Every star in the catalogue is bent by the Sun

In May 1919 two expeditions photographed the stars around the eclipsed Sun and found them pushed outward, away from the Sun’s edge, by about the 1.75 arcseconds that general relativity predicted — twice what a Newtonian particle of light would suffer. The result made Einstein famous, and it fixed the deflection of light in the public mind as something that happens at the edge of the Sun, during an eclipse, to stars that happen to be there.

It happens everywhere. The Sun’s gravity bends the light of every star in the sky, by an amount that falls with the angle between the star and the Sun but reaches zero only for a star exactly opposite. At right angles to the Sun the deflection is four milliarcseconds. That is a tiny angle — the width of a coin seen from a thousand kilometres — and it is two hundred times the precision with which a modern catalogue gives a star’s position. Every position in that catalogue has been corrected for it, and the correction changes with the time of year, because the Sun moves.

What began as one of the two classical tests of general relativity has become a routine correction, applied to billions of stars, and the routine is what made it one of the most precise tests there is.

Why it does not fall off quickly #

The deflection at the limb comes from a ray passing a solar radius from the Sun’s centre, and the textbook formula, $4GM/c^2b$ with $b$ the closest approach, falls as one over that distance. A naive extrapolation would put the deflection of a star ninety degrees from the Sun — whose light passes the Sun at a distance of one astronomical unit, two hundred solar radii — at a two-hundredth of the limb value, about nine milliarcseconds.

That extrapolation is wrong by a factor of two, and the reason is that the observer is not at infinity. The textbook formula is the total bending of a ray that comes from infinity and goes to infinity. The Earth sits only one astronomical unit from the Sun, so a ray reaching it from a star ninety degrees away has received only part of its bending — the part from infinity to its closest approach to the Sun, which is at the Earth itself. The full expression, for an observer at distance $r$ from the mass and a source at infinity at angle $\chi$ from it, is

$$\delta = (1+\gamma)\,\frac{GM}{c^2 r}\,\cot\frac{\chi}{2}.$$

At the limb, where $\chi$ is a quarter of a degree, the cotangent is huge and the formula reproduces 1.75 arcseconds. At ninety degrees the cotangent is one and the deflection is $(1+\gamma)GM/c^2r$: two times 2.04 milliarcseconds for the Sun at one astronomical unit, 4.07. At 180 degrees the cotangent is zero.

The parameter $\gamma$ is the reason the formula has a 1 plus something in it. In general relativity $\gamma$ is exactly one: the bending comes half from the curvature of time, which a Newtonian particle would also feel, and half from the curvature of space, which it would not. A theory of gravity that curved space less would have a smaller $\gamma$ and a smaller deflection, in exactly that proportion, at every angle. The dashed curve is $\gamma = 0$.

Why the bending is twice the old answer #

The factor of two between the dashed and solid curves has a history that is worth knowing, because it is the reason the deflection is a test at all.

Light bent by gravity was not Einstein’s idea. If light is a stream of particles moving at a finite speed, a mass should pull on them as on anything else, and the deflection of a particle passing the Sun’s limb at the speed of light follows from Newtonian mechanics: half the modern value, about 0.87 arcseconds with today’s solar radius. Johann von Soldner published the calculation in 1801, and it attracted little attention, because nobody could measure an angle that small near the Sun.

Einstein’s first relativistic calculation, in 1911, gave the same half value. It used only the equivalence of gravity and acceleration — the effect of gravity on the rate of clocks — which is the part of general relativity a Newtonian particle shares. An expedition to measure it at the eclipse of August 1914 in the Crimea was overtaken by the outbreak of war, and its astronomers were briefly interned. In 1915 Einstein completed the theory and found that space itself is curved near a mass, contributing a second, equal deflection. The prediction doubled to 1.75.

Had the 1914 expedition succeeded, it would have measured the larger value and contradicted the prediction then on record. The 1919 expeditions measured it after the prediction had been corrected, and their result distinguished general relativity from both Newton and the 1911 calculation. In the formula, the old answer is $\gamma = 0$ and the new one $\gamma = 1$, and every measurement since has been a measurement of how much space a mass curves, in units in which general relativity says one.

Four milliarcseconds against twenty microarcseconds #

The importance of a correction is its size compared with what is being measured, and the comparison is stark.

Annual aberration, from the Earth’s orbital velocity, moves every star by up to twenty arcseconds and is by far the largest correction. The Sun’s gravitational deflection comes next: ten milliarcseconds at forty-five degrees from the Sun, the closest an astrometric satellite looks, four at right angles and two at 135 degrees. Then come the quantities a catalogue exists to measure. The parallax of a star a kiloparsec away is one milliarcsecond. The drift of the quasars caused by the Sun’s acceleration about the Galaxy amounts to a twentieth of a milliarcsecond over a decade. A good modern position is known to a fiftieth.

A catalogue that ignored the deflection would misplace every star by more than its parallax, and would do so in a pattern centred on the Sun’s position at the moment of each observation — which, since the Sun moves round the sky in a year, would smear into the parallaxes themselves. The deflection is not a correction to be applied when convenient. It is part of the definition of where a star appears to be.

A star’s year #

Because the deflection points away from the Sun and depends on the angle to the Sun, a given star’s deflection changes through the year as the Sun moves round the ecliptic.

A star on the ecliptic passes the Sun once a year. Near conjunction it is deflected strongly, by ten milliarcseconds at the forty-five degrees where observation stops; at opposition, six months later, not at all. A star at the ecliptic pole is always ninety degrees from the Sun and is always deflected by 4.07 milliarcseconds, but the direction of its deflection — away from the Sun — rotates once a year as the Sun circles the pole. Stars at intermediate latitudes do something between.

An astrometric satellite never sees the deflection at its largest. Gaia scans the sky in great circles while its spin axis precesses around the direction of the Sun at forty-five degrees, so every observation it makes is between forty-five and 135 degrees from the Sun, where the deflection runs from ten milliarcseconds down to under two. Each star is observed about seventy times over the mission, at different angles from the Sun and in different scan directions, and each observation carries a different deflection — so a single star’s record already samples a stretch of the curve in the opening figure, and the whole catalogue samples it thousands of millions of times.

That annual pattern is what allows the deflection to be measured rather than merely corrected. Parallax also produces an annual pattern, displacing each star towards the Sun along the great circle through the Sun by an amount proportional to the sine of the angle between them. The deflection displaces it along the same great circle, away from the Sun, by an amount proportional to the cotangent of half the angle. The two patterns lie along the same direction and differ in their dependence on the angle, so they can be separated, but they partly trade against each other — and that trade is the main thing limiting how well the deflection, and with it $\gamma$, can be measured.

The planets do the same, nearby #

The Sun is not the only mass in the solar system, and at the precision of a modern catalogue the others matter too.

Jupiter bends starlight by sixteen milliarcseconds at its limb, Saturn by six, Venus by half a milliarcsecond, and the Earth, seen from a spacecraft a million and a half kilometres away at L2, by about the same. The Moon, from the Earth, manages a few hundredths. Because a planet is much closer to the observer than to its own distance from anything that matters, its deflection falls off simply as one over the angular distance once the star is a few planetary diameters away, and it matters only within a few minutes of arc to a degree of the body. Stars that pass close to Jupiter in the sky are rare, but a mission observing billions of stars over years sees thousands of such passages, and each must be corrected.

Jupiter is also oblate, and its flattened figure bends light slightly differently from a sphere of the same mass: the quadrupole adds a deflection about a seventieth of the monopole’s at the limb, falling off as the cube of the distance. It was proposed as a measurement in its own right — the bending of light by the shape of a mass, which no eclipse could ever have detected — and a stellar occultation by Jupiter’s limb observed with sufficient precision would measure it.

The correction that became a test #

Correcting every star for the deflection requires assuming a value of $\gamma$. Fitting $\gamma$ instead, as one more parameter alongside every star’s position, parallax and motion, turns the catalogue into a measurement of it.

Each measurement of a star’s position along the scanning direction of the satellite carries information about the deflection, and the deflection is proportional to $(1+\gamma)$. Counting measurements as independent, a mission making thirty million measurements at a milliarcsecond each — the scale of the Hipparcos mission in the early 1990s — would determine $\gamma$ to about a part in ten thousand. Hipparcos actually reported $\gamma$ equal to one to within three parts in a thousand, thirty times worse than the idealised count. The difference is the trade with parallax: since both effects displace a star along the great circle through the Sun, fitting them together inflates the uncertainty on each, and the count of independent measurements overstates how independent they are.

At the precision of the Gaia mission, with tens of billions of measurements at tens of microarcseconds, the same idealised count reaches parts in ten million, and even with the parallax trade-off the expected final precision is a few parts in a million — better than the current best measurement, a two-parts-in-a-hundred-thousand determination from the delay of radio signals to the Cassini spacecraft as it passed behind the Sun in 2002. A star catalogue made to map the Galaxy would then be the most precise test of the curvature of space by a mass.

Bending that brightens #

A deflected beam is also a focused one, and near a mass the focusing brightens what lies behind it. For the Sun seen from the Earth the effect is negligible for any real star, because the Sun’s focusing distance — where rays grazing its limb converge — is at 550 astronomical units, far beyond the Earth. For a mass seen from far beyond its focusing distance, the brightening becomes the dominant effect, and a star passing behind another star brightens by a factor that depends only on their alignment, which is how microlensing finds planets.

Between the two regimes is astrometric microlensing: a foreground star passing close in the sky to a background one shifts its apparent position by an amount set by the foreground star’s mass and distance, without brightening it noticeably. The shift is milliarcseconds at most, and it is the Sun’s deflection at ninety degrees transplanted to another star. In 2017 the shift was measured for a white dwarf passing in front of a background star, and it gave the white dwarf’s mass — 0.675 solar masses — with no binary companion and no model of its interior: a mass obtained the way the eclipse of 1919 obtained the curvature of space, by watching the light of something behind it move.

Gaia’s catalogue contains thousands of predicted close passages of nearby stars in front of distant ones, and the shifts of the background stars during those passages will weigh the foreground stars. The same correction applied to every star for the Sun becomes, for each of those passages, a measurement of a mass. Strong lensing by galaxies is the same effect again, at a scale where it produces multiple images and rings.

What was actually measured #

The eclipses. The 1919 measurements gave deflections at the limb of $1.98 \pm 0.12$ and $1.61 \pm 0.30$ arcseconds from the two expeditions, consistent with general relativity and inconsistent with the Newtonian half. Their precision was debated then and since; later eclipse expeditions improved it to about ten per cent, which is the limit of the technique, set by the atmosphere and by the difficulty of comparing plates taken months apart.

Radio interferometry. Quasars pass behind or near the Sun every year, and very-long-baseline interferometry measures their positions relative to each other to tens of microarcseconds. The deflection of radio waves by the Sun, measured over thousands of such passages, gives $\gamma$ to a few parts in ten thousand, independent of any optical catalogue.

Hipparcos and Gaia. Hipparcos fitted $\gamma$ in its global solution and found $0.997 \pm 0.003$. Gaia’s solutions correct every observation for the deflection by the Sun, the planets and several large moons, and its final catalogue is expected to fit $\gamma$ to a precision of parts in a million.

Light-time delay. The other relativistic effect of the same curvature, the delay of a signal passing a mass, measures $\gamma$ in the time domain rather than the angular one, and the Cassini measurement of $\gamma - 1 = (2.1 \pm 2.3)\times10^{-5}$ remains the standard against which the angular determinations are compared.

Where the formula stops #

Point masses. The Sun’s deflection is computed for a point mass. The Sun’s own oblateness adds a quadrupole term, utterly negligible away from the limb, and its rotation adds a frame-dragging term even smaller.

An observer at rest. The formula assumes a stationary observer. The Earth’s motion adds aberration of the deflected direction, a cross term of the deflection times the Earth’s speed as a fraction of light’s — under a microarcsecond, which a catalogue aiming at parts in a million in γ still has to carry.

Idealised forecasts. The γ figure counts measurements as independent and ignores the correlations with parallax and with the attitude of the spacecraft that a real solution must fit. Its curves are an upper bound on what a mission can do, and the Hipparcos point shows by how much a real solution falls short of it.

Representative planetary distances. Each planet’s deflection is drawn at one typical distance from the observer. Its limb value scales inversely with that distance, so Jupiter at opposition bends light more than Jupiter near conjunction.

A measurement that outgrew its test #

The general lesson is about how a physical effect changes status. The deflection of light began as a prediction distinguishing two theories, tested once with a precision of tens of per cent. As instruments improved it became a nuisance — a correction that had to be applied before other things could be measured — and then, because it had to be applied to everything at once, a measurement of its own cause, to a precision the original test could never approach. The precession of Mercury’s perihelion made the same journey, from anomaly to test to a term in every planetary ephemeris.

An effect that has to be corrected everywhere can be measured everywhere, and the sum of a billion small corrections is a far stronger test than one large measurement made under difficult conditions. The eclipse photographed the deflection where it is largest; the catalogue measures it where it is small and everywhere.

Still open: the frame the corrections are made in #

Every correction here — aberration, deflection, the planets’ contributions — is applied in a reference frame, and the frame itself is defined by distant quasars assumed to have no motion. Those quasars are not points, and the frame they define has to be tied to the frame defined by the stars, which are measured in a different way at different wavelengths. How the optical frame of the stars and the radio frame of the quasars are brought onto one system, and what the residual differences between them are measuring, is the question every relativistic correction ultimately depends on.