Consider a double star system under the influence of the gravitational force between the stars. Star 1 has mass m1 = 1.72 × 1031 kg and Star 2 has mass m2 = 1.60 × 1031 kg. Assume that each star undergoes uniform circular motion about the center of mass of the system (cm). In the figure below r1 is the distance between Star 1 and cm, and r2 is the distance between Star 2 and cm.

Question

Consider a double star system under the influence of the gravitational force between the stars. Star 1 has mass m1 = 1.72 × 1031 kg and Star 2 has mass m2 = 1.60 × 1031 kg. Assume that each star undergoes uniform circular motion about the center of mass of the system (cm). In the figure below r1 is the distance between Star 1 and cm, and r2 is the distance between Star 2 and cm.

If the stars are always a fixed distance s=r1+r2 = 3.05 × 1018 m apart, what is the period of the orbit (in s)?

Answer 1

To find the period of the orbit, we need to use Kepler's Third Law, which states that the square of the period of an orbit (T) is proportional to the cube of the average distance between the two objects (r).

First, we need to find the average distance between the stars, r. We can use the fact that the total distance between the two stars is given as s = r1 + r2. Rearranging the equation, we get r = s - r1.

Let's calculate r:

r = s - r1
r = 3.05 × 10^18 m - r1

Now, using Kepler's Third Law, we can write:

T^2 = k * r^3

Where k is a constant.

We can solve this equation for T^2:

T^2 = k * (3.05 × 10^18 m - r1)^3

Since we know the masses of the stars (m1 = 1.72 × 10^31 kg and m2 = 1.60 × 10^31 kg), we can calculate r1 using the formula:

r1 = m2 * s / (m1 + m2)
r2 = m1 * s / (m1 + m2)

Substituting the given values:

r1 = (1.60 × 10^31 kg * 3.05 × 10^18 m) / (1.72 × 10^31 kg + 1.60 × 10^31 kg)
r2 = (1.72 × 10^31 kg * 3.05 × 10^18 m) / (1.72 × 10^31 kg + 1.60 × 10^31 kg)

Now that we have r1 and r2, we can substitute them back into the equation for T^2:

T^2 = k * (3.05 × 10^18 m - [(1.60 × 10^31 kg * 3.05 × 10^18 m) / (1.72 × 10^31 kg + 1.60 × 10^31 kg)])^3

Simplifying this expression will give us the value of T^2:

T^2 = k * [(3.05 × 10^18 m * (1.72 × 10^31 kg + 1.60 × 10^31 kg) - 1.60 × 10^31 kg * 3.05 × 10^18 m)^3] / (1.72 × 10^31 kg + 1.60 × 10^31 kg)^3

Finally, we can take the square root of T^2 to find the period, T:

T = sqrt(k * [(3.05 × 10^18 m * (1.72 × 10^31 kg + 1.60 × 10^31 kg) - 1.60 × 10^31 kg * 3.05 × 10^18 m)^3] / (1.72 × 10^31 kg + 1.60 × 10^31 kg)^3)

The exact value for T can be obtained by substituting the appropriate values for k and evaluating this expression.