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.

AXIAL MOTION

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