1 "Michael Müdsam" Question about solar system donderdag 7 februari 2002 10:56 2 "Gordon D. Pusch" Re: Question about solar system donderdag 7 februari 2002 16:13 3 "Chris Hillman" Re: Question about solar system vrijdag 8 februari 2002 11:34 4 "Nicolaas Vroom" Re: Question about solar system dinsdag 19 februari 2002 11:36 5 "Chris Hillman" Re: Question about solar system woensdag 20 februari 2002 15:17 6 "Ahmet Gorgun" Re: Question about solar system donderdag 21 februari 2002 11:04 7 "Chris Hillman" Re: Question about solar system vrijdag 22 februari 2002 11:33 8 "Nicolaas Vroom" Re: Question about solar system zaterdag 23 februari 2002 12:58 9 "Ahmet Gorgun" Re: Question about solar system woensdag 27 februari 2002 13:35

Van: "Michael Müdsam"
Datum: donderdag 7 februari 2002 10:56

Hey,
regarding a planetary system like us with only periodical orbits. What is the condition that the system itself is periodical? One orbit is ok, clear, two orbits, if the time for one period are commensurable?
I think, our solar system is not (do not regard asteroids, comets or such things with non periodic orbit) periodical, but maybe a certain state (the coordinates of the planets and their moons) can be approximatively the same after a very long time? And tends this approximation maybe to zero? Do there exist reesults in Chaos theory?

Michael

Van: "Gordon D. Pusch"
Onderwerp: Re: Question about solar system
Datum: donderdag 7 februari 2002 16:13

 > Hey, regarding a planetary system like us with only periodical orbits. What is the condition that the system itself is periodical? One orbit is ok, clear, two orbits, if the time for one period are commensurable?

Correct.

 > I think, our solar system is not (do not regard asteroids, comets or such things with non periodic orbit) periodical,

On the contrary, it is almost maximally _IN-commensurable_. If you compute the ratios of planetary orbital periods, you will find that the only pairs of major planets that come close to being in a low-integer resonance are Jupiter and Saturn (1% off 3:2), and Uranus and Neptune (2% off 2:1) --- which is a good thing, since an orbital resonance without some form of dissipation can rapidly destabilize an N-body system!

(Neptune and Pluto are also close to a 3:2 resonance, but Pluto is too small for this resonance to be important.)

 > but maybe a certain state (the coordinates of the planets and their moons) can be approximatively the same after a very long time? And tends this approximation maybe to zero? Do there exist reesults in Chaos theory?

The solar system is thought to be ``mildly'' chaotic --- see: http://www.arXiv.org/abs/astro-ph/0111600

The evidences from the long-term numerical integrations are that the inner system is close to the borderline of chaos (Venus, Earth, and Mars are stable, but Mercury could potentially be ejected someday). The outer system planets have semimajor axes that are fairly stable (except for Pluto, which will probably be ejected in a few billion years), but their perihelion longitudes execute a chaotic ``pendulum-like'' libration.

The asteroids near the resonance gaps and the short period comets are chaotic because of their interactions with Jupiter.

-- Gordon D. Pusch

perl -e '\$_ = "gdpusch\@NO.xnet.SPAM.com\n"; s/NO\.//; s/SPAM\.//; print;'

Van: "Chris Hillman"
Onderwerp: Re: Question about solar system
Datum: vrijdag 8 februari 2002 11:34

 > regarding a planetary system like us with only periodical orbits. What is the condition that the system itself is periodical? One orbit is ok, clear, two orbits, if the time for one period are commensurable?

I suspect that the idea you really want here is the "Chinese remainder theorem", which is discussed in almost any book on number theory. Try for example this very readable undergraduate textbook:

author - {Ivan Niven, Herbert S. Zuckerman, and Hugh L. Montgomery},
title - {An Introduction to the Theory of Numbers},
publisher - {Wiley},
year - 1991}

 > I think, our solar system is not (do not regard asteroids, comets or such things with non periodic orbit) periodical, but maybe a certain state (the coordinates of the planets and their moons) can be approximatively the same after a very long time?

Gordon Pusch already pointed out that our solar system is certainly not periodic. This is true even if you only consider the motion of the major planets around the Sun; it is well known that even in Newtonian gravity, planets like Mercury precess in their orbits due to perturbations from other planets (among other things). In the case of Mercury, Venus, Earth, and Mars, most of the observed precession is accounted for by Newtonian physics, but small residual precessions can only be accounted for by gtr (or a still more complicated theory).

But to again broaden the context: the fundamental work of Poincare on the three body problem in Newtonian gravitation led to the creation of the modern theory of dynamical systems. One of the oldest and most successful branches of this huge body of work is ergodic theory, which is the study of long-term phenomena in rather general ("measure-theoretic") dynamical systems.

In ergodic theory, there is a fundamental theorem due to Poincare (later elaborated by Mark Kac) on recurrence which is extremely general. I'll try to state informally a version which applies to Hamiltonian dynamics in which the evolving configuration (e.g. of the positions and momenta of n particles, referred to Newtonian absolute space and absolute time) gives a curve in a "phase space" which is a torus (thus compact). If the evolution is ergodic, the Poincare-Kac theorem says that if we take a region R of volume vol(R), which is finite since the phase space is compact and thus has finite volume vol(S), then a generic trajectory will repeatedly revisit R with an expected recurrence time vol(S)/vol(R).

As this suggests, ergodic dynamical systems, while usually exhibiting chaotic behavior of one type or another, also exhibit a great deal of -statistical- regularity. Indeed, these strong statistical regularities are what lie behind the vast and powerful edifice of communication theory (aka information theory). The Poincare-Kac theorem is just the tip of the iceberg--- there is a very powerful theorem called the Szemeredi Regularity Theorem which also deals with recurrence phenomena. Indeed, there is a growing field called "ergodic Ramsey theory" which combines Ramsey theory (a combinatorial theory of "unavoidable coincidences") with probability theory to explain many classes of statistical/numerical "coincidences" which are not -unavoidable- but which have the property that -some- coincidence in the class is almost sure to occur.

For ergodic theory, Poincare-Kac recurrence and Szemeredi regularity, try

author - {M. Pollicott and M. Yuri},
title - {Ergodic Theory and Dynamical Systems},
publisher - {London Mathematical Society},
series - {Student Texts},
number - 40,
year - 1998}

author - {Karl Petersen},
title - {Ergodic Theory},
publisher - {University of Cambridge Press},
series - {Cambridge Series in Advanced Mathematics},
volume - 2,
year - 1983}

For Ramsey theory and another take on Szemeredi regularity, as well an intriguing phase transition which occurs in "random graph theory", try

author - {Bollob\'as, B\'ela},
title - {Modern Graph Theory},
series - {Graduate texts in mathematics},
volume - 184,
publisher - {Springer-Verlag},
year - 1998}

For the close relation between ergodic theory and statistical mechanical models like the Ising model, try Bollobas and also this book

author - {Gerhard Keller},
title - {Equilibrium States in Ergodic Theory},
publisher - {London Mathematical Society},
series - {Student Texts},
number - 42,
year - 1998}

 > And tends this approximation maybe to zero?

I think you might be groping toward the concept of "almost-periodicity".

There is a subfield of ergodic theory called symbolic dynamics which deals with dynamical systems defined in terms of certain compact metric spaces of infinite symbolic sequences under the "shift map", called "one-dimensional shift spaces", and various generalizations. Symbolic dynamics (especially the study of one-dimensional "shifts of finite type") is very closely related to communication theory. In "one-dimensional shift spaces" there is a notion of "almost-periodic" which traces its routes back to the notion of "almost-periodic functions" which were introduced by Harald Bohr (the mathematician brother of Niels), and which led in an independent development to the formulation by Yves Meyer of a theory which eventually gave rise to the theory of wavelets. It turns out that the well-known Penrose tilings are also symbolic dynamical systems in disguise, and they are almost-periodic. Indeed, if you photocopy a picture of a Penrose tiling onto a sheet of transparent plastic, and then try moving the copy over the original, you'll see that for particular "magic shifts", the copy and the original agree except on certain "strips". By choosing a sufficiently large and good "magic shift", you can make the area of the strips of disagreement as small as you like--- this is essentially the definition of "almost periodic". It turns out that to find the "magic shifts" which do the job, you need to look at certain continued fraction expansions and then in the case of the Penrose tilings, the magic shifts are very closely related to the Fibonacci sequence 1,1,2,3,5,8,13,... If you try the transparency experiment, you might be able to guess the pattern--- its easier to find this for yourself than to try to understand someone else's description of it, I think.

As this summary suggests, almost-periodicity is a very special property which not many dynamical systems possess. And the solar system is not almost-periodic. But nonetheless the phenomenon of "resonance" is very important in our solar system (and certainly in other solar systems as well). See for example this highly recommended article

author - {J. C. Lagarias},
title - {Number Theory and Dynamical Systems},
booktitle - {The Unreasonable Effectiveness of Number Theory}, editor - {Burr, Stefan A.},
series - {Proceedings of Symposia in Applied Mathematics}, volume - 46,
publisher - {American Mathematical Society},
year - 1991}

See also this very attractive and readable new textbook, which (AFAIK) is the first textbook to give due regard to the importance of resonance phenomena in our solar system:

author - {Carl D. Murray and Stanley F. Dermott},
title - {Solar system dynamics},
publisher - {Cambridge University Press},
year - 1999}

(If you look up none of the other books I mention in this post, you should at least look up these two!)

For continued fractions, try Ivan, Zuckerman, and Montgomery, or the Dover book by Khinchin, Continued Fractions. For symbolic dynamics, try

author - {Bruce P. Kitchens},
title - {Symbolic Dynamics: One-Sided, Two-Sided, and Countable State {M}arkov Shifts},
publisher - {Springer-Verlag},
date - 1998}

I hesitate to offer references for almost-periodicity, much less "magic shifts", but for some hints on another way in which the Fibonacci sequence arises in Penrose tilings, try

author - {Branko Grunbaum and G. C. Shephard},
title - {Tilings and Patterns},
publisher - {W. H. Freeman},
year - 1987}

 > And tends this approximation maybe to zero? Do there exist reesults in Chaos theory?

The ergodic theoretic phenomena mentioned above are certainly relevant to chaos theory. Less obvious is the fact that shifts of finite type lie hidden inside most naturally occuring chaotic dynamical systems, like the iteration of z -> z^2 + c, for a suitable complex constant c, or the "logistic map" or "cat map" or "Henon map", etc. (See the book by Kitchens for some hints about how this happens.) Many chaotic dynamical systems have a compact phase space with a dense set of -repelling- periodic orbits; this means that no matter where a trajectory takes the system, it lies near a periodic orbit which repels it (makes it exponentially diverge from that orbit), but because the phase space is -compact-, it can only wind up near another repelling periodic orbit. This is oversimplified, but this is the general idea as to why one should expect highly unpredictable "microbehavior" in chaotic dynamical systems. The shifts of finite type mentioned above have this character and thus provide simplified models of many "naturally occuring" chaotic dynamical systems. I stress again many chaotic systems are "statistically highly predictable" (ergodic) even though they exhibit "unpredictable microbehavior".

Another phenomenon which might interest you is the KAM theory, which concerns perturbations of Hamiltonian systems. For this see for example

author - {E. Atlee Jackson},
title - {Perspectives of Nonlinear Dynamics},
note - {Two Volumes},
publisher - {Cambridge University Press},
year - 1991}

author - {Robert C. Hilborn},
title - {Chaos and Nonlinear Dynamics: An Introduction for Scientists and Engineers},
publisher - {Oxford University Press},
year - 1994}

(KAM theory is quite difficult, and neither of these books attempt a rigorous discussion. I could give citations to books which attempt a more complete account, but that's probably inappropriate--- in this post, I tried to give citations to the most accessible/readable books I know which discuss a given topic.)

Chris Hillman

[Mod. note: MIME character replaced with dash -- mjh.]

Van:
Onderwerp: Re: Question about solar system
Datum: dinsdag 19 februari 2002 11:36

"Michael Muedsam" schreef in bericht news:...
 > Hey, regarding a planetary system like us with only periodical orbits.

SNIP

 > Do there exist results in Chaos theory?

The same issue is raised in the book:
"Pierre Simon Laplace" by Charles Coulston Gillispie At page 273 is written:
"More recently, it has been calculated in the light of chaos theory that the motions of the planets become unpredictable after some 100 million years." Two questions are answered by that sentence:
1. Can we predict the motions of the planets over a long period?
2. If no, is the reason the chaos theory?

I have a different opinion:
IMO the better model we use, the more objects we include the better computer we use the more accurate we can predict the motions of the planets.

The best model is GR. For an overview see:
http://lanl.arXiv.org/abs/gr-qc/?0106072 Numerical Relativity: A review
Authors: Luis Lehner

A different question is:
Is the solar system stable?

If you consider that our Sun is already 5 billion years old you can not claim that the Solar system is not stable. However that does not mean that the planets were and will always be the same.

You can read about that in Scientific American of September 1999 in an article by Renu Malhotra: Migrating Planets.

Is that behaviour explained by the chaos theory? IMO the model that you use is the most important.

Nick

Van: "Chris Hillman"
Onderwerp: Re: Question about solar system
Datum: woensdag 20 februari 2002 15:17

Nicholas Vroom in quoted from a book which is apparently a biography of Laplace:

 > "More recently, it has been calculated in the light of chaos theory that the motions of the planets become unpredictable after some 100 million years."

and then claimed

 > Two questions are answered by that sentence: 1. Can we predict the motions of the planets over a long period? 2. If no, is the reason the chaos theory?

I think you are reading too much into a comment in what is apparently a nontechnical book. To find out the true beliefs of people who work with dynamical systems, including celestial dynamics, and who study issues of stability and "chaos" (presumably meaning "sensitive dependence upon initial conditions"), you must study textbooks and monographs. The more accurate a picture you want of what scientists currently think they know about this stuff and why, the more books you'll have to read. This is a -huge- subject!

 > I have a different opinion: IMO the better model we use, the more objects we include the better computer we use the more accurate we can predict the motions of the planets.

I think that you have misunderstood the relevant concept of "stability" and also of "sensitive dependence on initial conditions". In my contribution to the thread in sci.physics.research titled "The Fallacy Of Chaos" [sic] I suggested a large number of books at various levels where you can learn about these concepts.

(I fear you might also be confusing notions of "numerical instability" of specific numerical methods with notions of "dynamical instability". Unfortunately :-/ modern notions of dynamical systems are so general that some notions of "dynamical instability" actually -do- capture some notions of "numerical instability", but never mind that--- I think it best that you regard these as completely separate notions!)

 > The best model is GR. For an overview see: http://lanl.arXiv.org/abs/gr-qc/?0106072 Numerical Relativity: A review Authors: Luis Lehner

This is indeed a nice survey of numerical relativity, but it has nothing to do with the stability of solar systems! I can't imagine why you thought it did.

The question asked by Laplace (is our solar system stable?) was posed in the context of Newtonian gravitation. If you are suggesting that using gtr in place of Newtonian gravitation might somehow restore stability, that is clearly incorrect.

(To see this, just recall that in Newtonian gravitation, an isolated "point mass" two-body system is periodic; not so in gtr! Of course, it is crucial to be able to estimate the characteristic time scales of various relevant effects in order to draw meaningful conclusions. In most solar systems, instability of the type studied by Poincare and his successors should appear well before orbital decay due to the emission of gravitational radiation becomes significant.)

 > A different question is: Is the solar system stable?

^^^^^^^^^

You surely realize that this question was first asked by Laplace? And that in the book you quoted from, Gillespie is presumably discussing the attempts by Laplace to answer his question? Laplace thought he had given a definitive answer, but Poincare proved otherwise. The work of Poincare on this question eventually led to the founding of the theory of dynamical systems in general, and nonintegrable Hamiltonian systems in particular.

 > If you consider that our Sun is already 5 billion years old you can not claim that the Solar system is not stable. However that does not mean that the planets were and will always be the same. You can read about that in Scientific American of September 1999 in an article by Renu Malhotra: Migrating Planets. Is that behaviour explained by the chaos theory? IMO the model that you use is the most important.

This seems very confused to me, and I can't make out what you are trying to say.

It is true that the question as asked by Laplace and studied by Poincare did not take account of the evolution of our Sun (indeed, in their day, nuclear physics and its consequences for stellar astrophysics was not even suspected). But this does -not- make Poincare's work irrelevant to understanding the dynamics of our solar system, if that is what you are claiming. Far from it.

I think that instead of questioning whether the appropriate mathematical models are being used by astronomers studying the long time behavior of our solar system (or bits of it, like the Earth-Moon subsystem), you need to begin by reading enough to understand the definitions of smooth, topological and measure-theoretic dynamical systems, as well as essential notions like bifurcations, linear vs. nonlinear perturbations, repellors vs. attractors, homoclinic vs. heteroclinic, conservative vs. dissipative, integrable vs. nonintegrable, resonances, ergodicity, topological mixing, recurrence, as well as various notions of dynamical stability. Next, you can study the basic theory of solar system dynamics, paying careful attention to the idealizations and assumptions involved at each place, and when these assumptions/idealizations are justified. You will find that in fact astronomers are generally careful to use appropriate idealizations in studying various different phenomena. And if you read intelligently, I suspect that you will ultimately find that the key to understanding the qualitative long-term evolution of our solar system is to compare the -characteristic time scales- of various phenomena which you will learn about in your reading in the theory of solar system dynamics.

For example: in the book by Murray and Dermott which I listed in my post (op. cit.), you will find very nice estimates of the time required for mode-locking of the orbital period around the Earth with the rotational period of our Moon, as well as estimates of the time period during which the Moon will spiral -outward- from the Earth due to tidal dissipation. You can also look in most modern gtr textbooks for a problem helping you to estimate the characteristic time required for significant orbital -inspiral- due to the emission of gravitational radiation. The result (for the Earth-Moon system):

T_(mode-locking) << T_(tidal friction outspiral)

I'll leave it to you to figure out where "the present epoch" and the expected lifetime of our Sun fit into this scheme, and also to figure out how the common "starting point" of these characteristic times is to be understood. Suffice it to say that if you do the computations correctly, the conclusions are unambiguous.

Chris Hillman

Van: "Ahmet Gorgun"
Onderwerp: Re: Question about solar system
Datum: donderdag 21 februari 2002 11:04

Another good post, Chris, loaded with interesting information and resources, pleasure to read. I agree with you that chaos theory is a huge field and one must learn the related academic language if one wants to discuss such things with other academics. Unfortunately this is also true in the field of history of science where Laplace is a huge field. Laplace's biographer, Charles Coulston Gillispie, is one of the experts in the field.

 > You surely realize that this question [Is the solar system stable?] was

strikes me as not historically correct. The legend propagated in textbooks and popular sources is that Laplace *proved* the stability of solar system thus saved the Newtonian theory.

Remember that at the time Newtonian theory was still open to debate and such topics as whether it was correct or not was still debated by the leading scientists. Clairaut, for instance, suggested that Newtonian inverse square law was wrong and to save the observations it must be supplanted by an additional inverse-cube term. Euler, d'Alembert, Lacaille, and many others debated the stability of the solar system and the Newtonian theory. A mine of information in this topic is the historian Curtis Wilson's book length paper entitled "Perturbations and Solar Tables from Lacaille to Delambre: the Rapproachement of Observation and Theory," in Archive for History of Exact Sciences, Volume 22 1980.

Wilson writes on page 197, for instance, "the idea of the stability of the solar system, with its parts undergoing innumerable superimposed harmonic oscillations in such a way as to be ever in equilibrium, and the idea of the possible sufficiency of universal gravitation to account for all the observed perturbations," was in the air before Laplace and Lagrange tackled them. Euler asked the same question but advocated some kind of catastrophism.

Gillespie, who unlike you and I, carefully and thoroughly studied Laplace's work, challenges the common myth that Laplace proved the stability of the solar system in his Traite du Mecanique Celeste.

On page 9 he writes that Laplace did not consider "apparent anomalies gathering toward a cosmic catastrophe ... on the contrary, he *assumed* the state of the universe to be steady."

The problem for him was "not whether the phenomena can be deduced from the law of universal gravity, but how to do it."

does not hold in light of historical evidence. Laplace simply assumed that solar system was stable.

Something that you might find interesting is that Laplace probably was the first one who attempted to quantize gravity. He "posited a corpuscle to be the bearer of gravitational force. In his analysis, the effect of weight in a particle of matter is produced by the impulse of such a gravitational corpuscle, infinitely smaller than the particle, moving toward the earth at some undetermined velocity...." page 33.

I am puzzled by your statement that

 > ... in Newtonian gravitation, an isolated "point mass" two-body system is periodic; not so in gtr!

There ought not be a difference in predictions of gtr and Newtonian gravity on such a simple fact as the two body system. They should make the same predictions. Furthermore, given the state of development of both theories, as mathematically mature, complex and modular theories, an expert can fine-tune these theories to predict both stability and instability of the solar system, but this would be just an academic exercise in futility, I believe.

Nicholas Vroom's statement that

 > > the better model we use, the more objects we include the better computer we use the more accurate we can predict the motions of the planets...

seems to make perfect sense to me. And this is what happens in practice. Both NASA development ephemerides and VSOP, that French version, get more and more accurate by applying this recipe.

--Ahmet Gorgun http://home.att.net/~agorgun/AG-01.htm

Van: "Chris Hillman"
Onderwerp: Re: Question about solar system
Datum: vrijdag 22 februari 2002 11:33

 > > You surely realize that this question [Is the solar system stable?] was
 > first asked by Laplace? strikes me as not historically correct.

Actually, I knew that, but I wanted to stress the really important point here (that modern theories offer previously undreamt-of insights), rather than the long and complicated historical development of these theories from very crude beginnings.

 > The legend propagated in textbooks and popular sources is that Laplace *proved* the stability of solar system thus saved the Newtonian theory.

The only legend I have heard is the one I mentioned, namely that Laplace -thought- he had proven the stability of motion of the major planets, but that Poincare found an error in his "proof".

Be this as it may, thanks for the correction and for the citations!

 > Something that you might find interesting is that Laplace probably was the first one who attempted to quantize gravity.

If I am not mistaken, the legend here say that it was -Newton- who first considered a corpuscular theory of gravitation, and that he decided his attempt wouldn't work, whereupon, the legend has it, he declined to feign hypotheses. (By implication, he left open the possibility that future developments would lead to a more fundamental theory of gravitation, which is of course the goal of the quantum gravity crowd. And no, I am not confusing Newton's alleged speculations about a corpuscular theory of gravitation with Newton's corpuscular theory of light.)

BTW, if I am not mistaken, it was Newton who pointed out that in his theory of gravitation, hypothetical infinite static configurations of point masses would presumably become nonstatic after a typical small perturbation, so he was apparently speculating about questions of stability even earlier than Euler.

 > I am puzzled by your statement that
 > > ... in Newtonian gravitation, an isolated "point mass" two-body system is periodic; not so in gtr!
 > There ought not be a difference in predictions of gtr and Newtonian gravity on such a simple fact as the two body system. They should make the same predictions.

Preconceptions are dangerous in science, and this particular preconception is unambiguously -wrong-. Its appearance here puzzles -me-, because this fact is very well known! There are -many- papers dealing with the cumulative effect of the emission of gravitational radiation on a two body systems in gtr; this is also a subject with a long history, including early controversies which have however long since been -decisively resolved-.

Also, anyone who reads Nobel Prize citations will know that this particular prediction of gtr has been confirmed -in detail- by observations of the Hulse-Taylor pulsar! This was big news at the time, and there is more good news: subsequent observations over two decades have given a much more accurate verification of the prediction. For a recent overview, see

Wrt what I said before about timescales: the important point here is that binary pulsars tend to be orbiting one another -much- more tightly than planets orbiting an ordinary star like our Sun. The effect of gravitational radiation is noticeable in the case of binary pulsars, but as I said, it can be expected to be negligible in solar system dynamics.

Perhaps it will also be helpful to point out that the "Newtonian approximation to gtr" (if that is what you were thinking of) involves assuming -slow motion- and -weak fields-; binary pulsars violate both assumptions!

 > Furthermore, given the state of development of both theories, as mathematically mature, complex and modular theories,

^^^^^^^

I have no idea what you mean by the word "modular" here. Be warned that in mathematical circles, a "module" is usually an R-module (a concept which is the common generalization of vector spaces and abelian groups), and referring to a "modular theory" usually indicates a connection with -modular arithmetic-. For example, "modular invariants" usually indicates the subring F[V]^rho(G) of polynomials over a finite field F which are invariant under the action induced by some representation rho of a group G on a finite dimensional vector space V over F.

 > an expert can fine-tune these theories to predict both stability and instability of the solar system,

If this were true, modern notions of stability would of course be vacuous. That alone ought to suggest that your assumption is not true: it's tantamount to suggesting that mathematicians as a group are -extremely- stupid.

 > but this would be just an academic exercise in futility, I believe.

Well, this is -precisely- why the modern theory of dynamical systems is far superior, in the context of questions of stability, to older and more naive ideas: it allows one to formulate such questions in a -precise- and -meaningful- way, and then, often, to answer them -unambiguously-!

It is true (if this is what you had in mind) that sometimes when two different notions of stability are both defined, they can give different results. However, when that happens, this information provides the user not with a "contradiction" but with valuable insight! The whole -point- here is that the modern theory of dynamical systems provides one with -useful insights- which simply would not be possible without using modern developments. This is precisely why so many mathematicians are currently working in the area of dynamical systems.

 > Nicholas Vroom's statement that
 > > > the better model we use, the more objects we include the better computer we use the more accurate we can predict the motions of the planets...
 > seems to make perfect sense to me. And this is what happens in practice. Both NASA development ephemerides and VSOP, that French version, get more and more accurate by applying this recipe.

Same comment: no-one is denying that better data and better numerical methods can be expected to lead to more accurate predictions of solar system dynamics, but this truism -completely misses the point-. Again, all I can do is point you and Vroom at various books where you can read about the relevant modern definitions and theorems. But I've already done that, so I'll quit here.

Chris Hillman

Van: "Nicolaas Vroom"
Onderwerp: Re: Question about solar system
Datum: zaterdag 23 februari 2002 12:58

Chris Hillman wrote:
 > Nicolaas Vroom quotedfrom a book which is apparently a biography of Laplace:
 > > "More recently, it has been calculated in the light of chaos theory that the motions of the planets become unpredictable after some 100 million years."
 > I think you are reading too much into a comment in what is apparently a nontechnical book.

The quote is (based on) by Lasker (1995) For more about Lasker see the literature list of the document mentioned by Gordon D. Pusch: Chaos in the Solar System http://www.arXiv.org/abs/astro-ph/0111600

See also: Newton's clock by Ivars Peterson Chapter 11 Celestial disharmonies page 249-256

 > To find out the true beliefs of people who work with dynamical systems, including celestial dynamics, and who study issues of stability and "chaos" (presumably meaning "sensitive dependence upon initial conditions")

Many issues are involved: stability, predictability, chaotic behaviour, initial conditions. IMO all model parameters (transfer function) also belongs in this list.

 > > I have a different opinion: IMO the better model we use, the more objects we include the better computer we use the more accurate we can predict the motions of the planets.
 > I think that you have misunderstood the relevant concept of "stability" and also of "sensitive dependence on initial conditions".

Stability starts with the Nyquist diagrams. See also my reply in the thread "The Fallacy of Chaos" in sci.physics.research In order to estimate if a process is stable you should know the transfer function including all the parameters.

However for the solar system this picture is quite different. A proto star is born in a contracting gas cloud.
As part of this process planets are born. Finally the stars explodes (supernovae) and becomes a white dwarf or a black hole.

The exact nature of this process is not the issue. The issue is: do you call this stable or non stable. What is for sure that a slightly different proto gas cloud will lead to a different planet configuration.
To simulate this whole process accurate is very difficult. (You need an astrophysical chemical model) On the other hand, in order to simulate the movement of objects involved alone, I doubt if the chaos theory is a strong contender.

You could call the movement of the Moon not stable because the Moon slowly moves away from Earth. The explanation is the tides. Not the chaos theory.

 > (I fear you might also be confusing notions of "numerical instability" of specific numerical methods with notions of "dynamical instability".

Numerical instability have to do with the tools we are using in our simulation. Dynamical instability with the process under study.

 > > The best model is GR. For an overview see: http://lanl.arXiv.org/abs/gr-qc/?0106072 Numerical Relativity: A review Authors: Luis Lehner
 > This is indeed a nice survey of numerical relativity, but it has nothing to do with the stability of solar systems! I can't imagine why you thought it did.

If you want to study the movement of Mercury you must use General Relativity.
One reason why I found this article interesting because it stretches the importance of initial conditions.

 > The question asked by Laplace (is our solar system stable?) was posed in the context of Newtonian gravitation. If you are suggesting that using gtr in place of Newtonian gravitation might somehow restore stability, that is clearly incorrect.

I definitely do not. Only to be more accurate. If you want to simulate over a period of 100 million years you must do that.

 > > A different question is: Is the solar system stable?
 > ^^^^^^^^^
In fact there are two very different questions: Is our solar system stable ?
Is a solar system (consisting of only 1 sun 8 planets and 100 asteroids) stable ?

In both questions the time scale is important. ie. 10 years, 10 million years, versus 10 billion years.

The first question addresses the issue what we observe ie the whole reality including the influence of other stars.
We observe that comets collide with the Sun.
(Or asteroids with other planets) If that is a lot, specific if you want to simulate over a long period than this must be taken into account.
This is the situation that the Scientific American article discusses.

The second question addresses the simulation of simplified cases. You have to be carefull with simulations of simplified systems specific over longer time periods because other factors can become important.

In case you want to prove that either is stable you must start with a clear definition what means stable.

 > You surely realize that this question was first asked by Laplace? And that in the book you quoted from, Gillespie is presumably discussing the attempts by Laplace to answer his question? Laplace thought he had given a definitive answer, but Poincare proved otherwise.

IMO you can not prove the first case mathematically because it is the reality. The model that someone uses should match that.

 > > You can read about that in Scientific American of September 1999 in an article by Renu Malhotra: Migrating Planets. Is that behaviour explained by the chaos theory? IMO the model that you use is the most important.
 > This seems very confused to me, and I can't make out what you are trying to say. It is true that the question as asked by Laplace and studied by Poincare did not take account of the evolution of our Sun (indeed, in their day, nuclear physics and its consequences for stellar astrophysics was not even suspected). But this does -not- make Poincare's work irrelevant to understanding the dynamics of our solar system, if that is what you are claiming. Far from it.

I would never claim that some one's work is irrelevant. On the other hand it is important to study based on current understanding which aspect is the most important.

One specific case to study is gravity assist. (This is also a simplified case) In order to demonstrate gravity assist you need 3 objects: A sun, A planet and a space ship or asteroide. The planet moves in a circle around the sun.

In general if the speed of the spaceship is just below the escape velocity and if the direction is radiaal outwards then the speed of the spaceship will diminish slowly, goes to zero, will increase again in opposite direction and finally the shapeship will crash into the star.

On the other hand if the space ship flies close to the planet three things can happen:
The space ship can collide with the planet.
The speed of the space ship can increase "in total" and the space ship will escape from the star.
The speed of the space ship can decrease "in total" and the space ship will crash almost immediately onto the star.

This scenario is dependent on the initial conditions of the 3 objects, on all parameters involved ie the masses of the 3 objects. on the model (Newton, GR) used in order to calculate its behaviour.

To call the result of this example stable or non stable IMO does not make much sense.
To call this example unpredictable (in line with the top quote) also does not make sense. As far as the numerical simulation is involved (for example) both step size required and accuracy are important. The following rule applies: The smaller the better.
To call this example chaotic also does not improve our understanding.

One important way to predict the future better is to improve the accuracy of our measurements and the duration involved. The chaos theory, in light of the above quote, will not prevent that.

I have a strong objection against the word chaos in astronomy because this seems to prevent scientific progress.

Nick.

Van: "Ahmet Gorgun"
Onderwerp: Re: Question about solar system
Datum: woensdag 27 februari 2002 13:35

[Mod. note: this seems to be tending in the direction of philosophy of science, which is not really on-topic for s.a.r. Followups should focus on the astrophysics or take it elsewhere -- mjh.]

"Chris Hillman" wrote in message news:mt2.0-20732-1014373987@star.bris.ac.uk... [...]
 > If I am not mistaken, the legend here say that it was -Newton- who first considered a corpuscular theory of gravitation, and that he decided his attempt wouldn't work, whereupon, the legend has it, he declined to feign hypotheses.(By implication, he left open the possibility that future developments would lead to a more fundamental theory of gravitation, which is of course the goal of the quantum gravity crowd. And no, I am not confusing Newton's alleged speculations about a corpuscular theory of gravitation with Newton's corpuscular theory of light.)

I am not familiar with Newton's corpuscular theory of gravitation. Newton was a master polemicist and a demagogue so I prefer to read and understand what his theory actually says rather than take Newton's word for it, ("I feign no hypotheses..." etc).

In fact, Newton *ascribes* a cause to gravity, his denials of it for politico-religious reasons are polemics. In Newton the cause of gravity is matter; motion (gravity) is proportional to matter. What Newton leaves open is not the cause of gravity, but how this occult gravity is communicated between matter.

By the way, Einstein accepted this Newtonian occultism and made gravity also proportional to matter but he tried to devise a non-occult mechanism of communication. But both theories are fundamentally identical because both make motion proportional to matter.

 >
 > > Furthermore, given the state of development of both theories, as mathematically mature, complex and modular theories,
 > ^^^^^^^ I have no idea what you mean by the word "modular" here. Be warned that in mathematical circles, a "module" is usually an R-module (a concept which is the common generalization of vector spaces and abelian groups), and referring to a "modular theory" usually indicates a connection with -modular arithmetic-[...]

By "modular" I meant something much simpler, probably closer to its meaning in computer science:

"A portion of a program that carries out a specific function and may be used alone or combined with other modules of the same program."

Applied to a physical theory this should read:

"A portion of a physical theory that saves a specific observation and may be used alone or combined with the other modules that save other observations within the same theory."

So if there are new observations new modules are created, or if an observation is discarded the module is also discarded and the theory stays always valid.

What I mean is illustrated in a recent article by A J Tolland (2/20/02) in sci.physics.research in the thread "Muon magnetism OK." http://www.lns.cornell.edu/spr/2002-02/msg0039717.html

Tolland is commenting on a statement that the validity of SUSY may be questioned because some experiment seems to contradict it:

"Brookhaven releases a premature and wrong analysis of its data, data indicates deviation from Standard Model, some SUSY theorist writes a paper showing that the phenomenon can be modeled with SUSY, a new more accurate analysis of the Brookhaven data comes out, deviation is no longer present, some SUSY theorist writes a paper showing that the phenomenon can be modeled with SUSY."

Here the job of the mathematician, or the theorist, is to save the observations by creating a new mathematical module in the theory, the addition or subtraction of this module does not invalidate the theory. This is the case for the Standard Theory, but what about gtr, is this also true for gtr?

 >
[...]
 > If this were true, modern notions of stability would of course be vacuous. That alone ought to suggest that your assumption is not true: it's tantamount to suggesting that mathematicians as a group are -extremely- stupid.
I don't know if this was meant seriously but it's not even worth replying to because I am not calling anybody stupid. Mathematicians can always explain a new observation by adding new mathematics to the theory, this seems like the standard procedure in the development of modern theories. I think this is related to the modern understanding of the relation between mathematics and physics, the structuralist approach, which ignores the nature of the object being studied but considers "only the system of relations embodied in them." (The quote is from Roberto Torretti, The Philosophy of Physics, p.4.)

The whole quote is:

--------------------------------------------

However, for the sake of understanding the use of mathematics in modern physics, it would seem that we need only pay attention to two general traits.

(1) Mathematical studies proceed from precisely defined assumptions and figure out their implications, reaching conclusions applicable to whatever happens to meet the assumptions. The business of mathematics has thus to do with the construction and subsequent analysis of concepts, not with the search for real instances of those concepts.

(2) A mathematical theory constructs and analyzes a concept that is applicable to any collection of objects, no matter what their intrinsic nature, which are related among themselves in ways that, suitably described, agree with the assumptions of the theory. Mathematical studies do not pay attention to the objects themselves but only to the system of relations embodied in them. In other words, mathematics is about *structure,* and about *types* of structure.
---------------------- end quote by Torretti

I think in terms of physics this is disturbing because physics must be about "real instances of those objects," not about "the system of relations embodied" in a fictional or mythological object.

I am interested in this topic because "the universe" that physicist studies is such a fictional object. By upholding this structuralist approach of the mathematician the physicist also ignores the question of whether or not what he is studying -- "the universe as a whole" -- exists as a scientific entity, or if it is simply a cosmos.

--Ahmet Gorgun

On Cosmos: http://home.att.net/~agorgun/CP-03.htm
On Newton: http://home.att.net/~agorgun/AG-09.htm

Created: 24 March 2002