Which expression correctly gives the orbital speed for one of the stars in a binary star system?

In a binary star system, stars orbit their common center of mass, and the orbital speed of one of the stars can be derived using principles from gravitational physics. The gravitational force provides the necessary centripetal force that keeps the star in orbit.

The correct expression for the orbital speed ( v ) of a star in a binary star system is derived from the balance of gravitational force and centripetal force. The force of gravity between two masses can be described by Newton's law of gravitation, given by the equation:

[ F = \frac{G M m}{R^2} ]

where ( G ) is the gravitational constant, ( M ) is the mass of the other star and ( R ) is the distance between the two stars.

For a star in orbit, the centripetal force required to maintain circular motion is provided by this gravitational force. The centripetal force can be expressed as:

[ F = \frac{m v^2}{R} ]

Setting these forces equal gives us:

[ \frac{G M m}{R^2} = \frac{m v^2}{R} ]

By simplifying the equation (canceling ( m ) from both sides and