[math-fun] Saari spiderweb exact solutions of Newton laws
It seems to me, Saari's "spiderweb" exact solutions may be mathematically interesting. This is in spite of any complaints I have re Saari's hype about dark matter, and his enormous misuse of the word "stability." Why are they interesting? First of all, if you have some matter, and it is subject only to Newton's laws, and it has total momentum=0 and nonzero total angular momentum=A, then if you make the matter lose energy subject to those 2 constraints, it will eventually reach a state where all the matter lies in one plane (orthogonal to the A vector). So "discs" shaped much like many galaxies, and "saturn ring" configurations, thus arise naturally. I think the importance of discs was first realized by P.S.Laplace (?). The Saari spiderwebs are exact solutions, and they do lie in a plane. So in that sense they seem realistic. Second, the Saari spiderweb rotates as a rigid body. This is completely unlike actual galaxies and saturnian rings. Actual disc galaxies rotate with fairly "flat velocity curves," meaning the velocity v of the rotating stuff (which lies in the plane and orthogonal to the radial vector) stays approximately constant as its radial distance to the center changes. This empirical fact, first highly publicized by astronomers Vera Rubin and W.Kent Ford in the 1970s, is attributed to a conjectural halo of invisible "dark matter." A rigid body, on the other hand, would have linear velocity curve v=const*r. And a uniform matter density in the plane, would have a velocity curve v(r) obeying const*r^2 / r^2 = v^2 / r i.e. a "square root curve" v = const*r^(1/2). [If the matter in the plane were to have an areal density(r) correct to cause a flat velocity curve then that density presumably would need to obey const/r^2 * integral(0 to r) density(r)*r*dr = v^2 / r i.e. the density would be density = const/r.] And finally, the Saturn rings should have velocity curve obeying v^2 / r = const / r^2 i.e. v = const*r^(-1/2) the inverse-square-root curve. So the collection of powers of r we have here is {-1/2, 0 1/2, 1} for velocity curves v=const*r^P. Henry Baker has asked: "what would Saari's exact solutions do in general relativity rather than Newton?" Or about GR and galaxies. Normally GR is considered pretty irrelevant to galaxies because almost all motions are way slower than lightspeed, and the Einstein cosmical constant is too small to affect things on galactic length scales, so Newton works quite well. I would presume that exact solutions of GR much like Saari spiderweb, should exist, but nobody ever found them and they are probably not expressible in close form. However, there are a sequence of better and better approximations to GR, called the "post-Newtonian" and "post-Minkowskian" approximations. The first is Newton's laws. I believe the second is used by NASA to simulate the solar system. It seems to me that within any one of those approximations, exact Saari spiderweb solutions should exist, and they should exhibit essentially the same degree of difficulty to find them numerically, as it is to find Saari's Newton-law spiderweb exact solutions numerically. Another interesting remark is this. Among all spiderweb solutions, some small subset of them ought to be distinguished as "optimum" in some respect. What kinds of optimality would you want, and what spiderwebs would result? Finally, there are some unusual galaxies out there. Take a look at "Hoag's object" https://en.wikipedia.org/wiki/Hoag%27s_Object ---- PS. The "bullet cluster" although it has been hyped as a "proof" of the existence of dark matter, can also be (and has been) argued as proving the quantitative failure of the dark matter model. So be careful, there is a lot of hype about dark matter supported by very shallow and poor analyses.
Thinking a bit more, Saari's spiderwebs have the property that all the masses are located at the intersection of a set of concentric circles and lines thru the center. However, with general relativity, or any approximate ("post Minkowski") form thereof involving "retarded" gravitation (traveling at lightspeed) THIS WILL NO LONGER BE THE CASE! Because the masses on some inner circle, will transmit their gravity to those on an outer circle after a DELAY during which there was some rotation, and so in order to "feel like" they had the same phase angles as the masses on the outer circle, they actually need to have a phase shift. So this is an important qualitative change in the nature of the exact solution caused by GR effects. And it perhaps is the case that the many phase-demands (which under Newton's laws were trivially satisfiable by making all phase shifts be 0) are too numerous to be satisfied at all, i.e. it might be the case that Saari spiderwebish solution CANNOT EXIST with retarded gravity, or anyhow are way less stable. So that's actually quite an interesting question to investigate, and it might end up further demolishing any Saari claims to having any relevance. And its even hairer than I just said. To make that clear, consider a SINGLE circle of planets equispaced. This has been called a "Klemperer rosette" https://en.wikipedia.org/wiki/Klemperer_rosette . Wikipedia says "such systems are definitely NOT stable: any motion away from the perfect geometric configuration causes an oscillation, eventually leading to the disruption of the system (Klemperer's original article also states this fact). This is the case whether the center of the Rosette is in free space, or itself in orbit around a star. The short-form reason is that any perturbation destroys the symmetry, which increases the perturbation, which further damages the symmetry, and so on." Wikipedia vaguely claims instability for every number of planets N>2, although it seems to think 6 planets ought to be less-unstable than usual. I don't think Wikipedia has any proof of any theorem on this, it is merely an empirical finding. (This is in contrast to Saari, who claims the rosettes ARE stable, for every N, because Saari in his paper simply redefines the word "stable" in a way I've never seen before in physics, and Saari just ignores the instability fact.) Anyhow, consider the Klemperer Rosette NOT with Newton but rather with retarded gravity. Due to the retardation, any planet "feels" the gravity from planets "ahead" of it in the orbit, as though they were angularly closer. It also feels gravity from the planets "behind" it, as though they were angularly further. With "retarded Newton gravity" it seems to me there ought to be a net torque, and angular momentum would NOT be conserved! This is ultimately due to the failure of Newton "equal & opposite reaction" law in the presence of retarded forces. This effect would immediately destroy the symmetry, the light planets would no longer be perfectly dovetailed between the heavy planets. (It would not destroy the symmetry in the case where the light and heavy planets have equal masses, but it still would destroy the constancy of the rotation speed.) I conclude that not only are these rosette solutions unstable, they with more-realistic laws of physics than plain Newton can no longer even be solutions at all. Furthermore with GR, any such system must emit gravitational waves and the orbits therefore will decay, which in turn would destroy the exactness of any Saari-esque solution, i.e. FACT: in GR, there is no time-invariant solution of N>1 point masses. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)
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Warren D Smith