A frame made of things that are not points

A position on the sky is a direction, and a direction is meaningless without something to measure it from. Astronomy’s answer has always been the same in form and different in substance: pick a set of objects, declare them fixed, and refer everything else to them.

The substance has changed three times. It was the fixed stars, until the fixed stars were found to move. It was the Earth’s own rotation axis and equinox, until those were found to precess, nutate and wobble. Since 1998 it has been a list of extragalactic radio sources, chosen because a quasar at a redshift of one has a proper motion of order a microarcsecond a century — smaller than anything measurable.

The difficulty is that a quasar is not a point.

The difficulty matters because of what the frame is for. Every proper motion in every stellar catalogue, every spacecraft navigation solution that uses an angle, every measurement of the Earth’s orientation, and every parallax is expressed in this frame. A systematic in it propagates into all of them at once, and it propagates as a pattern rather than as a scatter — which means it does not average away and cannot be found by observing more stars.

What the frame is made of #

The measurements are very long baseline interferometry. Two or more radio telescopes thousands of kilometres apart record the same source with hydrogen-maser timing, and the correlation of their signals gives the delay between the arrival of a wavefront at one and at the other. That delay is a projection of the baseline vector onto the direction of the source, so a night of observations of many sources on many baselines gives the sources’ directions and the baselines simultaneously.

The technique is the same one used to measure a spacecraft’s angular position against a quasar, run the other way round: there the spacecraft is unknown and the quasar is the reference, and here the quasars are what is being determined. Both are differential measurements over a baseline, and both derive their precision from the fact that almost everything which could go wrong — the clocks, the troposphere, the ionosphere — affects the two directions nearly equally and cancels in the difference.

The precision is extraordinary. A delay measured to a few picoseconds across a baseline of six thousand kilometres is an angle of a few tens of microarcseconds, and averaging thousands of observations of one source over decades reduces its formal error below a hundred.

What that precision immediately runs into is that the source has structure at exactly the same scale. A typical frame source is a compact core with a one-sided jet, and both components are resolved by the same baselines that measure the position. The “position” being measured is a weighted centroid, and the weighting depends on the frequency, the baseline, and what the source happened to be doing that year.

The size of the structure effect can be stated concretely. A core–jet source with a component ten milliarcseconds from the core, carrying a fifth of the flux, has a centroid displaced two milliarcseconds from the core — vastly larger than the formal error. What saves the frame is that the interferometry does not measure a centroid so much as fit a model, and a well-observed source can have its structure imaged and its core located within it. What is left after that is the motion of the core itself, which is smaller and is the floor in the first figure.

The floor also depends on the frequency, and it depends on it in a direction that is useful. A jet’s core is optically thick out to a distance that falls with frequency, so the apparent core position moves inward as the observing frequency rises — an effect called the core shift, of order a few hundred microarcseconds between two and eight gigahertz. It is a systematic that is measurable precisely because it has a known frequency dependence, and the current frames are built at multiple frequencies partly to constrain it.

It is worth noting how few sources actually define the frame. The current catalogue contains several thousand positions, of which a few hundred are defining — selected for long observing histories, compact structure and good sky distribution — and the rest are carried along. That asymmetry is deliberate: adding a poorly behaved source to the defining set degrades the frame for everybody, while adding it to the catalogue costs nothing. The distinction between a source whose position is measured in the frame and a source that helps define the frame is the same distinction as between a star observed by a survey and a star used to calibrate it, and it is the same discipline that separates a standard from a target.

Three numbers, and the whole difference between two catalogues #

When a frame is rebuilt — more sources, more observations, a better model of the troposphere — the new catalogue differs from the old one. The interesting question is what kind of difference it is.

To an excellent approximation it is a rotation. Three numbers describe the entire systematic difference between two realisations of a frame, and everything else is per-source noise.

The pattern is recognisable because a rotation is a very particular vector field on a sphere and almost nothing else looks like it. That is what allows a frame built at radio wavelengths to be tied to one built at optical wavelengths: observe a set of objects in both, fit for the three rotation angles, and apply them.

It is also why an unrecognised rotation is so dangerous. A frame rotating at some small rate relative to an inertial one imparts an apparent proper motion to every object in the sky, of the same magnitude and in a circulating pattern. That is exactly what a real, physical effect would look like if the frame’s axis happened to point somewhere interesting, and telling the two apart requires a physical argument rather than a statistical one.

And there is a fourth number in principle and it is deliberately not fitted: a global scale. A rotation preserves angles between sources and so is a genuine coordinate freedom; nothing else about a catalogue of directions is free, because a direction has no length. That is a useful discipline, because it means any difference between two catalogues that is not a rotation is a difference in the measurements rather than in the conventions, and there is nowhere to hide it.

The three-number description also has a limit that is worth stating, because it is the thing a careful comparison actually looks for. Two frames can differ by more than a rotation: a glide, which is the dipole pattern the secular aberration produces, and higher-order distortions described by the next terms in a vector spherical harmonic expansion. Modern frame comparisons fit rotation and glide together and then examine the residual harmonics, and a significant quadrupole term would mean something in the modelling is wrong — a zonal error in the troposphere model, say, or a declination-dependent systematic from the northern preponderance of radio telescopes. The expansion is the diagnostic, and the fact that the first two terms account for nearly all of the difference between successive catalogues is the evidence that the frame is well behaved rather than merely precise. It is the same manoeuvre as decomposing a gravity field into harmonics and asking which degrees carry the signal.

What was actually measured #

Three results anchor the argument, and each is a measurement of something the frame was supposed to make impossible.

The secular aberration drift. The solar system accelerates towards the Galactic centre, so the aberration of starlight changes slowly, and every source on the sky acquires an apparent proper motion of about five microarcseconds a year pointing at the Galactic centre. That is a dipole rather than a rotation, so it is a different vector field and separable from one. It has been detected in the radio frame and independently in the optical, and it is the largest known systematic in either — an entire sky drifting towards one point, which is the aberration essay’s subject seen from the frame’s side.

The frame tie between radio and optical. The optical astrometry mission that measured a billion stars had no direct connection to the radio frame, because it observes almost none of the radio sources — quasars are faint at optical wavelengths and the ones bright enough to observe are not the ones with clean radio structure. The tie was made through a few thousand sources observable in both, and its uncertainty, tens of microarcseconds in orientation and a few microarcseconds a year in spin, is a systematic on every optical proper motion in the catalogue.

Source structure, measured directly. Imaging the frame sources with the same interferometry that measures their positions shows the jets, and the position offsets predicted from the images correlate with the position residuals. The sources with the worst structure are excluded from the defining set, which is a selection rather than a correction — it improves the frame by discarding information rather than by using it.

The three numbers of the previous section are not equally hard to measure, and the largest of them is the one with a known cause. A uniform acceleration of the observer produces a pattern with a shape nothing else has, which is what makes it separable at all.

Where the picture stops #

There are three, and the first is the awkward one.

The frame is not inertial and cannot be shown to be. Its defining property is that the sources have no net rotation, and that is imposed by construction — the fit is constrained so that no global rotation is allowed. Whether the resulting frame rotates relative to a genuinely inertial one is not answerable with the sources themselves, because they are the only reference available. What can be checked is consistency: a frame built from one half of the sources should not rotate relative to one built from the other half, and it does not, at a few microarcseconds a year.

The optical and radio frames are measuring different physical things. The radio position is the base of a jet, whose location depends on the frequency because the core becomes optically thick at different distances from the black hole. The optical position is the accretion disc and the host galaxy’s nucleus. These can differ by hundreds of microarcseconds, systematically, in the direction of the jet, and the offset varies with the source’s activity.

And the whole structure rests on there being nothing further away. The frame is fixed because its objects are distant. If a source turns out to be closer than assumed — a Galactic object misclassified, or a gravitationally lensed image — it moves, and it moves in a way that is indistinguishable from a frame error until enough time has passed.

A fourth limit is worth recording because it is a practical one that repeatedly bites. The frame is a catalogue of positions with uncertainties, and it is used by software that mostly treats those positions as exact. A user who takes a frame source’s coordinates, propagates them with zero proper motion, and compares them against a measurement is implicitly claiming a precision the catalogue does not have, and the resulting residual is attributed to whatever is being measured. That is the same failure as treating a catalogue position as a fact rather than as a measurement, and it is the most common way a frame’s systematics leak into somebody else’s result.

Why the fixed points must be chosen rather than found #

The pattern here is one this collection meets whenever a measurement is relative rather than absolute, which is most of them.

There is no absolute position and no absolute direction. Every astrometric quantity ever published is a difference between one object and a set of others, and the set is chosen. What makes a good choice is not that the objects are truly fixed — nothing is — but that their motions are below the precision of the measurement being made, which is a statement about the instrument as much as about the objects.

That is why the reference frame has been rebuilt three times and will be again. Each rebuild happened when the instruments got good enough to see the previous set moving: the fixed stars moved when proper motions were measured, the equinox moved when nutation was modelled, and the quasars move now that microarcsecond interferometry exists. The next rebuild will be prompted by whatever is measurable when a hundred nanoarcseconds becomes routine.

There is a second observation, and it is the more useful one for a reader trying to judge a published astrometric result. The right question about a proper motion is never “how precise is it” but “relative to what”. A proper motion measured relative to background galaxies in the same image is nearly immune to frame errors and is limited by the galaxies’ own structure. One measured relative to the global frame carries the frame’s rotation uncertainty. One measured relative to nearby stars carries their motion. The three can differ by amounts that dwarf their formal errors, and papers that quote one and compare against another are common.

The optimistic reading is that each generation of frame is a measurement of the previous one’s assumptions. Where a star is depends on who is asking, and the answer has been refined by discovering, four times now, that the things being asked about were moving.

One closing remark on the phrase that gives the frame its name. It is called a quasi-inertial frame, and the qualifier is doing real work. An inertial frame is one in which a free particle moves in a straight line, which is a dynamical statement testable with dynamics. This frame is defined kinematically — by declaring a set of directions to be fixed — and the two definitions agree only to the extent that the objects chosen really are unaccelerated. The secular aberration drift is precisely the discovery that they are not, or rather that the observer is, and it is the one case where the kinematic and dynamical definitions have been compared and found to differ by a measurable amount. That the difference was predicted before it was measured, and comes out at the value the Galaxy’s rotation requires, is the strongest evidence available that the frame means what it is supposed to mean.

There is one more property of the frame that is worth holding onto, because it is what makes the whole construction defensible. A reference frame is not a claim that the objects in it are motionless; it is a definition, and the definition is checked by its own internal consistency rather than against anything external. What can be measured is whether the sources move relative to each other — whether the frame rotates, deforms, or drifts — and those are the quantities the analysis reports. An overall translation of the whole set is unobservable and uninteresting, because nothing is being measured against anything outside. So the frame’s quality is a statement about the residual scatter of a few hundred objects with respect to a fitted rigid body, and the individual sources’ wanderings enter only through how much of that scatter they contribute.

It is worth naming the trade that decides how many sources a frame should contain. A larger set averages down each source’s individual wandering and necessarily includes fainter, less well-observed and more variable objects, whose wandering is larger. So the frame’s accuracy is not monotonic in its size, and the optimum is a few hundred carefully vetted sources rather than the several thousand available — which is the choice the current catalogues make, and which is unusual in a field that otherwise takes every object it can get.

Where the ladder goes next #

The natural next rung is the tie itself: how two catalogues sharing a few thousand objects are brought onto one system, and why the residual after the rotation is fitted is the quantity that matters. Above it again sits the use the frame is put to — an angle measured against a quasar, which is how a spacecraft’s position is fixed to the same sky the stars are on.