The Celestial View From A Relativistic Starship: Part 4

Bussard Interstellar Ramjet
Bussard Interstellar Ramjet (source)

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):

View ahead, Beta=0
Click to enlarge

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):

View ahead, Beta=0.5
Click to enlarge

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:

View ahead, Beta=0.8
Click to enlarge

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.

Side view, Beta=0
Click to enlarge

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:

Side view, Beta=0.8
Click to enlarge

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:

Side view, Beta=0.95
Click to enlarge

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:

Crux, Beta=0
Click to enlarge

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:

Crux, Beta=0.5
Click to enlarge

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:

Crux, Beta=0.8
Click to enlarge

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:

Crux, Beta=0.95
Click to enlarge

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):

Crux, Beta=0.999
Click to enlarge

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.

6 thoughts on “The Celestial View From A Relativistic Starship: Part 4”

  1. I have read all 4 of these and found them very interesting. You have obviously done a lot of work in pulling all this together.

    However, I must admit that my mathematical incompetence has meant that a lot of it just zoomed over my head. (Almost as if it was travelling at β=0.99.)

    The Celestia views do help me grasp more of what you are trying to get into my thick head. I will sit down and reread them now and see if I can get a better grasp.

    So thanks for all your efforts.

  2. Yes, probably seeing the end result and then rereading will be useful.
    My efforts were largely for my own benefit, I must confess. I find this stuff interesting, and trying to produce clear visual aids combined with a minimum of mathematics often helps my own understanding.

  3. You’ve lost me ! I could follow up to a point then realised I was lost . I ‘m old fashioned and still like to have an open book, to study rather than a computer screen in the heat. . Not an excuse I know.

  4. I’m not really convinced that red or blue shifting a black body would in fact yield a black body curve of a different temperature. I can accept that is perhaps a useful approximation but the spectroscopist in me says it would become a shifted and stretched or compressed black body curve not something that would actually correspond to the emission spectrum of a stationary BB at a different temperature. (It matters for the design of solar cells to be used for gathering energy from the flux of incoming radiation. Because the energy bandwidth of a cool BB is less than that of a hotter one leading to the possibility of more efficient photovoltaic energy conversion.) Other than that though an amazing and very informative series!! Thank you very much for your efforts!

    If you’re interested there’s a related problem that’s been an interesting thought experiment to me for some time. Given an observer on a relativistic spinning disk (or space hab) what would light emitted from a series of lamps equally spaced around the rim of the disk look like? It seems to me that there ought to be a region in the center of the disk or on the rim where the light from the other side would no longer be detected (actually a dark region not just invisible). Could this be thought of as the reverse of a black hole event horizon? (In a black hole you can theoretically see outside but the outside cannot see in or through. While in this case you could not see a certain angular region but the non moving observer should still be able to see you. Sort of a optical corriolis effect.)

    1. Glad you found it interesting.
      The redshift transformation of a black-body spectrum actually turns out to be exact, a derivation that dates back to Richard Tolman’s Relativity, Thermodynamics and Cosmology in 1934. As an example, think of the Cosmic Microwave Background Radiation, which was emitted at a temperature of around 3000K and has been redshifted down to 2.7K, but still retains its black-body spectrum.
      And it has a slight temperature dipole of a few milliKelvin caused by the movement of the Earth relative to the CMBR frame of rest—a very mild example of the transformations I wrote about here.
      Maybe the best way to visualize this, without delving into the maths, is to look at log-log plots of the black-body spectrum like this one. In this sort of plot there’s a “universal” black body curve, and the difference between temperatures is just a horizontal (Wien’s Law) and vertical (Stefan-Boltzmann Law) shift in the curve. Because redshift changes every wavelength by the same multiplier, it shifts the black-body curve on this plot horizontally without changing its shape. And, as I showed in Part 3, the surface energy output is multiplied according to the fourth power of the redshift, which moves the shifted curve vertically so that it corresponds precisely with the black-body curve for a different temperature.

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.