And note that this would apply even to the moon — removing (say) water from ice at the poles and using it in a rocket to deliver material to lunar or other orbit would theoretically change its orbit around the earth, even if immeasurably. But surely no one would agree that the U.S. owns the moon because we pushed it a little bit with the Apollo ascent stages or slammed an upper stage into it to determine whether or not it had water.
So how much would the orbit need to be changed in order to make a claim on the whole body (assuming that a new orbit allowed such a claim under customary law)? Clearly, moving it from an earth-crossing heliocentric orbit (the proposed location of the Planetary Resources’ initial prospecting targets) to an orbit around the earth would be an indisputable new location. But what if only a few meters per second change of velocity were imparted as a result of the mining operations? Technically, it would now be in a new, artificial orbit, but it would be trivially different than the natural one. What would be the criteria by which a new orbit was declared? Would they be qualitative (e.g., orbiting a new body) and/or quantitative (minimum of X meters per second)? One way of deciding the question would be if the change was deliberate (e.g., attaching a rocket, or a electric mass driver that catapulted asteroidal material to change the momentum, with a prior notification of intent for new orbit) versus inadvertent (routine mining operations with no stated intent for orbit change).
Of course, asteroid ownership is a dual-edged sword. With power to move great objects comes great responsibility. Under the 1972 Liability Convention to the Outer Space Treaty, a company that moved an asteroid to the vicinity of earth, or even changed its orbit so that it eventually plowed into it, would be liable for the damage, which means that the U.S. as “the States Party” under which Planetary Resources would operate would have to regulate it. There is currently no legislation allowing the U.S. to do so, other than having to satisfy the State Department that there are no problems, in order for the FAA to issue a launch license. As they get closer to actually demonstrating such a capability, it’s likely that such legislation will be drafted and passed, to keep the country in treaty compliance (just as the Commercial Space Launch Act was passed in 2004 to allow the FAA to license commercial launch).
In any event, no one knows what the legal requirements and responsibilities are for now, and no one will until it is tested in a court of law, either federal or international. And in the absence of such knowledge, the company (and its competitors) operate in an area of legal uncertainty. For now, though, it doesn’t seem to be slowing them down.