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:

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:

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

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:

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:

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.

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:

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.

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.

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Mathematical notes

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

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

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

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

In the moving frame, η=1 at angle:

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

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