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A small population of asteroids pass by both the Earth and Mars in their orbits. So the idea is that a spacecraft containing Mars-bound astronauts could rendezvous with one of these objects as it goes by the Earth and travel with it until it nears the Red Planet.
In one version of the idea, the astronauts would actually dig a hole in the asteroid, put the spacecraft inside and cover it over with material from the asteroid. Within this protective burrow, the spacecraft would be shielded from cosmic rays during the six- to 10- month journey to Mars.