A particular star in a binary star system orbits the other in an elliptical orbit with a semimajor axis of 3 au and a period of 5 years. What is the sum of the masses of the two stars in the system?
A) 0.9
B) 1.1
C) 13.9
D) 0.07

Question

Quizplus · Accepted Answer

The sum of the masses of the two stars in a binary system can be calculated using Kepler's third law of planetary motion, which states that the square of the orbital period (P) of a planet (or star in this case) is directly proportional to the cube of the semimajor axis (a) of its orbit around the sun (or the other star in a binary system). The formula, when adapted for a binary star system and expressed in terms of the masses of the two stars (M1 + M2), is given by:$$ P^2 = \frac{4\pi^2}{G(M_1 + M_2)}a^3 $$Where:- $ P $ is the orbital period in years,- $ a $ is the semimajor axis in astronomical units (AU),- $ G $ is the gravitational constant, but when using AU for the semimajor axis, years for the period, and solar masses for the mass, $ G $ effectively becomes a constant that allows us to simplify the equation to $ P^2 = a^3 / (M_1 + M_2) $,- $ M_1 + M_2 $ is the sum of the masses of the two stars in solar masses.Given that $ a = 3 $ AU and $ P = 5 $ years, we can rearrange the formula to solve for $ M_1 + M_2 $:$$ 5^2 = \frac{3^3}{M_1 + M_2} $$$$ 25 = \frac{27}{M_1 + M_2} $$$$ M_1 + M_2 = \frac{27}{25} $$$$ M_1 + M_2 = 1.08 $$This value is closest to the given option B) 1.1, which is the correct answer. This calculation assumes the units are consistent with the simplified version of Kepler's third law for binary star systems, where the period is in years, the semimajor axis is in astronomical units, and the mass is in solar masses.

A Particular Star in a Binary Star System Orbits the Other