Two stars in a binary system orbit around their center of mass. The centers of the two stars are 7.78x1011 m apart. The larger of the two stars has a mass of 4.19x1030 kg, and its center is 1.75x1011 m from the system's center of mass. What is the mass of the smaller star?
m=M/10^30
4.19*1.75 = m*(7.78-1.75) = m * 6.03
so
m = 1.22
so
m = 1.22 * 10^30 kg
To find the mass of the smaller star in the binary system, we can use Kepler's Third Law of Planetary Motion, which also applies to binary systems.
Kepler's Third Law states that the square of the period of revolution of a planet around a star is directly proportional to the cube of the semi-major axis of its elliptical orbit. In simpler terms, it relates the orbital period of a planet (or star) to its distance from the center of mass.
In this case, since the larger star has a known mass (4.19x1030 kg) and its distance from the system's center of mass is given (1.75x1011 m), we can use this information to find the period of revolution.
Let's start by determining the period of revolution of the system (T). We can use the following formula:
T^2 = (4π^2 / G) * (r^3 / M)
where:
- T is the period of revolution
- π is a mathematical constant (approximately equal to 3.14159)
- G is the gravitational constant (approximately equal to 6.67430 × 10^-11 m^3 kg^-1 s^-2)
- r is the distance between the centers of the stars (7.78x1011 m)
- M is the mass of the larger star (4.19x1030 kg)
Now, let's calculate the period of revolution:
T^2 = (4π^2 / G) * (r^3 / M)
T^2 = ((4 * 3.14159^2) / (6.67430 × 10^-11)) * ((7.78x1011)^3 / (4.19x1030))
Solving this equation will give us the value of T^2, which is the square of the period of revolution.
Next, we need to determine the distance between the center of mass and the smaller star. Since we know the distance between the centers of the stars (7.78x1011 m) and the distance between the center of mass and the larger star (1.75x1011 m), we can subtract the distance to find the distance between the center of mass and the smaller star.
Distance to smaller star = Distance between the centers of the stars - Distance to larger star
Distance to smaller star = (7.78x1011 m) - (1.75x1011 m)
After obtaining the distance to the smaller star, we can then use Kepler's Third Law equation again:
T^2 = (4π^2 / G) * (r^3 / m)
where:
- T is the period of revolution (calculated earlier)
- 4π^2 / G is a constant
- r is the distance between the center of mass and the smaller star
- m is the mass of the smaller star (what we're trying to find)
Now, we can rearrange the equation to solve for the mass of the smaller star (m):
m = (4π^2 / G) * (r^3 / T^2)
Substituting the known values, we can calculate the mass of the smaller star.
To solve this problem, we can use the concept of center of mass and gravitational force.
The center of mass of a binary system is given by the equation:
m1 * r1 = m2 * r2
Where m1 and m2 are the masses of the stars, and r1 and r2 are the distances of each star's center from the system's center of mass.
In this case, we are given:
m1 = 4.19x10^30 kg (mass of the larger star)
r1 = 1.75x10^11 m (distance of the larger star's center from the system's center of mass)
r2 = 7.78x10^11 m (distance between the two centers of the stars)
Let's solve the equation for m2, the mass of the smaller star:
m2 = (m1 * r1) / r2
Substituting the given values:
m2 = (4.19x10^30 kg * 1.75x10^11 m) / 7.78x10^11 m
Calculating this expression:
m2 = 9.41313x10^29 kg
Therefore, the mass of the smaller star is approximately 9.41313x10^29 kg.