Re: Democratic candidates address the issue of global warming while Republicans snooze...

On Mon, 7 Jan 2008, Rumpelstiltskin wrote:

On Sun, 6 Jan 2008 23:36:56 -1000, "Alvin E. Toda" <aet@xxxxxxxx>
wrote:

On Sun, 6 Jan 2008, Rumpelstiltskin wrote:

There are greater worries about planetary orbits. It's an oldie but once astronomers were not sure that Saturn and Jupiter were in stable orbits. Since the many body problem is non-linear (varies as the square), it is an unsolvable problem.


I think it's generally agreed that solar systems are fundamentally unstable, though since there's no non-algorithmic solution to the three-body problem, so-far or ever, that can only be "opinion". Our system has apparently been around for four or five billion years in relative stability, but the fact that protocomets and asteroids are occasionally flung out of their stable positions in their belts by close gravitational contact or collision suggests how quickly a cascade of events might change things.

Whatever the solutions, stability can be established. It's just more difficult with an algorightmic approach because the approach gives little information on stability.

Yes the "cascade of events" is not really related to stability. The orbits of both bodies can be predictive of a collision if enough information is fed to the simulation. The simulation would show an intersection of the orbits at the point of collision at some time. The question with Jupiter and Saturn is whether the solution to the three body problem is unstable orbits. Using simulation only, all possible orbits would need to be simulated to answer the stability problem. The more standard procedure is to test the closed solution-- computable or not-- for stability critireon. This was difficult for Jupiter and Saturn because the infinite mathematical series of the function IIRC
was not shown to converge for like a hundred years after astronomers had computed the orbits (don't know the algorithm that they used).

Problem with a computer approximation of the orbits is that it is difficult if not impossible to simulate all the possible conditions of an orbit to see if it is stable. Orbits are a little different from weather. It is possible to make predictions of orbits. But the stability of an orbit with many bodies is another question completely.


Yes. Stephen Wolfram in "A New Kind of Science" was the one from whom I got the idea that "time" might be the steps of an algorithm, and any solution other than the actual working out of the orbits in the real world would, not only in practice but specifically "in principle", take more time than the actual physical working out. That's assuming, of course, that we really never can have a calculus-like solution to the three-body problem "in principle".

Here's what looks like the book online, though I'm not suggesting you read it online. You might glance at something that piques your interest, though this book is quite a project to tackle. I didn't finish it and I didn't come anywhere near mastering it, just out of laziness when I saw what a project it would be to really get an adequate understanding of it. http://www.wolframscience.com/nksonline/toc.html

Thanx but this is probably over my head. I'd rather let the mathematician or programmer compute a solution. With the powerful PCs that we have now, a lot can be done for very little cost.

Perturbations are another way of talking about
steps in the algorithm, except that as far as I
know it hasn't been compacted into a process
such as calculus but remains in the state of the
pre-calculus "delta process". If it can be

I'm not familiar with the methodology, but I would NOT
try to look at this problem as an approximation to
solving the problem, or an algorithm to compute
something. The conceptual problem is to take the close
solution and see if the series expansion of the
integral (may involve the Hamiltonian of the three
possible interactions?) converges and the properties of
the solution indicate stable orbital patterns for
Jupiter and Saturn.

You lost me on "Hamiltonian", I'll have to look
that up.

OK, after checking old-faithful Wikipedia it seems
that if I just think of it as analogous to Feynman's
"sum over histories", that will be close enough for
present purposes.

I'm just guessing at a possible solution by looking at a standard approach. It should lead to an integral solution which would require the a computer to solve. In general the solution is not a simple function. The integral solution is only useful for understanding the features of the orbit such as stability. A programmer would probably take a more direct approach and just use a suitable computer algorithm to solve the differential equations of motion.

The worse thing that can happen for planetary orbits would be to have mathematical non-linearities in the governing equations that can lead to chaotic orbits. Thankfully that's been shown not to happen. I hope we are not tempting global warming naysayers with this discussion.


Yeah, not "non-linear" but the linearity can get
so extraordinary, as with close gravitational
interactions of asteroids such as I mentioned above,
that if the orbits can't get non-linearly "chaotic",
they can get so out-of whack that they might as
well, for practical purposes, be thus. What if
there were a major interaction between two of the
biggest asteroids that sent one of them careening
out of the asteroid belt, which hit Mars and threw
Mars' orbit off enough that it could come under
the influence of Earth's gravity enough that its
path would eventually lead to the possibility of
collision with the earth?

A good point, but with orbits you can compute whether the collisions can take place IF you have enough information-- ie if you know which two asteroids will collide (that's why there is an effort to map all the asteroids. Future super computers may have the power to keep track of them all and predict future collisions). The orbits are not chaotic in the sense that one little tinny winny difference in the simulation can lead to radically different solutions.

For orbits, small differences in the boundary or initial conditions will lead to different solutions. The different solutions will track and not exhibit the chaos that solutions to weather predictions display. There's that famous butterfly diagram of the solution to simplified weather equations of the phase changes in the solution.

It shows an unpredictable bifurcation into two dominant modes of behavior. When will the switchover occur is the unpredictable part. We can know the conditions to set up a bifurcation, but we can never know which branch the bifurcation will take (it generally takes both, but it's unpredictable about the individual sample), or similarly we can never know exactly when in a time series of the evolution of the system that the switchover will happen. I don't know much about chaos. These are just some popular descriptions that you might find in a newspaper or magazine article.
.

Relevant Pages

  • Re: Hurricane Forecasting is inaccurate
    ... the phase space for the orbits of stars ... Just when you think you find some stability, ... The peloton must be a quasi-ergodic ensemble. ...
    (rec.bicycles.racing)
  • Re: Other spacecraft questions
    ... Why is the Juno spacecraft seemingly tied to this specific number of orbits, ... Solar would mean that it in fact should have a very long life as it won't ... The orbit round Jupiter is very elliptical and exactly timed. ...
    (sci.space.shuttle)
  • Re: One Face Gas Giant confusions
    ... orbits have turned up. ... We've had radial velocity data good enough to detect Jupiter mass ... I think people will report if they can show the orbit is closed, ...
    (rec.arts.sf.science)
  • Re: Double gas giants and a habitable world...
    ... Jupiter in mass and volume to a very first approximation, ... planet relationship. ... 4/1 resonance, thus orbits at roughly 2,5 times the distance. ... Gliese 876b is 2,6 Jupiter masses, and orbits at 0,207 au from star. ...
    (rec.arts.sf.science)
  • Re: Double gas giants and a habitable world...
    ... Jupiter in mass and volume to a very first approximation, ... planet relationship. ... 4/1 resonance, thus orbits at roughly 2,5 times the distance. ... unless the geometry puts the giants and inner habitable planet a lot closer ...
    (rec.arts.sf.science)