The Celestial View From A Relativistic Starship: Part 1

Bussard Interstellar Ramjet
Bussard Interstellar Ramjet (source)

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:

Aberration: Direction of shift

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:

Aberration: apparent magnification

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:

Aberration: rest frame view of sky

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:

Aberration: Apparent shift at 0.5c

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:

Aberration: apparent shift at 0.5c and 0.85c

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.


Mathematical notes

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

Relativistic aberration formulaThe 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.

Formula for r-prime, relativistic aberrationThis 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).

Polar formula for ellipse in terms of semiminor axis and eccentricitySo 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:Equivalences for relativistic aberration ellipseIn Cartesian coordinates with the z axis aligned with the velocity vector of the spacecraft, we get the transformation:

Cartesian transformation for relativistic aberrationThe 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.

3 thoughts on “The Celestial View From A Relativistic Starship: Part 1”

  1. Urrrr ! Had to read this several times to follow the examples. I loved the concepts you outlined. Astronomical physics is really interesting. Think if I hadn’t done Med. I’d like to have studied Astronomy

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