Final answer:
The distance of the barycenter from a star of three times the mass of the Sun in a binary star system with another star of one solar mass, 4 AU apart, is 1 AU. This is calculated using the center of mass formula where one star is three times the mass of the Sun and the other has a mass equal to the Sun's.
Step-by-step explanation:
The question is asking to determine the distance of the barycenter from a star of three times the mass of the Sun in a binary star system, where the other star's mass is equal to the Sun's, and they are 4 AU apart. The barycenter is the center of mass around which two bodies orbit due to gravitational interaction. To find the position of the barycenter, we can use the formula for the center of mass:
r = (m2 * d) / (m1 + m2)
Where m1 and m2 are the masses of the two stars, d is the distance between the two stars, and r is the distance of the barycenter from the more massive star.
In this situation, one star is three times the mass of the Sun (3M), and the other is equal to the mass of the Sun (1M), and they're 4 AU apart (d).
r = (1M * 4 AU) / (3M + 1M) = 4 AU / 4 = 1 AU
Therefore, the barycenter from a star of three times the mass of the Sun in this binary system is 1 AU away from it.