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The Celestial View From A Relativistic Starship: Part 2

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Bussard Interstellar Ramjet
Bussard Interstellar Ramjet (source)

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

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

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

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

Aberration: rest frame view of sky
Click to enlarge

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

Aberration: Apparent shift at 0.5c
Click to enlarge

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

Aberration: apparent shift at 0.5c and 0.85c
Click to enlarge

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

RELATIVISTIC DOPPLER

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

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

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

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

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

Relativistic Doppler in the moving frame
Click to enlarge

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

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

Relativistic Doppler translated to the rest frame
Click to enlarge

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

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

Percentage of sky blue-shifted (rest frame and moving frame)
Click to enlarge

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

Effect of beta on aberration and Doppler
Click to enlarge

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

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

Relativistic Doppler for various starting positions and various velocities
Click to enlarge

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

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

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

Aberration and Doppler: apparent shift at 0.5c
Click to enlarge
Aberration and Doppler: apparent shift at 0.5c and 0.85c
Click to enlarge

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

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

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

Mathematical notes

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

\Large \eta =\frac{1+\beta \cos \theta }{\sqrt{1-\beta ^2}}=\frac{\sqrt{1-\beta ^2}}{1-\beta \cos \theta '}

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

\Large \cos \theta _{1}=\frac{\sqrt{1-\beta ^2}-1}{\beta }

In the moving frame, η=1 at angle:

\Large \cos \theta' _{1}=\frac{1-\sqrt{1-\beta ^2}}{\beta }

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

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The Celestial View From A Relativistic Starship: Part 1

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Bussard Interstellar Ramjet
Bussard Interstellar Ramjet (source)

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

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

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

LIGHT ABERRATION

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

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

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

Aberration: Direction of shift

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

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

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

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

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

Effect of beta on aberration
Click to enlarge

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

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

Aberration: apparent magnification

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

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

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

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

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

Aberration: rest frame view of sky

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

Aberration: Apparent shift at 0.5c

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

Aberration: apparent shift at 0.5c and 0.85c

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

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

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

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

Mathematical notes

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

\Large \cos \theta '=\frac{\beta +\cos \theta }{1+\beta \cos \theta }

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

\Large r'=\frac{r\sqrt{1-\beta ^2}}{1-\beta \cos \theta '}

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

\Large r'=\frac{b\sqrt{1-e^2}}{1-e \cos \theta '}
\Large b=r\; \; \; a=\frac{r}{\sqrt{1-\beta ^{2}}} \;\;\;e=\beta

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

\Large (x',y',z')=\left ( x,y,\frac{z+\beta r}{\sqrt{1-\beta ^{2}}} \right )

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

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Coriolis Effect In A Rotating Space Habitat (Supplement)

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

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

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

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

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

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

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

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

AXIAL MOTION

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

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

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

Trajectories in a rotating habitat 6
Click to enlarge

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

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

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

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

ROLLING MOTION

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

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

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

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

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

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

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Territories That Crossed The Date Line: Part 2 – 1900 To Present

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180 meridian marker, Krasin Bay, Wrangel Island
Click to enlarge
The 180º meridian crossing Wrangel Island © The Boon Companion, 2016

 

The Date or Calendar Line is a modification of the line of the 180th meridian, and is drawn so as to include islands of any one group, etc, on the same side of the line.
When crossing this line on a westerly (true) course, the date must be advance one day; when crossing it on an easterly (true) course, the date must be put back one day.

Admiralty Hydrographic Department (1921)

In my last post on this topic, I brought the story up to the year 1900, when the International Date Line assumed a simple and elegant shape that was to persist, effectively unchanged, for almost a century—a zig-zag through the Bering Strait and around the Aleutians in the north, and a neat eastward shift around the Chatham Islands and Tonga in the south. But it actually took a while for the standard form of the Date Line to settle down into the shape I mapped out at the end of my previous post. So before I move on to other territories that have moved from one side of the Date Line to the other, I need to deal with the evolution of the shape of these northerly and southerly diversions.

TWO DIVERSIONS—PART 1. CHATHAM ISLANDS

Prior to 1910, the southern diversion around the New Zealand possession of the Chatham Islands followed a variety of curves, passing very close to the Chatham group. You can see an example in the chart from the paper “Where The Day Changes” (A.M.W. Downing. Journal of the British Astronomical Association, 1900), below:

Dateline detail (Chatham) from "Where The Day Changes" (1900)And some more in the plots from a 1921 article produced by the Royal Navy’s Hydrographic Department, which was later reproduces as “Notes On The History Of The Date Or Calendar Line” (New Zealand Journal of Science and Technology, 1930). Here’s a detail from the original 1921 version:Dateline detail (Chatham) from "Notes on the History of the Date or Calendar Line" (1921)

Most early versions of the Date Line curved around very close to the east side of the Chathams. One (the Admiralty version from 1892-1910) seems to slice right through the middle of the group. But there appears to be no evidence that the Chathams ever used any dating system but Asiatic, in synchrony with New Zealand.

In 1910 the Hydrographic Department standardized the shape of the southern diversion, so that its eastern boundary ran along the 172.5ºW meridian. That style was copied by other cartographers, and has persisted to the present day.

TWO DIVERSIONS—PART 2. WRANGEL ISLAND

The northern end of the Date Line also took an unusual and confusing course on some maps, before it finally settled to its current form. For a while, on Royal Navy charts, it was shown following the 180º meridian right through the middle of Wrangel Island, in the high Russian Arctic. Along the whole length of the Date Line, Wrangel was the only piece of land it was depicted as traversing. So what was going on there?

In their book Plotting The Globe, Avraham Ariel and Nora Ariel Berger offer the story of Vilhjalmur Stefansson’s doomed attempt to plant a colony on Wrangel Island in 1921. The Soviet Union, the United States and the United Kingdom all had competing claims to this desolate spot —but Stefansson intended to use his colony to claim the island for Canada. When the Canadian government wisely decline the option, Stefansson decided to claim it for Britain instead. Four out of five of his original “colonists” died—the fifth, an Iñupiat woman known as Ada Blackjack, was rescued in 1923. (The full story of this tragic fiasco is told in Jennifer Niven’s book Ada Blackjack.)

Cover of "Plotting The Globe", by Ariel and BergerAriel and Berger offer this episode as an explanation for the Admiralty’s routing of the Date Line through Wrangel, claiming that:

[The British government] did not want to wrangle with the Soviet Union about Wrangel Island. The British Admiralty, however, quickly shifted the date line from its position east of Wrangel Island—which made it completely Russian—to the 180th meridian. The island was sliced into two date zones, as an initial recognition of a British-Canadian claim to at least its eastern part.

After Stefannson’s venture failed, and the Russians established a presence on the island to bolster their own claim:

Reluctantly the Admiralty returned the date line to its pre-1921 position in the Chukchi Sea, well east of the island.

This makes a great story, but Ariel and Berger don’t provide any references to support it, and it actually doesn’t make much sense.

  • Military cartographers are well aware of the political implications of their charts, and are unlikely to jump the gun in the way described.
  • The Date Line is not a territorial claim—there has, for instance, been a country (Kiribati, see below) that spent more than a decade with the Date Line running right through its middle, and no-one ever suggested that this divided it into two countries.
  • No-one (including Stefansson) was ever claiming half the island—everyone wanted the whole of Wrangel. So splitting it down the middle exactly along the 180º meridian corresponded to no-one’s view of the situation.
  • Stefansson, never one to underplay his own importance, makes no mention of this Date Line shift in his book, The Adventure of Wrangel Island (1925).

And, tellingly, we know for a fact that the Admiralty’s Hydrographic Department was running the Date Line to join the 180º meridian south of Wrangel Island long before Stefansson’s adventure. In 1900, the article “Where The Day Changes” contained a plot of the Date Line according to the Hydrographer of the Navy, Admiral William Wharton—it’s the solid line in the detail below, and it shows the Date Line terminating due south of Wrangel, on the 180º meridian.

Dateline detail (Wrangel) from "Where The Day Changes" (1900)

(Another cartographer, Benjamin E. Smith, is responsible for the dashed line, which terminates east of Wrangel.)

And “Notes On The History Of The Date Or Calendar Line”, published by the Admiralty Hydrographic Department in November 1921, routes all of its versions of the historical Date Line, from 1892 onwards, straight across Wrangel on the 180º meridian:

Dateline detail (Wrangel) from "Notes on the History of the Date or Calendar Line" (1921)Now, Stefansson’s “colony” was established in September 1921, but he kept his territorial aspirations a secret, at first. The fact that he claimed Wrangel for Britain was not revealed until March 1922, in an article in the New York Times*, well after the Hydrographic Department produced the map above. So it’s evident that, whatever the reason for the Admiralty’s routing of the Date Line through Wrangel, it predated Stefansson’s claims, and therefore had nothing to do with them.

What were they up to, then? I think the way Wharton’s line terminates non-commitally south of Wrangel tells us the answer—the Admiralty were trying to avoid even the implication of a territorial claim. Terminating the Date Line on the 180º meridian south of Wrangel was one way of avoiding the issue. If pressed, running the line north along 180º degrees as if Wrangel didn’t exist was another way of doing so—deviating it to the west could be interpreted as supporting a British or American claim; to the east could be interpreted as a vote for the Soviet Union.

Once the British government’s official position was that Wrangel was Russian territory (which was established in 1924), the Admiralty were free to move the Line east of Wrangel to reflect political reality—and it has been there ever since.

Now. On with the story of territories that have changed their calendars and crossed the Date Line.

KWAJALEIN (1969 and 1993)
Kwajalein Atoll in the Marshall Islands
Click to enlarge
Original source

Kwajalein Atoll is one of the many atolls that make up the Republic of the Marshall Islands. The Marshall Islands came under United States administration at the end of the Second World War, as part of the Trust Territory of the Pacific, and Kwajalein island (the largest island in the atoll) became the site of a large American military base, while the USA used the Marshall Islands as a site for atomic bomb and ballistic missile testing.

And then a curious thing happened in 1969. The Marshall Islands made a slight adjustment in their time zone, from GMT+11 to GMT+12—at midnight on Wednesday, 30 September, the Marshallese put their clocks forward by an hour, to 01:00 on Thursday, 1 October. Except, that is, for the inhabitants of Kwajalein Atoll (predominantly military personnel and support staff), who put their clocks back by 23 hours, to 01:00 on Wednesday, 30 September, thereby repeating a day and shifting across the Date Line to GMT-12. Their clocks remained in synchrony with the rest of the Marshall Islands, but their calendar was now one day behind. This was done so that the Kwajalein military base had the same working week as colleagues on the American side of the Date Line—office hours falling into approximate alignment on five days out of seven instead of just four.

Few cartographers bothered to mark this on the map, but the Date Line had just developed a little isolated loop of American dates, surrounded on all sides by Asiatic calendar.

The Marshalls became self-governing in 1979, and then fully independent in 1986—but the US military base was still there, and Kwajalein Atoll was still a day out of synch with the rest of the country. In 1993, the Marshallese government made a request to the US for Kwajalein to return to the same date as the other Marshall Islands. So Friday, 20 August 1993 was followed by Sunday, 22 August 1993. The residents staged a celebratory two-mile run that started just before midnight on Friday, and so took a whole day to complete—it was called the “Run Around The Clock”. Synchrony with the working week in the rest of the USA was maintained by the sort of solution only the military can impose—the working week on Kwajalein now starts on Tuesday and ends on Saturday, with Sunday and Monday being the official weekend.

Since the Marshall Islands were part of the Spanish possessions that shifted from the east to the west side of the Date Line in the nineteenth century, Kwajalein is the only territory in the world to have crossed the Date Line three times.

EASTERN KIRIBATI (1994)

Kiribati is pronounced “Kiribas” (the letter t is pronounced “s” in Gilbertese, and a final i is silent). It’s an island nation that straddles the 180º meridian, composed of the Gilbert Islands (west of the meridian, and once owned by the UK), and the Phoenix and Line Islands (east of the meridian, once claimed by both the UK and USA). The Gilberts used to keep Asiatic dates, while the Phoenix and Line Islands observed American dates. When the Gilberts became independent in 1979, the USA signed over the Phoenix and (most of) the Line Islands to this newly formed Republic of Kiribati (“Kiribas” being as close as it’s possible to get to saying “Gilberts” in Gilbertese).

So, this new country had the Date Line running right through its middle. The working week east of the Date Line did not match the working week in the Gilberts, the main population centre. So after putting up with this ridiculous situation for more than a decade, Kiribati adjusted itself so as to observe the same date throughout its territory—eastern Kiribati moved straight from Saturday, 31 December 1994 to Monday, 2 January 1995. (Pause for the obligatory joke: Haven’t we all had a Hogmanay like that?) The neat and minimalist International Date Line that had lasted almost a century suddenly developed a huge eastward panhandle as it routed itself around the Line Islands and their eccentric GMT+14 time zone.

While you might think this was all fine and sensible, it provoked an outcry, simply because of its timing. The new millennium was looming, and a (fairly minor) tourist demand had been created to offer a view of the first sunrise of the year 2000 (since everyone had decided that 2000 was the year the new millennium began, despite the cries of the purists). Kiribati had just shoved the Date Line so far towards the rising sun that it now owned the territory on which that sunrise was going to happen (if we ignore Antarctica). Uninhabited Caroline Island, in the extreme south-east of the Line Islands, was accordingly renamed Millennium Island.

At this remove, it’s difficult to credit the fuss that was made about a perfectly reasonable date adjustment. Pitt Island, one of the Chatham Islands (see above), had previously been the place that would greet the millennial sunrise, by virtue of being tucked up against the southern deviation in the Date Line. But neither the Chathams nor the Line Islands are particularly accessible, and both lack the facilities to support any major influx of sunrise tourists. So there wasn’t some huge financial implication underlying all this. And yet appeals were made to the United Nations (who said, in effect, “Nothing to do with us”) and the Greenwich Observatory (who said, in effect, “Well, that’s interesting, but nothing to do with us”).

Sunrise line in the Pacific, 16:00 GMT, 31 December 1999 - Chatham and Millennium Islands marked
Here’s one I prepared earlier.
I produced this diagram for an article I wrote about the millennium sunrise, back in 1999. On the map, it’s 16:00 GMT, 31 December 1999, and most of the western Pacific is in darkness. But the sun has already risen on the morning of 1 January 2000 on the Chatham Islands and Millennium Island (marked).

Bizarrely, long after the event, Ariel and Berger devote a large part of a chapter in Plotting The Globe (2006) to exercising their outrage against Kiribati—expressions like “tricksters”, “chutzpah or desperation” and “has shown the world how the International Date Line can be prostituted” seem just a little overwrought, don’t they? They conclude that:

The international community has not taken the Kiribati adjustment seriously. World atlases still ignore Kiribati and show the International Date Line in that republic’s vicinity as it has been for the past century—a straight line congruent with 180º meridian.

I don’t know how many atlases they checked before they made this claim, but my Encyclopædia Britannica World Atlas from 2005 (the year before Plotting The Globe was published) certainly has the Kirbati deviation neatly plotted, as do most atlases published since.

SAMOA and TOKELAU (2011)
Samoa in 1900
Samoa in 1900
Original source

We left the Kingdom of Samoa, back in my previous post on this topic, having crossed the Date Line from west to east in order to improve trade relationships with the USA, and for its troubles having subsequently been divided up between Germany and the USA. From the start of the First World War, German Samoa passed into the hands of New Zealand, as the Western Samoa Trust Territory, and then became independent, as Western Samoa, in 1962. And then, in 1997, Western Samoa changed its name to just plain Samoa, much to the annoyance of American Samoa (still US territory), who felt that their neighbour wasn’t entitled to the full, unadorned title that had once applied to the original kingdom.

So the Samoa that now shifted back across the Date Line from east to west is actually only about half of the Samoa that originally shifted from west to east in 1892. The shift was again calculated to shift into alignment with the dates used by its main trading partners, which were now Australia and New Zealand. (In 2009, Samoa had made a shift from driving on the right to driving on the left—again, reflecting its ties to Australia and New Zealand, and falling into line with the practice of many of its island neighbours.)

The change took place at midnight on Thursday, 29 December 2011, which was followed by Saturday, 31 December.

The New Zealand dependency of Tokelau, three atolls with a total of just 1500 inhabitants, which lie immediately north of Samoa, elected to make the same calendrical switch on the same day. Tokelau’s transport hub is the island of Apia, in Samoa, and its administration is from New Zealand, so the shift made a lot of sense.

Interestingly, the Samoa-Tokelau Mission of the Seventh Day Adventist Church has refused to acknowledge the date change. This means that while Christian churches in Samoa and Tokelau are holding their Sunday services, Adventist churches are simultaneously observing their Saturday Sabbath—a situation that is probably unique in the world.

And that’s the story. So far.

Shifts in the International Date Line 1900 to present
Click to enlarge

* The article appeared on the front page of the New York Times of 20 March 1922:

STEFANSSON CLAIMS WRANGELL [sic] ISLAND FOR GREAT BRITAIN
The Expedition He Sent Out Last Fall Has Established Possession, Says Explorer.
TIMED TO FORESTALL JAPAN
Any Previous Claims of America or Britain Had Lapsed, He Holds.
NOW OFFERED TO CANADA
Stefansson Denies That Russia, to Whom the Island Is Allotted on Maps, Has Any Right to It.

The British flag has been raised by a party sent out by Vilhjalmur Stefansson on Wrangell Island, one of the most important islands in the Arctic region, because strategically it dominates Northeastern Siberia. The explorer now admits that when the little vanguard of his fifth and latest expedition, including citizens of the United States, landed on Wrangell Island, on Sept. 21 last, its mission was political as well as scientific.

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Territories That Crossed The Date Line: Part 1 – Up To 1900

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Magellan's ship Victoria (contemporary illustration)
The Victoria, the only surviving ship of Magellan’s fleet

Constrained by extreme necessity, we decided on touching at the Cape Verde Islands, and on Wednesday the 9th of July, we touched at one of those islands named St. James’s. […] In order to see whether we had kept an exact account of the days, we charged those who went ashore to ask what day of the week it was, and they were told by the Portuguese inhabitants of the island that it was Thursday, which was a great cause of wondering to us, since with us it was only Wednesday. We could not persuade ourselves that we were mistaken; and I was more surprised than the others, since having always been in good health, I had every day, without intermission, written down the day that was current.

Antonio Pigafetta, The First Voyage Round The World (c.1550)

Pigafetta sailed with Magellan’s small fleet, and was one of the few survivors who completed the circumnavigation. He recorded his experience in Italian (Relazione del primo viaggio intorno al mondo), and what he originally wrote has been pieced together from various surviving copies.

So he was one of the very first people to encounter a commonplace reality of modern long-distance travel. In the absence of an International Date Line (or even a general understanding of why a Date Line might be required), he found that his carefully recorded sequence of days and nights as experienced aboard ship had fallen out of synchrony with the days and nights of those who had stayed at home.

It works like this:
For a mythical observer sitting on the surface of the sun and watching the Earth revolve, we all of us go around the Earth once per day, carried around on its surface as it rotates from west to east. But anyone who travels across the surface of the Earth, making a circumnavigation from east to west, undoes one of the Earth’s rotations, and experiences one less day and night than those who stay at home. (This is why Pigafetta thought it was Wednesday when the Cape Verdeans thought it was Thursday.) Anyone who circumnavigates from west to east adds a rotation to the Earth’s natural period, experiences one more day and night, and comes back with their calendar a day ahead of those at home. (This famously supplied the twist at the end of Jules Verne’s novel, Around The World In Eighty Days.)

If we’re to keep everything straight, there needs to be a disjunction between time zones; a line at which we wind the calendar forward by a day when moving west, and turn it back by a day when moving east. And that’s the International Date Line, which runs down the middle of the Pacific, separating later dates in the west (“Asiatic” dates) from earlier dates in the east (“American” dates).

Interestingly, this International Date Line is nowhere defined in international law—you can’t look up a list of its precise coordinates.  In international waters, the nautical date line is well-defined—it follows the 180º meridian across the Pacific, and ships on the high seas will adjust their calendars as they cross that line. But in territorial waters, every sovereign territory chooses its own date, according to what’s most convenient. Since a number of countries in the Pacific lie directly on or close to the 180º meridian, the International Date Line must zig-zag back and forth around their territorial waters, according to which date they’re keeping. Cartographers plot a Date Line by choosing the most economical set of line segments they can find that keeps countries with Asiatic dates separate from those with American dates, and the precise choice of lines varies from one map-maker to the next.

From time to time, countries and territories have chosen to change the date they’re using, and the International Date Line then has to be flipped from one side of them to the other. And that’s what this post is about.

The Western calendar arrived in the Pacific aboard European ships. Some, mainly Spanish, arrived from the east (American) side of the ocean, and some arrived from the west (Asiatic) side. Those ships coming from the west had adjusted their clocks forwards as they sailed eastwards; those from the east had adjusted their clocks backwards as they sailed westwards—with the result that Asiatic and American time extended in a piecemeal fashion into the Pacific, and met up with their calendars a day out of synchrony. The date a territory first used depended on whether its first encounter with the Western calendar had come from the east or the west; the date it finally adopted depended on the dates its neighbours and trading partners were using. The following is a chronological list of those territories that started out with one date, and subsequently shifted to another.

PITCAIRN (between 1808-1814)

Pitcairn Island was colonized by mutineers from Captain William Bligh’s Bounty, along with their (not necessarily willing) Tahitian companions. The Bounty had entered the Pacific from the Asiatic side. The mutiny took place in the waters around what is now Tonga, after which the mutineers sailed their ship a long way east to Pitcairn, where they arrived in 1790. Their little colony was not discovered until 1808, when they were visited by the American sealing ship Topaz, which arrived from the American side of the Pacific. The next outsiders to visit the island arrived in 1814, aboard the British frigates Briton and Tagus. And this is when it gets interesting. In his narrative of the visit, Captain Philip Pipon of the Tagus records encountering the first-born son of the mutineer Fletcher Christian, whose name he gives as Thursday October Christian, named for the day and month of his birth. Whereas Marine Lieutenant John Shillibeer of the Briton, in his book A Narrative of the Briton’s Voyage to Pitcairn’s Island, describes meeting the same man, but gives his name as Friday Fletcher October Christian.

Thursday or Friday Christian, by John Shillibeer (1817)
Click to enlarge
Shillibeer’s illustration of Friday Fletcher October Christian (1817)

Thursday October Christian is certainly the name by which Fletcher Christian’s son was subsequently known. Did Shillibeer just make a mistake? This seems unlikely, since he spent a lot of time in Christian’s company, asking him questions and even making a drawing of him, which is reproduced in his book. So the man seems to have been called Friday, but to have adopted the name Thursday for use in later life. Which would make sense if the mutineers had carried their dates to Pitcairn from the west, and then subsequently found out from the American ship Topaz that their calendar seemed to be a day out of alignment. In which case Christian would have been born on Friday, 29 October 1790 by the Bounty‘s calendar, but on Thursday, 28 October by the Topaz‘s calendar. Pitcairn certainly now keeps the American dates appropriate for its position well east of the 180º meridian, but the story of Friday/Thursday Christian seems to suggest that it once used Asiatic dates, and made a change some time between the visit of the Topaz and the visit of the Tagus and Briton.

PHILIPPINES (1844)
Date Line from Meyers Konversations-Lexicon (1885)

The Philippines, despite their position very close to the Asian side of the Pacific, had their main calendrical contact with Spanish ships arriving from the Americas, with the result that the Philippines were, for three centuries, the tip of a peninsula of American dates which extended west into regions that otherwise observed Asiatic dates. Nineteenth-century maps show the Date Line with a huge westward bulge, encompassing not only the Philippines but other Spanish possessions in that part of the Pacific—the Marianas, the Carolines, the Marshalls and Palau. Here’s an illustration from Meyers Konversations-Lexicon, a German-language encyclopaedia, for instance. But the Date Line in this map is inaccurately placed at Alaska (see next entry), and locates many islands farther south, like Fiji and the Cook Islands, on the wrong side of the line. So it’s more of an artist’s impression than an accurate depiction.

By the nineteenth century, Latin America was becoming independent from Spain, while Philippine connections with Asia and Australia were increasing. So at midnight on Monday, 30 December 1844, the Philippines flipped to an Asiatic calendar, by simply omitting 31 December entirely, and beginning the New Year in synchrony with neighbouring territories on Wednesday, 1 January 1845. Documentation seems to be lacking, but it seems likely that the Marianas, Carolines, Marshalls and Palau (which were governed from the Philippines) made the transition on the same date, allowing the Date Line to spring almost entirely back to the middle of the Pacific.

ALASKA (1867)

Alaska was owned by Russia until the Alaska Purchase of 1867, when the United States acquired the territory for $7,2000,000. Up to that time Alaska had been using Asiatic dates, to match Russia, with the Date Line running along the border between Alaska and Canada. To shift the Date Line westwards, into the Bering Strait, Alaska needed to repeat a day, bringing it into synchrony with American dates. But it was more complicated than that—Russia and Alaska were still using the old Julian calendar, whereas the USA (and most of the rest of the world) had moved on to the Gregorian calendar.* By the nineteenth century, the Gregorian calendar was 12 days ahead of the Julian.

So at midnight on Friday, 6 October 1867 (Julian calendar, Asiatic date), Alaska prepared to have another Friday, 6 October (Julian calendar, American date), but transformed into Friday, 18 October (Gregorian calendar, American date). This combined shift, with a duplicate weekday but a different date, appears to be unique in the annals of Date Line crossings.

With the shift of Pitcairn, the Philippines and Alaska to geographically appropriate dates, it began to look possible to run the Date Line fairly neatly through the middle of the Pacific, without too many zig-zags. The idea was clearly voiced at the International Meridian Conference of 1884, by one of the British representatives, Lieutenant-General Richard Strachey:

I think that if the world were to adopt the meridian of Greenwich as the origin of longitude, the natural thing for it to do would be to have the international day, the universal day, begin from the 180th meridian from Greenwich—that is, to coincide with the Greenwich civil day. That meridian passes, as I said before, outside of New Zealand, and outside of the Fijee Islands; it goes over only a very small portion of inhabited country. It appears to me, therefore, that inasmuch as there must be an absolute break or discontinuity in time in passing round the earth—a break of twenty-four hours—it is much more convenient that this break should take place in the uninhabited part of the earth than in the very centre of civilization.

The Meridian Conference firmly established the Greenwich meridian as the zero for longitude, but never got around to doing anything about formally defining the Date Line. But more calendrical adjustment was about to happen in the Pacific.

SAMOA (1892)

During the nineteenth century, the Kingdom of Samoa was keeping Asiatic dates, which had been established by missionaries originating in Australia. Britain, Germany and the USA were tussling for influence in the region. In 1892, the Americans managed to persuade King Malietoa Laupepa that increasing trade from San Francisco would be well served if Samoa shifted to the American side of the Date Line. And in any case, Samoa’s longitude of 172ºW put it on the American side of the critical 180º meridian. So Samoa shifted from Asiatic to American dates by having two American Independence Days in succession, repeating Monday, 4 July 1892.
But the story wasn’t over for Samoa, yet. In 1899, shortly after Malietoa Laupepa’s death, Germany and the USA divided up the Kingdom of Samoa between themselves, with Germany claiming the western islands (German Samoa) and America the eastern (American Samoa). That division into two colonial territories would become relevant later.

Samoa in 1900
Samoa in 1900
Original source
COOK ISLANDS (1899)

The Cook Islands got their calendar from Australian missionaries, too, and so ran on Asiatic dates. But at 160ºW, even farther east than Samoa, they were a long way into the American side of the Pacific, and they decided to shift their calendar to match that of their neighbouring territories. They made the change by repeating Monday, 25 December 1899. According to James Michener, the Cook Islanders enjoyed their two Christmases so much that they announced they would have made the transition years previously, if only someone had come up with this brilliant plan earlier.

“MORRELL ISLAND” AND “BYERS ISLAND” (1900)
Cover of Lost Islands by Henry Stommel
Lost Islands, by Henry Stommel, gives much more information about the fictitious Morrell and Byers Islands, and the dubious story of Captain Benjamin Morrell, who gave detailed descriptions of both

These two islands were part of a vast archipelago, stretching from Hawaii to Japan, which appeared in early eighteenth-century charts. It was sometimes labelled the Anson Archipelago, and is distinguished by the fact that it was almost entirely imaginary or fictitious—very few of its islands actually existed, although all had been reported by more-or-less reputable mariners, sometimes on more than one occasion. But later in the century a lot of them were turning up absent when their location was revisited. So Captain Frederick Evans, newly appointed to the post of Hydrographer of the Royal Navy in 1875, went over the Pacific chart with a fine-tooth comb, and deleted no fewer than 123 of these islands (although a few had to be subsequently reinstated). Morrell and Byers Islands survived, temporarily, although marked “doubtful”. (In fact, almost everything about them seems to have been doubtful, from their existence and position to their correct names—Morrell also appears as Morell or Merrel, and Byers as Byers’s, Byers’, Byer’s, Byer and even Patrocinio.†)

Their existence (or otherwise) was significant to the position of the International Date Line because they lay just off the western end of the Hawaiian island chain, were both claimed by Hawaii, and were both plotted just on the western side of the 180º meridian —174.5ºE and 176ºE, respectively. So the Date Line took a little jog westwards between 24ºN and 36ºN, to keep these two non-existent islands in the same American date zone as Hawaii. But as dubiety about their existence increase, cartographers began to straighten out the Date Line in that vicinity. This was a slow and piece-meal process—the islands were gone from some charts by 1903, but were still marked in J.G. Bartholomew‘s Times Survey Atlas Of The World as late as 1922.

Nautical chart of Morrell and Byers Islands (1869)
Mor(r)ell and Byer(s) Islands (“P.D.” for “Position Doubtful”) in 1869
Source
Nautical chart of vicinity of Morrell and Byers Islands (1903)
Morrell and Byers Islands, gone by 1903. Notice how many others have vanished, too
Source
Morrell and Byers Islands in Times Atlas of 1922
Click to enlarge
Mor(r)ell and Byer(s) Islands in my copy of the Times Atlas of 1922

The Date Line shift associated with the vanishing of these islands is usually dated to 1910, based on a typewritten memorandum circulated by the Royal Navy Hydrographic Department in 1921,  which was reproduced in the New Zealand Journal of Science and Technology in 1930 as “Notes on the History of the Date Or Calendar Line“. This shows the course of the Date Line, as plotted by the Hydrographic Department over the course of the years, and the Morrell and Byers diversion is marked as persisting until 1910. (My picture below comes from a scan of the original, hand-illustrated Hydrographic Department document, which is clearer than the scanned version of the 1930 publication in my link above.)

Dateline detail (Hawaii) from "Notes on the History of the Date or Calendar Line" (1921)

But I’ve chosen 1900 as my cut-off date for Morrell and Byers because of an article about the International Date Line that appeared in the Journal of the British Astronomical Association in February 1900, written by A.M.W. Downing and entitled “Where the Day Changes“. It shows several plots of the Date Line, including one provided by Admiral William Wharton (Frederick Evans’s successor as Hydrographer of the Navy) which draws a straight line right past the fabled locations of Morrell and Byers. So it seems the Hydrographic Department had given up on that particular  Date Line diversion a little earlier than they subsequently recalled.

And of course it’s also a nice round number on which to pause this exposition. I’ll bring the story up to the present day in another post.

The story so far (click for an enlargement):

Shifts in the International Date Line up to 1900
Click to enlarge

* If you want to know more about the Julian/Gregorian calendar shift, see my post about February 30th.
† But we know that Morrell Island was named for its “discoverer”, Captain Benjamin Morrell, and (from Morrell’s memoir of his voyages), that Byers Island was named for James Byers, one of the owners of Morrell’s ship, Wasp. (Morrell used the form Byers’s Island.) Patrocinio is another illusory island, reported in the same vicinity by the Spanish Captain Zipiani in 1799. When Patrocinio could not subsequently be found at the coordinates Zipiani had set down for it, it was suggested that his was probably an earlier report of the island Morrell had named after Byers.

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Which Place Gets The Most Daylight?

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So this puzzle isn’t about sunshine (the amount of time the sun shines from a clear sky), or even about the intensity of sunlight (which decreases with increasing latitude), but about cumulative daylight—the length of time between sunrise and sunset in a given place, added up over the course of a year.*

It’s a surprisingly complicated little problem. I addressed it using an antique solar calculator I wrote many years ago, using Peter Duffett-Smith’s excellent books as my primary references:

Peter Duffett-Smith booksIt runs in Visual Basic 6, which means I had to open up my VirtualBox virtual XP machine to get it running again. The original program calculates the position of the sun by date and time for any given set of coordinates, and also works out the times of sunrise and sunset.

Solar calculator written in VB6You’ll see it gives sunrise and sunset times to one-second precision, which is entirely spurious—the refractive state of the atmosphere is so variable that there’s no real point in quoting these times to anything beyond the nearest minute. I just couldn’t bring myself to hide the extra column of figures.

Anyway, it was a fairly quick job to write a little routine that cycled this calculator through a full year of daylight, adding up the total and spitting out the results so that I could begin exploring the problem.

At first glance, it seems like there shouldn’t be any particular place that wins out. As the Earth moves around the sun, its north pole is alternately tilted towards the sun and away from it, at an angle of about 23.5º. If we look at a diagram of these two solstice points (which occur in June and December every year), there’s an obvious symmetry between the illuminated and unilluminated parts of the globe:

SolsticesBetween the solstices, the latitude at which the sun is overhead varies continuously from 23.5ºN (in June) to 23.5ºS (in December), and then back again:

Solar latitude by day of the year
Click to enlarge
Solar latitude by day of the year

So for every long summer day, there should be an equal and opposite long winter night. The short and long days should average out, during the course of a year, to half a day’s daylight per day—equivalent to 4280 hours in a 365-day calendar year.

And that would be true if the Earth’s orbit around the sun was precisely circular—but it isn’t. As I described in my first post about the word perihelion, the Earth is at its closest to the sun in January, and its farthest in July. Since it moves along its orbit more quickly when it’s closer to the sun, it passes through the December solstice faster than through the June solstice. This has the effect of shortening the southern summer and the northern winter. The effect isn’t immediately obvious in my diagram of solar latitudes, above, but it’s there—the sun spends just 179 days in the southern sky, but 186 days north of the equator.

This means that the total number of hours of daylight is biased towards the northern hemisphere. In the diagram below, I plot the hypothetical flat distribution of daylight hours associated with a circular orbit in purple, and compare it to the effect of Earth’s real elliptical orbit in green:

Hours of daylight by latitude (1)
Click to enlarge

So far, I’ve been treating the sun as if it were a point source of light, rising and setting in an instant of time. But the real sun has a visible disc, about half a degree across in the sky. This means that when the centre of the sun drops below the horizon, it’s only halfway through setting. Sunrise starts when the upper edge of the sun first appears; sunset finishes when the the upper edge of the sun disappears. So the extent of the solar disc slightly prolongs daylight hours, and slightly shortens the night.

At the equator the sun rises and sets vertically, and the upper half of the solar disc takes about a minute to appear or disappear. An extra minute of daylight in the morning, an extra minute of daylight in the evening—that’s more than twelve hours extra daylight during the course of a year, just because the sun is a disc and not a point.

And if we move north or south of the equator, the sun rises and sets at an angle relative to the horizon, and so takes longer to appear and disappear—adding more hours to the total daylight experienced at higher latitudes. There’s a limit to this effect, however. When we get to the polar circles, we run into the paired phenomena of the midnight sun and the polar night. There are days in summer when the sun never sets, and days in winter when the sun never rises.  The extent of the solar disc can make no difference to the length of daylight if the sun is permanently above the horizon, and it can add only a few hours to the total as the sun skims below the horizon at the start and end of polar night.  And as we move towards the poles, the midnight sun and polar night start to dominate the calendar, with only short periods around the equinoxes that have a normal day/night cycle. So although the sunrises and sunsets within the polar circles are notably prolonged, there are fewer of them.

Day and night above the Arctic Circle
Click to enlarge
Midnight sun and polar night begin to dominate over the normal day/night cycle as you go farther north (the situation is symmetrical in the southern polar circle, except with the times of polar night and midnight sun reversed)

So the prolongation of daylight caused by the rising and setting of the solar disc increases steadily with latitude until it peaks at the polar circles (around 66.5ºN and 66.5ºS), after which it declines again. Here’s a diagram of daylight hours predicted for a point-like sun (my previous green curve) with the effect of the solar disc added in red:

Hours of daylight by latitude (2)
Click to enlarge

And there’s another effect to factor in at this point—atmospheric refraction. As I described in my post discussing the shape of the low sun, light from the rising and setting sun follows a slightly curved path through the atmosphere, lifting the image of the sun by a little over half a degree above its real position. This means that when we see the sun on the horizon, its real position is actually below the horizon. This effect hastens the sunrise and delays the sunset, and it does so in a way that is identical to simply making the solar disc wider—instead of just an extra couple of minutes’ daylight at the equator, more than six minutes are added when refraction is factored in, with proportional increases at other latitudes. So here’s a graph showing the original green curve of a point-like sun, the red curve showing the effect of the solar disc, and a blue curve added to show the effect of refraction, too:

Hours of daylight by latitude (3)
Click to enlarge

The longest cumulative daylight is at the Arctic Circle, with latitude 66.7ºN experiencing 4649 hours of daylight in the year 2017. The shortest period is at the south pole, with just 4388 hours. That’s almost eleven days of a difference!

So is the answer to my original question just “the Arctic Circle”? Well, no. I have one more influence on the duration of daylight to deploy, and this time it’s a local one—altitude. The higher you go, the lower the horizon gets, making the sun rise earlier and set later. This only works if you have a clear view of a sea-level (or approximately sea-level) horizon—from an aircraft or the top of a mountain. Being on a high plateau doesn’t work, because your horizon is determined by the local terrain, rather than the distant curvature of the Earth. So although the south pole has an altitude of 2700m, it’s sitting in the middle of the vast polar plateau, and I think there will be a minimal effect from altitude on the duration of its daylight.

So we need to look for high mountains close to the Arctic Circle. A glance at the map suggests four mountainous regions that need to be investigated—the Cherskiy Range, in eastern Siberia; the Scandinavian Mountains; Greenland; and the region in Alaska where the Arctic Circle threads between the Brooks Range to the north and the Alaska Range to the south.

Arctic Circle
Click to enlarge
Source of base map

The highest point in the Cherskiy Range is Gora Pobeda (“Victory Peak”). At 65º11′N and 3003m, its summit gets 5002 hours of daylight—almost an hour a day of extra sunlight, on average.

But Pobeda is nudged out of pole position in the Cherskiy Range by an unnamed 2547m summit on the Chemalginskiy ridge, which lies almost exactly on the Arctic Circle, giving it a calculated 5006 hours of daylight.

There’s nothing over 2000m near the Arctic Circle in the Scandinavian Mountains, so we can skip past them to 3383m Mount Forel, in Greenland, at 66º56′N, which beats the Siberian mountains with 5052 hours of daylight.

Finally, the Arctic Circle passes north of Canada’s Mackenzie Mountains, and between the Brooks and Alaska Ranges. Mount Isto, the highest point in the Brooks Range, is 2736m high at 69º12′N, and comes in just behind Pobeda, with 4993 hours of daylight. Mount Igikpak, lower but nearer the Circle (2523m, 67º25′N), pushes past all the Siberian summits to hit 5010 hours. And in the Alaska Range is Denali, the highest mountain in North America. It is 6190m high, and sits at 63º04′N. It could have been a serious contender if it had been just a little farther north—but as it is it merely equals Igikpak, and falls short of Forel’s total.

So the answer to my question appears to be that the summit of Mount Forel, Greenland, sees the most daylight of any place on the planet.† I confess I didn’t see that one coming when I started thinking about this.

* “A year” is a slightly slippery concept in this setting. The sun doesn’t return to exactly the same position in the sky at the end of each calendar year, and leap years obviously contain an extra day’s daylight compared to ordinary years. Ideally I should have added up my hours of daylight over a few millennia—but I’m really just interested in the proportions, and they’re not strongly influenced by the choice of calendar year. So for simplicity I ran my program to generate data for 2017 only.

† What I wrote at the start of this piece, about spurious precision in rising and setting times, goes doubly for the calculations concerning altitude. These results are exquisitely sensitive to the effects of variable refraction, and my post about the shape of the low sun gives a lot more detail about how the polar regions are home to some surprising mirages that prolong sunrises and sunsets. I can’t hope to account for local miraging, or even to correctly reproduce the temperature gradient in the atmosphere from day to day. I think the best that can really be said is that some of the contenders I list will experience more daylight than anywhere else on the planet, most years, and that Mount Forel will be in with a good shot of taking the record for any given year.

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Signalling Mirrors

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Signalling mirror
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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.

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Perspective Tricks

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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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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

Tay Bridge Mural Plaque
Click to enlarge

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
Source

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

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Radiation Fog

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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
Click to enlarge
© 2016 The Boon Companion
Frost on pine needles
Click to enlarge
© 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
Click to enlarge
© 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
Click to enlarge
© 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
Click to enlarge
© 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.

Murmansk
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
Click to enlarge
© 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
Click to enlarge
© 2017 The Boon Companion
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Coriolis Effect In A Rotating Space Habitat

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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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge
Trajectories in a rotating habitat 9
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
Click to enlarge

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
(Source)

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

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

Note: Just to bring my points of reference into the 21st century, I’ll point out that the centrifuges that featured in the Endurance spacecraft from the film Interstellar (2014) and the Hermes from The Martian (2015) are intermediate in size between the two centrifugal habitats used in 2001: A Space Odyssey, which I’ve been using as examples. So we could expect Coriolis effects to feature reasonably prominently in either environment.
All these fictional centrifuges are to some extent unrealistic, at least in the short term, because they involve a lot of mass which would need to be moved to orbit and then moved around in space. Where centrifuges are proposed for prolonged space missions, as in Robert Zubrin’s Mars Direct project, they involve whirling a small habitat around on the end of a long tether—usually with a radius of gyration at least as large as the large Space Station V from 2001: A Space Odyssey. Coriolis deflection would therefore be potentially observable (for instance in dropping objects or throwing and catching), but there simply wouldn’t be room for the longer trajectories I’ve described here.

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

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

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

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