A delay that tells which way the gas is moving

The broad emission lines of an active nucleus are made by gas a few light-days to a few light-months from the black hole, outside the accretion disc whose own spectrum is a stack of temperatures, too close to resolve and too far to see anything else by. When the continuum from the accretion disc brightens, the gas responds a little later, and the delay is a light-travel time and therefore a length. Multiplied by the square of the line’s width and divided by the gravitational constant, it gives a mass — up to a factor that encodes everything unknown about the gas: whether it orbits in a disc or a sphere, at what inclination, whether it is falling in or blowing out. That factor is fixed by making a few dozen nearby nuclei agree, on average, with a relation measured in quiet galaxies, and it is uncertain for any single object by a factor of two or three.

The delay that gives the radius is a single number because it is measured on the whole line. The line is not a single thing. Its blue wing is light from gas approaching the observer and its red wing from gas receding, and each part of the line echoes the continuum at its own delay. Resolve the delay in velocity and a radius becomes a map.

Where an echo comes from #

The geometry of a delay is exact. The continuum flash leaves the centre in every direction; a parcel of gas at distance $r$ and angle $\theta$ from the line of sight re-emits when the flash reaches it, and its re-emitted light reaches the observer late by the extra path,

$$\tau = \frac{r}{c}\,(1 - \cos\theta).$$

Gas on the near side of the hole, between it and the observer, responds with no delay at all; gas directly behind responds after $2r/c$; everything else lies between. Surfaces of equal delay are paraboloids opening away from the observer with the hole at their focus, and a delay is a statement about which paraboloid the gas lies on.

The velocity is just as direct. The line-of-sight component of the gas’s motion shifts its emission by a Doppler amount, blue if the gas approaches and red if it recedes. So each parcel of gas occupies one point in a plane of delay and velocity, and the line’s response to a flash, followed over time and across the profile, fills in that plane. The velocity–delay map is the density of responding gas across it — the transfer function of the unresolved region, with the velocity axis kept instead of summed away.

The same construction is how planetary radar maps an asteroid it cannot resolve: echoes sorted by their round-trip delay and their Doppler shift, each surface element landing at one point in the plane, with a mirror ambiguity across the rotation’s equator that only a second viewing geometry breaks. In a nucleus the flash is the continuum’s variability, the echo is the line, and the ambiguity is the inclination.

Three arrangements, three maps #

The figure computes the map for three arrangements of the same gas at the same radii, and they separate completely.

A rotating disc puts gas on circular orbits. The fastest gas is nearest the hole and has the shortest delays, so the map’s wings — the high velocities — sit at short delay, and its core at long delay. The disc’s near side and far side have the same range of velocities, because rotation is symmetric about the line of sight, and the map is symmetric about zero velocity: an ellipse, or a set of nested ellipses, one for each radius.

Gas falling in reverses that symmetry. Infalling gas on the near side is moving towards the hole, which is away from the observer, so it is redshifted — and it has the shortest delays. Infalling gas on the far side moves towards the hole and so towards the observer, blueshifted, with the longest delays. The red side of the line responds first and the blue side last.

Gas flowing out is the mirror image: the near side approaches, blueshifted and prompt; the far side recedes, redshifted and late. The blue side leads and the red side lags.

Two numbers that cannot tell them apart #

What makes the map necessary is that neither of the quantities an ordinary campaign measures distinguishes the three.

The mean lag is the centroid of the whole map, and it depends on where the gas is rather than how it moves. All three arrangements put their gas between 10 and 30 light-days, and all three give a mean lag of about 22 days. An ordinary cross-correlation of the whole line’s light curve against the continuum’s returns that number and nothing else; it measures the radius correctly and says nothing about the kinematics.

The lag in each velocity bin does. The disc’s profile is an arch, longest at the centre and shorter in both wings. The inflow’s is a slope, from a few days in the far red wing to forty on the blue side; the outflow’s is the opposite slope. The asymmetry is tens of days across the line, which is large compared with the sampling a well-designed campaign achieves — a spectrum every day or two for several months — and it is the signature that velocity-resolved campaigns look for first.

The cost of seeing the slope is paid in photons and in nights. Splitting the line into eight velocity bins divides the flux in each by eight, so each bin’s light curve is noisier by nearly a factor of three than the whole line’s; and a lag difference of ten days across a line whose mean lag is twenty needs the light curves sampled at least every two days for several times the lag, without the seasonal gaps that a target setting behind the Sun imposes. That is why the first velocity-resolved lags came decades after the first lags, from campaigns that devoted a telescope to a handful of nuclei for a season each.

Nor does the profile. A radial flow looks the same in velocity whichever way it runs, so the infalling and outflowing gas give identical line profiles, point for point; only the order in which the two wings respond distinguishes them. The disc’s line is narrower in this example, because at 30° only part of each orbital speed lies along the line of sight — but a narrower line from a heavier hole’s disc and a broader one from a lighter hole’s radial flow can have the same width. The width is the product of the mass and a geometric projection, and the whole difficulty of black-hole masses at high redshift is that a single spectrum measures only the product.

The width of the map is the inclination #

For gas in a disc, the unknown that matters most is the inclination, and the map measures it.

A disc seen face-on shows only the small line-of-sight velocities from the tilts of its orbits, and its line is narrow; the same disc seen at sixty degrees spreads its full orbital speed across the line. The virial factor — the number that turns the lag and the line’s velocity dispersion into a mass, $M = f\,c\tau\sigma^2/G$ — therefore depends on the inclination steeply. For a thin disc it goes roughly as $1/(4\sin^2 i)$, bounded at small inclinations by the disc’s thickness; measured directly on the model clouds, it is 21.5 at fifteen degrees, 7.1 at thirty and 2.5 at sixty. The average factor applied to every quasar is an average over whatever inclinations the calibrating sample happened to have, and the scatter of a factor of two or three around it is mostly this.

The map breaks the degeneracy because its two axes scale differently with inclination. The delay axis depends on the radius and only weakly on the angle; the velocity axis depends on the orbital speed times the sine of the inclination. For a given mean lag, a face-on disc makes a map that is tall and narrow and an inclined one a map that is short and wide, and fitting the shape returns the inclination and the orbital speed separately. The width of the map compared with the width of the line is the factor $f$ for this one nucleus, which is the thing no amount of averaging over a sample can supply.

Real maps are mixtures #

Pure models are for seeing the signatures. Real broad-line regions are not obliged to be any one of them, and the maps measured so far rarely are.

A disc with some infalling gas gives a mostly symmetric map with a tilt — the blue side lagging the red by an amount that grows with the infalling share. That is the commonest shape among the few dozen nuclei with velocity-resolved lags: disc-like, with a modest asymmetry most often in the sense of inflow, sometimes in the sense of outflow, and in a few objects symmetric within the errors. The first well-resolved maps, from monitoring campaigns of nearby Seyfert galaxies with sampling of a day, showed exactly that: an elliptical envelope of rotation with the blue side delayed.

Classifying a map as one of three shapes wastes most of what it contains. The method that has replaced classification is forward modelling: a flexible parametric model of the region — a thick disc of adjustable opening angle and inclination, clouds on orbits ranging from bound ellipses to radial inflow or outflow, a radial distribution of adjustable shape — is used to compute the line’s light curve from the observed continuum’s, and the parameters are fitted to the observed spectra directly. Applied to the best-sampled campaigns, it returns an inclination, a black-hole mass and a virial factor for each object, with no appeal to any relation measured elsewhere. The factors it finds scatter by about the amount the calibration sample implied, and correlate with the inclination the model infers, which is the check the whole scheme needed.

There is one complication the mixtures make unavoidable. Different lines are emitted by gas in different ionisation states, and gas in different ionisation states lies at different radii. The same nucleus mapped in hydrogen’s Balmer line and in a higher-ionisation line of carbon or helium probes different parts of the region, and it can look like a disc in one line and like an inflow in another. That is not a contradiction; it is a region with structure in radius. It does mean that a virial factor measured in one line cannot be carried to another.

A picture to check the map against #

For forty years the geometry of the broad-line region was inferred and never seen. In 2018 an interferometer combining the four large telescopes of one observatory in the infrared resolved the broad-line region of the brightest quasar in the sky — an angular size of about ten microarcseconds, measured as a gradient of position across the line’s velocities, the red side offset from the blue on the sky. The gas was a thick rotating disc, and its fitted radius agreed with the radius from the quasar’s reverberation lag.

That comparison is more than a check. The interferometer measures an angle; reverberation measures a length; their ratio is a distance, with no chain of calibrations beneath it, the same trick as an eclipsing binary played on a quasar. It gives the Hubble constant, so far to about six per cent from a single object, and it gives the black-hole mass with the inclination measured rather than assumed. Where the two techniques have been applied to the same nucleus, the virial factor has been fixed for that object by geometry alone, and it has been consistent with the value the calibration sample implies — which is the first evidence from outside the method that the average factor applied to every quasar is roughly right.

Why the factor decides a question about the first billion years #

The factor could be treated as a nuisance, a constant that moves every mass up or down together. It stops being one when the masses are used for anything that depends on the ratio of one quantity to another, and the most consequential such use is the growth of the first black holes.

Quasars are observed at redshift seven, when the universe was about 750 million years old, with single-epoch masses of a billion solar masses. A black hole grows fastest when it accretes at the rate at which its own radiation pressure balances the infall, and at that rate its mass $e$-folds in about fifty million years for a typical radiative efficiency. Growing from a stellar-mass seed of a hundred solar masses to a billion takes sixteen $e$-folds — some 800 million years of uninterrupted accretion at the limit, which is more time than the universe had. Either the seeds were far heavier than stellar remnants, or the growth exceeded the limit for long stretches, or the masses are overestimated.

The third possibility is exactly the factor. A mass from a single spectrum is the width squared times a radius from the luminosity times $f$, and every one of those masses at redshift seven uses the average $f$ from nearby, modestly accreting nuclei. If the gas in a luminous quasar’s broad-line region is partly a wind rather than orbits — and the blueshifted, asymmetric profiles of its high-ionisation lines suggest it often is — then the width measures outflow speed as well as gravity, the appropriate $f$ is smaller, and the masses are too large. The same broad lines are used to compute the accretion rate relative to the limit, so an overestimated mass also makes a quasar look further below its limit than it is.

A factor of three in $f$ is a factor of three in the mass and one $e$-fold of growth, about fifty million years. That does not remove the problem of the earliest quasars, but it changes its size by a meaningful fraction, and it is the one term in the arithmetic that nothing observed at redshift seven can check. The relation between a black hole’s mass and its host galaxy’s stellar velocities, which is how the factor is calibrated, cannot be measured there either: the host’s stars are invisible behind the quasar. What can be measured there, with the most sensitive infrared spectrographs, is a velocity-resolved lag in a luminous quasar at intermediate redshift — which would say directly whether the gas that sets the width is orbiting.

What the figures leave out #

The model regions are clouds with no physics beyond kinematics: every cloud responds equally and instantaneously to the flash, whereas real gas responds by an amount that depends on its density, its optical depth and how much of its emission escapes towards the observer, so the near and far sides of a region can respond with different strengths. The inflow and outflow are pure radial free fall and free escape at the same speed; real flows have velocity laws of their own and are mixed with turbulence. The discs are Keplerian with a uniform tilt distribution. And the maps are drawn noiselessly from six thousand clouds, while a real map is recovered by inverting a noisy, gapped light curve — an ill-posed problem whose solutions depend on how they are regularised, which is the reason forward modelling has largely replaced direct inversion.

The three signatures survive all of that. Symmetry means rotation; a blue lag means inflow; a red lag means outflow; and the ratio of the map’s height to its width is the inclination.

Still open: whether one factor serves every quasar #

Velocity-resolved maps and dynamical models exist for a few dozen nearby, modestly luminous active nuclei, and interferometric images for a handful of the brightest. Every black-hole mass beyond redshift one, and almost all of them below it, comes from a single spectrum and the average factor. The maps have shown that the factor varies by a factor of several from object to object, mostly with inclination, and that the average is about right. What they have not shown is that the average measured on low-luminosity nuclei applies to luminous quasars at high redshift, whose broad lines are often blueshifted and asymmetric in ways that suggest winds rather than orbits, and whose accretion rates are far higher than those of any nucleus mapped so far. The same flash-and-echo geometry applied to a star’s line profile resolves a transiting planet’s path across a disc; applied to the most luminous quasars, it would say whether the gas that sets their masses is orbiting or leaving, and nobody has yet had a campaign long enough to find out.