Astronomy Q: are Klemperer rosettes stable?

A Klemperer Rosette is a hypothetical arrangement of astronomical bodies in a polygonal pattern around a common center. I’ve searched several sites on the subject, but I haven’t been able to find confirmation that at least some configurations are stable. I have found mention that some definitely aren’t stable. I’m intrigued by the idea that a dozen or more habitable worlds could share an orbit around a sun-(imagine being able to travel to a habitable planet for little more cost than that of breaking out of your home planet’s gravity well. The “Twelve Colonies” anyone?)- but is that the case?

IIRC, Klemperer’s original paper shows that several are. I dug it out once, out of curiousity, but I’m not sure where I’ve put it:
http://www.journals.uchicago.edu/cgi-bin/resolve?id=doi:10.1086/108686&erFrom=-3973302529684105097Guest

Some properties of rosette configurations of gravitating bodies in homographic equilibrium
Author(s) W. B. Klemperer
Identifiers The Astronomical Journal, volume 67 (1962), page 162

No. Klemperer rosettes are no stable, even for relatively short time periods. They aren’t statically stable, of course, because they rely on dynamic forces in a rotating reference frame to keep from collapsing and dynamically, the smallest perturbations will cause them to become radically, chaotically unstable within a few rotations. You can satisfy yourself of this by running a few simulations in Mathcad or any other numerical simulation package (as my officemate and I did to while away a slow, rainy afternoon last time the topic came up) and observing even the miniscule effect that floating point errors have on an otherwise perfect system. Oddly (or so it seems at first) systems with long periods and low masses don’t hold up long, largely because of the amount of lag any correcting or stabilizing force has, and of course fast moving systems come apart like a cheap gold watch.

In general, any system multibody system with M[sub]1[/sub] ~ M[sub]2[/sub] ~ … ~ M[sub]i[/sub] is going to be unstable over any reasonable period. A two-body system in which M[sub]1[/sub] ~ M[sub]2[/sub] can be stable, but secondary bodies that lie in an orbit influenced by both main bodies are not unless the main bodies like in a circular orbit about their barycenter and the secondaries fall into a specific resonance. If the two main bodies are sufficiently distant such that a smaller body (M[sub]3[/sub]<<M[sub]1[/sub]) does not enter the sphere of influence of the opposing main body (M[sub]2[/sub]) it can be stable though it will tend to be perturbed in the long term. If two main bodies are close enough and the third minor body is relatively distant, then it may orbit the main bodies as a collective mass with relatively small orbital perturbations such that the orbit can be considered stable if chaotic.

We expect most real world stable systems to be like the Sun/Earth/Moon System, in which M[sub]Sol[/sub]>>M[sub]Earth[/sub]>>M[sub]Luna[/sub]. The only exeception to this in the Solar System is Pluto and Charon, which are nearly the same mass and orbit each other as a doublet…but they are very, very far from the Sun, and are both likely captured Kuiper objects whose orbits were retarded by close passes to Neptune.

So, not only did Niven get the spelling wrong (he refers to them as “Kempeler rosettes”) but they also wouldn’t be a stable formation for the Fleet of Worlds. I suppose one could rationalize that the Puppeteers use their reactionless drives to keep the worlds on station; if you’re going to assume that the Ringworld engineers used Bussard ramjets to do the same for the Ringworld, and ignore the fact that the Earth is rotating the wrong direction, what’s a little stability between friends?

Stranger

I don’t understand “ignore the fact that the Earth is rotating the wrong direction.” What’s that about?

In the first edition of Ringworld, Larry Niven infamously has Louis Wu trying to escape the coming day by transferring from one time zone to another…going east. This was corrected in later editions.

Don’t get me started about the action of strong tidal forces on a long object in hyperbolic orbit…

Stranger

<bump>
Ok, so we have one post saying that Klemperer himself said they were stable, and another saying “no, they’re not!”. About the same result I got Googling various sites. Anyone else?

Have you seen this: http://www.burtleburtle.net/bob/physics/kempler.html ?

Is this the place to mention that I possess a signed, First Edition of Ringworld? Mistake and all, with a comment on same by Larry? :smiley: