A midplane that depends on how far it is averaged

The galactic coordinate system was fixed in 1958 by a pole taken from the radio map of neutral hydrogen and a zero of longitude at the radio source in the Galaxy’s centre, and its equator was drawn through the Sun because the Sun is where every observer of 1958 was. The Galaxy’s actual midplane lies a little below the Sun, so the plane of galactic latitude zero and the plane of the disc are not the same plane. Seen from the Sun, the midplane appears more than a degree below the galactic equator a kiloparsec away towards the centre, and several degrees below it a few hundred parsecs away.

How far below the Sun the midplane lies has been measured dozens of ways. Counting stars above and below the Sun at faint magnitudes, fitting the distribution of young clusters, mapping the gas: the answers scatter between about five and thirty parsecs, and the most careful recent determinations, from the vertical distribution of millions of stars measured by the Gaia satellite, give about twenty. The scatter is usually read as the measurements’ errors. The essay on the coordinate system ended by suggesting another reading: that the disc near the Sun is not flat, and that its midplane is not a single plane at the precision of a few parsecs.

This essay works that reading out. A rippled disc has no single midplane, and a survey that fits one measures something that depends on the survey’s own radius. The figures say how much, and they turn the scatter of published heights into something closer to a measurement.

What a midplane is #

A disc galaxy’s stars are held near its plane by the disc’s own gravity. Each star oscillates vertically through the plane, and the population’s vertical spread — its scale height, about three hundred parsecs for the old stars of the thin disc near the Sun — is set by the balance between their vertical motions and the disc’s pull. In a disc in equilibrium the density is symmetric about a plane, and that plane is the midplane: the place where the vertical force vanishes and the stars’ density peaks.

The Sun’s height above it is measured by asymmetry. If the Sun were in the midplane, a census of stars in a cylinder centred on it would find equal numbers above and below at every height. With the Sun twenty parsecs above, the census finds more stars below — the cylinder reaches deeper into the dense midplane on the southern side — and the size of the excess, at each height, gives the offset. With millions of stars the asymmetry is measured to a fraction of a per cent, and the statistical error on the Sun’s height is well under a parsec.

That procedure assumes something it does not state: that there is one plane of symmetry across the whole cylinder. A census within fifty parsecs and one within a kiloparsec are both averaging over a disc, and if the disc’s midplane bends between fifty parsecs and a kiloparsec, they are fitting different planes.

How the Sun’s height has been measured #

The first estimates came from counting stars of known types in the two galactic hemispheres. If the Sun sits above the plane, faint stars of a given type should be more numerous towards the south galactic pole, because the line of sight southward crosses more of the dense disc before leaving it. Early in the twentieth century this gave heights of tens of parsecs with large uncertainties, limited by how well the stars’ distances were known and by the patchy dust that hides stars unevenly in the two directions.

Later estimates used tracers whose distances were better known: young open clusters and the clouds of molecular gas they form from, Cepheid variables, the dust layer itself mapped in infrared light, and the neutral hydrogen of the 21-centimetre line that defined the 1958 pole. Each tracer gives a midplane and a height, and they do not agree. The young tracers, concentrated within a few hundred parsecs and following the gas, tend to give heights of five to fifteen parsecs; the old stars, counted to kiloparsecs, give twenty or more. The differences were long attributed to the tracers’ different selection effects and to the dust.

Gaia removed the distance problem for the old stars. With parallaxes for millions of stars within a few kiloparsecs, the north–south asymmetry can be measured at every height and every distance, and the Sun’s height comes out at twenty parsecs with a statistical uncertainty of a fraction of a parsec. The same data showed that the disc is not symmetric about any plane, which is what makes the precision difficult to interpret.

A disc that rings #

It does bend. The Gaia satellite’s measurements of the positions and velocities of a billion stars showed in 2018 that the disc around the Sun is not in vertical equilibrium. Plotted in the plane of vertical position and vertical velocity, the stars near the Sun trace a spiral — a phase spiral — which is the signature of a population disturbed out of equilibrium a few hundred million years ago and still winding up as stars with different vertical frequencies drift out of phase. Maps of the stars’ mean vertical positions and velocities over kiloparsecs show waves: regions where the disc’s midplane sits tens of parsecs above or below the average, alternating on scales of one to a few kiloparsecs.

The most likely cause is the Sagittarius dwarf galaxy, which has passed through the Galaxy’s disc several times on its decaying orbit and is being torn apart in the process. Each passage gives the disc a vertical kick that propagates outward as a bending wave, as a stone dropped in a pond makes a ring. The Large Magellanic Cloud, which pulls the Galaxy’s outer halo off-centre, adds a warp on larger scales. In the gas, a coherent wave nearly three kiloparsecs long, rising and falling by more than a hundred parsecs, runs through the nearest star-forming clouds a few hundred parsecs from the Sun.

The ripple is not a static shape but a wave in motion: each part of the disc is oscillating vertically, and the Sun’s own oscillation through the plane, with a period of about eighty million years, is one star’s share of the same physics. The figure draws an idealised version: one wave, with an amplitude of fifteen parsecs and a wavelength of two and a half kiloparsecs, the Sun sitting fifteen parsecs above the midplane directly beneath it. The real disc carries several waves at once, of different wavelengths and orientations. One is enough to show what a survey does with it.

The height depends on the radius #

Take the ripple and fit a flat midplane to the stars within a projected radius R of the Sun, as a survey of that radius would.

Within a hundred parsecs the ripple is flat, and every survey finds the Sun fifteen parsecs above the midplane beneath it, whatever part of the wave the Sun happens to sit on. Beyond a few hundred parsecs the survey’s average starts to include the ripple’s other phases, and the answer moves. If the Sun is on a crest, the midplane averaged over kiloparsecs is lower than the local one and the Sun’s height grows towards thirty parsecs; if the Sun is in a trough, it shrinks towards zero. On a node, where the ripple crosses the mean plane, the average stays where it was.

Every value along each curve is a correct measurement. It answers the question “how far above the midplane of the disc within R of the Sun is the Sun”, for the R the survey reached. The published values were made with tracers that reach different distances — nearby dwarfs, young clusters at a few hundred parsecs, giants at kiloparsecs — and the five-to-thirty-parsec spread they show is exactly the size this ripple produces between its local and its global answers.

A larger ripple #

The amplitude near the Sun is not well measured. Fifteen parsecs is at the low end of what the stellar maps show at a kiloparsec or two; further out, the waves are larger.

With a forty-parsec ripple the spread of correct answers becomes eighty parsecs, and in a trough the Sun’s height above the global mean plane becomes negative: the Sun is above the midplane beneath it and below the average midplane of the disc around it. That is not a contradiction. It is the statement that “above the midplane” is a phrase with a scale in it, and that the scale has to be quoted with the number.

A tilt the neighbourhood imposes #

A rippled midplane is also tilted, locally, by its own slope.

On a node, where the wave is steepest, a plane fitted to the nearest few hundred parsecs is tilted by about two degrees relative to the disc’s mean plane. On a crest or in a trough the local plane is level but curved. A survey fitting a plane locally measures the orientation of the ripple the Sun sits on, and only a survey reaching several kiloparsecs averages it away.

That puts an old observation in a new light. The young stars and gas within a few hundred parsecs of the Sun lie in a ring tilted by about twenty degrees to the galactic plane — the Gould Belt, catalogued since the nineteenth century and long thought to be a single expanding structure. It is now understood mostly as the projection of separate star-forming structures, including the wave in the gas. Its tilt is the extreme case of a local plane differing from the Galaxy’s: the youngest tracers follow the gas, and the gas follows the waves most closely.

Two directions, two latitudes #

The ripple also changes where the midplane appears on the sky.

For a flat midplane below the Sun, the midplane appears at a negative latitude that depends only on distance, the same in every direction, converging on the galactic equator as the distance grows. For a rippled one, it appears higher in one direction and lower in the opposite one. At a kiloparsec the difference in this example is a degree. A survey that measured the midplane’s latitude in many directions and fitted a flat plane would interpret the difference as a tilt of the reference frame — a rotation of the galactic pole — rather than a property of the disc.

Where a star is depends on who is asking, and the galactic frame adds a case to that list: where the disc is depends on how far away the asker looks. The coordinate system itself is unaffected. Its pole and origin are fixed in the modern quasar-based frame by convention, and the frame made of distant quasars does not care how the Galaxy bends. What changes is the meaning of latitude zero for anyone using it as a proxy for the disc.

Counting is no longer the limit #

For most of the twentieth century the Sun’s height was limited by the number of stars that could be counted with known distances. That is no longer true.

With a tracer population of one star per square parsec of disc, the statistical error on the midplane falls below a parsec at a survey radius of under two hundred parsecs, and keeps falling. The ripple’s contribution — the difference between the height a survey measures and the local value — grows with radius, crossing the statistical error at a few hundred parsecs for any ripple of more than a few parsecs. Beyond that radius the precision of a measurement of the Sun’s height is set entirely by the question of which plane is meant, and every additional star sharpens the answer to a question that is already ambiguous.

That is a general pattern in astronomy once a survey becomes large enough: the statistical error falls below the systematic one set by the model. Here the model is the assumption of a flat disc, and the systematic error is the disc’s ripple.

What the committee of 1958 could see #

The committee that fixed the galactic coordinates chose the pole from the neutral hydrogen because the gas, mapped by its radio emission across the whole disc, traced the plane over tens of kiloparsecs — far further than any stellar survey of the time — and a plane fitted over that scale averages over every local ripple. That was the right choice, and for the reason this essay draws: the larger the region a plane is fitted to, the less it depends on where the observer happens to sit. Their pole is the global plane’s pole to within a few tenths of a degree, and the Sun’s offset from that plane was a separate number, measured locally, which they did not try to build into the frame.

What they could not see was that the local number would turn out to have no single value. The model of the sky as a sphere seen from one point works because the stars are far; the model of the Galaxy as a plane seen from just above it works because the disc is thin. Neither model breaks. What breaks is the assumption that the disc’s thinness comes with a flatness on every scale, and that assumption entered the coordinates only through the one number — the Sun’s height — that was never part of the definition.

What it does to the weighing of the disc #

The midplane is not only a reference for coordinates. It is the plane about which the disc’s vertical equilibrium is assumed when the local density of matter — including dark matter — is measured from how far stars rise above the disc. That measurement compares the stars’ vertical velocities with their vertical spread, and both are measured relative to a midplane. A disc that is out of equilibrium, with its midplane displaced by tens of parsecs and the stars still oscillating about where it used to be, violates the assumption directly.

The size of the effect depends on how far the disturbance has phase-mixed. The phase spiral shows it has not finished. Estimates of the local dark-matter density from vertical equilibrium have shifted by tens of per cent as models have started to include the disequilibrium, and the galaxy measured from inside it turns out to be harder to weigh than one seen from outside for this reason among others: an observer inside a disc that is ringing cannot step back far enough to average the ringing out.

What the model leaves out #

One wave stands for many. The figures draw a single sinusoidal ripple along one direction. The disc’s real vertical structure is a superposition of waves of different wavelengths and orientations, a warp on larger scales and local features like the gas wave, and the curves’ detailed shapes — their overshoots at kiloparsec radii — belong to the idealisation.

The survey is a uniform disc. Real tracers are not uniformly distributed: young stars cluster along spiral arms, and faint stars are complete only to a magnitude limit that varies with direction. A survey’s effective radius is a weighted one, and different tracers weight the ripple differently.

The ripple is static. Bending waves propagate and wind up, and the Sun’s position relative to a crest changes over tens of millions of years. The Sun’s own vertical oscillation, with a period of about eighty million years, carries it through the ripple too.

The tracers share one midplane. Gas, young stars and old stars respond differently to a disturbance, and their midplanes need not coincide even where the disc is calm. The figures assume one midplane for everything.

Still open: how the disc’s ripples are arranged around the Sun #

The scatter in the published heights of the Sun above the midplane is too large to be measurement error alone and is about the size a modest corrugation would produce. Whether it is that, and where the Sun sits on the ripples, is now a question of mapping rather than counting: the vertical positions and velocities of the stars over the nearest few kiloparsecs, which Gaia’s later data releases will give for tens of millions more stars with better distances, show the wave pattern directly.

Once the pattern is mapped, the Sun’s height can be quoted the way it should be — as a function of scale, with the scale stated — and the local plane, the global plane and the 1958 equator can each be related to the others by numbers rather than by a single height and its error bar. The coordinate system will not change. What will change is the description of the disc it sits in, from a sheet with the Sun slightly above it to a ringing surface with the Sun riding one of its waves.