In a previous post describing the Coriolis effect, I mentioned its relevance to space travel—if a rotating habitat is being used to generate simulated gravity, Coriolis deflection can interfere with the performance of simple tasks and, at the extreme, generate motion sickness.

As an example of the sort of effect you could expect to encounter, I posted the following pair of diagrams:The first shows the trajectory of a dropped ball when observed from *outside* a rotating habitat (direction of rotation marked by the blue arrow), with the ball and the experimenter marked at four successive positions they occupy while the ball falls. The ball moves in a straight line, with a velocity determined by the rotation speed of the habitat at its point of release. The floor (and the experimenter) meanwhile move in a curved path, and they travel a little farther than the ball does during its time in flight.

The result, as observed by the experimenter rotating with the habitat, is shown in the second image—the ball appears to be deflected to the right as it falls. To explain this deflection, the experimenter invokes the Coriolis pseudo-force, which I explained in much more detail last time. This rightward deflection of moving objects occurs in all counterclockwise-rotating reference frames (leftward in clockwise-rotating frames).

Having prepared those two diagrams, I got to thinking about the range of possible trajectories one might encounter, while chucking a ball around in a rotating centrifuge. Out of curiosity, I put together some code to sketch the resultant trajectories for objects launched at any angle, with any speed. The results I’ll show are fairly generalizable—it turns out the trajectory depends only on the launch velocity as a proportion of the rotation speed of the habitat. Interesting things happen when the velocity is comparable to the speed of rotation of the habitat floor—at higher velocities trajectories become progressively flatter (and for our purposes, more boring).

First, a bit of terminology. Back in 1970, Larry Niven coined two useful words in his science fiction novel *Ringworld*, which dealt with a (very large!) rotating space habitat. Niven called the direction in which his habitat rotated *spinward*, and the opposite direction *antispinward*. So in the case of an object that’s simply dropped within the habitat, as in the situation diagrammed above, we can say that the object will always hit the floor antispinward of its release point.

Which means you need to impart a little spinward velocity to an object to get it to hit the floor directly below its launch point. Here’s a set of spinward trajectories, as observed in the rotating reference frame of the habitat, with each object being launched “horizontally” (that is, parallel to the part of the curved floor on which our experimenter is standing):

The curve labelled “0” is a launch with no horizontal velocity—just a simple drop, as previously illustrated. The curve labelled “1” is the trajectory of an object that has been thrown with an additional velocity equal to the speed of the habitat’s rotation at the launch height. The red curves up to “5” are objects thrown with twice, three times, four times and five times the local rotation velocity, and the blue curves subdivide the span from “0” to “1” into ten equal increments. At higher velocities, the object falls to the floor in a curve that doesn’t seem too counterintuitive compared to a standard gravitational field.

But if our experimenter turns the other way and throws objects to antispinward, more interesting stuff happens:

The curve labelled “0” is the same trajectory as before. The blue lines are the same increments in launch speed as in the previous diagram, but in this direction the rightward deflection of Coriolis is serving to *lift* each trajectory, so the object flies farther and swoops *around* the curve of the habitat before it strikes the floor. The green trajectory, with a launch velocity of magnitude 0.9 times the local rotation speed, is remarkable. It doesn’t just sweep out of sight to antispinward, it reappears from spinward and makes more than a complete circuit of the habitat before it hits the floor.

For clarity, I saved the red trajectories, with velocities from 1 to 5, for another diagram:

The trajectory labelled “1” simply stays at the same height constantly, in principle going round and round the habitat for ever at the same speed, buoyed up by Coriolis force. (If there’s any air in our habitat, of course, it would actually slow down and fall to the floor because of air resistance.) Trajectories with higher launch velocities become progressively flatter, but still exhibit upward curves.

So what’s going on with Trajectory 1? That object has been launched with a velocity that *exactly cancels the rotation speed* of the habitat. To an outside observer, such an object just hangs in space, stationary, while the habitat rotates around it, carrying the experimenter past the object repeatedly, once per rotation. To the same outside observer, all objects with blue or green antispinward trajectories in my diagram are actually floating slowly to *spinward*, having had some, but not all, of their rotation speed removed—but because the habitat and experimenter move faster to spinward, objects on these slow trajectories recede to *antispinward* in the rotating reference frame.

A diagram may help illustrate this. Here’s how a non-rotating observer sees the situation, when an object is thrown antispinward with a velocity less than the local rotation speed:

Now, back to something I mentioned earlier. To make an object land on the floor directly below its launch point, it needs to be given a little nudge to spinward as it’s released. The closer our experimenter is to the axis of the habitat (the higher above its floor), the more of a nudge the object needs, and the wider the curved trajectory it follows. Here are trajectories for objects launched from a variety of heights within the habitat:

Each of them curves steadily to the right, moving initially spinward and then returning antispinward.

The same thing happens if you launch an object “vertically” (that is, aiming directly towards the spin axis). For each height above the floor of the habitat, there is a unique launch velocity that will allow Coriolis to curve the trajectory around so that it strikes the floor directly below the launch point:

Interestingly, the launch velocity required in this situation initially increases as our experimenter climbs closer to the spin axis, but then *decreases* again at radii less than about 0.3 times the radius of the floor. But, as before, the trajectories get progressively wider as the experimenter climbs closer to the spin axis.

A corollary to all these spinward curves is that, if you want to throw an object up and catch it, you need to throw it a little antispinward of vertical. Its trajectory curves right on the way up and on the way down, and will return to your hand in a closed loop if you have thrown it correctly. The more speed you impart, the more antispinward you need to direct your throw, so we have a family of possible curves that will carry the tossed object back to its starting point:

If you get it wrong, and throw your object too far to antispinward, then the overhead loop may still occur, but the object won’t return to your hand, as in the green trajectory below:

All the trajectories in this diagram have the same launch speed, but different launch directions. The blue trajectory is the perfect throw-and-catch loop. The green trajectory still loops, but the object falls to antispinward. The red trajectory corresponds to a critical launch angle, at which the loop just disappears, leaving the object momentarily stationary in the rotating reference frame, just at the peak of its trajectory. At launch angles flatter than the critical angle, we get something like the black trajectory, in which the object simply rises and then falls again, without any fancy embellishments. It’s important to note that all of these trajectories involve objects that have been launched with antispinward velocities of lower magnitude than the local rotation speed at the launch point. To a non-rotating outside observer, they’re therefore still moving *spinward*, but more slowly than the habitat and experimenter are rotating, so they are moving *antispinward* in the rotating reference frame of the experimenter and the habitat.

What’s happening with the red trajectory is that the experimenter, by choosing an upward trajectory, has propelled the object to a small radial distance within the habitat, to a point where the slower rotation speed exactly matches the object’s slow spinward velocity. So as it passes through this point, the object is momentarily stationary relative to the rotating habitat.

In the green trajectory, the object is thrown higher, and its slow spinward velocity now *exceeds* the rotation speed in a region close to the spin axis. So although it moves antispinward relative to the rotating habitat when it’s close to the floor, it moves spinward relative to the habitat when it’s close to the axis—hence the looping trajectory in the rotating habitat frame.

That’s maybe a bit difficult to visualize, so here’s a picture of what the distribution of rotation speeds looks like in the rotating habitat:

So if the experimenter throws something upwards, it travels into regions that have a lower rotation speed because they’re nearer to the spin axis.

And, again, as with the simple horizontal throws, the trajectory of a thrown object is determined by summing the experimenter’s rotation speed and the launch velocity, like this:

In this case, the rotating experimenter throws an object up and antispinward, but the resulting velocity in the non-rotating frame is pointed up and spinward. Note that the experimenter is initially moving spinward faster than the thrown object is, so will see it recede to antispinward. But to an external, non-rotating observer, the situation looks like this:

At the peak of its trajectory, the object is able to outpace the habitat’s rotation, and so briefly moves *towards* the experimenter again, creating the loop that we saw appear in the rotating reference frame.

So that’s the theory. But would these trajectories be observable in any plausible rotating space habitats?

They would. My diagrams are actually roughly to scale for the small *Discovery* centrifuge that featured in the novel and film *2001: A Space Odyssey*.

As I discussed in a previous post about centrifugal force, that structure, 35 feet across, is probably about as small as a space centrifuge could be without causing serious motion sickness in its inhabitants because of Coriolis effect. In Arthur C. Clarke’s novel, it rotated at 6rpm, to produce the centrifugal equivalent of lunar gravity. In Stanley Kubrick’s film it was necessarily depicted generating the equivalent of Earth’s gravity, which would require it to rotate at about 13rpm. But the rotation speed turns out not to matter, because the centrifugal and Coriolis effects scale equally with angular velocity, so trajectories stay the same. If one of the *Discovery* astronauts dropped an object from a metre above the floor of the centrifuge, it would travel along a curve like the one I’ve illustrated above, landing about 75 centimetres antispinward of its release point. The only difference would be that it would fall more slowly in a centrifuge that was rotating slowly.

The effect is also immune to changes in linear scale—if we make the centrifuge twice as large and drop the object from twice the height, the shape of its trajectory will be the same, and it will land twice as far to antispinward.

This constancy with scaling also applies to trajectories that involve throwing an object—so long as the launch velocity keeps the same proportion to the rotation speed, the trajectory will be the same shape.

For the *Discovery* centrifuge, the rotation speed at floor level is 3.2 m/s (seven miles per hour) in the version that appears in the novel, and 7.2 m/s (16 mph) for the film version. So the astronauts could very easily throw objects into the various trajectories I’ve shown. For *2001*‘s larger space station, shown in the image at the head of this post, the rim speed is 15 m/s (34 mph) for the lunar-gravity version in the novel, and 37 m/s (84 mph) for the 1*g* version in the film. So it would take a fairly strong wrist, or a hand catapult, to launch an object so that it curved out of sight down the long circumferential corridor that featured in the film. If you dropped an object from a metre up in that environment, it would fall a mere eight centimetres to antispinward. *****

And there’s the problem—as the habitat gets larger, the human scale becomes proportionally smaller, so the Coriolis effects become less noticeable. On the scale of an O’Neill habitat, kilometres in diameter, the Coriolis deflection in a fall of one metre at the rim amounts to only a centimetre, and begins to get difficult to see; and the rotation speed at the rim is measured in hundreds of metres per second, so launching objects on interesting trajectories becomes problematic. In these large-scale habitats, the interesting stuff happens only near the axis (where rotation velocities are low), or on large scales (for instance, if an object falls from a great height).

Sadly, then, Tye-Yan “George” Yeh’s beautiful Coriolis fountain can only ever grace the smallest of rotating habitats.

**Postscript: **In response to some queries I’ve received, I’ve written a supplementary article discussing what happens to objects that are moving parallel to the habitat’s rotation axis, and also describing the effect of Coriolis on objects that are *rolling along a surface*, rather than thrown through the air. You can find that here.

**Post-postscript:** If you’re the sort of person who finds this post interesting, you might also be interested in my posts about Human Exposure to Vacuum, Parts One (theory) and Two (experimental evidence).

**Note:** Just to bring my points of reference into the 21st century, I’ll point out that the centrifuges that featured in the *Endurance* spacecraft from the film *Interstellar* (2014) and the *Hermes* from *The Martian* (2015) are intermediate in size between the two centrifugal habitats used in *2001: A Space Odyssey*, which I’ve been using as examples. So we could expect Coriolis effects to feature reasonably prominently in either environment.

All these fictional centrifuges are to some extent unrealistic, at least in the short term, because they involve a lot of mass which would need to be moved to orbit and then moved around in space. Where centrifuges are proposed for prolonged space missions, as in Robert Zubrin’s Mars Direct project, they involve whirling a small habitat around on the end of a long tether—usually with a radius of gyration at least as large as the large Space Station V from *2001: A Space Odyssey*. Coriolis deflection would therefore be potentially observable (for instance in dropping objects or throwing and catching), but there simply wouldn’t be room for the longer trajectories I’ve described here.

***** The deflection in the trajectory of a dropped object is a useful parameter to estimate the significance of Coriolis in a given habitat. As discussed in the text, it’s unaffected by rotation rate, and scales with the size of the habitat. If we drop an object from radius * r*, and it lands at radius

*, the magnitude of the antispinward deflection is given by:*

**R**\small R\left [ \sqrt{\left ( \frac{R^{2}}{r^{2}}-1 \right )} -\arccos \left ( \frac{r}{R} \right )\right ]

(Note that the arccos term needs to be in radians, *not* degrees.)

or

Fascinating pair of Coriolis articles. In a space-related college class, we’re looking at the use of a horizontal rotating torus (producing about 1 g) on the surface of the Moon, where we would have a perpendicular g/6 at right angles to the motions in 2D that you describe. Are you aware of any such problem being tackled? Have you done it yourself?

Also, would it be possible to have access to the code you wrote for this article? My intuition is that this would be superimposed on your solutions.

Looking forward to your answers here.

Sean

I’m glad you found it interesting.

Years ago, I designed a lunar “Paraboloid Park” to address the situation you describe—a paraboloid of rotation set vertically, apex down, and rotating around its axis of symmetry. Gravity at the axis (through which one could enter the park) is entirely lunar. Standing there, you would see the huge paraboloid landscape rotating around and above you. Stepping out on to the rotating surface of the paraboloid, the effective gravity vector would always be at right angles to the surface, comprised of the vector sum of lunar gravity and centrifugal gravity. Walking away from the axis, and up the curvature of the paraboloid, you’d move into areas of higher gravity, eventually reaching a zone in which the combined effect of lunar gravity and centrifugal gravity was 1

g—the environment of your much more compact torus.Of course, the consequences of mechanical failure in such an object would be catastrophic, but it was good fun to think about!

I haven’t ever done the Coriolis maths on such an environment, but it sounds like fun.

I’m afraid the code for these diagrams is a mess of custom routines and spreadsheets—the project grew rather haphazardly—so there’s nothing I could usefully pass on to you. My approach was fairly artisanal, anyway. I didn’t calculate any pseudoforces at all. I simply calculated the initial velocity vector in the non-rotating frame, and then calculated the position in the rotating frame at time intervals short enough to allow me to plot a smooth curve. I imagine your project will be aiming for something more elegant!

Excellent article. I’m trying to calculate the deflection of an object dropped from a height of 1m in a torus or radius 900m. So I assume r=899 and R=900. This gives a deflection of 2388m. Surely that’s not right. What am I doing wrong?

Thanks for the kind words.

I’d guess you’ve calculated using the arccos in degrees rather than radians, which would account for the large value you’ve derived. (Sorry that wasn’t clear in my original text – I’ve now added a note to make that requirement explicit.)

Using your figures, and expressing the arccos in radians, I’m getting a deflection of just over 3cm.

“But the rotation speed turns out not to matter, because the centrifugal and Coriolis effects scale equally with angular velocity, so trajectories stay the same.”

Actually, that’s not correct. “Centrifugal” / centripetal acceleration and force scale with the square of the angular velocity (A_cent = Ω²·R) whereas Coriolis acceleration and force scale with the angular velocity not squared (A_Cor = 2·Ω·v). It’s true that dropped balls with zero initial relative velocity v will follow the same relative trajectory independent of the angular velocity or “gravity” level, but for thrown balls with non-zero initial v, the shape of the trajectory depends on the ratio of the relative velocity v and the environment’s tangential velocity V. This is illustrated in figures 9-13 in this paper: http://www.artificial-gravity.com/AIAA-2006-7321.pdf

For more mathematical analysis, see: http://www.artificial-gravity.com/AIAA-99-4524.pdf

Thanks for the comment, but if you check my diagrams, you’ll see that they are deliberately generated for stipulated ratios of launch velocity in the rotating frame and rotation speed at the launch position in the inertial frame, to address the problem you mention. So, in accordance with what you point out, the depicted trajectories are independent of the rotation speed of the environment. If you spin the toy habitat in my diagram twice as fast, you must launch your ball twice as fast, and it follows the same trajectory twice as fast.

As I wrote shortly after the section you quoted, “This constancy with scaling also applies to trajectories that involve throwing an object—–

so long as the launch velocity keeps the same proportion to the rotation speed, the trajectory will be the same shape.”(My bold.)Sorry, I should have read more carefully before commenting. I missed the stipulation that “the trajectory depends only on the launch velocity *as a proportion of the rotation speed* of the habitat”

With the stipulation that launch velocity is a constant proportion of the rotation speed, so that v = Ω·k, then the Coriolis effect 2·Ω·v = 2·Ω²·k and it scales equally with the centrifugal effect Ω²·R, for any value of Ω, assuming that k and R are both constants or also scale together.

Thanks for coming back to resolve the issue—I appreciate your taking the time to do so.

Just to run on a little with Ted’s parameter k, above:

If we make R the radius of the habitat at which our projectile is launched, and use c to designate the relative velocity parameter with which I’ve been labelling my trajectory plots, then k = R·c. So the Coriolis pseudoforce on the projectile in flight is 2·Ω·v = 2·Ω²·k = 2·Ω²·R·c, which compares to the centrifugal pseudoforce at the launch radius R of Ω²·R. If we set c=1, then the coriolis pseudoforce is exactly twice the centrifugal force at the moment of launch.

If we launch “horizontally” to antispinward, Coriolis opposes centrifugal. The factor of 2 means that Coriolis undoes the floorward acceleration cause by centrifugal pseudoforce, and introduces a ceilingward acceleration that is exactly equal and opposite. This centripetal force is enough to keep the antispinward projectile with velocity parameter 1 floating along at the same distance above the floor, describing a circle around the habitat at constant radius R. Which is exactly what you see in one of my diagrams above.

In the inertial, non-rotating reference frame, we can understand this because the projectile has been launched with a velocity that exactly counters its rotational speed, so that (barring air resistance) it merely hovers in place while the habitat revolves around it.