The mass at the centre that is not stars

The centre of the Milky Way is eight kiloparsecs away, behind thirty magnitudes of visual extinction, and it contains an object that has been studied by watching individual stars orbit it for thirty years.

The measurement that results is the most direct in this field and one of the most direct in the whole collection. It uses Kepler’s third law, unmodified, applied to a single star.

The calculation #

Everything about the arithmetic is elementary, which is the point.

The semi-major axis is measured as an angle, 0.1251 arcseconds. Multiplying by the distance to the Galactic centre in parsecs converts it directly to astronomical units, because an arcsecond at a parsec is an astronomical unit — that is the definition of the parsec. So $a = 0.1251 \times 8280 = 1036$ AU.

The period is watched: the star has now completed more than one full orbit since monitoring began, so the period is not fitted from a partial arc but measured, at 16.05 years.

Kepler’s third law in solar units states $M = a^3/P^2$ with $a$ in AU, $P$ in years and $M$ in solar masses. That gives $1036^3/16.05^2 = 4.3$ million.

No modelling, no assumed profile, no equilibrium assumption. A star on an ellipse around a focus, and the two numbers that fix the ellipse.

Why the density is the argument, not the mass #

Four million solar masses in the middle of a galaxy is not by itself remarkable; the inner parsec of the Milky Way contains a dense star cluster of comparable mass. What is remarkable is the volume.

The orbit constrains the mass to lie inside S2’s periapsis distance of 120 AU. A mass of $4.3\times10^6$ suns inside a sphere of radius 120 AU is a mean density of about $5\times10^{15}$ solar masses per cubic parsec.

For comparison, the densest star clusters known reach $10^6$ to $10^7$ solar masses per cubic parsec. The measurement therefore exceeds the densest stellar system ever observed by nine orders of magnitude.

That gap is the argument. A cluster of ordinary stars at such a density would evaporate or collapse in far less than the age of the Galaxy through collisions and encounters; a cluster of stellar remnants or brown dwarfs runs into the same difficulty; a single degenerate object of four million solar masses cannot exist, since no equation of state supports anything above a few solar masses. Every alternative has been examined and the constraint is now so tight that the argument is essentially closed.

The observation behind the number #

The measurement is a triumph of instrumentation more than of physics, and it is worth stating what it took.

Wavelength. The extinction towards the Galactic centre is thirty magnitudes in the visual — a factor of $10^{12}$ — and about 2.5 magnitudes in the near infrared. So the work is done at two microns, where a tenth of the light gets through instead of none of it.

Angular resolution. S2’s orbit is a quarter of an arcsecond across. Resolving it against a crowded field of stars requires better than the atmosphere allows, which meant speckle imaging in the 1990s and adaptive optics from about 2002 — and, more recently, interferometry combining four telescopes, which reaches tens of microarcseconds.

Persistence. The orbital period is sixteen years, so measuring it required a programme that outlasted several generations of instrument. Two groups, in Germany and in California, ran independent campaigns for three decades and their results agree.

The interferometric measurements have gone further than the orbit. They detect the relativistic precession of S2’s orbit — the same advance of periapsis that Mercury shows, here twelve arcminutes per orbit rather than 43 arcseconds per century — and the gravitational redshift of the star’s light at periapsis. Both match general relativity.

What was there before the orbits #

The centre of the Galaxy was known to contain something unusual for decades before anybody could measure a stellar orbit, and the intermediate evidence is worth recording because each step was reasonable and none was conclusive.

A compact radio source. Sagittarius A* was found in 1974 as a bright, extremely compact radio source at the dynamical centre. Very-long-baseline interferometry then showed it to be smaller than an astronomical unit, which is a remarkable constraint on something producing that much radio power.

Its stillness. The source’s own proper motion is essentially zero — it moves less than a kilometre per second relative to the Galactic frame, in a region where every star is moving at hundreds. An object that refuses to be pushed about by its neighbours is heavy, and that argument alone put a lower bound of hundreds of thousands of solar masses on it before any orbit had been traced.

Gas velocities. The ionised gas streamers in the inner parsec move faster than the enclosed stellar mass can explain, which pointed the same way with the weaker assumption that gas follows gravity — weaker because gas also feels pressure, magnetic fields and radiation, none of which a star does. Each of these was consistent with a black hole and none excluded a dense cluster. The orbits did, by adding the one thing the others lacked: a small radius with a large mass inside it.

Doing it for a galaxy that cannot be resolved #

Individual stellar orbits are available for exactly one galaxy. For everything else, the black hole’s mass has to be inferred from the aggregate motion of the stars near the centre, and that only works if its gravity dominates them.

The radius inside which it does is the sphere of influence, $r_h \approx GM_{\bullet}/\sigma^2$, and for a $10^8$ solar-mass hole in a galaxy with a 200 km/s dispersion that is about ten parsecs. At the distance of the Virgo cluster it subtends a tenth of an arcsecond, which is why these measurements need space-based or adaptive-optics spectroscopy and why the sample of galaxies with well-measured black-hole masses is a hundred or so rather than thousands.

What the correlation is thought to mean #

The M–σ relation was found independently by two groups in 2000 and immediately reoriented the subject, because a tight correlation between a small object and a large one demands a mechanism.

The favoured account is self-regulation through feedback. As the hole accretes, it radiates; the radiation and the winds it drives push on the surrounding gas; when the hole is large enough, that push is sufficient to expel the gas from the bulge’s potential well, at which point accretion stops. The condition that the energy or momentum output balances the binding energy of the bulge gives a relation of about the observed slope.

That account has a satisfying consequence. It explains not just the correlation but the direction of the causation — the hole’s growth stops when it has affected its surroundings, so the final mass is set by the surroundings’ depth. And it connects to why the most massive galaxies are red and to why the luminosity function has a knee: the same feedback that sets the hole’s mass suppresses star formation in the galaxy.

Whether it is right is not settled. A rival account requires no feedback at all: if galaxies grow mainly by mergers, and both black holes and bulges add up in the same mergers, then the central limit theorem alone drives the ratio towards a constant with small scatter. That mechanism produces a correlation without any physical interaction whatever, which is an uncomfortable but genuine alternative.

The scale of the thing, in units that mean something #

Numbers of this size resist intuition, so it is worth converting them twice.

The Schwarzschild radius of $4.3\times10^6$ solar masses is $2GM/c^2 = 1.3\times10^7$ kilometres, which is 0.085 astronomical units — about a fifth of Mercury’s orbit. The event horizon of the Milky Way’s central black hole would fit comfortably inside the orbit of the innermost planet.

Its angular size at eight kiloparsecs is ten microarcseconds. That is the angular diameter of an orange on the Moon, and it is the resolution that imaging the shadow required — which is why the observation needed a telescope the size of the Earth, built by combining radio dishes on several continents.

And S2 at periapsis is travelling at 7,700 kilometres per second, which is 2.6 per cent of the speed of light. That is by a wide margin the fastest orbital motion of a star known, and it is the reason the relativistic corrections are measurable at all.

What the picture cannot show #

An orbit measures a mass inside a radius, not a black hole. Everything above is a statement about mass and volume. The identification with a black hole comes from the absence of alternatives at that density, not from any positive observation of an event horizon in this measurement. That positive observation has since been made by very-long-baseline interferometry, which images the shadow cast by the horizon — but it is a different measurement, and the orbit alone does not make it.

The distance enters as the cube. The mass scales as $a^3$ and $a$ scales as the distance, so a one per cent error in $R_0$ is a three per cent error in the mass. The improvement in the black hole’s mass over the last two decades is mostly an improvement in $R_0$.

And the M–σ figure draws a model population. The points are drawn from the published relation with its measured intrinsic scatter, and the slope is fitted back off them. What is real is the relation’s parameters; the individual points are what such a relation looks like, not a catalogue.

What one of these does when it is fed #

Almost every galaxy has one, and almost all of them are quiet — Sagittarius A* radiates about $10^{-9}$ of what its mass would allow. When one is fed, the result is the most luminous sustained phenomenon in the universe.

The energy source is gravitational and its efficiency is extraordinary. Matter spiralling in through an accretion disc radiates something like ten per cent of its rest-mass energy before it crosses the horizon, against the 0.7 per cent that hydrogen fusion extracts. Accretion is, by a wide margin, the most efficient energy-release mechanism known that does not involve annihilation.

There is a natural ceiling on the rate. As the luminosity rises, radiation pressure on infalling material grows until it balances gravity; above that point the flow is pushed back. The Eddington limit that results is proportional to the mass alone,

$$L_{\text{Edd}} = \frac{4\pi G M m_p c}{\sigma_T} \approx 1.3\times10^{31}\,\frac{M}{M_\odot}\ \text{W},$$

which for a $10^8$ solar-mass hole is $10^{39}$ watts — some ten trillion suns, from a region the size of the solar system.

That number is the reason quasars are visible across the observable universe, and the reason the feedback account above is plausible: an object radiating at that level, sitting at the centre of a galaxy, is not a passive resident.

Weighing one that is too far away to resolve #

The sphere of influence argument above rules out orbital or dispersion measurements for anything beyond about a hundred megaparsecs. Quasars are found at redshifts of six and beyond, and their masses are quoted routinely — so something else is being measured.

The method is reverberation mapping, and it substitutes a light-travel time for an angular size. An active nucleus varies: the continuum from the inner disc brightens and fades on timescales of days to weeks. The broad emission lines, produced by gas further out that is photoionised by that continuum, follow the same variations with a delay, and the delay is the light-crossing time of the intervening distance. Cross-correlating two light curves therefore measures a radius in light-days, from photometry alone, with no resolution requirement at all.

The line’s width supplies the speed. Putting them together,

$$M_{\bullet} = f\,\frac{R\,\Delta v^2}{G},$$

which is the virial theorem with a geometry factor $f$ absorbing the unknown shape and inclination of the emitting region.

That factor is the whole difficulty, and it is calibrated rather than derived. Nobody knows the geometry of the broad-line region — whether it is a flattened rotating structure, a wind, or something less tidy — so $f$ is fixed by requiring that reverberation-mapped galaxies obey the same M–σ relation as the ones measured from stellar dynamics. Its value comes out near 4 to 5, with a scatter that is itself a factor of two or three.

The consequence is worth stating plainly, because it is the shape of the whole subject. The masses of the most distant black holes rest on a factor calibrated against nearby galaxies, whose masses rest on stellar dynamics inside a sphere of influence, whose scale rests on the one galaxy where a single star’s orbit has been traced. Every quoted quasar mass is at the far end of a chain that begins with S2, and the chain is three links long.

The light that says they grew by eating #

There is one further measurement, made in 1982 and requiring no individual mass at all, and it is the reason the accretion account is believed rather than merely available.

Add up all the light ever emitted by quasars — the integrated luminosity of the quasar population over cosmic time, which is measurable from their counts and their luminosity function. If that light was produced by accretion at ten per cent efficiency, it corresponds to a definite mass of material having been swallowed, and therefore to a definite mean density of black-hole mass in the universe today.

Independently, add up the black holes in nearby galaxies, using the M–σ relation and the local galaxy population. That gives a mean density directly.

The two agree, to within the uncertainties, at a few times $10^5$ solar masses per cubic megaparsec. The mass that is there now is the mass the light says was eaten, which establishes that supermassive black holes acquired essentially all of their mass by radiatively efficient accretion rather than by mergers of smaller ones or by collapsing that way to begin with. It also pins the efficiency: the agreement fails if the figure is much below ten per cent, which is an argument for the holes being rapidly rotating, since a maximally spinning hole’s innermost stable orbit is closer in and the efficiency correspondingly higher.

An argument that compares two integrals over entirely different observations, and constrains the spin of objects nobody can see, is a fair example of what this field has to do instead of experiments.

The relation is a fit with three numbers in it, and each of them is worth moving, because the argument the relation is used for depends on a different one each time.

The generalisation #

The structure of the argument is one that recurs and is worth extracting: a mass measurement becomes an identification only when the volume is also constrained.

The mass of the Sun does not tell anyone what the Sun is. The mass of the Sun inside a radius of 700,000 kilometres, giving a mean density of 1.4 grams per cubic centimetre, says it is a fluid rather than a rock. The mass of a white dwarf inside an Earth-sized radius says the electrons are degenerate. The mass of a neutron star inside ten kilometres says the nuclei have merged.

In every case the mass alone is compatible with several things and the density is compatible with one. That is why the periapsis distance of S2 matters as much as its period, and it is why the next improvement in this measurement will come from a star with a smaller orbit rather than from a better mass.

And the relation itself drawn through a larger sample, since what a bigger survey buys is worth being explicit about.

Where the ladder goes next #

The next rung is what these objects do when they are being fed: active galactic nuclei, quasars, and the accretion physics that makes a region the size of the solar system outshine a galaxy of a hundred billion stars.

Later rungs on this anchor: the Eddington limit and what sets the maximum accretion rate; the sphere of influence and why the measured sample is so small; the relativistic effects in S2’s orbit and what they test; the event-horizon images and what they show; the seeds of supermassive holes and the difficulty of growing them fast enough to power the earliest quasars; and the coalescence of two of them after a galaxy merger, which is the loudest event in the low-frequency gravitational-wave sky.