The other twenty arcseconds

A star’s position on the sky is not one number pair but a small annual loop, and the loop that everybody expects — parallax, the projection of the Earth’s own orbit onto the sky — is not the largest one. For every star known, a second loop is bigger. For the nearest of them it is 27 times bigger; for a typical naked-eye star, a thousand times.

That second loop is aberration, and its size is the same for every object in the universe.

The same ellipse from a different vector #

Parallax and aberration are both consequences of the Earth’s orbit, and they differ in which property of the orbit they respond to.

Parallax responds to position. The Earth is somewhere, the star is somewhere else, and the direction from one to the other changes as the Earth moves — by an angle $\varpi = 1\,\text{AU}/d$, which is small in proportion to the star’s distance and vanishes for an infinitely distant one.

Aberration responds to velocity. A telescope moving through space has to be tilted slightly forward to catch light arriving from a fixed direction, by an angle whose tangent is the ratio of the telescope’s speed to the light’s. This is the classical picture — the raindrops on a moving train, the umbrella held into the direction of travel — and the relativistic treatment changes the second-order terms without changing the first. It is the same first-order ratio that appears in the shift of a spectral line, turned across the line of sight instead of along it. The angle is

$$\kappa = \frac{v}{c} = \frac{29.7847\ \text{km/s}}{299{,}792.458\ \text{km/s}} = 20.4955''.$$

Nothing about the star enters. Not its distance, not its mass, not what it is. A quasar at redshift 2 and Proxima Centauri are displaced by the same 20.5 arcseconds, because both are displaced by a property of the observer.

The two ellipses have the same shape for a reason that is worth naming rather than noticing: both are the projection of a circle in the ecliptic plane onto the celestial sphere at ecliptic latitude $\beta$, so both have axis ratio $\sin\beta$ exactly. One circle is the Earth’s orbit and the other is the Earth’s velocity going round the same way. Projection does not care which.

Quadrature is what separates them #

If two effects draw the same shape at different sizes, one might be mistaken for the other with a scale error. What makes them unmistakable is the phase.

Parallax is maximum when the Earth is furthest to one side of the star, which is when the Earth’s position is at right angles to the direction of the star. Aberration is maximum when the Earth’s velocity is at right angles to that direction. On a nearly circular orbit velocity leads position by exactly a quarter turn, so the two ellipses are traced 90° out of step.

That three-month offset is the whole content of Bradley’s discovery, and it is why he could not read what he had found as parallax. He had chosen γ Draconis because it passes almost exactly overhead from London, so that atmospheric refraction — which shifts everything towards the zenith by an amount depending on altitude — is negligible and cannot be blamed. He built a zenith sector to measure the star’s distance from the vertical to a fraction of an arcsecond. And over the winter of 1725–26 the star moved: south by 20 arcseconds between September and March.

Parallax would have put the maximum displacement in December and June. This was in March and September. Bradley spent nearly two years failing to explain it, tried nutation of the Earth’s axis and rejected it, and — according to the story his colleagues told — settled the matter while sailing on the Thames, watching a pennant on the mast swing as the boat came about. The wind had not changed. The boat’s motion had, and the pennant showed the sum.

What was actually measured #

The observation is a sequence of angles between a star and a plumb line, made with a fixed telescope pointed at the zenith, over a period of years. Everything else is inference.

Bradley’s zenith sector was a 12.5-foot telescope mounted vertically against a graduated arc, and it did one thing: it measured how far from the vertical a star passed. That is a single angle, in one coordinate, and it is enough because the star is nearly overhead, so a displacement in declination is very nearly a displacement in position. His stated precision was about half an arcsecond and his measured amplitude was $20.2''$; the modern value is $20.4955''$.

From that one number two independent things follow, and it is worth separating them because the second is what makes the observation famous.

The Earth moves. This is the direct inference — the fact that an observer inside a rotating sky had until then been assumed to be moving rather than shown to be and it required no assumption at all beyond the finite speed of light. Before 1728 there was no observational proof of the Earth’s orbital motion — the Copernican system was overwhelmingly accepted and it was accepted on grounds of simplicity, because the parallax that would have demonstrated it was below every instrument ever built. Aberration is the first measurement that could not be produced by a stationary Earth.

The speed of light, from the Earth’s orbital speed. Or, taken the other way, the Earth’s orbital speed from the speed of light. $\kappa = v/c$ is one equation in two quantities, and in 1728 neither was known independently in absolute terms — Rømer had timed Jupiter’s moons and got light’s speed in units of the astronomical unit, which is what aberration also delivers. The two agreed, which was the first time two utterly different methods had given the same value for anything of the kind.

Where the model stops #

The annual term is not all of it. The Earth’s velocity has three parts and each produces its own aberration. The annual term, 20.5 arcseconds, is the orbital motion. The diurnal term is the observer’s rotation with the Earth’s surface — at most $0.32''$ at the equator, falling as $\cos(\text{latitude})$, and routinely corrected for in modern astrometry. And there is a secular term from the Sun’s own motion through the Galaxy at about 220 km/s, which is $151''$ — seven times the annual term — but constant on any human timescale and therefore absorbed into the catalogue positions rather than observed as a variation.

The orbit is not circular. At $e = 0.0167$ the Earth’s speed varies by 3.4 per cent over the year, so the aberration ellipse is not exactly an ellipse: it has a small term with the period of the anomalistic year, the elliptic aberration, of amplitude about $0.34''$. It is a fixed function of the date, has nothing to do with the star, and is applied as a correction.

The classical formula is first order. The relativistic aberration formula differs from $v/c$ at order $(v/c)^2$, which for the Earth is $10^{-8}$ radian — about $0.002''$. That is below Bradley’s precision by two orders and above Gaia’s by two, so it is a term that had to be added at exactly one point in the history of the subject and has been in every reduction since.

And an aberration is not a displacement of the star. This is the point where the model’s honesty matters most. Nothing about the star’s position has changed. What changed is the direction in which the light arrives at a moving instrument, which is a fact about the instrument’s frame. Two observers at the same place at the same instant, moving differently, measure different positions for the same star — and neither is wrong.

The reason parallax took another century #

Bradley’s result has an epilogue that is more instructive than the discovery. Having found aberration, he corrected for it, and looked again for parallax in his residuals. He found none — and correctly concluded that γ Draconis’s parallax was below his precision of about half an arcsecond, which put a lower bound on the star’s distance of 400,000 astronomical units — the first useful statement about a stellar distance ever made from data.

That was itself a substantial result and it was received as a disappointment. It also, in the residuals of the residuals, produced nutation: an 18.6-year oscillation of the Earth’s axis with an amplitude of $9.2''$, driven by the precession of the Moon’s orbital plane. Bradley published it in 1748 after observing for a full cycle, which is the correct thing to do and is why the term is not called something else. Parallax was finally measured in 1838 — by Bessel, for 61 Cygni, at $0.314''$ — 110 years after Bradley bounded it. The reason for the delay is exactly the ratio in the first figure. To detect parallax at a tenth of an arcsecond, aberration at twenty arcseconds must be modelled to a part in two hundred, nutation to a part in a hundred, refraction to a part in ten thousand of its zenith value, and the instrument’s own flexure to below all of them. The smaller effect could not be seen until the larger one was understood, and the larger one was found by accident while looking for the smaller.

The telescope filled with water #

The raindrop analogy is a good one and it has a leak in it, and finding the leak took a century and produced one of the most elegant null results in the history of the subject.

If aberration is the tilt required because the light takes time to cross the telescope while the telescope moves, then filling the telescope with water should change it. Light travels through water at $c/1.33$, so it spends a third longer inside the tube, the tube moves a third further while it does so, and the required tilt should be a third larger.

George Biddell Airy built the experiment in 1871: a zenith sector with a water-filled tube, pointed at γ Draconis, the same star Bradley had used. The measured aberration was exactly the same as in air, to the precision of the instrument.

That result was a serious embarrassment for the theories then available. In an ether theory the null result requires the water to drag the ether along with it by a specific partial amount — Fresnel’s drag coefficient, $1 - 1/n^2$ — which had been introduced decades earlier for unrelated reasons and which works, in the sense that the two effects cancel to first order. It is a fitted quantity with no derivation, and having to invoke it to explain why an experiment found nothing is the mark of a theory in difficulty.

Special relativity gives the answer directly and without a free parameter. Aberration is not a statement about the medium the light crosses at the end of its journey; it is a statement about the transformation between two frames in relative motion, and the direction of arrival is fixed before the light enters the tube. What the water does to the light inside the instrument is refraction, which bends the ray on entry in exactly the way that cancels the extra transit time. The tilt is $v/c$ and the $c$ is the speed of light in vacuum, whatever the telescope is full of.

The experiment is therefore a measurement of a frame transformation disguised as a measurement of an optical instrument, and its null result was one of the constraints special relativity had to satisfy — alongside Michelson and Morley’s, published sixteen years later and far better known.

The pattern over the whole sky #

Every figure here follows one star, and it is worth stepping back to what the effect does to the sky as a whole, because the shape is not the one a first reading suggests.

Aberration displaces every object towards the point the observer is moving to — the apex. The size of the displacement is $\kappa\sin\theta$, where $\theta$ is the angle between the object and the apex, so it is zero at the apex itself, zero at the opposite point, and maximum on the great circle 90° from both.

That is not a rigid rotation of the sky. A rotation moves every point by the same angle about an axis; this moves points by different amounts depending on where they are, so the pattern of stars is distorted rather than merely displaced. Angular separations between stars near the apex are stretched apart, separations near the antapex are compressed, and a constellation straddling the 90° circle is sheared.

The distortion has one property that saves the astrometry: it is conformal. Angles between directions at a point are preserved even though separations are not, so the sky is not sheared in the sense that a small circle becomes an ellipse — a small circle stays a circle, of a different size. That is a consequence of the transformation being a Möbius map of the celestial sphere, which is the deeper statement about what a change of velocity does to a sky, and which reduces to the $\kappa\sin\theta$ dipole for a speed as small as the Earth’s.

The practical consequence is that aberration cannot be absorbed into a coordinate system. A catalogue position is a direction in a frame with a specified origin and a specified state of motion, and the international celestial reference frame specifies both. Every observation made from the ground is transformed into that frame by removing the observatory’s own velocity, and the removal is per-star because the displacement is.

It is also why the frame’s definition names a set of quasars rather than a set of stars. A quasar has no measurable proper motion, so the only thing that moves its catalogued position is the observer — which makes the residual motions of the defining sources a direct check that the transformation has been done correctly.

What the picture cannot show #

The velocity. Every figure here draws positions on the sky, and aberration is a response to a vector that lives in three-dimensional space and never appears in an angle plot. The quadrature between the two ellipses is the shadow of that vector’s relation to the position vector, and the relation itself is off the page.

The Sun’s own motion. The secular aberration of $151''$ towards the solar apex is seven times everything drawn here and it is not drawn, because it does not vary. A figure of annual loops is a figure with a large constant subtracted, and the constant is inside the catalogue coordinates every position is measured against.

The scale. The parallax ellipse in the first figure is drawn 955 times smaller than the aberration ellipse, which is honest and is why it is nearly invisible. A figure in which both are legible would be a figure in which the ratio — the entire reason one was found and the other was not — has been thrown away.

Where the ladder goes next #

Later rungs on this anchor: the relativistic aberration formula and the second-order terms Gaia has to carry. Secular aberration drift, the slow change in the apparent positions of quasars as the solar system accelerates around the Galaxy, which has now been measured at about 5 microarcseconds a year and is a direct detection of the Sun’s galactic acceleration. The light-time correction, which is a different effect that looks similar and is often conflated with this one. The Gaia astrometric solution, in which aberration, parallax, proper motion and perspective acceleration are fitted simultaneously to $10^9$ objects. And the terrestrial analogue — the aberration of a radio telescope in very long baseline interferometry, where the same $v/c$ appears as a delay rather than an angle.

Bradley’s instrument had one axis and measured one angle. It produced, in twenty years, the first proof that the Earth moves, an independent determination of the speed of light, the discovery of nutation, and a lower bound on the distance to a star. The parallax he was looking for was not among them, and when it did arrive it started a ladder that now reaches to the edge of the observable universe.