A binary system has a period of 5 years and semi-major axis of 3 AU. The primary star is 2 times the mass of the secondary, what is the approximate ate mass of the secondary star in terms of solar masses?
Group of answer choices
3.2 solar masses
1 solar mass
0.36 solar masses
0.52 solar masses
choose the best answer
The period of a binary system can be determined using Kepler's third law, which states that the square of the period is directly proportional to the cube of the semi-major axis:
T^2 = (4π^2 / GM) * a^3
where T is the period, G is the gravitational constant, M is the total mass of the system, and a is the semi-major axis.
We can rearrange this equation to solve for the total mass of the system:
M = (4π^2 / G) * (a^3 / T^2)
Given that the period (T) is 5 years and the semi-major axis (a) is 3 AU, we can substitute these values into the equation.
M = (4π^2 / G) * (3^3 / 5^2)
The mass of the primary star is given as 2 times the mass of the secondary star.
Let's assume the mass of the secondary star is x solar masses. Then the mass of the primary star would be 2x solar masses.
Therefore, the total mass of the system can be expressed as:
M = x + 2x
M = 3x
Substituting this value into the equation:
3x = (4π^2 / G) * (3^3 / 5^2)
We are looking for the approximate mass of the secondary star, which is x.
Dividing both sides of the equation by 3:
x = (4π^2 / G) * (3^3 / 5^2) / 3
We can now calculate this approximate value using the given values and the necessary constants.
Using G ≈ 6.674 × 10^-11 m^3 / (kg * s^2) and 1 AU ≈ 1.496 × 10^11 meters:
x ≈ (4π^2 / 6.674 × 10^-11) * (3^3 / (5^2 * (1.496 × 10^11)^2)) / 3
x ≈ 0.52019
Therefore, the approximate mass of the secondary star in terms of solar masses is 0.52 solar masses.