The factor that multiplies every quasar mass

The first rung of this anchor got a length out of a delay: the broad lines follow the continuum by days to weeks, the delay is a light-travel time, and a length inside an unresolvable object is an extraordinary thing to have. Multiply it by the square of the line width and divide by the gravitational constant, and the units are a mass.

The units are a mass. The quantity is not, quite. What comes out of that arithmetic is

$$M_{\rm BH} \;=\; f\,\frac{c\,\tau\,\Delta V^2}{G}$$

and $f$ is a dimensionless number of order unity that encodes everything about the geometry and the kinematics of the emitting gas — how flattened it is, how it is inclined, whether the motions are orbits or a wind, and which part of the line profile the width was measured on. Nothing in the observation constrains it. The product $c\tau\Delta V^2/G$ is called the virial product, and it is what a reverberation campaign actually measures.

The distinction the bracket makes is the essay’s subject. A scatter can be averaged down by observing more objects. A factor cannot.

From a campaign to a catalogue #

Reverberation mapping is expensive. It needs a spectrum every few days for months, on one object, and the delay it returns is a few tens of days for a modest active nucleus and hundreds for a luminous quasar.

What turns a few hundred campaigns into a hundred thousand masses is an empirical relation between the lag and the luminosity, which is what the first rung ended on. The physics behind it is almost too simple: the broad-line gas sits where the ionising flux gives the right ionisation state, and a fixed flux at a distance means the distance goes as the square root of the luminosity.

A single-epoch mass is then: measure the continuum luminosity from the spectrum, get a radius from the relation, measure the width of a broad line in the same spectrum, and multiply. One exposure, one mass, and no waiting.

Why the region is where it is #

The half-power is worth pausing on, because it is the one part of this chain that is a derivation rather than a fit, and because it explains what the scatter is scatter in.

The state of photoionised gas is governed by the ionisation parameter — the ratio of the density of ionising photons arriving to the density of particles waiting for them. Gas at a given ionisation parameter emits a given line strongly; gas an order of magnitude either side does not. The photon density at distance $R$ from a source of ionising luminosity $L$ goes as $L/R^2$, so a fixed ionisation parameter at a fixed gas density means $R \propto \sqrt{L}$ exactly. The broad-line region is not a place with edges; it is wherever the gas happens to be that satisfies that condition, and it moves outwards as the square root of the luminosity because the condition does.

Two things follow. The first is that the relation should hold across the whole quasar population without a break, which is roughly what is observed over four decades of luminosity. The second is that its scatter is a scatter in gas density and in the shape of the ionising continuum — two properties that differ from object to object and that nothing in a survey spectrum reveals. That is why the 0.19 dex is not going to be reduced by better photometry.

There is also a check available inside a single object, and it works. Different lines have different ionisation parameters, so they should reverberate at different radii — higher-ionisation lines closer in, moving faster. Measured, they do: in the best-studied nuclei the high-ionisation lines return shorter lags and broader profiles, and the product of lag and width squared is the same for all of them. That consistency is the strongest evidence that the motions are gravitational rather than a wind, and it is a test the method could have failed.

The four widths #

The terms deserve to be named individually, because they fail in different ways.

The radius. The relation’s scatter is partly real and partly an artefact of the continuum measurement: at low luminosity the host galaxy’s starlight contributes most of the light in the aperture, and subtracting it needs an image the survey may not have. Objects whose luminosity is overestimated get radii and masses that are too large.

The line width. A broad line is measured on a profile blended with narrow-line emission at its core and with a forest of iron lines across its wings, and the answer depends on whether the width is taken as the full width at half maximum or as the second moment of the profile. Those two differ by a factor of about two for a typical profile, and they differ differently for different profile shapes — so the choice is not a convention that cancels. The width enters the mass squared, which doubles whatever error it carries.

The variability. The luminosity in the survey spectrum is the luminosity on one night. The object was brighter or fainter when the relation was calibrated for it, and a quasar varies by a few tenths of a magnitude on the timescales that matter. This is the smallest term and the only one that would shrink with repeated observation. It is also the one that carries a subtlety: the gas takes a light-crossing time to respond, so the radius appropriate to tonight’s spectrum is the radius set by the luminosity of some weeks ago, and a rapidly varying object is being assigned a radius it has not yet reached.

The orientation. And the largest.

A flattened broad-line region is not an assumption made for convenience. The evidence for it is that radio-loud objects seen face-on, identified as such by their beamed jets, systematically show narrower broad lines than those seen edge-on — a correlation between an orientation indicator and a line width that has no explanation if the gas moves isotropically. The consequence is that a single $f$ is a population average over an unknown viewing angle, and that a mass derived with it is wrong by a factor that depends on which way the object happens to point.

A broad-line region that is a flattened rotating structure presents a line width that depends on how it is inclined: face-on, the orbital motion is across the line of sight and the line is narrow; edge-on it is broad. The mass inferred from a fixed $f$ therefore scales roughly as $1/\sin^2 i$, and the spread of inclinations across a sample is a spread in mass of about a third of a dex.

Where the factor comes from #

The four widths above all assume $f$ is right. Establishing that it is right is a separate exercise, and it involves no quasars.

Quiescent galaxies with resolvable centres have black-hole masses from stellar or gas dynamics — orbits, measured, in a potential. Those masses correlate tightly with the bulge’s stellar velocity dispersion, which is the same quantity a galaxy’s own support is read from, through a relation that spans three decades in mass. The calibration is to assume that the active galaxies obey the same relation, and to choose the single $f$ that puts them on it.

The correlation being used has its own history worth a sentence, because it is not obvious that it should exist at all. A black hole’s gravitational influence extends over a few parsecs; a bulge is thousands. The hole knows nothing about the bulge’s dispersion and the bulge knows nothing about the hole’s mass, and yet the mass at the centre that is not stars tracks that dispersion to the fourth power with less than a third of a dex of scatter. Whatever produces that relation is a coupled history rather than a force, and using it as a calibration means importing whatever that history was.

Two consequences follow immediately and both are awkward.

A third consequence is quieter and affects what the sample can be. The calibration needs a bulge velocity dispersion, which needs stellar absorption lines, which needs the nucleus not to outshine its host. That restricts the calibrating sample to the least luminous active galaxies — Seyferts rather than quasars — and the factor fitted on them is then applied to objects a thousand times brighter, whose broad-line regions are a hundred times larger and whose accretion rates are entirely different. Whether $f$ is the same for both is not known and is not testable with the data that exist. It is assumed, and the assumption is the same shape as the one that lets two colours and almost nothing between them stand in for a galaxy’s whole history.

The first is that any evolution in the M–σ relation itself is absorbed into $f$ rather than measured. A study asking whether black holes grew before their bulges or after uses masses whose zero point was fixed by assuming the relation does not evolve. The circularity is not fatal — the evolution being sought is in the ratio at high redshift and the calibration is local — but it is the kind of dependence that has to be stated, and often is not.

The second is that the scatter of the relation used for the calibration, about 0.29 dex, sets a floor on how well $f$ can be determined from a sample of a few dozen. That floor is the factor-of-three bracket in the first figure.

What is actually measured #

Assembling the chain end to end makes the length of it plain. A survey records a spectrum. From it come a continuum flux and a line profile. The flux becomes a luminosity through a distance, which comes from a redshift and a cosmology. The luminosity becomes a radius through a relation calibrated on a few hundred campaigns. The profile becomes a width through a decomposition into broad, narrow and iron components. Radius times width squared over $G$ becomes a virial product. And the virial product becomes a mass through a factor fitted at redshift zero against a correlation whose own masses came from a different technique entirely.

Six steps, of which two are empirical relations and one is a fitted constant. Only the first two are measurements in the sense that an instrument recorded them; the rest are conversions, and each conversion was established on a sample of objects that is not the object being measured. It is worth saying plainly that this works — the masses are not nonsense, they order objects correctly, and they reproduce independent estimates where those exist. But an ordering good to a factor of three is what it is, and the quantities most often computed from these masses are the ones least tolerant of it. The Eddington ratio is a luminosity divided by a mass, so it inherits the whole 0.42 dex. The black-hole mass function at high redshift is a steeply falling distribution convolved with that width, which scatters the numerous small objects up into the rare large bins and produces an apparent excess of the most massive holes that is entirely a smearing artefact.

That last effect has a name in every field that measures a steep distribution badly, and it is the same one behind a dispersion inflated by orbits nobody resolved and behind the corrections that a luminosity function with a knee in it needs before it can be believed at its bright end.

The size of it can be estimated in one line. A mass function falling as a power law of index $-3$, convolved with a Gaussian of 0.42 dex, is inflated at any given mass by $\exp\!\big[(\ln 10)^2 \sigma^2 \alpha^2/2\big]$ — a factor of about 3.5. So the number of black holes above a given large mass, read straight off a single-epoch catalogue, is several times the true number, and the discrepancy grows with mass. Every claim that the most massive black holes at high redshift are too massive for their host galaxies has to survive that correction before it means anything, and the correction depends on the width of a distribution which is itself uncertain.

There is an independent check on the whole enterprise and it is a weak one. A tidal disruption flare puts a ceiling on a mass, because a hole above about 10⁸ solar masses swallows a solar-type star whole and produces no flare at all; and the masses of the galaxies that do flare, from the M–σ relation, are consistent with that ceiling. Consistent is what it is: a bound on a population, not a measurement of an object.

The history is short and the method is old #

Reverberation as an idea dates from 1982, when Blandford and McKee wrote down the transfer function and pointed out that a delay is a length. It took until the 1990s for the observations to exist — a spectrum every few days for a season, on the same object, needs a telescope nobody wants to give up, and the campaigns that produced the first reliable lags were international consortia sharing small instruments precisely because no large one could be booked that way.

The virial factor’s calibration against the M–σ relation came later still, around 2004, and it was proposed as a stopgap. It is still the method. Nothing in the two decades since has replaced it, and the reason is instructive: replacing it means measuring $f$ for individual objects, which means resolving the broad-line region, which is what the whole method exists because nobody can do.

The generalisation #

The structure worth extracting is the difference between an error that averages down and one that does not, and how easily the two are added together.

Everything in the first figure’s quadrature is a scatter: observe more quasars and the mean mass of a population is determined better. The bracket underneath is not. It is one number applied to every object, so a sample of a hundred thousand determines it exactly as well as a sample of one — which is to say, not at all. Quoting a mass function with error bars that shrink as the square root of the sample size, when the dominant uncertainty is a multiplicative constant, produces a result whose stated precision is a statement about the catalogue’s size and nothing else.

The same shape is everywhere in this collection. The Hubble constant’s two measurements disagree by far more than either’s statistical error because each carries a calibration. A cluster weighed three ways gives three answers whose differences are not scatter. In every case the diagnostic is the same question: would this error be smaller if the observation were repeated on a different object?

Where the ladder goes next #

The next rung takes the same delay and resolves it in velocity. Measuring the lag separately in the blue wing, the core and the red wing of a broad line gives the delay as a function of the line-of-sight velocity, and that two-dimensional map does distinguish a rotating disc from an inflow from an outflow — the thing the centroid cannot. It exists for about a dozen objects, it is the only route to $f$ for an individual nucleus, and what it has found is that the same object can look like different geometries in different lines.

Further rungs on this anchor: the radius–luminosity relation’s breakdown at high accretion rate, where the most luminous objects have lags shorter than the relation predicts; continuum reverberation between wavebands, which maps the accretion disc rather than the line region and returns a disc several times larger than a standard model allows; the use of quasars as standard candles built on that same relation; and the microlensing of a lensed quasar, which measures the disc’s size by an entirely unrelated route and agrees with the reverberation answer no better than a factor of a few.