Re: [Phys-l] Fun orbital problem

[Ludwik Kowalski <kowalskil@mail.montclair.edu>]

On Jan 2, 2008, at 1:35 PM, Carl Mungan wrote:

> > Apropos of nothing in particular, here's a fun problem whose
> > solution  may raise more questions in your mind:
> >
> > Three particles interact only gravitationally, follow circular
> > orbits, and remain at all times collinear.  The radii of two of the
> > orbits is unity.  What is the radius of the third?
> >
> > John Mallinckrodt
> > Cal Poly Pomona
>
> I think this is a solution:
>
> http://usna.edu/Users/physics/mungan/Scholarship/BinaryOrbit.pdf
>
> but let me know if you had something else in mind.

1) Thanks for sharing, Carl.
2) The title of the above web-page is "Synchronous Orbit of a Satellite 
about a Binary Star." This is consistent with the way in which the 
problem is introduced. You wrote: "Two equal-mass stars circle each 
other a distance R apart. A much lighter third body is in line with 
them (but not between them) and orbits them synchronously. What is the 
radius x of its orbit?" . . .

3) After showing that x should be close to 1.2 you make the following 
statement: "What we have calculated here is the second/third Lagrange 
point (well known to be unstable) for an equal-mass astronomical 
system."

4) In my opinion (3) and (2) contradict each other. If the satellite 
was orbiting the binary stars synchronously then the state of the 
motion would be stable. The three objects would be always on the same 
line. If this were true (which is not the case) then the motion would 
be stable (would remain periodic cycle after cycle after cycle. But in 
(3) you say, that the state of motion is is known to be unstable.

5) Do agree that there is a contradiction? If so then how should it be 
resolved? This kind of questions belong to critical thinking, not to 
scholasticism.

6) How do you distinguish a stable state of motion from an unstable one?
_______________________________________________________
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/