[math-fun] stable time-invariant configurations of planets (sci-fi myth); flaw in my argument
And also, there are no "attractor trajectories" either. That is, in a Hamiltonian system, there is no energy (or entropy) loss mechanism, so no trajectory can attract all nearby ones. However, in the solar system there are slight non-Hamiltonian effects (beyond merely Newton laws) such as tidal friction losses. It is only as a result of these, that certain "attractor" configurations, such as trojans, and also such as 2:1 resonances, tidal locking of Earth's moon (does not rotate) have happened. To return to the original problem -- which we now see is more interesting that I thought -- PUZZLE: find every configuration of N point masses, which (1) form a stationary configuration in rotating reference frame, and which (2) is stable. FACT 1: The N bodies must all lie in a single plane perpendicular to the rotation axis since otherwise could decrease energy by "flattening" configuration into said plane. FACT 2: My repaired proof, if no further problems spotted, shows N<=6. FACT 3: Requirement (1) is a set of polynomial equations which can be solved by Grobner basis, elimination, etc methods, but it might get pretty damn difficult especially if you are naive about it. Apparently Lagrange settled the "2 and a half bodies" problem (2 bodies, plus one more of negligible mass) but I do not think even the 3-body case has been done?
On 1/11/2013 1:39 PM, Warren Smith wrote:
And also, there are no "attractor trajectories" either. That is, in a Hamiltonian system, there is no energy (or entropy) loss mechanism, so no trajectory can attract all nearby ones.
However, in the solar system there are slight non-Hamiltonian effects (beyond merely Newton laws) such as tidal friction losses. It is only as a result of these, that certain "attractor" configurations, such as trojans, and also such as 2:1 resonances, tidal locking of Earth's moon (does not rotate) have happened.
To return to the original problem -- which we now see is more interesting that I thought --
PUZZLE: find every configuration of N point masses, which (1) form a stationary configuration in rotating reference frame, and which (2) is stable.
What do you mean by 'stable' if not 'moves in small bounded region when slightly perturbed"? Brent Meeker
FACT 1: The N bodies must all lie in a single plane perpendicular to the rotation axis since otherwise could decrease energy by "flattening" configuration into said plane.
FACT 2: My repaired proof, if no further problems spotted, shows N<=6.
FACT 3: Requirement (1) is a set of polynomial equations which can be solved by Grobner basis, elimination, etc methods, but it might get pretty damn difficult especially if you are naive about it. Apparently Lagrange settled the "2 and a half bodies" problem (2 bodies, plus one more of negligible mass) but I do not think even the 3-body case has been done?
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A number of years ago, I engaged in a thought experiment (which I might have reported on math-fun) of very small perturbations of a geosynchronous orbit. Case 1 (planar). We have 2 geosynchronous satellites which are in nearly circular orbits, and both have the same major & minor axes, except that the orbits are 180 degrees out of phase; i.e., when one satellite is at perigee, the other is at apogee, and vice versa. Furthermore, both satellites travel more-or-less together, with the one speeding up when it is closer to the Earth than the other, and slowing down when it is further from the Earth than the other. I believe that in the frame of reference of one satellite, the other appears to be "orbiting" it, even though the gravitational attraction of the two is completely negligible. Case 1 could be further extended with N satellites in close proximity, which appear to be "orbiting" around some center. Perhaps some interesting use could be made of such orbiting cyclic configurations -- e.g., a very large structure connected by links which don't need to be particularly strong (at least in compression). While it isn't necessary for these orbits to be equatorial (geosynchronous) orbits, I would imagine that there would be interesting usages from such orbits. Case 2 (non-planar). We have 2 geosynchronous satellites whose circular orbits cross in an extremely small angle, so that the satellites never get very far from one another. In this case, the satellites appear to be orbiting one another in a plane perpendicular to the plane of the orbits, and perpendicular to the direction of travel of the satellites. The orbits are also ever-so-slightly elliptical, so that the satellites don't run into one another where the planes of their orbits cross. This also causes one satellite to speed up a tiny bit relative to the other, and then fall back behind. Once again, we could have N nearby satellites engaging in a similar dance, but this one is a bit more complicated than case 1. We could also utilize this configuration for some large, loosely coupled structure that utilized not particularly strong links. Bottom line: a space "station" need not be a fixed/rigid body, but might be a collection of loosely coupled objects in very nearby orbits, connected by wires/tubes that are strong in tension but weak in compression. At 02:26 PM 1/11/2013, meekerdb wrote:
What do you mean by 'stable' if not 'moves in small bounded region when slightly perturbed"?
Brent Meeker
Something like this is already in existence. The GRACE two-satellite array is mapping Earth's gravitational field. It can measure the thickness of ice sheets, the content and flow of water in aquifers and oceans, and magma within the Earth. http://www.csr.utexas.edu/grace/ http://en.wikipedia.org/wiki/Gravity_Recovery_and_Climate_Experiment -- Gene
________________________________ From: Henry Baker <hbaker1@pipeline.com> To: meekerdb <meekerdb@verizon.net>; Warren Smith <warren.wds@gmail.com> Cc: math-fun <math-fun@mailman.xmission.com> Sent: Friday, January 11, 2013 3:09 PM Subject: Re: [math-fun] small orbit perturbations
A number of years ago, I engaged in a thought experiment (which I might have reported on math-fun) of very small perturbations of a geosynchronous orbit.
Case 1 (planar). We have 2 geosynchronous satellites which are in nearly circular orbits, and both have the same major & minor axes, except that the orbits are 180 degrees out of phase; i.e., when one satellite is at perigee, the other is at apogee, and vice versa. Furthermore, both satellites travel more-or-less together, with the one speeding up when it is closer to the Earth than the other, and slowing down when it is further from the Earth than the other.
I believe that in the frame of reference of one satellite, the other appears to be "orbiting" it, even though the gravitational attraction of the two is completely negligible.
Case 1 could be further extended with N satellites in close proximity, which appear to be "orbiting" around some center.
Perhaps some interesting use could be made of such orbiting cyclic configurations -- e.g., a very large structure connected by links which don't need to be particularly strong (at least in compression).
While it isn't necessary for these orbits to be equatorial (geosynchronous) orbits, I would imagine that there would be interesting usages from such orbits.
Case 2 (non-planar). We have 2 geosynchronous satellites whose circular orbits cross in an extremely small angle, so that the satellites never get very far from one another. In this case, the satellites appear to be orbiting one another in a plane perpendicular to the plane of the orbits, and perpendicular to the direction of travel of the satellites. The orbits are also ever-so-slightly elliptical, so that the satellites don't run into one another where the planes of their orbits cross. This also causes one satellite to speed up a tiny bit relative to the other, and then fall back behind.
Once again, we could have N nearby satellites engaging in a similar dance, but this one is a bit more complicated than case 1. We could also utilize this configuration for some large, loosely coupled structure that utilized not particularly strong links.
Bottom line: a space "station" need not be a fixed/rigid body, but might be a collection of loosely coupled objects in very nearby orbits, connected by wires/tubes that are strong in tension but weak in compression.
At 02:26 PM 1/11/2013, meekerdb wrote:
What do you mean by 'stable' if not 'moves in small bounded region when slightly perturbed"?
Brent Meeker
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Thanks, Gene! However, 1) these satellites are in _exactly the same orbit_, except that one trails the other by ~220km (currently 288km, according to the GRACE web page); and 2) they aren't connected by anything -- e.g., a wire or fiber optic link. Here's a quote from the GRACE FAQ sheet: "The two identical satellites orbit one behind the other in the same orbital plane at an approximate distance of 220km (137 miles). As the pair circles the Earth, areas of slightly stronger gravity (greater mass concentration) will affect the lead satellite first, pulling it away from the trailing satellite, then as the satellites continue along their orbital path, the trailing satellite is pulled toward the lead satellite as it passes over the gravity anomaly. The change in distance would certainly be imperceptible to our eyes, but an extremely precise microwave ranging system on GRACE is able to detect these minuscule changes in the distance between the satellites." --- There was a NASA program that talked about "arrays" of (10's ?; 100's ?; of) mini-satellites only a few pounds each. But I still don't think that they were intended to connect physically with one another. At 03:28 PM 1/11/2013, Eugene Salamin wrote:
Something like this is already in existence. The GRACE two-satellite array is mapping Earth's gravitational field. It can measure the thickness of ice sheets, the content and flow of water in aquifers and oceans, and magma within the Earth.
http://www.csr.utexas.edu/grace/ http://en.wikipedia.org/wiki/Gravity_Recovery_and_Climate_Experiment
-- Gene
________________________________ From: Henry Baker <hbaker1@pipeline.com> To: meekerdb <meekerdb@verizon.net>; Warren Smith <warren.wds@gmail.com> Cc: math-fun <math-fun@mailman.xmission.com> Sent: Friday, January 11, 2013 3:09 PM Subject: Re: [math-fun] small orbit perturbations
A number of years ago, I engaged in a thought experiment (which I might have reported on math-fun) of very small perturbations of a geosynchronous orbit.
Case 1 (planar). We have 2 geosynchronous satellites which are in nearly circular orbits, and both have the same major & minor axes, except that the orbits are 180 degrees out of phase; i.e., when one satellite is at perigee, the other is at apogee, and vice versa. Furthermore, both satellites travel more-or-less together, with the one speeding up when it is closer to the Earth than the other, and slowing down when it is further from the Earth than the other.
I believe that in the frame of reference of one satellite, the other appears to be "orbiting" it, even though the gravitational attraction of the two is completely negligible.
Case 1 could be further extended with N satellites in close proximity, which appear to be "orbiting" around some center.
Perhaps some interesting use could be made of such orbiting cyclic configurations -- e.g., a very large structure connected by links which don't need to be particularly strong (at least in compression).
While it isn't necessary for these orbits to be equatorial (geosynchronous) orbits, I would imagine that there would be interesting usages from such orbits.
Case 2 (non-planar). We have 2 geosynchronous satellites whose circular orbits cross in an extremely small angle, so that the satellites never get very far from one another. In this case, the satellites appear to be orbiting one another in a plane perpendicular to the plane of the orbits, and perpendicular to the direction of travel of the satellites. The orbits are also ever-so-slightly elliptical, so that the satellites don't run into one another where the planes of their orbits cross. This also causes one satellite to speed up a tiny bit relative to the other, and then fall back behind.
Once again, we could have N nearby satellites engaging in a similar dance, but this one is a bit more complicated than case 1. We could also utilize this configuration for some large, loosely coupled structure that utilized not particularly strong links.
Bottom line: a space "station" need not be a fixed/rigid body, but might be a collection of loosely coupled objects in very nearby orbits, connected by wires/tubes that are strong in tension but weak in compression.
At 02:26 PM 1/11/2013, meekerdb wrote:
What do you mean by 'stable' if not 'moves in small bounded region when slightly perturbed"?
Brent Meeker
participants (4)
-
Eugene Salamin -
Henry Baker -
meekerdb -
Warren Smith