In the earth-asteroid two-body problem, is it possible for the asteroid to be captured into an elliptical orbit around earth?

Question

Consider an asteroid of mass $m$ rushing towards the earth. What is the mathematical condition under which the asteroid will be captured into an orbit around the earth (given that the asteroid and the earth are the only astrophysical objects in the universe)?

Additions after existing answer According to the current answer, in such a situation, asteroid capture into a stable orbit is not possible. I know that stable elliptical orbits in an inverse square force field are allowed only if the total energy $E$ of the asteroid (its kinetic energy + potential energy is the gravitational field of the earth) is negative ($E<0$).

Can we conclude that this condition is not satisfied for the situation at hand? Will the asteroid hit the earth or escape it completely in a hyperbolic orbit?

Answer 1

A passing asteroid will not go into a closed orbit around the earth unless it can lose energy by, for example, splitting in two, grazing the atmosphere or interacting gravitationally with a third body such as the moon.