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Time Periods of Revolution of Two Stars

Let's discuss a problem where we find out the time periods of revolution of two stars.

The problem: Time Periods of Revolution of Two Stars

The masses of two stars are (m_1) and (m_2) and their separation is (l). Determine the period (T) of their revolution in circular orbits about a common centre.
Since the system is closed, the stars will rotate about their common centre of mass in concentric circles. The equation of motion for the stars will have the form $$ m_1\omega_1^2l_1=F$$ and $$ m_2\omega_2^2l_2=F......(1)$$
Here (\omega_1) and (\omega_2) are the angular velocities of rotation of the stars, (l_1) and (l_2) are the radii of their orbits, (F) is the force of interaction between the stars, equal to (\frac{Gm_1m_2}{l^2}) where (l) is the seperation between the stars and (G) is the gravitational constant.
By the definition of centre of mass,
$$ m_1l_1=m_2l_2$$
$$l_1+l_2=l...... (2)$$
Solving equations 1 and 2 together, we get
$$ \omega_1=\omega_2=\sqrt{\frac{G(m_1+m_2)}{l^3}}$$
The required period of revolution of these stars is $$ T=2\pi l\sqrt{\frac{l}{G(m_1+m_2)}}$$

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