to have all copies of the journal destroyed. He basically succeeded, but not without creating a minor scandal in elite mathematical circles across the Continent. (The Wikipedia entry on Poincare tells a much less interesting, and less accurate, version of the story.)

However, just because the general solution to the three-body (and more-body) problem is chaotic, doesn’t mean we can’t find special exact solutions in highly-symmetric conditions, and that’s just what Cris Moore and Michael Nauenberg have recently been doing. The image at the top really is an exact solution to twenty-one equal-mass objects moving in a figure-eight under their mutual gravitational attraction. They’re moving in a plane, of course, but that’s not strictly necessary; here’s a close relative of the figure-8, perturbed outside the plane.



From there you can just go nuts; here’s an example with twelve objects orbiting with cubic symmetry — four distinct periodic paths with three particles each.



Knowledge of this exact solution, plus $3.50, will get you a grande latte at Starbucks. Mathematicians have all the fun.

Oh sure, I’ve seen the problem. Hell - I’ve done it!

There are plenty of symmetric solutions, crazy unstable solutions, beautiful orbits that trace out flower patterns. So many solutions have been computed.

Give the article a try

I think it would be well worth it for you. At the very least, it will update you on some of the current developments, computations.

In this paper, 6 bodies have their stable liberation points computed - not a epicycle in sight!

[Tom Bishop]

Yep. Those are the orbits Newton's gravity produces. And here are over a thousand more solutions, as described by New Scientist. Read carefully:

Infamous three-body problem has over a thousand new solutions - New Scientist

For more than 300 years, mathematicians have puzzled over the three-body problem – the question of how three objects orbit one another according to Newton’s laws. Now, there are 1223 new solutions to the conundrum, more than doubling the current number of possibilities.

No single equation can predict how three bodies will move in relation to one another and whether their orbits will repeat or devolve into chaos. Mathematicians must test each specific scenario to see if the objects will stay bound in orbit or be flung away.

The new solutions were found when researchers at Shanghai Jiaotong University in China tested 16 million different orbits using a supercomputer.

All the fresh orbits found are periodic. This means that each object, whether it’s a planet or a proton, ends up where it first began its orbit, with their paths forming three intertwined, closed loops.

“It is impressive that they’ve made the list a lot longer,” says Robert Vanderbei at Princeton University in New Jersey – though he adds that there is “basically an unlimited number of orbits”, so it may be overkill if anyone sought to find them all.

Perhaps the most important application of the three-body problem is in astronomy, for helping researchers figure out how three stars, a star with a planet that has a moon, or any other set of three celestial objects can maintain a stable orbit.

But these new orbits rely on conditions that are somewhere between unlikely and impossible for a real system to satisfy. In all of them, for example, two of the three bodies have exactly the same mass and they all remain in the same plane.

Knot-like paths

In addition, the researchers did not test the orbits’ stability. It’s possible that the tiniest disturbance in space or rounding error in the equations could rip the objects away from one another.

“These orbits have nothing to do with astronomy, but you’re solving these equations and you’re getting something beautiful,” says Vanderbei.

Aside from giving us a thousand pretty pictures of knot-like orbital paths, the new three-body solutions also mark a starting point for finding even more possible orbits, and eventually figuring out the whole range of winding paths that three objects can follow around one another.

“This is kind of the zeroth step. Then the question becomes, how is the space of all possible positions and velocities filled up by solutions?” says Richard Montgomery at the University of California, Santa Cruz. “These simple orbits are kind of like a skeleton to build the whole system up from.”

As suggested by the above article, the field of Celestial Mechanics is still on step zero—the stone age. The found orbits are nothing like heliocentric astronomy and there will be an attempt to use them as a skeleton to "build the whole system up from."

From the linked source paper at the bottom of the article:



All highly symmetric orbits, as Dr. Carroll described.

Symmetrical orbits with two of three masses being identical which "have nothing to do with astronomy" and which "are somewhere between unlikely and impossible for a real system to satisfy."

The phys.org article Scientists discover more than 600 new periodic orbits of the famous three-body problem describes the discovery of other orbits:

"These 695 periodic orbits include the well-known figure-eight family found by Moore in 1993, the 11 families found by Suvakov and Dmitrasinovic in 2013, and more than 600 new families reported for the first time."

Again, highly symmetrical orbits.

[BRrollin]

Yep. Those are the orbits Newton's gravity produces. And here are over a thousand more solutions, as described by New Scientist. Read carefully:

Infamous three-body problem has over a thousand new solutions - New Scientist

For more than 300 years, mathematicians have puzzled over the three-body problem – the question of how three objects orbit one another according to Newton’s laws. Now, there are 1223 new solutions to the conundrum, more than doubling the current number of possibilities.

No single equation can predict how three bodies will move in relation to one another and whether their orbits will repeat or devolve into chaos. Mathematicians must test each specific scenario to see if the objects will stay bound in orbit or be flung away.

The new solutions were found when researchers at Shanghai Jiaotong University in China tested 16 million different orbits using a supercomputer.

All the fresh orbits found are periodic. This means that each object, whether it’s a planet or a proton, ends up where it first began its orbit, with their paths forming three intertwined, closed loops.

“It is impressive that they’ve made the list a lot longer,” says Robert Vanderbei at Princeton University in New Jersey – though he adds that there is “basically an unlimited number of orbits”, so it may be overkill if anyone sought to find them all.

Perhaps the most important application of the three-body problem is in astronomy, for helping researchers figure out how three stars, a star with a planet that has a moon, or any other set of three celestial objects can maintain a stable orbit.

But these new orbits rely on conditions that are somewhere between unlikely and impossible for a real system to satisfy. In all of them, for example, two of the three bodies have exactly the same mass and they all remain in the same plane.

Knot-like paths

In addition, the researchers did not test the orbits’ stability. It’s possible that the tiniest disturbance in space or rounding error in the equations could rip the objects away from one another.

“These orbits have nothing to do with astronomy, but you’re solving these equations and you’re getting something beautiful,” says Vanderbei.

Aside from giving us a thousand pretty pictures of knot-like orbital paths, the new three-body solutions also mark a starting point for finding even more possible orbits, and eventually figuring out the whole range of winding paths that three objects can follow around one another.

“This is kind of the zeroth step. Then the question becomes, how is the space of all possible positions and velocities filled up by solutions?” says Richard Montgomery at the University of California, Santa Cruz. “These simple orbits are kind of like a skeleton to build the whole system up from.”

As suggested by the above article, the field of Celestial Mechanics is still on step zero—the stone age. The found orbits are nothing like heliocentric astronomy and there will be an attempt to use them as a skeleton to "build the whole system up from."

From the linked source paper at the bottom of the article:



All highly symmetric orbits, as Dr. Carroll described.

Symmetrical orbits with two of three masses being identical which "have nothing to do with astronomy" and which "are somewhere between unlikely and impossible for a real system to satisfy."

The phys.org article Scientists discover more than 600 new periodic orbits of the famous three-body problem describes the discovery of other orbits:

"These 695 periodic orbits include the well-known figure-eight family found by Moore in 1993, the 11 families found by Suvakov and Dmitrasinovic in 2013, and more than 600 new families reported for the first time."

Again, highly symmetrical orbits.

Oh! No no no, I see what’s happened. It’s not that it’s in “the Stone Age,” it’s just that the solutions are sensitive to initial conditions. This is why you get so many solutions. The equations can tell you how ANY three bodies will behave, and there isn’t (/edit meant to write: IS) literally millions of ways that can happen, depending on how the system started out.

This is indeed fascinating. I suppose they hoped that nature would be nice and only permit a few. But that is not what happened.

Hence, they are able to model any 3 body system across all times with the initial conditions known.

Knowing the initial conditions of the solar system isn’t too easy

But this is not the fault of the THEORY. It’s just our own ignorance of things a long time ago, and the misfortune of having systems that can behave quite different if you get the initial conditions even a bit wrong.

Have you studied chaotic systems? It is a fascinating mathematical field. It doesn’t mean that the system is poorly defined, or unknown. It just means a high sensitivity to initial conditions.

[stack]

As suggested by the above article, the field of Celestial Mechanics is still on step zero—the stone age. The found orbits are nothing like heliocentric astronomy and there will be an attempt to use them as a skeleton to "build the whole system up from."

'Stone Age' astrophysics predicted the when and where of the collision for each of the fragments. The operative word is 'where'. You said that FET has no theories about Jupiter. That's fine. But that means FET could never have predicted the 'where' the collisions occurred on Jupiter. FET would need to know the size of Jupiter, not to mention calculating it's gravitational pull among other things.

You can go on and on about the N-Body problem as you always seem to default to, but you can't predict the 'where'. Heliocentric astrophysics can and did. That puts a solid point in the Helio win column.

[BRrollin]

Yep. Those are the orbits Newton's gravity produces. And here are over a thousand more solutions, as described by New Scientist. Read carefully:

Infamous three-body problem has over a thousand new solutions - New Scientist

For more than 300 years, mathematicians have puzzled over the three-body problem – the question of how three objects orbit one another according to Newton’s laws. Now, there are 1223 new solutions to the conundrum, more than doubling the current number of possibilities.

No single equation can predict how three bodies will move in relation to one another and whether their orbits will repeat or devolve into chaos. Mathematicians must test each specific scenario to see if the objects will stay bound in orbit or be flung away.

The new solutions were found when researchers at Shanghai Jiaotong University in China tested 16 million different orbits using a supercomputer.

All the fresh orbits found are periodic. This means that each object, whether it’s a planet or a proton, ends up where it first began its orbit, with their paths forming three intertwined, closed loops.

“It is impressive that they’ve made the list a lot longer,” says Robert Vanderbei at Princeton University in New Jersey – though he adds that there is “basically an unlimited number of orbits”, so it may be overkill if anyone sought to find them all.

Perhaps the most important application of the three-body problem is in astronomy, for helping researchers figure out how three stars, a star with a planet that has a moon, or any other set of three celestial objects can maintain a stable orbit.

But these new orbits rely on conditions that are somewhere between unlikely and impossible for a real system to satisfy. In all of them, for example, two of the three bodies have exactly the same mass and they all remain in the same plane.

Knot-like paths

In addition, the researchers did not test the orbits’ stability. It’s possible that the tiniest disturbance in space or rounding error in the equations could rip the objects away from one another.

“These orbits have nothing to do with astronomy, but you’re solving these equations and you’re getting something beautiful,” says Vanderbei.

Aside from giving us a thousand pretty pictures of knot-like orbital paths, the new three-body solutions also mark a starting point for finding even more possible orbits, and eventually figuring out the whole range of winding paths that three objects can follow around one another.

“This is kind of the zeroth step. Then the question becomes, how is the space of all possible positions and velocities filled up by solutions?” says Richard Montgomery at the University of California, Santa Cruz. “These simple orbits are kind of like a skeleton to build the whole system up from.”

As suggested by the above article, the field of Celestial Mechanics is still on step zero—the stone age. The found orbits are nothing like heliocentric astronomy and there will be an attempt to use them as a skeleton to "build the whole system up from."

From the linked source paper at the bottom of the article:



All highly symmetric orbits, as Dr. Carroll described.

Symmetrical orbits with two of three masses being identical which "have nothing to do with astronomy" and which "are somewhere between unlikely and impossible for a real system to satisfy."

The phys.org article Scientists discover more than 600 new periodic orbits of the famous three-body problem describes the discovery of other orbits:

"These 695 periodic orbits include the well-known figure-eight family found by Moore in 1993, the 11 families found by Suvakov and Dmitrasinovic in 2013, and more than 600 new families reported for the first time."

Again, highly symmetrical orbits.

I’d like to draw attention to a few things. First the article:

“This is kind of the zeroth step. Then the question becomes, how is the space of all possible positions and velocities filled up by solutions?” says Richard Montgomery at the University of California, Santa Cruz. “These simple orbits are kind of like a skeleton to build the whole system up from.”

The second is your comment on the above:

“As suggested by the above article, the field of Celestial Mechanics is still on step zero—the stone age. The found orbits are nothing like heliocentric astronomy and there will be an attempt to use them as a skeleton to "build the whole system up from."

The zeroth step is in reference to finding ALL analytical solutions, not the entire field of celestial mechanics, as you inferred.

This zeroth step apparently is being used to build up an understanding of all analytical solutions, not the relatively simple orbits of our solar system.

As I mentioned before, and provided evidence in support, those simple orbits have already been modeled both computationally and analytically - see my two posts on 3 body evidence.

In my opinion, there can be a danger in conflating the two efforts of computational and analytical methods, or generalizing the 3 body problem to celestial mechanics as a whole.

[Tom Bishop]

The zeroth step is in reference to finding ALL analytical solutions, not the entire field of celestial mechanics, as you inferred.

That is incorrect. Those are all numerical solutions they are finding.

https://wiki.tfes.org/Three_Body_Problem#Analytical_Vs._Numerical

Analytical Vs. Numerical

Q. I think those quotes are talking about analytical solutions. There are working numerical solutions...

A. This is a misconception which stems from some sources which state that there are no analytical solutions, only numerical solutions. This might cause a casual reader to assume that there must be solutions in which the conventional systems of astronomy work. While it is true that the analytical approach of creating an equation to predict future positions based on initial conditions is much more difficult, the working 'numerical solutions' are the special cases described above -- the figure eight and other highly symmetric configurations.

The "numerical solutions" are symmetrical and require at least two of the three bodies to be of the same mass.

Over a Thousand New Solutions - New Scientist

From the New Scientist article Infamous three-body problem has over a thousand new solutions we read:

  “ Perhaps the most important application of the three-body problem is in astronomy, for helping researchers figure out how three stars, a star with a planet that has a moon, or any other set of three celestial objects can maintain a stable orbit.
But these new orbits rely on conditions that are somewhere between unlikely and impossible for a real system to satisfy. In all of them, for example, two of the three bodies have exactly the same mass and they all remain in the same plane. ”

Clicking on the arxiv.org source at the bottom of the that article takes us to the paper The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum, where we see at the bottom of p.1:

  “ Therefore, without loss of generality, we consider m1 = m2 = 1 and m3 is varied. ”

Elsewhere in the paper it describes:

  “ Thus, we further integrate the motion equations by means of “Clean Numerical Simulation” (CNS) [17–20] with negligible numerical noises in a long enough interval of time ”

Over 600 New Orbits

Similarly, the phys.org article Scientists discover more than 600 new periodic orbits of the famous three-body problem describes the discovery of other symmetrical orbits:



  “ These 695 periodic orbits include the well-known figure-eight family found by Moore in 1993, the 11 families found by Suvakov and Dmitrasinovic in 2013, and more than 600 new families reported for the first time. The two scientists used the so-called clean numerical simulation (CNS), a new numerical strategy for reliable simulations of chaotic dynamic systems proposed by the second author in 2009, which is based on a high order of Taylor series and multiple precision data, plus a convergence/reliability check. ”

Figure Eight

The famous symmetrical Figure Eight problem was discovered numerically:

http://numericaltank.sjtu.edu.cn/three-body/three-body.htm

  “ The famous figure-eight family was numerically discovered by Moore [10] in 1993 and rediscovered by Chenciner and Montgomery [11] in 2000. ”

1349 New Families

Over a thousand new periodic orbits of a planar three-body system with unequal masses

  “ Here, we report 1349 new families of planar periodic orbits of the triple system where two bodies have the same mass and the other has a different mass. ”

Further down in the same paper, in the section "Numerical searching for periodic orbits" we verify that these are numerical simulations:

  “ As mentioned by Li and Liao (2017), many periodic orbits might be lost by means of traditional algorithms in double precision. Thus, we further integrate the equations of motion by means of a "clean numerical simulation" ”

We see that these special solutions are the numerical solutions. Just where are the solutions with different masses and non-symmetrical configurations? Opponents are unable to show that there are  non-symmetrical configurations, or that the Sun-Earth-Moon system can be simulated by the Three Body Problem.

[BRrollin]

The zeroth step is in reference to finding ALL analytical solutions, not the entire field of celestial mechanics, as you inferred.

That is incorrect. Those are all numerical solutions.

https://wiki.tfes.org/Three_Body_Problem#Analytical_Vs._Numerical

Analytical Vs. Numerical

Q. I think those quotes are talking about analytical solutions. There are working numerical solutions...

A. This is a misconception which stems from some sources which state that there are no analytical solutions, only numerical solutions. This might cause a casual reader to assume that there must be solutions in which the conventional systems of astronomy work. While it is true that the analytical approach of creating an equation to predict future positions based on initial conditions is much more difficult, the working 'numerical solutions' are the special cases described above -- the figure eight and other highly symmetric configurations.

The "numerical solutions" are symmetrical and require at least two of the three bodies to be of the same mass.

Over a Thousand New Solutions - New Scientist

From the New Scientist article Infamous three-body problem has over a thousand new solutions we read:

  “ Perhaps the most important application of the three-body problem is in astronomy, for helping researchers figure out how three stars, a star with a planet that has a moon, or any other set of three celestial objects can maintain a stable orbit.
But these new orbits rely on conditions that are somewhere between unlikely and impossible for a real system to satisfy. In all of them, for example, two of the three bodies have exactly the same mass and they all remain in the same plane. ”

Clicking on the arxiv.org source at the bottom of the that article takes us to the paper The 1223 new periodic orbits of planar three-body problem with unequal mass and zero angular momentum, where we see at the bottom of p.1:

  “ Therefore, without loss of generality, we consider m1 = m2 = 1 and m3 is varied. ”

Elsewhere in the paper it describes:

  “ Thus, we further integrate the motion equations by means of “Clean Numerical Simulation” (CNS) [17–20] with negligible numerical noises in a long enough interval of time ”

Over 600 New Orbits

Similarly, the phys.org article Scientists discover more than 600 new periodic orbits of the famous three-body problem describes the discovery of other symmetrical orbits:



  “ These 695 periodic orbits include the well-known figure-eight family found by Moore in 1993, the 11 families found by Suvakov and Dmitrasinovic in 2013, and more than 600 new families reported for the first time. The two scientists used the so-called clean numerical simulation (CNS), a new numerical strategy for reliable simulations of chaotic dynamic systems proposed by the second author in 2009, which is based on a high order of Taylor series and multiple precision data, plus a convergence/reliability check. ”

Figure Eight

The famous symmetrical Figure Eight problem was discovered numerically:

http://numericaltank.sjtu.edu.cn/three-body/three-body.htm

  “ The famous figure-eight family was numerically discovered by Moore [10] in 1993 and rediscovered by Chenciner and Montgomery [11] in 2000. ”

1349 New Families

Over a thousand new periodic orbits of a planar three-body system with unequal masses

  “ Here, we report 1349 new families of planar periodic orbits of the triple system where two bodies have the same mass and the other has a different mass. ”

Further down in the same paper, in the section "Numerical searching for periodic orbits" we verify that these are numerical simulations:

  “ As mentioned by Li and Liao (2017), many periodic orbits might be lost by means of traditional algorithms in double precision. Thus, we further integrate the equations of motion by means of a "clean numerical simulation" ”

We see that these special solutions are the numerical solutions. Just where are the solutions with different masses? Opponents are unable to show that there are solutions with different masses, that there are non-symmetrical configurations, or that the Sun-Earth-Moon system can be simulated by the Three Body Problem.

I did not say they weren’t numerical solutions. I said they are being used as a zeroth order step in finding all analytical solutions. My other statements this hold unchanged.

[Tom Bishop]

I did not say they weren’t numerical solutions. I said they are being used as a zeroth order step in finding all analytical solutions. My other statements this hold unchanged.

It says nothing about that.

The found orbits that are symmetrical and of identical masses are numerical solutions, not analytical solutions. Analytical solutions have nothing to do with this, and are not even attempted.

The problem with Newtonian gravity is that it can only produce these weird orbits. Hence, the 'zeroth step' on figuring it all out.

[JSS]

The problem with Newtonian gravity is that it can only produce these weird orbits. Hence, the 'zeroth step' on figuring it all out.

Where do you get that Newtonian gravity can only produce 'weird' orbits?  I have yet to see any published paper say that numerical methods are invalid. Numerical integration is used all over science and engineering and is just another valid tool. All kinds of problems that don't have exact solutions use numerical methods to solve them. Citation is needed.