Bigger Systems Are More Stable

Howdy folks :)

Well, I am finally able to answer Vagueofgodalming's question. He's right, if there is more spacing between planets the system is much more stable.

It took a while to get a baseline simulation that worked. I thought it would be simple, double the distances and divide the velocities by the square root of two. Trouble is, I did that for the Sun, too, which caused an immediate crash.

When I ran the simulation for the larger system with the largest mass on the outside (Neptune) it lasted 2.86 million years. Unfortunately, it crashed as I was preparing my diabetes medicine, which involves a syringe, so there's a couple of hundred thousand years over which it could have crashed. That's quite a bit longer than the closer simulation.

When I made the inside (Jupiter) mass the largest (8 Jmasses) it ran quite stably for over 11 million years, when I decided to cut it off. What this tells me is that as expected, the larger mass being on the inside makes it more stable.

When I did a simulation with all masses at 3.75 J's it fell apart really fast, like in 86000 years. So I ran it over and it fell apart in 1.88 million years. Jupiter was ejected, Saturn had a highly eccentric orbit from a little over Jupiter's (doubled) distance down to about the altitude of our real system's Mars. Earth survived.

I can't get the clock stopped for when I start these simulations, so the initial conditions are always very different. So, watching these systems decay is like watching for a radium atom to decay. There's an average life, but it could go off at any time.

If you could do thousands and thousands of simulations, then you would get an average lifetime of these systems, just like you can get an average half-life of exploding atoms. But simulations take a long time. That's impractical. So, I'm doing snapshots.

It seems that these systems are least stable when all the planets are the same mass. That might not be what natural systems are like, in fact I would imagine that usually the largest planet is on the inside. But it occured to me that an artificial system with the masses all the same is the best for getting a clear vision of what is happening with these gravitational interactions. That way you don't have to imagine the effect of the masses.

Watching these things evolve I have noticed that the strongest perturbations occur when the inner planet is at aphelion, excuse me, apastron, and the outer planet is at periastron at the same time and the same place. Thus they are closest together. The most stable configuration is when all the planets have circular orbits. The systems hum along quite nicely when that happens.

But, eventually one of the orbits will get eccentric. Then it will start precessing. At that point it starts to affect the other planets and they get eccentric and start precessing, too. Pretty soon, the periastron of the outer object is lined up with the apastron of the inner object, so there is some definite point at which they are closest together.

At this point "resonances" kick in. We've all heard of the famous resonances like Pluto's, where Pluto make two orbits for every three Neptune makes (talking about the "real" planets here). Of course there are all kinds of resonances in between the 3:2's or 5:2's or whatever. But when the resonances start pumping energy between the objects, this changes the orbital periods. So usually the resonance goes away.

When you do these simulations, ideas occur to you about how to observe them or streamline them or manipulate them. I looked at these things and the orbits seemed to swell and shrink. I hit on an observing technique tonight. You can turn the trails that the objects leave behind them on or off. You can also turn off the plotting function so that the machine does not have to create 30 displays per second and so runs faster. I hit on a sequence; turn the trails off and then the objects are displayed as dots. Then turn the plotting function off and the last positions are frozen in place. Then turn the trails back on while it's still in "don't plot" mode; when you resume the plot mode, the dot is still there and so you can see whether the orbit has shrunk overall or expanded, whether it has gained or lost energy.

There is a certain amount of total energy in this sytem and that is a conserved quantity. A very early result I noticed is that the orbit of Jupiter (the "fake" one with 3.75 Jupiter masses) shrank just a little and the other three objects moved out quite a bit. That makes sense; the inner object moves the fastest and so has the most energy.

But a little while later, all the orbits shrank. I still havent' figured that one out, but I did notice (due to panicky searching) that in this case, all the orbits were quite circular. I think there must be some mathematical formula there, but I don't know what it is.

Addressing the results of these simulations to the Main Question, I'm going to keep the #13 square in the betting pool for how many "terroids" Kepler will find. But I have moved to thinking now that that is probably pessimistic, rather than optimistic, i.e. my estimate of the number of terroids has gone up.

Hooray!

:)

-Michael C. Emmert