Calculating the Minimum Orbit Radius for a 'Chainworld' System

[HarryWertM]

Trying to compute a 'Chainworld' with simplest possible Newtonian physics but still lost with my lousy math. By 'Cainworld' I mean series of man-made, worlds inspired by classic novel "Ringworld", which orbit around a common center of mass in one, common, perfectly circular orbit. They are held apart by centrifugal force and pulled together by Newtonian gravity. In 'Ringworld' , there is a star at the center of mass. I want no star, just independent 'worlds', a 'chain' of cities held in place only by gravity and acceleration.

Say you start with two cities with masses equal. Both M2. They orbit a common center of gravity with radius R. Now consider three cities with the same total mass as the first two. Do they orbit with smaller radius? What is the smallest possible orbit for unlimited number of cities with total mass equal the original total mass? Surely it has something to do with the gravitational constant, G and is some simple function of our total mass!?

 

[mgb_phys]

They aren't in orbit.
If they are just spinning you can have them spin at whatever speed you want.
The 'gravity' felt on their surface will only depend on the diameter of the ring and the speed, the limit being the tension in the cables

 

[Borek]

As far as I remember such systems (several bodies orbiting their center of mass, all at the same distance from the center) are very unstable. Only binary systems are stable. That means that if you have stable system composed of three bodies, two of them are in relatively tight orbit (one binary system) and - as seen from the third body - they can be treated as one body, thus you have something like "higher level" binary system made of one single body and binary system. Four bodies are either "three level" binary system, of two binary systems making third one - and so on.

 

[SpectraCat]

As far as I remember such systems (several bodies orbiting their center of mass, all at the same distance from the center) are very unstable. Only binary systems are stable. That means that if you have stable system composed of three bodies, two of them are in relatively tight orbit (one binary system) and - as seen from the third body - they can be treated as one body, thus you have something like "higher level" binary system made of one single body and binary system. Four bodies are either "three level" binary system, of two binary systems making third one - and so on.

Yeah .. I agree with mbg. It's basically identical to setting a chain-spinning in free fall. The centripetal acceleration will cause it to assume a circumference-minimizing shape .. i.e. a circle. The effects are going to be weird though, because there will be gravity from the planets, as well as from the centripetal acceleration. Are you going to let the planets spin on their axes as well? That is, are they each going to rotate around the cable?

Also, where is the light/energy going to come from?

 

[Borek]

OP asked for no cable.

'chain' of cities held in place only by gravity and acceleration.

 

[Janus]

http://en.wikipedia.org/wiki/Klemperer_rosette

 

[SpectraCat]

OP asked for no cable.

Ok .. I see what you are saying, and I guess it is the strictly correct verbatim interpretation, since the cables would indeed add a restoring force. Since he said "chain" I was assuming that the "links" were, well, linked somehow, and he meant that the configuration of the chain was determined by acceleration and gravity.

 

[DaveC426913]

http://en.wikipedia.org/wiki/Klemperer_rosette

Ah. Beat me to it.

What you are looking for is called a Klemperer Rosette. Not surprisingly, it too is featured in the Ringworld books.

 

[HarryWertM]

Thank you Borek, Dave & Janus! NEVER would have found 'Klemperer'.

Any ideas about my 'limit case? Near infinite number of bodies [[rocks?]] with same total
mass as original two? I figured three bodies with same total mass as first two would orbit in smaller radius than the two. So I am guessing the more 'bodies' the smaller the radius for constant total mass. Is simple function for limit radius?

 

[DaveC426913]

Thank you Borek, Dave & Janus! NEVER would have found 'Klemperer'.

Any ideas about my 'limit case? Near infinite number of bodies [[rocks?]] with same total
mass as original two? I figured three bodies with same total mass as first two would orbit in smaller radius than the two. So I am guessing the more 'bodies' the smaller the radius for constant total mass. Is simple function for limit radius?

You could look up some of the further reading references at the bottom of the Wiki page. One is a Java Applet simulator.

 

[Borek]

Klemperer Rosette! Damned dementia.