Category Archives: Phenomena

March equinox, 2019 — Click to enlarge
Prepared using *Celestia*

I’m posting this on March 20, the date of the first equinox of the year. In the northern hemisphere, we call it the spring or vernal equinox, because it marks the start of astronomical spring in northern latitudes. (The meteorological seasons follow the calendar months, so meteorological spring started on March 1.) Of course, for people who live in the southern hemisphere the same moment marks the onset of astronomical autumn—so it’s becoming more customary to refer to this equinox as the March or northward equinox, according to the month in which it occurs and the direction in which the sun is moving in the sky, thereby avoiding the awkward association with a specific season. Correspondingly, the other equinox is designated the September or southward equinox.

At the equinoxes, the sun stands directly above the Earth’s equator. Three months later, it reaches its most northerly or southerly excursion in the Earth’s sky, and begins to move towards the equator again, until another equinox occurs, six months after the previous one.

With the sun over the equator, the division between day and night runs through both poles. So every line of latitude is (almost) evenly divided between day and night, and (pretty much) everyone on Earth can expect to experience (something pretty close to) 12 hours of daylight and 12 hours of darkness around the time of the equinox. Hence the name, which is derived from Latin æquus, “equal”, and nox, “night”.

The previous paragraph is thick with disclaimers because an exact division into equal periods of day and night applies only to a strictly geometric ideal, in which a point-like sun illuminates an Earth with no atmospheric refraction. In the real world, the sun is about half a degree across, so it continues to shed daylight even when the centre of its disc is below the horizon. And atmospheric refraction serves to lift the solar disc into view even when it is, geometrically speaking, below the horizon. (I’ve written about these effects in more detail in my post about the shape of the low sun, and my calculation of which place on Earth gets the most daylight.) Both these effects serve to extent the period of daylight. At the equator, their combined effect means that the equinoctial day is almost quarter of an hour longer than the equinoctial night. And the effect increases the farther from the equator you travel, because the sun rises and sets on a more diagonal trajectory relative to the horizon. At the extreme, we find that the equinoctial sun is visible above the horizon at the north and south poles simultaneously, skimming along just above, and almost parallel to, the horizon. This year, the sun will rise at the north pole in the evening of March 18; it won’t set at the south pole until the very early morning of March 23. (Both according to Greenwich Mean Time.) So, counterintuitively, both poles are experiencing 24-hour daylight at the time of the equinox.

Now let’s consider the timing of the equinox. The image at the head of this post is the sun’s view of the Earth on 20 March 2019, at 21:59:34 GMT. You can see from the reflected highlight in the Pacific Ocean that the sun is shining directly down on the equator, somewhere in the Pacific to the east of the Date Line.

West of the Date Line, a new day has already begun, so for anyone in a time zone more than two hours ahead of Greenwich, this equinox is occurring on March 21. But here in the UK, which keeps GMT in March, we haven’t had an equinox on March 21 since 2007, and we won’t have another until 2102. In fact, all our March equinoxes will occur on March 20 until 2044, when we’ll start seeing them fall on March 19, one year in four.

What’s happening to make the dates shift like that?

The problem is that the average length of a tropical year (the time between one equinox and its equivalent the following year) is 365.2422 days. So in a sequence of 365-day years, the seasons will come around 0.2422 days (5 hours 49 minutes) later each year. The date of the equinoxes would very quickly run ahead through the calendar, if it weren’t for leap years. During a 366-day year, the equinox arrives 18 hours 11 minutes earlier than it did the previous year, because the extra day of February 29 has shoved the calendar date ahead by 24 hours, outstripping the movement of the equinox.

So the GMT timing of the March equinox looks like this, for the forty years spanning 2000 *:

March equinox times, 1980-2020 — Click to enlarge

A sawtooth pattern, made up of three steps forward, totalling 17 hours 26 minutes, followed by one jump back of 18 hours 11 minutes. So the equinoxes actually drift back through the calendar year, at a rate of about 45 minutes every four years. Hence the fact we haven’t seen a March 21 equinox in the UK for more than a decade, and will start seeing March 19 equinoxes in a couple more decades.

And that was the problem with the old Julian calendar, and its regular repeating pattern of leap years. The seasons drifted steadily earlier in the calendar. The problem was addressed with the introduction of the Gregorian calendar in 1582, which drops three leap years in four centuries. Centuries not divisible by 400 are not leap years—so we dropped a leap year in 1700, 1800 and 1900, but had one in 2000. And we’ll drop leap years in 2100, 2200 and 2300. (I wrote more about the Gregorian calendar reform in my post concerning February 30.)

The interruption to the sawtooth regression of the equinox relative to the calendar will look like this in 2100:

March equinox times, 2090-2110 — Click to enlarge

That extra forward drift is cumulative over the three centuries of dropped leap years. Here’s what the equinox timing looks like, on leap years between 1600 and 2400:

Leap-year March equinoxes, 1600-2400 — Click to enlarge

Here we see the steady backwards drift during each century, as shown in my first chart. Then a jump forward at the turn of the century when the leap day is omitted, as was shown in my second chart. If we omitted the leap day every century, it’s evident that the trend would carry the time of the equinox steadily forward relative to the calendar. But by observing a leap day in 2000 we allowed the backward drift of the equinox to continue uninterrupted from the 1900s into the 2000s, undoing the forward drift incurred by the three missed leap days.

It’s neat, isn’t it? But there’s still a very slight mismatch. The average length of a Gregorian calendar year is 365.2425 years, a little longer than the tropical year of 365.2422. So there’s a very slow backward drift of the equinox relative to the calendar. Compare, for instance, the peak immediately after 1900 to the peak after 2300. The 1904 leap year equinox fell on March 21, whereas the one in 2304 will occur late on March 20. There were twelve March 21 equinoxes in a row (1900 to 1911, inclusive) at the start of the twentieth century. There will be just four (2300 to 2303, inclusive) at the start of the twenty-fourth.

Still, not to be sniffed at. That’s the longest run of March 21 equinoxes that will ever happen in the future, at least until the Gregorian calendar is revised in favour of something more accurate. Mark it on your calendar.

* All my figures for equinox timings come from Jean Meeus’s incomparable Astronomical Tables Of The Sun, Moon And Planets.

Phenomena, Reading

The Myth Of The Starbow

19th September 2018 Oikofuge Leave a comment

Thus, with all Einstein numbers of flight [velocity as a proportion of the speed of light] greater than 0.37 a major dark spot will surround the take-off star, and a minor dark spot the target star. Between the two limiting circles of these spots, all stars visible in the sky are coloured in all the hues of the rainbow, in circles concentric to the flight direction, starting in front with violet, and continuing over blue, green, yellow and orange to red at the other end.

E. Sänger “Some Optical And Kinematical Effects In Interstellar Astronautics” Journal of the British Interplanetary Society (1962) 18(7): 273-7

Above is one of the earliest descriptions of the appearance of the sky as seen from a spacecraft travelling at close to the speed of light, written more than half a century ago. It predicts something remarkable—that the sky would be dark both ahead of and behind the spaceship, and between these two extensive discs of darkness a rainbow would appear. One of the best illustrations of this phenomenon that I’ve found appears on the cover of Frederik Pohl’s 1982 science fiction novel, Starburst, shown at the head of this post. (This is both unexpected and ironic, for reasons I’ll reveal later.)

Now, I’ve recently invested four posts in systematically piecing together the appearance of the sky from a spacecraft moving at close to the speed of light. If you’re interested, the series begins here, builds mathematical detail over the second and third posts, and draws it all together, with illustrations, in the final one. Using the equations of special relativity for aberration and Doppler shift, and applying them to black-body approximations of stellar spectra, I was able to come up with some pictures using the space simulator software Celestia.

Here’s a wide-angle view of the sky ahead seen when moving at half the speed of light:

Sky view ahead at 0.5c — Click to enlarge

And a tighter view at 0.95 times light speed:

Sky view ahead at 0.95c — Click to enlarge

And at 0.999 times light speed:

Sky view ahead at 0.999c — Click to enlarge

No sign of Sänger’s “minor dark spot” ahead, and no real indication of a rainbow. The stars appear hot and blue ahead, in a patch that becomes more concentrated with increasing speed, and that central area is surrounded by a scattered rim of red-shifted stars, shading off into darkness all around. At very high velocity, the blue patch begins to fade. (For a detailed step-by-step explanation of all this, see my previous posts, referenced above.)

What’s going on? Well, Sänger made an embarrassing mistake:

For simplicity’s sake we may assume that the stars in the sky, as seen from the space vehicle when at rest, are all of a medium yellow colour of perhaps λ₀ = 5900Å.

He modelled all the stars in the sky as if they emitted light at a single wavelength, like a laser! Unsurprisingly, when these monochromatic stars were Doppler-shifted, they passed through all the colours of the rainbow before disappearing into ultraviolet wavelengths (ahead) or infrared (behind). Hence the dark patches fore and aft of Sänger’s speeding spacecraft, and the rainbow ring between.

But of course real stars emit light over a range of wavelengths, with peak emissions that vary according to their temperatures. As I explained in previous posts, when real stars are Doppler-shifted they change their apparent temperature, so the stars ahead of our spacecraft appear to get hotter, while those behind appear cooler. Hot stars may look white or blue, but never violet. Cool stars may be yellow or orange or red, or faded to invisibility, but there is no temperature at which they will appear green. And the fact that stars of different temperatures are scattered all across the sky means that Doppler shift can’t ever produce the concentric circles of colour that Sänger imagined. Sänger’s rainbow is a myth, based on a fatally erroneous assumption (“for simplicity’s sake”) that really should have been picked up by reviewers at the British Interplanetary Society.

Sänger’s idea would have vanished into appropriate obscurity, were it not for the fact that science fiction writer Frederik Pohl was a member of the British Interplanetary Society, and received its monthly journals. Writing about it later, Pohl mistakenly recalled reading Sänger’s article in another BIS publication, Spaceflight. (BIS members received one publication as part of their membership, and could pay to receive the other, too—it seems likely Pohl subscribed to both.) He later described his encounter with Sänger’s article like this:

Before I had even finished it I sat up in bed, crying “Eureka!” It was a great article.

“Looking For The Starbow” Destinies (1980) 2(1): 8-17

Pohl loved this image of a rainbow ring, and called it a “starbow”. He went on to feature the starbow in an award-winning novella, “The Gold At The Starbow’s End” (1972):

The first thing was that there was a sort of round black spot ahead of us where we couldn’t see anything at all […] Then we lost the Sun behind us, and a little later we saw the blackout spread to a growing circle of stars there.
[…]
Even the stars off to one side are showing relativistic colour shifts. It’s almost like a rainbow, one of those full-circle rainbows that you see on the clouds beneath you from an aeroplane sometimes. Only this circle is all around us. Nearest the black hole* in front the stars have frequency-shifted to a dull reddish colour. They go through orange and yellow and a sort of leaf green to the band nearest the black hole* in back, which are bright blue shading to purple.

If you’re on the alert, you’ll notice that Pohl got the colours the wrong way around—Sänger’s prediction placed red behind and violet ahead (not Pohl’s “purple”, which is a mixture of red and blue).

When Pohl’s novella was published as part of a collection, its striking title was used as the book title, and Pohl’s description (including the reversed colours) leaked into the cover art of one edition:Pohl was a skilled and popular writer, and he cemented the erroneous “starbow” into the consciousness of science fiction readers.

Also in 1972 the space artist Don Davis, cooperating with the starship designer and alleged translator of Ice-Age languages (among many other things) Robert Duncan-Enzmann, produced a distinctly weird New Age image of the starbow, which you can find here. I have scant idea what that’s about.

But then, in 1979, along came John M. McKinley and Paul Doherty, of the Department of Physics at Oakland University, Michigan. They had a computer, and they were unconvinced by Sänger’s identical monochromatic stars. They instead modelled the real distribution of stars in Earth’s sky, approximating each one as a blackbody radiator of the appropriate temperature, and applying the necessary relativistic transformations:

One prediction for the appearance of the starfield from a moving reference frame has been circulated widely, despite physically objectionable features. We re-examine the physical basis for this effect. […] We conclude with a sequence of computer-generated figures to show the appearance of Earth’s starfield at various velocities. A “starbow” does not appear.

“In search of the ‘starbow’: The appearance of the starfield from a relativistic spaceship” American Journal of Physics (1979) 47(4): 309-15

The physicist (and science fiction writer) Robert L. Forward mischievously forwarded a preprint of McKinley and Doherty’s article to Pohl. And Pohl, tongue firmly in cheek, described this experience in the Destinies article I quoted above:

… “there is no starbow,” they conclude. True, they then go on to say, “we regret its demise. We have nothing so poetic to offer as its replacement, only better physics”—but what’s the good of that?

Only slightly chastened, Pohl later went on to expand the novella “The Gold At The Starbow’s End” into a frankly-not-very-good novel, Starburst, the cover of which appears at the head of this post, resplendent with a starbow. I find it difficult to imagine the confusion that might have led to that cover, given that Pohl had removed the starbow from his narrative, while managing to give McKinley and Doherty a very slight (but distinctly ungracious) kicking in the rewrite:

Right now we’re seeing more in front than I expected to and less behind. Behind, mostly just blackness. It started out like, I don’t know what you’d call it, sort of a burnt-out fuzziness, and it’s been spreading over the last few weeks. Actually in front it seems to be getting a little brighter. I don’t know if you all remember, but there was some argument about whether we’d see the starbow at all, because some old guys ran computer simulations and said it wouldn’t happen. Well, something is happening! It’s like Kneffie always says, theory is one thing, evidence is better, so there! (Ha-ha.)

As the cover of Starburst suggests, the starbow was just too good an image to die easily, and few science fiction readers (or writers) read the American Journal of Physics. Undead, the starbow continued to trudge forward—a zombie idea. In September 1988, Robert J. Sawyer had a short story published in Amazing Stories, entitled “Golden Fleece”. It scored the coveted cover illustration for that month:

It’s a slightly confusing image, illustrating a key event in the story. The vehicle in the foreground is a shuttle-craft, which is escaping from the large spacecraft in the background, a relativistic Bussard interstellar ramjet travelling from right to left. And there’s a starbow! And it’s the wrong way round again, with red at the front! I haven’t read Sawyer’s original short story, but I have read the 1990 novel of the same name, in the form of its 1999 revised edition:

The view of the starbow was magnificent. At our near-light speed, stars ahead had blue-shifted beyond normal visibility. Likewise, those behind had red-shifted into darkness. But encircling us was a thin prismatic band of glowing points, a glorious rainbow of star—violet, indigo, blue, green, yellow, orange and red.

I don’t mean to single Sawyer out, because lots of authors were still invoking the starbow in their writing, but his 1999 novel is the most recent persisting version of the starbow I’ve turned up so far, particularly notable because it recycled the Amazing Stories cover art:Twenty years after McKinley and Doherty wrote “We have nothing so poetic to offer as its replacement, only better physics”, the starbow lived on.

And you can still find it—order a starbow painting by Bill Wright on-line, here.

Note: I happened upon Stephen R. Wilk’s How The Ray Gun Got Its Zap, which I’ve previously reviewed, while searching for references to the starbow. Wilk’s chapter “The Rise And Fall And Rise Of The Starbow” overlaps with some of what I’ve written here, but also discusses starbow-like manifestations in film.

* Pohl’s “black holes” are the patches of sky devoid of visible stars ahead of and behind the narrator’s spaceship, as predicted by Sänger, not the astronomical objects of the same name.

Phenomena

The Celestial View From A Relativistic Starship: Part 4

8th August 2018 Oikofuge 4 Comments

This series of posts is about what the sky would look like to an observer travelling at close to the speed of light. In Part 1, I described the effects of light aberration on the apparent position of the stars; in Part 2, I introduced the effects of Doppler shift on the frequency of the starlight; and in Part 3 I described the effect that Doppler shift would have on the appearance of real stars.

In this post, I’m planning to pull all that together and show you some sky views I’ve generated using the 3-D space simulator Celestia. To do this, I had to write some code to rewrite Celestia‘s stars and constellation-boundaries databases, using the various aberration and Doppler equations I’ve previously presented. The result was a set of Celestia databases that reproduce the appearance of the sky for an observer moving at high velocity—allowing me to exploit all Celestia‘s rendering capabilities to produce my final graphics.

The final hurdle on the way to producing my sky views was to decide how to convert the Doppler-shifted energy spectrum of a star into the corresponding visual appearance. The human eye is not equally sensitive to all visible wavelengths, and that has to be taken into account when converting power (in watts) to luminous flux (in lumens). But the eye’s sensitivity also changes in response to differing light levels—the retinal cone cells which give us our photopic (daytime, colour) vision have a different sensitivity profile from the rod cells that give us scotopic (nighttime, black-and-white) vision.

There are two classic papers dealing with the sky view from a relativistic spacecraft. McKinley and Doherty (1978)* make the visual conversion using a model of scotopic vision, with peak sensitivity at a wavelength of 500nm, whereas Stimets and Sheldon (1981)^† make the conversion using an approximation to photopic visual sensitivity, with a peak at 555.6nm. You might imagine that McKinley and Doherty have the right idea, applying scotopic vision to a problem involving the visibility of the stars. Unfortunately, the stellar visual magnitude scale is calibrated by neither photopic nor scotopic vision, but by an instrument called a photometer, counting the number of photons that pass through a filter that approximates the sensitivity of the human eye. The old visual standard was provided by the Johnson V-band filter, but newer star surveys (like Hipparcos and Tycho) have used filters with slightly different passbands. The resulting differences are tiny compared to the variable sensitivity of the human eye, however.

Here are standard curves for scotopic and photopic sensitivity, compared to the V-band filter curve:

Luminous efficiency curves compared to V-band filter — Click to enlarge

Although the V-band peak lies intermediate between scotopic and photopic, the bulk of the curve lies within the photopic range, and well away from scotopic. This is confirmed when I generate black-body bolometric corrections (the difference between bolometric magnitude and visual magnitude) using the three different curves above:

Bolometric corrections generated from photopic, scotopic and V-band — Click to enlarge

Photopic vision turns out to be a very good match for the V-band. Scotopic vision, with its increased blue sensitivity, peaks at higher temperatures, and is actually inconsistent with the standard visual magnitude scale. So McKinley and Doherty’s results are unfortunately skewed away from the conventional visual magnitude scale, assigning blue stars inappropriately bright visual magnitudes, and red stars inappropriately dim magnitudes. This has consequences for anyone using their formulae—for instance John O’Hanley’s excellent Special Relativity site, which in its “Optics and Signals” section does something very similar to what I’m doing later in this post, but with results that are skewed by use of McKinley and Doherty’s formula to convert bolometric magnitude to visual magnitude.

To generate my Celestia views, I used the V-band profile.

First up, a series of views ahead of our speeding spacecraft, which is travelling directly out of the plane of the solar system, towards the constellation Draco. Throughout these images, you can orientate yourself using the superimposed grid. It’s Celestia‘s built-in ecliptic grid, but I’ve modified the latitude markings to show the angle θ′ instead—the angle measured between a sky feature and the dead-ahead direction. So θ′=0° directly ahead, and 180º directly astern, with the 90º position at right angles to the line of flight. The outermost ring in the views that follow is at θ′=30°. I’ve set Celestia to show star brightness with scaled discs, and to display colours according to black-body temperature. (In reality, the colours of dimmer stars would not be evident—they would appear white.) Stars are displayed down to a magnitude limit of 6.5, which is in the vicinity of the commonly quoted cut-offs for naked-eye visibility (although under ideal conditions some people can do much better).

Here’s the stationary view, for orientation (I’m afraid you’ll need to click to enlarge most of these images to appreciate what they show):

Draco occupies much of the view, with the Pole Star, Polaris, visible to the right.

Now here’s the view for a spacecraft in the same position, travelling at 0.5 of the speed of light (hereafter, I’ll quote all velocities in this form, using the symbol β):

Even unenlarged, you can see that many more stars are visible in the same visual area as the previous image. The constellations have shrunk under the influence of aberration, and many invisibly dim stars have been brightened by Doppler shift so as to become visible. Celestia‘s automatic labelling system has stepped in to add names and catalogue numbers to the brightest. The fact that the light from stars ahead is being shifted towards the blue end of the spectrum is highlighted by the brightening of μ Cephei, the “Garnet Star”, which is a red giant deserving of its nickname, but which now appears blue-shifted to an apparent temperature of 5800K, making it appear yellow-white. The brightening effect of blue-shift is most marked for cool stars—for instance, the cool carbon star Hip 95154 has become a very marked presence in Draco, having brightened through 4.4 magnitudes.

Here’s the view at β=0.8:

The sky is now so densely populated with bright blue stars that I’ve turned off Celestia‘s naming function to prevent clutter. Here and there are bright orange-yellow stars—these are in fact cool red stars like Hip 95154, which initially brighten dramatically, through several magnitudes, when blue-shifted.

Now, β=0.95:

View ahead, Beta=0.95 — Click to enlarge

I’ve had to turn off constellation names now, but a look to the right of frame reveals Orion just coming into view, although the red star Betelgeuse in one of Orion’s shoulders is now blue in colour. A little inwards from Orion is Taurus, with its orange giant star Aldebaran similarly blue-shifted. Taurus gives you the marker for the zodiac constellations, which are arrayed in a circle just 20º away from the centre of the view.

And finally, β=0.999:

View ahead, Beta=0.999 — Click to enlarge

The blue-shifted region has now shrunk to a diameter of 34º, although it contains most of the stars in the sky. The bright yellowish star on the right, lying just beyond the θ′=20º circle as a very noticeable outlier within the red-shifted zone, is Canopus, a star familiar to those in the southern hemisphere. In the rest frame, Canopus lies only about 14º from the dead-astern position of our spacecraft.

At this velocity, the total number of visible stars has begun to decline, as has the overall brightness of the blue-shifted patch—most stars are now so strongly blue-shifted that their visible light is fading away, as described in Part 3.

Here’s a graph of the number of visible stars (visual magnitude<6.5) in the whole sky as velocity increases. The dashed line marks the number of visible stars in the blue-shifted region ahead:

Overall count of stars with V less than 6.5, and stars in the blue-shifted region, against beta — Click to enlarge

Unsurprisingly, stars in the blue-shifted region dominate the star count as velocity increases. In this dataset (the Tycho-2 star catalogue as prepared for Celestia by Pascal Hartmann, which contains more than two million stars), the star count peaks at β=0.97. At this velocity 99% of the visible stars are in the blue-shifted region, which occupies just 10% of the sky.

Here’s the curve for the integrated star magnitude of the whole sky, and for the blue-shifted region:

Integrated sky magnitude for V less than 6.5, and magnitude of blue-shifted region, against beta — Click to enlarge

The overall brightness of the sky, and of the blue-shifted region, peaks at a slightly higher velocity than the star count—in this dataset, at β=0.98. The difference is because of cool stars joining the edge of the blue-shifted region and undergoing marked brightening, which temporarily offsets the gradual fading out of strongly blue-shifted stars in the middle of the forward view.

Now, a look to the side of the spacecraft, to illustrate how the stars in this view thin out and fade away. Firstly, the view when β=0. The direction of travel is towards the top of the image.

Orion is visible at bottom of frame, with the zodiac constellations of Gemini and Taurus occupying the θ=90º position.

Now, β=0.5:

View to the side, Beta=0.5 — Click to enlarge

Aberration has carried Orion towards ecliptic north, where it straddles the transition from blue-shift to red-shift, at θ′=74º.

And β=0.8:

The whole view is now red-shifted, with the blue/red transition out of sight at θ′=60º. The southern constellation of Columba has now crept into view, considerably enlarged. Sirius, at top of frame, remains bright as it edges towards the blue-shift region, although moderate red shift has dropped its apparent temperature from 9200K to 7800K.

Finally, for this series, β=0.95:

Very few stars are visible—either aberration has carried them into the blue-shifted region ahead, or they are strongly red-shifted to invisibility. The constellation boundaries visible in this view separate Columba (at top) from Puppis and Carina (left) and Pictor (right).

As a final exercise, I’m going to follow the progress of a single constellation as it undergoes aberration and Doppler shift. I’m going to use the southern hemisphere constellation of Crux, the Southern Cross, which in the rest frame lies well in the rear view from our spacecraft, at θ=140º. Its four prominent stars are: two hot blue giants, Mimosa (β Crucis) and δ Crucis; one hot blue subgiant, Acrux (α Crucis); and one cool red giant, Gacrux (γ Crucis). All the blue stars have temperatures over 20000K, which places them above the 16350K threshold discussed in Part 3, meaning that they will initially brighten with red shift. The red giant has a temperature of 3400K, so it can be expected to brighten dramatically when blue-shifted, and to dim equally dramatically when red-shifted.

Here’s the rest-frame view, with the stars labelled:

Now, here’s the same constellation at β=0.5. I’ve kept the size of the field of view the same, and merely shifted it to follow the movement of the constellation under aberration:

The constellation is larger, and now positioned at θ′=115º. All the blue stars have grown brighter by about half a magnitude under the influence of red shift, whereas the red giant Gacrux has fallen in brightness by 0.8 magnitudes.

Here’s β=0.8, with the width of the field of view set to the same as previously:

The constellation is at θ′=85º, placing it in the side view from the spacecraft, at its maximum magnification by aberration, and its maximum red shift. Acrux and Mimosa are very slight brighter, their apparent temperatures still above the 16350K threshold; δ Crucis is very slightly dimmer, its apparent temperature having dropped to 13000K. And Gacrux has dropped in brightness by another half magnitude. But at higher velocities the constellation will move into regions of lower red-shift, so trends will now reverse.

Here’s β=0.95:

The constellation is crossing θ′=50º, and is just about to enter the blue-shifted region at θ′=44º. It’s now almost as close to the dead-ahead direction in the moving frame as it was to the dead-astern direction in the rest frame. Its Doppler shift is therefore close to 1, and its appearance in terms of size, colour and brightness is returning to approximately what it was in the rest frame.

Finally, here’s β=0.999 (I’ve had to zoom in fivefold compared to the previous images, to pick Crux out of the blue-shifted clutter):

The constellation is now only 7º away from the forward direction, and is strongly blue-shifted. Gacrux is now an extremely bright blue star, whereas the blue giants are blue-shifted to such high temperatures that their visible output has declined dramatically.

So that’s about it for the appearance of the stars. With velocities greater than 0.999, the blue-shifted area become more compact, and the number of visible stars gradually diminishes. At β=0.99999, only about 5000 visible stars are packed into an area 9º across, with an integrated visual magnitude of -5. The rest of the sky is dark.

But beyond β=0.99999, something interesting happens. As the number of visible stars continues to fall, a fleck of red appears, just a few arc-minutes across. It becomes white, and then blue, and brightens to an astonishing visual magnitude of -26—as bright as the sun seen from Earth. It’s the Cosmic Microwave Background (CMB), blue-shifted into the visible spectrum, and it won’t begin to fade until the velocity is over 0.9999999 (seven 9’s!).

The appearance of the CMB reminds us that there are many things in the sky that are not stars—I haven’t simulated the appearance of galaxies (including our own Milky Way), or of the cold clouds of dust and gas between the stars, which will be blue-shifted to visible wavelengths before the CMB.

Maybe another time …

* Stimets RW, Sheldon E. The celestial view from a relativistic starship. Journal of the British Interplanetary Society 1981; 34: 83-99.
^† McKinley JM, Doherty P. In search of the “starbow”: The appearance of the starfield from a relativistic spaceship. American Journal of Physics 1979; 47(4): 309-16.

Phenomena

The Celestial View From A Relativistic Starship: Part 3

25th July 2018 Oikofuge Leave a comment

This is the third of a series of posts about what the sky would look like for the passengers aboard an interstellar spacecraft moving at a significant fraction of the speed of light, like the Bussard interstellar ramjet above.

In the first post, I wrote about light aberration, which will cause the apparent direction of the stars to be shifted towards the direction of the spacecraft’s line of flight. In the second post, I discussed the relativistic Doppler shift, which will cause the stars concentrated ahead to undergo a spectral shift towards shorter, bluer wavelengths, while the stars astern become red-shifted. I introduced the parameter η (eta) the relativistic Doppler factor, which I promised would have extended relevance to this section, in which I’m going to discuss the effects of aberration and Doppler on the appearance of the stars.

If you’ve read the previous posts, you’ll recall the terminology, but here’s a quick recap. We call the measurements made by an observer more or less at rest relative to the distant stars the “rest frame” (for our purposes, the Earth is pretty much in the rest frame). The observations we’re interested in are those from a spacecraft which has a large velocity relative to the rest frame. That’s the moving frame, and its velocity is customarily given as a fraction of the speed of light, and symbolized by β (beta). Of particular interest is the angle between the line of the spacecraft’s flight and the position of a given star. In the rest frame, that measurement is symbolized by θ (theta). Aberration transforms that measurement to a smaller angle in the moving frame, which we symbolize by θ′. It’s a convention in Special Relativity to mark variables in this way—the simple symbols used in the rest frame are marked with a prime mark (′) when they’re transformed into the moving frame.

One graph and one diagram can summarize much of what happened in the preceding posts:

Effect of beta on aberration and Doppler — Click to enlarge

In the graph we see how increasing values of β cause the stars to shift progressively farther towards the spacecraft’s dead-ahead position, θ′=0°. In doing so, stars to the rear of the spacecraft in the rest frame are displaced into the forward view, and eventually pass from a region of red shift into one of blue shift.

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

In the diagram, we see how aberration shifts the apparent position of a sphere of stars in the rest frame to an ellipsoid in the moving frame, with the ellipsoid becoming more elongated at higher velocities. Not only does the angular position of the stars change, but their apparent distance does, too. And the change in distance is proportional to the Doppler parameter η—a star with a blue shift that doubles the frequency of its light will also be moved to double the apparent distance by the effects of aberration.

After that summary, I now want to discuss how the stars will actually look, when their visual appearance has been transformed by Doppler and aberration as described above. This involves a digression on the subject of the black body radiation spectrum—it’s a good first approximation to the electromagnetic radiation profile emitted by stars, and has the advantage that it can be easily treated mathematically, which is not true of the rather lumpy radiation distribution of real stars. However, if all you want is the executive summary, you can reasonably skip ahead to THE APPEARANCE OF THE STARS.

BLACK BODY RADIATION

I’m not going to describe black body radiation in detail. There’s a superficial treatment here, and a more mathematical description here. Suffice it to say that there’s a mathematical formula which describes the amount and spectral distribution of energy a black body radiates at any given temperature, and that stars are (to a first approximation) black body radiators. So we can use the black body formulae to look at what happens when the light from the stars is transformed by a Doppler shift.

Here are some typical black body radiation curves, plotted against wavelength, using intensity units that don’t matter for our purposes. The violet and red vertical lines mark out the range of the spectrum of visible light. To the left of violet, ultraviolet and X-rays at short wavelengths; to the right of red, infrared and radio waves at long wavelengths.

As we heat an object, two things happen to its black body radiation curve—the area under the curve (the total energy) gets larger, in proportion to the fourth power of the temperature; and the peak in the curve shifts to shorter wavelengths, in inverse proportion to the temperature (which means the frequency goes up, in direct proportion to the temperature). So there’s a pretty simple relationship between temperature and radiant energy.

But we’re more interested in what happens in the visible band. You can see that the amount of energy in that range goes up with increasing temperature. So our radiation source gets visibly brighter as its temperature rises.

And you can see that the shape of the curve crossing the visible band changes with temperature—at 4000K, red wavelengths predominate; at 7000K, blue predominates. So a black body (and therefore, to a first approximation, a star) follows a characteristic trajectory through colour space (called the “Planckian locus”) as it changes temperature:

Planckian locus — Click to enlarge (source)

So we have the familiar sequence of red, orange, yellow, white and blue-hot, which is reflected in the colours of the stars.

Finally, notice that black bodies are not very efficient producers of light—you can see that the 4000K and 5500K sources are putting out more energy in the infrared than in the visible. At temperatures above 7000K, the radiation output is dominated by a huge spike in the short wavelengths. In fact, 7000K is close to being as efficient as a black body gets at producing visible light. The human eye’s sensitivity varies at different wavelengths and in different lighting conditions, but here’s the “luminous efficacy” curve for black body radiation in photopic vision—the sensitivity of an average human eye in daylight:

Luminous efficacy curve of black body radiation — Click to enlarge

You can see that black bodies with temperatures below 2000K are pretty rubbish at producing visible light. The curve then rapidly spikes to a maximum near 7000K, before declining in an exponential decay at higher temperatures. While an increase in temperature will always produce an increase in visible light emission, it does so with less and less efficiency at high temperatures.

This variation in luminous efficacy shows up when we look at the magnitude scale used to measure the brightness of stars. The total energy output is measured by the star’s bolometric magnitude; its visible brightness by the visual magnitude. By convention, the visual and bolometric magnitudes of a star are about equal for temperatures in the vicinity of 7000K, near the peak of black body luminous efficacy. But above and below that temperature, the visual and bolometric magnitudes diverge, the difference between the two being called the bolometric correction.

For historical reasons, stellar magnitudes are measured on a logarithmic scale, with a change of 5 magnitudes reflecting a 100-fold change in brightness. And for really annoying historical reasons a decrease in magnitude reflects an increase in brightness—a star of magnitude -1 is brighter than a star of magnitude 0, which is in turn brighter than a star of magnitude 1. So visual magnitudes are always greater than bolometric magnitudes, either side of the 7000K peak.

If we take a big chunk of something that behaves as a black body radiator, and heat it up so it progressively brightens, this is what happens to its bolometric and visual magnitudes (I’ve flipped the vertical axis to match intuition—magnitude values decrease towards the top, reflecting increasing brightness):

Bolometric versus Visual surface brightness for increasing temperature — Click to enlarge

The red bolometric line rises steadily on the logarithmic plot, at a slope of -10 magnitudes/decad (that is, it brightens through ten magnitudes for each tenfold increase in temperature). But the orange visual magnitude line starts low on the chart (reflecting the poor luminous efficacy of 3000K black bodies), rises to kiss the bolometric line at about 7000K, and then settles into a less steep rise. At high temperatures it becomes effectively straight, brightening at just -2.5 magnitudes/decad.

THE APPEARANCE OF THE STARS

So in what follows, I’m going to treat the stars as if they are perfect black body radiators, which is a reasonably approximation that allows simple mathematical treatment.

We know that the frequency of all electromagnetic radiation emitted by a star will be Doppler-shifted by a factor of η for an observer aboard our spacecraft. That implies that the energy of each photon will be changed by a factor of η. The number of photons received in a given time period will also be changed by a factor of η, meaning that the energy received in a given time period will vary with η².

And we know that the apparent distance to the star will be changed by a factor of η, which means its apparent angular diameter will change in proportion to 1/η, and its angular area in proportion to 1/η². So, compared to the rest frame, the spacecraft observer receives η² times the energy from 1/η² times the area—implying that the radiance of the star (it surface energy output) varies as η⁴.

The frequency shift of η combined with the radiance change of η⁴ means that a given black body spectrum in the rest frame is Doppler-shifted to another black body spectrum in the moving frame—one that has a temperature of T′=ηT. The overall effect of the Doppler shift is simply to change the apparent temperature of the star!

So now we know that the Doppler-shifted colour of a star will still lie on that Planckian locus of black-body colours, simply shifting up or down the curve according to the value of η.

But the changing apparent size of the star, with total energy received by the moving observer varying as η² rather than η⁴, means I have to redraw my curves of bolometric and visual magnitude:

Apparent bolometric and visual magnitudes under Doppler + aberration — Click to enlarge

We’re heating the same black body radiator as in the previous example, so that its surface brightness increases, but varying its distance from us in proportion to the temperature—just as happens when Doppler effect and aberration work together on a star.

Now the bolometric (total energy) line has half its previous gradient, rising at just -5 magnitudes per decad (a change of five magnitudes for each 10-fold increase in temperature). The bolometric correction remains the same at every temperature, so the visual magnitude curve stays the same distance below the bolometric curve as it did in the previous graph, but that now means it takes a down-turn at higher temperatures, eventually dimming at 2.5 magnitudes per decad.

So a sufficiently large decrease in the Doppler factor η will reduce a star’s apparent temperature enough to red-shift into invisibility; but a sufficiently large increase in η will increase a star’s apparent distance enough to blue-shift it into invisibility (despite its increasing surface brightness). How quickly these effects happen depends on the real temperature of the star, as measured in the rest frame. Here are plots of the change in visual magnitude against η for black body radiators of various temperatures. I’ve placed the spectral class of a corresponding star in brackets:

Visual magnitude change under Doppler + aberration — Click to enlarge

A cool 3000K M star will brighten dramatically, by five magnitudes, under blue shift; but it will dim equally dramatically under red shift. K, G, F (not shown) and A stars will brighten less on blue shift, but are slightly more resistant to the dimming effect of red shift. A star with a temperature of 16350K (around spectral class B4) is a special case—it will grow dimmer with either blue shift or red shift. Stars hotter than B4, like the 40000K O-class star shown here, will initially grow brighter under red shift, but only by half a magnitude or so.

So it works like this:

Any star will reach its maximum brightness when it has been Doppler-shifted to an apparent temperature of 16350K.
The slope on the red-shift side of 16350K is very steep—stars will change visual magnitude dramatically for relatively small changes of η in that region.
The slope on the blue-shift side of 16350K is relatively gentle—large changes in η result in fairly modest changes in visual magnitude.

So that’s it. Over three posts I’ve got to the point where we can predict the apparent position, colour and visual brightness of the stars as seen from a rapidly moving spacecraft.

My next post on this topic will exploit the 3-D space simulator Celestia to generate some views of the real sky, showing how it all fits together.

Go to the next post in this series

Phenomena

The Celestial View From A Relativistic Starship: Part 2

11th July 2018 Oikofuge Leave a comment

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:

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

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.

Go to the next post in this series

Mathematical notes

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

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

In the moving frame, η=1 at angle:

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

Phenomena

The Celestial View From A Relativistic Starship: Part 1

27th June 2018 Oikofuge 3 Comments

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

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

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

LIGHT ABERRATION

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

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

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

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

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

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

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

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

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:

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

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

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

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

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

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

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

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

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

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

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

Go to the next post in this series

Mathematical notes

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

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

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

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

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

Phenomena

Coriolis Effect In A Rotating Space Habitat (Supplement)

23rd May 2018 Oikofuge 3 Comments

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

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

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

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

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

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

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

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

AXIAL MOTION

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

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

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

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 spinward, then the Coriolis force will act upward—the ball will become lighter, and may therefore experience less friction and roll farther, but it won’t experience sideways deflection. Likewise, a ball rolled to antispinward will experience downward Coriolis, will become heavier, and may roll less far. But again, no sideways forces are generated. And that’s true of any combination of axial and spinward/antispinward velocities for an object rolling on a horizontal surface—its apparent weight may change, but its direction won’t.

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

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

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

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

Phenomena

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

28th February 2018 Oikofuge Leave a comment

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:

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:

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.)

Ariel 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.

(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:

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)

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.

Phenomena

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

21st February 2018 Oikofuge Leave a comment

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, 15 October 1790 by the Bounty‘s calendar, but on Thursday, 14 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)

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.

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.)

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.

Phenomena, Software

Which Place Gets The Most Daylight?

3rd January 2018 Oikofuge Leave a comment

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:

It 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.

You’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:

Between 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:

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.

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.

The Oikofuge

Category Archives: Phenomena

Equinox

The Myth Of The Starbow

The Celestial View From A Relativistic Starship: Part 4

The Celestial View From A Relativistic Starship: Part 3

BLACK BODY RADIATION

THE APPEARANCE OF THE STARS

The Celestial View From A Relativistic Starship: Part 2

RELATIVISTIC DOPPLER

The Celestial View From A Relativistic Starship: Part 1

LIGHT ABERRATION

Coriolis Effect In A Rotating Space Habitat (Supplement)

AXIAL MOTION

ROLLING MOTION

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

TWO DIVERSIONS—PART 1. CHATHAM ISLANDS

TWO DIVERSIONS—PART 2. WRANGEL ISLAND

KWAJALEIN (1969 and 1993)

EASTERN KIRIBATI (1994)

SAMOA and TOKELAU (2011)

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

PITCAIRN (between 1808-1814)

PHILIPPINES (1844)

ALASKA (1867)

SAMOA (1892)

COOK ISLANDS (1899)

“MORRELL ISLAND” AND “BYERS ISLAND” (1900)

Which Place Gets The Most Daylight?

A discursive blog on various topics of minor interest