The sky crowded forward by the observer's speed

Every star in the sky traces a small ellipse over a year, twenty arcseconds in radius, because the Earth’s motion around the Sun tilts the direction from which starlight appears to arrive. Bradley found it in 1728, explained it as the tilt a runner gives an umbrella in vertical rain, and wrote down its size as the ratio of the Earth’s speed to the speed of light. That ratio is about one part in ten thousand, and at the precision of his instrument nothing more was needed.

The ratio is the first term of a series. Aberration is a property of how directions transform between observers in relative motion, and the exact law comes from special relativity: a source at an angle θ from the direction of motion, as seen by an observer at rest relative to the sources, appears to an observer moving at speed βc at an angle θ′ given by

$$\cos\theta' = \frac{\cos\theta + \beta}{1 + \beta\cos\theta}.$$

At small β this reduces to Bradley’s displacement, $\beta\sin\theta$, towards the direction of motion. At large β it does something the small-angle picture never suggests: it maps the whole sphere of the sky onto itself, pulling everything towards one point.

Why the exact law is not the umbrella #

The umbrella picture adds velocities: the light’s velocity and the observer’s, as vectors, and reads the direction off their sum. That is Galilean addition, and it gives a displacement whose tangent is $\beta\sin\theta/(1+\beta\cos\theta)$. The relativistic formula agrees with it to first order in β and differs at second order, because velocities do not add as vectors when one of them is the speed of light. Light moves at c in every frame, so the transformed direction must be a unit vector in the moving frame as well, and the Lorentz transformation achieves that by contracting the component along the motion.

At the Earth’s speed the Galilean and relativistic formulae differ by a term of the same order as the second-order term itself, a few tenths of a milliarcsecond, so the choice between them is a choice that only the most precise catalogues can detect. At a tenth of the speed of light they differ by degrees. The Galilean version gathers the stars forward by a different amount at every angle, and its version of the forward crowding comes out wrong at second order — a small symptom of treating light as though its speed could depend on who is measuring it.

The distinction was, for a century, a problem rather than a refinement. If aberration is the umbrella effect, it should depend on how fast light travels inside the telescope, and a telescope filled with water should see a larger aberration. It does not — the famous null result the earlier discussion of Bradley’s ellipse recounts — and the explanation that the moving water drags the light along by exactly the right fraction was one of the loose ends that special relativity eventually tied off. In the relativistic account aberration is not about the light’s journey down the tube at all. It is about how two observers moving relative to each other assign directions to the same light, and the medium inside the telescope, being at rest relative to the telescope, cannot enter.

Half a milliarcsecond that a catalogue must carry #

At the Earth’s orbital speed the second-order term is small, and it is no longer negligible.

The second-order term is $\tfrac{1}{4}\beta^2\sin 2\theta$, which for the Earth peaks at half a milliarcsecond. Bradley could measure about a second of arc; half a milliarcsecond is two thousand times smaller. For the first two and a half centuries of positional astronomy the term did not exist in any practical sense.

For space astrometry it is fifty times the precision of the best measurements. A catalogue whose positions are good to ten microarcseconds cannot be reduced with the first-order formula, and the reduction therefore uses the exact expression — not an expansion to a chosen order, but the formula itself, applied to the spacecraft’s velocity relative to the solar system’s barycentre. The velocity has to be known well enough that the first-order term, which is twenty arcseconds, is right to ten microarcseconds: a part in two million, or about a centimetre a second out of thirty kilometres a second. That is a demand on the spacecraft’s orbit determination as much as on its optics, and it is met by the radio tracking that measures a spacecraft’s velocity from a Doppler shift.

The same reduction includes the other effects that the first-order picture keeps separate. The bending of starlight by the Sun, which reaches milliarcseconds even far from the Sun, is applied in the same relativistic framework; and the slow change in the observer’s velocity as the solar system orbits the Galaxy makes the aberration itself drift, which is a measurement of the Sun’s galactic acceleration. At microarcseconds, aberration is not a correction applied to positions but part of the definition of what a position is — a direction as seen by an observer in a specified state of motion.

That shift is recorded in the system of astronomical constants itself. For most of the twentieth century the constant of aberration — the twenty-arcsecond amplitude of the annual ellipse — was a fundamental constant, quoted to many figures and used to reduce every observation. It no longer has that status. The velocity that sets the aberration is taken directly from the numerical ephemeris of the solar system, for the observer’s actual position and time, and the aberration is computed from it exactly. What was once a measured constant of nature is now a derived quantity of a model, which is what it always was.

Crowded towards the apex #

At speeds approaching light’s, the second-order term stops being small and the map of the sky changes character.

The forward displacement is largest for stars at right angles to the motion, and it moves each of them towards the direction of travel by an angle that grows with speed. Stars that were behind the observer move forward too, less far. The effect on the sky as a whole is a compression towards the apex and a rarefaction away from it, and the number of stars per unit solid angle changes accordingly.

The density follows from the Jacobian of the mapping. A small patch of sky of solid angle dΩ in the rest frame appears with solid angle dΩ′ in the moving frame, and the ratio is

$$\frac{\mathrm{d}\Omega}{\mathrm{d}\Omega'} = \frac{1-\beta^2}{(1-\beta\cos\theta')^2}.$$

Directly ahead that is $(1+\beta)/(1-\beta)$ and directly behind its inverse, and over the whole sphere it integrates to exactly one: no star is created or lost, only moved. At a tenth of the speed of light the forward sky is twenty-two per cent denser; at half, three times.

The word apex has an older astronomical meaning that is worth keeping separate. William Herschel in 1783 found the direction in which the Sun is moving among the nearby stars — towards the constellation Hercules — from the pattern of their proper motions: stars ahead appear to spread apart and stars behind to converge, because the Sun is travelling through them. That is a parallax effect, and it depends on the stars’ distances. Aberration’s apex is the same direction for the same motion, but its crowding does not depend on distance at all: the most distant quasar is displaced exactly as much as the nearest star. The two effects of one velocity separate cleanly, because one falls with distance and the other does not.

The cone that holds half the sky #

The most compact way to state the effect is the size of the region into which half the sky’s stars are gathered.

Half the stars lie in the rest frame’s forward hemisphere, bounded by the circle at ninety degrees from the motion. That circle maps to the apparent angle whose cosine is β. At half the speed of light it is sixty degrees; at nine-tenths, twenty-six; at 0.99, eight. A traveller at relativistic speed sees most of the sky’s stars in a shrinking cone ahead and a sparse, darkening sky behind.

The stars ahead are not only more numerous. Their light is blueshifted by the Doppler factor, which directly ahead is $\sqrt{(1+\beta)/(1-\beta)}$ — 1.73 at half the speed of light, 4.4 at nine-tenths — and brightened by the combination of the blueshift and the crowding, while the stars behind are reddened and dimmed. The same Doppler shift that measures a star’s radial velocity here reshapes the colours of the whole sky, and the reshaping is continuous: every direction has its own shift, set by its angle from the motion.

A tenth of light speed is not far off #

Relativistic travel is not only a thought experiment. Proposals to send gram-scale probes to the nearest stars, pushed by laser light on sails that need no propellant, aim at a fifth of the speed of light, and at that speed the sky is already substantially rearranged. A star at right angles to the probe’s motion appears 11.5 degrees further forward; the sky directly ahead is half again as dense as at rest and the sky behind two-thirds as dense; the target star itself, directly ahead, does not move, but every star used to navigate by does.

A probe that steered by comparing the positions of guide stars with a catalogue would have to apply the full formula to every one of them, with its own velocity as the input — which it would not know exactly, since its velocity is what the laser gave it. Turned around, the stars’ displacements measure that velocity: an observer who knows where the stars really are can read its own speed and direction from how the sky has been distorted, without any external reference. Aberration at high speed is a speedometer as well as a nuisance, in the same way that at the Earth’s speed it measured the Earth’s motion before anything else did.

The ring that is not there #

That continuity disposes of a popular image. It was once suggested, and widely illustrated, that an observer at high speed would see the stars gathered into a ring of rainbow colours around the direction of travel — a starbow — with red on one side of the ring and blue on the other, because stars ahead would be shifted into the ultraviolet and stars behind into the infrared, leaving a band of visible starlight between.

The calculation does not produce a ring. The aberration moves stars smoothly towards the apex, with no discontinuity and no accumulation at any particular angle — the density rises monotonically from behind to ahead. The Doppler shift changes smoothly with angle too. A real stellar spectrum is broad, and shifting it by a factor of two moves some of the star’s light into the visible from the infrared as fast as it moves other light out into the ultraviolet, so stars do not disappear from view when they are shifted. What a relativistic traveller would see is a sky whose stars are concentrated ahead, brighter and bluer there, and fainter and redder behind, graded smoothly between. Calculations that include real stellar spectra, first done carefully in the 1970s, found no ring.

The episode is a useful caution about intuition built on the first-order picture. At small β, aberration is a small displacement and the Doppler shift a small change of colour, and combining two small effects gives a small effect. At large β both are large and they are the same transformation seen two ways, and only the full calculation says what their combination looks like.

The forward crowding in the sky already observed #

None of this needs relativistic travel to be observable. The solar system moves at about 370 kilometres a second relative to the microwave background, β of about a thousandth, and at that speed the crowding is a tenth of a per cent. For stars that is invisible. For a catalogue of hundreds of thousands of distant galaxies and quasars, spread over the whole sky, a tenth of a per cent more of them per square degree in one direction than in the opposite one is a measurable dipole in the counts.

That is one of three independent ways of measuring the same velocity — from the temperature of the microwave background, from the distortion of its anisotropies, and from the counts of distant sources — and whether the three agree is a question the observations have not yet settled. The aberration that Bradley measured as a twenty-arcsecond ellipse for one star becomes, at the other extreme of the same formula, a statistical property of the whole extragalactic sky.

The same transformation, seen from the other end #

Everything here has been about an observer moving past stationary sources. The transformation is symmetric, and its other face is familiar in a different context: a source moving at close to the speed of light towards an observer emits light that, in the observer’s frame, is concentrated into a narrow cone around its direction of motion. The half-sky cone of a fast observer becomes the beam of a fast emitter, with the same opening angle, whose cosine is β, and the same Doppler brightening inside it.

That is relativistic beaming, and it shapes what is seen of the fastest objects in the universe. The jets launched from the neighbourhoods of the black holes that power quasars move at close to the speed of light, and a jet pointed nearly at the observer is brightened by a large power of its Doppler factor while its twin, pointed away, is dimmed below detection — which is why such jets so often appear one-sided. The narrow beams that make the flashes of gamma-ray bursts so bright are the same geometry with larger Lorentz factors still. The inference from the brightness of such sources to their intrinsic power therefore carries the beaming factor as a multiplier, and a factor that multiplies every inferred quantity is exactly the kind that is hardest to measure independently.

What the formula leaves out #

The formula assumes the sources are infinitely distant, so that the only effect of the observer’s motion is on the direction of arriving light. For nearby stars the observer’s changing position matters too, which is parallax, and for a moving observer the two are separated by their different dependence on distance — parallax falls with distance, aberration does not. It also assumes the observer moves uniformly. An accelerating observer, like one on the Earth or on a spacecraft in orbit, has an instantaneous velocity that changes, and the aberration follows the instantaneous velocity; the acceleration itself does not add a term, a subtle point that follows from the equivalence of instantaneous inertial frames and that space astrometry’s reductions build in.

And the stellar positions in the figures are uniform on the sky. Real stars are concentrated towards the Milky Way, so the view from a relativistic observer depends on the direction of travel relative to the Galaxy; a traveller heading along the galactic plane would see the plane’s stars crowded forward into a narrowing band, one heading towards a galactic pole would see the plane’s stars gathered from the whole horizon into a ring — a real ring, made by the Galaxy rather than by the optics, and a different thing from the starbow.

Still open: whether the frame of the stars is the frame of the sky #

Every statement here refers to an observer moving relative to the sources, and the sources are assumed to share a common rest frame. For stars in the Milky Way they do, roughly; for distant galaxies the rest frame is assumed to coincide with the rest frame of the microwave background, which is the frame in which the universe’s expansion looks isotropic. That assumption is testable with aberration itself, by measuring the solar system’s velocity relative to each and comparing — and the comparison is not straightforward, because the measured velocities do not obviously agree. Whether the universe has a single rest frame to the precision that its most distant contents can now test is a question that aberration, extended from one star to the whole sky, is well placed to answer.