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Re: [Phys-l] Fun orbital problem



On Jan 4, 2008, at 1:25 PM, John Denker wrote:

On 01/04/2008 01:07 PM, Ludwik Kowalski wrote:
Assuming Carl is away from the Internet access, I just simulated his
situation with I.P. The two stars have identical mass of 2e30 kg; the
distance between them is 1.5*10^15 m. The stars orbited exactly the
same circle many times, as expected. Then the simulation was stopped
and a much less massive satellite (m=2*10^25 kg) was added at r=
3.595e^15 m. The three particles were initially on the horizontal
line,
at locations -1.5e15, +1.5e15 and 3.595e15 meters. The speeds of large
stars were 149.1 m/s (clockwise) while the speed of the satellite was
357.36 m/s (also clockwise).

Under such initial conditions, the three particles were initially
collinear. But not for too long. After about one T (period for each
star) the satellite started approaching the nearest star. Then it
started orbiting around it. But this also did not last too long. It
less than 2*T it hit the small circle representing the star which was
initially far away from it.

If you are interested in stability, this is your chance to
investigate stability.

It looks like there is roundoff error in the initial conditions
on the order of few parts in 10^4 since I doubt that 149.1 m/s
is the exact answer. So run the simulation several times with
various initial conditions, perturbed by something on the order
of parts in 10^4, and see what effect the perturbations have on
the trajectory.

Thanks for the suggestion; I will do this later.
1) I did calculate initial conditions at the level of 10 digits, as suggested by JohnD. But this did not produce a very significant difference with respect to what is described above. The time needed to noticeably destroy the initial collinearity of three objects was slightly longer. The collision between the satellite and the star occurred after 1.26e14 seconds, instead of 1.22e14 seconds, as yesterday. These times are slightly shorter than 2*T, where T is the period during which each star traces its steady circular orbit.

2) The question addressed to Carl would be eliminated if calculations with much smaller rounding errors produced much more stable orbit if the satellite. The configuration at z=1.198406145 (defining the original location of the satellite, as described by Carl) cannot be called stable, as he does. I am a little disappointed. And I hope that Carl is well. He will probably comment on the concept of motional stability soon.

3) What is the criterion for distinguishing stable motion from unstable motion? This is not a scholastic question.

_______________________________________________________
Ludwik Kowalski, a retired physicist
5 Horizon Road, apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at http://csam.montclair.edu/~kowalski/cf/