Two stars in a binary system orbit around their center of...

Two stars in a binary system orbit around their center of mass. The centers of the two stars are $7.17 \times 10^{11} \mathrm{m}$ apart. The larger of the two stars has a mass of $3.70 \times 10^{30} \mathrm{kg},$ and its center is $2.08 \times 10^{11} \mathrm{m}$ from the system's center of mass. What is the mass of the smaller star?

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Key Concepts

Center of Mass

In any system of particles or bodies, the center of mass is the unique point where the weighted relative position of the distributed mass sums to zero. In orbital systems, the positions of bodies relative to the center of mass obey the relation that the product of a body’s mass and its distance from the center of mass is constant for all bodies in the system.

Binary Star Systems

Binary star systems consist of two stars that orbit around a shared center of mass. The dynamics of their orbits are governed by mutual gravitational attraction and the relative masses of the stars, with the more massive star being closer to the center of mass compared to the less massive star.

Mass-Distance Relationship in Orbiting Systems

The mass-distance relationship in orbiting systems states that the product of a star’s mass and its distance from the center of mass is equal for both bodies. This inverse relationship implies that a more massive body will orbit closer to the center of mass, and it forms the mathematical basis for determining unknown masses in binary systems.

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