Question

In a binary star system, if the orbital period of the secondary star doubles while the semi-major axis remains constant, by what factor must the total mass of the system change according to Newton’s version of Kepler’s Third Law?

Accepted Answer

Newton&#x27;s version of Kepler&#x27;s Third Law is expressed by the formula P squared equals four times pi squared times the semi-major axis cubed, divided by the gravitational constant times the total mass of the system. In this equation, P represents the orbital period, which is the time it takes for one star to complete an orbit, and the semi-major axis represents the average distance between the centers of the two stars. Because the semi-major axis remains constant, the term four times pi squared divided by the gravitational constant times the semi-major axis cubed functions as a constant value. This means that the square of the orbital period is inversely proportional to the total mass of the system. If the orbital period doubles, the square of the period increases by a factor of four, since two squared equals four. To maintain the equality of the equation while the period squared increases by a factor of four, the total mass in the denominator must decrease by a factor of four. Therefore, the total mass of the system must change by a factor of one-fourth, or 0.25.

Home → All Courses → Science and Mathematics Courses → Astronomy → Flashcard

In a binary star system, if the orbital period of the secondary star doubles while the semi-major axis remains constant, by what factor must the total mass of the system change according to Newton’s version of Kepler’s Third Law?