Klemperer Rosettes

Klemperer Rosettes are many moons forming a circle around a planet, with all moons having the same period. Also often called Kemplerer Rosettes by Larry Niven and others, they were first described by W.B. Klemperer in The Astronomical Journal, vol. 67, number 3 (April, 1962), on pages 162-7, "Some Properties of Rosette Configurations of Gravitating Bodies in Homographic Equilibrium".

Click on images to start them moving, or to stop them. It helps your CPU if you only have one running at a time.

Here's one big world and two smaller ones, all in the same orbit. The smaller ones are at the Lagrangian points of the bigger one. Lagrangian points are points around a large object where small objects have no net acceleration -- if you put them in the Lagrangian points they just stay there.

Here's one big world and two smaller ones, all in the same orbit. The smaller ones are in orbits about the Lagrangian points of the bigger one. I started them out at 90 degrees to the middle one. This displays once per orbit, so the worlds should appear to stand still except for how they move relative to one another. It's doing 20 orbits per second.

Do Lagrange points (and similar things) happen in real life? Here is the outer solar system, displayed once per Pluto's orbit. Neptune and Pluto are in a 3::2 resonance. There are other objects called "Plutinos" that have similar orbits. (This does 1000 years per second or so, depending on your processor speed.)

Here is the outer solar system again, except displayed once per Neptune orbit.

Here we have an ellipse and two worlds orbiting about its Lagrangian points.

The Lagrangian points are stable for an ellipse. (The Lagrangian points are still 60 degrees leading and trailing, and the Lagrangian points track elliptical orbits just like the main orbit.) Here they are, the non-strobe version.

This is the strobe version of the Lagrange points for an ellipse. The worlds are displayed slightly more than once per orbit. It's doing 20 orbits per second.

The Lagrangian points are stable for an ellipse, but as you can see, not as stable as in a circle. I originally claimed the leading position (green) is more stable than the trailing (purple). However, the physics is reversible (if you reverse all directions, then trailing becomes leading and leading becomes trailing, and the orbits will stay the same), so they have to be equally stable. Any apparent asymmetry must be coincidental or an artifact of my (true, nonreversible) simulation. It starts both side worlds out at 90 degrees from the central world. 20 orbits per second again.

I originally guessed the Lagrangian points would be on the central planet's elliptical orbit, leading and trailing by 1/6th of an orbit. Here's a simulation of that. The leading world's orbit (green) becomes steadily more elliptical as the trailing world's orbit (purple) becomes steadily more circular. That is, until leading becomes trailing and trailing becomes leading. Eventually they come back to where they started. So this is stable, after a fashion. This is not a strobe version.

Here's a strobe effect of the leading & trailing by 1/6 an orbit thing. 20 orbits per second.

Tag-team orbits, as in some small moons of Saturn. Strobe, displayed once per year.

Here's 6 earths with a central sun. Stable, in an odd sort of way. This one eats a lot of CPU. It displays once every orbit so the worlds appear to be standing still (except for how they move relative to one another). 50 orbits per second.

Here's 7 earths with a central sun and a lot of wobble. Warren D Smith says that H. Salo and C. F. Yoder wrote that for 2 through 8 there is a unique nonequally-spaced stable configuration for the moons, and also for 7+, equally spaced is a stable configuration. I haven't looked at exactly stable configurations, I'm only looking at what wobbly configurations oscillate around.

Here's 7 earths with a central sun, finely balanced.

Here's 8.

Here's 8, which is really 9 with one omitted. 7 seems to like to clump to one side then spring back, but 8 really does like to remain 8 evenly spaced moons.

Here's 9, with a little wobble added.

Here's 9, with a lot of wobble added.

Here's another Klemperer rosette, of 24 earths this time. 12 to 24 seems to be very stable. The simulation is rigged to display once per orbit, so the planets will stand still (except for how they move relative to one another). This eats a lot of CPU. Kinda boring, isn't it? 50 orbits per second.

OK, that last one wasn't really a Klemperer Rosette. Grant Hutchinson points out that the original paper said: "So much for the trivial case of the completely regular polygon, which is by no means the only configuration which exhibits symmetry about each radius vector. Such symmetry is also possessed by a peculiar family of geometrical configurations which may be described as 'rosettes'. In these an even number of 'planets' of two (or more) kinds, one (or some) heavier than the other, but all of each set of equal mass, are placed at the corners of two (or more) interdigitating regular polygons so that the lighter and heavier ones alternate (or follow each other in a cyclic manner)." Here's a true Klemperer Rosette, 12+12 worlds again. That looks stable too.

48 earths, however, is not stable. 48 worlds of 2/3 the mass of earth each would be stable. (Here you see the worlds orbiting like normal, I'm not doing that one-display-per-orbit strobe effect.) It takes I think eight orbits to fall apart. I added some error in the z-direction below the precision of the orbit placements, making this not exactly symmetric nor planar.
After this has fallen apart, try to imagine living on one of these worlds. A trip around the sun takes a year. The newspaper headlines would be really exciting, no?

24 earths in a circle again, but with no sun, and with the masses scaled up (or equivalently the velocities scaled down) so that the worlds stay in orbit. No strobe effect. (This is accurate only until the pattern decays. The popcorn effect is due to inaccurate simulation.) (No Jupiter in this one.)

And finally, everyone's favorite question about Klemperer Rosettes. Is Larry Niven's Puppeteer solar system stable? That was five worlds orbiting one another without a central sun.

If I were an engineer of an intelligent, paranoid, and manipulative race (and I am), I would not choose to build my own solar system this way.

Exploring orbits with Javascript
Orbit simulation source code
Cruithne, earth's second moon
MOMA (Museum Of Mom's Art)
Table of Contents

Here's one big world and two smaller ones, all in the same orbit. The smaller ones are at the Lagrangian points of the bigger one. Lagrangian points are points around a large object where small objects have no net acceleration -- if you put them in the Lagrangian points they just stay there.
Here's one big world and two smaller ones, all in the same orbit. The smaller ones are in orbits about the Lagrangian points of the bigger one. I started them out at 90 degrees to the middle one. This displays once per orbit, so the worlds should appear to stand still except for how they move relative to one another. It's doing 20 orbits per second.
Do Lagrange points (and similar things) happen in real life? Here is the outer solar system, displayed once per Pluto's orbit. Neptune and Pluto are in a 3::2 resonance. There are other objects called "Plutinos" that have similar orbits. (This does 1000 years per second or so, depending on your processor speed.)
Here is the outer solar system again, except displayed once per Neptune orbit.
Here we have an ellipse and two worlds orbiting about its Lagrangian points. The Lagrangian points are stable for an ellipse. (The Lagrangian points are still 60 degrees leading and trailing, and the Lagrangian points track elliptical orbits just like the main orbit.) Here they are, the non-strobe version.
This is the strobe version of the Lagrange points for an ellipse. The worlds are displayed slightly more than once per orbit. It's doing 20 orbits per second.
The Lagrangian points are stable for an ellipse, but as you can see, not as stable as in a circle. I originally claimed the leading position (green) is more stable than the trailing (purple). However, the physics is reversible (if you reverse all directions, then trailing becomes leading and leading becomes trailing, and the orbits will stay the same), so they have to be equally stable. Any apparent asymmetry must be coincidental or an artifact of my (true, nonreversible) simulation. It starts both side worlds out at 90 degrees from the central world. 20 orbits per second again.
I originally guessed the Lagrangian points would be on the central planet's elliptical orbit, leading and trailing by 1/6th of an orbit. Here's a simulation of that. The leading world's orbit (green) becomes steadily more elliptical as the trailing world's orbit (purple) becomes steadily more circular. That is, until leading becomes trailing and trailing becomes leading. Eventually they come back to where they started. So this is stable, after a fashion. This is not a strobe version.
Here's a strobe effect of the leading & trailing by 1/6 an orbit thing. 20 orbits per second.
Tag-team orbits, as in some small moons of Saturn. Strobe, displayed once per year.
Here's 6 earths with a central sun. Stable, in an odd sort of way. This one eats a lot of CPU. It displays once every orbit so the worlds appear to be standing still (except for how they move relative to one another). 50 orbits per second.
Here's 7 earths with a central sun and a lot of wobble. Warren D Smith says that H. Salo and C. F. Yoder wrote that for 2 through 8 there is a unique nonequally-spaced stable configuration for the moons, and also for 7+, equally spaced is a stable configuration. I haven't looked at exactly stable configurations, I'm only looking at what wobbly configurations oscillate around.
Here's 7 earths with a central sun, finely balanced.
Here's 8.
Here's 8, which is really 9 with one omitted. 7 seems to like to clump to one side then spring back, but 8 really does like to remain 8 evenly spaced moons.
Here's 9, with a little wobble added.
Here's 9, with a lot of wobble added.
Here's another Klemperer rosette, of 24 earths this time. 12 to 24 seems to be very stable. The simulation is rigged to display once per orbit, so the planets will stand still (except for how they move relative to one another). This eats a lot of CPU. Kinda boring, isn't it? 50 orbits per second.
OK, that last one wasn't really a Klemperer Rosette. Grant Hutchinson points out that the original paper said: "So much for the trivial case of the completely regular polygon, which is by no means the only configuration which exhibits symmetry about each radius vector. Such symmetry is also possessed by a peculiar family of geometrical configurations which may be described as 'rosettes'. In these an even number of 'planets' of two (or more) kinds, one (or some) heavier than the other, but all of each set of equal mass, are placed at the corners of two (or more) interdigitating regular polygons so that the lighter and heavier ones alternate (or follow each other in a cyclic manner)." Here's a true Klemperer Rosette, 12+12 worlds again. That looks stable too.
48 earths, however, is not stable. 48 worlds of 2/3 the mass of earth each would be stable. (Here you see the worlds orbiting like normal, I'm not doing that one-display-per-orbit strobe effect.) It takes I think eight orbits to fall apart. I added some error in the z-direction below the precision of the orbit placements, making this not exactly symmetric nor planar. After this has fallen apart, try to imagine living on one of these worlds. A trip around the sun takes a year. The newspaper headlines would be really exciting, no?
24 earths in a circle again, but with no sun, and with the masses scaled up (or equivalently the velocities scaled down) so that the worlds stay in orbit. No strobe effect. (This is accurate only until the pattern decays. The popcorn effect is due to inaccurate simulation.) (No Jupiter in this one.)
And finally, everyone's favorite question about Klemperer Rosettes. Is Larry Niven's Puppeteer solar system stable? That was five worlds orbiting one another without a central sun. If I were an engineer of an intelligent, paranoid, and manipulative race (and I am), I would not choose to build my own solar system this way.