A rosette the centre's other mass would turn backwards

One star at the centre of the Milky Way has been followed round a complete orbit, and its period and size weigh the mass it orbits at about four million suns, packed into a volume too small to be anything but a black hole. The star, S2, comes within 120 astronomical units of the black hole every sixteen years, moving at nearly three per cent of the speed of light, and at that distance the black hole’s gravity is no longer quite Newton’s. The orbit does not close: each time round, the point of closest approach has moved forward in the direction of the star’s motion.

That advance is the same effect that Mercury shows in the solar system, forty-three arcseconds a century, here twelve arcminutes an orbit. It was measured by interferometry of S2’s position through the 2018 pericentre and announced in 2020, agreeing with general relativity to about twenty per cent. As a test of relativity it is modest — binary pulsars test the same physics far more precisely. What makes it interesting is a second effect that would push the orbit the other way.

If anything else with mass is spread through S2’s orbit — stars too faint to see, the black holes and neutron stars left by earlier generations of massive stars, a concentration of dark matter — then the force on S2 is not purely that of a point mass, and an orbit in such a force does not close either. It precesses backwards. Relativity turns the rosette forward; extended mass turns it back; the measured rosette is the sum. A precession that agrees with relativity is therefore also a measurement that very little else is there.

The rosette relativity draws #

At first order beyond Newton, general relativity adds to the attraction of a point mass a small extra pull that falls as the fourth power of the distance and is proportional to the square of the orbit’s angular momentum. An orbit under that force is an ellipse whose pericentre advances each revolution by

$$\Delta\omega_{\rm GR} = \frac{6\pi G M}{c^2 a (1-e^2)} .$$

For S2, with a semi-major axis of 1,036 astronomical units and an eccentricity of 0.884, the advance is 12.2 arcminutes per orbit. The figure turns the orbit forward by twenty, forty and sixty orbits to make the rosette visible; in reality the turn per orbit moves the far end of the ellipse by only seven astronomical units, which at the Galactic centre’s distance of 8.3 kiloparsecs is less than a milliarcsecond on the sky. The rosette would take nearly eighteen hundred orbits — twenty-eight thousand years — to close.

The high eccentricity is what makes the measurement possible at all. The advance depends on the semi-latus rectum $a(1-e^2)$, which for S2 is a fifth of the semi-major axis: the star spends most of its time far out, where relativity is negligible, and plunges through a pericentre where the correction is strong enough to matter.

Why spread mass turns it backwards #

An orbit closes exactly in only two force laws: the inverse square and the linear spring. Any other central force makes the orbit precess, and the sense of the precession follows from how the force compares with the inverse square. A mass distributed through the orbit pulls on the star with the mass enclosed inside its current distance, which grows outward; near pericentre the star feels less of it than at apocentre. The force falls off more slowly than the inverse square, and an orbit in such a force regresses — its pericentre moves backwards, against the direction of motion. The precession in a galaxy’s disc, which makes stellar orbits rosettes that turn backwards, is the same effect on a larger scale.

Relativity’s extra term is the opposite: a force steeper than the inverse square near pericentre, which makes the orbit advance. The two compete, and a modest amount of spread mass can cancel the relativistic advance entirely.

Where the mass sits decides the sign #

It is worth being careful about which spread mass turns an orbit backwards, because the answer depends on where the mass is, and the history of celestial mechanics contains the opposite case. A mass lying entirely inside the orbit’s pericentre — a ring, or a flattened distribution close to the central body — does not make the force fall off more slowly across the orbit. Seen from outside, it looks like a point mass plus a small quadrupole, and a quadrupole aligned with the orbit’s plane adds a force that falls off faster than the inverse square. An orbit outside such a mass advances.

That was the reasoning behind Le Verrier’s search for an unseen planet, or a ring of small bodies, inside the orbit of Mercury in the 1850s. Mercury’s perihelion advanced by about forty-three arcseconds a century more than the pull of the known planets explained, and matter inside Mercury’s orbit would have produced exactly such an advance. Nothing was found, and the forty-three arcseconds were eventually explained by relativity. A flattened Sun would have done the same thing, which is why the solar oblateness was argued over for decades after relativity had apparently settled the matter.

At the Galactic centre the situation is the reverse. The black hole is effectively a point, and the mass in question — stars, remnants, dark matter — is spread through the volume the orbit sweeps, from inside the pericentre out past the apocentre. A star at apocentre feels the pull of everything inside nearly two thousand astronomical units; at pericentre, of everything inside a hundred and twenty. The enclosed mass grows outward along the orbit, and that is the arrangement that turns an orbit backwards. The sign of an unexplained precession says not only how much unseen mass there is but where it sits relative to the orbit — inside the pericentre, or spread through the orbit — and that is a distinction no measurement of the total mass could make.

The Moon offers a third version of the same lesson. Its perigee advances by a full revolution in under nine years, not because of anything inside its orbit but because of the Sun’s tidal pull from outside, which changes sign through each month. External, internal and distributed perturbations all make orbits precess, and in each case the rate and the sign are a measurement of the perturbing mass’s position as much as its size.

How little it takes #

The competition can be computed by integrating S2’s orbit with both effects. The extended mass is modelled as a cusp, with density falling as a power of the distance from the black hole — about $r^{-7/4}$ is the profile a relaxed cluster of stars settles into round a black hole, and shallower profiles are possible if the cluster is not relaxed.

A thousandth of the black hole’s mass, four thousand suns spread through the orbit, takes the advance from twelve arcminutes to four. A sixth of a per cent cancels it altogether; a per cent reverses it to more than a degree an orbit. The shallower cusp acts a little more strongly at the same enclosed mass, because more of its mass lies just inside the pericentre distance, where the star feels its absence most sharply. The Newtonian term is linear in the enclosed mass and the relativistic term is fixed, so the net precession falls in a straight line with the mass — straight on a linear plot, a cliff on the logarithmic one.

That is why S2’s precession is so sensitive. The quantity measured, a few arcminutes, is the difference between two contributions of similar size, one of which is completely known — the black hole’s mass and the orbit’s shape fix it — and one of which is the unknown.

What the measurement allows #

The interferometric measurement through the 2018 pericentre gave the precession as 1.10 times relativity’s prediction, with an uncertainty of 0.19.

The precession could be lower than relativity’s prediction by about a quarter before the measurement objected at two standard deviations. That allows an extended mass of at most a few hundredths of a per cent of the black hole’s mass inside S2’s orbit — between one and two thousand solar masses, depending on how it is distributed. Analyses by the team that made the measurement, using the full orbit fit rather than this simple comparison, quote a limit of a few thousand solar masses, about a tenth of a per cent, within S2’s apocentre. The two agree in order of magnitude, and the details depend on the assumed profile.

That bound constrains several things at once. The stars visible near the black hole account for a small fraction of it. The population of stellar-mass black holes that should have sunk to the centre by dynamical friction over the Galaxy’s life — expected to form a dense cluster because they are heavier than ordinary stars — has to hold no more than a couple of thousand solar masses inside the orbit. And the spike of dark matter that a slowly growing black hole would pull in after it, which in some models reaches densities far above the halo’s, is constrained directly: a spike massive enough to matter inside a thousand astronomical units would have stopped the rosette turning.

The terms that need stars further in #

The measured precession is dominated by one term. Others are present and much smaller.

A spinning black hole drags the space around it, adding a precession that depends on the spin and on the orientation of the orbit relative to the spin axis; and the spin flattens the hole’s external field, adding a quadrupole term. At S2’s distance both are tiny: frame dragging is a two-hundredth of the main term and the quadrupole a forty-thousandth. These are the terms that would measure the black hole’s spin and test the theorem that a black hole’s quadrupole is fixed by its mass and spin alone, and they are out of reach with S2.

Stars closer in would do better. The relativistic precession per orbit goes as the inverse of the orbit’s size, and per year as the inverse five-halves power, because closer orbits are also faster; frame dragging rises more steeply still. A star with a semi-major axis of a few hundred astronomical units would precess by tens of arcminutes per year and would show frame dragging within a few orbits. Such stars are expected — the density of stars round the black hole rises inward — but they would be faint, and the crowding near the black hole makes finding them hard. Fainter stars on smaller orbits have been reported in the latest interferometric images, and their orbits are being followed.

The other relativistic signal #

S2’s pericentre passage gives a second test that does not need a whole orbit. Light leaving a star deep in a gravitational well is redshifted, and a fast-moving clock runs slow; both add to the ordinary Doppler shift of the star’s spectrum.

At pericentre the combined effect is two hundred kilometres a second, in a star whose line-of-sight velocity swings through several thousand; it was measured in 2018 by two independent groups within months of the passage, and agrees with relativity to about ten per cent. The redshift tests how clocks run in a strong field; the precession tests how orbits respond to it, and accumulates over a whole revolution. The redshift could not tell an extended mass from none, because the extended mass is far too small to change the potential at pericentre; the precession can, because it accumulates the extended mass’s effect over the whole orbit. The two measurements are complementary in exactly the way a pulsar binary’s several relativistic effects are: each tests a different part of the same theory, and only their agreement together fixes the system.

Thirty years of watching one star #

The measurement is the end of a long programme. Two groups, one working at the European Southern Observatory’s telescopes in Chile and one at the Keck telescopes in Hawaii, began following the stars near the Galactic centre in the early 1990s with speckle imaging — very short exposures that freeze the atmosphere’s blurring — and continued with adaptive optics as it became available. Their positions for S2 were good to a few milliarcseconds, enough to trace the ellipse and weigh the central mass by Kepler’s third law, but not enough to see a precession of a milliarcsecond per orbit.

What changed the precision was interferometry. Combining the light of the four eight-metre telescopes of the Very Large Telescope coherently, over baselines of up to a hundred and thirty metres, gives positions of S2 relative to the radio source at the black hole’s position good to about fifty microarcseconds — the width of a coin on the Moon. With that, the 2018 pericentre passage could be followed in enough detail to fit the orbit’s orientation before and after closest approach and see that it had turned. The 2020 Nobel Prize in Physics recognised the discovery of the compact mass, and the precession measured in the same year was one of the results that showed what the compact mass was.

It is unusual for a single object to carry so much of an argument. The weighing of the central black hole, the dynamical mass that makes it the best-measured supermassive black hole, the gravitational redshift, the relativistic precession and the bound on extended mass all rest principally on the orbit of S2, with the orbits of other nearby stars as corroboration. The precision of each is limited not by the theory but by how many pericentre passages have been observed — two so far with any precision, one with interferometry.

What the integration leaves out #

The orbit is integrated in its own plane with relativity at first post-Newtonian order, as an extra radial force; higher orders change the precession by parts in a thousand at S2’s pericentre, below the measurement’s precision. The extended mass is a smooth, spherical cusp with a single power-law profile normalised at S2’s semi-major axis. A real cluster is lumpy — a few hundred stellar black holes, each of ten solar masses, scattered through the orbit — and a lumpy distribution adds random perturbations to S2’s orbit that could, over many orbits, be detected in their own right. The black hole’s mass and distance are held fixed, while in the real orbit fit they are free parameters correlated with the precession, which is part of why the published bound differs from the simple one drawn here.

The frame-dragging and quadrupole terms are given by their leading-order magnitudes for an orbit perpendicular to the spin, without the dependence on orientation that the real effects carry.

Still open: a second orbit through pericentre #

S2 next passes pericentre in 2034. A second passage observed with interferometric precision would roughly halve the uncertainty on its precession, and the measurement of the extended mass would improve with it. If the precession then came out significantly below relativity’s prediction — at, say, eight arcminutes rather than twelve — it would not be read as a failure of relativity, which is far better tested elsewhere. It would be read as a weighing: a measurement of a few thousand solar masses of something unseen, spread through the innermost thousand astronomical units of the Galaxy.

Which something it would be is the open question. Stellar black holes sinking to the centre, a dark-matter spike, a population of faint stars below the detection limit — each predicts a different radial profile and therefore a different signature in the orbits of stars at different distances. With S2 alone the profile cannot be separated from the mass. With several stars on orbits of different sizes, each precessing by an amount set by the mass inside its own orbit, the profile would be measured shell by shell. The rosettes of the innermost stars are the instrument; whether there are enough bright enough stars close enough in to read it is what the next decade of observations at the Galactic centre will find out.