Suppose that a binary star system consists of two stars of equal mass. They are observed to be separated by 380 million kilometers and take 9.0 Earth years to orbit about a point midway between them. What is the mass of each?
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I've worked this over so many times, yet every answer seems off. Right now, I want to say it's 1.604 x 10^28, but I've only got two more chances to submit.
To determine the mass of each star in the binary star system, we can use Kepler's Third Law. This law relates the orbital period and the distance between the stars to their masses.
Kepler's Third Law states that the square of the orbital period (T) is proportional to the cube of the semi-major axis (a) of the orbit. In this case, the semi-major axis is the distance between the stars, which is given as 380 million kilometers.
To calculate the mass of each star, we need to determine the gravitational constant (G), which is approximately 6.67430 x 10^-11 m^3 kg^-1 s^-2.
Let's convert the given distance of 380 million kilometers to meters:
380 million kilometers = 380,000,000,000 meters.
Now, we can use the following formula to find the mass of each star:
M = (4π²a³) / (GT²)
where M is the mass, a is the distance (semi-major axis), G is the gravitational constant, and T is the orbital period.
Plugging in the values into the formula:
M = (4π² * (380,000,000,000)^3) / (6.67430 x 10^-11 * (9 * 365.25 * 24 * 60 * 60)^2)
Note that we also convert the orbital period of 9 Earth years into seconds.
Evaluating this expression:
M ≈ 1.603 x 10^29 kg
Therefore, the mass of each star in the binary star system is approximately 1.603 x 10^29 kg, not 1.604 x 10^28 kg as you initially suspected.