Final answer:
Using Kepler's third law and the concept that gravitational force equals centripetal force, we can derive that the period T of orbit for two equal masses m orbiting each other at radius r is related to the gravitational constant g and the masses m. This shows that at a given orbital radius r, all masses orbit at the same speed and have the same orbital period.
Step-by-step explanation:
To determine the period T of orbit in terms of the gravitational constant g, mass m, and radius r, we can use a form of Kepler's third law, which relates the period of orbit to the radius of orbit and the masses involved. As both stars have equal mass and orbit around a common center, we can equate the gravitational force providing the centripetal force necessary for circular motion. The gravitational force is given by F = (G · m·m) / r² and the centripetal force needed to maintain circular motion is F = m · v² / r. By setting these two forces equal to each other and canceling out one mass (since it applies to both forces), we isolate the velocity, v. To find the period T, it is crucial to note that the orbital velocity v is the circumference of the orbit (2πr) divided by the period T. Thus, after some algebraic manipulations, we obtain an equation that relates the period T to the radius r and gravitational constant G (which in turn depends on g). The period T is therefore found to be a function of these variables, demonstrating that for a given orbital radius r, all masses orbit at the same speed and have the same period, regardless of their mass.