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In a binary star system, why does $$e_1 = e_2 = e$$

where $e_1$,$e_2$, and $e$ are the eccentricities of the three orbits of $m_1$,$m_2$, and the reduced mass, respectively.

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Isn't it just conservation of momentum? Without any external forces, the centre of mass of the binary system must stay in the same place.

For example, imagine that you had a binary system, where one star had a circular orbit and the other, with equal mass, was eccentric. Clearly, as the orbits proceeded, the centre of mass, which is half way between the stars, would have a position which oscillates with time. But with no external forces applied to the binary system, this is forbidden by the conservation of linear momentum.

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