Reduced Mass Coordinates: Isolated Two-body Problem

According to Newton's law of gravitation, the Sun and Earth exert equal and opposite forces on each other. However, the center of mass of the Earth-Sun system lies very close to the Sun. The Sun has negligible relative velocity with respect to this center of mass, while Earth possesses significant relative velocity, allowing it to revolve around the Sun.

Reduced mass is a quantity that represents the effective mass of a two-body system. It is calculated from the individual masses of both bodies and allows the two-body problem to be mathematically converted into an equivalent one-body problem. This simplification makes it easier to analyze the motion of interacting bodies under central forces like gravity.

When the center of mass is taken as the origin of an isolated system, the velocities of both bodies can be expressed relative to the center of mass velocity. This transformation converts the equations of motion from a two-body system into a single equation involving reduced mass. The system then behaves as one particle with reduced mass acting at the center of mass.

Reduced mass enables the solution of two-body gravitational problems by reducing them to single-body problems. Instead of solving two separate equations of motion for each body, a single equation involving reduced mass describes the system's behavior. This approach is particularly valuable for isolated two-body systems where only mutual gravitational interactions occur.

When the center of mass is considered the origin, it divides the line joining two particles in the inverse ratio of their masses. A more massive body lies closer to the center of mass, while a less massive body is positioned farther away. This geometric relationship ensures that the center of mass remains the balance point of the system.

The reduced mass concept is crucial because it allows astronomers and physicists to treat complex two-body orbital systems as single-body problems. This simplification is essential for analyzing planetary orbits, satellite motion, and binary star systems. By using reduced mass, calculations become tractable while maintaining physical accuracy for circular orbits and critical velocity for satellites.

The system must be isolated with no external forces acting on it, and only mutual interactions between the two bodies are present. Under these conditions, the system behaves as a single particle with reduced mass. The center of mass remains stationary or moves at constant velocity, allowing the relative motion between bodies to be described by a single equation of motion.