Instability of a Rigid Untethered Ring Around a Planet

[PvtRyan96]

I had this discussion while driving home to California from a trip to Washington state with a friend.

We were discussing the stability of a completely rigid, untethered, ring-structure around Earth, and I did not know how to explain to him that such a thing must be tethered by rigid towers lest one portion of it begin a runaway acceleration towards the planet's surface, crashing into the ground. Certainly, this is the case, right? There is no restoring force to bring the ring back to its original position when even the slightest drift occurs, and, even if the ring were to be spinning in geostationairy orbit, that does not impart any additional stability, does it? The apsis wouldn't suddenly begin rotation around the planet, so, since the perigee is closer to the surface and experiencing a stronger gravitational pull than the rest of the rigid ring, it would overall experience a net force leading to the perigee slamming right into the ground, as far as I am aware.

Anyway, if I'm correct, I'd like to have some kind of thought experiment or layman's terms explanation to convey the information I can't easily explain to him.

This is my first thread, also, so I apologize for any silent rules about this process that I am ignorant of.

 

[Filip Larsen]

Welcome to PF!

Such a "free floating" ring with central gravitational mass will indeed be unstable as can be seen from energy considerations. A fairly detailed derivation of the stability for the Niven Ring should carry over to the setup you describe (I have not read the paper in full, but I did some similar calculations on the Niven Ring years back so I concur with its conclusion about stability).

 

[Ibix]

A hand-waving plausibility argument is this. Let the ring lie in the x-y plane, centred on the origin. Obviously a mass located at the origin can feel no force because of the symmetry. Now displace the mass in the +y direction. Obviously there's still no net force in the x direction due to the symmetry. But what about the y direction? You can see that there's less ring further in the +y direction than there is in the -y direction. But on average, the parts of the ring on the +y side are closer to the mass than the parts on the -y side. One would expect these two effects to cancel to some degree, so it's not totally implausible that they cancel out.

 

[CWatters]

All of the above (and the paper Filip mentions) make perfect sense, but I notice they all ignore angular momentum. For example if part of the ring is displaced towards the planet then that part should try to speed up. Does that offer any hope of stability?

 

[Filip Larsen]

All of the above (and the paper Filip mentions) make perfect sense, but I notice they all ignore angular momentum. For example if part of the ring is displaced towards the planet then that part should try to speed up. Does that offer any hope of stability?

No hope there. Angular momentum will act to stabilize the ring form itself and its rotation axis (if the rotation is pure), but will not on average change anything regarding translatory stability. If the ring can be modeled as being rigid, then the rotational and translatory dynamics can be completely separated at all times around the centre of mass.

 

[PvtRyan96]

Welcome to PF!

Such a "free floating" ring with central gravitational mass will indeed be unstable as can be seen from energy considerations. A fairly detailed derivation of the stability for the Niven Ring should carry over to the setup you describe (I have not read the paper in full, but I did some similar calculations on the Niven Ring years back so I concur with its conclusion about stability).

Ah, very interesting. I've since shown that read to the one I was having the discussion with and he seems to understand now, so thank you.

All of the above (and the paper Filip mentions) make perfect sense, but I notice they all ignore angular momentum. For example if part of the ring is displaced towards the planet then that part should try to speed up. Does that offer any hope of stability?

The nature of it being a rigid ring means that no one part of the structure can move faster than another, so every part of the ring must necessarily be traveling at the same velocity around the center of the ring at all times. This is why a solid ring would fail where a collection of gas/dust would be very stable, since individual objects in free-fall are allowed to accelerate as they approach their apsides, as far as I'm aware.

 

[Janus]

In 1859 James Clerk Maxwell addressed this problem in connection with the structure of Saturn's rings. He proved that such a solid ring would be unstable, and thus the rings must be made up of many separate bodies.

 

[CWatters]

Interesting. Presumably something similar applies to a Dyson Sphere?

Edit: Of course it does. There would be no net force between the sphere and the star so the sphere can just drift.