Re: [math-fun] triple star
Bill: I did some reading & searching about these kinds of issues several years ago. Answer: triple systems _are_ highly unstable, and _do_ fly apart. As you point out, the "stable" ones are hierarchical -- there's such a vast scale difference between the masses, that the small perturbations take a long time to build up to something meaningful. There's another problem with stars that isn't true of point masses -- you can bring point masses arbitrarily close together, but when you do this with stars, they get pulled apart like taffee (sp?). I think that the star systems that we see in reasonable quantities are those which _have_ managed to remain "stable" over a reasonable period of time. So to find the "unstable" ones, you have to look quickly before they fall apart. There's been some progress in improving simulations to preserve important quantities -- e.g., energy, angular momentum, etc. I don't know whether it's possible to _guarantee_ such preservation to a large number of digits, though. Hopefully, someone on this list will give a better report on the state of the art of conservation of "constants of integration" in modern simulation algorithms. I have a small simulation on my Palm Pilot (called "asteroids" or some such). I downloaded it from one of the Palm Pilot freeware download sites. (BTW, I'd love a Palm Pilot version of the original MIT space war, if anyone is aware of such a thing.) ------ At 11:52 AM 6/27/03 -0700, R. William Gosper wrote:
http://www.freewebz.com/vitaliy/triApplet/triGrav.html is an interesting three-body simulation applet with colored, fading trails. Unfortunately, you don't get to play with the initial conditions, which are equal (point) masses, zero net momentum, zero net angular momentum (wrt origin), fixed or slightly randomized initial positions, and slightly randomized initial velocities. Chaos quickly magnifies the initial perturbations into qualitatively different runs.
The bodies spend some of their time "braiding", but most of their time approximating two "two body problems", with a pair of bodies in a tight orbit simulating a double mass about which the third body describes a larger orbit that frequently crosses the screen boundaries. However, due to the zero initial angular momentum, these larger orbits are highly eccentric and terminate after only one period in a sort of "elastic collision" which shuffles the orbital relationships.
Sooner or later you have to restart because one of these collisions leaves two bodies in such a tight, low energy orbit that the liberated kinetic energy imparts the third body with escape velocity, the tight pair flying off in the opposite direction. (Occasionally, there is not quite escape velocity, and the bodies return after several minutes.)
So why don't triple star systems fly apart, too? One obvious difference is that "collisions" are vastly less frequent in 3-space than 2-space. Another is that heavenly masses are typically highly unequal (e.g., Sun, Earth, Moon). Another possibility is spurious energy buildup in the simulation from accumulated Runge-Kutta errors. A sensitive test of this would be if we could initialize to the recently discovered (barely) stable braiding orbit.
But I think the main source of instability is the absence of net angular momentum. My guess is that the author nulled out the angular momentum because stability is visually boring compared to braiding and "colliding", even at the cost of impermanence. Or perhaps someone has found a chaotic triple star that proves me wrong? --rwg PS: Does anyone know of tumbling brick/asteroid animation on the Web?
Henry>Answer: triple systems _are_ highly unstable, and _do_ fly apart. But not always. There was a recent claim that three equal masses can "braid" forever, even with perturbations as "large" as 10^-5.
As you point out, the "stable" ones are hierarchical -- there's such a vast scale difference between the masses, that the small perturbations take a long time to build up to something meaningful.
What I wish I could try is two equal masses in tight circles simulating a double mass making a circle of r/2 (r large) with the third mass at r, with the signs of "both" (all three) angular velocities equal. Even assuming small initial noncircularities and nonplanarities, do you think this system will eventually decay? I guess we have to make the Newtonian approximation, else even the two body problem decays.* How many bodies are necessary/sufficient to disambiguate time? E.g., if we observe sixteen swarm around for a while, whereupon fifteen crystallize into a motionless triangle while the sixteenth departs like a shot, our suspicions might be aroused, especially if it was the white one. Reconsider now the low angular momentum case, which we see usually flies apart fairly quickly. This seems to disambiguate the sign of time, even though the physics is reversible. I.e., if you time-reverse a "fly-apart" event, two stars in tight orbit are cruising along when this rogue star comes out of nowhere and "collides inelastically" with the pair, converting its kinetic energy to the gravitational potential of the bound triple system. A likely story. Yet that is exactly what happens in the "almost fly-apart" scenario where the single and the pair don't quite escape, and eventually recombine. Furthermore, if you run the "likely story" for a while, it will usually (eventually) fly back apart as in the forward time case. So the apparent time disambiguation seems to be an illusion due to insufficient duration of observation. ?
There's another problem with stars that isn't true of point masses -- you can bring point masses arbitrarily close together, but when you do this with stars, they get pulled apart like taffy.
And if there's no lower limit on how tightly two can orbit, there is no limit on the energy with which the third can be ejected. (Moments after I typed this, the applet obliging exhibited its most violent ejection in my experience--two bodies nearly superposed, with no visible orbital motion, shot off to the left, the third twice as fast to the right, leaving afterimagess about a quarter inch apart. Minor puzzle: Why was there no noticeable vertical vibration in the tight pair proportional to their perceptible horizontal separation? I can think of three explanations: 1. A stroboscopic coincidence; 2. Only one body moves during a given time step; 3. The bodies were displayed superimposed, but at slightly different times, tricking my motion-tracking into seeing an illusory spread. (I failed to check the afterimages before their dimness surpassed mine.) --rwg *Who did this to Pope? Nature and nature's laws lay hid in night, God said, "Let Newton be," and all was light. It did not last; the devil howling "Ho! Let Einstein be!" restored the status quo.
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Henry Baker -
R. William Gosper