AND HERE IS THE MESSAGE COMPOSED EARLIER THIS MORNING:
In playing with a superb educational tool called Interactive Physics, I
simulated a simple solar system. It is a single planet (mass m)
traveling around the sun (mass M>>m) along a circular trajectory. The
four parameters (G, M=2e30 kg,, R=1.5e11 m and the calculated v=2.98e4
m/s) were specified and the RUN button was clicked. The screen started
showing repeatable motion along the expected circle. No one but a
super-pessimist would say that such idealized system, or our real solar
system, is unstable..
Pretending to be a super-pessimist, and trying to make a point, I
disturbed the trajectory for a short time period. More specifically the
mass of the sun was increased for the duration of about 5% of one
period. Then the mass was restored. The result was not unexpected.
Restoring the mass did not restore the orbit; the new orbit became
elliptical. The difference between the long semi-axis of the ellipse
and R depended on the duration of the disturbance and on the solar mass
during the disturbance. That is what one would expect, I have no doubt
that a home-made simulation program, based for example, on Euler's
method, would produce similar results. The elliptical trajectory is
also highly repeatable but the period of repetition is longer that it
was for the circular orbit. All this seems to be consistent with
Kepler's laws (except transition from one repeatable motion to
another). Nothing was significantly different when durations of
simulation steps was reduced by the factor of four. Only durations of
simulations became four time longer.
The situation, however, seems to be paradoxical. On one hand we know
that undisturbed cycles are highly reproducible, on the other hand, we
see that a disturbance-due change, for example in the period T, is not
corrected after the disturbance is removed. Consider circular orbits of
a common conical pendulum. The period of it's cycles is
highly-repeatable. Suppose the mass of the earth is increased. The g
becomes larger than 9.8 m/s^2 and the period becomes shorter. Then the
mass of the earth is restored. Wouldn't this lead to the restoration of
the original period? Yes it would.
So what is my point? I want to know how to use simulations, or any
other simple method, in a disagreement about stability or instability
of a solar system with two identical planets, instead of one. Three
body systems, I was told, are highly unstable. I am confused by the
apparent paradox; what we know to be stable seems to be unstable. I
must be wrong somewhere, in dealing with a simple two-body system.
Where am I wrong?
THIS TWO BODY SYSTEM WOULD PROBABLY BE EASIER TO ANALYZE THAN A FOUR
BODY CIRCULAR SYSTEM DESCRIBED IN MY PREVIOUS MESSAGE. BUT THE QUESTION
IS THE SAME. HOW TO CONVINCE ONESELF ABOUT STABILITY OR INSTABILITY.
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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/