If the Sun were orbited by a star of 1.8 solar masses at Jupiter's distance of 5.2 au (or, more precisely, the Sun and the other star were orbiting each other 5.2 au apart), what would be the orbital period of the system? (It might be useful to compare the answer with the actual orbital period of Jupiter, 11.9 years.)

Question

Quizplus · Accepted Answer

The orbital period can be calculated using Kepler's third law, which states that the square of the orbital period (T) is directly proportional to the cube of the semi-major axis (a) of its orbit, with the proportionality constant being the same for all objects orbiting the same primary. However, when considering a binary system of two stars, the formula needs to account for the total mass of the system rather than just the mass of the primary. The formula becomes $T^2 = \frac{4\pi^2}{G(M_1 + M_2)}a^3$, where $M_1$ and $M_2$ are the masses of the two stars, $G$ is the gravitational constant, and $a$ is the semi-major axis (in this case, 5.2 AU). Given that the total mass is 1 + 1.8 = 2.8 solar masses, and using the fact that Jupiter's period is 11.9 years at the same distance from a single solar mass, we can infer that the period will be shorter due to the increased mass. The exact calculation would show that the period is significantly less than Jupiter's, leading to the correct answer of 7.1 years, which accounts for the increased gravitational pull from the additional mass, shortening the orbital period.

If the Sun Were Orbited by a Star of 1