Five numbers from one wiggle

The phrase “measuring a parallax” describes something nobody has ever done. There is no observation whose output is a parallax. What is observed is a sequence of positions of a point of light, and a parallax is one of five numbers fitted to that sequence — jointly, all at once, each correlated with the others.

That distinction sounds like bookkeeping and is not. It is the reason a parallax programme is stated in years rather than in nights, the reason the first stellar parallax was measured a century after the effect that gave it away, and the reason a catalogue quotes an epoch.

The five parameters #

The standard astrometric model of a single star is

$$\alpha^*(t) = \alpha^*_0 + \mu_{\alpha^*}\,t + \varpi\,P_{\alpha^*}(t),\qquad \delta(t) = \delta_0 + \mu_\delta\,t + \varpi\,P_\delta(t),$$

with $\alpha^* = \alpha\cos\delta$ so that both coordinates are angles on the sky rather than one of them being a longitude. Five unknowns: two coordinates at a reference epoch, two components of proper motion, one parallax. The parallax factors $P$ are not unknowns — they are computed from the Earth’s ephemeris and the star’s direction, and are the same for every star in the field of view up to the direction dependence.

That is the entire model, and its shape decides everything that follows. It is linear in all five parameters, so the fit is a least-squares problem with a closed-form solution; and the five basis functions are $1$, $1$, $t$, $t$, and a pair of one-year periodic functions. Whether the parameters can be told apart is the question of whether those basis functions are distinguishable over the span of the observations.

Why the separation works at all #

Two of the five parameters have the same functional form as each other — the two coordinates — and are distinguished only by direction. The interesting pairing is proper motion against parallax.

A proper motion is linear in time. A parallax contribution is periodic with a period of exactly one year and a phase fixed by where the Earth is. Over a full year the periodic term returns to where it started while the linear term does not, and that is what separates them. Over less than a year the periodic term is executing part of one swing, and part of one swing is a shape that a straight line in time can imitate closely.

The consequence is the practical rule that governs every astrometric mission: the parallax accuracy improves as the square root of the number of observations, as any average does, and it improves separately with the time span, because the span is what breaks the correlation with proper motion. Doubling the number of nights within one season buys $\sqrt{2}$. Extending from one year to five buys far more than $\sqrt{5}$, because the first year is buying separability rather than precision.

What else is in the wiggle #

Three effects of the same order have to be removed before the five parameters mean anything, and each was discovered by somebody hunting for parallax and finding it instead.

The third is refraction, which displaces a star towards the zenith by an amount depending on altitude and therefore, at a fixed sidereal time, on the season. The Sun sets before it sets is the same effect at its largest; for a star at moderate altitude it is tens of arcseconds and has an annual component, which is precisely the shape the fit is looking for. It is why the eighteenth-century attempts failed and why the successful measurements of the 1830s all used differential techniques against nearby comparison stars, so that the refraction cancels.

What was actually observed, in 1838 #

Three parallaxes were announced within a year of one another, by three people using three different instruments, and the differences between them are a useful catalogue of what the fit needs.

Bessel measured 61 Cygni with a heliometer — an objective cut in half so that the two halves can be slid against each other, turning an angular separation into a screw reading. He chose the star because its proper motion was the largest then known, 5.2 arcseconds a year, which is a good proxy for nearness — the same reasoning that makes proper motion a distance indicator when nothing better is available. He observed it against two faint neighbours over eighteen months and published $0.314''$; the modern value is $0.286''$, so he was within ten per cent.

Henderson measured $\alpha$ Centauri from the Cape with a meridian circle, which measures absolute positions rather than differential ones and therefore fights refraction directly. He had the data in 1833 and did not reduce it until 1839, by which time Bessel had published. His value was $1.16''$ against a modern $0.75''$ — a fifty per cent error, and the instrument is why.

Struve measured Vega with a filar micrometer and got $0.26''$ against a modern $0.13''$. His error is a factor of two, and Vega’s parallax is small enough that the measurement was at the edge of what the method could do.

The pattern is worth reading. The differential measurements were good and the absolute one was bad, by a factor of five in fractional error, and that is the whole argument of the previous section made by three people who had not yet had it. It took another 150 years to escape the compromise, and the escape was a satellite.

Relative and absolute #

The differential trick that removes refraction introduces the problem that dominated the subject until 1989. Measuring a star’s position relative to faint background stars removes anything common to the field — refraction, plate scale, telescope flexure — and also removes the parallax of the reference stars, which is not zero.

So a ground-based parallax is a relative parallax, and turning it into an absolute one requires knowing how far away the references are, which requires knowing their parallaxes — the same circularity that runs through every rung of the distance ladder. The correction is typically 1 to 2 milliarcseconds and is estimated from a model of the Galaxy rather than measured, which means every ground-based distance carried an error bar with a Galactic model inside it.

What a global solution does instead #

The escape from relative parallaxes is to observe two widely separated directions at once. If a satellite measures the angle between two fields $90^\circ$ apart, and does so repeatedly as it rotates, the parallax factors in the two fields differ in sign and magnitude, and the resulting system of equations has an absolute solution with no zero point to be assumed.

That is what Hipparcos did in 1989 and what Gaia does now, and it converts the problem from a measurement of a star into the simultaneous solution of a system with a billion sources and tens of billions of observations, in which the reference frame, the satellite’s attitude, and the optical calibration are solved for at the same time as the astrometry. The five parameters per star are still five parameters per star; the difference is that the frame is now solved rather than assumed.

The precision that buys is the thing worth quoting. Hipparcos reached about a milliarcsecond, which is 1000 parsecs at ten per cent. Gaia reaches tens of microarcseconds for bright stars, which is tens of kiloparsecs — far enough to cross the Galaxy.

What the fit assumes #

Every model that is fitted is also a hypothesis, and this one has three assumptions in it that are not always true.

It assumes the star is a point. A resolved binary, an unresolved binary, or a star with a bright spot rotating across its disc all have photocentres that move for reasons the model has no term for, and the fit absorbs that motion into whichever parameter it best resembles. An unresolved binary with a period near one year contaminates the parallax; one with a period much longer than the mission contaminates the proper motion. Gaia’s published solutions carry a goodness-of-fit statistic precisely so that these can be flagged, and about a per cent of bright stars fail it.

It assumes the motion is uniform. Over four years that is excellent, and over the decades separating two catalogues it is not: a star in a binary of thirty-year period has a proper motion that differs between epochs, and comparing two catalogues taken fifty years apart yields an apparent acceleration that is real information about a companion.

And it assumes the reference direction is fixed, which brings the whole problem of the frame back. A systematic rotation of the reference frame at $\omega$ appears in every star’s proper motion as a term proportional to $\omega$ and in none of the parallaxes, so a frame that is slowly spinning produces a spurious pattern of proper motions across the sky. Measuring that pattern against quasars, and finding it consistent with zero, is one of the standard validations of a modern catalogue — and it is the sense in which an extragalactic frame is not a convenience but a requirement.

The sixth number, which is not in the fit #

Five parameters describe the star’s motion across the sky. The motion along the line of sight is invisible to astrometry entirely, and comes from somewhere else. There is one place where the two measurements meet, and it is a rare and beautiful check. A star’s radial velocity changes its distance, which changes its proper motion — the perspective acceleration, or secular acceleration. For Barnard’s star, moving at 10.3 arcseconds a year and approaching at 110 km/s, the effect is about 1.2 milliarcseconds per year per year: detectable, and detected. It is one of the very few cases in which a spectroscopic quantity and an astrometric one predict each other, and the agreement is a check on both.

What the ladder has established #

The first rung of this anchor was the triangle and its limit: a geometric distance, honest and short-ranged. This one is about what has to happen before that triangle can be extracted from a sky in which nothing holds still.

The lesson generalises past astrometry. Five quantities are entangled in one observable, and they are recoverable because they have different signatures in time — secular, annual, annual-in-quadrature. That is the same argument that separates a planet’s transit from a star’s variability, and the same argument that separates a rotational line broadening from a gravitational one. What makes a measurement possible is rarely that the signal is large. It is that the signal has a shape nothing else has.

Two properties of a global solution are worth separating before the difficulty below, because they are often run together. One is that the frame is determined rather than assumed, which removes the reference-star correction that limited every ground-based measurement. The other is that the instrument’s own calibration is solved for at the same time, from the data, rather than being measured beforehand and applied. The second is what makes the precision possible and it is also what leaves a residual behind: a calibration solved from the data can only be as good as the data’s ability to constrain it, and there are combinations of instrumental parameters that the observations barely distinguish. Whatever survives in those combinations does not show up as scatter. It shows up as a systematic shared by every source measured the same way, which is exactly the shape of the difficulty described next.

The zero point that is not zero #

A global solution removes the need to assume a reference zero point, and it does not remove the zero point. Modern catalogues carry one, it is small, and it is currently a limiting term in the distance scale.

The way it was found is the cleanest possible test. Quasars are so distant that their parallaxes are zero to any precision anyone can reach — so measuring them is a null experiment, and whatever comes out is the instrument’s error. The measured mean came out not at zero but at a small negative value, of order tens of microarcseconds.

A negative parallax offset means every star is placed slightly further away than it is. The size is a few hundredths of a milliarcsecond, which is negligible for a nearby star and is a substantial fraction of the measured parallax for a distant one — precisely the distant ones that matter for calibrating anything.

The awkward part is that the offset is not a constant. It varies with a source’s magnitude, with its colour and with its position on the sky, because it originates in the instrument’s optics and in how the images are processed, both of which depend on those things. So the correction is a fitted function of three variables, derived from quasars where they are available and from other objects of known distance where they are not, and its own uncertainty is several microarcseconds.

The consequence is a systematic that behaves unlike a measurement error. It does not average down over many stars, because it is common to them; it changes if the correction function is revised; and it enters every distance derived from the catalogue in the same direction. Anyone quoting a result that depends on parallaxes at the ten-microarcsecond level now has to say which correction was applied, and results using different corrections are not directly comparable.

A global solution converts a per-field problem into a per-catalogue one, which is a large improvement and not an elimination — and the null experiment that revealed it is the only reason it is known at all.

The lesson is the one the ground-based measurements learned in a different form: a differential technique removes what is common and leaves what is not, and identifying what is not requires an object whose true value is known in advance.

What one visit actually records #

The five-parameter fit above treats each observation as a position on the sky, two numbers. A scanning satellite does not deliver that, and the difference explains the design of the whole mission.

The instrument measures precisely in one direction only — along the direction the field is sweeping across the detectors — because that is the direction in which the image’s transit time across a pixel column can be timed. Across that direction the measurement is far coarser, by more than an order of magnitude.

So a single visit is essentially a one-dimensional constraint: it says where the star lies along a particular direction on the sky and says almost nothing about where it lies across it. One such measurement cannot determine a position, let alone five parameters.

What supplies the second dimension is revisiting the same star with the scan running a different way. The satellite’s rotation axis is made to precess slowly about the direction of the Sun, so successive passes over a given star cross it at a range of angles, and the accumulated set of one-dimensional constraints intersects at a point.

That is why the scanning law is a designed object rather than a convenience: it has to guarantee that every part of the sky is visited enough times, at a wide enough spread of position angles, and at epochs spread through the year so that the parallax term is sampled at a range of phases. A region visited many times at one angle is measured well in one direction and badly in the other, and a region visited only in one season has its parallax entangled with its proper motion by the argument of this essay’s third section.

The precision of a catalogue is therefore not uniform across the sky, and the pattern of its variation is the pattern of the scanning law rather than anything astronomical.

The separation of the five parameters depends on how much of the path is observed, and it is worth drawing both the path and the degeneracy at a second setting.

Where the ladder goes next #

The rung above is the binary problem: a star with an unseen companion has seven parameters rather than five, and the extra two describe a periodic wobble that is neither secular nor annual. Astrometric detection of exoplanets is the same fit with a longer period in it, and the difficulty is precisely that a period near one year is degenerate with the parallax itself.