Coriolis Effect In A Rotating Space Habitat (Supplement)

Space Station V from 2001 A Space OdysseyI’ve received a few enquiries in response to my post “Coriolis Effect In A Rotating Space Habitat”, concerning something I didn’t address at the time—what happens to the trajectory of objects moving parallel to the axis of rotation. (Though I did mention this topic in passing in my post about the Coriolis effect in general.) So that’s what I’m going to write about here. And after discussing that, I’ll talk a bit about the trajectory of rolling objects, which is another thing science fiction writers sometimes get wrong.

If you’ve arrived here directly, then you’re probably going to need to at least read the original Coriolis Effect In A Rotating Space Habitat post before returning here (there’s a link at the end of that post that will get you back); in fact, going all the way back to my Coriolis post could be useful, and you might even want to cast an eye over Saying “Centrifugal” Doesn’t Mean You’re A Bad Person. They’re all relevant to the topic of habitats that generate artificial gravity by rotating.

My previous post on this topic provided a lot of diagrams of the trajectories of objects moving in the plane of rotation of the habitat. That’s because there is no Coriolis pseudoforce acting on any velocity parallel to the rotation axis. To illustrate this, let’s go back to two diagrams from previous posts, which portray the trajectory of a dropped object.

First, we have the view of a stationary observer outside the habitat:

Coriolis in a space station 1The dropped ball retains the rotation velocity it had at the moment it was released, and follows the red trajectory. The person who dropped the ball meanwhile continues to rotate with the station (I’ve marked four successive positions of ball and rotating observer.)

From the rotating observer’s point of view, the ball’s trajectory looks like this:

Coriolis in a space station 1The rotating observer invokes two pseudoforces to explain this. One is centrifugal force, pulling the ball directly towards the floor; the other is Coriolis force, which (under anticlockwise rotation) deflects the ball to the right whenever it has any velocity in the plane of rotation.

Now, before going any father, lets name the different directions in our rotating habitat. We have spinward and antispinward, which are in the direction of rotation and against the direction of rotation, respectively. We have up and down, which are towards the axis and towards the floor of habitat, respectively. Centrifugal force is always directed down. Any movements in any combination of these directions (which lie in the plane of the screen in my diagrams) will be deflected by Coriolis force—rightward if the rotation is anticlockwise, leftward if the rotation is clockwise. The direction parallel to the axis of rotation (in or out of the screen in my diagrams) is axial, and movements in this direction experience no Coriolis deflection.


Now imagine that the observer in the diagrams above has not dropped the ball, but has instead fired it axially towards you—straight out of the plane of the screen, parallel to the rotation axis. There are no new forces invoked by this extra direction of movement—the ball will fall to the floor at the same rate, under centrifugal force, and will experience the same deflection in the plane of rotation, caused by Coriolis force. So with the ball coming directly towards you, its trajectory will look exactly the same as in the diagrams above. If you are rotating with the habitat, the ball will fall downward and antispinward as it moves towards you. If the ball is fired towards you very quickly, then it won’t have time to fall very far, or be deflected very far, and it will probably hit you, just a little below and antispinward of target. If it is fired very slowly, it will fall to the floor before it gets to you, landing some way antispinward of its launch point. At intermediate velocities, it will whisk by on your antispinward side before hitting the floor behind you.

And this is true of all the diagrams I produced for previous posts. In any of them, you can imagine that the object has some axial velocity, without that changing the trajectory you see projected on to the plane of rotation (which is the plane of the screen). If the object has a large axial velocity, it will travel a long way parallel to the axis before it completes the trajectory illustrated; if it has a low axial velocity, it will complete the evolution I’ve shown without travelling very far parallel to the axis.

For example, imagine that you and the observer are standing some distance apart, with purely axial separation. The observer wants to hit you in the head with a thrown ball. To do that, he needs to launch the ball with some axial velocity (so that it moves towards you), and an upward velocity (so that it doesn’t fall to the floor before it reaches you). Those components are familiar from throwing a ball in a real gravity field on Earth. But in a rotating environment, he also needs to throw the ball to antispinward, so that the Coriolis deflection will bring it around in a loop as it travels. From your point of view, as the ball comes sweeping towards your head, it will appear to follow one of a family of curves that look like this:

Trajectories in a rotating habitat 6
Click to enlarge

These are exactly the same curves that are required if the observer simply wants to toss the ball in the air and catch it. Except, this time, he has added an axial velocity that brings the ball into contact with your head just as it has completed its loop in the plane of rotation.

This is tricky. If he throws the ball with high axial velocity (so it reaches your head quickly) he needs to direct it only a little upward and antispinward—it will follow the short, interior loop as it comes towards you. If he throws the ball with low axial velocity, he needs to prolong its time in the loop, so he must throw it with a higher antispinward and upward velocity—the long, outer loop in the diagram.

We’re used to this, in the vertical direction—we throw balls fast and low or slow and high. But the inhabitants of rotating environments will need to adjust the antispinward component of their launch velocity, too, if their thrown ball is to arrive on target. And the antispinward and upward components will vary independently from habitat to habitat, according to how their sizes and rotation speeds differ. Games like cricket and baseball will (quite literally) take on a whole new dimension.*

Notice that Coriolis only appears if there is some velocity in the up/down or spinward/antispinward directions. That’s unavoidable with an object moving along a free trajectory (like a thrown ball) because all objects in a rotating reference frame will experience an apparent centrifugal force that accelerates them downward.


But what happens if you just roll an object along a horizontal surface? Science fiction writers occasionally invoke a sideways Coriolis deflection in this scenario, so that a rolling ball will follow a curved path across the floor, for instance. (I noticed Alastair Reynolds doing this in The Prefect, which I reread recently, but he’s only the first writer who springs to mind—there are many.) However, it doesn’t work that way.

If we roll a ball in the purely axial direction, then it will not be influenced by Coriolis forces at all, because it is not moving either up/down or spinward/antispinward. If we roll a ball antispinward, then the Coriolis force will act upward—the ball will become lighter, and may therefore experience less friction and roll farther, but it won’t experience sideways deflection. Likewise, a ball rolled to spinward will experience downward Coriolis, will become heavier, and may roll less far. But again, no sideways forces are generated. And that’s true of any combination of axial and spinward/antispinward velocities for an object rolling on a horizontal surface—its apparent weight may change, but its direction won’t.

That’s also true if we roll something down an inclined ramp facing in the spinward or antispinward direction. The up/down movement will generate a Coriolis force, but it will act to either lift the object away from the ramp (if moving antispinward) or press it against the ramp (if moving spinward)—again, no sideways component.

The only time sideways deflection of a rolling object occurs is if the object is rolling downhill in an axial direction, in which case it will experience the antispinward Coriolis force experienced by a dropped object. So general downhill rolling trajectories become complicated, with a combination of antispinward deflection and changing apparent weight continuously influencing how the rolling object and the ramp interact with each other.

Crown green bowls is played on a surface with a slight dome in the middle, and bowls that are biased in their weight distribution. Think how much extra fun that would be, in a rotating habitat.

* Bowlers and pitchers already use the aerodynamics of a spinning ball to shape laterally curved trajectories. What they might be able to do with Coriolis stirred into the mix defies imagination (at least, my imagination for the time being).

9 thoughts on “Coriolis Effect In A Rotating Space Habitat (Supplement)”

  1. I have read this a few times to get my head around the various forces involved – and still have to think hard about every action described. But it is good to get my, creaky, mind working again, so thanks for the explanation.

    I noticed that the makers of “The Expanse” had a bit of fun with these forces in an early episode in Series One. The character, Miller(?), was drinking from a falling stream of water, that definitely didn’t have a straight path.

    1. Yes, The Expanse works harder than most at trying to get things right. In my original “Coriolis” post I embedded a gif of a drink being poured in the first season:
      Pouring a drink in The Expanse
      The “James S.A. Corey” writing duo invoke Coriolis fairly frequently in the novels, though not as a plot element in my reading so far. But they (like so many others) get it a little wrong – on a couple of occasions they claim that the magnitude of the force varies with distance from the axis, which seems intuitive, but is wrong.

  2. Oops should have taken your advice and read/re-read your older post. Anyway it is nice seeing the water “trick” again. And, at least trying to get things right is something as a lot of SF shows don’t even bother about.

  3. There were a couple of moments in the Expanse television series where gunfire happened and I’ve always had the intuition (thus likely wrong) that bullet trajectories in smaller habitats would be slightly different depending on the spinward and antispinward (and angle relative to them) that would be important for longer trajectories. Tell me how I’ve got that wrong?

  4. You haven’t got it wrong–Coriolis will deflect the bullet. It’s just that the amount of deflection is inversely proportional to the velocity, so very fast-moving objects are deflected much less than objects that are moving slowly (that is, with a speed comparable to the local rotation speed).

    Interestingly, a bullet is actually subject to a greater Coriolis acceleration than (say) a thrown ball. But it completes its trajectory so quickly that the acceleration doesn’t have time to produce a large deflection.
    It goes like this, for any given rotating environment:
    1) Coriolis acceleration is proportional to (and at right angles to) the velocity of the moving object
    2) The deflection caused by this acceleration is proportional to the acceleration and to elapsed time squared
    3) But the elapsed time in flight is inversely proportional to velocity
    4) Putting all that together means that deflection is proportional to [velocity times one over velocity squared], which is just one over velocity

    So it turns out to be fairly easy to shoot someone, but much harder to hit them with a tennis ball at the same range.

  5. Not a true physics wonk, so please correct me if I am wrong, but it seems that an object rolling spinward along the inside skin of a rotating vessel ought to be subject to greater, not less, “downward” force, and vice versa; my reasoning being that, relative to the axis, the angular velocity of the object would effectively increase. If the object were rolled anti-spinward, then its total rotational velocity wrt the axis of rotation would decrease by its velocity relative to the rotating hull. If you were to roll a ball anti-spinward with the same velocity as the hull is rotating in the opposite direction, then the ball would effectively be stationary wrt the axis of rotation and would experience no “centrifugal” or “downward” pseudoforce. Am I misunderstanding your use of the terms “spinward” and “anti-spinward”?

  6. No, you’ve picked up an error on my part—I got my “spinward” and “antispinward” consistently crossed in the second half of this post, for some reason.
    So you’re absolutely right. In the inertial frame a ball’s movement to spinward will increase the centripetal force it experiences, as you say, which translates into a “downward” Coriolis force arising from its speed in the rotating reference frame. And the other way around when moving to antispinward.
    Thanks for picking that up—I’ve fixed the text now.

  7. How could one compensate for some of the coriolis effects for a spinning space habitat, besides just increasing your distance from the axis (analogous to compensating for dispersion in a reflecting telescope)? Having pairs linked counter rotating (like with O’Neil’s Island One, coaxial or nestled) might prevent gyroscopic effects, precession … but perhaps by adding some other angular momentum vector??

    1. No getting away from Coriolis, I’m afraid. Moving further from the axis makes absolutely no difference, despite what some science fiction authors seem to think—the Coriolis acceleration depends only on the rotation rate and the velocity relative to the rotating reference frame, not on the radius of revolution. And while having linked contrarotating habitats can give the whole structure zero angular momentum, the individual habitats will still experience Coriolis.

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