In my previous post, I described the visual appearance of the starry sky for an observer moving at a substantial fraction of the speed of light—for instance, aboard a working Bussard interstellar ramjet, like the one pictured above.

I’ll recap the terminology I established in that post, which comes from Special Relativity. We call the viewpoint of an observer who is effectively stationary relative to the distant stars the “rest frame”. The “moving frame” is, as you might guess, the viewpoint of an observer who is travelling with an appreciable velocity relative to the rest frame. This relative velocity is given as a fraction of the speed of light, and symbolized by β (beta).

For the travelling observer, the aberration of light causes a shift in the apparent position of the stars, moving them across the sky towards the direction of travel. The relevant angle is the angle between the direction of travel and the star’s location, symbolized by θ (theta) in the rest frame, which aberration converts to a smaller angle, θ′, in the moving frame.

If a sphere of stars surrounds a rest frame observer, like this:

it will be transformed into an ellipsoid for an observer moving through the same location at half the speed of light, with each star shifted parallel to the line of flight:

and into an even more stretched ellipsoid at 85% of lightspeed:

So that’s aberration. The other important phenomenon to address is Doppler shift.

RELATIVISTIC DOPPLER

Like aberration, the Doppler effect (named for the physicist Christian Doppler) is something that should be familiar from everyday life. The siren of a police car or ambulance sounds more high-pitched when it is approaching than when it is receding. The distance between successive wavefronts of the sound is reduced by the vehicle’s velocity towards us, and then increased by its velocity of recession. As the vehicle passes us, there’s a moment when we are at 90º to its line of travel and we hear the sound of the siren with exactly the frequency at which it was emitted.

The same thing happens with light waves—the light from an approaching object is shifted towards the higher-frequency blue end of the spectrum (a “blue shift”), while the light from a receding object is shifted in the other direction (“red shift”). But (as with light aberration) we can’t use the same simple geometry to predict the behaviour of light—Special Relativity intrudes again. This time, we must allow for the fact that a moving observer measures time as running more slowly in the rest frame. An observer on a speeding spacecraft therefore does not see the original colour of light from a star that is at 90º to the spacecraft’s line of flight. The slowing of clocks predicted by Special Relativity means that the star’s light is red-shifted in this position (so-called “transverse Doppler”), and the boundary between red-shift and blue-shift always lies a little ahead of the spacecraft.

What we need to calculate is the relativistic Doppler factor, which is symbolized in various ways by different authors. I’m going to use the symbol η (eta). Eta is the multiplication factor for the frequency of light observed—if η>1, the light is blue-shifted; if η<1, the light is red-shifted. When η=1, the light is received at the same frequency at which it was emitted.

The value of η depends on the two variables β (the moving observer’s velocity as a fraction of the speed of light), and θ′ (the angle in the spacecraft’s sky between the direction of flight and the object being observed).

Here’s a plot of how η varies between θ′=0º (dead ahead) and θ′=180º (dead astern), for three different values of β:

We can see that, as predicted, there is always red shift at the 90º position (transverse Doppler due to relativistic time dilation). And the point in the sky at which red shift switches to blue shift is progressively farther forward for higher values of β—the faster the spacecraft flies, the smaller the region ahead in which blue shift occurs. But the faster the ship moves, the more strongly blue-shifted are objects ahead, and the more strongly red-shifted are objects astern. In fact, there’s a precise inverse relationship—if the frequency of light coming from dead ahead is doubled, the frequency of light coming from dead astern is halved.

So that’s the situation as seen in the sky of the speeding spacecraft, which is distorted by the effects of aberration. But it’s instructive to convert from θ′ back to θ (the corresponding angle in the rest frame). Here’s the relationship between η and θ for the same three values of β:

Although the blue-shifted region as seen from the spacecraft gets smaller ahead with increasing velocity, it actually includes progressively larger regions of the sky as seen from the rest frame. In fact, there’s another nice symmetry—the angle θ′ at which η=1 in the moving frame converts to θ = 180º – θ′ in the rest frame.

Here’s the proportion of sky (by area) affected by blue shift, for the moving frame (solid line) and rest frame (dashed line). It could equally well depict the red-shifted proportions, with the moving frame dashed and the rest frame solid.

So with increasing velocity, aberration moves more and more stars into the forward, blue-shifted region, even though that blue-shifted region is shrinking. Here’s the diagram of aberration effects I used in my previous post, except this time with the regions of red- and blue-shift marked on it:

We can see that, with increasing velocity, stars are continuously crossing from behind the spacecraft to enter the blue-shifted region ahead. At the light-speed limit, the whole sky ends up in the forward blue-shifted area, which has shrunk to a dimensionless point dead ahead.

And here are some “Doppler trajectories” for stars at various locations in the rest frame:

The line markers are for the same values of β along each trajectory. To indicate their meaning I’ve tagged them with a small “c“, the conventional symbol for the speed of light, but I’ve labelled only the 90º and 170º curves, to avoid visual clutter. We can see that a star which is in the θ=90º position is immediately incorporated into the blue-shifted region of the moving frame. As β increases, it moves farther forward in the spacecraft’s sky, and becomes increasingly blue-shifted. But a star at θ=170º, close to being astern of our spacecraft, requires a very high velocity to bring it into the θ′=90º position, and then an even greater velocity before it moves into the (now very small) blue-shifted region ahead. And notice that for each star the maximum red-shift occurs as it passes through θ′=90º.

Now, there’s a very satisfying relationship between η and the aberration ellipsoids I derived in the previous post and reproduced at the top of this one. If an object has distance r in the rest frame, it has distance r′=ηr in the moving frame. For example, if an object appears twice as distant due to aberration, its light will be blue-shifted to twice the frequency.

So we can immediately mark up the aberration ellipsoids with an indication of red- and blue-shift. The parts of the ellipsoids that fall inside the sphere of stars observed in the rest frame must be red-shifted, because r′<r, and so η<1. And the parts that fall outside the sphere must be blue-shifted, because r′>r, and so η>1.

That’s neat, isn’t it? Notice how the longer ellipsoid produced by a greater velocity has fewer red-shifted stars in the rear view. Notice the topmost red arrow, which shows a star that is red-shifted at half the velocity of light, but which becomes blue-shifted at 85% of lightspeed. And notice that all the rearward stars are at their closest (and, we now know, most red-shifted) as they pass through the 90º position, with the stars that are farthest astern necessarily passing closest of all and therefore experiencing the greatest red shift. It all hangs together.

And because the apparent distance of an object is proportional to η, its apparent diameter is inversely proportional to η, and its angular area is proportional to 1/η². The η value turns out to be the key to a great deal about the appearance of the sky from our speeding spacecraft.

There will be more about that next time, when I deal with how the visual appearance of the stars is changed by blue- or red-shift.

Mathematical notes

The relationship between the relativistic Doppler factor η, velocity β and viewing angles θ (rest frame) and θ′ (moving frame) is:

The angle at which η=1 in the rest frame is given by:

In the moving frame, η=1 at angle:

So cos θ = -cos θ′, and θ = 180º – θ′.

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This is another one of those topics (like Coriolis effect and human vacuum exposure) that many science fiction writers seem to know enough about to include it in their stories, but not quite enough to get right.

So in this post (and an estimated three subsequent posts) I’m going to write about what the starry sky would look like if you were travelling at some significant fraction of the speed of light—for instance, if anyone ever built a working Bussard interstellar ramjet (like the one pictured above). In a later post, I’ll show you some illustrative images I’ve generated using Celestia’s star databases and a bit of brute-force coding. I’ll try to keep it maths-free, but will post a few relevant equations in footnotes, for anyone who’s interested in playing with this idea themselves.

Two things transform your view of the Universe if you travel close to the speed of light—aberration and Doppler shift. In this post I’ll deal with aberration; in the next post I’ll describe the Doppler effect.

LIGHT ABERRATION

The basic principle underlying light aberration is evident to anyone who has ever run through rain, or driven through snow. If you run through vertically falling rain, your front will get wetter than your back, because the rain will appear to be slanting in from ahead of you as you run. And if you have ever driven in a snowstorm at night you’ll  recall how, no matter which direction you drive in, the gently falling snow seems to be coming at you almost horizontally, straight towards the windscreen. This happens because your horizontal velocity adds to the velocity of the falling precipitation, as a vector, producing a resulting velocity that is slanted towards your direction of motion, like this:

The same thing happens to the light from distant stars. In fact, the Earth moves fast enough in its orbit that light aberration was detected telescopically as long ago as the early eighteenth century by James Bradley, who noticed that the light from the star Eltanin (Gamma Draconis) came from slightly different directions throughout the course of the year. (The same turned out to be true for every other star too, of course.)

Similarly, an observer aboard a spacecraft in rapid motion will see a distorted view of the sky, with all the stars in the sky apparently shifted in position towards the line of flight of the spacecraft:

All the displacement is parallel to the line of flight of the spacecraft, and the faster the spacecraft’s motion, the greater the angular shift. But we can’t calculate the angle using the simple geometric construction above, because we need to take into account the coordinate effects of Special Relativity, too. Once we do that, we find that the closer the ship comes to the speed of light, the more extreme is the forward displacement of the stars, until at the unattainable limit of light-speed itself, the whole sky would appear to be displaced into a dimensionless point, dead ahead of the speeding spacecraft.

Now it’s time to introduce some symbols and terminology. What we’re doing is comparing the “normal” view of the sky (as seen by an observer approximately at rest relative to the background stars) with the view seen by an observer who is in rapid motion relative to that stationary observer—speeding past in an interstellar spacecraft, for our purposes here. The observer who is unmoving relative to the distance stars occupies what’s called the “rest frame”; the observer on the spacecraft occupies the “moving frame”. By convention in Special Relativity, variables measured in the rest frame (such as angles, distances and times) are assigned simple letters or numbers, whereas the corresponding measurements made in the moving frame are identified by the addition of a prime mark (′).

The velocity of the moving frame relative to the rest frame is conventionally given as a ratio to the speed of light, and designated with the symbol β (beta). So β=0 when our spacecraft is at rest among the stars, and β=1 at the unattainable limit of light-speed velocity.

Now let’s designate the angle between a given star and the line of flight of the spacecraft as θ (theta). θ=0º for an object dead ahead, θ=180º for an object dead astern, and θ=90º for an object directly to the side. According to convention we’ll use plain θ for the position of the stars as seen in the rest frame and θ′ for the corresponding angle seen in the moving frame of the spacecraft.

Now I can plot a graph showing the effect of β on θ′, for the full range of starting values of θ:

You can see that as β increases, all the stars appear to crowd toward θ′=0. Stars that are evenly spaced around the sky for an observer at rest will appear to a travelling observer to be strongly compacted ahead, and thinly spread astern.

This process of compacting ahead and thinning astern has interesting consequences if we consider what happens to the appearance of an extended object, like a constellation. Here’s an example, projected on to the spherical sky:

As a constellation to the rear of the spacecraft is displaced forward in the sky by increasing velocity, its stars will follow initially diverging course, and will pull apart, making the constellation look bigger. This process continues until the ship is moving fast enough to displace the apparent position of the constellation to θ′=90º, after which point the stars will start to converge on each other again, and the constellation will shrink. It will reach its original size when it is displaced to a position in which it appears as far from the forward direction as its rest-frame position is from the rearward direction (that is, when θ′ = 180º-θ). On farther displacement (by increasing velocity), the constellation will then shrink to a smaller size.

So constellations that lie in the forward view in the rest frame will always appear to shrink as they are displaced forward by aberration. But constellations that are to the rear of the spacecraft will appear to grow in size, up to a maximum which will occur at some critical value of β when the constellation has been displaced so that it appears directly sideways from the line of flight. And the farther to the rear the constellation lies in the rest frame, the higher the value of β needed to bring into this position.

What applies to constellations applies to any extended object, include the disc of a star if viewed through a very powerful telescope—stars up ahead will appear to dwindle in size when viewed from a moving spacecraft; stars astern will get bigger (up to a maximum size when they appear to be at the θ′=90º position).

If they get smaller, they look farther away; if they get bigger, they look closer. And if observers aboard the spacecraft tried to measure the distance to the stars using parallax (for instance, by flying two spacecraft in tandem and comparing observations), they’d find that light aberration affected the parallax measurements so as to confirm the distances deduced from apparent size. The shrunken stars ahead really would appear to be proportionally farther away; the enlarged stars visible on either side really would appear closer.

All of this complication conspires to produce a rather simple and pleasing result. Suppose we start with a sphere of evenly spaced stars distributed around our spacecraft in the rest frame, like this:

Then from the point of view of a moving spacecraft at the same location is space, the stars will appear to be displaced so that the original sphere turns into an ellipsoid, with the spacecraft at one focus, like this:

I’ve marked the direction of displacement with red arrows for a few stars; you should be able to see how the general trend works. The faster the spacecraft moves, the more elongated the ellipse. The one above is for half the speed of light, β=0.5. Below, I’ve added the ellipsoid for a spacecraft moving at β=0.85:

If you look at the uppermost red arrow, you can see how a star which is a little behind the spacecraft in the rest frame will appear to be directly to one side of a the spacecraft if it is moving at β=0.5, and slightly ahead of the spacecraft if it is moving at β=0.85. And as its position changes, it will appear to get nearer to the spacecraft at β=0.5, and then farther away again at β=0.85.

Bear in mind that this is what the sky looks like to observers aboard their spacecraft. Special Relativity tells us that coordinate distances shrink both ahead of and behind a moving observer—but the aberration calculations tell us that this coordinate change is actually obscured by the shift in the apparent direction of light.

Bear in mind also that these diagrams are strictly accurate only as comparisons between the view of a stationary observer and of an observer on a spacecraft sweeping past the same point in space. If we want to think about how some particular sky view will warp as a spacecraft accelerates from rest to high velocity, we need to take into account the movement of the spacecraft relative to the stars, too. That’s perhaps a topic for another post, but not something to introduce at this point. The discussion here gives a general impression of how the celestial view would be distorted for any spacecraft.

That’s it for now. So far, I’ve dealt with how the location of the stars appears to change if we move with high velocity. In the next post, I’ll deal with how Doppler effect changes their colours and brightness.

• Go the next post in this series

Mathematical notes

The relationship between θʹ and θ depends (as seen in my graph) on β.

The apparent radial distance to the star in the moving frame (rʹ) depends on its radial distance (r) in the rest frame, and on β and θʹ.

This bears a close resemblance to the polar equation of an ellipse with one focus at the origin, given in terms of semiminor axis (b) and eccentricity (e).

So a sphere of stars at distance r from the spacecraft in the rest frame will appear in the moving frame to be displaced into an ellipsoid with semiminor axis (b), semimajor axis (a) and eccentricity (e) given by:In Cartesian coordinates with the z axis aligned with the velocity vector of the spacecraft, we get the transformation:

The x and y coordinates (in a plane transverse to the line of flight) are unchanged by the transformation, confirming that the apparent displacement of the stars due to aberration is purely parallel to the line of flight, as depicted in my diagrams.

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I’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:

The 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:

The 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:

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.

ROLLING MOTION

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 spinward, 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 antispinward 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 spinward) or press it against the ramp (if moving antispinward)—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 spinward Coriolis force experienced by a dropped object. So general downhill rolling trajectories become complicated, with a combination of spinward 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).

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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 1g 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 R, the magnitude of the antispinward deflection is given by:

R.[sqrt(R²/r² – 1) – arccos(r/R)]

where arccos is in radians.

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