Space is ridiculously vast, even within the Solar system. Mind-bogglingly so, to the point that even most upgraded human minds still can't grasp the utter emptiness. It's the biggest obstacle to interplanetary travel, never mind interstellar. Given the dimensions of the asteroid belt, and the actual mass involved (an estimated 3.6 x 10^21 kilograms, or only about four percent of the mass of Luna), an asteroid might go a very long time indeed without seeing a neighbor. The biggest exception comes in the form of asteroid families, which are formed when one strikes another and the shards go off in the same direction. These two rocks are merely a close pass, already getting farther and farther apart.
Asteroid sighted, the Joyeuse's automated systems leap into action. Despite herself, Sitri feels a quick thrill of adrenaline as ship operations steps up. Reaction mass burns at a fantastic rate as the ship orients into a matching vector on the nearest asteroid, the alchemy tanks coming dangerously close to bottoming out. This takes almost a full Earth day, and the maneuvers alternately press Sitri flat as a pancake or make her guts flipflop like a circus performer on meth.
Once in proximity, a set of tethered drones – little more than fancy drills – drift across the gap between ship and rock, and impact on protected undersides. Diamond-tipped spikes jam into the rock, then wind in tight to hold steady in the asteroid's microgravity. Core samples are drilled out over the next day or so, shipped up the tether back into the Joyeuse, and analyzed for mineral content while Sitri prepares the official claim report marking the asteroid as the property of StarGen Aerospace. All she needs to actually do is include a sample analysis and sign off as witness with her biometrics, a formality agreed-upon by all the major aerospace corporations and enforced down the chain. Otherwise, every corporation would have simultaneously filed claim on the basis of unprovable “unmanned probe” visits, turning the interplanetary gold rush into a corporate bloodbath. And despite the relative emptiness of the belt, 4% of the Moon's mass is still plenty to go around. So far.