Suppose that a binary star system consists of two stars of equal mass. They are observed to be separated by 330 million kilometers and take 3.0 Earth years to orbit about a point midway between them. What is the mass of each?

3.3E25

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To determine the mass of each star in the binary system, we can use Kepler's Third Law of Planetary Motion, also known as the Law of Harmonies. This law states that the square of the orbital period of a planet (or in this case, stars) is directly proportional to the cube of the semi-major axis of its orbit.

The equation for Kepler's Third Law can be written as:

T^2 = (4π^2 / G(M1 + M2)) * a^3

where T is the orbital period (in this case, 3 Earth years), G is the gravitational constant, a is the average separation between the stars (in this case, 330 million kilometers converted to meters), M1 and M2 are the masses of each star.

First, let's convert the separation between the stars from kilometers to meters:

330 million kilometers = 330,000,000,000 meters

Now, we can substitute the given values into the equation:

(3 years)^2 = (4π^2 / G(M1 + M2)) * (330,000,000,000 meters)^3

Simplifying the equation further, we have:

9 = (4π^2 / G(M1 + M2)) * (330,000,000,000)^3

Next, we can rearrange the equation to solve for the masses of the stars (M1 and M2):

M1 + M2 = (4π^2 / G) * (330,000,000,000)^3 / 9

Finally, we divide the value obtained by 2 since the stars are of equal mass:

M1 = M2 = [(4π^2 / G) * (330,000,000,000)^3 / 9] / 2

By plugging in the values for π (approximately 3.14159) and G (approximately 6.67430 × 10^(-11) N(m/kg)^2), we can calculate the masses of the stars.