A baseline that grows four astronomical units a year

A trigonometric parallax is measured against a baseline of two astronomical units, the diameter of the Earth’s orbit, and the baseline is the whole limitation of the method. Watching a star for longer averages down the noise in the measurement but never lengthens the triangle: after ten years the Earth is back where it started, ten times over.

The Sun is not where it started. It moves through the stars around it at about 19.5 kilometres a second, towards a point in the constellation Hercules called the solar apex, and that motion carries the whole solar system 4.1 astronomical units a year. After a quarter of a century the Sun is a hundred astronomical units from where it was, and the nearby stars, seen from the new position, have shifted against the distant background by fifty times their annual parallax.

That shift is secular parallax, and it has a catch large enough to explain why it is not the method every distance is quoted from. The stars move too. A nearby star has its own velocity relative to the mean of its neighbours, typically twenty to forty kilometres a second, which is as large as the Sun’s — so in any one star’s proper motion the part caused by the Sun’s travel and the part caused by the star’s own cannot be told apart. The long baseline gives a distance only to a group of stars whose own motions can be assumed to cancel, and only on average.

The longest baseline available #

The arithmetic is short. A velocity of one astronomical unit a year is 4.74 kilometres a second, so the Sun’s 19.5 kilometres a second is 4.11 astronomical units a year. A star at distance $d$ parsecs, seen at an angle $\lambda$ from the apex, appears to drift away from the apex at

$$\mu_{\odot} = \frac{4.11\,\sin\lambda}{d}\ \text{arcseconds per year},$$

the same expression as an annual parallax with the baseline replaced by the distance the Sun has travelled. The factor $\sin\lambda$ is geometry: a star directly ahead or directly behind shows no sideways drift at all, and a star at right angles to the motion shows the full amount.

Two things about this are better than an annual parallax. The baseline grows without limit, so the angle to be measured grows in proportion to the time span, and an old catalogue position compared with a new one gives a drift of decades. And the signal is a proper motion rather than a small ellipse to be separated from one, which the astrometric solution shows is the better-conditioned of the two measurements: proper motions improve as the three-halves power of the observing span, parallaxes only as its square root.

What is worse is everything about the stars.

A reflex that only a crowd can show #

The opening figure is the problem drawn honestly. Each star’s motion across the sky is split into two components: $\upsilon$, along the great circle through the apex, and $\tau$, at right angles to it. The Sun’s travel contributes to $\upsilon$ alone, as a drift away from the apex of $V_\odot\sin\lambda$ divided by the distance. Each star’s own random velocity contributes equally to both.

For a group at three hundred parsecs with twenty-five kilometres a second of random motion per star, the random part of one star’s proper motion is larger than the whole of the Sun’s reflex. Nothing about an individual star says which of its motion came from where.

A group can do what a star cannot. If the stars’ own motions are random — no preferred direction, no correlation with position on the sky — then averaging the $\upsilon$ components over many stars at similar distances leaves only the Sun’s reflex. Fitting the sine curve to two hundred stars gives its amplitude to about twelve per cent, and the amplitude is the Sun’s speed divided by the group’s mean distance. The Sun’s speed is known from radial velocities, which measure it in kilometres a second without any distance at all, so the mean distance follows.

The precision falls as one over the square root of the number of stars, exactly as an average of independent errors should, and the floor it falls towards is set by the ratio of the stars’ random speed to the Sun’s. A group whose stars move at twice the Sun’s speed needs four times as many members for the same answer. Measurement error hardly enters: the proper motions in this figure are measured perfectly, and the twelve per cent is entirely the stars’ own motion.

The other half of the proper motion #

The $\tau$ components carry no reflex, and that turns out to be useful rather than wasteful. Their spread is the stars’ random motion, measured as an angle per year. The spread of the same stars’ radial velocities is the same random motion measured in kilometres a second by the Doppler shift. If the random motions have the same size in every direction, the two spreads are one quantity in two units, and the conversion factor between them is the distance.

This is statistical parallax, and it is a genuinely independent measurement. Secular parallax compares a mean proper motion with the Sun’s speed; statistical parallax compares a spread of proper motions with a spread of radial velocities, and the Sun’s motion enters only as a correction removed from the radial velocities first. For two hundred stars the two spreads are each known to about five per cent and their ratio to about seven, so the distance from this route is somewhat better than the one from the reflex. A modern treatment fits both at once, with the group’s mean velocity, its velocity ellipsoid and its mean absolute magnitude as parameters of one likelihood, and that combined fit is what the name usually means in practice.

The comparison with the moving-cluster method is worth making because the two are mirror images. A star cluster’s members share one velocity, so their proper motions converge on a point and the geometry gives each member’s distance. Statistical parallax needs the opposite: stars whose velocities are as different as possible, and isotropically so. The cluster method uses a shared motion; this one uses a shared disorder.

The Sun’s speed is measured the same way #

Both routes need the Sun’s motion in kilometres a second, and it is measured from the same stars with the roles reversed. Radial velocities need no distance: a star ahead of the Sun approaches it, on average, and a star behind recedes, so the mean radial velocity of a sample spread over the sky follows $-V_\odot\cos\lambda$. Fitting that cosine to a few hundred stars gives the Sun’s speed and the direction of the apex together, in kilometres a second, before any distance is known.

That ordering is what makes the method work. The direction of the apex can be found from proper motions alone, as Herschel found it, because the drift away from the apex is visible in angle whatever the distances. Its speed needs the Doppler shift. The reflex in proper motion then needs only one more number, the distance, and that is the number it returns.

The speed depends on which stars it is measured against, and that is not a nuisance but a measurement. Against young, hot stars the Sun moves at about 15 kilometres a second; against old, cool dwarfs at more than 20, because older populations have been stirred to larger random motions and lag further behind the Galaxy’s rotation. The figure of 19.5 is the value against the mixture of stars in a typical catalogue, and a secular parallax must use the Sun’s speed relative to the group it is applied to — which is why a careful analysis fits the group’s own mean motion rather than borrowing a standard one.

What each assumption costs #

Both distances rest on assumptions about the stars that no amount of data removes.

The motions are random. Secular parallax assumes the group has no net motion of its own relative to the Sun’s reference. A group that shares a stream motion — a moving group, a population that lags the Galaxy’s rotation — has its mean velocity added to the reflex, and the distance comes out wrong by the ratio. The cure is to solve for the group’s mean motion as well, which uses the radial velocities and costs precision.

The motions are isotropic. Statistical parallax assumes the random velocities have the same spread in every direction. Disc stars do not: their velocity ellipsoid is elongated towards the Galactic centre, with a radial spread roughly twice the vertical one, so a group seen in one part of the sky has a larger $\tau$ spread than a group seen in another. A sample spread over the whole sky averages over the ellipsoid; a sample concentrated in one direction inherits its shape as a distance error. Modern fits solve for the ellipsoid’s axes, which is possible because different parts of the sky see it from different angles.

The group is at one distance, or the spread is known. Every star in these figures is at the same distance. A real sample has a spread, and both methods return a weighted mean whose weighting depends on how the sample was chosen — the problem the distance-from-inverted-parallax argument found for individual stars, appearing here for groups. A sample chosen by apparent brightness is brighter than its population, and the correction for that enters the absolute magnitude derived from the mean distance.

These are the reasons the methods are used for calibration — of a class of stars, assumed to share a luminosity — and not for the distance of any one object. The question asked of a statistical parallax is never “how far is this star?” but “how luminous is this kind of star?”, and the answer is an absolute magnitude for the class.

A century of distances from the Sun’s motion #

The solar apex was found before any stellar parallax was measured. William Herschel in 1783 took the proper motions of seven bright stars, noticed that they tended to diverge from one region of the sky and converge on the opposite one, and placed the apex in Hercules — within a few tens of degrees of the modern position, from seven stars.

Through the following century the reflex became the main source of distances beyond the reach of the annual parallax. Kapteyn’s model of the Galaxy, at the start of the twentieth century, was built on the mean distances of stars grouped by apparent magnitude and proper motion, obtained this way, because the annual parallaxes of the time reached no further than a few tens of parsecs with any reliability.

The most consequential use was in 1913, when Hertzsprung applied a statistical parallax to thirteen Cepheid variables. None of them was close enough for an annual parallax, and Leavitt’s relation between their periods and their apparent brightness in the Small Magellanic Cloud was a relation without a zero point. The statistical parallax supplied one, and with it the first distance to another galaxy. It was several times too small, partly because the extinction by interstellar dust was not yet known and partly through the small sample — the calibration of the relation was argued about for most of the following century.

The RR Lyrae variables of the Galactic halo were the last major case. Statistical parallaxes of halo RR Lyrae through the 1990s consistently gave absolute magnitudes near +0.7 to +0.8, a fifth to a quarter of a magnitude fainter than several other calibrations, and the difference — about ten per cent in every distance built on the stars, including the ages of the oldest clusters — was argued over for two decades. Direct parallaxes have since settled most of it, and the comparison is the clearest available test of what the method’s assumptions about isotropy and sample selection cost in practice.

Where the Earth’s baseline took over #

The two methods fail differently, and the figure draws where each is better. The annual parallax of a group averages the measurement errors of its members, so its fractional error grows in proportion to distance: the angle falls and the error does not. The secular parallax’s error is almost independent of distance, because the reflex angle and the stars’ random motions shrink together; it has a floor set by the ratio of random to solar speed, and rises only when the proper-motion measurement error, multiplied by the distance, becomes comparable to the stars’ own motions.

With ground-based astrometry — parallaxes to about eight milliarcseconds, proper motions to about three milliarcseconds a year from long plate baselines — the two errors crossed at about two hundred parsecs. Beyond that, for a hundred stars, the Sun’s baseline gave the better mean distance, and that is why the period-luminosity relation, the RR Lyrae calibration and much of the Galaxy’s structure rested on it.

Hipparcos moved the crossing out by nearly a factor of ten, to about a kiloparsec and a half, which already covered the brightest calibrators. Gaia, at a few hundredths of a milliarcsecond for bright stars, moved it to fifty-five kiloparsecs, outside the Galaxy’s disc and past the reach of that precision for faint stars. Within the Milky Way, for any group of stars bright enough for Gaia’s best astrometry, the annual parallaxes averaged now beat the Sun’s motion. The method that set the distance scale for a century is now mainly a cross-check — and a useful one, because its errors come from the stars rather than from the instrument, and the zero-point offset in every annual parallax does not enter it at all.

A faster Sun for the halo #

The reflex depends on how fast the Sun moves relative to the group, and for one population that is much faster than 19.5 kilometres a second. The stars of the Galactic halo do not share the disc’s rotation, so relative to them the Sun moves at roughly the disc’s circular speed, over two hundred kilometres a second.

The random motions of halo stars are larger too, a hundred kilometres a second or more, but the ratio of random to solar speed — the quantity that sets the secular floor — is smaller than for disc stars. That is why the method was applied to halo RR Lyrae and why it worked as well as it did: the reflex is large and clean against a population with no rotation of its own. It is also why the halo’s own small net rotation matters so much: any mean motion the halo has relative to an assumed non-rotating frame goes straight into the distance scale.

A baseline pointed at galaxies #

The Sun has a larger motion still: 370 kilometres a second relative to the cosmic microwave background, measured from the dipole that motion imprints on the background’s temperature. That is 78 astronomical units a year, and it should make every galaxy in the sky drift away from the apex of that motion by 78 microarcseconds a year divided by its distance in megaparsecs.

The arithmetic is the same as for the stars with every number scaled. Galaxies have their own peculiar velocities of a few hundred kilometres a second, comparable to the reflex, so only averages over many galaxies give a distance. The signal at a hundred megaparsecs is under a microarcsecond a year, and a galaxy is an extended, irregular object whose centre is far harder to locate than a star’s. The figure’s curves are for proper-motion precisions of a hundred, ten and one microarcseconds a year per galaxy, over a hundred thousand galaxies in a shell: only the best of them reaches a hundred megaparsecs with a useful error.

What makes the idea worth drawing is what it would measure. A distance from the Sun’s motion against the background is geometric, like any parallax, and it does not pass through the chain of calibrations whose steps disagree about the expansion rate. It would be a distance to the Hubble flow with no Cepheid and no supernova in it.

Where the model stops #

Gaussian, isotropic motions. Every star in these figures draws its random velocity from one round distribution. The real disc’s velocity distribution is elongated, has a vertex deviation, and is lumpy with moving groups, and every one of those shapes leaks into one or other distance.

One distance per group. The samples sit at a single distance. Real groups have a spread, and the method returns a mean whose weighting depends on how the stars were chosen.

A perfect apex. The Sun’s speed and direction are treated as known. They are measured from the same kind of data, and their errors enter every secular parallax in proportion.

No Galactic rotation. At a few hundred parsecs the differential rotation of the disc adds a pattern of proper motions — Oort’s constants — that varies with Galactic longitude, and a real analysis fits it alongside the reflex.

The same construction, three times #

Secular parallax, statistical parallax and the moving-cluster method are one idea applied to three assumptions about how a group of stars moves: that the group’s mean is at rest, that its spread is the same in every direction, or that its members share one velocity. Each turns a proper motion, which is an angle per year, into a speed by appeal to something measured in kilometres a second, and the ratio is a distance.

When a measurement gives an angle per unit time and a speed is known independently, the ratio is a distance, and the whole art is in arranging for the speed to be known. The Earth’s orbit supplies one for a single star by moving the observer a known amount. The Sun’s motion, and the stars’ own disorder, supply it only for a crowd — and for a century the crowd was further away than any single star could be measured.

Still open: a distance to the expansion with no standard candle in it #

The secular parallax of galaxies has not been measured. Current astrometric catalogues contain the positions of millions of galaxies, but their precision for extended sources is far worse than for stars, and estimates suggest that a detection with present data would be marginal at best. Whether a future astrometric mission, or a radio array measuring compact galactic nuclei over decades, can reach the microarcsecond-a-year precision the figure asks for is open — and if one can, it would give the expansion rate a geometric distance that no step of the present chain of calibrations touches.