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, we can use the concept of center of mass.
The center of mass of a system is the point where the total mass of the system can be considered to be concentrated. In a binary star system, the center of mass lies between the two stars, and the distance from each star to the center of mass depends on their masses.
Let's define a few variables:
- M1 = mass of the larger star
- M2 = mass of the smaller star
- d = distance between the centers of the two stars
- r1 = distance of the larger star's center from the center of mass
- r2 = distance of the smaller star's center from the center of mass
According to the law of conservation of momentum, the center of mass of a system remains stationary unless acted upon by an external force. Since the two stars are orbiting around their center of mass, we can equate the gravitational forces between the two stars to keep the system in balance:
M1 * (G * M2/d^2) = M2 * (G * M1/d^2)
Here, G denotes the universal gravitational constant.
Cancelling out G/d^2 and rearranging the equation, we get:
M1 / M2 = r2^2 / r1^2
Given that M1 = 3.50 * 10^30 kg, r1 = 3.00 * 10^11 m, r2 = 7.80 * 10^11 m, we can substitute these values into the equation:
3.50 * 10^30 kg / M2 = (7.80 * 10^11 m)^2 / (3.00 * 10^11 m)^2
Simplifying this equation, we can solve for M2:
M2 = (3.50 * 10^30 kg * (3.00 * 10^11 m)^2) / (7.80 * 10^11 m)^2
Calculating this expression, we find:
M2 ≈ 2.22 * 10^30 kg
Therefore, the mass of the smaller star in the binary system is approximately 2.22 * 10^30 kg.