Problem1) two stars orbit in circular orbits due to their mutual gravitational f

Question

Problem1) two stars orbit in circular orbits due to their mutual gravitational force. Which of the following is true, including the reasoning.

A) the star that's farther out must orbit at a lower speed and a lower angular velocity because of Kepler's third law.

B) both stars must orbit at the same radius and the same speed because of momentum conservation.

C) Both stars must be opposite each other across the common orbit center, so that the gravitational force points the same direction as the centripetal force. This means they both orbit at the same speed.

D) Both stars must be opposite each other across the common orbit center, so that the gravitational force points the same direction as the centripetal force. This means they both orbit at the same angular velocity.

E) one star must be stationary and the other star orbit around it. The sitiuation is just like a planet orbiting a star or a moon orbiting a planet.

Explanation / Answer

Answer to the question:

True:   D) Both stars orbit facing each other because the only force present is the gravitational, and it acts all the time along the line joining the two stars, so they will be always facing each other. This force provides the centripetal (centripetal means: toward a center) acceleration that makes the stars remain in a closed orbit. Since they are facing each other all the time, the angular speed for both is the same: in equal times, the line joining the stars sweeps the same angle.

The common orbit center is the mass center of the two star system. It lies on the line joining the two stars, nearer the star with more mass. It is around this center of mass that both spin.

The rest of the options are false:

A) (False or rather ambiguous) It’s not clear the statement . “ . . the star that's farther out . . . ” because it is not specified with respect to what point it is “ . . . farther . . .”

B) (False) The same observation as in A). Momentum conservation specifies that the momentum rate of change are opposite and there are other ways to allow this to be accomplished, being the case of equal radius and speed, only valid for equal mass stars, and in the special case of circular orbits.

C) (Part is true and part is false) It is true that : “Both stars must be opposite each other across the common orbit center, so that the gravitational force points the same direction as the centripetal force.” but is false that : “ . . . they both orbit at the same speed. . . .” because the radius of the orbit around this orbit center is shorter for the star with more mass and larger for the star with lesser mass, so they cannot have the same speed.

E) (False) One star cannot be stationary because there is a force acting on it, so there is change of velocity and consequently a change in its position.

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