Question

A double star is a system of two stars moving around the centre of inertia of the system due to gravitation. Find the distance between the components of the double star, if its total mass equals $M$ and the period of revolution $T$.

Answer 1

The double star can be replaced by a single star of mass $\frac{m_1{m}_2}{m_1+m_2}$ moving about the centre of mass subjected to the force $\frac{\gamma m_1{m}_2}{r^2}$. Then
$T=\frac{2\pi r^{\frac{3}{2}}}{\sqrt{\frac{\gamma m_1{m}_2}{\frac{m_1{m}_2}{m_1+m_2}}}}=\frac{2\pi r^{\frac{3}{2}}}{\sqrt{\gamma M}}$
So, $r^{\frac{3}{2}}=\frac{T}{2\pi }\sqrt{\gamma M}$
or, $r={\left(\frac{T}{2\pi }\right)}^{\frac{2}{3}}(\gamma M{)}^{\frac{1}{3}}=\sqrt[3]{3\gamma M{\left(\frac{T}{2\pi }\right)}^2}$