Question

Two stars of mass m and M, separated by a distance d, revolve in circular orbits about their center of mass. Show that each star has the same period and find it.

Answer 1

The time period of each star is T = 2×π×d×√d / √G(m+M).

Step-by-step explanation:

As, the time period is a time required for a star to complete its one revolution around the center of mass:

So, the formula to calculate the time period is:

T = 2×π×r / v

Here, 'r' is the center of mass of each star having mass 'M' and 'v' is the orbital velocity.

The center of mass of two star having mass 'M' and 'm' is,

r = M×d / (m+M)

Here, 'd' is the distance between two stars

As, we know that these two stars revolving due to the mutual gravitation attraction, so the centripetal force towards the center of mass is equal to the gravitational force,

(m×v^2) / r = G×M×m / d^2

'G' is the universal gravitational constant.

put r = M×d / (m+M) in the above equation:

(m×v^2) / M×d / (m+M) = G×M×m / d^2

v^2 = G×M^2 / d×(m+M)

v = M√G / √(d×(m+M))

put this in T = 2×π×r / v

T = 2×π×r / M√G / √(d×(m+M))

T = 2×π×(M×d/(m+M)) / M√G/√(d×(m+M))

So, the time period of each star is:

T = 2×π×d×√d / √G(m+M)