A third of an arcsecond set by where the telescope stands

Light arrives at a moving telescope from a direction tilted towards the telescope’s motion. The tilt is the ratio of the telescope’s speed to the speed of light, and Bradley found it in 1727 as a twenty-arcsecond ellipse traced by every star over a year, the signature of the Earth’s orbit round the Sun. That is not the only motion a telescope shares. It rides the Sun round the Galaxy, it rides the Earth round the Sun, and it rides the surface of the Earth round the Earth’s axis once a sidereal day. Each motion produces its own aberration, with its own size and its own clock.

The last of these is the smallest that anybody has to correct. At the equator the ground moves east at 465 metres a second, which is one part in 644,000 of the speed of light, or 0.32 arcseconds. At the latitude of most European observatories it is two-thirds of that; at the poles it is nothing. It is called diurnal aberration, and it has three properties none of the larger aberrations has. It depends on where the telescope is. It changes over a day rather than a year. And it is different, at the same instant, for two observers looking at the same star from different places — which makes it the term that a network of telescopes has to get right before it can compare anything.

This essay follows that third of an arcsecond through the instruments that have had to deal with it: the meridian circles of the nineteenth century, for which it was a delay in a clock reading; the space telescopes in low orbit, for which the same effect is sixteen times larger and reverses in forty-six minutes; and the wide fields, for which the part that matters is not the shift of the whole field but the way it stretches.

The motions a telescope shares #

The aberration of a moving observer is set by the velocity, and the velocities involved span three orders of magnitude.

The largest is never seen. The Sun’s motion round the Galaxy, about 230 kilometres a second, displaces every star by up to two and a half arcminutes towards the point the Sun is heading for — but that point moves by less than a millionth of a degree a year, so the displacement is constant over all of recorded history and is absorbed, without anyone noticing, into the catalogue positions of the stars. An aberration is only observable when the velocity that causes it changes direction, and what is seen of the Sun’s galactic motion is not its aberration but the slow curvature of its orbit, measured as an acceleration of microarcseconds a year.

The Earth’s orbital velocity reverses every six months, so its twenty-and-a-half-arcsecond aberration is seen in full. The rotational velocity reverses every twelve hours, so its aberration is seen in full too, and at a third of an arcsecond it was large enough to matter for every meridian instrument from Bradley’s onward. A space telescope in a low orbit adds a fourth velocity, seven and a half kilometres a second, reversing every forty-six minutes.

The whole sky leans towards the east #

The Earth turns from west to east, so a point on its surface moves towards the east point of its own horizon. That point is the apex of the diurnal aberration: every star is displaced along the great circle joining it to the east point, towards the east point, by $\kappa_d \sin\theta$, where $\theta$ is the star’s angular distance from the east point and

$$\kappa_d = \frac{\omega_\oplus R_\oplus \cos\varphi}{c} = 0.320'' \cos\varphi .$$

The pattern is the same as that of any aberration, drawn on the observer’s own horizon. Stars near the east and west points are not moved at all, since the motion is along their line of sight. Stars ninety degrees from the east point — which is the whole of the meridian, from the north point through the zenith to the south point — are moved by the full amount. On the eastern half of the sky the stars are pushed down towards the eastern horizon, and on the western half they are pushed up and over towards it, so that the entire sky appears to have been tipped very slightly to the east.

That tip is not the same thing as the Earth’s rotation, which carries the stars from east to west once a day. It is the opposite direction and a far smaller angle: a correction to where a star is seen at a given moment, not a motion of the star. But its geometry rotates with the observer, which is why its effect on the equatorial coordinates of a star — the coordinates a catalogue uses — changes through the day.

The annual ellipse on a daily clock #

In right ascension and declination the shift takes a compact form. For a star at hour angle $H$ and declination $\delta$, observed from latitude $\varphi$,

$$\Delta\alpha\cos\delta = \kappa_d \cos H, \qquad \Delta\delta = \kappa_d \sin H \sin\delta .$$

As the star turns with the sky, its hour angle runs through a full circle in a sidereal day, and the displacement runs round an ellipse.

These are the same ellipses Bradley found for the annual aberration, with the Earth’s spin in place of its orbit and the celestial pole in place of the ecliptic pole. A star at the celestial pole traces a circle, because the observer’s velocity turns through every direction perpendicular to the line of sight once a day. A star on the celestial equator traces a line, because the velocity’s component across the line of sight swings east and west but never north or south. In between, an ellipse whose flattening is the sine of the declination. The analogy is exact because the geometry is identical: an observer moving at constant speed round a circle, seen against a sphere of stars, displaces each star round an ellipse whose shape is the projection of that circle.

The difference is the size of the ellipse and the clock it keeps. The annual ellipse is sixty times larger and takes a year; the diurnal ellipse is a third of an arcsecond and takes a day. An astronomer measuring a star at a single instant sees one point on each.

A transit that runs late #

For most of the history of fundamental astrometry, positions were measured with a transit instrument or meridian circle — a telescope that could swing only in the plane of the meridian. The right ascension of a star was the sidereal time at which it crossed the central wire, read from a clock, and its declination was the angle read off a graduated circle at that moment.

On the meridian the hour angle is zero, so the diurnal aberration there is entirely in right ascension: the star is displaced east by $\kappa_d \sec\delta$ in hour angle, which means it arrives at the wire late. Converted to time,

$$\Delta t = \frac{0.320''}{15} \cos\varphi \sec\delta = 0.0213\,{\rm s}\,\cos\varphi\sec\delta .$$

For an equatorial star seen from Greenwich the delay is thirteen milliseconds, a fiftieth of an arcsecond in the star’s position — small, but larger than the random error of a good nineteenth-century transit and systematic, always in the same direction. Towards the pole it grows as the secant of the declination, because a given angular displacement east corresponds to a larger change in hour angle where the hour circles crowd together. Polaris, two-thirds of a degree from the pole, crosses the meridian more than a second late at latitude fifty-two degrees. A second of time is fifteen arcseconds of hour angle; at Polaris’s distance from the pole that is a fifth of an arcsecond on the sky, which is the whole of the diurnal aberration appearing as a large error in a coordinate that has become sensitive to it.

The circumpolar stars were exactly the ones used to determine the instrument’s orientation — its azimuth error, the small angle by which the transit’s plane missed the true meridian — because their slow motion and large secants made them sensitive to it. A diurnal aberration left uncorrected would have been absorbed into the azimuth and then propagated, multiplied by a different secant, into every other star. The reductions of the great nineteenth-century catalogues applied the correction transit by transit, as a term of hundredths of a second subtracted from the clock reading, and the term was large enough to appear in the tables printed for the purpose.

It does not affect the declination measured on the meridian, since its north–south component vanishes there. For any instrument that measures away from the meridian — an altazimuth, a modern telescope with a camera, an interferometer — both components are present, and their size depends on the hour angle at which the observation was made.

Two observatories, two skies #

The annual aberration is the same for every observer on Earth to within a part in a few thousand, because they all share the Earth’s orbital velocity. The diurnal aberration is not. At a given instant an observatory in Chile and one in Hawaii, separated by a third of the way round the Earth, are moving in directions that differ by a hundred and twenty degrees, and the same quasar is displaced by different amounts in different directions for each. Two observatories at opposite ends of a diameter of the equator see the same star at positions that differ by up to two-thirds of an arcsecond.

For an individual telescope that is a correction like any other, applied once. For a network of telescopes combining their signals, it is part of the geometry. A very-long-baseline interferometer measures the delay between a wavefront’s arrival at two antennas thousands of kilometres apart, to a few picoseconds, and the stations’ velocities enter that delay directly: the wavefront reaching the second antenna has to be compared with the position that antenna had when the wavefront arrived, not when it arrived at the first. The delay model used for geodetic and astrometric interferometry carries the stations’ rotational velocities explicitly, and dropping them would produce errors many thousands of times larger than the measurement precision. No interferometer would call this “diurnal aberration” — it is a term in a delay — but it is the same physical fact, that the light’s direction depends on the velocity of whoever receives it.

Satellite and lunar laser ranging meet it from the other side. A reflector on a moving satellite returns the light to where the station will be, not where it was, and the offset the reflector has to be built with is set by the satellite’s velocity relative to the station — a velocity that includes the station’s own rotation. The rotation’s contribution is small beside the satellite’s orbital speed, but it varies with the station’s latitude and with the direction of the pass, which is part of why one reflector cannot serve every station equally.

The same effect, from orbit #

A telescope in a low orbit moves at seven and a half kilometres a second, sixteen times the speed of the equatorial ground, and the direction of that motion turns through a full circle every ninety-odd minutes.

For a target in the plane of the orbit, the aberration swings by more than five arcseconds either way within an orbit. A space telescope pointing at it by gyroscopes alone would see the target drift across its detector by ten arcseconds every forty-six minutes. The Hubble Space Telescope’s fine guidance sensors lock onto guide stars, which share the same aberration, so the drift of the whole field is followed automatically; what the guidance cannot follow is the part that differs across the field.

Diurnal aberration on the ground and orbital aberration in space are the same phenomenon, with the observer’s local velocity different by a factor of sixteen and the period different by a factor of fifteen. The ground term is slow, and a correction computed once per exposure is sufficient. The orbital term changes appreciably during a single exposure of a few minutes, so a space telescope’s astrometric software has to compute the aberration as a function of time through the exposure — and every image the telescope takes carries, in its header, the aberration factor that applied while it was taken.

The part that cannot be pointed away #

Aberration is not a rigid shift. A star in the direction of motion is not displaced at all, and one ninety degrees from it is displaced by the full amount; between, the displacement varies smoothly. Across a small field, the variation shows up as a change of scale: the field is compressed along the direction of motion when pointing towards it and stretched when pointing away, by a fraction of order $v/c$. A rigid shift is removed by re-pointing or by registering on a reference star. A change of scale is not.

For a ground telescope the rotational part of this is a few milliarcseconds across a degree, and it changes so slowly that the plate constants fitted to each image absorb it along with the scale change that refraction produces. For a telescope in low orbit it is about a milliarcsecond across a two-arcminute field, which is the precision Hubble’s astrometry now reaches with careful calibration — so the scale of every Hubble image breathes by tens of parts per million over an orbit and has to be corrected exposure by exposure. For a telescope sharing the Earth’s orbital velocity, as Gaia does, the differential term across its fields is of order a hundred milliarcseconds and is part of the full relativistic model of every observation.

This differential part is the sense in which aberration is an astrometrist’s problem rather than a pointing engineer’s. The pointing problem — where to aim — is solved by guide stars, which aberrate with the target. The astrometric problem — where each star in a field is relative to the others — inherits the velocity’s variation across the field, and that cannot be solved by aiming better.

Why the third of an arcsecond was found late #

Bradley’s theory of aberration contained the diurnal term implicitly from the start: any velocity of the observer produces it. But his instruments measured declinations to about a second of arc, and the diurnal term, at a fifth of an arcsecond in England and zero in declination on the meridian, was below what he could see. The term was written into the reduction of meridian observations as instruments improved through the first half of the nineteenth century, and it became routine once transit timing reached hundredths of a second, which happened when the chronograph replaced the observer’s ear-and-eye method of estimating transits against a beating clock.

It is a common pattern in positional astronomy. A correction is predicted as soon as the physics is written down and becomes necessary only when the instruments catch up with it. Annual aberration was found before parallax because it was larger; diurnal aberration was handled after parallax because it was smaller; and the bending of starlight by the Sun, four milliarcseconds at ninety degrees from it, was included in catalogue reductions only when space astrometry made a milliarcsecond the working unit. Each was in the physics long before it was in the reductions, and each arrived in the reductions at the moment it first exceeded the random error.

What the figures leave out #

The figures treat the Earth as a sphere rotating uniformly, with the observer at sea level. The equatorial radius, the flattening and the observer’s height change the rotational speed by parts in a thousand, which changes the diurnal constant by the same fraction — below a milliarcsecond everywhere. The variation of the Earth’s rotation rate, by milliseconds a day, is smaller still.

The aberration is computed to first order in $v/c$. For the rotational velocity, second-order terms are parts in $10^{12}$ and of no consequence; for the Earth’s orbital velocity they are a milliarcsecond and are included in modern reductions, and the fully relativistic form is needed when the velocity is a substantial fraction of the speed of light. Diurnal aberration also adds to the annual: the observer’s velocity is the vector sum of both, and the two aberrations add as vectors to first order.

And the low orbit is circular and the telescope’s attitude is held fixed. A real orbit has an eccentricity, a real spacecraft has a pointing jitter, and the guide stars themselves are measured through optics with their own colour-dependent distortion; each adds terms to the scale correction that are calibrated rather than computed.

Still open: the observatory in the solution #

The diurnal aberration is a correction that is perfectly known: the Earth’s rotation is measured to a part in $10^{9}$, the position of every observatory to centimetres, and the speed of light is defined. In that sense nothing is open about it. What has changed is where it sits. For a meridian circle it was a correction applied to a measurement. For an interferometer or a satellite-ranging network it is a term inside the model that is fitted to the measurements, alongside the positions of the stations, the orientation of the Earth and the length of the day.

That raises a question that has no single answer. When the rotation, the station positions and the source positions are all solved for at once, a small error in the model of the rotation becomes partly an error in the source positions and partly an error in the stations, and how it divides depends on the geometry of the network. The celestial reference frame built from radio sources and the terrestrial frame built from stations are tied together through exactly these velocity terms. Whether the two frames, now each realised to tens of microarcseconds, can be held consistent at the next factor of ten — when a network’s own motion through the sky is a term to be estimated, not a correction to be subtracted — is a question about how well the observatory knows where it is going, and it is the third of an arcsecond, at a millionth of its size, that asks it.