Astronomy & Celestial Mechanics
the sky, computed
The observed sky

A second that depends on where the clock is

A clock in orbit runs fast, a clock at the bottom of a gravity well runs slow, and a clock at rest far from the Sun runs faster than anything on Earth by fifteen parts in a thousand million. Astronomy therefore has four different seconds, two of them longer than the others, and an ephemeris has to declare which one it is tabulated against.

Assumes Timescales and Ephemerides.

16 min read 7 figures The sky as a clockMeasuring the unreachable

Six kinds of second have already appeared in this collection, and every one of them was a different way of counting. This essay is about a different problem entirely: four seconds that count the same way and are not the same length.

A clock measures the interval along its own worldline and nothing else. Two clocks in different places, moving differently, in different gravitational potentials, accumulate different amounts of it, and the difference is not an error in either of them.

That is a familiar statement and it has a consequence that is easy to underestimate: a time coordinate for the solar system cannot be the reading of any clock. It has to be a construction, agreed upon, whose relation to each real clock is specified. Astronomy uses four such constructions, and the differences between them are large enough that using the wrong one is a gross error rather than a refinement.

TCB has gained 23.5 seconds on TT since 1977. How far four of the solar system's time scales have drifted apart, against years since they were set equal. TT is the time a clock on the Earth's geoid keeps, and it is flat here by definition. TCG is the time a clock at rest just outside the Earth's gravitational well would keep, and it runs faster by seven parts in ten thousand million — 0.0220 seconds a year. TCB is the time a clock at rest outside the Sun's well would keep, and it runs faster by fifteen parts in a thousand million, or 0.4893 seconds a year. TDB is TCB with that rate divided out so that it stays within milliseconds of TT, which is what makes it usable as an ephemeris argument and what makes it a coordinate rather than a proper time. The differences are constants and they are not corrections: an ephemeris tabulated against one of these and evaluated against another is wrong by the whole of this plot.
Fig. 1 How far four of them have drifted apart since they were set equal in 1977. The scale a clock on the Earth’s geoid keeps is flat by definition. A clock at rest just outside the Earth’s well runs faster by seven parts in ten thousand million; one at rest outside the Sun’s well runs faster by fifteen parts in a thousand million, which is nearly half a second a year. The fourth scale is the third with that rate divided out, so that it stays usable as an argument to an ephemeris.

The situation is not a peculiarity of relativity. Any measurement made from inside a system needs a convention about what is being held fixed, and the older astronomical time scales had exactly the same character for a different reason: ephemeris time was defined by the Earth’s orbit because the Earth’s rotation had turned out to be irregular, and it was a coordinate rather than a clock reading in the same sense. What relativity added was that the discrepancy is now calculable in advance rather than discovered afterwards.

The two rate constants, and where they come from

Every difference in the figure comes from one expression. To the accuracy that matters, a clock’s rate relative to a coordinate time is

dτdt=1−Uc2−v22c2,\frac{d\tau}{dt} = 1 - \frac{U}{c^2} - \frac{v^2}{2c^2},

with UU the gravitational potential at the clock and vv its speed in the coordinate frame. Both terms slow the clock, and both are small: on the Earth’s surface the potential term is about 7×10−107\times10^{-10} and the rotational velocity term about 1.2×10−121.2\times10^{-12}.

The Earth’s constant. Averaged over the Earth’s surface — more precisely, on the geoid, the surface of constant effective potential that sea level defines — the sum of the two terms is LG=6.969290134×10−10L_G = 6.969290134\times10^{-10}. That number is not measured any more; it is defined, and Terrestrial Time is defined as the coordinate time that runs slow by exactly that fraction relative to a clock at rest at infinity in the Earth’s frame.

The Sun’s constant. The corresponding quantity for the solar system, averaging the Earth’s orbital speed and the Sun’s potential at the Earth, is LB=1.550519768×10−8L_B = 1.550519768\times10^{-8} — twenty-two times larger, because the Sun’s well is much deeper than the Earth’s and the orbital speed is much larger than the rotational one.

A clock gains nothing at 3169 kilometres and 38 microseconds a day higher up. How fast a clock in a circular orbit runs compared with one on the rotating geoid, in microseconds a day, against altitude. Two effects compete and they have opposite signs: being higher in the potential makes a clock run fast, and moving faster makes it run slow. In low orbit the speed term wins and a clock loses; the two cancel exactly at 3169 kilometres; above that the potential term wins and the gain rises towards a ceiling set by the depth of the Earth's well. A navigation satellite at 20,200 kilometres gains 38.6 microseconds a day, which is 11 kilometres of ranging error if it is not corrected, and it is corrected by offsetting the clock's frequency before launch. Everything in the solar system's time scales is this calculation done for a different pair of places.
Fig. 2 The expression evaluated for orbits at different altitudes. The two terms have opposite effects and they cancel exactly at about 3,200 kilometres: below that a clock loses because it is moving fast, above it a clock gains because it is high. A navigation satellite at 20,200 kilometres gains 38.6 microseconds a day, which is 11 kilometres of ranging error, and it is corrected by offsetting the clock’s frequency before launch rather than in the receiver.
TCB has gained 23.5 seconds on TT since 1977. How far four of the solar system's time scales have drifted apart, against years since they were set equal. TT is the time a clock on the Earth's geoid keeps, and it is flat here by definition. TCG is the time a clock at rest just outside the Earth's gravitational well would keep, and it runs faster by seven parts in ten thousand million — 0.0220 seconds a year. TCB is the time a clock at rest outside the Sun's well would keep, and it runs faster by fifteen parts in a thousand million, or 0.4893 seconds a year. TDB is TCB with that rate divided out so that it stays within milliseconds of TT, which is what makes it usable as an ephemeris argument and what makes it a coordinate rather than a proper time. The differences are constants and they are not corrections: an ephemeris tabulated against one of these and evaluated against another is wrong by the whole of this plot.
Fig. 3 The same drift extended over a century and a bit. The lines are exactly straight, because the rate differences are defined constants rather than measured quantities that might vary, and by the end of the twenty-first century the barycentric scale will be a minute ahead of the terrestrial one. Nothing about that is a problem so long as it is declared; it becomes a problem the moment a number is passed between two pieces of software that assume different scales, because the difference is far larger than anything either of them is trying to compute.

It is worth putting the two constants side by side, because their ratio says something physical. The Earth’s term is dominated by the planet’s own gravitational potential at its surface; the Sun’s term is dominated by the Sun’s potential at the Earth’s orbit and by the Earth’s orbital speed, and those two contribute in the ratio two to one. The Sun’s well is shallower per unit distance than the Earth’s is at its surface, and it wins by twenty-two to one anyway, because the Earth’s radius is small. A clock on the surface of the Earth is deeper in the Sun’s potential well than in its own, by a factor of about twenty — which is the same statement as the Sun’s tide-raising being weaker than the Moon’s while its pull is far stronger, seen in a different currency.

Why there are four and not two

The four scales are two pairs, and the reason for the pairing is a compromise between rigour and convenience.

TCG and TCB are the honest ones. They are the coordinate times of the geocentric and barycentric reference systems respectively, defined so that the equations of motion take their proper relativistic form. Nothing is fudged and nothing is rescaled.

Their inconvenience is that neither has the same rate as a clock anybody owns. A TCB second is longer than an SI second measured on the Earth’s surface by fifteen parts in a thousand million, so a mass or a length expressed in TCB units differs from the same quantity in ordinary units, and every constant in an ephemeris has to be rescaled. The differences accumulate: TCB has run about twenty-three seconds ahead of TT since 1977.

TT and TDB are the convenient ones. TT is the geocentric coordinate time rescaled so that its rate matches a clock on the geoid, which is where clocks actually are. TDB is the barycentric coordinate time rescaled so that its rate matches TT on average, which keeps the two within a couple of milliseconds of each other forever.

The rescaling is a linear change of unit, so nothing is lost — but it means that TDB and TCB, though they measure the same physical thing, differ by a factor, and that constants tabulated in one system are wrong in the other by that factor. The mass of the Sun expressed as GM⊙GM_\odot in TDB-compatible units differs in the eighth digit from the same quantity in TCB units, and both appear in the literature.

There is a fifth scale that belongs on the list and is not in the figure, because it is a different kind of object: International Atomic Time, TAI, which is not a definition but a realisation. It is a weighted average of some four hundred atomic clocks in about eighty laboratories, computed monthly, and TT is defined to be TAI plus exactly 32.184 seconds. The offset is an accident of history — it is the difference that existed between atomic time and the older ephemeris time when the two were connected in 1977 — and it is preserved so that the older series remain usable. A time scale of that importance carrying a constant chosen for continuity rather than for principle is normal rather than unusual, and the same is true of every epoch and zero point in astronomy.

What the difference actually does

Be concrete about the consequences, because they are not subtle.

Milliseconds, on the scale of the difference between TT and TDB. The two differ by a periodic term of amplitude 1.66 milliseconds, dominated by the annual variation in the Earth’s distance from the Sun and its orbital speed. That term is not a rate difference — it averages to zero — and it is the whole of what most users of an ephemeris need to apply. For pulsar timing it is essential: a pulsar’s arrival times are referred to the barycentre, and a 1.66-millisecond periodic error in the time argument is enormous against timing residuals measured in microseconds.

Seconds, on the scale of TCB against TT. Twenty-three seconds and growing by half a second a year. An ephemeris evaluated at a TT epoch when it was tabulated against TCB is being asked for a position twenty-three seconds in the past, and the Moon moves half a kilometre in that time.

Parts in 10810^8, on the scale of the constants. The astronomical unit, the solar mass parameter, and the light-time for unit distance all take slightly different numerical values in TDB units and TCB units. Mixing them produces errors of a few metres in a planetary position — small, systematic, and impossible to diagnose from the residuals because they look like an error in the constant itself.

TT − UT1 = 69.364 s at 2020, from three unrelated causes. The four time scales an astronomical calculation passes through, at 2020, in order of their offset from TAI and not to scale — UTC and UT1 differ by 0.18 s out of a 69 s span, which on any proportional axis puts their two rows inside one pixel of each other. Every offset is printed beside its own row. Three different KINDS of number are stacked here and telling them apart is the whole point. TT − TAI = 32.184 s is a definition, chosen once so that Terrestrial Time continued the ephemeris time it replaced; it will never change. TAI − UTC = 37 s is a count, of leap seconds inserted one at a time by a committee, and it changes by exactly one whenever they decide. UT1 − UTC = −0.18 s is a measurement of where the Earth actually is, published weekly and never allowed past 0.9 s, which is what the leap seconds are for. Add them and ΔT = TT − UT1 = 69.364 s, the same quantity the ΔT curve reaches 69.4 s for at this epoch — and every one of the 5.7 hours of it at the far end of that curve accumulated one leap second's worth at a time.
Fig. 4 The other difference, the one visible in an almanac: the offset between Terrestrial Time and the time the rotating Earth keeps. It is not a relativistic effect at all. Its three contributions are the arbitrary offset chosen when atomic time was defined, the leap seconds inserted since, and the slow lengthening of the day — and it is the quantity that makes a computed eclipse time useless without a value of ΔT.
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.
Fig. 5 The scale of the effect that all of this sits underneath. The rotating Earth has lost more than five hours against a uniform time scale since antiquity, which dwarfs every relativistic term in this essay by five orders of magnitude — and it is not a rate offset but an accumulating, irregular, unpredictable quantity that has to be measured rather than computed. The two problems are completely different in character: the relativistic differences are exact constants, and this one is a record of friction in the oceans.

Nanoseconds, on the scale of what is currently detectable. Millisecond pulsars are timed to tens of nanoseconds over decades, and a solar-system ephemeris error or a time-scale error shows up in their residuals as a signal common to every pulsar in the array. That sensitivity is what makes a timing array a detector and it is also what makes it an instrument for checking the reference frames it is built on. The order of operations matters here: the array cannot measure a gravitational-wave background without first modelling the time scale and the ephemeris to better than the signal, and the two problems are solved together rather than in sequence.

What was actually measured

Almost everything above is definition rather than measurement, and it is worth separating the two carefully.

The constants are measured, then frozen. LGL_G was determined from the Earth’s gravity field and rotation, and once the number was good enough it was adopted as exact by convention. Improving the measurement now changes the geoid’s potential rather than the definition of TT — which is the right way round, because a time scale that changed whenever geodesy improved would be useless.

The rate difference is measured directly, and the measurement is routine. Every satellite navigation system carries clocks whose relativistic offset was computed before launch and applied as a frequency offset. The first such satellite, launched in 1977, carried a switch: the frequency offset could be enabled or disabled, precisely because the effect had not been verified for that orbit. It was enabled after twenty days, and the measured rate agreed with the prediction.

And the difference between the scales is verified by consistency. A pulsar timing array observes many millisecond pulsars over decades and fits each one’s arrival times against a solar-system ephemeris and a time scale. An error in the time scale is common to every pulsar, while each pulsar’s own noise is not, so the array can measure a systematic in the reference time scale — and it does, at a level of tens of nanoseconds. That is a time standard constructed from objects tens of thousands of light years away, checking one constructed from atomic clocks in laboratories.

A clock gains nothing at 3169 kilometres and 38 microseconds a day higher up. How fast a clock in a circular orbit runs compared with one on the rotating geoid, in microseconds a day, against altitude. Two effects compete and they have opposite signs: being higher in the potential makes a clock run fast, and moving faster makes it run slow. In low orbit the speed term wins and a clock loses; the two cancel exactly at 3169 kilometres; above that the potential term wins and the gain rises towards a ceiling set by the depth of the Earth's well. A navigation satellite at 20,200 kilometres gains 38.6 microseconds a day, which is 11 kilometres of ranging error if it is not corrected, and it is corrected by offsetting the clock's frequency before launch. Everything in the solar system's time scales is this calculation done for a different pair of places.
Fig. 6 The same curve with the crossover marked explicitly and a geostationary satellite added. The gain approaches a ceiling because the potential term saturates once the clock is well outside the Earth’s well while the velocity term keeps falling — so everything above a few Earth radii gains within a factor of two of the same amount, and the interesting structure is all below one Earth radius of altitude.
A metre along the line of sight and 4 kilometres across it. Above: the round-trip light time of a ranging signal, against distance. It is a straight line of slope one, which is the point — the delay is the distance, with no model and no calibration between them, and a code correlated to a few nanoseconds gives the radial coordinate of a spacecraft at Mars to about 1 metre. Below: what the same tracking session gives across the line of sight. The angle is good to 50 nanoradians — the Doppler noise alone would give 12, and the difference is the troposphere, the station coordinates and the solar plasma, none of which average down — and an angle times a distance is a length that grows: at Mars's 0.52 au it is 4 kilometres, 3890 times the radial error. The error region of an interplanetary spacecraft is therefore not a sphere but a pancake, thin along the line of sight and enormously wide across it, and every operational consequence follows from that shape: a Mars arrival is aimed at a corridor in the plane of the sky rather than at a distance, the last approach manoeuvres are almost entirely lateral, and the observation that improves the fit most is never another day of Doppler but a single differenced measurement against a quasar. The cheapest observable measures the coordinate that matters least.
Fig. 7 Where the abstraction meets a measurement. A round-trip time to a spacecraft is a difference of two clock readings at the same station, so most of the time-scale question cancels — which is exactly why radiometric ranging is such a robust observable. What does not cancel is the rate at which the station’s clock runs relative to the coordinate time the trajectory is integrated in, and that rate is the subject of this essay. Every deep-space navigation solution carries the conversion explicitly.

Where the picture stops

There are three, and the second is the one that will move next.

The expression is first order. Terms of order c−4c^{-4} have been neglected, and for the Earth they amount to parts in 101810^{18} — below the best clocks today and not below the best clocks of the next decade. The formal definitions are written to higher order than the expression above, precisely because the definitions have to outlast the instruments.

Optical clocks have made the geoid the limiting quantity. A clock good to 10−1810^{-18} is sensitive to a height change of one centimetre, so comparing two such clocks in different laboratories now measures the difference in their gravitational potentials better than geodesy does. The relation between TT and a laboratory clock is limited by knowing where the laboratory is in the potential, not by the clock — a complete reversal of the situation the definitions were written for.

And the whole framework assumes the solar system is isolated. The barycentric system is defined with respect to a metric that ignores the Galaxy, and the Galactic potential contributes a rate offset of order 10−610^{-6} — far larger than anything in this essay. It is unmeasurable, because it is common to everything in the solar system and there is nothing outside to compare against, and it is absorbed silently into the definition of the second. The scales are internally consistent rather than absolute.

There is a fourth limit that is worth separating because it is about scope rather than accuracy. Everything here concerns the solar system. A time scale for a pulsar timing array, or for a cosmological calculation, has different requirements again — the first needs a barycentric scale stable over decades at the nanosecond level, and the second is not a time scale at all but a coordinate in a homogeneous universe, related to a clock only through a model. The word “time” is doing several different jobs across this collection, and the four scales here are the local resolution of one of them.

Why a coordinate is not a clock

The recurring difficulty in all of this is a conceptual one and it is worth stating plainly, because most of the practical errors come from getting it wrong.

A proper time is a physical quantity: it is what a clock reads, and it is unambiguous. A coordinate time is a label, chosen for convenience, whose only requirement is that the equations of motion be simple in it. TCB is not the time anybody experiences anywhere; it is the time coordinate in which the solar system’s equations of motion take their standard form.

Confusing the two produces a specific and recognisable error: applying a rate correction twice, or not at all, because the quantity being corrected was already a coordinate. It is the same category of mistake as confusing where a planet is with where it is seen, and it has the same signature — a fit that converges, absorbs the error into a parameter, and reports small residuals.

The confusion is made easier by the fact that the four scales are all called seconds and all agree to a part in 10810^8. Nothing about the arithmetic warns anybody. A programme that takes a date in one scale and hands it to a routine expecting another produces an answer that is right to seven digits, and the eighth digit is where the science is. That is the general shape of a units error in this subject — not a catastrophic factor but a small, systematic, plausible offset — and it is why the modern practice is for every ephemeris and every timing package to carry the scale explicitly in the data rather than in the documentation. The table a modern ephemeris is ships with software rather than as numbers for exactly this class of reason.

The reason the subject bothers with four scales rather than one is that both requirements are real. A coordinate time has to be relativistically correct or the dynamics is wrong; a practical time scale has to tick at the rate of the clocks that realise it or every constant in the subject has to be rescaled. There is no single quantity that does both, and the four scales are the record of that.

One remaining observation about why the definitions look the way they do. Each of the four scales was designed to make one particular class of calculation simple, and the four classes are genuinely different: laboratory metrology wants a scale that matches its clocks, geocentric orbit determination wants a relativistically correct geocentric coordinate, solar-system dynamics wants a barycentric one, and almanacs want something close to what a clock on the ground reads. A single scale satisfying all four would have to be a compromise, and a compromise would be wrong for each of them by a different amount. Four exact scales with exactly specified relations is a better answer than one approximate scale, and the price is that everybody has to say which one they are using — in the same way that a magnitude has to say which light it was measured in.

There is one practical point worth adding, because it is where the distinction most often causes real damage. The offsets between the timescales are not constant: the largest term is periodic with the Earth’s orbit, at about 1.7 milliseconds amplitude and a one-year period, from the Earth’s changing distance from the Sun and its changing speed. A programme that uses the wrong timescale therefore acquires not a constant offset — which any fit absorbs — but an annual sinusoid of a couple of milliseconds. That is exactly the shape and roughly the size of several signals people look for, including the parallax term in a pulsar timing solution and the light-travel term across the Earth’s orbit. A timescale error does not look like an error; it looks like a detection, with the right period and a plausible amplitude, and it is caught by noticing that its phase is the same for every source on the sky.

Where the ladder goes next

The obvious next rung is the practical one: how a time scale is realised rather than defined — the ensemble of atomic clocks, the weighting, and the fact that the best available realisation of TT is recomputed retroactively each year and is not the one that was distributed at the time. Further up sits the transfer problem: getting a time scale from a laboratory to a spacecraft eight light-hours away, which is what a navigation link actually measures.

Where this goes next

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. Mars to five metres and Neptune to five thousand kilometres, in the same file. Present-day heliocentric position uncertainty for each planet, in kilometres, with the range component marked separately below it. The two differ because a transponder measures a distance along the line of sight and says nothing about the two directions across it, so a planet with an orbiter is known radially some 17 times better than it is known altogether. Neptune is 10⁶ times less well determined than Mars and only 20 times further away, which is the whole point: the accuracy is a property of the observations, not of the geometry. Mars has carried a transponder almost continuously since 1976; Neptune has been visited once, in 1989, and everything else known about it is meridian-circle astrometry covering 1.07 of one orbit. The two ice giants are the only entries here whose ephemerides are still limited by nineteenth-century technology, and the only cure is a spacecraft. The table that is a fit A planetary ephemeris is not evaluated from Kepler's laws and is not evaluated from a theory. It is a numerical integration whose starting conditions were least-squares fitted to a century and a half of observations, and its accuracy is a property of those observations rather than of the mathematics. Three ways for a timing model to be wrong, and three shapes that say which. Timing residuals over 6 years for the Crab pulsar, one curve per kind of error in the model, in microseconds. A position error of 1.2 mas leaves a sinusoid of period exactly one year — measured off the drawn curve as 1.000 — with amplitude (a/c)·δθ·cos β = 2.7 μs, because the error is being projected onto a baseline that is the Earth's own orbit and nothing about the pulsar. An unmodelled proper motion of 0.9 mas/yr leaves the same sinusoid with an envelope growing linearly: twice as large at 6 years as at 3. An error of one part in 10⁹ in Ṗ leaves a parabola, the second integral of a frequency drift, whose second derivative is constant to 4e-12 across the span — and that fractional error is deliberately minute, because anything larger produces a residual thousands of times the other two and draws them as flat lines. The shapes do not resemble each other, which is the whole reason a pulsar is an instrument rather than a clock: fitting them simultaneously delivers a position, a proper motion and — from the annual curvature term, not drawn here — a parallax, all from the arrival times of pulses and no image of anything. What is left when every known shape has been removed is the science: glitches, red noise, and the correlated residual between pairs of pulsars that a timing array exists to find. A clock read against a model of everything in between A pulsar delivers arrival times and nothing else. Everything else is a model, and what is measured is the difference between the model and the arrivals — a residual whose shape says which term is wrong, and whose annual sinusoid is a position measured from pulses rather than from an image.

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.

Barycentric coordinate timeClock rateCoordinate timeEphemeris timeGeoidGravitational redshiftProper timeRelativistic correctionTerrestrial timeTime dilation