The bend a moving planet gives

The bend Newton got half right measures the deflection of light by the Sun: 1.75 arcseconds for a ray grazing the solar limb, twice what a Newtonian corpuscle would suffer, and confirmed at an eclipse in 1919. The formula is the same for any body. A ray passing a mass $M$ at closest distance $b$ is turned through $4GM/c^2 b$, so the deflection scales with the mass and falls inversely with the distance of approach. Jupiter has a thousandth of the Sun’s mass and a tenth of its radius, and a ray grazing its edge is bent by a hundredth as much: sixteen thousandths of an arcsecond.

That is still a measurable angle. Radio interferometry, which links dishes thousands of kilometres apart to act as one telescope the size of a continent, measures the positions of quasars to a few millionths of an arcsecond. On 8 September 2002 Jupiter passed within 3.7 arcminutes of a quasar called J0842+1835, a fixed point of radio light billions of light years away, and Edward Fomalont and Sergei Kopeikin arranged for the Very Long Baseline Array and the Effelsberg telescope to track the quasar’s apparent position throughout. Measuring the deflection was not the purpose. They wanted something smaller, which arises because Jupiter moves.

A bend a hundred times smaller #

The deflection by a planet falls off with distance just as the Sun’s does, and it is easily tabulated.

At the limb Jupiter bends a ray by 16.3 milliarcseconds, Saturn by 5.8, the Earth by 0.57. The quasar of September 2002 passed 14.2 Jupiter radii from the planet’s centre, where the deflection was 1.14 milliarcseconds. The line through it is the same $1/b$ that holds for the Sun, because outside a body the deflection depends only on the body’s mass, as the delay that is not a bend shows for the related delay in a ray’s travel time. Jupiter’s flattening adds a quadrupole term near the limb, about a quarter of a milliarcsecond at the edge, which falls as the cube of the distance and is negligible by fourteen radii.

The scale of the angles needs a word. A milliarcsecond is the angle subtended by a coin seen from three thousand kilometres; a microarcsecond is a thousand times smaller again. Those are not angles any single telescope can resolve. How far apart two things have to be gives a telescope’s resolution as the wavelength divided by its aperture, and for a radio wavelength of a few centimetres a single dish resolves only arcminutes. Linking dishes across a continent makes an aperture of eight thousand kilometres, which resolves about a milliarcsecond, and the centroid of a bright compact source can be measured to a small fraction of its resolution.

A quasar pushed aside #

The measurement itself was a differential one: not the absolute direction of the quasar, which the atmosphere and the instruments would corrupt by far more than a milliarcsecond, but its position relative to two other quasars a few degrees away, which suffered the same corruption and almost none of Jupiter’s deflection.

Jupiter moved across the sky past the quasar at about thirty arcseconds an hour, a motion mostly due to the Earth’s own orbit, and over the day of closest approach the quasar’s apparent position was pushed away from the planet, radially, by an amount rising to 1.14 milliarcseconds at closest approach and falling on either side. Exaggerated sixteen hundred times, the figure shows the apparent track bulging away from the planet. The bulge was measured, and it agreed with general relativity’s prediction for a planet at rest.

The interesting part is the phrase “at rest”. In the fifty minutes or so that the quasar’s light took to travel from Jupiter to the Earth, Jupiter moved about forty thousand kilometres along its orbit — more than half its own radius. So there is a question about which position of Jupiter the deflection is computed from: where the planet was when the light passed it, or where it is when the light arrives at the telescope. The two differ by a small fraction of the deflection, and the difference has a definite direction, so it shows up as a small change in the direction of the displacement that grows and fades as Jupiter goes by.

A millionth of an arcsecond, against the air #

An interferometer measures direction by timing. The same wavefront reaches two dishes at slightly different moments, and the difference, multiplied by the speed of light, is the projection of the baseline between them on the direction of the source. For a baseline of eight thousand kilometres, a microarcsecond of direction corresponds to a timing difference of about a tenth of a picosecond, or a path difference of forty micrometres. Anything that delays the wave by forty micrometres more at one dish than at the other mimics a microarcsecond shift.

Almost everything does. The water vapour in the troposphere above each dish delays radio waves by centimetres, varying from minute to minute as weather passes. The ionosphere delays them by an amount inversely proportional to the square of the frequency, and so does the plasma of the solar wind and of Jupiter’s own magnetosphere, which the quasar’s light crossed on its way past. The dishes’ positions drift with the solid Earth tide and with the continental plates, and the hydrogen masers that time each station wander.

The answer is to measure differences. Switching rapidly between the target quasar and a reference quasar a degree or two away, the array measures the same atmospheric delays on both, because the two lines of sight pass through nearly the same air, and subtracts them. Using two reference sources on either side lets a residual gradient across the sky be fitted and removed as well. Observing at more than one radio frequency separates the plasma delay, which depends on frequency, from the geometric one, which does not. What remains after all of it is a relative position, the target’s displacement from where the reference sources say it should be, and the 2002 experiment measured that to about ten microarcseconds per hour of tracking.

That is the sense in which a gravitational deflection was measured “against a fixed point”. The quasars are billions of light years away and do not move measurably on the sky. Jupiter passes in front of one of them, and the displacement it causes is measured relative to its neighbours, which Jupiter barely touches: at two degrees from the planet its deflection is about forty microarcseconds and nearly uniform across a small patch, and the analysis models it explicitly. The quasar’s displacement is then the deflection, the velocity-dependent term and the measurement noise, and nothing else of comparable size.

The bend depends on when the planet was there #

General relativity gives a definite answer, and it can be seen by analogy with electromagnetism. The field that points where the charge is now shows that the electric field of a uniformly moving charge points not at the retarded position, where the charge was when the field at the observer was “sent”, but at the position extrapolated to the present. Retardation and the velocity dependence of the field cancel exactly for uniform motion. Gravity in general relativity behaves the same way to the relevant order: the gravitational field felt by a photon passing Jupiter points at where Jupiter is at that moment, the moment of passing.

The photon then travels on to the Earth, which takes time $D/c$, where $D$ is the distance from Jupiter to the Earth. By the time it arrives, Jupiter has moved on by $vD/c$. The observer, looking at Jupiter, sees it where it was when the photon left its neighbourhood, because the light from Jupiter itself took the same time to arrive. So the deflection points away from where Jupiter is seen in the sky, and not away from where it “really” is at the moment of observation. The difference between those two computations is the term the experiment was designed to see, and it is of order the deflection times $(v/c)(D/b)$: the planet’s speed as a fraction of light’s, multiplied by the ratio of its distance from the Earth to its distance from the ray. That ratio is large — about nine hundred here — which is why a velocity ratio of four parts in a hundred thousand gives a term as big as one twenty-fifth of the deflection. In this simplified geometry, with Jupiter’s motion taken square to the line of sight, the term is 46 microarcseconds at its largest.

The measured value was 51 ± 9 microarcseconds. The deflection points away from Jupiter’s position at the time the light passed it, as general relativity says, and not away from its position at the time of observation. That much nobody disputes.

What the number was said to measure #

The dispute is over how to describe the result. Kopeikin had argued, before the event, that the delay entering the term is the time for gravity to propagate from Jupiter to the photon, so that measuring the term measures the speed of gravity. The way to make that quantitative is to parameterise the delay by a speed $c_x$ and ask which value the data prefer.

On that reading the experiment gave $c_x = 1.06 \pm 0.21$ times the speed of light, and it was announced in January 2003 as the first measurement of the speed of gravity. Clifford Will, Stuart Samuel, Hideki Asada, Steven Carlip and others objected. Their argument, made in several equivalent forms, is that in general relativity the delay that enters the term at first order in $v/c$ is the light-travel time from Jupiter to the Earth, so the parameter $c_x$ is the speed of light, measured — if that is the word — through the aberration of Jupiter’s position. To see the speed of gravity, a theory has to be written in which gravity propagates at some speed $c_g$ different from $c$, and the deflection computed in it. When that is done, the difference between $c_g$ and $c$ affects the deflection only at second order, $(v/c)^2$, far below what the 2002 measurement could see.

The argument is not about the data; it is about what the parameter $c_x$ means, and that depends on the family of theories in which the measurement is embedded. Kopeikin’s parameterisation put a speed into the formula at the place where general relativity has the light-travel time, and labelled it as gravity’s. His critics’ parameterisation started from a theory with a separate gravitational speed and found that the first-order term did not depend on it. Both agree that the measurement is a successful confirmation of general relativity, and both agree on the formula. They disagree about which alternative it rules out.

The lesson is general. A measured number is a number of a parameter, and a parameter is a place in a family of theories where the measured theory is one point. The same data, embedded in two families, constrain two different things. Here one family was constructed so that the measurement constrains it, and the other was the family that physicists actually wanted constrained, and the two did not coincide. The speed of gravity is now known extremely well from a different experiment altogether: in 2017 the gravitational waves from a neutron-star merger arrived 1.7 seconds before the gamma rays from the same event after 130 million years of travel, so the two speeds agree to a few parts in $10^{15}$.

Why the cancellation happens #

That the field of a uniformly moving mass should point at its present position, despite gravity taking time to travel, looks like a paradox. It is the same paradox as for the moving charge, and resolved the same way. A field computed from the retarded position alone would point backwards, and a body orbiting another would feel a force tilted against its motion that would make the orbit unstable within a few thousand years — Laplace’s old argument, from 1805, that gravity must travel at least millions of times faster than light. The resolution is that in both electromagnetism and general relativity the field depends on the source’s velocity as well as its retarded position, and for uniform motion the velocity-dependent part exactly redirects the field towards the extrapolated present position. What carries information about the delay is acceleration: a charge that turns must glow, and a mass that accelerates radiates gravitational waves, at second order.

Jupiter is not moving uniformly; it is orbiting the Sun. But its acceleration over the hour that matters changes its velocity by a tiny fraction, and the effect of that on the deflection is far below the microarcsecond level. To the precision of the experiment Jupiter is a uniformly moving lens, and its field at the photon points at its position at the moment the photon passes.

The relation to the speed of light is then a matter of geometry. When the source is not heard all at once is the general statement for any moving emitter: what reaches an observer at one moment left the source at a time that depends on the distance. Jupiter’s light and the quasar’s light that grazed Jupiter left the neighbourhood of Jupiter together and arrived together, so the deflection points away from where Jupiter is seen. That is a statement about light.

The terms a moving planet’s bend is built from #

The disagreement is easier to see when the deflection is taken apart into its pieces, each with its own size and its own dependence on how close the ray passes.

The largest is the ordinary bend, proportional to the planet’s mass and falling as one over the impact parameter. Next comes the velocity-dependent term, the one the experiment was designed to see: it is the bend times $v/c$ times the ratio of the distance to the Earth to the impact parameter, and so it falls as the square of the impact parameter, more steeply than the bend itself. Jupiter’s flattening, measured by its quadrupole moment $J_2$, adds a term that falls as the cube; at fourteen radii it is a tenth of a microarcsecond and negligible, though at the limb it is a quarter of a milliarcsecond. General relativity’s second-order bend, proportional to the square of the mass, is a few millionths of a microarcsecond at fourteen radii. And a term of order $(v/c)^2$ times the bend, which is where a theory in which gravity propagated at a different speed from light would first change the answer, is smaller still — seven orders of magnitude below the term that was measured.

Laid out this way, the dispute’s resolution is a matter of reading the figure. The 2002 measurement, with an uncertainty of about ten microarcseconds, could see the bend and the velocity term and nothing else. The velocity term is set by the light-travel time from Jupiter to the Earth. The terms that would constrain the speed of gravity are invisible, not because the experiment was badly done but because they are millions of times smaller. A measurement can only constrain what lies above its own noise, and the place a parameter sits in the hierarchy of terms decides whether any foreseeable measurement can reach it.

The same hierarchy explains why light deflection by the Sun is such a precise test of the curvature of space and so poor a test of gravity’s speed. The Sun’s bend is a hundred times Jupiter’s, and the Sun moves slowly relative to the solar system’s centre of mass, so its velocity terms are small; its deflection measures the first, monopole term — the coefficient that the half of the bend a slow body never feels traces to the curvature of space — very precisely, and has nothing to say about anything that enters at higher order in velocity. Jupiter, moving faster and with a larger ratio of distance to impact parameter, made the velocity term large enough to see, which is why it was the planet chosen.

A planet is not a point, and a pass is not a straight line #

The figures treat Jupiter as a point mass moving in a straight line across the sky at a constant thirty arcseconds an hour, with its velocity taken perpendicular to the line of sight; the real geometry had Jupiter’s velocity partly along the line of sight and its apparent motion set by the Earth’s orbit as much as its own, which is why the computed term of 46 microarcseconds is a simplified-geometry estimate rather than the published prediction. The planet’s quadrupole and the deflection by its moons are omitted; both are below a few microarcseconds at fourteen radii. The radio measurements were also affected by the plasma of the solar wind and of Jupiter’s magnetosphere, which refract radio waves by amounts depending on frequency, and which were corrected by observing at several frequencies. The terms figure takes Jupiter’s flattening at its measured $J_2$ and its speed at 13 km/s for every term, and orders the second-order terms by their leading coefficients only.

Still open: what a measurement at second order would see #

A test of the speed of gravity through light deflection would need to see the second-order term, of size $(v/c)^2$ times the deflection, which for Jupiter is a few parts in $10^{9}$ of 16 milliarcseconds — far below any foreseeable measurement. The gravitational waves from merging neutron stars settled the speed of gravity directly. What deflection experiments still constrain is the parameter that measures how much space curvature a unit of mass produces, known from radio timing of the Cassini spacecraft to a few parts in $10^5$, and the velocity-dependent term itself, which tests the way the deflection follows a moving lens. Whether either departs from general relativity at a level a future measurement could reach — a part in a million in the curvature, a per cent in the velocity term — is open, and theories that modify gravity at large distances generally predict that something should.

The habit worth carrying away is to ask what family a parameter lives in. A ray passing Jupiter is bent by $4GM/c^2b$ and the bend points away from where Jupiter was when the light went by — fifty minutes before the light arrives, a difference of fifty microarcseconds that was measured in 2002 — but that delay is the light-travel time from Jupiter to the Earth, and a different speed for gravity would show only at second order. The experiment confirmed general relativity; what it was a measurement of depends on which alternatives one chooses to imagine.