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If you had a hypervelocity star travelling through space incredibly quickly, would the hypervelocity star's Oort cloud remain intact? Would the high speed of a hypervelocity star shed the comets and dwarf planets in its Oort cloud due to the star's high speeds, or would the Oort cloud get stretched into a teardrop shape? Could the Oort cloud remain unaffected?

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There is no problem with orbiting a star that moves very fast. Remember that space is mostly empty, so there is nothing like air resistance or friction that would tend to deform the orbit.

Stars that become hypervelocity stars likely tend to lose their Oort clouds. They gain their velocity by passing very close to a massive body and getting a velocity increment, for example by having a secondary star or planet ripped away by tidal forces. That already doesn't bode well for the stability of any Oort cloud.

Simulations show that planets tend to be ejected from hypervelocity systems unless they are very tightly bound, since they have lower mass and respond more strongly to the velocity changes. This suggests that the lightly bound and small bodies in the Oort cloud will be scattered - likely at even higher velocities.

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Hypervelocity stars are moving at up to 1000 $km/s$ relative to the general movement of stars in their part of the galaxy (which is typically 100 $km/s$ relative to one another or the galactic centre). The Oort cloud of a star is a collection of rocky and icy bodies orbiting it up to 1 light year or so away.

Those bodies are too dense and massive to be deflected much by passing through the interstellar gas at these kinds of speeds and they are so widely spread that they could pass through the Oort cloud of another star without much risk of collisions, There are two things that might affect them, though:

  1. Gravitational interactions with stars they pass could strip some Oort cloud bodies, or send then into the system as comets, or whatever

  2. Whatever caused this star to become a hypervelocity star in the first place (eg a close encounter with a supermassive black hole) might well have disrupted the Oort cloud, possibly fatally.

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