Orbital Mechanics

In the case of a spherical galaxy with a uniform matter distribution, we can apply the concepts of gravitational force and orbital mechanics. The force on a star of mass at a distance from the galactic center is due to the gravitational attraction from the mass inside that radius. 1. Gravitational Force: According to Newton's law of gravitation, the gravitational force acting on the star is: where is the mass inside the radius , which for a uniform distribution is proportional to . Since , the effective mass inside radius is given by: where is the total mass of the galaxy. 2. Centripetal Force: The centripetal force required for circular motion is: 3. Equating the Forces: For stable orbital motion, these two forces must be equal: Simplifying the equation, we get: Thus, the orbital speed is proportional to . Since the question asks for the relationship at a specific radius , we substitute to obtain: This matches the expression in option (A). Therefore, the correct answer is option (A).