The throw a spinning station bends

The angle that throws furthest begins the study of projectiles with the simplest case there is: a ball thrown on level ground in uniform gravity, landing furthest when launched at forty-five degrees, its path a parabola. Every essay that follows about projectiles is a variation on that parabola — with drag, with a slope, with a target, on a curved Earth — and the parabola itself is never in question, because gravity is a force on the ball, pulling it down at every moment of its flight.

There is a place where gravity can be imitated almost perfectly at rest and not at all in flight. A large wheel-shaped station in space, spun about its axis, presses its rim against anyone standing on it, and the push needed to keep them going round the circle is $\Omega^2 R$, which can be made equal to the Earth’s $g$. Turning is an acceleration is the whole principle: anyone standing on the rim of a turning wheel must be pushed by the floor to be turned, and the push feels exactly like weight. Spin a ten-metre station at nearly ten turns a minute and its floor presses with one g. Water stays in a glass. A pendulum hangs. But a thrown ball does something no ball on the Earth has ever done, because in the station nothing is pulling it.

Three throws in a spinning room #

Once the ball leaves the hand, the floor is no longer pushing it, and there is no other force. It moves in a straight line at constant speed, by Newton’s first law, with the velocity it had at the moment of release. That velocity is its velocity relative to the hand plus the velocity of the hand itself, which was going round with the floor. The ball flies across the inside of the station along a chord of the rim’s circle, and lands where the chord meets the rim. By then the floor has turned, and the person who threw it has been carried round.

Seen from inside the station, where the thrower stands, each ball traces a curve that looks rather like a parabola and is not one. A ball thrown at forty-five degrees in the direction the station is spinning — spinward — falls short: 1.52 metres, against the Earth’s 2.55 for the same throw. The same throw antispinward carries long, 3.06 metres. A ball thrown straight up to about the height of a ceiling does not come back to the hand; it lands 1.30 metres ahead of the thrower, in the spinward direction. None of those effects is small. In a station this size, a game of catch would be a different game for each player.

The reason the two forty-five-degree throws differ is clear from outside. Thrown spinward, the ball’s speed relative to the floor adds to the floor’s own speed, and the ball moves round the station faster than the floor; it is in effect in a slightly higher orbit, pressed outward harder, and it meets the rim sooner. Thrown antispinward, it subtracts from the floor’s speed; the ball goes round more slowly than the floor, is pressed outward less, and stays up longer. The “gravity” the ball feels in the station’s frame depends on how fast it moves along the floor, which is not true of anything on the Earth.

The same throws from outside #

Drawn by an observer who is not turning with the station, the picture has no curves in it at all.

The rim moves at nearly ten metres a second, so every ball leaves the hand moving sideways at that speed whatever direction it is thrown in, and the throw itself only adds a few metres a second. Each ball crosses the circle on a straight chord. The upward throw is the clearest case: the ball’s chord begins at the thrower’s position and ends some way round the rim, after 0.81 seconds; meanwhile the thrower has walked — been carried — 8.05 metres round the curved floor. A chord is shorter than the arc it cuts across, so the ball, moving along the chord at the thrower’s speed, gets slightly ahead of the thrower and lands spinward.

This is the straight-line motion that the ball a turntable will not throw off shows from the other side: a marble placed on a spinning turntable at rest relative to the room does not move, and the turning table makes it appear to circle. In the station, the floor is the turntable and the thrown ball is the marble, released with the floor’s speed. The difference is that the station’s floor is a rim, which the ball eventually meets, and the place it meets it is decided entirely by geometry.

The forces that are not there writes Newton’s law in a rotating frame and finds three extra terms: the centrifugal term, which pushes outward and here plays the part of gravity; the Coriolis term, which pushes sideways on anything moving and depends on its velocity; and a third that matters only if the rotation changes. In the station’s frame the ball’s curved path is the sum of the first two acting on a body that is in fact force-free. The centrifugal term alone would give a parabola, as gravity does. The Coriolis term is what makes the spinward throw short and the antispinward one long, and what carries the upward throw ahead. In a small station it is as large as the centrifugal term: the ratio of the two is the throw’s speed to the floor’s, five metres a second against ten.

The dropped ball that lands behind #

A ball that is not thrown but simply let go behaves in the same way. Held at rest relative to the station at some height above the floor, it is moving round with the station at the speed of that height, which is slower than the floor’s, since points nearer the axis move more slowly.

Released, it flies straight at that slower speed, along a tangent to the circle it had been travelling, until the rim comes round to meet it. Meanwhile the floor beneath its release point has moved on further, so the ball lands behind it, antispinward. From two metres in a ten-metre station it lands more than a metre behind. The fall also takes longer than on the Earth: 0.76 seconds against 0.64, because a ball released above the floor never had as much “weight” as one standing on it — its speed round the axis was lower, and the push it would have needed to keep going round at that height was less than the floor’s.

On the Earth the corresponding effects exist, with the signs reversed, and are tiny. The ball thrown up that lands to the west follows them: a ball dropped down a hundred-metre mine shaft lands a couple of centimetres to the east, and one thrown up a hundred metres lands a few centimetres to the west. On the Earth “up” is away from the axis of rotation, whereas in a station “up” is towards it, so in the station dropped balls land antispinward and thrown balls spinward, the opposite of the Earth’s pattern. The difference in size is entirely the difference in scale. The Earth is six thousand kilometres in radius and turns once a day; the station is ten metres and turns ten times a minute.

How big a station has to be #

The obvious remedy is to make the station larger and spin it more slowly, which keeps the floor at one g and reduces the speed of the rim relative to the speed of a throw.

The improvement is slower than intuition suggests. A ball tossed up a metre misses the hand by almost a metre in a ten-metre station, by 37 centimetres in a hundred-metre station, by 12 centimetres at a kilometre and by nearly four centimetres in a station ten kilometres across. The miss falls only as the inverse square root of the radius, because the rotation rate needed for one g falls as the inverse square root too: a station a hundred times larger turns only ten times more slowly, and its Coriolis effects are only ten times weaker. Even the vast cylindrical habitats Gerard O’Neill proposed in the 1970s, several kilometres in radius, would have thrown balls visibly off the mark at short range.

Designers of rotating habitats have argued for spin rates below about two turns a minute, which for one g means a radius of at least two hundred metres, and the reason usually given is not projectiles but the inner ear. A person who turns their head in a rotating room feels a Coriolis acceleration in the fluid of their semicircular canals, which produces dizziness and nausea, and experiments in rotating rooms in the 1960s found most people adapted to rates of a few turns a minute only after days. Projectiles and ears report the same thing: in a rotating frame, anything that moves feels a sideways force proportional to the rotation rate and its own speed.

A range that depends on where it is aimed #

On the Earth a throw at a given speed and elevation lands the same distance away whatever compass direction it is thrown in, to the precision anyone would notice. In a small station the direction matters as much as the elevation.

Thrown along the station’s axis — across the width of the wheel rather than round it — a ball is in the plane of no rotation for its sideways component, and its range of 2.32 metres is close to the Earth’s, a little short because its velocity round the station is the floor’s and its centrifugal “gravity” stays fixed while the floor drops away from under it along a curve. Spinward and antispinward the ranges part, from 1.52 to 3.06 metres. The asymmetry is the station’s spin, measured by a thrown ball: everywhere a throw can reach, which on the Earth is the region inside a paraboloid of revolution about the vertical, is in the station a lopsided region elongated antispinward.

A sport played in such a station would have to be designed around this. A basketball shot, analysed in the throw most likely to go in for a hoop on the Earth, would need different aims in different directions on a court whose long axis ran round the station, and the best players would be those who had learned the station’s spin rather than the Earth’s gravity. Engineers designing ball games for proposed habitats have suggested aligning courts along the axis, where the effects are smallest.

The throw that never lands #

One throw in the station has no counterpart on the Earth at all. A ball thrown horizontally antispinward at the speed of the floor itself — just under ten metres a second in the ten-metre station — leaves the hand with no velocity whatever in the outside observer’s frame. The thrower’s motion and the throw cancel exactly. The ball then hangs motionless in space at the height it was released, and the station turns around it. Seen from inside, the ball flies round the whole station at ten metres a second at head height, passing overhead every six seconds, and never comes down. A ball thrown a little slower or a little faster than that drifts slowly down, since its speed in the outside frame is only the small difference, and a body moving slowly in a straight line takes a long time to reach the rim.

Thrown spinward instead at the floor’s speed, the ball moves at twice the floor’s speed and needs four times the inward push to stay at the rim; released from the floor, it is pressed into it with four g. A runner running spinward round the rim of a small station is heavier, and one running antispinward lighter: at the floor’s own speed in the antispinward direction, a runner would be weightless and float off the floor. In a ten-metre station that is a sprinter’s speed, and running round the rim against the spin would be the way to fly.

These are statements about straight lines seen from a turning room. A body free of the floor is moving on a straight line in the outside frame, or not moving at all, and in the station’s frame a body at rest in space appears to circle the axis once a turn. The deflection that closes on itself shows the related case on a rotating planet, where gravity balances the centrifugal term and a free body’s straight line appears as a small circle traced at twice the frame’s rate. The station’s crew would see free bodies go round on such inertial circles whenever they let anything go, and the circles would be as natural to them as falling is to anyone on the Earth.

What artificial gravity is and is not #

The equivalence principle, which the floor that cannot be told from gravity develops from Einstein’s thought experiment of a lift accelerating in empty space, says that no local experiment can distinguish a uniform gravitational field from a uniformly accelerating frame. A spinning station is not uniformly accelerating. Its floor accelerates towards the axis everywhere, but in a direction that changes from place to place, and anything that moves relative to it moves into regions where the required acceleration is different. The station is a rotating frame, not an accelerating one, and the Coriolis term is the local experiment that tells the difference.

Standing still, a crew member feels only the floor’s push and cannot distinguish it from weight, except by the gradient: the push is proportional to distance from the axis, so in a ten-metre station a person’s head, nearly two metres nearer the axis than their feet, weighs a fifth less than it would on the floor. Moving, they feel the Coriolis term at once. Throwing a ball is simply the most visible way of moving. The illusion of gravity holds for things at rest and fails for things in flight, because gravity acts on the ball in flight and the station’s floor does not.

A rigid wheel with no air in it #

The station is a perfect cylinder of the stated radius spinning at constant rate, with air inside moving round with it; in a real station the air would exert a small drag that slowly brings a thrown ball to rest relative to the air, and so to the floor, which reduces but does not remove the effects at the speeds of a throw. Throws are launched from floor level and caught at floor level, so heights are measured from the floor; a throw from shoulder height, or caught in the hand, changes every number without changing the pattern. The “on the Earth” comparisons ignore the Earth’s own rotation, which is negligible at these ranges. Station radii are those of proposals and illustrations, not of any station built; the only crewed rotating experiments so far have been short tethered spins of spacecraft and centrifuges on the ground.

Still open: how much spin people will tolerate #

The size of a habitat is set by the rotation rate its occupants can live with, and that is not well known. Experiments in slowly rotating rooms on the Earth, the longest lasting a few weeks, found that people adapt to about three or four turns a minute after some days, and that higher rates are increasingly troublesome. No one has lived for months in a rotating environment in weightlessness, where the absence of the Earth’s real gravity alters how the inner ear responds, and the answer may be quite different there. Whether a few turns a minute is tolerable — allowing a station of a hundred metres — or whether one turn a minute is needed — demanding a kilometre — is the largest uncertainty in the cost of artificial gravity, and the ball game is the least of what depends on it.

The habit worth carrying away is to ask what is pushing a body in flight. In a station spun to give one g, nothing pushes a thrown ball: it flies a straight chord at its own velocity plus the floor’s, and the turning room makes the chord look like a parabola — short spinward, long antispinward, an upward throw landing a metre ahead and a dropped spanner a metre behind in a station ten metres across, with every effect shrinking only as the square root of the size. Artificial gravity is a floor, not a field, and a ball in the air knows the difference.