A frame fixed before its centre was found

Every astronomical catalogue gives positions in equatorial coordinates, right ascension and declination, tied to the Earth’s equator and nowadays to a frame defined by distant quasars. For anything to do with the Milky Way itself — the distribution of stars in the disc, the rotation curve, the structure towards the centre — astronomers convert those positions into galactic coordinates, longitude $l$ and latitude $b$. The galactic equator runs along the middle of the Milky Way’s band of light, latitude zero, and longitude is counted from the direction of the Galaxy’s centre.

That description makes the system sound like the Galaxy’s own, defined by its geometry the way the celestial equator is defined by the Earth’s rotation. It is not. It is a convention agreed by a committee of the International Astronomical Union in 1958, from the best data available then, and frozen. Everything learned about the Galaxy since has been expressed in a frame that was fixed before the Galaxy’s centre was identified, before the Sun’s height above the disc was measured, and before the radio source at the centre was resolved into the black hole now known to be there. The frame is still exactly what it was, and the Galaxy has turned out to be a little different.

Three planes on one sky #

The relation between the two systems is a single rotation, three fixed numbers that carry a direction from one set of axes to the other. The rotation is easiest to see as the familiar circles drawn on one map.

The galactic equator is a great circle inclined at about 63 degrees to the celestial one, crossing it in the constellations of Aquila and Monoceros. Its pole, the north galactic pole, lies in Coma Berenices, in a part of the sky where the view out of the disc is clear and the distant galaxies of the Virgo and Coma clusters crowd the field. The centre lies in Sagittarius at a declination of nearly minus 29 degrees, which is why it is best seen from the southern hemisphere and why the brightest part of the Milky Way never rises high from northern Europe. The ecliptic, the Earth’s orbital plane, makes a third great circle at about 60 degrees to the galactic one.

The three circles are tied to three physical planes: the Earth’s equator, which precesses; the Earth’s orbit, which barely moves; and the Galaxy’s disc, which does not move at all on any timescale that matters for coordinates. The first two are defined by dynamics measured continuously to microarcseconds. The third was defined once, by a vote, from a map of hydrogen gas.

A pole from the hydrogen, an origin on a radio source #

The 1958 system replaced older galactic coordinates whose pole had been estimated from the distribution of stars, which is badly confused by dust in the plane. The new pole came from radio maps of the 21-centimetre line of neutral hydrogen, which penetrates the dust freely and showed the gas layer as a thin sheet across the whole sky. The sheet’s plane, fitted to the hydrogen within a few kiloparsecs of the Sun, defined the galactic equator, and its pole was fixed at a round position in the equatorial coordinates of the day, the 1950 epoch: right ascension 12 hours 49 minutes, declination +27.4 degrees. From then on those numbers were not an estimate but a definition.

For the origin of longitude the committee chose the direction of the radio source Sagittarius A, the brightest and most compact radio source in the direction of the centre, which was believed to mark the centre itself. The zero of longitude was placed through it, at a position given to an arcminute or so, and the galactic equator was fixed as the great circle through the Sun perpendicular to the pole. The equator passes through the Sun by construction: galactic latitude is measured from where the observer is.

The system was later carried forward into each new equatorial reference frame by computing the rotation matrix that reproduces the 1958 definition, and today it is defined by a rotation from the quasar-based frame, three numbers accurate to the tenth of a milliarcsecond. That precision is the precision of the convention. The physical planes and points the convention was meant to describe are known far less precisely, and they turn out not to coincide with it.

The plane’s tilt is a matter of the date #

Before turning to the Galaxy, the equatorial side of the relation has a large and slightly surprising consequence. The celestial pole is not fixed. It precesses around the ecliptic pole once in about 25,800 years, at the obliquity of 23.4 degrees, while the galactic pole stays where it is. So the angle between the celestial equator and the galactic plane — 62.87 degrees today — is a function of the date.

The swing is large. The galactic pole lies about 60 degrees from the ecliptic pole, and as the celestial pole circles the ecliptic pole the angle between it and the galactic pole runs from 60 minus 23.4 to 60 plus 23.4 degrees. About six and a half thousand years ago the angle was at its smallest, 37 degrees. In about six and a half thousand years it will be at its largest, 84 degrees: the Milky Way will lie at nearly right angles to the celestial equator, with its pole within seven degrees of the equator. Over a precession cycle, the arch of the Milky Way across the sky of any given latitude changes its posture completely.

This is the reason every equatorial catalogue must state its epoch, and the reason the galactic system is carried by a rotation from a fixed modern frame rather than by a pole quoted in the coordinates of the day. A direction fixed in the Galaxy has equatorial coordinates that change by tens of degrees over the precession cycle, which is a statement about the Earth’s axis, not about the Galaxy.

The centre is not at the centre #

The radio source chosen as the origin in 1958 was later resolved into several components: a supernova remnant, a spiral of ionised gas, and, discovered in 1974, a compact source too small and too bright to be anything but an accretion flow onto something very massive. That source, Sgr A*, has since been shown by the orbits of stars around it to mark a black hole of four million solar masses — the dynamical centre of the Galaxy, as far as anything is. It is not at the origin of galactic coordinates.

Sgr A* sits at a longitude of minus 0.056 degrees and a latitude of minus 0.046 degrees: about four arcminutes from the origin, which at the centre’s distance of 8.2 kiloparsecs is about ten parsecs. Ten parsecs is small compared with the Galaxy and large compared with the black hole’s sphere of influence, about two parsecs. Studies of the inner few parsecs therefore routinely work in coordinates centred on Sgr A* rather than on the galactic origin, and convert between the two with an offset that the convention itself does not contain.

The latitude offset says something more. If the galactic equator were parallel to the Galaxy’s true midplane, and the Sun sat 20.8 parsecs above that midplane — the value measured from the vertical distribution of nearby stars — then the centre would appear 0.146 degrees below latitude zero, almost nine arcminutes. It appears at less than three. The 1958 equator is therefore tilted relative to the true midplane by about a tenth of a degree, the difference between the plane of the local hydrogen the committee mapped and the plane through the Sun and the centre. A tenth of a degree is far below anything the 1958 data could have detected, and far above the precision with which the convention is now specified.

A plane with no single latitude #

The Sun’s height above the midplane has a consequence that makes galactic latitude strange to use for nearby objects. Latitude zero is a plane through the Sun. The midplane is a plane about twenty parsecs below it. Seen from the Sun, the midplane is not a great circle at latitude zero: it appears at a latitude that depends on how far away one looks.

A kiloparsec away towards the centre, the midplane lies more than a degree below latitude zero. A few hundred parsecs away it lies several degrees below. Only at distances comparable to the centre’s does the midplane converge on the galactic equator, and beyond the centre it crosses it. A young star cluster at three hundred parsecs that lies exactly in the disc’s midplane appears at a latitude of about minus four degrees, and an analysis that counted stars “in the plane” by selecting latitudes near zero would miss it. Studies of the vertical structure of the disc’s gas layer, whose thickness is a few hundred parsecs, must correct for the Sun’s offset explicitly. Studies of the stars that measure the Galaxy from inside do the same, converting each object to a height above the midplane rather than trusting its latitude.

The general point is that latitude, like any angular coordinate centred on the observer, mixes direction with the observer’s position. The celestial equator escapes this because the Earth’s centre is at the origin of geocentric coordinates and the distances to the stars dwarf the Earth’s size. For the Galaxy the observer’s offset from the plane is a few thousandths of the distance to the centre, enough to tilt the apparent plane by degrees nearby.

The Sun does not stay where latitude zero was drawn #

The Sun’s height is not even fixed. The Sun moves through the Galaxy with a small velocity relative to the average motion of the stars around it, the local standard of rest, and part of that velocity, about 7.25 kilometres a second, is towards the north galactic pole. Gravity from the disc pulls it back, and it oscillates through the midplane like a ball through a stretched sheet.

For a local density of about a tenth of a solar mass per cubic parsec, the vertical oscillation has a period of about 84 million years, and the Sun’s present position and velocity give it an amplitude of about a hundred parsecs. The Sun crossed the midplane a few million years ago going north and will reach its greatest height in about twenty million years, a hundred parsecs above the disc. The galactic coordinate system will still have its equator through the Sun’s position in 1958, since the pole is fixed in space and the origin moves with the observer. But the Sun will be several times further from the midplane than it is now, and the apparent latitude of the midplane in every direction will have shifted by degrees.

The period of this oscillation has been proposed as a pacemaker for terrestrial events, from mass extinctions to cratering rates, on the grounds that passing through the dense midplane might perturb the Oort cloud more strongly. The evidence for any such periodicity in the geological record is weak, and the period itself depends on the local density, which is uncertain by tens of per cent through its dark-matter component. The oscillation is real; its imprint on the Earth is not established.

A standard of rest that nothing occupies #

The galactic frame carries velocities as well as directions, and its velocity conventions have the same character as its positional ones. Stellar motions near the Sun are expressed as components $U$, $V$ and $W$ along the directions towards the centre, along the direction of Galactic rotation, and towards the north galactic pole — the galactic coordinate axes, used as a basis for velocity. They are measured relative to the local standard of rest, defined as the velocity a star would have on a perfectly circular orbit through the Sun’s present position.

No star is on that orbit, and the standard is defined by a model rather than by an object. The Sun moves relative to it at about 11 kilometres a second inwards, 12 in the direction of rotation and 7 towards the north galactic pole, values found by examining how the average motions of groups of stars depend on their velocity dispersion and extrapolating to a group with none. The rotation speed of the standard itself, about 230 kilometres a second, is measured separately and with a larger uncertainty, from the rotation of the disc around the Sun’s position and from Sgr A*'s apparent motion across the sky, which is the Sun’s orbital motion reflected.

The pattern is the one that runs through the whole system. A convention is chosen because some physical reference is needed and none is directly observable: the centre, the plane, the circular orbit. The convention is then fixed, and the physics is expressed as offsets from it. The Sun’s motion of 7 kilometres a second towards the pole is what drives the vertical oscillation above. It is an offset of the Sun from a standard of rest, measured in a frame whose equator was drawn through the Sun.

Positions converted, and the angle that comes with them #

Converting a position from equatorial to galactic coordinates is a rotation. Converting a motion across the sky needs one more number, the angle at the star between the directions to the two poles — the celestial and the galactic — which turns a proper motion’s components in right ascension and declination into components in longitude and latitude. That angle varies over the sky, from zero along the great circle through both poles to large values near the galactic equator, and it is the source of a common error: a proper motion converted by rotating its components with the wrong angle has its direction wrong while its size is right.

The astrometric catalogues now give positions and motions for more than a billion stars, all in the equatorial frame. Every study of the Galaxy’s kinematics begins by converting them, and the conversion is exact because the frame is a definition. That exactness is what the fixed convention buys. The numbers that come out are offsets from a plane and a point that the Galaxy does not quite contain, and the analysis has to restore the Sun’s height and the centre’s offset itself.

Why the convention is not updated #

It would be straightforward to define a new galactic system with its origin on Sgr A* and its equator in the true midplane through the Sun’s position. It has not been done, for the reason that coordinate systems in astronomy are defined, not measured. A system that moved whenever the Galaxy’s structure was better understood would make every catalogue a moving target, every comparison between epochs a reanalysis, and every published position ambiguous unless it named its version. The 1958 system’s value is that it has never moved. Its offsets from the physical centre and plane are small, known, and the same in every paper.

That is a choice the celestial sphere’s coordinates have always faced. Any system of coordinates privileges some point and some plane, and where a star is depends on who is asking. The equatorial system privileges the Earth and is redefined only when its realisation improves. The galactic system privileges a committee’s best estimate of the Galaxy’s geometry in 1958, and is never redefined at all. Using it well means knowing which of its features are physics and which are history: the pole is approximately the disc’s axis; the origin is approximately the centre; the equator passes exactly through the Sun, which is the one feature that is not approximately anything about the Galaxy.

What the figures assume #

The rotation between the equatorial and galactic systems is the standard one in the modern quasar frame, and the positions of the pole and the origin follow from it exactly. The precession figure treats the celestial pole as circling the ecliptic pole uniformly at a fixed obliquity, ignoring the slow drift of the ecliptic itself and the obliquity’s own variation of a degree or so over tens of thousands of years. The midplane is modelled as a flat plane through Sgr A* with the Sun 20.8 parsecs above it. The real disc is warped at large radii and corrugated on scales of kiloparsecs, so “the” midplane is itself a fit whose answer depends on which stars or gas are used. The vertical oscillation is that of a harmonic potential from a uniform slab, adequate for heights of a hundred parsecs but not for the detailed shape of the disc’s potential, and it ignores the Sun’s simultaneous radial and azimuthal motion.

Still open: where the midplane is #

The Sun’s height above the midplane has been measured many ways — from counts of stars above and below, from the distribution of young clusters, from the gas — and the results scatter between about five and thirty parsecs. The most careful recent determinations from the vertical distribution of millions of stars give twenty parsecs or so, and the tilt of the galactic equator that follows from it is the tenth of a degree drawn above. But the midplane traced by old stars, by young stars and by gas need not be the same plane: the disc is not in perfect equilibrium, it rings with vertical waves set up by the passage of the Sagittarius dwarf galaxy, and the Sun may sit on the crest of one of them.

If the disc near the Sun is corrugated, the Sun’s height above “the” midplane depends on the scale over which the midplane is averaged, and the tilt of the 1958 equator relative to the disc depends on it too. Mapping the disc’s vertical structure over kiloparsecs in every direction, which the astrometric surveys now make possible, will say whether the Galaxy has a midplane at the precision of a few parsecs at all. The coordinate system will not change in response. What will change is how far from its equator the Galaxy is understood to be.