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Why do astronauts float on the ISS?

Short answer

Because the Station and its crew fall together. At about 400 km altitude, Earth's gravity is only slightly weaker than at ground level, but the ISS moves fast enough to circle the planet in about 90 minutes: its constant fall curves its path, and on board nothing presses against anything else anymore.

Why do astronauts float on the ISS?
Astronaut Peggy Whitson floats in the Quest/Airlock during Expedition Five on the ISS

An orbit is a fall that keeps missing

The Station has not escaped gravity: at about 400 km altitude, Earth's pull is only slightly weaker than at ground level. What changes is speed. The ISS circles the planet in about 90 minutes, so its fall toward Earth's center only curves its path instead of bringing it back down. Structure, cabin air, objects and crew all fall together, at the same rate. The Moon obeys the same principle.

No more weight, because nothing pushes back anymore

The sensation of weight comes from contact: floor, seat, soles of feet. When everything falls together, that contact disappears and bodies drift. The consequences take up a good deal of the work on board: body fluids shift toward the chest and face, the spine relaxes, weight-bearing muscles and bones stop being used, which is why daily exercise sessions are required. Nothing stays put anywhere either, which explains the straps, the velcro and the handrails everywhere.

It's called microgravity, and that's not a minor detail

Free fall on board is not perfect. Friction from residual atmosphere, orbit boosts, fans, spacecraft docking and crew movements produce small accelerations. Some experiments are sensitive to this and travel in damped racks. This is also why astronauts strap in before maneuvers: during a thrust, acceleration briefly restores an up and a down. The word weightlessness describes the sensation, while microgravity describes the actually measured environment.

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