Two small dense stars rotate about their common centre of mass as a binary system with the period 1 year for each. One star is of double the mass of the other and the mass of the lighter one is 1/3 of the mass of the sun. The distance between the stars is xR if distance between the earth & the sun is R, find x.

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Solution

Let, a be the distance between stars and their masses are $\frac{2 M}{3}, \frac{M}{3}$ . M is mass of sun.

$ω_{s} = ω_{e} = \frac{2 π}{365 \times 24 \times 3600} = (\frac{G M}{R^{3}})^{0.5}$

consider the binary star system, Gravitational force = centripetal force

$\frac{G (\frac{2 M}{3}) (\frac{M}{3})}{a^{2}} = \frac{M}{3} (ω_{s})^{2} \frac{2 a}{3}$

$\frac{G M}{a^{2}} = \frac{G M a}{R^{3}} \to a = R$

So, $x = R$ .

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Let, a be the distance between stars and their masses are 2M3,M3. M is mass of sun. ωs=ωe=2π365×24×3600=(GMR3)0.5 consider the binary star system, Gravitational force = centripetal force G(2M3)(M3)a2=M3(ωs)22a3 GMa2=GMaR3→a=R So, x=R.