The Celestial View From A Relativistic Starship: Part 3

Bussard Interstellar Ramjet
Bussard Interstellar Ramjet (source)

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.

Black body spectra
Click to enlarge

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.

Planckian locus
Click to enlarge (source)

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.

(Be the first)

Leave a Reply

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