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I wonder if is possible to find a binary system in which one star is orbiting the center of mass in one direction and the latter orbiting in the opposite direction.

Would this transgress angular momentum conservation?

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The question as it was originally written, and still is written as of 4:21 PM EST October 20, 2020 is ambigious, since "orbiting in the opposite direction" has two possible meanings, and there is a different answer for each meaning of "orbiting in the opposite direction".

The answer depends on what is meant by "orbiting in the opposite direction".

In a binary star system, both stars orbit around the center of mass of the system.

To make the system as simple as possible imagine that the two stars have equal mass and thus that the center of gravity is always halfway between them. No continue making it simple by assuming that the ellipitcal orbits of the two happen to be perfectly circle.

Thus you can imagine or draw a circle for the orbit, with the two stars directly opposite each other, and the center of gravity of the system (not counting the gravityof any hopotheticla planets in the system) being in the center of the circle, directly between the two stars. As the stars move, the line between also moves, and the line always passes though the center of gravity between them, and the two stars are always exactly on opposite sides of the center of gravity.

Imagine that you are viewing this double stars system from a distance along a line which passes thrugh the center of gravity and is perpendicular to the plane in which the two stars orbit around each other and the center of gravity.

If one of the stars is moving clockwise from your point of view, the other star will also have to be moving clockwise from your point of view. If one of the stars is moving counterclockwise from your point of view, the other star will also have to be moving counterclockwise from your point of view. They have to do so in order to remain on opposite sides of the center of gravity between them.

So both stars in a binary system are always moving in the same direction, if "in the same direction" means both moving clockwise or counterclockwise.

But from your point of view "above" or "below' the plane in which the two stars orbit, if one star is moving toward your right the other star has to move toward your left, in order for them to stay on opposite sides of the center of gravity between them. From your point of view "above" or "below' the plane in which the two stars orbit, if one star is moving toward your left the other star has to move toward your right, in order for them to stay on opposite sides of the center of gravity between them.

So both stars in a binary system are always moving in opposite dirctions, if "in opposite directions" means that their diections of travel are 180 degrees apart.

So depending on what is meant in the original question by "orbiting in the opposite direction" the answer is:

  1. No, the two stars in a binary system can never orbit in the opposite direction. Both stars must always orbit in the same direction, either clockwise or counterclockwise.

or:

  1. Yes, the two stars in a binary system must always orbit in the opposite direction. If one star is moving right from a point of view, the other must move left from that point of view. If one star is moving left from a point of view, the other must move right from that point of view.

And you can probably find online programs to simulate the orbits of binary stars and see for yourself how they move.

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Stars don't "orbit the centre of mass", they orbit each other.

What is true is that in a two-body system, the orbit of each star is an ellipse with the centre of mass at the focus. What is not true is that the acceleration is towards the centre of mass. There may be nothing at the centre of mass to attract! The acceleration of each star is directed towards the other star. This is the "law of gravity". So it is quite accurate to say "Each star orbits the other star".

An orbit in which one star moves clockwise and the other moves anti-clockwise would break the law of gravity, because the acceleration would not be directed towards the other star.

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  • $\begingroup$ Both of the stars may be orbiting clockwise or counterclockwise, but that is not the same thing as orbiting in the same direction. The two stars have to remain on opposite sides of the center of gravity between them. If both stars are headed toward the same direction they can't remain on opposite sides of the of the center of gravity.. $\endgroup$ – M. A. Golding Oct 2 '20 at 19:53
  • $\begingroup$ Indeed, this is a consequence of the fact that "stars don't orbit their centre of mass" $\endgroup$ – James K Oct 2 '20 at 20:09
  • $\begingroup$ Since the centre of mass is on a direct line between the two stars, then the acceleration is towards the centre of mass. $\endgroup$ – ProfRob Dec 17 '20 at 19:08

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