Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: [Phys-l] Fun orbital problem

On Jan 2, 2008, at 10:53 PM, Ludwik Kowalski wrote:

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?

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.
_______________________________________________________
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/