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Hyper velocity stars are starts that travel at a much faster rate then the Galactic keplerian velocity. Each HVS is thought to have started out as a member of a binary, which approaches a black hole and was tidally torn apart. We will first calculate the distance from the black hole at which the rapture occurs. Suppose we have a symmetric binary, i.e. two equal mass stars (both of mass $ m $) in a circular orbit around each other, separated by a distance $ a $. The distance at which the force between the two stars is the same order of magnitude as the shear from the black hole is

$ \frac{G M m}{r^2} \frac{a}{r} \approx \frac{G m^2}{a^3} \Rightarrow r = a \left( \frac{M}{m} \right)^{1/3} $

The difference in potential energies between the two stars is

$ \Delta E \approx v^2 \approx \frac{G M}{r} \frac{a}{r} \approx \frac{G M}{r} \left( \frac{M}{m} \right)^{1/3} $

Hence the terminal velocity of the escaping companion is

$ v \approx \sqrt{\frac{G M}{r}} \left(\frac{M}{m} \right)^{1/6} $

For two solar mass stars approaching a $ 10^6 $ solar mass black hole, the terminal velocity can be 10 times higher than the escape velocity.