Final answer:
In a binary star system, despite having different masses, both stars will take the same amount of time to complete one full orbit around the center of mass.
Step-by-step explanation:
In a binary star system, two stars orbit their common center of mass. The star with the higher mass will be closer to the center of mass and will have a smaller circular orbit compared to the less massive star, which will have a larger orbit. Importantly, both stars will take the same amount of time to complete their orbits around the center of mass. This is due to the fact that in a binary system, the period of the orbit, which is the time taken to complete one full orbit, is the same for both stars regardless of their masses.
Newton's reformulation of Kepler's third law, which relates the orbital period to the semimajor axis of the orbit and the masses of the two stars, confirms this. According to this law, D³ = (M₁ + M₂)P², where D is the semimajor axis, M₁ and M₂ are the masses of the stars, and P is the orbital period. This equation means that the period is related to the total mass of the system, not the individual masses of the stars.
Therefore, the correct answer to the question is c) Both stars take the same time to orbit.