Where a star is depends on who is asking

Point at a star and the direction is a fact about the sky. Say where it is and the answer immediately becomes a fact about two things: the star, and whoever is pointing.

“Forty degrees up, in the south-east” is a complete and useful description, and it is worthless to anyone standing anywhere else, or standing in the same place an hour later. “Right ascension 5 hours 55 minutes, declination +7° 24′” is complete and useful to everyone forever, and tells nobody where to look.

Both are needed, both are used constantly, and the whole of positional astronomy sits in the conversion between them — which turns out to be a single rotation whose angle is a clock reading.

Two frames, two things that stay still #

A coordinate system on a sphere needs a pole and a zero of longitude. There are two natural choices, and they disagree about which things are moving.

Horizon coordinates take the pole to be the zenith, the point directly overhead, and measure altitude up from the horizon and azimuth around from north. This is the frame in which a telescope on an altazimuth mount points, in which an observer decides whether an object is up, and in which the atmosphere sits — refraction and extinction depend on altitude and on nothing else.

Equatorial coordinates take the pole to be the celestial pole, the point the sky turns about, and measure declination up from the celestial equator and right ascension around from the vernal equinox. This is the frame the stars are catalogued in, because in it they do not move.

The two poles are separated by the co-latitude, $90° - \varphi$. That single number is the whole geometric relationship, and it produces the most quoted fact in the subject: the celestial pole stands due north at an altitude exactly equal to the observer’s latitude. At the equator it is on the horizon; at the North Pole it is overhead; in London it is 51.5° up.

The consequence: what rises, and what never sets #

Because a star’s declination is fixed and the horizon’s tilt is set by the latitude, whether a star ever rises or ever sets is decided by an inequality between two numbers.

A star is circumpolar — never sets — if

$$\delta > 90° - \varphi,$$

and never rises if $\delta < -(90° - \varphi)$. At latitude 52° that means everything north of declination +38° is always up and everything south of −38° is never visible, leaving 76° of the sky that rises and sets and 14° of it permanently below.

This is why observatories are placed where they are, and it is a harder constraint than it looks. From Mauna Kea at +19.8° the invisible cap is 20° across and includes nothing much; from the South Pole, half the sky is permanently invisible and the other half never sets, which makes it superb for long uninterrupted monitoring and useless for anything in the northern sky.

The same inequality run the other way is a navigation method. Measure the altitude of Polaris and, to within the degree by which Polaris misses the pole, that is the latitude — a technique available to anyone with a sextant and no clock at all, and the reason latitude was a solved problem for centuries while longitude was not — the latter needing a clock the sky itself does not provide.

The rotation, and why it needs a clock #

Converting between the frames requires the co-latitude and one more number, because the equatorial frame is turning underneath the horizon frame. The extra number is the hour angle: the angle, measured westward along the celestial equator, from the observer’s meridian to the object.

$$H = \text{LST} - \alpha,$$

where $\alpha$ is the right ascension and LST is the local sidereal time — which is simply the right ascension currently on the meridian. With $H$ in hand, the conversion is spherical trigonometry:

$$\sin a = \sin\delta\sin\varphi + \cos\delta\cos\varphi\cos H,$$

$$\cos a \sin A = -\cos\delta\sin H,$$

with $a$ the altitude and $A$ the azimuth measured from north through east. Those two lines are the entire content of the transformation, and they contain exactly three inputs: the object’s declination, the observer’s latitude, and the hour angle.

So a position in the sky is a position, a latitude and a time. Drop any one and the conversion cannot be done. The equatorial frame is the one in which a catalogue can exist; the horizon frame is the one in which a telescope can point; and a clock is what stands between them.

Sidereal time is not clock time #

The sidereal day is the rotation period of the Earth with respect to the stars, and it is 23 h 56 m 04.09 s — three minutes and fifty-six seconds shorter than the solar day.

The reason is that the Earth is also going around the Sun, so in one rotation with respect to the stars it has moved about a degree along its orbit and must turn nearly four minutes further to bring the Sun back to the meridian. Over a year that accumulates to exactly one extra rotation: the Earth turns 366.24 times with respect to the stars while turning 365.24 times with respect to the Sun. The practical consequence is that a given star rises four minutes earlier each night, two hours earlier each month, and returns to the same place at the same clock time once a year — which is why the constellations are seasonal, and why an observing plan is made against sidereal time rather than against a watch.

The mount is the coordinate system, in metal #

The two frames have physical incarnations, and the history of telescope building is largely an argument between them.

An equatorial mount tips one of its two axes to point at the celestial pole. The instrument then tracks a star by turning one axis at a constant rate, once per sidereal day, and the field of view does not rotate. That is an enormous simplification: a clock drive is a motor turning at a fixed speed, and it was buildable in 1824, when Fraunhofer put the first one under the Dorpat refractor. Every large telescope for the next century and a half was equatorial.

An altazimuth mount turns about the vertical and about the horizontal. It is far cheaper and stiffer — the load is carried straight down, so the structure does not have to hold a heavy tube out on a tilted axis — but tracking requires both axes to move at continuously varying rates, and the field of view rotates as well, so the instrument needs a third motion to hold it still. None of that was practical before computer control.

The crossover came in 1975 with the six-metre BTA-6, and every large telescope since has been altazimuth. The frames did not change; the cost of converting between them did, from impossible to a few floating-point operations per second. The altazimuth mount has one genuine defect that the geometry forces and no engineering removes. At the zenith, the azimuth axis must turn arbitrarily fast to follow an object through the overhead point — the same coordinate singularity that breaks a set of orientation angles at their degenerate configuration. Every altazimuth telescope has a keyhole a degree or two across directly overhead in which it cannot observe, and the finest seeing in the sky is in the middle of it.

What was actually measured #

Neither frame is directly observed. What a meridian instrument measures is a transit: the instant at which an object crosses the local meridian, and its altitude at that instant. Two numbers, one of them a time.

From those two, the equatorial coordinates follow. The altitude at transit gives the declination immediately, $\delta = \varphi - 90° + a$ for an object south of the zenith. The time of transit gives the right ascension, because at transit $H = 0$ and therefore $\alpha = \text{LST}$. That is the entire method, and it is how every fundamental star catalogue from Flamsteed’s to the mid-twentieth century was built: a telescope that moves in one plane only, a clock, and a great many nights.

The precision of the method is limited by things that have nothing to do with the geometry. Refraction lifts an object’s apparent altitude by 34 arcminutes at the horizon and about 1 arcminute at 45° — larger than the effect being measured, and dependent on temperature and pressure, so it is modelled and subtracted rather than avoided. Aberration displaces every star by up to 20.5 arcseconds in the direction of the Earth’s motion, a purely kinematic effect of the observer’s velocity. Nutation wobbles the pole by 9.2 arcseconds with an 18.6-year period, from the Moon’s orbital precession. Parallax displaces nearby stars by under an arcsecond. Every one of these has to be applied before an observed direction becomes a catalogue position, and the sequence in which they are applied is itself a standard. The modern replacement measures the frame rather than the objects in it. The International Celestial Reference Frame is defined by the positions of 4,536 extragalactic radio sources — quasars, far enough away that they have no measurable proper motion — fixed by very-long-baseline interferometry to about 30 microarcseconds. It is not tied to the equator or the equinox at all; those are now derived from it rather than defining it. Gaia has tied the optical sky to that radio frame to about 20 microarcseconds, so a modern catalogue position is a direction in a frame that no longer refers to the Earth in any way.

That is a genuine inversion of the old logic. The equatorial frame was originally defined by the Earth’s rotation axis and orbit, both of which move; it is now defined by objects outside the galaxy, and the Earth’s axis is one of the things measured against it.

The same star at the same declination, seen from three more latitudes, is the whole of what the transformation does.

The star has three more motions, and none of them is the sky turning #

A catalogue position is a direction in a fixed frame, and it still is not constant, because the star itself moves and so does the observer. Four displacements have to be separated, and they have four different periods, which is what makes the separation possible at all.

Diurnal rotation — a full turn every sidereal day. This is the one the horizon frame exists to describe and the equatorial frame exists to remove.

Annual parallax — an ellipse of semi-major axis $p$, once a year, the size of which is the distance. Under an arcsecond for every star.

Annual aberration — an ellipse of semi-major axis 20.5 arcseconds, also once a year, but ninety degrees out of phase with parallax, because it depends on the Earth’s velocity rather than its position. It is twenty to a hundred times larger than the parallax it sits on top of.

Proper motion — a straight line, accumulating without limit. Barnard’s Star moves 10.4 arcseconds a year, which is a quarter of a degree per century.

The phase difference is what disentangles the two annual terms, and it is also the reason parallax was so hard to find. Bradley was hunting for parallax in γ Draconis in 1725, found an annual ellipse of the right period and the wrong phase, and spent two years working out that he had discovered something else — the finite speed of light, showing up as a displacement in the direction the Earth was travelling. The parallax he was after is 22 milliarcseconds, and he could not have seen it.

The generalisation: a frame is chosen by what stays still in it #

The rule that decides which coordinate system to use is not aesthetic. It is that a frame is worth having when the thing being studied does not move in it, and the two sky frames are the clearest possible illustration.

Atmospheric effects — refraction, extinction, seeing, sky brightness — are functions of altitude, so they are described in horizon coordinates and are simple there and hideous in equatorial coordinates. Stellar positions are constant in equatorial coordinates and vary continuously in horizon coordinates. Neither frame is more correct; each is the one in which a particular set of facts is short.

The subject uses at least three more for the same reason. Ecliptic coordinates, with the pole perpendicular to the Earth’s orbit, are the natural frame for solar-system bodies, which all sit near the ecliptic plane — the planets’ inclinations are single-digit degrees, so a table in ecliptic latitude is a table of small numbers, as the Sun’s own path makes plain. Galactic coordinates, with the pole perpendicular to the Milky Way’s plane, are the natural frame for anything in the galaxy, and the concentration of stars toward zero galactic latitude is the whole of galactic structure in one number. Supergalactic coordinates do the same for the local sheet of galaxies. Each of them is one rotation away from the others, each of them makes some catalogue short and some other catalogue long, and the software that converts between them is a solved problem that everybody nonetheless still gets wrong occasionally, usually by mixing epochs.

Where the model stops #

The sphere is not a place. Both frames are directions only. The celestial sphere carries no distances, so two objects adjacent on it may be a light-year and a billion light-years away, and neither coordinate system contains any hint of that.

The equatorial frame is not fixed. Precession moves the equinox 50.3 arcseconds a year and nutation wobbles it; “right ascension and declination” without an epoch is not a position. This is why every catalogue coordinate is labelled J2000 or ICRF, and why the old B1950 catalogues cannot be used directly.

The horizon frame is not fixed either. It turns with the Earth, so a horizon coordinate without a timestamp is a position for one instant only.

Neither frame is inertial. Both rotate, so anything dynamical — an orbit, a trajectory — must be computed in a third frame and converted for display. That conversion is where a great many small errors live.

The figures draw both grids as though they were equally real. They are drawn on one sphere at one instant, which is exactly the moment at which the distinction the essay is about disappears. A minute later the two grids have moved by a quarter of a degree relative to one another, and nothing on a static page can show that. One consequence of the whole chain is worth stating for anybody using published coordinates. A position is not a property of a star; it is a property of a star, a date, a frame, an observer’s location and a set of models for everything in between. Two catalogues that disagree are usually not disagreeing about the sky — they are answering slightly different questions, and reconciling them means finding which of the five differs rather than which of the two is wrong.

The five are also in a fixed order of size, which is what makes the reconciliation tractable: a disagreement of arcminutes is a frame or an epoch, one of arcseconds is refraction or aberration, and one of milliarcseconds is a model of something subtler.

Two more settings vary the other two arguments the transformation takes.

The ladder from here #

Later rungs on this anchor: the spherical-trigonometry identities derived from the rotation matrix rather than quoted. Refraction, and why the Sun is fully below the horizon when it appears to touch it. Aberration, and Bradley’s discovery of it while looking for parallax. Nutation, and the 18.6-year signature of the Moon’s nodes. Ecliptic and galactic frames, and the rotations between all five. The ICRF, and what it means for a reference frame to be defined by quasars. Sidereal time in its three flavours — mean, apparent and UT1 — and the fact that the Earth’s rotation is not uniform.

The pleasing part of this structure is what it says about the pole star. Polaris is not special; it is a second-magnitude star that happens to sit near a point defined entirely by the Earth’s axis. The point is real and the star is a coincidence — and a temporary one, because the point moves.