Asteroid capture can happen when an asteroid approaches a large planetary body.
Typically asteroids that approach close enough to a planet are thrown out into space or impact the body. In rarer instances, the asteroid is captured in orbit around the planet[1]. This is possible with any planetary body given the right conditions.
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Asteroid capture happens when an asteroid has enough velocity to keep "missing" the planet it is falling towards but not enough velocity to escape orbit, in other words, when it reaches a stable orbit around it. Asteroids have natural properties that may lend themselves to be captured. The properties of an asteroid that are most significant to this process are its mass and relative velocity to the planet in question. The mass of the planet is also a key variable.
An asteroid with too much velocity will pass by the planet with a hyperbolic orbit, shooting it out into space. Asteroids that do not have enough velocity will fall into the planet in an impact event.
The earth-moon system creates a particularly unstable situation whereby the orbit of the asteroid is destabilized. However, with the right orbital stabilization techniques, it may be possible to keep an asteroid in orbit above the earth.
Spontaneous Capture relates to a method whereby an asteroid's velocity is not changed and it is captured by the planet.
Aerocapture relates to a method of slowing down an object, which could theoretically be an asteroid. It would require slowing down the asteroid with the atmosphere of the nearby planet.
The problems surrounding causing a man-made capture scenario are many. These stem from the fact that we do not have enough detailed information on the orbital elements of our solar system. Even the slightest variations in estimates for planetary velocity and mass can cause multiple earth radii of change in the asteroid deflection.
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