A clock whose zero is moving

A clock is two things: something that repeats, and a mark on the dial to count the repetitions from. A sidereal clock’s repetition is the Earth’s rotation, which is as steady as anything on the planet. Its mark is the vernal equinox — the point on the sky where the Sun crosses the celestial equator going north in March — and sidereal time is defined as how far that point has turned past the observer’s meridian. Right ascension, the sky’s longitude, is counted from the same mark.

The mark is not fixed. The equinox is where two great circles cross, the celestial equator and the ecliptic, and both circles move: the equator because the Earth’s axis is torqued by the Sun and the Moon, and the ecliptic because the planets tug at the Earth’s orbit. When the crossing point moves along the equator, every sidereal clock on the Earth is reset by that much, whatever the planet underneath is doing.

The four-minute difference between the sidereal and solar days came from counting turns, and a note at its end recorded that the sidereal day is 8.4 milliseconds shorter than the day measured against a direction that does not move. That note is the first of three motions of the mark. This essay separates all three and shows where each one sits in the formulas an observatory actually uses.

A second that comes and goes with the Moon’s node #

Start with the largest of the three, which is also the one that repeats.

The Earth is not a sphere. Its equatorial bulge is a ring of extra mass tilted 23.4 degrees to the ecliptic, and the Sun and the Moon pull on that ring and try to twist it into their own planes. The steady part of the twist is precession, which swings the pole round a 26,000-year circle. The unsteady part is nutation, and the wobble inside the wobble is its geometry. The largest piece of nutation comes from the Moon, whose orbit is itself tilted five degrees to the ecliptic and whose nodes circle backwards in 18.61 years. When the Moon’s ascending node lies near the vernal equinox, the Moon’s orbit is tilted about 28.6 degrees to the equator; nine years later, about 18.3. The torque on the bulge rises and falls with that tilt, and the equator nods.

A nod of the equator slides its crossing with the ecliptic back and forth along the ecliptic by an angle $\Delta\psi$, the nutation in longitude, whose main term is 17.2 arcseconds. Projected onto the equator, which is the direction right ascension and sidereal time are measured along, that slide becomes

$$E_e = \Delta\psi\cos\varepsilon,$$

the equation of the equinoxes — the difference between sidereal time counted from where the equinox really is and sidereal time counted from where it would be if it only precessed. Converted to time at fifteen arcseconds a second, the main term is ±1.052 seconds.

That second is the whole of the hero figure’s swing, and it is larger than anything else in this essay by two orders of magnitude. A star timed across the meridian today crosses up to a second away from the time a smoothly running sidereal clock predicts, and the difference is not an error in the clock or in the rotation. It is a change in what the clock is counting from.

It was found by a man who was not looking for it. James Bradley had discovered the aberration of starlight in 1728 from the positions of one star through one year, and he kept measuring the same stars with the same zenith telescope afterwards. The annual ellipse he had explained did not close exactly; a residual of a few arcseconds wandered from year to year, and he recognised that its period matched the circling of the Moon’s nodes. He waited for a whole cycle before publishing, and announced nutation in 1748, after nineteen years of observations. The second of sidereal time in the hero figure is that residual, projected onto the equator.

A second sounds like nothing to worry about, and for most of the history of the subject it was worth a great deal. A navigator who fixed longitude from the stars was converting a time into an angle at fifteen arcseconds a second, so a second of error in sidereal time was fifteen arcseconds of longitude — close to half a kilometre at the equator. Every nautical almanac therefore tabulated the correction, and every observatory that distributed time kept apparent and mean sidereal time as two separate quantities.

The faster wiggles on top of it #

The node term is not the only nutation, and a shorter span shows what rides on it.

The half-yearly term is the Sun’s version of the same torque. Twice a year the Sun is at its greatest distance from the equatorial plane, at the solstices, and twice it lies in it, at the equinoxes, so its twist on the bulge rises and falls twice a year: ±0.081 seconds. The fortnightly term is the Moon doing the same thing twice in each month: ±0.014 seconds. Both are genuine, both are in every precise ephemeris, and neither is visible to a transit timed to a tenth of a second.

The four terms drawn here are the four largest. The full series used by the international conventions has well over a thousand, and it takes most of them to reach the level of a microsecond. That does not change the shape of the story: the equation of the equinoxes is a sum of periodic terms set by the Sun and the Moon, with one term, the node, carrying nearly all of the amplitude.

The steady part, and where it went #

Take the nutation away and a smooth sidereal time remains — mean sidereal time, counted from the mean equinox. That equinox still moves. It regresses along the equator, westward, at about 46.1 arcseconds a year in right ascension, as the pole traces its slow circle.

That regression is what separates the sidereal day, one turn relative to the mean equinox, from the stellar day, one turn relative to a direction with no motion along the equator. The equinox moves to meet the meridian, so it is reached slightly early and the sidereal day is the shorter of the two. Eight milliseconds a day accumulates to three seconds a year and to 307 seconds a century, and that number — 4612 arcseconds a century, divided by fifteen — is the linear term in the formula every observatory uses to turn a date into a sidereal time.

A quadratic term that is not the Earth #

The formula is worth writing out, because it contains a surprise. In the IAU 2006 conventions, Greenwich mean sidereal time is the Earth rotation angle plus a polynomial in time:

$$\mathrm{GMST} = \theta_{\rm ERA} + 0.014506'' + 4612.156534''\,t + 1.3915817''\,t^2 - 0.00000044''\,t^3 - \ldots$$

with $t$ in Julian centuries from J2000. The rotation angle $\theta_{\rm ERA}$ is linear in time by definition. So everything that makes sidereal time depart from a straight line is in the polynomial, and it is natural to guess that the quadratic term is the Earth’s rotation slowing down under tidal friction. That guess is wrong, and the reason is the whole point of the formula.

The polynomial is GMST minus the rotation angle, which is to say it is the accumulated motion of the equinox along the equator, and nothing else. The Earth’s rotation does not appear in it at all: the slowing of the rotation — about 1.75 milliseconds a century in the length of the day, which over two and a half thousand years has lost the Earth hours — is carried separately, in the difference between the rotation’s own time scale and the uniform time scale of the ephemerides. The quadratic term is the precession accelerating. The rate at which the equinox regresses is itself changing, slowly, because the obliquity is decreasing by nearly half an arcsecond a year and because the ecliptic the Sun and the Moon are torquing against is being turned by the planets. A regression that speeds up accumulates as $t^2$.

The two plots put the term in proportion. Across the whole history of telescopic astronomy it amounts to a tenth of a second, and it only becomes seconds over a millennium. It matters now because the rotation angle is measured, by interferometry against distant quasars, to a few microseconds of time, and a formula that is wrong by a tenth of a second in its zero point is not a formula that such a measurement can be compared with.

What the acceleration does to the length of a day #

A clock whose zero regresses at a changing rate has a day whose length changes, even if the rotation underneath does not.

The change is five microseconds a century, three hundred and forty times smaller than the tidal lengthening of the day and exactly as real. It is a property of the reference, and it would be there on a planet whose rotation was perfect. The two numbers are worth keeping apart because they are routinely confused: the sidereal day grows longer mostly because the Earth is slowing, and its difference from the stellar day grows larger only because the equinox is speeding up.

Why the equinox was ever the zero #

It is fair to ask why anyone counted from a point with so many motions, and the answer is that for most of the history of the subject there was nothing better to count from. A zero point for the sky has to be something that can be located by observation, and before radio astronomy the only large-scale structure that could be located from the ground to a fraction of an arcsecond was the Sun’s path. The equinox is where that path crosses the equator, and it can be found by timing: measure the Sun’s declination around the start of spring on successive days, and interpolate to the moment it passes through zero. No star has to be assumed fixed, and every observatory in the world finds the same point.

The stars were the less trustworthy reference, not the more. Each has its own motion across the sky, and until the twentieth century nobody knew which stars moved least. So the equinox’s precession, its acceleration and its nutation were the price of a zero point that could be measured, and paying it through a formula was a reasonable bargain. It stopped being one only when the quasars supplied directions that could be measured more precisely than the equinox and had no motion to correct at all.

A zero point defined not to move #

Every correction so far has had the same cause. The zero of the clock is a point on the equator that moves along the equator, and each of its motions — steady, accelerating and nodding — reappears as a term in sidereal time that has nothing to do with the rotation it is supposed to measure.

The fix adopted by the International Astronomical Union in 2000, and given its present names in 2006, is to count the rotation from a different point. The celestial intermediate origin is a point on the equator defined kinematically: as the equator moves, the origin moves with it but never slides along it. Its motion is always perpendicular to the equator, so at every instant it has no component in the direction a rotation is measured. The angle between that origin and a meridian fixed to the Earth is the Earth rotation angle, and because nothing about its zero point moves in the relevant direction, it is exactly a linear function of the Earth’s rotation:

$$\theta_{\rm ERA} = 2\pi\,\big(0.7790572732640 + 1.00273781191135448\,D\big),$$

with $D$ the number of days of the Earth’s own time scale since J2000. The second number in that expression is the ratio of the stellar day to the mean solar day, and it differs from the sidereal ratio of the four-minute essay in the eighth decimal place — which is the 8.4 milliseconds, now absent from the clock.

Sidereal time did not disappear. It is still needed to point a telescope at a right ascension counted from the equinox. What changed is its status. It is now derived: the rotation angle, less a single quantity called the equation of the origins, which is the angle along the equator between the intermediate origin and the true equinox and which carries both the accumulated precession and the nutation. The quantity that is measured and modelled as the Earth’s rotation is the one with no equinox in it, and the equinox’s three motions are applied afterwards, as a correction for a choice of coordinates.

The change reached the definition of time as well. Universal time, UT1, was for most of its history defined through sidereal time, and so it inherited every motion of the equinox that the formula for sidereal time had to remove. Since 2003 it has been defined instead as a linear function of the Earth rotation angle, with the constant chosen so that nothing changed at the switch. The astronomical time scale that tracks the Earth’s rotation is now, by definition, the rotation and nothing else, and the long formula with its quadratic term has moved from the definition of time into the conversion to coordinates.

The observations had already moved there. The rotation is determined by very-long-baseline interferometry against quasars, which define a frame made of objects too distant to show any motion. Nothing in that measurement refers to the Sun’s crossing of the equator in March. Counting the rotation from the equinox would mean computing an equinox, with all its motions, in order to subtract it again.

The same idea, three centuries earlier #

The move from sidereal time to the rotation angle has a precedent that every sundial owner has met. The true Sun is a bad clock: it moves unevenly along an ecliptic tilted to the equator, and the time it keeps drifts by up to sixteen minutes from a uniform one. The fix, adopted in the eighteenth and nineteenth centuries, was a fictitious mean Sun that moves uniformly along the equator, and a correction — the equation of time — to get from one to the other.

The equation of the equinoxes is the same device applied to the other clock. A real reference point moves irregularly along the equator; a fictitious one is defined that does not; and a correction with the word equation in its name carries the difference. What is striking is the scale. The Sun’s reference point moves by degrees and the correction is minutes; the equinox moves by arcseconds and the correction is a second. The ratio between them is the ratio between a year and a nineteen-year nod, and the two corrections were invented for exactly the same reason: nobody can build a clock that runs at the rate of a point that does not move uniformly.

What the figures leave out #

The nutation is truncated. Four terms carry the equation of the equinoxes to about a hundredth of a second, which is enough to show its shape and not enough to use it. The full series has well over a thousand periodic terms and a few very small non-periodic ones.

The polynomial is a fit. Its coefficients come from a precession theory adjusted to decades of observation, and the higher terms are not physically meaningful beyond a few thousand years. The ten-century figure is drawn to show the form of the quadratic term, not to predict sidereal time in the year 3000.

Polar motion is absent. The Earth’s crust wanders relative to its rotation axis by a few metres, which changes where a meridian points and enters the conversion from the rotation angle to a telescope’s pointing. It does not change the rotation angle’s zero point, which is why it has no place in these figures.

And the rotation itself is taken as given. Every figure here holds the Earth’s rotation fixed in order to isolate the motion of the reference. The rotation is not fixed: it fluctuates by milliseconds over seasons and decades and slows over centuries, and those changes are measured precisely because a reference that does not move has been separated from them.

Still open: when a coordinate becomes a date #

Everything above concerns what the sidereal clock counts from. The other half of its use is what it is counted against: a sidereal time is the right ascension on the meridian, and a solar time is how dark the sky is. The two drift apart by one turn a year, so a right ascension is not only a position but a season — the time of year at which an object is on the meridian at midnight, and the span of nights on which it is high enough and the sky dark enough to observe it at all. Where a star is depends on who is asking; when it can be seen depends on both clocks at once.