The black holes that can be weighed are the heavy ones

Roughly a hundred supermassive black holes have masses measured directly, from the motions of stars or gas orbiting close to them, and nearly everything else known about the population rests on those hundred. They define the correlation between a black hole’s mass and the velocity dispersion of its galaxy’s bulge, with a scatter of about three-tenths of a dex, and that correlation is then used to estimate the masses of millions of black holes that can never be weighed directly — in distant quasars, in galaxies too faint for detailed spectroscopy, in the populations whose mergers set the rate of recoils and the amplitude of the gravitational-wave background.

A direct measurement has a precondition. The orbits that weigh a black hole must be close enough to it that its gravity dominates, which means inside its sphere of influence, and the sphere has to be resolved on the sky. That condition depends on the black hole’s mass. At a given distance and a given galaxy, a heavier black hole has a larger sphere and is easier to weigh, and a lighter one may not be weighable at all.

A sample defined by a threshold on the quantity being measured is a biased sample. The question is only how biased.

The radius a black hole rules #

The sphere of influence is the radius inside which a black hole’s gravity dominates the stars’. A practical definition sets the black hole’s mass equal to the mass that would produce the galaxy’s velocity dispersion at that radius, giving

$$r_{\rm infl} = \frac{GM_\bullet}{\sigma^2}.$$

Inside it, stars orbit the black hole like planets orbit the Sun, their speeds rising inward as one over the square root of radius. Outside it, they orbit the galaxy’s distributed mass, and the black hole is a small perturbation.

For the Milky Way’s black hole, four million solar masses in a bulge with a dispersion of about a hundred kilometres a second, the sphere is under two parsecs in radius. For the black hole in M87, six and a half billion solar masses in a galaxy with a dispersion of over three hundred, it is about two hundred and fifty parsecs. The two black holes differ in mass by a factor of more than a thousand and in the size of the region they rule by a factor of more than a hundred, because the dispersion rises with the mass.

What a telescope sees is the angle. At the distance of the Virgo cluster, about seventeen megaparsecs, M87’s sphere subtends three arcseconds — easily resolved from space and from the ground. A typical black hole in a galaxy with a dispersion of seventy kilometres a second, one and a half million solar masses with a sphere of about a parsec, subtends a hundredth of an arcsecond at the same distance, below the resolution of any telescope now operating at optical or infrared wavelengths. The figure draws the angles against distance for three galaxy types: each is a straight line falling as one over distance, and each crosses each resolution limit at a distance that depends steeply on the galaxy.

A horizon that grows as the dispersion to the power 2.24 #

Turning that around gives each type of galaxy a horizon: the greatest distance at which its black hole can be weighed by a given telescope.

The black-hole mass rises as the dispersion to the power 4.24 along the mean relation, and the sphere’s radius as the mass divided by the dispersion squared, so the horizon rises as the dispersion to the power 2.24. At Hubble’s resolution of a tenth of an arcsecond, the black holes of the most massive galaxies are weighable out to about seventy megaparsecs; those of galaxies with dispersions of seventy kilometres a second only out to about three. The volume searched for each scales as the cube, so the most massive galaxies are sampled from a volume twenty thousand times larger than the smallest.

That alone would distort the sample’s make-up — it is why the measured sample is dominated by massive ellipticals, and why the low-mass end of the relation rests on a handful of nearby galaxies. But the distortion that matters most is not between galaxy types. It is within each type.

The sample is the upper tail #

At fixed dispersion, black-hole masses scatter about the mean relation by a factor of two or three. A black hole twice as heavy as average for its galaxy has a sphere of influence twice as large, and can be weighed twice as far away — in eight times the volume. A survey that weighs every black hole whose sphere it can resolve therefore finds, at each dispersion, an over-representation of black holes above the relation, drawn from the larger volume in which only they are resolvable.

The opening figure is that effect simulated. Four thousand galaxies are spread uniformly through a sphere of 120 megaparsecs, with dispersions between 60 and 350 kilometres a second and black-hole masses scattered by 0.45 dex about a relation of slope 4.24. The five hundred or so whose spheres exceed a tenth of an arcsecond are kept. Fitted to those alone, the relation comes out with its normalisation 0.6 dex — a factor of four — higher than the truth at 200 kilometres a second, and with a slope near three rather than 4.24, because the bias is worse at low dispersions where fewer galaxies are resolvable at all.

The size of the effect in the simulation depends on the scatter assumed, and the scatter is itself measured from the biased sample — which understates it, because the sample is missing the lower half of the distribution at most dispersions. A relation that is too high, too shallow and too tight is exactly what selection on the upper tail produces, and those are the three properties that need checking in the real one.

This argument was made quantitatively in 2016, by comparing galaxies with dynamically measured black holes against large samples of galaxies of the same stellar mass and dispersion, without black-hole measurements. The galaxies with measured black holes turned out to have systematically higher dispersions for their stellar mass than galaxies in general — they are not typical — and correcting for the resolution selection lowered the inferred normalisation of the relations by factors of several. The size of the correction is disputed, because it depends on the scatter and on how the selection is modelled, and the direction is not.

Why the slope comes out flat and the scatter tight #

The bias is not a constant offset, and the shape it takes is the important part. A galaxy with a dispersion of three hundred kilometres a second has a sphere of influence resolvable out to seventy megaparsecs, so within a survey volume of a hundred megaparsecs most such galaxies are weighable whatever their black hole’s mass, and the sample is nearly complete. A galaxy with a dispersion of eighty is weighable only within a few megaparsecs if its black hole is average, and within ten or twenty if its black hole is four times heavier — so among such galaxies the survey finds almost exclusively the heavy ones. The offset therefore grows towards low dispersion, and a line fitted through a sample that is nearly unbiased at the top and strongly biased at the bottom comes out shallower than the truth.

The scatter suffers in the same direction. At each dispersion, the selected black holes are drawn from the upper part of the distribution, and the upper part of a distribution is narrower than the whole. The measured scatter about the measured relation is therefore smaller than the true scatter, and a population argument that uses it — how many galaxies of each kind host a black hole of each mass — inherits a distribution that is both too high and too narrow.

There is also a subtler coupling. The velocity dispersion itself is measured within an aperture, often the half-light radius, and is a statement about the whole galaxy’s orbits, not about the black hole. Galaxies with unusually high dispersions for their stellar mass are more compact, and a compact galaxy at a given distance has a brighter, more easily measured nucleus. The selection on the sphere of influence and the selection on the galaxy’s own compactness push the same way, and separating them requires samples of galaxies with and without measured black holes, matched in everything else.

What an unresolved sphere does to a mass #

Selection is one consequence of the sphere of influence. The other is what happens to a mass measurement when the sphere is only marginally resolved, and it explains why measured masses for the same black hole have sometimes changed by factors of two.

The orbits measure the total mass inside each radius. The black hole’s mass is that total minus the mass of the stars, and the stars’ mass is their light times a mass-to-light ratio that has to be fitted alongside the black hole. Inside the sphere of influence, the black hole dominates the enclosed mass and the stellar term is a small correction. At the sphere’s own radius, by its definition in an isothermal galaxy, the stars inside it weigh twice as much as the hole. At three times the radius, they weigh six times as much, and a ten per cent error in their mass-to-light ratio becomes a sixty per cent error in the black hole.

The dark matter enters the same way. Most early stellar-dynamical models left the galaxy’s dark halo out, because inside a few kiloparsecs stars dominate. For M87 that omission mattered: the stellar mass-to-light ratio was being fitted to motions over a range of radii where the halo contributed, the fit pushed the stars’ mass-to-light ratio up to compensate, and the black hole, which is the total minus the stars, came out too light. Including the halo in 2009 roughly doubled the black hole’s mass, to six billion solar masses — the value the Event Horizon Telescope’s image of the shadow, ten years later, would confirm from a ring whose size depends on almost nothing but the mass.

Sharper telescopes move the bias #

The obvious remedy is better resolution, and it helps less than it seems.

At a tenth of an arcsecond, the resolvable sample sits about 0.4 dex above the relation in the most massive galaxies and nearly a dex above it in the smallest it reaches. At a hundredth of an arcsecond — the resolution of a forty-metre telescope with adaptive optics — the bias in the most massive galaxies all but vanishes, because their spheres are resolvable to distances beyond any reasonable survey volume and every black hole in that volume is weighed. But the sample now reaches galaxies with smaller dispersions, which were not in it before, and there the same selection operates: only the heavier holes are resolvable, and the offset is again several tenths of a dex.

The pattern is general. Any threshold on a quantity correlated with the measurement produces a sample biased at its threshold, and improving the instrument moves the threshold without abolishing it. It is the logic of Malmquist bias applied to masses instead of luminosities: a flux-limited survey contains the most luminous objects at each distance, and a resolution-limited one contains the most massive black holes at each dispersion. The cure in both cases is the same — model the selection and fit the population through it — and it requires knowing the scatter, which the biased sample cannot measure.

What was actually measured #

Four techniques weigh black holes inside their spheres of influence, and each has its own reach.

Stars. The dispersion and rotation of stars measured from spectra across the nucleus, modelled with orbit-superposition methods that fit the whole distribution of orbits. This is the workhorse, applicable to any galaxy with a bright enough nucleus, limited by resolution and by the degeneracy with the stellar mass-to-light ratio and the dark halo.

Gas. The rotation of a disc of gas around the black hole, from emission lines or, increasingly, from molecular lines observed with millimetre interferometers at resolutions of hundredths of an arcsecond. Simple to model when the disc is regular, and impossible when it is not — and regular discs are commoner in some kinds of galaxy than others, which is a selection of its own.

Masers. Water masers in a thin disc of molecular gas a fraction of a parsec from the black hole, observed with radio interferometry at resolutions of a ten-thousandth of an arcsecond, whose velocities and positions trace a Keplerian rotation curve directly. The maser disc in NGC 4258 gives a black hole of forty million solar masses to a few per cent, and a geometric distance to the galaxy. Only a few dozen such discs are known, nearly all in a particular kind of active galaxy.

Individual stars. Only in the Milky Way, where single stars are followed around complete orbits and the mass comes from Kepler’s third law with no modelling at all.

The resulting sample is about a hundred galaxies, heavily weighted towards nearby massive ellipticals and bulges, with a few dozen spirals and very few dwarfs. Every relation derived from it carries the selection of each technique and, on top of all of them, the sphere-of-influence threshold.

Black holes weighed in time instead of space #

There is one way to measure black-hole masses that needs no spatial resolution at all, and it shows how far the local selection reaches.

In an active nucleus, gas orbiting within a few light-days to light-months of the black hole emits broad lines whose widths are the gas’s orbital speeds. When the central source flickers, the lines respond after a delay equal to the light-travel time to the gas, and that delay measures the size of the region without resolving it. Size and speed together give a mass by the virial theorem, for quasars at any distance — but only up to a factor that encodes the geometry and orientation of the gas, which the delay does not measure.

That factor is calibrated by requiring the reverberation masses of nearby active galaxies to lie on the same relation with dispersion as the dynamically weighed black holes. If the dynamical sample sits above the true relation, the calibration inherits the offset, and the factor that multiplies every quasar mass comes out too large by the same amount. The selection of a hundred nearby galaxies by the size of their spheres of influence thereby reaches into every black-hole mass measured at high redshift.

The one break in that chain is interferometry. An array of large telescopes combined at near-infrared wavelengths resolves structure no single mirror can, and it has begun to resolve the broad-line regions of bright quasars directly, measuring their angular sizes and velocity gradients and so giving masses whose geometry is measured rather than calibrated. A few such masses exist so far. Where they can be compared with the calibrated ones, they test the whole chain, the selection included.

Where the simulation stops #

A synthetic population. The galaxies are spread uniformly in volume with dispersions uniform in logarithm, and the masses scattered about a straight relation with constant scatter. The real local volume is clumpy, its galaxies have a dispersion function falling steeply at high dispersion, and the scatter may depend on galaxy type. The size of the bias would change; its existence and direction would not.

A sharp threshold. A real measurement does not switch from impossible to perfect at one angle. Marginally resolved spheres give masses with large uncertainties and systematic errors from the stellar and dark contributions, and how they are treated — included, excluded or used as upper limits — is part of how a sample is selected.

An isothermal galaxy. The mass-share figure assumes stars distributed as an isothermal sphere. A galaxy with a flattened core has fewer stars inside the sphere of influence and a cleaner measurement; one with a steep cusp has more.

A threshold on the measured quantity #

The general statement is about samples defined by their measurability. A sample selected by a threshold on anything correlated with the quantity being measured is a sample of that quantity’s upper tail at the threshold, and relations derived from it describe the tail rather than the population. That is true of flux-limited surveys, of planet searches with a detection threshold, of clusters selected by their gas, and of black holes selected by the size of the region they rule.

Whether a quantity can be measured is itself a function of the quantity, and a sample built from the measurable cases inherits that function. The correction always needs the distribution’s width, which the selected sample systematically underestimates.

The other reading is about how far a calibration travels. The relation between black-hole mass and dispersion is used to calibrate the factor that turns every quasar’s line width into a mass, and through it the masses of black holes at every redshift. A bias in the local sample’s normalisation propagates into every one of those masses, into the black-hole mass function across cosmic time, and into the predicted amplitude of the gravitational-wave background from merging supermassive binaries — which, when pulsar-timing arrays first measured it, came out at the high end of the predictions, closer to what the biased local relation implies than to what the corrected one does.

Still open: a relation measured where the selection does not reach #

The way out is a sample defined by something other than the black hole’s mass — every galaxy in a volume, weighed or bounded. Interferometers at infrared wavelengths have begun to resolve the regions of fast-moving gas around distant quasars directly, giving black-hole masses at redshifts where no sphere of influence could ever be seen, and molecular-gas interferometry is reaching dwarf galaxies whose black holes were beyond optical telescopes. Whether the relation’s normalisation, slope and scatter survive in samples selected some other way — and whether the scatter is in fact the factor of two the selected sample says or something larger — is the measurement the next generation of large telescopes is designed to make, galaxy by galaxy, down to the smallest dispersions where the selection bites hardest.