The binary star that would be seen twice

Speeds that refuse to add found that velocities in relativity combine by a rule that never exceeds the speed of light, and that the quantity which does add is the rapidity. The drag that was only an addition found that rule hidden in Fizeau’s measurement of light in moving water, the space that speeds live in drew the curved geometry the rule implies, and the river that sound cannot swim up followed the rule’s Galilean cousin to a horizon for sound. The speed that cannot be measured one way found that only the round-trip speed of light is measurable without a convention.

All of those take for granted the postulate that started the subject: that the speed of light does not depend on the motion of its source. It is not obvious. A ball thrown forward from a moving train leaves the hand at the throw’s speed plus the train’s, and in the decade after 1905 several physicists — Walther Ritz most seriously — proposed that light behaves the same way, and that the null results of the ether experiments could be explained by light always travelling at $c$ relative to whatever emitted it. That emission theory agrees with the Michelson–Morley experiment, which uses a single source moving with the apparatus, and it is not refuted by any experiment in which source and detector share a frame. This essay follows the argument Willem de Sitter gave in 1913 that disposes of it, the loophole that reopened the question fifty years later, and the X-ray stars that closed it.

Light that catches up with itself #

Suppose light leaves a source at $c + kv$, where $v$ is the source’s velocity towards the observer and $k$ measures how much of it the light inherits: $k = 1$ for Ritz, $k = 0$ for relativity. Consider a star in a binary orbit, whose velocity along the line of sight swings between $+K$ and $-K$ once per orbital period. Light it emits at time $t$ travels a distance $D$ to the Earth and arrives at $t + D/(c + kv(t))$, which for small $kv/c$ is

$$t_{\text{arrive}} \approx t + \frac{D}{c} - \frac{kD\,v(t)}{c^2}.$$

The last term is the whole effect: light sent while the star is approaching arrives early, light sent while it is receding arrives late, by an amount proportional to the distance. For a near star it is a small shuffle. For a far one it grows without limit.

The figure plots arrival time against emission time. While the star is near, arrival follows emission with some bunching and stretching. The shuffle grows with distance until, at a critical distance, the curve goes momentarily flat — light from a stretch of the orbit arrives all at once. Beyond that, the curve folds back on itself: light emitted later, when the star was coming towards the Earth and its light was faster, overtakes light emitted earlier, and at certain moments light from three different points in the orbit reaches the telescope together. The critical distance is where the slope of the arrival curve first reaches zero, which is where $kD$ times the star’s greatest acceleration equals $c^2$:

$$D_c = \frac{c^2}{kK\omega}.$$

The orbit an observer would reconstruct #

Astronomers do not see arrival times directly; they see a star’s spectrum shifting back and forth — the note that changes on approach, in light — and read off its velocity along the line of sight as a function of when the light arrives. That is the curve an emission theory would distort.

For light that travels at $c$ whatever its source, a circular orbit gives the familiar sine wave. At six-tenths of the critical distance, in an emission theory, the same circular orbit would give a lopsided curve — a slow swing one way and a sharp swing back — which an astronomer who took the speed of light for granted would fit with a highly eccentric orbit that is not there. At twice the critical distance the curve doubles back on itself, and at some moments the spectrum would show two or three velocities at once, as though the star were in several places: a single star seen as a ghostly multiple.

The idea of using double stars to test the emission theory was suggested by the American physicist Daniel Comstock in 1910, while Ritz’s theory was still under active discussion; de Sitter’s contribution in 1913 was to make it quantitative, to show how large the distortion would be for real binaries at their real distances, and to observe that the distortion was simply absent. The test appealed because it needed no apparatus: the stars had been making the measurement for as long as their light had been travelling, and spectroscopists had already recorded the result without knowing what question it answered.

This is de Sitter’s argument. Spectroscopic binaries do not do this. Their velocity curves are fitted by Kepler orbits, circular ones where tides have circularised them, and no binary shows a star seen in two places. The numbers show how strong the argument appears to be.

For a ten-day binary whose stars move at a hundred kilometres a second, the critical distance is four parsecs, thirteen light years. Almost every such star is further away than that. Algol, the eclipsing binary in Perseus, has a period of under three days, an orbital speed along the line of sight of 44 kilometres a second, and lies 28 parsecs away — eleven times past its critical distance. In an emission theory its light would be thoroughly scrambled; instead its eclipses recur like clockwork and its velocity curve is a clean sine wave. Capella, with a period of 104 days and closer, lies inside its critical distance, and would show only a mild distortion. The argument does not need any subtle measurement: it needs only that distant binaries look like binaries.

A circular orbit disguised as an eccentric one #

The distortion has a precise form, and it is one every astronomer already knows. Measure the orbit’s progress by its phase angle $\phi$, running once round per period. For a circular orbit the star’s line-of-sight velocity is $K$ times a sine of $\phi$, and the arrival time, in the same units, is $\phi - \eta\sin\phi$, where $\eta = D/D_c$ is the distance measured in critical distances. That is Kepler’s equation — the relation between the mean anomaly and the eccentric anomaly of an orbit whose eccentricity is $\eta$.

The resemblance is more than a coincidence of form. An eccentric orbit adds to the sine wave of a circular one a second harmonic, at twice the orbital frequency, with an amplitude proportional to the eccentricity; the emission distortion, to first order, adds a second harmonic too, with an amplitude of half of $k\eta$. So a circular orbit seen through an emission theory would be fitted, by an astronomer who assumed light travels at $c$, with an eccentric orbit of eccentricity about $k\eta/2$. That is why the lopsided curve in the figure above looks like an eccentric orbit. The folding at the critical distance is where the phase relation, like Kepler’s equation at an eccentricity of one, stops being invertible.

This turns de Sitter’s argument into a measurement with a number attached. Close binaries are circularised by tides: stars that orbit each other in a day or two raise tides on each other that damp any eccentricity over millions of years, and their orbits are measured to be circular to high precision. Her X-1’s orbit, timed by its pulses, has an eccentricity below about one part in ten thousand. In an emission theory it would have an apparent eccentricity of $k$ times eight thousand. The two agree only if $k$ is below about $10^{-8}$, and combining the three X-ray pulsars gives Brecher’s two parts in a thousand million.

There is a pleasing irony in this. The tool that rules out the emission theory is the same Kepler’s equation that the emission theory’s distortion reproduces, applied to stars that tidal physics independently guarantees to be circular. The argument needs no assumption about what the orbits “ought” to look like beyond the fact that tides circularise close binaries, which is observed directly in eclipsing systems whose eclipses are evenly spaced.

The gas between the stars #

The argument looked conclusive for half a century. In 1962 J. G. Fox pointed out that it rested on an unexamined assumption: that the light reaching the telescope is the light the star emitted.

It is not. Light crossing interstellar space passes through a thin gas of electrons and atoms, and every one of them, driven by the passing wave, radiates a wave of its own. In any medium the light that propagates is the sum of the incoming wave and all those re-radiated waves, and the sum has the property — the Ewald–Oseen extinction theorem — that the original wave is cancelled and replaced by the combined wave after a certain distance, the extinction length. The replacement travels at the speed set by the medium. If light from a moving source did start out at $c + kv$, it would lose that extra speed once it had crossed an extinction length of gas, and thereafter carry no record of its source’s motion.

For the interstellar medium the extinction length at visible wavelengths is less than a parsec. So visible light from Algol, 28 parsecs away, would have forgotten its source’s speed long before arriving, whatever the truth about emission, and de Sitter’s argument, made with visible light, tests nothing. The same objection applies to the earlier laboratory tests that used starlight or light that had passed through glass or air. A clean test needs light that keeps its speed for the whole distance.

X-ray stars #

The extinction length grows in inverse proportion to the wavelength, so shorter-wavelength photons keep any speed their source gave them much further. X-rays of a few kiloelectronvolts keep it for more than a kiloparsec — comparable with the distances of the brightest X-ray binaries in the Galaxy. And some of those binaries carry a clock. Her X-1, Cen X-3 and SMC X-1 are neutron stars in close orbits round ordinary stars, pulsing in X-rays every second or so as they spin, and the arrival times of their pulses trace their orbits: every pulse is a time stamp from a known point in the orbit.

The figure computes the leverage. In an emission theory Her X-1, six and a half kiloparsecs away in a 1.7-day orbit, would lie sixteen thousand times past its critical distance; Cen X-3, forty thousand; SMC X-1, in a neighbouring galaxy, a hundred and twenty thousand. Their pulses would arrive in a hopeless tangle. They arrive instead in the smooth sequence of an ordinary Kepler orbit, fitted to a few parts in a hundred thousand. So the fraction $k$ of the source’s speed that light could carry must be small enough that $k$ times those factors is smaller still. In 1977 Kenneth Brecher used these three systems to bound it below two parts in a thousand million.

That bound is five orders of magnitude tighter than the best laboratory measurement. In 1964 T. Alväger and colleagues at CERN measured the speed of gamma rays from neutral pions decaying in flight at 99.975 per cent of the speed of light, and found them travelling at $c$ to within about one part in ten thousand of the pions’ speed: a direct test at a source moving almost as fast as possible, over a flight path of tens of metres. The binaries use sources moving at a thousandth of that speed, and win by a factor of a hundred thousand, because their light has been travelling for twenty thousand years and the effect accumulates with distance. It is the same leverage that the photons that raced for seven billion years used on a different question.

What the argument actually rules out #

It is worth being exact about what the binaries show, because the question is easy to state loosely. They do not show that the one-way speed of light is $c$ in every direction; the speed that cannot be measured one way found that no experiment can, without a convention for synchronising clocks. They show that light from sources moving at different speeds travels at the same speed — that two photons emitted at the same place, one from a source approaching and one from a source receding, stay together for thousands of light years. That is a measurable statement, independent of synchronisation, because it compares two signals over the same path, and it is exactly Einstein’s second postulate in the form that has physical content — the assumption everything from an exchange of pulses builds the whole of special relativity on.

The emission theory had one more virtue that its defeat takes away. It explained the null result of the Michelson–Morley experiment without any change to space and time: light always moves at $c$ relative to the apparatus that emitted it, so of course the interferometer sees no ether wind. Once the binaries rule it out, the null result needs another explanation, and the one that remains is the relativity of simultaneity and the contraction of lengths. The binaries are, in that sense, a necessary half of the evidence for special relativity: Michelson and Morley showed that the speed of light does not depend on the observer’s motion through the ether, and the binaries that it does not depend on the source’s.

Where the argument stops #

The analysis treats the extinction length as a sharp cutoff, which it is not: re-emission is gradual, and a small part of any original speed difference would survive a few extinction lengths, which is why Brecher chose systems within about one extinction length and computed what fraction could survive. It assumes the interstellar gas is the only medium; the binary’s own stellar wind, and the gas close to the X-ray source, are much denser and have their own, shorter extinction lengths, a point that critics of the X-ray argument raised and that subsequent analyses have addressed by the high energies of the photons used. It assumes $k$ is a constant, the same for every source speed and every wavelength, and a theory in which light inherited its source’s speed only at some wavelengths or only above some speed would need its own test. None of the attempts to construct such theories has survived the combination of binary, laboratory and particle-physics evidence, but the tests constrain them one parameter at a time.

What the pictures cannot show #

The figures draw a single star’s light arriving along a single line of sight, ignoring the companion, the eclipses and the finite size of the stars. They show the folding of arrival times as a clean curve, when a real observation would record the combined light of several points in the orbit arriving together, blended in a spectrum or a light curve. The extinction figure takes a single average density for interstellar gas, which varies by factors of a hundred between hot, thin regions and cold, dense clouds, and a single formula that is a simplification of how the extinction theorem works in a real, partly ionised medium. And none of the figures can show the historical oddity that the argument was accepted as decisive for fifty years before anyone checked whether the light it relied on was the light the star had sent.

Still open: how far a photon remembers its source #

The extinction theorem itself raises a question that is not fully settled: what exactly is the light that arrives from a distant source, if every photon of the original beam has been absorbed and re-emitted many times over? In the classical theory the answer is clear — the arriving wave is the sum of the source’s wave and every re-radiated wave, and the sum travels at the medium’s speed. In a quantum description the same physics is written in terms of photons scattered forward coherently by the medium, and the question of whether a photon that arrives “is” the photon that left, with some memory of its source’s motion, is not one the theory answers or needs to. Experiments with gamma rays from astrophysical sources moving at relativistic speeds — the jets of active galaxies and gamma-ray bursts — now test the independence of light’s speed from its source over cosmological distances and at energies where extinction is negligible, and have found no dependence at all.

The habit worth carrying away is to ask whether a test’s signal has been through something that would erase it. De Sitter’s binaries should have shown, if light’s speed added its source’s, stars seen in two places at once — but visible light forgets its source’s speed within a parsec of interstellar gas, so the argument only became a test when it was made with X-rays, which remember for kiloparsecs, and then it bounded the inherited fraction below two parts in a thousand million. A null result is only as good as the path the signal took to reach the detector.