Astronomy & Celestial Mechanics
the sky, computed
The observed sky

A wobble that should have stopped

The Earth's rotation pole wanders across its own crust in a circle a few metres wide. A rigid Earth would do it in 305 days; it takes 433, and the difference is a measurement of the planet's elasticity. At the observed damping it should have died out within a human lifetime, and it has not.

Assumes Precession, Oblateness and Moment of inertia.

16 min read 9 figures The sky as a clockSmall perturbations, large consequences

Precession and nutation are motions of the Earth’s rotation axis in space, driven by the couple the Sun and Moon exert on the equatorial bulge. This essay is about a third motion which is not driven by anything, does not move the axis in space at all, and is measured in metres.

A rigid body that is not spinning exactly about a principal axis wobbles. The axis of rotation traces a cone about the axis of symmetry, at a rate set entirely by the body’s own shape — no external torque required, and no change to the angular momentum vector, which stays fixed in space while the body moves around it. For the Earth the relevant number is its dynamical ellipticity, H=(C−A)/C=0.0032737H = (C-A)/C = 0.0032737, and the free period follows immediately:

TEuler=1H sidereal days=305 days.T_{\rm Euler} = \frac{1}{H}\ \text{sidereal days} = 305\ \text{days}.

The Earth does it in 433.

A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 150 milliarcseconds and the annual wobble at 90. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically.
Fig. 1 Left, the path of the rotation pole across the Earth’s own crust over thirteen years, as the sum of two circular motions: the 433-day Chandler wobble and a forced annual one. The spiral is a beat, and its period measured off the drawn path is 6.4 years against the 6.39 the two frequencies require. Right, the same path’s distance from the mean pole against time. The whole excursion is about ten metres of ground.

The 42 per cent that is a measurement

The discrepancy between 305 and 433 days is not an error. It is the Earth failing to be rigid, and the failure is quantitative.

When the rotation pole moves relative to the body, the centrifugal bulge moves with it — and a deformable Earth’s actual bulge follows, partially. The redistributed mass changes the moments of inertia, and the changed moments alter the wobble period. The correction depends on how much the body yields, which is the Love number k2k_2 for the whole planet, and on how much the oceans yield, which is a separate and comparably large effect since the oceans redistribute themselves in response to the changed centrifugal potential.

Working the correction out and requiring it to give 433 days from 305 returns a whole-Earth k2k_2 of about 0.30. That value is consistent with what is measured from the solid Earth tide, which is a completely different observation — a periodic deformation forced by the Moon rather than a free oscillation of the planet — and the agreement is the reason the interpretation is secure.

The moment-of-inertia factor against core size, for five density contrasts. What a moment of inertia can say. The vertical axis is C/MR², the polar moment divided by what a hoop of the same mass and radius would have, and for a uniform sphere it is exactly 2/5 — the value both ends of every curve return to, because a body with no core and a body that is entirely core are both uniform. In between, the ratio dips: a moment weights mass by the square of its distance from the axis, so moving density inward lowers it, and the deeper the dip the more differentiated the body. The five curves are five core-to-mantle density ratios, and the minimum moves down and inward as that ratio grows — a denser core reaches its greatest effect at a smaller radius, because beyond that the core is so much of the body that the whole thing looks uniform again — 1.5 gives 0.370 at 75 per cent of the radius, 2 gives 0.348 at 72 per cent of the radius, 3 gives 0.317 at 69 per cent of the radius, 5 gives 0.279 at 65 per cent of the radius, 10 gives 0.231 at 59 per cent of the radius. Two things the figure makes visible are worth more than the numbers. The relation is not invertible: one measured factor is met by two core sizes on each curve and by a whole family of curves, so a moment of inertia alone never gives a core radius — it gives a constraint that a second measurement has to be combined with. And the whole diagram lives between 0.4 and about 0.15, which is a narrow range for so much physics; distinguishing a large core from a small one means measuring C/MR² to a per cent or two, and every technique for doing so is a way of watching the body turn.
Fig. 2 The other quantity in the period formula. The dynamical ellipticity is a difference of two principal moments of inertia divided by one of them, so it is a statement about how the Earth’s mass is distributed both radially and with latitude. The radial part — how concentrated the mass is towards the centre — is what fixes the overall moment of inertia coefficient, and for the Earth it is 0.3307 rather than the 0.4 a uniform sphere would give. Both numbers enter the wobble, and both come from independent measurements.

The beat

There are two polar motions of comparable size and they are not the same kind of thing.

The Chandler wobble is free: nothing drives it, and its period is a property of the Earth. The annual wobble is forced: the seasonal redistribution of air, water and ice moves mass around on the surface, changing the moments of inertia at a period of exactly one year.

Two circular motions of similar amplitude and periods of 433 and 365.25 days beat against each other. The beat period is

1∣1/365.25−1/433∣=2334 days=6.4 years,\frac{1}{|1/365.25 - 1/433|} = 2334\ \text{days} = 6.4\ \text{years},

and the polar path alternates between a wide spiral, when the two are in phase, and a nearly closed small circle when they oppose. That six-year modulation is the most obvious feature of the record and it was the first thing measured about polar motion after the wobble itself.

The discovery, and why it took until 1891

The wobble is small — the pole moves by a few tenths of an arcsecond, which is a few metres — and detecting it required latitude measurements good to a fraction of that, made continuously, at more than one place.

The reasoning that made it findable is worth stating. If the rotation pole moves within the Earth, then every observatory’s latitude changes, because latitude is defined relative to the rotation axis. The changes at two observatories on opposite sides of the planet must be equal and opposite: when the pole moves towards one, it moves away from the other. That anti-correlation is the signature, and it distinguishes a real polar motion from a systematic error in either instrument, from local refraction, or from a station moving.

An anti-correlated pair of latitude records is therefore the minimum experiment, and once the effect was established a network was built to monitor it — six stations on one parallel, chosen so that the pole’s coordinates could be extracted from their combined latitudes without any station being privileged. That network ran for most of a century and its records are still the long baseline against which the modern radio measurements are checked.

The 305-day prediction had been in the literature since Euler and had been looked for and not found, which is a useful cautionary example: the search had been for a signal at the predicted period, and the signal was at a period 42 per cent longer. What made the difference was analysing the record for whatever periodicity was in it rather than testing for the one expected.

The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close.
Fig. 3 The much larger motion the small one sits on top of. The celestial pole traces a circle of 23.4 degrees radius among the stars once in 25,772 years, driven by the couple on the equatorial bulge. That is a motion of the axis in space and it does not move the pole across the Earth’s surface at all. The Chandler wobble is the opposite: the axis stays put in space and the Earth moves around it, by a few metres.

What is actually measured

Polar motion is measured by comparing a set of positions on the Earth with a set of directions in the sky, repeatedly.

The primary technique is very long baseline interferometry: radio telescopes thousands of kilometres apart observe the same quasar, and the difference in arrival time of the wavefront delivers the projection of the baseline vector onto the source direction. Quasars are distant enough to have no measurable proper motion, so the frame they define is fixed — which is the sense in which where a star is depends on who is asking, and any change in the delay is a change in the Earth. The measurement is good to about a tenth of a milliarcsecond, which is three millimetres on the ground, and it is made every day. Satellite laser ranging and the global navigation satellite systems contribute the same parameters at higher cadence and lower long-term stability, and the published Earth orientation parameters are a combination.

The damping problem

Every free oscillation decays. The Chandler wobble’s decay is set by how much energy the Earth dissipates as it deforms, expressed as a quality factor. Estimates of the wobble’s QQ from the width of the spectral peak fall in the range of about 50 to 100, which corresponds to a damping time of a few decades to about a century.

The wobble has been observed continuously since the 1890s, and it has not decayed. Its amplitude varies — it was very small for a couple of years around 1930, and it has been unusually small again recently — but it recovers, which a freely decaying oscillation does not do.

Something is exciting it, continuously, at a rate that on average balances the damping. Finding what took a century.

What the excitation is

The candidate had to satisfy two conditions: it had to have power at 433 days, and it had to be able to change the Earth’s moments of inertia or apply an equatorial torque by enough.

Earthquakes were the favourite for a long time and do not work: the redistribution of mass in even the largest earthquakes is orders of magnitude too small, and the timing of large events does not correlate with jumps in the wobble.

The answer, established once ocean models became good enough to compute it, is fluctuating pressure at the bottom of the ocean, with atmospheric pressure fluctuations contributing about a third. Neither is periodic. Both have broadband power, some of which lands at the wobble’s frequency, and a resonantly driven oscillator picks out its own frequency from a broadband input and ignores the rest.

That last point is the reason the excitation was hard to identify. There is no 433-day signal anywhere in the atmosphere or the oceans, and there does not need to be: a lightly damped resonance driven by noise oscillates at its own frequency with an amplitude set by the noise power at that frequency, and shows nothing at the frequencies where the noise is loudest.

The excitation budget can be checked rather than merely asserted. Ocean and atmosphere models compute the mass redistribution and the surface stresses independently of any wobble measurement, and the resulting excitation function can be integrated forward through the wobble’s own equation of motion. The predicted amplitude and phase track the observed ones over the decades where both are available, to within the models’ own uncertainties. That is the closure of the argument: an input measured elsewhere, put through a known transfer function, reproducing an output measured here.

A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 220 milliarcseconds and the annual wobble at 60. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically.
Fig. 4 The same construction with the free wobble much larger than the annual one, which is roughly what the record looked like in the middle of the twentieth century. The beat is still there and still at 6.4 years, because the beat period depends only on the two frequencies — but the path no longer collapses to a small circle when the two oppose, because they no longer cancel. The observed amplitude ratio has varied over the century of record, and the near-disappearance of the wobble around 1930 was a period when the two were comparable and out of phase.

The pole that is not coming back

Superimposed on the wobble is a slow drift of the mean pole itself — the centre the wobble circles about is moving, at about ten centimetres a year, and it has been doing so for as long as the record exists.

For most of the twentieth century the drift was towards the seventieth meridian west, in the direction of Hudson Bay, and its cause is the same one that lifts Scandinavia: the crust is still rebounding from the removal of the last ice sheets, mass is still flowing back into the regions the ice depressed, and a redistribution of mass on that scale moves the rotation axis relative to the crust.

That is a measurement of mantle viscosity by a route with no seismology in it. The drift rate depends on how fast the mantle flows, so matching the observed rate constrains the viscosity of the material a thousand kilometres down — which is the same quantity the rebound of a coastline constrains, measured on a global rather than a regional scale.

Then, around the year 2000, the drift changed direction. It turned eastward by nearly ninety degrees, and the turn was abrupt on the timescale of the record.

The explanation is that a new mass redistribution has become comparable to the old one. Ice loss from Greenland and from Antarctica, and the depletion of continental groundwater, are moving enough mass to shift the pole — and the resulting drift, computed independently from gravity-field measurements of where the water has gone, reproduces both the magnitude and the direction of the change.

A measurement made since the 1890s to keep track of where a telescope is pointing has become a record of how much ice has melted, and it is one of the few geophysical records long enough to show the change against a well-characterised background.

The beat depends on the ratio of the two amplitudes, and the record shows both of them varying.

A wobble that should have stopped seventy years ago. Left, the path of the Earth's rotation pole across its own crust over 13 years, as the sum of two circular motions: the 433-day Chandler wobble at 60 milliarcseconds and the annual wobble at 90. The spiral is a beat, and its period measured off the drawn path is 6.39 years against the 6.39 the two frequencies require. Right, the same path's radius against time. Two numbers in this figure are the argument. The first is the Chandler period itself: a rigid Earth of dynamical ellipticity 0.0032737 would wobble freely at 305 days, and the observed 433 is 42 per cent longer because the Earth deforms under its own wobble and the oceans move with it — the period is a measurement of the planet's elasticity, made by watching a free motion rather than by forcing anything. The second is the damping: at a quality factor of about 100 the wobble should decay in 38 years, and it has been running for as long as anyone has watched. Something is exciting it continuously, and the excitation is fluctuating pressure at the bottom of the ocean and in the atmosphere. What the figure cannot show is the excitation itself, which is not periodic and is only visible statistically.
Fig. 5 The same two motions with the Chandler amplitude reduced below the annual one, which is roughly what the record showed around 1930 when the wobble nearly died. The beat pattern inverts: the annual term dominates and the six-year envelope becomes a modulation on it rather than the other way round.
The wobble inside the wobble. Left: the two components of nutation over 10 years, from the four largest terms of the standard series. The long wave is the regression of the Moon's node in 18.613 years, which is where nearly all of it comes from; the ripple on it is the semi-annual solar term at 1.3″ and the semi-monthly lunar one at 0.2″. Right: the loop the pole actually traces over one node cycle, in arcseconds on the sky, with the mean pole at the centre. The loop is 14.89″ by 19.90″ — taller than it is wide, because the longitude term is foreshortened by sin ε while the obliquity term is not, which is the one thing a schematic of this is always drawn getting wrong. Over the same 18.6 years precession itself carries the pole 936″ along its circle, 47 times the loop's own height, so nutation is a wobble on a path and not a path. It is nonetheless 99,480 times the 0.2 mas astrometry of a modern catalogue, which is why a position has to say whether it is referred to the mean pole or the true one.
Fig. 6 And the third motion of the same axis, over ten years rather than forty. Nutation is forced rather than free, so its amplitude and phase are predictable to microarcseconds from the Moon’s orbit — the exact opposite of the Chandler term, and the reason the two are separated before anything is analysed.

Why the period is not quite constant either

A further complication, and a real one. The observed Chandler period is not exactly 433 days, and the observed amplitude and phase both wander. For a resonance driven by noise, that is expected: the phase of a noise-driven oscillator performs a random walk, and an apparent period estimated from a finite stretch of record differs from the true one by an amount that depends on how much the phase happened to drift.

So the widely quoted “433 days” is an estimate from a century of data with a formal uncertainty of a couple of days, and attempts to extract a secular change in the period — which would be a change in the Earth’s own elasticity — have to contend with a wander that mimics one.

The same wander sets a limit on what the quality factor can be known to. A resonance’s QQ is ordinarily read from the width of its spectral peak, and the peak here is broadened by two things at once: the dissipation, which is what is wanted, and the phase random walk, which is not. The two cannot be separated from the shape of the peak alone, and the published range of 50 to 100 is wide for exactly that reason rather than because the record is short. Estimates made by fitting the excitation and the response jointly — using the ocean and atmosphere models to supply the input rather than treating it as unknown — are tighter, and they sit near the low end, which implies a damping time closer to thirty years than to a century.

That matters for the picture the essay has been building. A shorter damping time means the wobble must be re-excited more vigorously, so the ocean-bottom pressure fluctuations have to supply more power at 433 days than a weakly damped oscillator would need. The budget still closes, but with less margin, and the near-disappearance around 1930 stops looking like an accident of phase and starts looking like an interval in which the driving genuinely fell away for a few years.

The wobble inside the wobble. Left: the two components of nutation over 40 years, from the four largest terms of the standard series. The long wave is the regression of the Moon's node in 18.613 years, which is where nearly all of it comes from; the ripple on it is the semi-annual solar term at 1.3″ and the semi-monthly lunar one at 0.2″. Right: the loop the pole actually traces over one node cycle, in arcseconds on the sky, with the mean pole at the centre. The loop is 14.89″ by 19.90″ — taller than it is wide, because the longitude term is foreshortened by sin ε while the obliquity term is not, which is the one thing a schematic of this is always drawn getting wrong. Over the same 18.6 years precession itself carries the pole 936″ along its circle, 47 times the loop's own height, so nutation is a wobble on a path and not a path. It is nonetheless 99,480 times the 0.2 mas astrometry of a modern catalogue, which is why a position has to say whether it is referred to the mean pole or the true one.
Fig. 7 The motion this is not, drawn for contrast. Nutation is forced, at periods set by the Moon’s node and the Sun’s longitude, and it moves the axis in space rather than in the body. Its largest term has an amplitude of about nine arcseconds — three hundred metres at the pole, thirty times the Chandler wobble — and it is predictable to microarcseconds because its drivers are orbital. The free wobble is a hundredth the size and cannot be predicted at all beyond a few years.

What the parameters are for

The measurement is published daily and it is worth saying who needs it, because the applications explain why the precision is pushed as hard as it is.

Every observation made from the Earth’s surface is recorded in a frame attached to the crust and has to be expressed in one attached to the sky, and the transformation between them is precession, nutation, the Earth’s rotation angle, and polar motion — five parameters, of which two are polar motion. Get them wrong and a position on the sky is wrong by the corresponding amount.

For most astronomy a tenth of an arcsecond is irrelevant. For three applications it is not.

Spacecraft navigation is the first. A deep-space trajectory is measured by ranging from stations whose positions are known in the terrestrial frame and whose targets are described in the celestial one, so an error in the transformation is an error in the target’s position of the same size — which at Mars is several kilometres.

Satellite geodesy is the second and it is circular in an informative way. The global navigation systems determine positions on the ground by reference to satellite orbits, and the orbits are computed in a celestial frame; so the systems need the orientation parameters, and they also contribute to measuring them. The circularity is handled by solving for everything at once.

The third is the definition of time. Universal time is defined by the Earth’s rotation angle, which is one of the same five parameters, and its difference from atomic time is what leap seconds correct. The same daily solution that returns polar motion returns that difference.

A quantity of a few metres, measured on a planet, is a required input to pointing at another one — and the requirement is why a network of radio telescopes observes quasars on a schedule rather than only when somebody wants an image.

What cannot be predicted, and why that is the unusual part

There is a second feature of these parameters that separates them from everything else in the transformation between the two frames, and it is worth stating because it inverts the usual arrangement in astronomy.

Precession and nutation are driven by the gravitational torques of the Moon and Sun on the equatorial bulge. Those are orbital, so they are predictable — a model published today gives the orientation of the pole in 2050 to microarcseconds, and nobody needs to observe anything to use it.

Polar motion and the rotation angle are not predictable at all beyond a few weeks. The excitation is atmospheric and oceanic, so predicting the Earth’s orientation a month ahead requires predicting the weather a month ahead, and that is the same problem with the same limit.

So two of the five quantities come from a theory and three come from a measurement made continuously, and the products are published as a daily series with short predictions attached whose uncertainty roughly doubles within a fortnight. A manoeuvre planned three weeks out uses a predicted orientation; one executed today uses a measured one.

The inversion is the interesting part. In most of this subject the model is trusted and the observation is the uncertain thing being fitted. Here the observation is routine and cheap, and the future is what cannot be supplied at any price — which is why the schedule of quasar observations cannot lapse, and why a gap in it is not recoverable afterwards.

Two more readings, of the quantity that sets the period and of the equilibrium the same axis has elsewhere.

The moment-of-inertia factor against core size, for five density contrasts. What a moment of inertia can say. The vertical axis is C/MR², the polar moment divided by what a hoop of the same mass and radius would have, and for a uniform sphere it is exactly 2/5 — the value both ends of every curve return to, because a body with no core and a body that is entirely core are both uniform. In between, the ratio dips: a moment weights mass by the square of its distance from the axis, so moving density inward lowers it, and the deeper the dip the more differentiated the body. The five curves are five core-to-mantle density ratios, and the minimum moves down and inward as that ratio grows — a denser core reaches its greatest effect at a smaller radius, because beyond that the core is so much of the body that the whole thing looks uniform again — 2 gives 0.348 at 72 per cent of the radius, 10 gives 0.231 at 59 per cent of the radius. Two things the figure makes visible are worth more than the numbers. The relation is not invertible: one measured factor is met by two core sizes on each curve and by a whole family of curves, so a moment of inertia alone never gives a core radius — it gives a constraint that a second measurement has to be combined with. And the whole diagram lives between 0.4 and about 0.15, which is a narrow range for so much physics; distinguishing a large core from a small one means measuring C/MR² to a per cent or two, and every technique for doing so is a way of watching the body turn.
Fig. 8 The moment-of-inertia factor against core size for the two density contrasts that bracket a rocky planet. The Chandler period depends on the dynamical ellipticity, which depends on this factor, so the same interior model that dates the Earth’s differentiation also sets the free wobble’s period.
Where a spin axis can sit still. Level curves of the Colombo Hamiltonian for a spin axis precessing at α cos ε about an orbit normal that is itself precessing at rate g about the invariable pole, with an orbit inclination of 1.5 degrees and α/g = 2. The horizontal axis is the longitude of the spin axis measured from the orbit's node and the vertical axis is the obliquity; the curves are traced by finding where the Hamiltonian crosses each level on a grid rather than drawn as ellipses that look right. The 4 marked points are the Cassini states, the obliquities at which the two precessions keep step so that the axis holds a fixed geometry — 1.5°, 59.5°, 60.5°, 179.5° — and each one is checked to satisfy α sin ε cos ε + g sin(ε − I) = 0 to a part in a billion. How many there are is not a matter of degree: four exist when α/g exceeds the three-halves power of the sum of sin I and cos I each raised to two thirds, which here is 1.135, and two when it does not, so a planet whose spin slowly changes can find two of its equilibria annihilate. Closed curves around a state are libration, and an axis on one of them oscillates in obliquity for ever without drifting; the curves that run the full width are circulation. What this cannot show is the real Solar System, which has not one orbital precession frequency but a dozen, and it is their overlap rather than any one of them that makes an obliquity chaotic.
Fig. 9 The Colombo Hamiltonian, whose level curves show where a spin axis can sit still relative to a precessing orbit. The Earth’s axis is not near any of these states and the Moon’s is, which is why the Moon has a fixed obliquity of one and a half degrees and the Earth’s wanders.

Where the ladder goes

The Chandler wobble is the free oscillation of the whole planet in one degree of freedom, and the same measurement programme detects others. The free core nutation is a second free mode, in which the liquid core’s rotation axis is offset from the mantle’s; it has a period of about 430 sidereal days in space, close enough to the annual retrograde nutation to be resonantly amplified by it, and its detection is a direct measurement of the flattening of the core–mantle boundary.

The other direction is towards the rest of Earth orientation as a geophysical instrument. Length of day varies by milliseconds on a decade timescale, far more than the atmosphere can explain, and the residual is angular momentum exchanged between the mantle and the liquid core — which makes a clock comparison into a measurement of a flow nobody can see, and is a cousin of weighing a planet’s core by whether its heavy material sank, in the same spirit as reading a planet’s core off its libration.

Where this goes next

The path of the celestial pole over 25,772 years. The circle the Earth's rotation axis traces among the stars, at a radius equal to the obliquity, with the bright stars that fall near it and the years at which each is closest. Polaris is the pole star for a few centuries either side of now, and nothing else on the circle is nearly as close. The pole star has a shelf life, and the sky has a slow hand The Earth's axis traces a circle among the stars once every 25,772 years. Polaris is at the pole now, was not four thousand years ago, and will not be in two thousand more. The wobble inside the wobble. Left: the two components of nutation over 40 years, from the four largest terms of the standard series. The long wave is the regression of the Moon's node in 18.613 years, which is where nearly all of it comes from; the ripple on it is the semi-annual solar term at 1.3″ and the semi-monthly lunar one at 0.2″. Right: the loop the pole actually traces over one node cycle, in arcseconds on the sky, with the mean pole at the centre. The loop is 14.89″ by 19.90″ — taller than it is wide, because the longitude term is foreshortened by sin ε while the obliquity term is not, which is the one thing a schematic of this is always drawn getting wrong. Over the same 18.6 years precession itself carries the pole 936″ along its circle, 47 times the loop's own height, so nutation is a wobble on a path and not a path. It is nonetheless 99,480 times the 0.2 mas astrometry of a modern catalogue, which is why a position has to say whether it is referred to the mean pole or the true one. The wobble inside the wobble Precession and nutation are the same torque. The difference is that the Moon's orbital plane turns once in 18.6 years, so part of the pull oscillates instead of accumulating — and a catalogue position is not a direction until it says which pole it is measured from. A clock that has lost 5.7 hours in 2720 years. ΔT = TT − UT1, the accumulated difference between a uniform time scale and the Earth's own rotation, from 700 BC to 2020, on a logarithmic scale. The points are the published record; the smooth curve is the parabola 31.94·u² fitted to the entries at or before 1000, with u in centuries from 1820. It is a parabola and not a line because the Earth is not merely slow, it is slowing: a day lengthening at a constant rate makes a clock fall behind by the integral of the lag. That coefficient is a length-of-day rate and nothing else — 31.94 = ½ × r × 36525 days per century gives r = 1.75 ms per century, against the 1.78 ms computed from tidal angular momentum with no eclipse anywhere in the derivation. At 700 BC the offset is 5.7 hours, which is 85° of the Earth's rotation, and that is why the ancient measurements are records of the place a total eclipse was seen from rather than of the hour: the hour was never written down accurately enough to matter, and the shadow's track on the ground was. Six kinds of second The Earth is a clock that loses, and it has lost five and a half hours since 700 BC. That number was measured from the places ancient eclipses were seen from, not the times they were seen at — because the record carries a longitude and no clock.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Angular momentumBeat frequencyChandler wobbleDynamical ellipticityEarth orientation parametersEulerian periodFree precessionLength of dayLove numberMoment of inertiaOcean bottom pressurePolar motionQuality factorVery long-baseline interferometry