Two stars in a binary system orbit around their center of mass. The centers of the two stars are 7.80 1011 m apart. The larger of the two stars has a mass of 3.50 1030 kg, and its center is 3.00 1011 m from the system's center of mass. What is the mass of the smaller star?

To find the mass of the smaller star in the binary system, we can use the concept of the center of mass and apply Newton's law of gravitation.

First, let's denote the mass of the larger star as M1 and the mass of the smaller star as M2.

According to the problem, the larger star has a mass of 3.50 * 10^30 kg, and its center is 3.00 * 10^11 m from the system's center of mass. We can consider this distance as the radius of the orbit of the larger star around the center of mass.

The total mass of the system (Mtotal) can be expressed as:

Mtotal = M1 + M2

The distance between the centers of the two stars is given as 7.80 * 10^11 m. Since the center of mass is the point where the stars orbit around, it lies at the midpoint between the two star centers.

Given that the distance from the system's center of mass to the larger star's center is 3.00 * 10^11 m, we can calculate the distance from the system's center of mass to the smaller star's center (d2) as:

d2 = (total distance between star centers) - (distance to larger star's center)
= (7.80 * 10^11 m) - (3.00 * 10^11 m)
= 4.80 * 10^11 m

Now, we can apply the concept of the center of mass. The distance of each star to the center of mass is inversely proportional to its mass.

M1 * d1 = M2 * d2

Substituting the given values:

(3.50 * 10^30 kg) * (3.00 * 10^11 m) = M2 * (4.80 * 10^11 m)

Simplifying the equation:

10.5 * 10^41 kg * m = 4.80 * 10^11 m * M2

Dividing both sides by 4.80 * 10^11 m:

M2 = (10.5 * 10^41 kg * m) / (4.80 * 10^11 m)
≈ 2.1875 * 10^30 kg

Therefore, the mass of the smaller star is approximately 2.1875 * 10^30 kg.