Category Archives: Phenomena

Signalling Mirrors

Signalling mirror
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I found this object a couple of months ago, prominently poised on a rock in the broken ground above the big bulldozed path on Beinn Bhuidhe. (Yes, I do occasionally climb a hill without telling you about it.) It was marked with rainwater and bird droppings, but cleaned up remarkably well once I got it home.

It’s a signalling mirror, of a very particular variety that I’ll come back to later, but I suspect from its location that whoever lost it had been using it as a more conventional mirror—adjusting an errant contact lens, perhaps, or making sure they’d rubbed in their sunscreen evenly.

Signalling with a mirror is a very slightly technical exercise. Movie characters can simply pluck up a random bit of broken glass and us it to attract the attention of a passing helicopter in short order. But the problem lies in knowing where the reflected beam of sunlight is going—easy enough to flick a spot of light around on a nearby wall, but how can you direct it specifically towards a target moving across the empty sky?

The simplest solution is to direct the sunlight on to something nearby which you have aligned with the distant object of your signalling. Hold up two fingers at arm’s length, frame the object between your fingertips, and use the mirror to reflect the sunlight on to your fingers.  Simple!

Finger sighting with signalling mirror
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The problem with this approach is (as my diagram shows) that the mirror is off your line of sight—the line from your eyes to your fingers extends to the helicopter, but the line from the mirror to your fingers doesn’t. So you need to keep the mirror as close as possible to your eyes, and your fingers as far as possible from your face, in order to minimize the angular error. If you place the mirror directly below your sighting eye, then the error will be in an upward direction, so painting the reflected spot up and down the length of your fingers should ensure that your target receives an intermittent flash.

So, the next elaboration is to use a purpose-built mirror with a hole in the middle. That way, you can look through the hole at your framing fingers, and know that when the fingers are illuminated, the distant helicopter is, too. In practice, this is a little tricky—once you’re peering through the hole in the mirror, the limited field of view makes it difficult to find the reflected spot of light and direct it on to your fingers.

What we need is some way of telling by looking at the back of the mirror where the reflected light is going. Then we can dispense with the fingers, and simply look through the sighting hole at the target while steering the reflected beam towards it.

The simplest way of doing this is using a double-sided mirror with a hole in the middle. When you hold it up to the sun, it casts a shadow with a bright spot in the middle, corresponding to the hole. This shadow may be projected somewhere on your body, if the sun is roughly ahead of you, but it may be off to one side. The first thing to do is to find that shadow and its bright central spot—you may need to hold a hand out to one side or the other. Next, tilt the mirror so that you can view the mirror’s shadow and central spot reflected in the back surface of the mirror. Next, align the central hole in the mirror so that you can see your distant target through it.

Got all that? Good. Now gently adjust the tilt of the mirror (without letting the target slip out of view in the central hole) so that the reflection of the mirror hole seen in the back of the mirror disappears into the hole in the centre of the mirror, superimposing it on your target.

Double-sided signalling mirror
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You end up with light reflecting along equal and opposite paths from the front and back of the mirror, as shown above. Simple geometry now guarantees that your reflected light beam is striking the target.

Here are some photos which may make the process clearer. I use a Deutsche Grammophon CD for this purpose, because their top surface has a mirror finish, providing me with a ready-made perforated double-sided mirror.

Here’s the CD. You can see my shirt (and camera) reflected in its back surface, and the shadow of the CD is on my shirt at the lower right rim:

Two sided signalling mirror 1
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Now I’ve tilted the CD to make the shadow entirely visible, with the reflected image of its bright central spot directly above the hole in the CD:

Two sided signalling mirror 2
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All I need to do is to steer the bright central spot into the hole in the CD, and the other side of the CD will be reflecting light towards whatever I can see through the hole. Below, I’ve got the reflected hole superimposed on the real hole, and through the hole you can make out the bright spot of light projected on to the building beyond:

Two sided signalling mirror 3
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This is the sort of signalling mirror that was provided in life-boats during World War II, except the sighting hole was small and cross-shaped for accurate use. Here’s the instructional video, which lets you see the device in action:

Still a bit of a palaver though, isn’t it? The next step in the development of the signalling mirror came just after the war, with the incorporation of retroreflective beads into the design. Retroreflective beads work in the same way raindrops do when they form rainbows. Light from the sun enters the drop, is refracted, reflected and refracted again, and so comes back out of the drop at an angle of about 42º to the direction at which it entered.

Raindrop refractionThe higher the index of refraction, the closer the light comes to bouncing straight back where it came from—retroreflection.

Retroreflector refractionIn practice, retroreflective beads generally have a reflective coating at the back, which allows the reflected light to be coloured and scattered over a reasonably wide viewing angle, and also allows the use of cheaper glass with a lower refractive index. In this form, they’re used to coat the surface of road signs and those reflective safety garments that appear to “light up” in the headlights of your car.

The little mirror I found has retroreflective beads distributed in a grid around the sighting hole in the corner. I can show them up if I take a flash photograph at an angle to the mirror surface—the beads suddenly glow brilliant white:

Signalling mirror, flash photo
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So the beads around the sighting hole are reflecting the sunlight back on itself. How does that help? Well, there’s a layer of glass across the front surface of the sighting hole, and a small proportion of the retroreflected light will bounce off the back of the glass, pass through the holes in the grid that supports the beads, and arrive at your eye as you view the back of the mirror. This gives you a little spot of light (a faint doubly-reflected image of the sun) at the edge of the sighting hole, which you can move around by tilting the mirror. Tilt the mirror until the spot is central in the sighting hole, and you’re looking directly down the beam of sunlight being reflected from the mirror on the other side. The geometry looks like this:

Retroreflective signalling mirror
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Here’s my best shot at showing you what it looks like. My found mirror turned out to be pretty rubbish for this purpose, so I’m using one of my own.

Retroreflector signalling mirror
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The retroreflector beads are distributed in the ring of small holes around the central sighting hole. It’s impossible to get a camera close enough to the sighting hole to show the full effect, but you can see that the lower three holes in the ring are brightly illuminated by the sun, whereas the rest are dim. If I move my eye in close to the sighting hole, that patch of bright illumination remains the same angular size, and so resolves itself into a small bright image of the sun, positioned in one of the ring holes. Then, by tilting the mirror, I can steer this bright image into the central sighting hole. In the image, the mirror needs to be tilted back slightly, to lift the reflected light into the line of sight.

This arrangement, patented by Richard S. Hunter in 1946, finally gives you a signalling mirror you can always use with one hand. Which is pretty useful if you’re injured, or trying desperately to hold the bow of a lifeboat into the oncoming waves.

Perspective Tricks

Okay, one last time.
These are small, but the ones out there are far away.

Father Ted, “Hell” (1996)
Graham Linehan, Arthur Mathews

I was recently reminded of Father Ted explaining perspective to Father Dougal (is it really more than twenty years ago?) when I happened on a bit of art under the Tay Road Bridge. The pillars that support the approach roads have been painted in patches of bright colour, like this:

Tay Bridge Mural 1
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Motorists on the underpass shoot past this with just a flicker in their peripheral vision. But if you go and stand in the right place, and get the columns aligned in just the right way, you see this:

Tay Bridge Mural 2
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That’s neat, isn’t it? I really wanted the direction in which the word SOUTH is revealed to be, well, south; but it’s closer to south-west. The geometry of the pillars makes it impossible to do the trick on a north-south axis. Curiosity made me walk down to the other end of this area, to stand next to the door that gives access to the pedestrian walkway on the bridge above, and sure enough there’s a NORTH as well, though one inconveniently placed pillar makes it impossible to see the whole word:

Tay Bridge Mural 3
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But that inconvenient pillar at least supports a plaque with the artistic credits and a bit of history:

Tay Bridge Mural Plaque
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And one of the “unused” pillars (blocked from view by the “R” in NORTH, above) supports a portrait of William Fairhurst, the designer of the bridge:

William Fairhurst, Tay Bridge Mural

Now, to get those two words spelled out evenly on columns that are very different distances from the viewer, the artists needed to do a trick with perspective—the nearby letters are much smaller than those farther away. If you take a look at my first photograph, you’ll see that the “UT” of SOUTH occupies about a third of the column height, while the “S” takes up about three-quarters of its column and even spills on to the ceiling.

So, as Father Ted might explain: big things far away look just like small things nearby.

This perspective effect is exploited in an optical illusion called the Ames Room, after its inventor, ophthalmologist Adelbert Ames, Jr.

Ames realized that you could make a trapezoidal room look like a rectangular one, if you made a distant corner taller than a closer corner, and confined the viewer’s position by making them look through a peep-hole. Once you’ve set up the illusion (and perhaps made it more compelling by including trapezoidal windows, doors and floor tiles), anyone who stands in the room will seem to vary in height, according to whether they stand in the far corner (in which they look small in proportion to the height of the room), or the near corner (in which they look taller).

Here’s a classic photograph of how it looks when three men of similar height line up along the far wall of an Ames Room:

Ames Room
(Notice how the position of their feet reveals that they’re actually standing on a sloping floor.)

And here’s the geometry of the illusion, as seen from above:

Geometry of an Ames Room

You can get a better feel for the three-dimensional shape of an Ames Room from this video demonstration  by neuroscientist Vilayanur S. Ramachandran, which featured in the BBC’s special-effects documentary The Computer That Ate Hollywood (1998):

Ramachandran suggests that the illusion is maintained by the fact that we’re used to seeing rectangular rooms, and there’s some support for that point of view from the fact that people who live in a less “carpentered world” (and therefore rarely encounter rectangular rooms) are less susceptible to the illusion. *

But you don’t need the whole room—even quite simple visual cues can create an Ames Room effect. A couple of cue objects of appropriate visual size, combined with an apparent horizontal alignment, will do the trick. This visual trickery was used frequently in Peter Jackson’s Lord Of The Rings films, to make the hobbits appear smaller than the humans. For instance, here is Elijah Wood (playing the hobbit Frodo) apparently sitting side by side on a cart with Ian McKellen’s Gandalf, as seen in the film:

Frodo and Gandalf

And here’s the real shape of the cart:

Ames cart from Lord Of The Rings

Jackson’s special effects engineers even managed to create an “Ames Room” that adjusted itself to maintain the illusion as the camera position changed:

The actors not only had to deliver their lines while looking past each other, but while they, along with parts of the set, were moving around on a platform as the camera moved.

So that’s how you make something look small by pretending it’s closer than it actually is. The reverse technique was also frequently used in film-making—making something look huge by pretending it’s farther away than it actually is. It was a common trick, in the days before CGI, to place a model in the foreground that was merged with live action in the background.

For example, here’s a classic scene from Steven Spielberg’s Close Encounters Of The Third Kind (1977)—the discovery of the lost ship S.S. Cotopaxi in the Gobi Desert:

Cotopaxi in the Gobi, Close Encounters of the Third Kind

The ship is actually a detailed model, a few metres long, and the people, camels and helicopters are several hundred metres beyond it—the smooth featurelessness of the intervening sand makes the distance impossible to judge.

And then there’s the trick of making something look far away by building it smaller than normal. That could be as simple as tapering the width and height of a fake street built on a sound stage, to give the impression that it extends farther into the distant than it actually does. But the classic example of this technique appeared in the background of the final scene of Casablanca (1942):

Final scene, Casablanca

That aeroplane parked on the tarmac behind Claude Rains, Humphrey Bogart and Ingrid Bergman is a Lockheed Electra 12A—or at least, a half-scale model of one, built from wood and cardboard. The people who are seen walking around the aircraft were all adults of short stature, hired specifically to give a misleading scale to the plane, to make it look farther away than it actually is. And the implausible “Casablanca mist” serves to conceal the fact that the whole scene was shot indoors, with just enough room for the scaled-down aircraft.

Finally, symmetry suggests I should have a category for “making something look closer by building it larger than normal”. For some reason, there doesn’t seem to have been any cinematographic demand for that technique …

* V. Mary Stewart. A Cross-Cultural Test of the “Carpentered World” Hypothesis Using The Ames Distorted Room Illusion International Journal of Psychology 1974; 9: 79-89

Radiation Fog

Radiation fog
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© 2016 The Boon Companion

Radiation fog sounds like something that might occur during a nuclear winter, but it’s not that kind of radiation.

The radiation here is heat radiation—infrared wavelengths radiated by the ground during the night, particularly when the skies are clear.

Usually, the air temperature gets lower as you get higher—a rising packet of air expands and cools as it moves into lower pressure at higher altitude; conversely, a falling packet of air contracts and warms. This is called the lapse rate.

But if the ground radiates away its heat into a clear sky on a long night, it can get colder than the overlying air. By conduction, the chill of the ground spreads to the air immediately above it. If there’s a wind, this cooled air is moved away and stirred into the general atmosphere, but in the absence of wind it can form a puddle of cold air, underlying the warmer atmosphere above. This is called a temperature inversion, because it reverses the normal progression from warm air low down to cold air higher up.

If this puddle of cold air cools below its dew point, water droplets will condense out—some can settle on the ground surface as dew or (if the ground is cold enough) even frost.

Dew on grass
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© 2016 The Boon Companion
Frost on pine needles
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© 2016 The Boon Companion

But some can also condense on particles of dust in the air, forming fog. And that’s radiation fog.

Ideal conditions for its formation are long, cold nights, clear skies and little wind—the sort of conditions that accompany high pressure in autumn and winter.

Although it’s called fog, it may actually be mist—the technical difference being that fog reduces visibility to less than one kilometre, whereas mist is thinner and permits visibility beyond a kilometre. In either case, it’s really just cloud at ground level.

And when the sun rises, we see wisps of radiation fog drifting past, gradually thinning and dissipating as the sun warms the air and ground.

Radiation fog clearing
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© 2016 The Boon Companion

Radiation fog often happens on low ground between hills, where it’s called valley fog. The sheltered valley protects the fog from the wind, and air that cools on the high valley sides  flows downhill to the valley bottom, where it enhances the puddle of cold air. Valley fog can persist for days during stable periods of high pressure—the roof of reflective fog prevents the sun warming the ground below by day, and the cold valley sides recharge the supply of cold air by night.

Valley fog in Strathmore
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© 2016 The Oikofuge

These sheltered puddles of cold air can form on a remarkably small scale—here’s a little patch of persistent frost I found, reflecting a pool of cold air that had accumulated overnight in a small dip in the lee of the trees:

Frost in the Sidlaws
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© 2016 The Oikofuge

It used to be that we could easily see the temperature inversion conditions even without the presence of fog. In the days when more people burned coal and wood for warmth, warm packets of smoke-filled air would rise from their chimneys, climb through the cold air, and then suddenly stall and stop rising when they encountered the layer of warmer air above the inversion. Winter valleys would be roofed with a thin pall of smoke, trapped at the top of the inversion. And, of course, if you were unlucky enough to live in an industrial town at the foot of a valley, a temperature inversion could trap pollution from factory chimneys for days, causing lung disease and deaths.

Click to enlarge
© 2011 The Boon Companion

That’s a rarer sight, nowadays, but I managed to capture the tell-tale sign of a temperature inversion above the smoke from heather-burning in the Angus glens, early this spring:

Smoke shows temperature inversion over Strathmore
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© 2017 The Oikofuge

The other main type of fog is advection fog, when moist air is pushed over colder land or water by the wind. It’s a common occurrence in these parts, with damp air from the North Sea pushing up the estuary on an easterly wind. The Scots word for the result fog is haar. So to finish with, here’s some haar photographed by The Boon Companion recently. I’ll perhaps write more on that topic in another post.

Advection fog in Tay estuary
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© 2017 The Boon Companion

Coriolis Effect In A Rotating Space Habitat

Space Station V from 2001 A Space Odyssey

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:Coriolis in a space station 1Coriolis in a space station 1The 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):

Trajectories in a rotating habitat 1
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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:

Trajectories in a rotating habitat 2
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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:

Trajectories in a rotating habitat 3
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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:

Trajectories in a rotating habitat 8
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Trajectories in a rotating habitat 9
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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:

Trajectories in a rotating habitat 4
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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:

Trajectories in a rotating habitat 5
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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:

Trajectories in a rotating habitat 6
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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:

Trajectories in a rotating habitat 7
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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:

Trajectories in a rotating habitat 10
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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:

Resultant velocities

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:

Trajectories in a rotating habitat 11
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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. Space Station V interior, 2001: A Space OdysseyIf 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.

Fountain under Coriolis force, by Tye-Yan "George" Yeh
Coriolis fountain, drawn by Tye-Yan “George” Yeh (Source)

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.



1960s roundaboutWhen I was a solitary, bespectacled and distinctly oikotropic child growing up in Dundee, I was fascinated by the little roundabout, like the one pictured above, in the play area of my local park. While my compatriots were trying their best to kill or maim themselves by using the swings, the slide and the witch’s hat for purposes their designers had neither intended nor dreamed of, I would spin up the roundabout, climb aboard, and try to kick the central pillar.

What was fascinating was that I kept missing. Despite my well-known inability to kick a football, even I shouldn’t have missed a thick metal pillar from a distance of less than a foot. But some mysterious force kept taking hold of my foot as soon as I moved it,  and shoving it sideways. The deflection was there on the backswing, too—if my foot was displaced to my right when I kicked, it was displaced to my left when I swung it back. And I could reverse the direction of deflection by reversing the rotation of the roundabout. Anticlockwise displaced my foot right; clockwise displaced it left. If I turned sideways and swung my foot back and forth tangential to the rotation, the displacement was still there, and in exactly the same direction relative to my body. With a bit of thought, it was evident that no matter what direction I swung my foot while standing on an anticlockwise-spinning roundabout, the foot was always shoved to the right of its direction of travel. Clockwise rotation brought left displacement.

It was almost eerie, and it lodged so firmly in my mind that, a decade later, when my physics teacher described Coriolis force to our little class, I said “Ooooooh! I see!” very loudly and appreciatively. (This was as well received as you might imagine. Which is to say, not.)

Coriolis force (named for Gaspard-Gustave de Coriolis) is a companion to centrifugal force, which I’ve written about before. They’re both often called pseudo-forces, because they only show up if we adopt a rotating reference frame—I described this in detail in my previous post, Saying “Centrifugal” Doesn’t Mean You’re A Bad Person. They’re also called inertial forces, because they arise from the tendency of objects in motion to  follow straight-line paths. Just staying stationary relative to a rotating reference frame (as I did when I stood on my whirling roundabout) means that you experience centrifugal force—what is stationary to you and the roundabout is a continuously curving path in the non-rotating world, and so you experience a constant inertial outward tug as your body tries to fly off in a straight line.

In contrast to centrifugal force, which is there all the time, Coriolis only shows up when you move around. A route that is a straight line across a turning roundabout is a curved path in the non-rotating world, and so whenever you move around in a rotating reference frame you feel a new, nagging inertial tug, trying to drag you away from the “straight” line you want to follow, and to make you move instead in a straight line relative to the outside world (which is a curved path relative to the roundabout).

Where does Coriolis come from? Go back to my attempts to kick the central pillar of the roundabout as it rotates anticlockwise. As my foot swings through a short fore-and-aft kick, aimed directly at the pillar, the roundabout rotates—turning my body anticlockwise and shifting me bodily to the right. My foot, moving with its original velocity, misses the pillar to the right, and is now swinging diagonally relative to my body. But as far as I’m concerned, stationary relative to my rotating reference frame, some mysterious force has seized my foot and curved its trajectory to the right.

Coriolis on a roundabout 1 If I turn and kick outwards, the same shift and rotation occurs, and my foot swings rightwards relative to my body again.

Coriolis on a roundabout 2Face in the direction of spin, or against the direction of spin—it’s always the same rotation and shift. Coriolis on a roundabout 3In fact, it doesn’t matter where I stand on the roundabout, or which direction I kick in—the same rotation and shift is always there, so the kick always deviates in the same direction, and by the same amount.Coriolis on a roundabout 4

In fact, the magnitude of the Coriolis acceleration depends on only two things—it increases the faster your reference frame rotates, and it increases as you move faster relative to the rotating frame. And the direction depends on only one thing—it is always at right angles to the direction of movement, and it pushes rightwards in anticlockwise-rotating systems, and leftwards in clockwise-rotating systems. (And, of course, if you lay underneath my anticlockwise roundabout and looked up at it from below, it would be rotating clockwise to you, and all the deviations in my diagrams would be leftwards.)

All of this applies in the plane of rotation. If you move parallel to the axis of rotation (in and out of the computer screen in my diagrams) then you’re not shifting around in the plane of rotation, and there’s no Coriolis effect. If I’d had the presence of mind to try some squats while on my roundabout, my head wouldn’t have been deflected from its vertical path as I bobbed up and down. (I’m actually slightly ashamed of my eight-year-old self for not coming up with that pretty obvious experiment at the time.)

As I mentioned in my previous post about inertial forces, Coriolis is a bit of a problem for long-duration space exploration. It would be nice to create spinning habitats that simulated gravity using centrifugal force, like the ones featured in the film 2001: A Space Odyssey:

But if they were as small as the one Arthur C. Clarke and Stanley Kubrick but in the spaceship Discovery, above, they’d need to spin fairly fast to produce a useful level of simulated gravity, which would produce significant Coriolis effects. Any movements in the plane of rotation (“up” and “down” in the centrifugal gravity, or around the curve of the habitat in either direction) would be deflected, making it difficult to perform physical tasks. Perhaps more importantly, the liquid in the semi-circular canals of the inner ear would suffer Coriolis deflection whenever the astronauts moved their heads, causing motion sickness and dizziness. Early studies in the 1960s suggested that any habitat rotation rate over 6 rpm would induce disabling motion sickness, but that limit seems to have been too conservative—with slow adaptation, rates of 10-20 rpm may be acceptable*. This means that the compact Discovery centrifuge in Kubrick’s film is more plausible now than when it was first released—surely a first for physical special effects in science fiction films.

For another demonstration of Coriolis deflection, here’s the sort of thing that would happen if you dropped something in a rotating space station. Coriolis in a space station 1Viewed from outside the station, the object would start to travel in a straight line as soon as you released it, and would continue to move in that direction until it hit the deck. Meanwhile, you would rotate alongside it as the station rotated. The diagram shows your position and that of the dropped object in four successive snapshots.

And here’s what it would look like to you, standing inside the station:

Coriolis in a space station 1

[If you’re interested in finding out more about the trajectories of objects launched in various directions inside a rotating space habitat, I’ve now written an entire post on that topic.]

Science fiction writers love this stuff, though they sometimes (well, often) get the detail wrong. In his novel 2312, Kim Stanley Robinson had Coriolis acting on movements parallel to the rotation axis of his habitat; and in the novel series The Expanse, the two authors who write as James S.A. Corey depict the Coriolis force varying from place to place within their rotating habitat.

That said, there was a valiant attempt to depict Coriolis in action in the television series of The Expanse. Here it is, deflecting a drink being poured:

Pouring a drink in The Expanse

The trajectory isn’t correct, but it’s still a thing of beauty.

But does Coriolis have any effect outside of the playground and science fiction? Very much so. Planet Earth behaves like a gigantic and slowly revolving roundabout—turning anticlockwise as you look down on the north pole, and clockwise as seen from over the south pole. So Coriolis forces deflect to the right in the northern hemisphere, and to the left in the southern hemisphere. But they don’t do very much at all near the equator—moving north or south at the equator takes you mainly parallel to the Earth’s axis, so doesn’t produce much Coriolis deflection. Moving west or east produces a larger Coriolis force, since you’re moving in the plane of rotation, but it is directed vertically, making you either a little heavier or a little lighter, rather than deflecting your direction of motion.

But away from the equator it’s Coriolis that generates the circulation of ocean currents, which tend to turn rightwards in clockwise gyres in the north, and leftwards in anticlockwise gyres in the south.

Ocean surface currents

The major wind systems are deflected by Coriolis too, drifting westwards as they approach the equator, and eastwards as they recede from it.

Global atmospheric circulation

And tropical storms and other low pressure regions  spin counterclockwise in the north and clockwise in the south, because of Coriolis. As air flows towards the centre of the low pressure area, it is deflected sideways and ends up flowing around the central low pressure, constantly being pulled inwards by the pressure gradient, but outwards by Coriolis.

Flow around a low pressure system in the northern hemisphere
Winds around a low pressure system in the northern hemisphere

All these are big things, you’ll notice. The Earth rotates slowly, the Coriolis forces are correspondingly weak, and they therefore need to operate for some time before they build up a noticeable deflection. This is all summed up by a ratio called the Rossby number (named for Carl-Gustaf Rossby):

Ro = U/Lf

U is the characteristic velocity and L is the characteristic length of the phenomenon you’re interested in. The f is something called the Coriolis parameter, which for situations that apply to the Earth’s rotation equals 2Ω sinφ, where Ω is the rotation rate of the Earth, and φ is the latitude (to allow for the fact that Coriolis is less effective at the equator, for reasons given above). At mid-latitudes f is equal to about 0.0001.

A small Rossby number (around one or lower) tells us that the length scale of our phenomenon is large enough in comparison to the velocity that Coriolis will have enough time to build up a significant deflection. A high Rossby number, on the other hand, tells us that Coriolis simply doesn’t have time to work, and that other forces in the environment will likely overwhelm its effect. The big oceanic and atmospheric phenomena I mentioned above operate at velocities on the order of ten metres per second, over distances of hundreds of kilometres, which places them at a range of Rossby numbers between 0.1 and 1, where Coriolis is significant.

Which brings me to bathtubs and toilet bowls, and the assertion that these objects drain in opposite directions in opposite hemispheres. Here, we have a phenomenon on the scale of a metre, with velocities of around a metre per second. That’s a Rossby number of ten thousand. The water flow is simply too fast for the scale, and Coriolis has no chance to produce a significant deflection—instead, the rotation of the drain vortex will depend on local factors like the shape of the bath, the swirl produced when the plug is pulled, and the direction in which the flush pipe of the toilet points.

But if you visit the equator, you’ll find people who appear to be demonstrating the Coriolis effect, on water draining from bowls placed just a few metres either side of the line.  Now, remember, the equator is the worst place on Earth to try to demonstrate this, because Coriolis forces operate so weakly there—ten metres north or south of the equator the Coriolis parameter (described above) is about 2.5×10-10, giving a Rossby number for a draining bowl of around four billion. That’s four billion. I think we can all agree that’s significantly greater than one.

But here’s a demonstration, from Michael Palin’s television series, Pole to Pole:

And another:

Remarkable, eh? Especially since the water drains the wrong way for the phenomenon they’re purporting to demonstrate. If Coriolis were at work, the drainage should behave like the flow into a low pressure system—anticlockwise in the north, clockwise in the south. Perhaps the demonstrators were confused into reproducing the rotation of the ocean gyres—but those are driven by tides and winds, not by a great big plughole in the middle of the Pacific.

So what you’re actually seeing demonstrated are two ways to generate whatever drainage swirl you want. In the first example, take a square bowl and make sure you turn to face your audience with the same rotation you want the water to follow—your turn will set the water swirling in the desired direction. In the second, change the shape of the bowl to favour a particular direction of drainage—for instance, by painting a big thick spiral of paint on the bottom of it.

Finally, here’s another, who gets an award for at least making the water swirl in the right direction, but a demerit for being so transparent about how she’s manufacturing the flow with her big bucket of water and her off-centre pouring style:

* Clément GR, Bukley AP and Paloski WH (2015) Artificial gravity as a countermeasure for mitigating physiological deconditioning during long-duration space missions. Front. Syst. Neurosci. 9:92.

If you’re thinking back to the fact that high velocity incurs high Coriolis force, you’re maybe wondering why the Rossby number favours low velocities. What’s happening is that high velocities get you to your destination faster, giving less time for the Coriolis forces to work. The distance travelled at a given velocity is proportional to the elapsed time, but the distance travelled at constant acceleration is proportional to time squared. Doubling the velocity gets you to your destination in half the time, and so the doubled Coriolis acceleration causes only half the deflection. (If I’d fired a gun at the central pillar of the roundabout, instead of kicking it, I’d have hit my target easily. There might have been other problems, though.)

Sun Dogs

Sunrise in St Andrews
Click to enlarge
© The Boon Companion, 2016

Ever since the success of her Clatto Swan photograph, The Boon Companion has been intermittently getting out of her warm bed at some truly God-forsaken hours to photograph sunrises. She recently took some early morning photos on the beach at Saint Andrews. She’ll be a bit annoyed with me for having chosen a glary one to post here—she has some much better views of the same scene. But I have my reasons.

Just right of centre of the frame there’s a little patch of colour in the sky, which could easily be mistaken for lens flare. It appears in the same place in all the views in this direction, but I chose the frame above  because it gives the best indication of where the sun is, showing that the little wisp of colour is at the same vertical height above the horizon. And it’s possible to zoom in on it, which is not a feature of lens flare:

Sundog in St Andrews
Click to enlarge
© The Boon Companion, 2016

A single patch of cloud is alive with light. Here it is in close up:

Sundog 2
Click to enlarge

What The Boon Companion had captured and brought home for me is a sun dog—or a parhelion, to give it its scientific name. They usually come in a pair, equally spaced either side of the sun and moving with it, like hunting dogs flanking a hunter. Unfortunately, the companion to this sun dog was out of sight behind higher ground to the right of the picture.

Sun dogs are formed by light refracting through hexagonal ice crystals, pretty much exactly in the way made familiar (to those of a certain generation) by Hipgnosis’s iconic cover design for Pink Floyd‘s 1973 album The Dark Side Of The Moon.

The Dark Side Of The MoonIf you cut the corners off that triangular prism, you have a hexagonal prism, which is the shape of the ice crystals that cause sun dogs.PrismLight goes in at one face, is deflected by refraction, and leaves by another face, being deflected again. There are various ways of describing the light path through prisms of various shapes, but I’m only going to talk about this one, in which the light enters one face of the hexagonal prism and leaves through what’s called an “alternate” face—the next face but one. Let’s call this route through the crystal the parhelion path, for ease of reference. And I’ll call the two relevant faces the entry face and the exit face.

The symmetrical situation I’ve depicted produces the minimum total deflection. For an ice crystal, that happens when the incident ray hits the surface of the prism at an angle of 41º, where an angle of zero degrees would mean that the ray hit the prism at right angles to its surface. (Don’t blame me, I didn’t invent the convention.) It is deflected by about 11º as it enters the crystal, and another 11º when it leaves, making a total deviation of 22º. Since that’s the minimum deviation, that’s the horizontal angle relative to the sun where we’ll start to see light that has been deflected along the parhelion path.

Parhelion light path
Click to enlarge
How ice crystals deflect sunlight to form a sun dog

(Actually, that’s only true when the sun is on the horizon, but it’s a good enough approximation for low solar altitudes. The situation at higher altitudes is fearsomely complicated, so I’ve relegated it to a Note and a link at the end of this post.) As Hipgnosis correctly showed, red (long-wavelength) light is deflected less than violet (short-wavelength) light, so we’ll see red light showing up closer to the sun than violet. The difference in minimum deflection amounts to about a degree between the longest and shortest visible wavelengths.

What about the deflection of light when it strikes prisms that are a little rotated relative to the nice symmetrical position in my diagram? We can make a graph showing how deflection varies with angle of incidence (remember, zero degrees incidence means entering the crystal at right angles to its surface; 90º incidence is a light ray that grazes along parallel to the surface). Here’s the graph, for violet, green and red wavelengths of light:

Light deflection in parhelion, by angle of incidence
Click to enlarge

That’s interesting, isn’t it? Between about 25º and 65º incidence, the deflection stays about the same, within a few degrees. So a beam of sunlight hitting an array of randomly rotated hexagonal ice crystals will generate a lot of light coming out in the vicinity of the 22º minimum angle, and then a smear of light out towards a maximum deflection of about 43º. (There’s no deflection at all below an angle of incidence of about 13º—light that comes in at a steeper angle to the entry face ends up being totally internally reflected at the exit face, so it can’t emerge on the parhelion path.)

So that’s the basic explanation for a patch of colour appearing in the sky 22º away from the sun—the cloud in the photograph contains hexagonal ice crystals. But why does the sun dog appear at the same height as the sun?

The crystals that create the sun dog are in the form of flat hexagonal plates, falling horizontally, like falling leaves:

Flat hexagonal prism It turns out that means the flat crystals must be between about  0.025mm and 0.25mm across—smaller, and they never get themselves orientated in the turbulent air; larger, and they tend to rotate end-over-end around a diagonal axis, rather than falling flat.

As these horizontal crystals fall level with the sun, they’re neatly orientated to send light towards our eyes. But because they naturally wobble a little, we also see refracted light coming from slightly tilted crystals that are a little higher or a little lower in the sky—the more the falling crystals wobble, the more vertically smeared the sun dog appears.

There’s more, though. The brightness of the sun dog depends on how much light gets through the crystals.  Some light gets reflected away from the parhelion path, at both the entry and exit faces, and that varies according to the angle at which it strikes each face. If we sum these reflection effects, we can plot the amount of light transmission through a triangular prism for various angles of incidence. It turns out that we get good transmission in the middle 25º-to-65º range that we saw producing a concentrated patch of light around 22º from the sun, with less light getting through at the extremes:

Proportion of unreflected light in parhelion, by angle of incidence
Click to enlarge

That’s what happens in a triangular prism, like the one in the Hipgnosis picture. Things are more complicated in a hexagonal prism, because not every ray that starts off along the parhelion path can find its way across the crystal to the exit face.

For the symmetrical orientation that produces the minimum angle of deflection, things work well—all the light that goes in one face can come out the other, barring the effects of reflection already discussed:Hexagon 1

But for other angles of incidence, only some light that enters the crystal along the parhelion path is on a trajectory that connects with the exit face:Hexagon 2

In effect, the aperture through which light gets to the sun dog is much diminished at angles that are not close to the minimum deflection angle. (Just make a mental note of that idea of a reduced aperture—it’s going to reappear in a different guise later.) When we put together the effects of reflection with the effects of this “face shuttering”, we find there’s a neat spike of transmission at the angle of incidence that corresponds to minimum deflection:

Total transmission of light in parhelion, by angle of incidence
Click to enlarge

So it’s clear that the larger angular deviations associated with extreme crystal rotation end up contributing very little light to the sun dog. And so we have a very good explanation of why the sun dog appears as quite a discrete patch of light in the vicinity of 22º from the sun, rather than extending into a long smear out towards the maximum of 43º.

But what about the distribution of colours? Working out the exact colour of the sun dog at different angles from the sun involves plotting the intensity of light at various wavelengths, over various angles. Now that we know deflections close to 22º are the important ones, I’m just going to graph three representative wavelengths (red, green, blue) over the range 21º to 25º:

Light intensity in parahelion, three wavelengths, refraction only
Click to enlarge

Now there’s a problem. Although, as I’ve shown, the brightness of the sun dog must fall off rapidly at large angular deviations, in our area of interest, a few degrees across, it doesn’t seem to decline much at all. There’s an initial spike of red at 22º, but when green appears, the red is still present. Those two colours together make yellow. And when blue appears at 22.5º, it has to compete with all the yellow light that’s still hanging around. When the detailed calculations are carried out, using more wavelengths and factoring in the colour perception of human eyes, they confirm first impressions from the graph above. Because the longer (red, yellow) wavelengths are still hanging about out beyond 22.5º where the shorter wavelengths begin to appear, the short-wavelength colours (blue, violet) should never become visible—they should simply cancel down to white. On the basis of refraction alone, the sequence of colours in a sun dog should go (from closest to farthest from the sun): red, orange, pale yellow, very pale greenish-yellow, white.

And sometimes that’s what we see. Here’s one that seems to follow the predicted sequence:

Sundog 4
Public domain

But sometimes not. Here’s the Boon Companion’s sun dog photo again, this time flipped left and right so that it’s orientated the same way as my graph and the image above:

Sundog 3
Click to enlarge

Beyond the hint of greenish-yellow, there’s definitely some pale blue. Some sun dogs show even more extensive colours:

Bright spectral sun dog
Click to enlarge
Public domain picture by Petr Kratochvil

So calculations performed using refraction match the appearance of some, but not all, sun dogs. What’s going on? One of the first people to think about this was S.W. Visser, in a paper (1.3MB pdf) published in the Proceedings of the Royal Netherlands Academy of Arts and Sciences, in 1917. (I used Visser’s data to plot my graph above.)

What Visser realized was that the ice crystals involved were small enough to cause significant diffraction. When light passes through a small aperture, it spreads out on the far side—the smaller the aperture, the greater the spreading. It also develops a characteristic pattern of light and dark streaks, called an interference pattern. Here’s a typical pattern of intensity for two sizes of aperture:

Diffraction fringes, two different slit widths
Click to enlarge

The blue line shows the interference pattern for green light passing through a 100μm aperture; the red line is the same light through a 25μm aperture. These apertures are in the vicinity of the size of ice crystals that produce sun dogs, so each ice crystal is a tiny aperture that causes diffraction and interference in the light that passes through it. And if all the ice crystals producing a sun dog are about the same size, then the diffraction and interference from all the crystals will add together and have an effect on its appearance.

When Visser did the calculations for the appearance of a parhelion with diffraction and interference superimposed on the effects of refraction, the data looked like this:

Light intensity in parhelion, three wavelengths, including diffraction
Click to enlarge

I’ve plotted on the same scale as before for comparison, although Visser’s data don’t extend beyond 23º. Several things are happening to change the shape of the light intensity curves:

  • Diffraction is broadening the main peaks of the curves, so that light starts to appear closer to the sun than is predicted by refraction alone.
  • The crystals that contributed to the long rightward tails of the curves are the ones that exhibit “face shuttering”—and the associated small apertures cause fierce diffraction, which spreads their light broadly and thinly over several degrees, so that it gets lost against the sky.
  • So instead of the long superimposed tails that combined to form a bright white patch of light, we see the intensity of each colour of light rise to its own peak and then fall off again—not quite separately, but not nearly as intermixed as is predicted by refraction alone.

When Visser did the detailed calculations, it turned out that blue does get a chance to be a dominant colour after all, albeit somewhat diluted by the lingering remnant of red and green wavelengths in the vicinity of 22.5º.

Visser’s calculations are for a uniform population of large crystals, with faces a quarter of a millimetre across, right on the stability limit. A mixed population of sizes would result in a lot of overlapping peaks, washing out the colour separation; a population of smaller crystals would give wider diffraction curves, again intermixing the colours to a greater extent. So Visser is giving us something like the best-case scenario for sun dog formation.

Put it all together and, just by looking at the sun dog picture brought home by the Boon Companion, we can tell that the cloud contains:

  • Flat hexagonal plate crystals; which are
  • Falling in a horizontal orientation; and
  • Oscillating gently. They are
  • Of approximately similar sizes (because blue is visible); and
  • They’re fairly large (because diffraction hasn’t severely intermixed the colours)

Not a bad series of inferences to be able to draw from a little patch of light in the sky.

Note: The 22º parhelion angle (and other parhelic angles discussed in the text) is strictly correct only for the case in which the sun is on the horizon, so that its light travels horizontally through the horizontally orientated crystals. When the sun is higher in the sky, its light necessarily travels through the crystals on a sloping path, which makes their prism angle appear broader than 60º, and pushes the sun dogs farther from the sun. Here’s a plot of where the sun dogs actually appear, against solar altitude:

Sundog position versus sun altitude
Click to enlarge

The sun dogs change shape and become more diffuse as the sun gets higher in the sky. They’re rarely observed above 40º, and optically impossible above 61º. For more on the complicated mathematics of sun dogs, see Roland Stull’s excellent free on-line textbook, Practical Meteorology. You want Chapter 22, Atmospheric Optics (1MB pdf).

Fall streaks

Fallstreaks at sunset, Juan-les-Pins
Click to enlarge
© The Boon Companion, 2016

At first glance, that’s just some nice sunset cloud over Juan-les-Pins in France. But there’s quite a lot going on in that picture, which you’ll see better if you click to enlarge it. (I’ll post some zoomed views of sections of the whole image below.)

The sun has just set, off to the left of the photograph. Its light is making it easier to figure out the heights of the various cloud layers. At right, we have some deep red clouds—low enough to be illuminated only by the sunset glow. At the left side of the picture, about halfway up, we have a line of clouds that look bright yellow—they’re a bit higher, and still seeing the setting sun. And at top left there are some fluffy white wisps—the highest clouds of all, still receiving full daylight illumination.

If I zoom in on the low, red clouds first, you’ll get a better view of what’s going on:

Fallstreaks detail 1
Click to enlarge

There are vertical streaks descending from the base of the cloud, which are then curving noticeably leftwards.

These downward extensions from the cloud are called fall streaks. As the air cools with sunset, the clouds are starting to release a little rain, but it’s evaporating before it reaches the ground. Another name for this phenomenon is virga, from the Latin for “rod”, and you can see the rod-like central cores in the fall streaks.

Why are they curving? The raindrops are accelerated vertically by gravity, but also pushed sideways by the wind. The larger the drop, the faster and more nearly vertically it falls. Small drops drift downwards at lower speeds, buoyed by air resistance, and therefore end up being pushed sideways more for a given vertical descent. But the drops are evaporating as they fall—big drops at the top of the fall streak turn into small drops lower down, so we see the streak bending more and more downwind the farther it falls below the parent cloud.

OK. If we look over at the yellow clouds now, we see something different:

Fallstreaks detail 2
Click to enlarge

More fall streaks, but pointing in the opposite direction. So the wind must be blowing in a different direction where these clouds are—either because it varies with height, or because there’s some local swirl caused by the land to the right of the picture.

Now look up at the highest, whitest clouds—they look like altocirrus to me, so will contain ice crystals rather than rain. They also have developed fall streaks, but rather remarkable ones:

Fallstreaks detail 3
Click to enlarge

The streaks go left initially and then turn abruptly right, making a neat right-angle in the air. Those falling ice-crystals aren’t doing much evaporating, since they seem to be travelling in pretty straight lines, but they have certainly fallen across a very abrupt transition in wind direction between the high cirrus and the lower clouds.

So there’s often quite a remarkable show going on up there, if you’re paying attention.

Fall streaks can produce some remarkable effects, if they rain out locally from an extended sheet of cloud:

Hole-Punch Cloud
Click to enlarge
Original photo by H. Raab. Used under Creative Commons Attribution-Share Alike 3.0 Unported licence

A little area of supercooled liquid droplets in the cloud layer has converted to ice, and that area is snowing out in the form of virga, leaving a patch of blue sky behind. These are called fallstreak holes, or hole-punch clouds.

Why do they form? The seem to be associated with aircraft passing through the cloud layer. The airflow around the plane’s wings (and propellers) causes a little local expansion of the air, which causes it to cool—if that’s enough to cause freezing, a fall streak is induced. And the vertical movement of air associated with the raining out of that initial fall streak causes a wave of up-and-down movement in the surrounding cloud, allowing the freezing effect to propagate outwards to produce a neat, almost circular hole. *

And if an aircraft hangs around in the vicinity of the cloud layer, rather than simply ascending or descending through it, you can get some rather spectacular linear hole-punch clouds, like the one in this video (posted by a birder, which explains the owl that appears in the bottom right corner):

* Heymsfield et al. Formation and Spread of Aircraft-Induced Holes in Clouds. Science 2011; 333: 77-81

Saying “Centrifugal” Doesn’t Mean You’re A Bad Person

Despite its daunting size, the huge structure was in fact a very simple machine, essentially a massive slingshot exploiting the rotation of the KBO to hurl objects into space. Slugs of refined, processed matter were loaded into open-topped buckets at the KBO’s surface. For the first hundred kilometres, they were hoisted up the length of the flinger by electric induction motors, until they passed through a point at which gravitational and centripetal effects were exactly balanced. After that, the flinger’s own rotation did the rest of the work.

Stephen Baxter & Alastair Reynolds The Medusa Chronicles (2016)

Baxter and Reynolds are describing the mining of a Kuiper Belt Object in the outer solar system. This asteroid-like body is spinning on its axis, and a tall tower (the “flinger”) has been erected on its surface at its equator. The tower is so high that the KBO’s rotation swings its upper end around at faster than orbital velocity. So the tower is being twirled around like a stone on the end of a string. If you can move buckets of refined ore up the tower far enough against the KBO’s slight gravity, they’ll soon get to a point at which they are impelled  up to the top of the tower and launched into space without any more input of energy. That transition point corresponds to the middle bucket in my little diagram:KBO and launch tetherIt’s the point at which a bucket, released from the tower, would just hang around in orbit, right where it was. It’s also the point Baxter and Reynolds describe as the “point at which gravitational and centripetal effects [are] exactly balanced”.

But wait. We all know that gravity is a central force—it pulls inwards, and in my little diagram it’s pulling the buckets to the left, down the tower, towards the centre of the KBO. But centripetal means “centre-seeking”—a centripetal force is one that pulls inwards, towards the centre. So gravity is a centripetal force, and there simply can’t be a point at which “gravitational and centripetal effects” balance, because they’re the same thing.

What Baxter and Reynolds meant to say was that gravity and centrifugal effects are exactly balanced, because centrifugal means “centre-fleeing”. Where gravity and centrifugal effects balance, the middle bucket in my diagram experiences no net inward or outward force, and stays in orbit. Lower down the tower, gravity wins, so the buckets need to be pushed up that section. Higher on the tower, centrifugal force wins, and the buckets slide higher, against gravity, and eventually fly off into space.

Now, between them, Baxter and Reynolds have multiple degrees in maths, engineering and physics. They know this stuff. Why then did they choose the word centripetal instead of centrifugal? I suggest that it was because, in some quarters, the use of the word centrifugal is thought to mark you out as someone ignorant of physics. It’s toxic. Famous physicists and astronomers who make public statements using the word “centrifugal” find themselves being loftily denounced on social media. So writers will try to work around it, even when it means writing something nonsensical instead.

For generations, guilt about the word “centrifugal” was one of the few things people took away from their physics classes at school. I can still picture Mr Anderson (a very fine physics teacher) tapping his desk with the corner of the blackboard eraser (yes, it was that long ago) and intoning: “There’s! No! Such! Thing! As! Centri! Fugal! Force!”

He was making an important point, which is this:

One orbital cycle
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If we spin something around in a circle (a stone on a string, a satellite in orbit), the only force it experiences is centripetal.

An object will move in a straight line unless acted on by a force. To make it move in a circle, it has to be pulled out of its straight-line path and made to accelerate constantly towards the centre of the circle. The centripetal force to generate that acceleration is provided by tension in a string (for the swinging stone), or by gravity (for the orbiting satellite). The centrifugal component is revealed to be just the tendency of the circling object to head off in a straight line, tangent to the original circle, as soon as the centripetal force is released (by letting go of the string, for instance). There’s no force pulling the object outwards. Hence my physics teacher’s emphatic attempt to drive the maxim, “There’s no such thing as centrifugal force,” into our reluctant little heads.

But what happens if we climb inside a rotating reference frame and rotate along with it? What does physics look like in that situation? Here’s Stanley Kubrick‘s gorgeous (and, for our purposes, accurate*) evocation of life inside a spaceship centrifuge, for 2001: A Space Odyssey:

It’s worth just clicking on the movie to see how eerie it looks. Of course if we step outside the centrifuge (and listen again to my physics teacher hammering away at his desk), then we can see that the jogging astronaut is whirling around in circles, and would be heading off in a straight line if the floor of the centrifuge wasn’t applying a centripetal force to his feet.

But from inside, with our viewpoint anchored to rotate along with the centrifuge, there certainly seems to be a force sticking the astronaut to the floor, doesn’t there? Indeed, if Isaac Newton had lived his life inside some gigantic space-borne centrifuge, without ever knowing that he was rotating, he’d have been able to formulate his Laws of Motion  just fine, but with the addition of a centrifugal force. (He’d also have need to add another force,  acting to deflect objects in motion relative to the rotating coordinates of his centrifuge—that one is called Coriolis force, and it’s the topic for another post, I think. [I’ve now written a post about Coriolis in general, and another about its application to rotating space habitats in particular.])

So if we choose to do physics in a rotating reference frame, then we find we have these extra forces to contend with—centrifugal and Coriolis.

Of course, they’re rather odd forces—they only crop up because we’ve chosen to use a particular kind of accelerating reference frame rather than an inertial reference frame. And they act to produce a specific acceleration, irrespective of mass—as if the force tuned itself to match the mass of the object it had to accelerate. For this reason they are sometimes called “pseudo-forces”, “fictitious forces” or simply “effects”. But when you figure with them, they work just like real forces. And, interestingly, gravity works just like one of these “pseudo”-force, always producing a specific acceleration—heavy objects fall no faster than light objects. That fact provided Einstein with the insight that led to General Relativity, and a way of treating gravity as being the result of a specific choice of accelerating reference frame. That insight is now a century old, but we’re strangely free of physics teachers hammering on their desks, saying, “There’s! No! Such! Thing! As! Gravity!”

But are there really situations where we, for preference, adopt a rotating reference frame? There sure are, and you’re sitting in one right now. For most purposes we treat the Earth as if it were stationary, despite our knowledge that it rotates. And that means that people whose job it is to calculate trajectories for rockets and missiles do so relative to a “stationary” Earth, while factoring in the effects of centrifugal and Coriolis forces. And meteorologists routinely deal with Coriolis force as it deflects air masses moving across a “stationary” Earth. There’s also sometimes benefit to be had in celestial mechanics, from adopting a rotating reference frame. The effects of gravity and centrifugal force are mathematically combined into a surface of “effective potential”, over which objects move subject to Coriolis force. That’s what’s happening in this contour map of the Lagrange points of the Earth and Sun, for instance:

Lagrange points
NASA / WMAP Science Team

So it’s all about adopting an appropriate reference frame—centrifugal and Coriolis forces are required in a rotating reference frame, forbidden in a non-rotating frame. Baxter and Reynolds, in my opening quote, were free to invoke centrifugal force as they followed the buckets up the length of the rotating flinger. But they seem to have become so nervous at the prospect of typing “centrifugal” that they just stuffed in the word centripetal instead, hoping no-one would notice.

So it’s fine to say “centrifugal”, as long as you are talking about a rotating reference frame. And actually, most times people use the word, they don’t really nail down the reference frame tightly enough to lay themselves open to justified criticism, anyway.

Finally, just in case you’re still anxious about this, I’m going to haul a few physics textbooks off the shelves, and take a look at their indexes:
Murray & Dermott, Solar System Dynamics. Four pages mentioning centrifugal acceleration, three centrifugal force, and four centrifugal potential. None concerning centripetal.
French, Newtonian Mechanics. One page on centrifugal force, one on centrifugal potential energy, and (by way of balance) two on centripetal acceleration.
Frautschi et al., The Mechanical Universe. Nine pages on centrifugal force, three on centripetal acceleration and four on centripetal force.
Feynman, The Feynman Lectures On Physics. Two pages on centrifugal force (in the second, as part of a discussion of how pseudo-forces arise from coordinate choices).

It really is okay to say “centrifugal”.

* Kubrick’s centrifuge film-set rotated to keep the astronaut at the bottom during filming, so it necessarily depicted a centrifugal force of one Earth gravity. This would be difficult to achieve with a centrifuge just 35 feet in diameter, the value given in the novel and depicted in the film (see here for a painstaking effort to retrieve the diameter of Kubrick’s centrifuge from movie footage and production stills). Such a centrifuge would need to complete 13 rotations per minute. Coriolis forces would be strong at that speed of rotation—deflecting limb movements and (more importantly) inducing abnormal fluid shifts in the semicircular canals of the inner ear during head movements. It might just be possible to adapt to the motion sickness induced in such a rapidly rotating environment—see Clément et al. for a recent review—but slower would be better.
In the novel of 2001: A Space Odyssey, Arthur C. Clarke addressed this problem by having the Discovery centrifuge rotate at 6 rpm, producing an approximation to lunar gravity. Clarke based his rotation speed on the hard data available in the 1960s, which suggested that 6 rpm was the limit of human tolerance. Whether lunar gravity is high enough to maintain long-term health is unknown—but there would seem to be some physiological wiggle room that would allow the centrifuge depicted in 2001 to rotate fast enough to maintain health, but not so fast as to be nauseating for its occupants.

Reflections In A Spiral Mirror

The title of this post looks like it could be the name of a concept album by a pretentious prog-rock band. But it’s completely literal—I came across the spiral mirror in question while walking back from Tralee into Benderloch the other day. It was an outdoor ornament of the kind that seems to be called a “spiral wind twister”, and it was conveniently dangling from a sign beside the road.Spiral Wind TwisterI had a camera in my pocket, so I had the chance to photograph a puzzling little phenomenon that was first pointed out to me by Dave Hewitt and Chris Tyler.Reflection in Spiral Wind Twister

If you look closely, there are two odd things about my reflection in the spiral mirror of the wind twister—one is that I’m turned sideways; the other is (as you can see from the readable text on the sign behind me) that I’m not mirror-reversed. A mirror that rotates but doesn’t reverse! I don’t know about you, but I certainly didn’t see that one coming.

It must be something to do with the complex shape of the reflective surface. It takes a moment to tease this out, but the spiral produces a mirror that is convex on one axis and concave on another. The two axes are roughly at right angles to each other, and at forty-five degrees to the vertical. I’ve marked the concave axis in red on the enlargement below—you can see the edges of the mirror curling to face you at the top and bottom of that line. The convex axis is in green, with the mirror curving away from you in both directions along that axis.Mirror axes of Spiral Wind Twister

To start working out what’s going on, I’m going to look at examples of mirrors that are either purely convex or purely concave. Here’s a photo of the cover of a book reflected in a rather spiffy Venetian convex mirror:Reflection in a convex mirrorThe convexity lends a fish-eye distortion to what is otherwise just a conventional mirror-image reflection. There’s nothing new going on there, which suggests we can ignore the convex component of the spiral mirror when we try to tease out the cause of its remarkable reflection.

Now here’s the same book reflected in a concave shaving mirror, with the (zoomed) photograph taken from a couple of metres away.Reflection in a concave mirrorAs you can see, concave mirrors do something interesting. When you’re close to them they act as magnifying mirrors. But if you step back outside the focus of the mirror, they turn into inverting mirrors.

This is what’s going on:Inverted reflection in concave mirrorThe curve of the mirror means you need to look up to see the reflection of your feet, and down to see the reflection of your face. Likewise, you need to look left to see your right hand, and right to see your left hand. In effect, a uniformly concave mirror flips the standard mirror image both left-to-right and top-to-bottom. Below, I’ve marked those rotation axes with black lines superimposed on a standard mirror image of the book:Transformation of mirror reflection by concave mirror

These two reflections turn out to be the equivalent of a 180° rotation—the image in the concave mirror is still mirror-reversed, but now it’s upside-down, too.

But the wind twister was concave in only one direction. Suppose we had a mirror that was only concave from top-to-bottom. There are metal mirrors like that behind the elements in old-fashioned two-bar electric heaters, like this ancient example I lugged out of the attic for illustrative purposes:Cylindrical mirror in electric fire

A cylindrical concave mirror like that will generate only a top-to-bottom image flip, and not a right-to-left flip:Transformation of mirror reflection by horizontal cylindrical mirrorAh-ha! The top-to-bottom flip, combined with the usual mirror-reversal, gives us an inverted but unreversed final image. Now we’re getting someplace.

Finally, we just need to remember that the concave curve of the wind twister is orientated diagonally, so it’s going to do a diagonal flip on the reflected image. Like this:Transformation of mirror reflection by tilted cylindrical mirrorTa-da! I’ve finally reconstructed the 90°-rotated but unreversed image from the wind twister. Who’d have thought you get so much out of peering closely at a garden ornament?

Note: There’s an old puzzle: Why does a mirror reverse left and right but not top and bottom? I’ve skated around that, above, by talking only about “the usual mirror reversal”. But, actually, a mirror doesn’t reverse left and right at all—it reverses front and back. The person you look at in the mirror has head, feet, left hand and right hand in the same places as you do, but is facing in the opposite direction. It’s keeping left and right in the same place while reversing front and back that turns your reflection into a mirror image.


Having recently criticized Tristan Gooley’s explanation of the tides, I felt obliged to try to do better myself. It’s a tricky job, and there are many partial and misleading explanations out there. So here goes.

Tides happen to anything that is orbiting in a gravitational field. I’m going to hone down on the Earth in a minute; but first, an orbit:

Radial acceleration in orbit
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The orbiting body (“the satellite”) would travel in a straight line if no force was being applied to it. But it is under the influence of the gravity of the central body (“the primary”). The force of gravity pulls the satellite into a curved path around the primary. For a range of speeds, this causes the satellite’s path to curve enough to make it loop right around the primary and then repeat itself. (If the satellite is moving too slowly,  its curved path will come close enough to hit the primary; too quickly, and the loop will never close, allowing the satellite to escape.)

In the diagram, the satellite has precisely the right speed to move in a circular orbit at a constant distance from the primary. The primary’s gravity pulls the satellite radially inwards with a force of constant magnitude. This generates an acceleration of constant magnitude, indicated by the blue arrow. Since the acceleration is always at right angles to the satellite’s motion, the satellite’s speed doesn’t change, only its direction of travel. This induces a constant curvature in the satellite’s path, which makes it circle endlessly.

One orbital cycle
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For simplicity, I’m going to deal with only circular orbits from now one, but the logic of the tides applies equally well to all orbits.

The model of a satellite whirling in circles around a stationary central body is good enough for any satellite with a mass that’s very low compared to its primary—like the International Space Station  in orbit around the Earth, for instance. But if the satellite’s mass is comparable to the primary’s, then the primary has to follow an orbit too:

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While the satellite follows a large circle, the primary moves in a small circle so that the two bodies staying exactly opposite each other on either side of their common centre of gravity, which is called the barycentre (from Greek barys, “heavy”). I’ve marked it with a little cross in the diagram. Since primary and satellite both complete one orbit in the same time, the primary has a lower speed and a smaller radial acceleration, which is provided by the weaker gravity of the less massive satellite. Like a fat man balancing a child on a see-saw, the more massive primary, huddled close to the barycentre, is in balance with the lightweight satellite moving in its more distant orbit.

The Earth-Moon system has a barycentre that is actually inside the Earth. While the Moon sweeps out its month-long orbit, the Earth describes a gentle wobble during the same time period, with its centre alway on the opposite side of the barycentre from the Moon:

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The centre of the Earth is therefore always accelerating gently in the direction of the Moon as it moves around its small balancing orbit.One cycle of Earth's barycentric orbit

What may not be intuitively obvious is that at any given moment every point on the Earth’s surface and within its bulk must have exactly the same acceleration (in magnitude and direction) as the centre of the Earth does. If that didn’t happen, then the various bits of the Earth would acquire relative velocities, and the Earth would change shape. (To be strictly accurate, a little relative acceleration is allowed, as the Earth flexes under the influence of the Moon’s gravity, but the net acceleration must average out to zero over time.)

General acceleration of Earth
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Even with that logic in place, it’s still a little difficult to see immediately why a point on the Earth’s surface on the opposite side of the barycentre from the centre of the Earth should be accelerating away from the barycentre, when the centre of the Earth is accelerating towards it.

The explanation is that every point on the Earth is tracing out its own circle in space, the same size as the Earth’s orbit around the barycentre, but displaced from it. To see how that works, let’s stop the rotation of the Earth (diagrammatically) and trace the path of a single point on its surface (marked in purple) during the course of a month.

Circular path of point on Earth's surface
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The acceleration of the purple point is always directed towards the centre of its own (purple) circle, even though it may be directed away from the barycentre. The rotation of the Earth doesn’t make any difference to this argument—the instantaneous accelerations remain the same, they’re just handed off to different points on the surface of the Earth as it rotates.

(If you’re having trouble visualizing the circular movement of the non-rotating Earth depicted in the diagram, put a coin flat on a table, put your finger on the coin, and slide the coin around in a small circle.)

So the blue acceleration arrows show what the Earth is actually doing during the course of a lunar orbit. But does the Moon’s gravity apply forces in the right direction, and of the right magnitude, to make the Earth accelerate smoothly throughout its volume in this way?

No, it doesn’t. There are two problems:
1) The Moon’s gravity decreases with distance. While it pulls on the centre of the Earth with just the right force to induce the necessary acceleration to keep the Earth in its orbit around the barycentre, it pulls a little harder on the near side of the Earth, and a little too weakly on the far side.
2) The Moon’s gravity is a central force—it radiates out from the centre of the Moon. So it’s directed a little diagonally when it pulls on parts of the Earth that don’t lie exactly on the line connecting the centres of the Earth and Moon.

That’s all shown in this diagram, with the green arrows representing the force of the Moon’s gravity laid on top of the blue arrows representing the true acceleration:

General acceleration of Earth + lunar gravity
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There’s a mismatch, everywhere but at the centre of the Earth, and the difference between the applied force  and the necessary force (for uniform acceleration) must be generated by internal forces within the substance of the Earth. The nature of the mismatch between applied force and real acceleration is shown with red arrows below:

Mismatch between general acceleration and lunar gravity
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These residual forces are called tidal forces, and so at last I’ve arrived at the cause of the tides. The Earth is being stretched along an axis that runs through the Moon and the barycentre, and squeezed inwards in a plane at right angles to that axis. (Even though I’ve built this argument around the Earth and its small barycentric orbit, this is a completely general result—it applies to all bodies in orbit around other bodies. They all experience tidal forces of this sort. In fact, it should be evident that it applies equally to bodies that aren’t even in orbit, but are just falling towards each other, or even sitting next to each other—all that’s required for these internal tidal forces to show up is for an object to be maintaining its shape against the forces produced by a central gravitational field.)

Now, if the Earth was a hunk of solid metal, held together by its internal chemical bonds, it would develop a bit of tension along the “stretch axis”, and compression in the “squeeze plane”. Those internal forces would oppose the tidal forces, and ensure that all the parts of the Earth moved together with uniform acceleration.

But objects on the scale of planets aren’t held together primarily by chemical bonds—what keeps them together is their own gravity, and they settle into an equilibrium shape that evens out internal pressures. The red arrows in the diagram show that the Moon’s gravity opposes the Earth’s own gravity along the “stretch axis”, and supplements it in the “squeeze plane”. This slight alteration in the local gravitational force means that the solid body of the Earth shifts slightly in shape in order to equalize its internal pressures.

The same thing happens to the oceans—they pile up under the reduced gravity of the “stretch axis”, and squash down under the increased gravity of the “squeeze plane”:

Tidal forces and tidal bulges
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And that’s where tides come from, and why there are two tidal bulges in the ocean, one under the Moon and one opposite it.

As the Earth rotates, it carries us past each tidal bulge in turn, so there are two high tides per day. Or, actually, not quite. By the time the Earth has completed one full rotation, the line between Earth and Moon has shifted a little, and the tidal bulge has shifted with it. The Earth therefore needs to rotate for another 50 minutes at the end of each day, in order to catch up with the position of the tidal bulges:

Tides come later every day
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So instead of experiencing a high tide every 12 hours, we get one every 12 hours and 25 minutes.

The situation is actually (you guessed it) a little more complicated—the presence of landmasses distorts the even flow of water suggested in my diagram; the Sun produces its own tidal bulges; and the inclination of the Moon’s orbit to the Earth’s equator introduces its own complexities.

Those are topics for another day.