The pole star has a shelf life, and the sky has a slow hand

The pole star is not a fixture. It is the star that happens, at the moment, to lie nearest the point the sky turns about — and the point moves.

It moves slowly and it moves a long way. The celestial pole traces a circle of radius 23.4° among the stars, once every 25,772 years, and Polaris occupies the useful position for a few centuries either side of now. Four thousand years ago the pole star was Thuban, in Draco. In twelve thousand years it will be Vega, which is the brightest star anywhere near the circle and will be a far better pole star than Polaris ever was.

The torque, and why a spinning body responds sideways #

The cause is a torque, and the reason it produces a circle rather than a topple is the least intuitive thing in rotational mechanics.

The Earth is not a sphere. It bulges at the equator by 21.4 kilometres, a flattening of one part in 298, thrown out by its own rotation. The Sun and the Moon both pull on that bulge, and because the bulge is tilted 23.4° to the ecliptic, the pull is stronger on the near side of the ring than the far side, which is exactly the situation the shell theorem does not cover. That difference is a tidal effect — a difference of two gravitational pulls rather than a pull — and its net result is a torque trying to twist the equatorial plane into line with the ecliptic.

A stationary Earth would respond by tipping over. A spinning one does not. A torque applied to a spinning body changes its angular momentum in the direction of the torque, and since the angular momentum points along the spin axis, the axis moves perpendicular to both the torque and itself. It goes round instead of over.

That is gyroscopic precession, and the rate follows from the ratio of the torque to the spin angular momentum. A larger torque precesses faster; a faster spin precesses slower. For the Earth the numbers give 50.29 arcseconds per year, or one turn in 25,772 years.

The division of labour between the two perturbers is worth having. The Moon supplies about 68% of the torque and the Sun about 32%, in the ratio their tidal fields stand in — which is the same 2.2-to-1 that makes the Moon the senior partner in the ocean tides. Precession is the same physics as the tides, applied to the solid bulge rather than to the water.

Where the pole has been, and where it goes #

The circle is fixed in space and the stars are, to good approximation, fixed on it, so the pole’s history and future are a matter of reading off positions.

Polaris is currently 0.66° from the pole and closing; it reaches about 0.45° in 2100 and then begins to recede. Before it, Thuban held the position around 2800 BC, and it was close — about 0.1°, considerably better than Polaris manages. Between the two there was a long stretch with no bright star anywhere near the pole at all, and the navigators of that era had to work from the pole’s position relative to a pair of stars rather than from a star at it.

The southern hemisphere is in that condition now. The south celestial pole has no bright star near it and has not had one for a long time, which is why southern navigation used the Southern Cross as a pointer rather than any single star.

What moves with the pole #

The pole is not the only thing carried round. The whole coordinate grid goes with it, and the consequences reach further than astronomy.

The equinoxes move. The March equinox is where the Sun crosses the celestial equator going north, which is an intersection of two planes — and one of the planes is tilting. The crossing therefore slides westward along the ecliptic at 50.29 arcseconds a year, which is where the name precession of the equinoxes comes from. It has moved about 30° since the constellation boundaries were fixed, which is why the astrological sign the Sun is “in” on a given date and the constellation it is actually in front of have parted company by a full sign.

The year has two lengths. The tropical year — equinox to equinox — is 365.2422 days. The sidereal year — one circuit against the fixed stars — is 365.2564 days. The difference of 20 minutes is precession, and the choice between them is the choice between a calendar that keeps the seasons and one that keeps the stars. The Gregorian calendar tracks the tropical year, which is why the seasons stay put and the constellations slowly do not.

Every catalogue needs a date. A star’s right ascension and declination are measured against a grid that is moving, so coordinates without a stated epoch are meaningless. The current standard is J2000.0, and converting between epochs is a rotation that every piece of astronomical software performs constantly — the same reason an orbital element set is useless without a date.

What was actually measured #

Precession was discovered from data, by someone with no possible theory of what could cause it, and the discovery is arguably the finest piece of work in ancient astronomy.

Hipparchus, around 130 BC, compared his own measurements of stellar longitudes against Babylonian and earlier Greek observations from roughly 150 years before. He found that the stars’ longitudes had all increased by about the same amount — his figure was at least one degree per century, against the modern 1.396 — while their latitudes had not changed. A uniform shift in one coordinate and not the other is exactly what a rotation of the coordinate grid about the ecliptic pole produces, and he identified it correctly as a motion of the equinox rather than of the stars.

The method deserves attention because it is the general shape of a great deal of astronomy. He was extracting a 26,000-year cycle from a 150-year baseline — a bit over half a percent of one period — and it worked because the quantity being measured accumulates linearly while the errors do not. A systematic drift of 2° is detectable against a measurement precision of perhaps 20 arcminutes; the same drift over a single year would have been 50 arcseconds and invisible to him.

The modern measurement is done against quasars. Very-long-baseline interferometry fixes the positions of a few hundred extragalactic radio sources to about 20 microarcseconds, and the Earth’s orientation relative to that frame is determined daily. Precession is not inferred from it — it is read directly, along with the smaller and faster wobbles superimposed on it.

Those wobbles are worth naming because they were also found before they were explained. Nutation is a nodding of the axis with an amplitude of 9.2 arcseconds and a principal period of 18.6 years, discovered by James Bradley in 1728 during the same twenty-year observing campaign that produced stellar aberration. Its period is that of the regression of the Moon’s orbital nodes, which is what fixes the geometry of the lunar torque, and Bradley waited a full 18.6 years before publishing to be sure the cycle closed.

The rate is not constant, and the axis is not the only thing precessing #

Two refinements matter, and both are measured.

The precessional rate itself drifts. The Earth’s spin is slowing from tidal friction by about 1.8 milliseconds per century, and a slower spin precesses faster; the Sun’s and Moon’s distances change slowly too. The current rate of 50.29 arcseconds a year is increasing by about 0.0002 arcseconds per year per year, which is small and is included in every modern model.

More substantially, the Earth’s orbit precesses as well. The gravitational pull of the other planets slowly rotates the ecliptic plane itself, at about 0.47 arcseconds a year — planetary precession, as against the lunisolar precession of the axis. The two combine into the observed general precession, and they are separate physical effects with separate causes that happen to move the same coordinate.

The combination has a consequence for the seasons. The direction of the Earth’s perihelion also advances, at about 11.6 arcseconds a year, and combining that with the precession of the equinox gives the precession of the perihelion relative to the equinox — one full cycle in about 20,900 years. That is the period on which the season containing perihelion changes, and it is one of the three Milankovitch cycles that modulate the ice ages. Northern winter currently coincides with the close approach; in ten thousand years it will coincide with the far one, and northern winters will be measurably harsher.

Precession as a dating method #

A cycle that is slow, steady and enormous in total is useless for telling the time and excellent for telling the century, and the archaeological use of precession runs on exactly that.

An alignment built to point at something in the sky is pointing somewhere else a few thousand years later, by an amount the geometry gives exactly. If a structure’s orientation can be measured and the intended target identified, the construction date follows.

The best-supported case is the Egyptian pyramids of the Fourth Dynasty, whose sides are aligned to true north with remarkable accuracy — the Great Pyramid’s is within about three arcminutes. The alignments are not perfect and, more usefully, they are systematically wrong by amounts that change monotonically from one pyramid to the next. If the surveyors used a pair of circumpolar stars that were simultaneously vertical, precession would carry that pair’s alignment slowly off true north at a computable rate, and the residuals would drift accordingly. Fitting the observed drift dates the sequence to within a few decades, independently of any historical chronology.

The method’s weakness is the same as its strength. It requires knowing which target was intended, and a wrong identification produces a confident and wrong date shifted by whatever offset the misidentification implies.

The same circle read at two other epochs makes that concrete.

For most of the cycle there is no pole star at all #

The essay’s title takes for granted that the pole has a star, and the more surprising fact is how rarely that is true.

Polaris is currently 0.66° from the celestial pole, and it is second magnitude. That combination — bright enough to find without effort, close enough that its diurnal circle is smaller than the width of two fingers at arm’s length — is what makes it a navigational instrument rather than a curiosity, and it is close to the best it will ever be: the pole passes within about 0.45° of Polaris around the year 2100 and then begins to move away.

Run the circle round and look at what else is available. Thuban, in Draco, was within 0.1° of the pole around 2700 BC and is fourth magnitude — very close and rather faint, which is why the Egyptian alignments that used it needed a pair of stars and a plumb line rather than a single sighting. Vega comes within about 5° in twelve thousand years’ time, which is bright and useless: a star five degrees off the pole traces a ten-degree circle every night and cannot be mistaken for a fixed point.

For the remainder of the cycle the pole passes through stretches of sky with nothing brighter than fourth magnitude anywhere near it. Adding it up, the fraction of the 26,000 years during which a naked-eye star sits within a degree or two of the pole is perhaps a fifth.

So the present arrangement is a coincidence of the era rather than a feature of the sky, and it has a historical consequence. Northern navigation by a fixed pole star is a technique that has been available for the last few thousand years and will not be available in a few thousand more — and the southern hemisphere, which has no bright star near its pole now, has had to navigate by the geometry of the Southern Cross throughout, which is the technique everyone would use if Polaris were not there.

The generalisation: any spinning oblate body does this #

Precession is not a fact about the Earth. It is what happens to any spinning body that is not spherical and is torqued, and the astronomical instances span an enormous range.

Mars precesses with a period of about 171,000 years, slower than the Earth’s because it has no large moon to supply most of the torque — and, as a consequence, its obliquity is not stabilised and has wandered chaotically over tens of degrees.

A satellite in low Earth orbit precesses too, but the roles are exchanged: it is the orbit that is torqued by the Earth’s bulge rather than the Earth being torqued by the satellite. The node regresses by several degrees a day, and choosing an inclination that makes the rate exactly one turn per year is how a Sun-synchronous orbit works.

A pulsar in a binary precesses under general-relativistic spin–orbit coupling — geodetic precession — and for the Hulse–Taylor system the rate is about a degree a year, large enough that the beam has swung measurably across the line of sight over the decades it has been watched.

A gyroscope in orbit does the same thing, and Gravity Probe B was built to measure it: four quartz spheres, the roundest objects ever manufactured, spinning in a drag-free satellite, precessing by 6.6 arcseconds a year from the curvature of spacetime and 0.039 arcseconds a year from the Earth’s rotation dragging spacetime around with it. Both were measured, and both agreed with the prediction.

The common thread is that a spinning body’s axis is a direction that responds to torque by moving at right angles to it. Once that is internalised, the pole star’s expiry date, the drift of the zodiac, the design of imaging satellites and a test of general relativity are the same fact in five settings.

Two things the circle does not contain are worth drawing, because both are motions of the same axis on quite different timescales.

Where the model stops #

Rigid Earth. The Earth is not rigid: it has a fluid core, oceans and an atmosphere, and each responds to the torque differently. The observed nutation differs from the rigid-body prediction by about a milliarcsecond, and modelling that difference is how the core’s flattening was determined.

Two perturbers. The Sun and Moon supply nearly all of it; the planets contribute the separate and smaller planetary precession, and Venus and Jupiter dominate that.

Constant obliquity. The obliquity oscillates between 22.1° and 24.5° on a 41,000-year cycle, so the radius of the circle in the first figure is itself slowly changing.

Fixed stars. The stars have proper motions, and over 26,000 years those are not negligible. The dates on the first figure assume the stars stay where they are, which over a full cycle is wrong by a degree or more for the nearer ones — Vega’s proper motion alone will carry it about 1.8° in twelve thousand years.

The figure has a limitation that is easy to miss and is the whole difficulty of the subject. It draws the pole’s path as a circle traced over 26,000 years, which invites reading it as a motion. Nothing on the page moves at a rate any observer experiences: 50 arcseconds a year is a fortieth of the Moon’s diameter, per year, and no single lifetime contains a visible change. The circle is a summary of an effect that must be inferred from records, and the reason Hipparchus is the one who found it is that the records were the only instrument that could.

One more epoch covers the part of the cycle in which the pole has no bright star anywhere near it.

The ladder from here #

Later rungs on this anchor: the torque on the equatorial bulge computed. Gyroscopic precession from the angular momentum equation. Lunisolar against planetary precession, and general precession as their sum. Nutation, and the 18.6-year term. The tropical and sidereal year, and calendar design. Epochs, and the transformation between them. The precession of the perihelion, and the Milankovitch cycles. Chandler wobble and polar motion, which are the Earth moving relative to its own axis rather than the axis moving relative to the stars. Geodetic and frame-dragging precession. And the obliquity’s chaotic history on Mars, where nothing stabilises it.

Hipparchus’s star catalogue, from which he found precession, was lost. Its contents survive because Ptolemy copied them into the Almagest three centuries later — and added 2°40′ to every longitude to bring them up to his own date, which is precisely the correction precession requires.