The position closure throws away

A phase that survives what corrupts it makes a point in passing and then leaves it: the closure phase’s immunity and its blindness are the same fact. A quantity that cannot be spoiled by a per-antenna error cannot see anything that resembles one, and shifting the whole source on the sky resembles one exactly — because a translation multiplies each visibility by a phase linear in the baseline vector, and the phases it adds sum to zero around any triangle.

So an image built from closure quantities has a structure and no position. It floats.

Getting the position back means abandoning the trick rather than improving it.

What the reference is doing #

The scheme is simple to state. Observe the target for a minute or two, swing the whole array to a compact source of known position a degree or so away, observe it briefly, swing back. Repeat for hours.

The calibrator is a point source at a known place, so its measured visibility phase is entirely instrumental and atmospheric. Interpolating that phase across the gaps and subtracting it from the target’s leaves the target’s own phase — including the part that says where it is.

Three things have to hold for this to work, and each of them is a design constraint rather than a detail.

The calibrator must be close enough that the atmosphere above it resembles the atmosphere above the target. What resembles means is the structure function, and that is the hero figure.

The switch must be fast enough that the atmosphere has not changed between one visit and the next. Above a certain cadence the interpolation is valid; below it, it is an extrapolation.

And the calibrator’s position must be known in an absolute frame, which pushes the problem one step back: the whole chain rests on a catalogue of distant quasars whose positions are defined rather than measured, and the reference frame is a human construction with a definition that has been revised four times — the same difficulty in kind as a magnitude that has to say which light it means, where the scale is defined by a set of objects rather than by a unit.

Why the exponent is five sixths #

The atmosphere’s effect on a phase is described by its structure function: the mean squared difference in phase between two points separated by a distance $d$. For turbulence with a Kolmogorov spectrum this is

$$D_\phi(d) = 6.88\left(\frac{d}{r_0}\right)^{5/3},$$

where $r_0$ is the coherence length — the separation at which the difference reaches about a radian.

Two separations enter a referenced observation. Looking at a calibrator an angle $\theta$ away pierces the turbulent layer at a horizontal distance $h\theta$ from the target’s own line of sight, where $h$ is the layer’s height. And visiting it every $\tau$ seconds means the layer has been blown past by a distance $v\tau/2$ between measurements, where $v$ is the wind aloft.

Both are distances, both enter the same structure function, and the residual phase after referencing is the square root of their sum. So the phase residual grows as $d^{5/6}$, and the astrometric error is the phase residual converted through $\lambda/2\pi B$.

The five-sixths is gentler than one, which is the single piece of good news: doubling the calibrator separation costs a factor of 1.8 rather than 2. It is also steeper than a half, so there is no regime in which the separation stops mattering.

The optimum in the cadence #

Switching faster always reduces the temporal term, and it is not therefore always better.

Every swing of the array costs time — tens of seconds of slewing and settling, during which nothing is recorded — and every visit to the calibrator is time not spent on the target. At a cadence of ten seconds a substantial fraction of the observation is spent pointing at something nobody is interested in, and the target’s signal-to-noise falls in proportion.

So the cadence is chosen where the marginal reduction in the systematic equals the marginal increase in the statistical error, which for a bright target is fast and for a faint one is slow. The right switching rate depends on the target’s brightness, which is an unusual thing for a calibration procedure to depend on.

There is a way round part of it, and it is the reason modern arrays do better than the scaling suggests. If a second receiver can observe the calibrator simultaneously — either because the array has enough beams, or because the calibrator is close enough to fall in the same primary beam, or because the same antennas are split — the temporal term vanishes entirely and only the spatial one remains. In-beam referencing is the best case available and it requires a calibrator within a few arcminutes, which is a matter of luck.

How far a parallax reaches by this route #

It is worth pricing the capability against the alternative, because the two instruments that measure parallaxes do not overlap.

A parallax is an annual wobble of amplitude $1/d$ arcseconds at a distance of $d$ parsecs. An astrometric precision of ten microarcseconds per epoch therefore detects a parallax at ten per cent for a source at ten kiloparsecs, and the technique has been demonstrated at that level.

That reach is comparable with the best optical astrometry and it applies to a completely different set of objects. Optical astrometry works on stars — a billion of them — and fails wherever the line of sight is dusty, which is exactly the plane of the Galaxy where the interesting structure is. Dust makes everything look further away, and toward the Galactic centre it removes thirty. Radio wavelengths are untouched by it.

So the two techniques are complementary in the useful sense: one measures a vast number of stars in the transparent directions and the other measures a few hundred objects in the opaque ones. The few hundred are masers in star-forming regions, which sit in the spiral arms, which is why a technique that measures so few objects has had so large an effect on the picture of the Galaxy’s shape.

The comparison also shows what the referencing buys. A single-dish radio telescope has a beam of arcminutes and could not measure a parallax at all. The gain from an array is the ratio of a baseline to a dish, ten thousand kilometres to a hundred metres — five orders of magnitude — and the gain from referencing is the difference between a position that floats and one that does not. Neither is optional and neither is sufficient alone.

What it has been used for #

Microarcsecond astrometry is a narrow capability with a few results that nothing else could have produced.

Parallaxes across the Galaxy. The trigonometric parallax is the only distance with no assumption in it, and referenced interferometry measures it for radio-bright objects — masers in star-forming regions, pulsars, X-ray binaries — at distances of kiloparsecs. A ten-microarcsecond parallax is a distance of a hundred kiloparsecs in principle; in practice the technique has mapped the spiral arms of the Milky Way directly, by measuring distances to the masers in them rather than inferring them from a rotation model.

Proper motions of galaxies. The transverse velocities of the Local Group’s members were unmeasurable until the positions of compact sources in them could be followed for a decade. That measurement is what turned the Andromeda collision from a radial-velocity extrapolation into a trajectory with a tangential component in it.

And the reference frame itself. The catalogue of quasar positions that defines the celestial coordinate system is built this way, and every optical astrometric mission is tied to it. The chain runs backwards from the thing this essay is about: the frame is defined by sources whose positions were measured by referencing them to each other, and the internal consistency of that is the frame’s accuracy — which is why the distance is not one over the parallax has a counterpart here: a position is not one measurement but a solution in which everything is fitted at once.

What was actually measured #

A phase difference between two directions on the sky, measured on each baseline, accumulated over hours, and converted into an angular offset through the baseline’s length in wavelengths.

Four systematics dominate, and none is the one the hero figure draws.

The troposphere’s dry component. Most of the delay through the atmosphere is not turbulent and not wet; it is the bulk refractivity of dry air, which depends on the pressure at each site and on the elevation of the observation. It is modelled rather than measured, and an error in the model is an error in the position that varies through the night as the source rises.

The ionosphere. At centimetre wavelengths the ionospheric delay is comparable with the tropospheric one and it scales as the inverse square of the frequency, so it can be removed by observing at two frequencies. That is standard and it is not free — it halves the bandwidth at each.

The calibrator’s own structure. A quasar is assumed to be a point at a fixed position and is neither: its radio emission is a core and a jet, the jet brightens and fades, and the centroid moves. Milliarcsecond-level wander in the reference is a microarcsecond-level error in everything referenced to it, and it is the limiting systematic for the longest programmes.

And the station positions. Converting a phase into an angle requires knowing the baseline vector to a fraction of a wavelength, which for a millimetre array over intercontinental baselines is a fraction of a millimetre out of ten thousand kilometres — and the Earth is not rigid, so the baseline is a function of the tides, the loading of the oceans and the motion of the plates. Those positions are themselves measured by the same technique, on the same sources, in the same solution — so the geometry and the astronomy are solved together and neither is prior to the other.

The two ways of using a reference, and when each wins #

There is a third scheme between the two this essay has set against each other, and naming it makes the trade legible.

Closure alone is immune and blind. It needs no reference, works at any wavelength on any target bright enough, and produces a structure floating on the sky.

Phase referencing recovers the position and imports the structure function. It needs a compact calibrator nearby and the ability to switch faster than the atmosphere changes.

And relative astrometry within one field is the best of both, when it is available. If two sources lie inside the same primary beam — a maser and a background quasar, or two components of one system — the atmosphere above them is very nearly identical and the separation between them is measured with almost no residual at all. No switching is required, the spatial term is tiny, and the precision reaches the thermal limit.

What that scheme cannot give is an absolute position, only a separation. Which is enough for a parallax, since a parallax is a changing separation, and enough for an orbit, since an orbit is a changing separation too — and not enough for a reference frame.

So the quantity being measured decides which scheme is needed, rather than the precision required. A programme measuring how something moves relative to a neighbour can have the atmosphere almost for free; a programme measuring where something is cannot, and the difference between the two is not accuracy but kind.

Where the model stops #

The turbulence is not one layer. The model behind the hero figure puts all of the disturbance at one height with one wind, and a real atmosphere has several layers moving at different speeds in different directions. The five-thirds exponent survives that; the coefficient does not, and it varies between sites by more than an order of magnitude and between nights at one site by a factor of a few.

The structure function saturates. Kolmogorov turbulence has an outer scale, beyond which the phase differences stop growing. For separations above that scale the error flattens rather than continuing to climb, which is a mercy and is not in the drawn curves — and where the outer scale is, for the wet component above a particular site, is not well measured.

Elevation is missing. Every quantity here is quoted at the zenith. An observation at forty degrees elevation looks through 1.6 times as much atmosphere, and the target and calibrator are at slightly different elevations, which introduces a systematic that does not average away because it has the same sign all night.

And the calibrator is treated as a point. It is not, and its structure is the systematic that outlasts all the others.

What a microarcsecond is #

The unit is worth making concrete, because the numbers in this essay are otherwise abstract.

One microarcsecond is the angle subtended by a human hair at a distance of ten thousand kilometres, or by a coin on the Moon. At the distance of the Galactic centre it is one and a half astronomical units — so an astrometric precision of ten microarcseconds locates a star near the centre of the Galaxy to within about fifteen astronomical units, and following one over a few years measures its orbit.

The corresponding path-length difference is what the instrument has to hold. An angle on a baseline corresponds to a delay equal to their product, which for ten thousand kilometres and ten microarcseconds is half a millimetre — and since the measurement is a phase, the fringe that difference is counted against is a wavelength, a millimetre or a centimetre. So the measurement is a fraction of a fringe on a baseline that changes by metres over a day as the Earth turns and deforms.

Everything in the systematics section is about that reconciliation: a geometric model of the Earth good to millimetres, an atmospheric model good to a few, and a reference whose own position is defined rather than measured. The precision is not achieved by measuring anything finely; it is achieved by differencing two things that share almost everything, which is what the whole of this is about and is why the residual is a structure function rather than an instrumental error.

The generalisation #

The shape to carry away is about what a differential measurement buys and what it necessarily spends.

The closure identity established a principle: when a measurement is corrupted by an unknown, look for a combination the unknown cannot enter. Closure phase is the clean case. What this adds is the other half of the same sentence — the combination that the unknown cannot enter is a combination that cannot see whatever resembles the unknown, and the experimenter has to decide in advance which quantity is worth losing.

That decision is not a technicality and it recurs everywhere. A colour index removes an unknown throughput and cannot measure a flux. A flux ratio removes an unknown distance and cannot measure a distance. A closure phase removes an atmosphere and cannot measure a position. In each case the quantity discarded is the one that transforms the same way as the nuisance, and the way to find out what a clever differential technique has given up is to ask what else would have looked like the thing it removed.

The second reading is about the alternative. Phase referencing is the other strategy: rather than cancelling the nuisance, measure it somewhere else and subtract. That always works in principle and its error is always set by how much the nuisance differs between there and here — which is a structure function, and structure functions rise. A differential correction is only as good as the correlation between the two measurements, so the whole design problem becomes finding a reference that is close in whatever variable the nuisance varies fastest in. Here that variable is angle on the sky; elsewhere it is time, temperature, or position on a detector, and the arithmetic is identical.

Still open: what a position is measured against #

What comes next is the frame. Every position in this essay is a position relative to a calibrator, and the calibrator’s own position is relative to a catalogue, and the catalogue is defined by a set of sources chosen to be distant enough to have no measurable motion. What happens when they turn out to have one — when a quasar’s jet moves, or when the whole catalogue rotates with respect to the distant universe — is a question about a coordinate system rather than about an instrument, and the answer has been revised each time the measurements improved.

Beside it lies the pairing these essays have kept separate: an image gives a structure with no position and a referenced measurement gives a position with a poor image, and combining the two is how a source’s motion and its shape are followed together. That is what has been done for the stars orbiting the mass at the centre of the Galaxy, and it is the most demanding application either technique has.