A binary star system consists of two stars of masses m1m1m_1 and m2m2m_2. The stars, which gravitationally attract each other, revolve around the center of mass of the system. The star with mass m1m1m_1 has a centripetal acceleration of magnitude a1a1a_1. Note that you do not need to understand universal gravitation to solve this problem.

The scenario described is a binary star system where two stars orbit their common center of mass due to gravitational attraction. The motion of each star can be explained by considering Newton's laws of motion and the centripetal force required for circular motion.

Since the stars orbit the center of mass of the system, each star will feel a centripetal acceleration towards this point. Let's denote the distance of each star from the center of mass as r1 for the star with mass m1, and r2 for the star with mass m2.

The magnitude of the centripetal force acting on each star must be provided by the gravitational force between the two stars. The gravitational force is given by: \[ F = \frac{G \cdot m_1 \cdot m_2}{(r_1 + r_2)^2} \]

But since we are focused on the relation between the masses and the acceleration a1 of star m1, we can consider the formula for the centripetal acceleration a on any mass m orbiting at a distance r from the center of mass, which is: \[ a = \frac{F}{m} \]

Since the star with mass m1 experiences a centripetal acceleration a1, we can write: \[ a_1 = \frac{F}{m_1} = \frac{G \cdot m_2}{(r_1 + r_2)^2} \]

Note that we don't need a similar expression for a2 because your question doesn't mention it specifically. Also, notice how the square of the sum of the distances appears in the denominator, which comes from the inverse square law of gravitation, and r1 is canceled out because we divided the gravitational force by m1.

To find out the precise distances r1 and r2, or to solve for other unknowns, we might need more information and possibly equations that relate to the conservation of momentum and the geometry of the system.

Extra: In a binary star system, the center of mass (often called the barycenter) is the point where the two stars effectively balance each other because of their masses. For example, if two stars have equal masses, the center of mass will be located exactly in the middle at an equal distance from both. However, if one star is more massive than the other, the center of mass will be closer to the more massive star.

The concept of centripetal acceleration is important for understanding the motion of objects moving in circles or curves. It's the acceleration that acts toward the center of the circle and is necessary to change the direction of the object's velocity without changing its speed. It's not a force in itself but rather the result of a force, such as gravity in this case, acting perpendicular to the object's direction of motion.

In the context of the universe, understanding binary star systems helps astronomers learn about the distribution of mass in galaxies and the dynamics of celestial objects. Observing and analyzing the motion of stars within such a system can also give clues about the presence of additional planets or other unseen masses due to the gravitational effects they have on the visible stars.