Final answer:
The speed with which the four identical stars orbit their common center of mass in the given arrangement is found using Newton's law of universal gravitation and the centripetal force formula, resulting in the orbital speed v = √(GM/D).
Step-by-step explanation:
To find the speed at which four identical stars of mass M are rotating around their common center of mass as they occupy the corners of an enormous square of side length D, we apply principles from celestial mechanics. The stars will experience a gravitational force towards the center which will provide the necessary centripetal force to keep them in a circular orbit around the center of the square.
The gravitational force acting on each star due to the other three stars can be simplified given that these forces will act along the diagonals of the square. As all stars have the same mass and distance from the center, we use Newton's law of universal gravitation and the formula for centripetal force to set up the equation:
GMm/D² = mv²/D, where G is the gravitational constant, M is the mass of the star, m is also the mass of the star (which ends up cancelling out), D is the distance to the center of the square, and v is the orbital speed of each star.
Solving for v we find that the orbital speed is v = √(GM/D). This is the speed of each star as they orbit the common center of mass.