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Can an asteroid pass near Earth at a right velocity and angle to bend its trajectory and make an orbit?
Note: I know about objects such as 2016 HO3, which are quasi-satellites of Earth due to Lagrange points. I'm asking whether an asteroid can become an actual satellite of Earth.
Our Moon makes this possible. An asteroid with a low $V_\infty$ with respect to Earth making a close flyby of the Moon in the right direction could get into a distant orbit around the Earth. That object would likely continue to encounter the Moon, and could be ejected again.
It is important to realize that things cannot just drift into orbit without something to slow them down. This results from conservation of energy. An object moving towards Earth will gain momentum due to gravity as it approaches. Unless something else interferes to slow the object down (like a gravitational assist by the Moon), that kinetic energy will also necessarily be enough to allow the object to escape Earth's gravitational pull.
Aerobraking would not work. If the asteroid shaved off enough speed to put it into orbit, the orbit would be highly eccentric resulting in repeated encounters with the atmosphere and eventual impact.
Capture inherently requires three bodies. By far the easiest way is with the moon being the third body. There is another capture scenario, though--take two asteroids that are orbiting each other. They approach very slowly, when they pass Earth they are moving just over escape velocity. They pass very close, the one moving forward at that point hits the atmosphere and goes splat. This leaves the other one with a periapsis that is dangerously low and an apoapsis that is high enough to be unstable.
Very unlikely.
One interesting thing about orbital trajectories is that they are reversible in time. You can watch a rocket take off from earth and enter orbit, and if you want to know how to land it, you can imagine doing the exact same maneuvers but in reverse. This is handy to get an intuitive grasp on what's possible. If an asteroid could come from deep space and get captured in earth's orbit spontaneously, that would mean that it's possible to do the reverse - have an object in orbit around earth that gets spontaneously launched into deep space completely for free without spending any fuel at all. The only way anything like this would be possible is with a very precise gravitational interaction with a third body such as the moon, but that would have to occur just right to capture an object or launch it out of orbit. Any object approaching earth from deep space is moving at escape velocity, and will need a way to actively slow down to enter earth's orbit.
+1 related: slowest crash on the Moon? and Have there been any documented mini-moons since 2006 RH120?
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It really depends whether you're willing to consider collisions, and how large an asteroid.
An object of sufficient size impacting the Earth would shear off a good amount of material and disturb the orbit of the Earth. This could potentially slow down the impacting object enough while still keeping it distinct from the Earth, causing it to orbit around the Earth.
I believe the currently predominant theory is that this is how the Moon was formed.
