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Question

# Two equal-mass stars maintain a constant distance apart of $8.0 \times 10^{11} \mathrm{~m}$ and revolve about a point midway between them at a rate of one revolution every $12.6 \mathrm{yr}$. $(a)$ Why don't the two stars crash into one another due to the gravitational force between them? $(b)$ What must be the mass of each star?

Solution

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(a) The stars are in free fall one towards another but they have high enough tangential velocity that they orbit around eachother. The force of attraction between them (gravitational force one star exerts on the other one) is enough to change the direction of their velocity towards the center of their orbit i.e. it plays the role of the centripetal force.

(b) From the time of revolution

$T=12.6\text{ yr.}=3.97\cdot10^{8}\text{ s},$

we can find the speed of revolution as

$v=\frac{\pi d}{T},$

where the diameter of their orbit $d$ is equal to the distance between them. According to the universal law of gravity given in eq. (5-4) in the book, the magnitude of the force each star exerts on the other one is

$F=G\frac{m^2}{d^2}.$

This force play the role of the centripetal force so it must be

$G\frac{m^2}{d^2}=\frac{mv^2}{r}=m\frac{2v^2}{d}=m\frac{2\pi^2d}{T^2}.$

Solving for $m$ we find that the mass of each star is

$\boxed{m=\frac{2\pi^2d^3}{GT^2}=9.6\cdot10^{29}\text{ kg}.}$

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