Consider a double star system under the influence of the gravitational force between the stars. Star 1 has mass m1 = 2.32 × 10^31 kg and Star 2 has mass m2 = 1.76 × 10^31 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 = 3.43 × 1018 m apart, what is the period of the orbit (in s)?
To find the period of the orbit, we can use the concept of centripetal force in circular motion.
The gravitational force between the two stars acts as the centripetal force, keeping them in their circular orbits. The centripetal force can be calculated using the equation:
F = (G * m1 * m2) / r^2
Where F is the gravitational force, G is the gravitational constant (6.67 × 10^-11 N * m^2 / kg^2), m1 and m2 are the masses of the stars, and r is the distance between the stars.
In this case, the total distance between the stars s = r1 + r2 is given. We can assume that the distance between the center of mass and each star is equal, so r1 = r2 = s/2.
Therefore, the centripetal force acting on each star is:
F = (G * m1 * m2) / (s/2)^2
F = (4 * G * m1 * m2) / s^2
The centripetal force can also be expressed as:
F = m * (v^2 / r)
Where m is the mass of each star and v is the velocity of each star in its orbit.
We can equate the two expressions for the centripetal force:
m * (v^2 / r) = (4 * G * m1 * m2) / s^2
We can cancel out the mass of the stars from both sides:
(v^2 / r) = (4 * G * m1 * m2) / (m * s^2)
Since the velocities of the stars are the same, we can use the same velocity term for both stars.
Now, let's solve for the period of the orbit. The period, T, is the time it takes for one complete revolution around the center of mass.
T = (2πr) / v
Substituting the value of v from the equation above:
T = (2πr) / √((4 * G * m1 * m2) / (m * s^2))
Plugging in the given values:
m1 = 2.32 × 10^31 kg
m2 = 1.76 × 10^31 kg
s = 3.43 × 10^18 m
First, we need to find the value of r:
r = s/2
r = (3.43 × 10^18 m)/2
r = 1.715 × 10^18 m
Now, we can calculate the period:
T = (2π(1.715 × 10^18 m)) / √((4 * G * 2.32 × 10^31 kg * 1.76 × 10^31 kg) / ((2.32 × 10^31 kg + 1.76 × 10^31 kg) * (3.43 × 10^18 m)^2))
Evaluating the expression will give us the period of the orbit in seconds.