In a binary star system where one star has mass M and the other has mass 3M, which expression describes the speed of the larger star?

In a binary star system, the gravitational forces and the resulting motion of the stars are governed by their masses and the distance between them. In this case, you have a smaller star with mass M and a larger star with mass 3M. To determine the speed of the larger star, it is important to apply the concept of gravitational dynamics and center of mass.

The key concept is that both stars orbit their common center of mass. The speed of either star in a binary system can be derived from the gravitational force acting between them and is influenced by their distance from the center of mass. The larger star's speed is influenced not only by the gravitational pull from the smaller star but also by the distances of both stars from the center of mass.

The gravitational potential energy in this system can be expressed using the formula involving the gravitational constant G, the mass of the stars, and the distance between them (in terms of ra and rb, where ra is the distance from the smaller star to the center of mass and rb is the distance from the larger star to the center of mass). The total distance between the two stars is the sum of these distances, so we can denote the total distance as ra + rb.

The expression that correctly describes the speed of