Where does an object dropped in space go?

What happens, step by step.

1. 01

   In orbit, nothing really floats

   Inside the station, everything seems weightless. In fact, gravity at 420 km is still 88% of its value at the ground: the station and everything in it fall around the Earth at 7.66 km/s, one orbit every 93 minutes. A released object falls exactly like the station, which is why it seems to hang still. But the slightest push puts it on a slightly different orbit, and that difference draws its path as seen from the station.

2. 02

   Method 1: the straight line

   Without gravity, an object pushed at Δv would go straight on, covering Δv × t. That holds for a minute or two: beyond that, the error grows as the square of time, and after a quarter of an orbit the straight line is completely wrong. Thrown forward, the object even ends up behind.

3. 03

   Method 2: Clohessy-Wiltshire (Hill)

   In 1878, George Hill wrote down the motion of the Moon as seen from a frame rotating with the Earth; in 1960, Clohessy and Wiltshire adapted these equations to the rendezvous of two satellites. We work in the station’s frame (V-bar forward, R-bar towards the zenith, H-bar to the side) and assume the gap is small compared with the radius of the orbit. The equations become linear and can be solved by hand: thrown forward at Δv, the object climbs to 4Δv/n and then falls back 6π·Δv/n per orbit (n: angular rate of the orbit); thrown upward, it traces an ellipse twice as long as it is tall and returns to its starting point after one orbit; thrown sideways, it oscillates and crosses the station’s orbit every half orbit. It is the basic tool for rendezvous, approaches and deployments.

4. 04

   Method 3: the two-body problem

   The station and the object are each computed on their exact Keplerian orbit around a round Earth, by numerical integration (Dormand-Prince Runge-Kutta, order 5, adaptive step), then their difference is expressed in the station’s frame. For small throws and short times, it matches Clohessy-Wiltshire to within a few centimetres. When the object drifts hundreds of kilometres away, CW, written along straight axes, no longer follows the curvature of the orbit: the error is first of all geometric.

5. 05

   Method 4: a flattened Earth, and air

   The Earth bulges at the equator (the J2 term): the plane of the orbits turns by about 5° a day at a 51.6° inclination. Since the station and the object turn almost together, this barely changes the gap between them. Air changes everything: at 400 km, its density is still around 10⁻¹² kg/m³, enough to brake. What matters is the ballistic coefficient, mass divided by Cd × area: a bag or a CubeSat (40 to 55 kg/m²) is braked proportionally more than the station (≈ 150 kg/m²), sinks faster and, on a lower orbit, moves faster and gets ahead.

6. 06

   Why the prediction stays uncertain

   Air density depends on the Sun: at 420 km, it is 10 to 20 times higher when the Sun is active, and it can double within hours during a geomagnetic storm. The area an object presents to the air changes as it tumbles. And the station itself manoeuvres. The close-approach calculations used by operators therefore start from fresh tracking data and are redone regularly, with an explicit error margin.