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Re: [Phys-l] Binary star question



On Dec 29, 2007, at 8:40 AM, John Denker wrote:

On 12/29/2007 06:47 AM, Savinainen Antti wrote:

I have a problem with one detail in quite a simple exercise.
It goes like this. Two stars are revolving in circular orbits
around their common center of mass. The masses are given and it
is told that their mutual distance remains constant (this distance also given). The period is asked; with the above mentioned
assumption they have the same periods.

Now this is not hard to do and I have no problems with solving
the exercise. I just can't see why their mutual distance remains
constant. I suspect it follows from a conservation law (angular momentum?).

Before asking why something is true, we should ask whether
it is true.

The statement that the distance remains constant is either
tautological or false.
-- If we are told that for this problem the distance is
constant /by hypothesis/ then that's true /by hypothesis/.
It's true for this problem, but not in general.
-- If the problem asserts that in nature the eccentricity
is always zero, then it's just wrong. You can have any
eccentricity you want, from zero on up.
e = 0 circle
e in (0,1) ellipse
e = 1 parabola
e > 1 hyperbola


The physics here is well known. Keplerian orbits. An obvious
way to set up a highly eccentric orbit is to start with two
stars far apart, nearly at rest (i.e. verrry small angular
momentum) and then just let go. The result is a verrry thin
ellipse.

Using Interactive Physics, I was simulating four stars revolving on a circular orbit. Nothing special. Initial conditions are as below:

Star 1 x=0, y=1.5e15m, vx=130 m/s and vy=0
Star 2 x=1.5e15m, y=0, vx=0 m/s and vy=-130 m/s
Star 3 x=0, y=-1.5e15m, vx=- m/s and vy=0
Star 4 x=-1e15m, y=0, vx=0 m/s and vy=130 m/s

The speeds needed for the common circular orbit is close to 292 m/s. For speeds zero stars accelerate toward each others, at speeds larger than 292 m/s they start spiraling away from the circular orbit and distances from the origin (x=y=0) start increasing (for a while). For speed less that 292 they also spiral away from the circular orbit but this time distances from the origin start decreasing. At v=130 m/s, as in my illustration, the stars spiral move toward the origin along the elliptically-looking orbits. The speed is increasing. The impression is that they will collide. But, as we know, that is not what happens. Instead of colliding with each other at the center, the stars reach a distance of a minimum approach (about 1/3 of R, where R is the radius of the circular orbit at 292 m/s) and start moving away from each other. The trajectories again become elliptical-like and speeds are decreasing. The minimal speed is at r=R, where stars again start moving toward each other, along the same trajectories as before. I am tempted to use the terms "perihelion and "aphelion" because no object is located in the center of mass. (I could have placed another star center but I didn't.)

An observer attached to the rotating frame of reference x', y' will see each star oscillating back and forth, more or less like a ball bouncing from a floor, along the rotating semi-axis on which it is located. First its potential energy decreases while its kinetic energy increases; then the potential energy start increasing at the expense of kinetic energy. In the inertial frame (on my screen) of tracks of instantaneous positions of four stars produce a nice flower-like picture. A picture of a slightly more complex flower (with six petals instead of four) would be produced by a similar set of six stars. I wish I could save images of tracks as jpg files. Then I would attach one of them.
P.S.
That is not a message I was composing this morning, when the message posted by Savinainen was posted. But it allows me to ask the same question, using a more complicated picture. How to convince myself that a system of four stars, orbiting along the same circles (in this case with v=292 m/s) is either stable or unstable? The system is highly periodic, each cycle on the screen is predictably identical. Does it mean that the system is stable. Please help. Note that my question, like Savinainen's question addresses a simple case of circular orbiting.
P.P.S.
I strongly recommend Interactive Physics; it is a great educational tool.
_______________________________________________________
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/