The satellites launched into the wrong orbit

The clock that is wrong in two directions found that a navigation satellite’s clock is pulled two ways at once: it runs slow because it moves, as the clock that has to slow showed every moving clock must, and fast because it sits higher in the Earth’s gravity than a clock on the ground, as the clock that runs slow lower down showed every deeper clock must. For a GPS satellite the two come to a net gain of 38 microseconds a day, and the clocks are deliberately set slow before launch to cancel it. In passing, that essay noted a further correction that cannot be set once and forgotten: on an orbit that is not quite circular, the clock’s height and speed vary round each revolution, and its offset wobbles by a few tens of nanoseconds.

For navigation the wobble is a nuisance, corrected by a formula every receiver applies. It is also a measurement waiting to be made. A constant offset between a satellite clock and a ground clock tells nothing about relativity, because nobody knows the satellite clock’s exact frequency well enough to separate an effect of gravity from an error of manufacture. A periodic signal, repeating every orbit with an amplitude fixed by the orbit’s shape, is different: it cannot be mimicked by a clock simply ticking at a slightly wrong rate. The trouble is that navigation satellites are put on nearly circular orbits precisely to keep the wobble small. Until 2014 none had an orbit eccentric enough to make the wobble worth measuring.

A frozen fuel line #

On 22 August 2014 a Soyuz rocket launched the fifth and sixth satellites of Europe’s Galileo navigation system from French Guiana. The final stage, a Fregat upper stage, was supposed to place them on a circular orbit 29,600 kilometres from the Earth’s centre. A fuel line in the Fregat had been routed next to a cold helium line, the hydrazine in it froze, and the stage fired in the wrong direction. The satellites were left on an elliptical orbit, tilted at the wrong inclination, with a closest approach far too low for their antennas and solar panels to be used as designed.

Over the following months the operators used the satellites’ own small thrusters to raise the low point of the orbit, and by early 2015 both were on orbits of eccentricity about 0.156, semi-major axis 27,977 kilometres and period 12.9 hours. They could be switched on and used, with care, and they carried exactly what a relativity experiment needed: passive hydrogen masers, among the most stable clocks ever put in orbit, and laser retroreflectors that let their positions be measured to centimetres from the ground.

The orbit is not dramatically elongated to the eye. Its closest point is about 37 per cent nearer the Earth’s centre than its farthest, and by Kepler’s second law the satellite moves 37 per cent faster there. That is enough, for clocks that keep time to a part in $10^{14}$, to turn a correction into a signal.

Two slowings that move together #

The rate of a clock in orbit, compared with an ideal clock far from the Earth, is

$$\frac{d\tau}{dt} = 1 - \frac{GM}{rc^2} - \frac{v^2}{2c^2},$$

the first correction from the gravitational potential at distance $r$, the second from the speed $v$. On an elliptical orbit both $r$ and $v$ change, and they change together: the satellite is fastest exactly where it is lowest.

That is the opposite of what happens between a satellite and a ground clock. Comparing a high orbit with the ground, height and speed act against each other: the satellite is higher, which speeds its clock, and faster, which slows it. Round a single eccentric orbit they act together. At closest approach the satellite is both deeper in the potential and moving faster, and both slow its clock; at the far point it is both higher and slower, and both speed it up. The two effects reinforce, and the rate swings by the sum of their swings.

The size of each is fixed by the orbit. The speed on an ellipse follows from the energy equation, $v^2 = GM(2/r - 1/a)$, as the velocity that always lies on a circle drew it, so both corrections are functions of $r$ alone. Integrating them round the orbit, using the eccentric anomaly $E$ that Kepler introduced to time an ellipse, each produces a periodic offset of the same size:

$$\Delta\tau_{\text{grav}} = \Delta\tau_{\text{speed}} = -\frac{\sqrt{GMa}}{c^2}\,e\sin E,$$

so that the total periodic offset is

$$\Delta\tau = -\frac{2\sqrt{GMa}}{c^2}\,e\sin E.$$

The reason the halves are equal is the conservation of energy. On a Kepler orbit $\tfrac12v^2 - GM/r$ is the same at every point, so whatever $GM/r$ gains as the satellite falls towards closest approach, $\tfrac12v^2$ gains exactly as much. The two clock corrections are those two quantities divided by $c^2$, and their variations round the orbit are therefore identical, point by point, not only on average. The orbit converts depth into speed one for one, and the clock is slowed by each in the same proportion. The equality of the two halves is not a coincidence of these numbers but a property of every Kepler orbit, and the computation behind the figures checks it by integrating the two rates numerically round the orbit: they match each other and the closed form to a part in a thousand.

The wobble a clock cannot fake #

The stranded satellites’ clocks fall behind by 367 nanoseconds a quarter-orbit after closest approach, catch up, run 367 nanoseconds ahead a quarter-orbit before the next, and return — every 12.9 hours, indefinitely. A GPS satellite with an eccentricity of 0.01 shows the same pattern at a sixteenth of the size. A satellite on its intended near-circular Galileo orbit shows almost none.

That is what makes the wobble a test rather than a calibration. Everything about it — its amplitude, its period, its phase relative to the satellite’s position on its orbit — is computed from the orbit, which laser ranging measures independently to a few centimetres. The clock’s own imperfections show up differently. An error in its frequency is a constant rate and is removed with the constant part; a slow drift as the maser ages is a smooth trend over weeks; random noise averages down. Only a disturbance that repeats once per orbit, at the right phase, could imitate the relativistic signal, and the analysis has to look for such disturbances explicitly — the satellites’ temperatures, for instance, vary round the orbit as they move in and out of sunlight, and a clock’s frequency depends slightly on its temperature.

The test then asks a precise question. A violation of the equivalence principle would change the clock’s response to the gravitational potential by some small fraction $\alpha$, multiplying the gravitational half of the wobble by $1 + \alpha$ while leaving the speed half — pure special relativity — untouched. A measured wobble whose amplitude matches the prediction to a part in $10^5$ bounds $\alpha$ to about twice that, because only half the signal is affected.

The correction every receiver already makes #

The formula is not new to navigation. The interface specification of the American GPS system has, since its first versions, told every receiver to add a relativistic correction to each satellite’s broadcast clock, $F\,e\sqrt{a}\,\sin E$ with $F = -2\sqrt{GM}/c^2 = -4.442807633\times10^{-10}$ seconds per square root of a metre — the same expression drawn here, written in the units a receiver uses. A satellite with an eccentricity of 0.02 would otherwise carry a timing error of up to 46 nanoseconds, which at the speed of light is fourteen metres of range, far more than the system’s error budget allows. Galileo receivers apply the same correction, and once the stranded satellites’ orbits were known and broadcast, receivers could use their signals with the larger wobble removed in the same way.

So the relativistic wobble had been corrected billions of times a day for decades by the time anyone set out to measure it. That is the familiar pattern of relativity in engineering, and the clock that is wrong in two directions followed it for the constant offset: a prediction becomes a correction, and a correction that works is evidence, but evidence at the level of the system’s tolerance and no better. Turning it into a precise test needs a large signal, a good clock and an analysis devoted to nothing else.

What the satellites were carrying #

Each Galileo satellite carries two kinds of atomic clock: rubidium vapour clocks and passive hydrogen masers. A hydrogen maser keeps time with the transition between the two hyperfine levels of hydrogen’s ground state, the flip of the electron’s spin relative to the proton’s at 1.42 gigahertz — the same transition that makes the 21-centimetre line of radio astronomy. Atoms in the upper level are stored in a coated bulb inside a microwave cavity, and the cavity’s response, locked to the atoms, sets the frequency. In the passive version flown on Galileo, the cavity is probed rather than allowed to oscillate on its own, which makes the clock smaller and more robust while keeping a stability of about a part in $10^{14}$ over the hours of an orbit.

That stability, rather than the maser’s absolute accuracy, is what the test needs. The constant part of the clock’s rate is fitted and thrown away; what matters is that the clock does not wander by more than a fraction of a nanosecond over the few hours in which the wobble swings through hundreds. A rubidium clock is good enough for navigation and noticeably worse for this purpose, and the analyses used the masers whenever they were operating.

The same instruments can be turned the other way. If the wobble’s amplitude is trusted, any residual in a satellite clock’s behaviour round its orbit is information about the clock — its sensitivity to temperature, to magnetic field, to the satellite’s attitude. The relativistic signal becomes a calibration source for the clock, in the reversal that the clock that measures a height described for gravitational shifts on the ground.

How large the wobble can be #

The amplitude is strictly proportional to the eccentricity and grows only as the square root of the orbit’s size. The eccentricity is therefore what matters, and the stranded satellites’ accident gave them about fifteen times the eccentricity of a typical navigation satellite. No navigation satellite was ever meant to have such a wobble; a mission designed to test the effect would have chosen an eccentric orbit deliberately, as Gravity Probe A’s single rocket flight did in 1976.

The size of the signal sets the scale of the experiment. A hydrogen maser in orbit is stable to about a part in $10^{14}$ over a few hours, which over half an orbit corresponds to a fraction of a nanosecond of accumulated error; the relativistic wobble is a thousand times larger. A single orbit therefore shows the wobble clearly, and averaging over thousands of orbits measures its amplitude to a small fraction of itself. The averaging is the same patience that the wait that sets the precision found behind every precise clock: a measurement repeated $N$ times, with independent noise, gets better as $\sqrt{N}$.

Forty years of one number #

The gravitational slowing of clocks was first measured on the Earth, in 1960, by Robert Pound and Glen Rebka, who sent gamma rays 22.5 metres up a tower at Harvard and detected their shift in frequency to about ten per cent; a refinement five years later reached one per cent. In 1976 Gravity Probe A carried a hydrogen maser on a rocket to an altitude of 10,000 kilometres and back, comparing it continuously with masers on the ground during the two-hour flight, and confirmed the prediction to 1.4 parts in $10^4$. For forty-two years nothing did better.

In 2018 two groups, one in Bremen and one in Paris, published analyses of about a thousand days of data from the two stranded satellites, each confirming the gravitational slowing to about two to three parts in $10^5$ — some five or six times better than Gravity Probe A. Neither group had flown anything. The satellites, the clocks, the laser ranging and the orbit were all accidents of a launch failure and the infrastructure of a navigation system, put to a use nobody had planned.

The result also sits in a longer argument. The floor that cannot be told from gravity began this site’s account of why clocks must run slow in a gravitational field: a clock at rest in gravity is equivalent to one accelerating in free space, and the slowing follows from that equivalence alone. Measuring the slowing precisely is therefore a test of the equivalence principle in a form no dropped object can test, involving clocks rather than masses. The stranded satellites’ bound on $\alpha$ is a bound on how differently a hydrogen maser’s ticking responds to gravity from what its falling would imply.

Where the comparison is more than two clocks #

The analysis is not a simple comparison of two clocks. The satellite’s clock is compared with clocks on the ground through the navigation system’s own radio links, which pass through the ionosphere and troposphere; the ground clocks are themselves at different heights and on a rotating Earth, the corrections that the two clocks that flew in opposite directions had to make for aircraft. The orbit is not a perfect Kepler ellipse: the Earth’s equatorial bulge, the Moon and the Sun all perturb it, slowly turning its long axis, and the analysis follows the real orbit from laser ranging rather than an ideal one. Even the solar radiation pressure on the satellite’s panels shifts its orbit slightly, and with it the predicted wobble.

An ideal ellipse round a point mass #

The figures treat the satellite as moving on a fixed Kepler ellipse round a spherical Earth, with clock rates computed to first order in $GM/rc^2$ and $v^2/c^2$, and the ground clock as an ideal clock at the Earth’s surface. The real analysis uses the measured orbit, the Earth’s full gravity field and the rotating geoid, and fits the clock’s own frequency, drift and temperature dependence together with the relativistic amplitude; those details move the result at the level of its uncertainty, not of its size. The history figure shows each experiment’s stated uncertainty and makes no attempt to compare their methods, which differ greatly; the last bar is a design goal, not a result. The domain is clocks in Earth orbit with eccentricities up to a few tenths, where the first-order expressions are accurate to better than a part in a billion of the rate.

Still open: how far a clock in orbit can push the test #

The stranded satellites are still in orbit and still collecting data, and the limit of their test is now set less by the clocks than by the clocks’ sensitivity to temperature and by how well that can be modelled. The next step is a clock built for the purpose. The ACES ensemble, a cold-atom clock and a hydrogen maser carried to the International Space Station in 2025, aims to compare its clocks with ground clocks to a few parts in $10^6$ of the gravitational shift, and proposals for optical clocks on eccentric orbits — deliberately chosen this time — aim at parts in $10^8$ or better. How small a violation of the equivalence principle could be hidden in clocks, and whether any theory predicts one large enough to find, is not known. What is known is that the test, once posed, needs no more than a good clock on an orbit that is not round.

The physics of the wobble is short. A clock on an orbit of eccentricity $e$ is slowed by both the potential, $-GM/rc^2$, and its speed, $-v^2/2c^2$, which are deepest together at closest approach; each contributes a periodic offset $-(\sqrt{GMa}/c^2)\,e\sin E$, together $-2\sqrt{GMa}\,e\sin E/c^2$ — ±367 ns every 12.9 hours for the stranded Galileo satellites — and three years of that wobble confirmed the gravitational half to about 2.5 parts in $10^5$, six times better than any experiment before. An orbit that was a navigation failure became a relativity experiment because it was not round.