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:

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:

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:

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:

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:

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:

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:

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.

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:

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.

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:

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:

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):

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:

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:

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