Astronomy & Celestial Mechanics
the sky, computed
Galaxies

A size measured from a delay

An active nucleus at redshift two is a point source in every telescope ever built, and its central mass is measured anyway. The continuum varies, the broad lines follow days later, and the lag is a light-travel time — which is a length, recovered from two light curves that share no wavelength.

Assumes Galactic nuclei, Accretion and Virial theorem.

19 min read 10 figures The sky as a clockLight is the only messengerMeasuring the unreachable

Every mass in this collection is measured from something orbiting. A planet’s mass comes from a moon’s period, a star’s from a companion’s, a galaxy’s from its rotation, a cluster’s from the speeds of its members. In each case something is going round something else and the period and the size of the orbit are both known.

For a black hole at the centre of a distant galaxy, one of those two is missing. The gas is moving — its emission lines are thousands of kilometres a second wide — and the size of the region it moves in is unknown, because at a redshift of two the entire nucleus subtends less than a millionth of an arcsecond and no telescope will ever resolve it.

The size is measured anyway, and not with a telescope. It is measured with a clock.

A 20-day delay, recovered at 20 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 51 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 20 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic.
Fig. 1 The measurement. Above: a quasar’s continuum, varying irregularly, and its broad emission line following the same variations later and more smoothly. Below: the cross-correlation of the two, which peaks at the delay. That delay is a length — twenty light-days is three and a half thousand astronomical units — and it has been measured for an object whose angular size is a hundred-thousandth of an arcsecond, with a photometer and a calendar.

The mechanism

An active nucleus has a central engine, an accretion disc that emits the continuum, and a surrounding region of dense gas that the continuum photoionises. That gas re-emits in broad permitted lines — hydrogen Balmer lines, Mg II, C IV — and it does so in response to the continuum reaching it.

When the continuum brightens, the lines brighten. Not immediately: the gas has to be told, and the message travels at the speed of light. If the gas sits at radius RR, the response is delayed by something of order R/cR/c.

So the measurement is: monitor the continuum and a line for months, cross-correlate the two light curves, and read off the lag. Multiply by cc and the answer is a radius.

The irregularity of the source is what makes it work, and it is worth stopping on. A periodic source would give a cross-correlation with many equal peaks and no way to choose between them. Quasar continua vary as a damped random walk — no characteristic period, correlations decaying over months — and a random signal has a unique cross-correlation peak. The whole method rests on active nuclei being erratic.

A 5-day delay, recovered at 6 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 5 days, so it is later and smoother — it varies only 90 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 6 days. That number is a length: 5 light-days is 1.3·10¹⁴ metres, or 866 astronomical units, and it has been measured for an object that subtends 8.7·10⁻⁶ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic.
Fig. 2 A five-day lag recovered from the same length of campaign, and it comes back at six. That one-day error is the cadence: the light curves are sampled every two days, and a cross-correlation cannot locate a peak much better than the spacing between the points that define it. So the fractional precision of the method falls apart at short lags rather than at long ones — a twenty-day lag is measured to five per cent and a five-day lag to twenty. Since the lag scales with the square root of the luminosity, that puts the faint nuclei out of reach for reasons of sampling, at the same time as the bright ones are out of reach for reasons of campaign length.

What the delay is actually a delay of

The gas is not at one radius. It occupies a region, and different parts of it are at different light-travel delays — so the line’s response to a sharp continuum flash is not a delayed copy but a smeared one.

The smearing is described by a transfer function: the distribution of delays weighted by how much line emission comes from each. The line light curve is the continuum convolved with it, which is why the line curve in the first figure is not only later but visibly smoother, varying only about half as much.

The two textbook cases are worth having because they say what a lag can and cannot determine.

A thin spherical shell of radius RR gives a response that is flat between zero and 2R/c2R/c. Light from the near face arrives with no delay at all, light from the far face is delayed by two crossing radii, and every delay in between is equally likely because equal solid angles cover equal ranges of cos⁡θ\cos\theta.

A thin inclined ring of the same radius gives a double-horned response, piled up at both ends, because the projection of a circle onto the line of sight is stationary there.

Two geometries, two shapes, and the same centroid at 20.0 days. The delay distribution — the transfer function — for two arrangements of gas at the same radius of 20 light-days. A thin spherical shell gives the flat response: light from the near face arrives with no delay, light from the far face is delayed by twice the light-crossing radius, and every delay between is equally likely because equal solid angles cover equal ranges of cos θ. A thin ring inclined at 70° gives the double-horned shape: the projection of a circle onto the line of sight is stationary at the two ends, so the response piles up there. The shapes are completely different and their centroids are the same, 20.0 against 20.0 days. That is the systematic error of the whole method stated as a picture. A cross-correlation measures the centroid, so it measures the radius correctly for either geometry — and the mass needs the radius and the relation between the measured line width and the true orbital speed, which depends entirely on which of these two the gas actually is. That relation is the virial factor, it is a factor of about 3 wide, and recovering it needs the whole transfer function rather than its centroid — which needs light curves ten times better sampled than the ones that give the lag.
Fig. 3 The ring at seventy degrees rather than thirty, which is nearly edge-on. The two horns move apart to the full range of delays the shell allows, because a ring seen edge-on projects onto the line of sight over its whole diameter — and the centroid does not move at all. That is the property the method depends on and it is worth seeing twice: the shape of the response carries the inclination, the centroid carries only the radius, and a cross-correlation returns the second. The inclination that this figure cannot report is precisely the quantity hidden inside the virial factor ff, which is why ff is calibrated on a population instead of measured on an object.
Two geometries, two shapes, and the same centroid at 20.0 days. The delay distribution — the transfer function — for two arrangements of gas at the same radius of 20 light-days. A thin spherical shell gives the flat response: light from the near face arrives with no delay, light from the far face is delayed by twice the light-crossing radius, and every delay between is equally likely because equal solid angles cover equal ranges of cos θ. A thin ring inclined at 30° gives the double-horned shape: the projection of a circle onto the line of sight is stationary at the two ends, so the response piles up there. The shapes are completely different and their centroids are the same, 20.0 against 20.0 days. That is the systematic error of the whole method stated as a picture. A cross-correlation measures the centroid, so it measures the radius correctly for either geometry — and the mass needs the radius and the relation between the measured line width and the true orbital speed, which depends entirely on which of these two the gas actually is. That relation is the virial factor, it is a factor of about 3 wide, and recovering it needs the whole transfer function rather than its centroid — which needs light curves ten times better sampled than the ones that give the lag.
Fig. 4 Two geometries, two completely different shapes, and the same centroid. A cross-correlation measures the centroid, so it measures the radius correctly for either — which is exactly why the geometry survives the measurement as an unknown. Recovering the shape rather than the centroid requires light curves an order of magnitude better sampled, and it is what a handful of campaigns have set out to do.

From a radius to a mass

Put the radius together with the width of the line — the gas’s velocity — and the virial theorem gives a mass:

M=f R Δv2G.M = f\,\frac{R\,\Delta v^2}{G}.

That is the same reasoning that turns a velocity dispersion into a mass for a cluster of galaxies, and it carries the same difficulty in a sharper form: the factor ff.

ff absorbs everything about the geometry and the kinematics of the region — whether the gas is a flattened rotating disc or an isotropic swarm, how it is inclined, and how the observed line width relates to the actual orbital speed. For a randomly oriented isotropic distribution it is one; for a disc seen at thirty degrees it is several; and it cannot be measured for any individual object, because measuring it needs the transfer function that the centroid throws away.

What is done instead is to calibrate ff as a population average. The handful of nearby galaxies whose black holes can be weighed from resolved stellar dynamics also lie on a relation between black hole mass and bulge velocity dispersion; requiring the reverberation masses to lie on the same relation fixes the mean ff, at about 4.3 for line widths measured as full width at half maximum.

Every quasar mass in every catalogue is therefore a measurement multiplied by a number that was fitted to a different sample. The scatter of ff from object to object is estimated at a factor of two to three, and it is the dominant systematic in the entire field.

What varies, and why that is a measurement in itself

The variability is treated above as a convenience. It is also a constraint, and it was the first thing anyone learnt from quasars.

A source cannot vary coherently on a timescale shorter than its own light-crossing time, because different parts of it would be out of causal contact and their variations would average away. So an object that doubles its brightness in a day is smaller than a light-day across.

Applied to quasars in the 1960s that argument produced a result nobody wanted: an object outshining an entire galaxy, confined to a region smaller than the solar system. There is no known way to make that much light out of that small a volume except by accretion onto a compact object, and the whole subject follows from the arithmetic of a light curve.

The same argument bounds the disc: the continuum’s variability timescale of days puts the emitting region at light-days, and the broad-line region’s lag puts it at light-weeks — a factor of ten separation that is the reason the delay is measurable at all.

The relation that turns dozens into a hundred thousand

Measuring a lag takes years of monitoring, and it has been done for perhaps a hundred nuclei. There are hundreds of thousands of quasars with spectra.

The bridge is an empirical relation between the measured radius and the continuum luminosity,

R∝L0.53,R \propto L^{0.53},

with a scatter of about 0.13 dex. The exponent is close to a half for a reason that is nearly geometrical: the broad-line gas sits where the ionising flux takes a particular value, and a fixed flux at radius RR from a source of luminosity LL requires R∝LR\propto\sqrt{L}. The measured 0.53 rather than 0.50 is a real and small departure.

With that relation, a single spectrum becomes a mass. The continuum luminosity gives the radius; the line width gives the velocity; the virial expression gives the mass. No monitoring, no cross-correlation, one night’s observation.

Slope 0.52, scatter 0.24 dex, and a mass from one spectrum. Broad-line-region radius against continuum luminosity, for 42 objects drawn from the published relation R ∝ L^0.533 with its measured scatter of 0.3 dex, and refitted here: the line through them has slope 0.523 and the points scatter by 0.243 dex about it. The exponent is close to a half for a reason that is almost geometrical — the gas sits where the ionising flux takes a particular value, and a fixed flux at distance R from a source of luminosity L means R goes as √L. This relation is what turned a decades-long monitoring programme into a survey tool. Measuring a lag needs years of light curves and has been done for a few dozen nuclei; the relation converts a single spectrum into a radius through the luminosity and into a velocity through the line width, and a hundred thousand black holes have been weighed that way. At the 4000 km s⁻¹ typical of a broad line, a nucleus at L₅₁₀₀ = 10⁴⁴ erg s⁻¹ comes out at 4.5·10⁸ solar masses. What that number carries with it is the virial factor: the geometry of the region, unmeasured for any individual object, a factor of about 3 wide, and calibrated as a population average against the handful of galaxies whose black holes can also be weighed from resolved stellar motions. Every quasar mass in every catalogue is a measurement times a number that was fitted to a different sample.
Fig. 5 The same relation with the scatter more than doubled, to what it would be if the calibrating sample were drawn from a wider range of accretion states than it is. The slope survives — a power law fitted through four decades of luminosity is robust against scatter — and the usefulness does not: a factor of two in radius at fixed luminosity is a factor of two in every mass derived from it, on top of the virial factor’s own factor of two to three. The tightness of the real relation is therefore load-bearing for the whole single-epoch industry, and it is measured on the few dozen objects with lags rather than assumed.
Slope 0.53, scatter 0.11 dex, and a mass from one spectrum. Broad-line-region radius against continuum luminosity, for 42 objects drawn from the published relation R ∝ L^0.533 with its measured scatter of 0.13 dex, and refitted here: the line through them has slope 0.529 and the points scatter by 0.105 dex about it. The exponent is close to a half for a reason that is almost geometrical — the gas sits where the ionising flux takes a particular value, and a fixed flux at distance R from a source of luminosity L means R goes as √L. This relation is what turned a decades-long monitoring programme into a survey tool. Measuring a lag needs years of light curves and has been done for a few dozen nuclei; the relation converts a single spectrum into a radius through the luminosity and into a velocity through the line width, and a hundred thousand black holes have been weighed that way. At the 4000 km s⁻¹ typical of a broad line, a nucleus at L₅₁₀₀ = 10⁴⁴ erg s⁻¹ comes out at 4.5·10⁸ solar masses. What that number carries with it is the virial factor: the geometry of the region, unmeasured for any individual object, a factor of about 3 wide, and calibrated as a population average against the handful of galaxies whose black holes can also be weighed from resolved stellar motions. Every quasar mass in every catalogue is a measurement times a number that was fitted to a different sample.
Fig. 6 The relation, with its scatter, refitted from the points drawn. That is how a hundred thousand black holes have been weighed — and the transformation is worth naming for what it is: a relation measured on dozens, applied to a hundred thousand. Every single-epoch mass carries the 0.13 dex scatter of this relation, the factor of two to three in the virial factor, and whatever systematic the calibrating sample has, none of which appears in the formal error bar a fitting routine returns.

What was actually observed, and what was assumed

It is worth separating the layers, because they are usually quoted as one number.

Observed: two light curves, in two different bands, over months. Their cross-correlation. A line width. A continuum flux.

Inferred with no free parameter: the lag, and therefore R=cτR = c\tau. This is as close to a direct measurement as anything in extragalactic astronomy gets — a time converted into a length by a constant.

Assumed: that the gas is gravitationally bound and virialised, that the line width reflects its orbital speed, and that the geometry factor ff has the population’s mean value.

Calibrated elsewhere: ff itself, from the dynamical relation above.

The first two are strong and the last two are not, and the honest statement of a quasar mass is that its radius is measured to ten per cent and its mass to a factor of three.

There is one internal check available, and it works. Several nuclei have lags measured in more than one line — Hβ, He II, C IV — and the lines with the higher ionisation potential come from smaller radii and show shorter lags. Plotting each line’s lag against its width gives a slope of −1/2-1/2, which is exactly what a Keplerian velocity field requires. The gas is orbiting, and that is a measurement rather than an assumption.

Why the masses matter

Three things depend on them, and each is a reason the factor of three is worth arguing about.

The growth history of black holes. The mass function of quasars as a function of redshift, integrated over time, has to match the mass locked up in the local relation — an argument that ties accretion to the population and that fails if the masses are systematically wrong.

The Eddington ratio. A luminosity divided by the Eddington luminosity of the measured mass says how fast an object is being fed relative to the maximum. Quasars come out near a tenth to a whole Eddington unit, which is the observation that black holes grow in short bright episodes rather than slowly.

And the growth timescale. At the Eddington limit a black hole’s e-folding time is about forty-five million years divided by the radiative efficiency, so growing from a stellar-mass seed to a billion solar masses takes of order a billion years of continuous accretion. Quasars are observed at redshift seven, when the universe was less than eight hundred million years old — which is the sharpest problem the masses create, and which is why an error of a factor of three in them matters.

The lines are not interchangeable

Single-epoch masses are quoted from whichever line happens to be in the observed spectral window, and at high redshift that is not a choice.

Hβ is the calibrated line: nearly all reverberation lags are measured in it, and the radius–luminosity relation is its relation. It leaves the optical window at about redshift 0.9.

Mg II takes over to about redshift 2.5, and C IV beyond that. Neither has anything like the same calibration, and C IV is known to be problematic: its profile shows a blueshifted component that behaves like an outflow rather than like bound gas, so its width is not simply an orbital speed. Masses from C IV and from Hβ for the same object can differ by a factor of several, and which is right is not settled.

There is a way to break the deadlock and it is being built rather than argued about. Infrared spectrographs put Hβ back in the observable window out to redshift four, so the same line can be measured in the same object as C IV and the two masses compared directly rather than through a population. The samples where that has been done are small — tens of objects — and they show a scatter of about 0.4 dex between the two lines with no single offset that would remove it, which is worse than a calibration error and better than a random one: the disagreement correlates with the blueshift of the C IV profile, which is exactly what an outflow contribution should do. Correcting for it is now standard and is itself a fitted relation with its own scatter, so the honest statement is that a high-redshift mass carries an uncertainty of a factor of three that no amount of signal-to-noise will reduce.

The consequence is that the highest-redshift masses — the ones the growth-time problem turns on — are the least trustworthy ones in the catalogue. That is the usual shape of things in this subject and it is worth saying plainly rather than burying in an error bar.

Slope 0.53, scatter 0.11 dex, and a mass from one spectrum. Broad-line-region radius against continuum luminosity, for 42 objects drawn from the published relation R ∝ L^0.533 with its measured scatter of 0.13 dex, and refitted here: the line through them has slope 0.529 and the points scatter by 0.105 dex about it. The exponent is close to a half for a reason that is almost geometrical — the gas sits where the ionising flux takes a particular value, and a fixed flux at distance R from a source of luminosity L means R goes as √L. This relation is what turned a decades-long monitoring programme into a survey tool. Measuring a lag needs years of light curves and has been done for a few dozen nuclei; the relation converts a single spectrum into a radius through the luminosity and into a velocity through the line width, and a hundred thousand black holes have been weighed that way. At the 8000 km s⁻¹ typical of a broad line, a nucleus at L₅₁₀₀ = 10⁴⁴ erg s⁻¹ comes out at 1.8·10⁹ solar masses. What that number carries with it is the virial factor: the geometry of the region, unmeasured for any individual object, a factor of about 3 wide, and calibrated as a population average against the handful of galaxies whose black holes can also be weighed from resolved stellar motions. Every quasar mass in every catalogue is a measurement times a number that was fitted to a different sample.
Fig. 7 The same relation read with an eight-thousand-kilometre-a-second line width instead of four thousand, which is roughly the difference between an Hβ profile and a C IV one in the same object. The mass goes as the square of the width, so the whole mass scale moves by a factor of four — from a relation whose radius axis has not changed at all. That is the arithmetic behind the disagreements quoted above, and it shows where they come from: not from the radius, which is the measured half, but from the assumption that a line width is an orbital speed. For C IV, with its blueshifted outflow component, that assumption is the one in doubt.

The lag measured across the line

There is a refinement of the technique that recovers part of what the centroid throws away, and it costs signal-to-noise rather than a new instrument.

Instead of measuring one light curve for the whole line, measure several — one for each velocity bin across the profile. The blue wing, the core and the red wing each have their own light curve, and each has its own lag.

The pattern of lags against velocity is a kinematic diagnostic with no ambiguity in it. If the gas is rotating, the two wings come from opposite sides of the same orbit and their lags are equal, while the core comes from material moving transversely and is at a similar radius — so the lag is symmetric about line centre. If the gas is falling inwards, the redshifted material is on the near side moving away from the observer and is therefore closer, so the red wing responds first and the lags are asymmetric one way. If it is flowing outwards, the blueshifted material is on the near side and the asymmetry reverses.

Three geometries, three signatures, and they are distinguishable from data that a well-executed monitoring campaign already contains.

What the campaigns have found is a mixture. Some objects show clean rotational symmetry, which is the case the virial mass estimate assumes and is reassuring where it holds. Some show a clear inflow signature. A few show outflow, which is the awkward case, because material leaving the system is not gravitationally bound and its width is not an orbital speed.

The measurement is demanding. Splitting one line into five velocity bins divides the photons by five and the lag precision by more than that, so it requires campaigns several times longer than a plain lag needs — which is why it has been done well for perhaps a dozen objects rather than for the hundred with measured lags.

There is a second and cheaper diagnostic hiding in the same data, and it costs nothing at all. The width of the line changes as the continuum varies: when the source brightens, the ionised region moves outwards, and gas at a larger radius moves more slowly — so a brightening should be followed by a narrowing, on the same lag.

That anticorrelation between luminosity and line width is observed, and its slope is close to the minus one-half a Keplerian velocity field requires. It is the same check the multiple-line comparison makes, performed within a single line and within a single object, and it is the strongest available evidence that the width being fed into the virial expression is an orbital speed rather than something else.

The information that would fix the field’s dominant systematic is in the data and is expensive to extract, and the number of objects for which it has been extracted is small enough that the population correction still has to be applied blind.

What the method cannot do

It needs the object to vary, and to vary enough. The most luminous quasars vary least, in fractional terms, and have the longest lags — so the objects whose masses matter most for the growth argument are the ones the method works worst on.

A 20-day delay, recovered at 20 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 20-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 39 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 20 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic.
Fig. 8 The favourable case, for contrast: the same lag from a continuum whose correlations decay in twenty days rather than ninety. The light curve has far more independent features in the same two hundred and twenty days, so the cross-correlation has far more to work with and its peak is sharper. That is the quantity a campaign is really buying — not the length of the record but the number of distinguishable events in it — and it is why a faint, rapidly varying Seyfert yields a cleaner lag in one season than a luminous quasar does in five.

It needs the monitoring to outlast the lag. A lag of a year needs several years of well-sampled data. Campaigns of that length exist for a few dozen objects.

That requirement is harsher than it sounds, because the lag is not the only thing that grows with luminosity. The response is spread as well as delayed, and the spread grows in the same proportion.

Two geometries, two shapes, and the same centroid at 60.0 days. The delay distribution — the transfer function — for two arrangements of gas at the same radius of 60 light-days. A thin spherical shell gives the flat response: light from the near face arrives with no delay, light from the far face is delayed by twice the light-crossing radius, and every delay between is equally likely because equal solid angles cover equal ranges of cos θ. A thin ring inclined at 30° gives the double-horned shape: the projection of a circle onto the line of sight is stationary at the two ends, so the response piles up there. The shapes are completely different and their centroids are the same, 60.0 against 60.0 days. That is the systematic error of the whole method stated as a picture. A cross-correlation measures the centroid, so it measures the radius correctly for either geometry — and the mass needs the radius and the relation between the measured line width and the true orbital speed, which depends entirely on which of these two the gas actually is. That relation is the virial factor, it is a factor of about 3 wide, and recovering it needs the whole transfer function rather than its centroid — which needs light curves ten times better sampled than the ones that give the lag.
Fig. 9 The same two geometries at sixty light-days rather than twenty, which is roughly where the broad-line region of a luminous quasar sits. Both transfer functions keep their shapes and stretch by a factor of three: the shell’s flat response now runs from zero to four months, and the disc’s peak is correspondingly broader. A campaign has to resolve a feature this wide, not merely a delay this long.

So the cost of measuring a luminous object is cubic rather than linear in the awkwardness: the lag is three times longer, the response is three times broader, and the fractional variability that has to be detected across it is a third as large. Three of the four ways the problem gets harder come from the same increase in radius, and the fourth — the redshift stretch below — is independent of it.

The lag is measured in the observed frame. At redshift two everything is stretched by a factor of three, so both the lag and the monitoring baseline are three times longer, and the effective survey requirement grows with the population being surveyed.

A 20-day delay, recovered at 14 days from two curves that share no wavelength. Above: a quasar's continuum, drawn as a damped random walk with a 90-day damping time, and the broad emission line responding to it. The line curve is the continuum convolved with a top-hat response of half-width 20 days, so it is later and smoother — it varies only 89 per cent as much, because at any instant it is an average of the continuum over a range of light-travel times. Below: the cross-correlation of the two, which peaks at 14 days. That number is a length: 20 light-days is 5.2·10¹⁴ metres, or 3463 astronomical units, and it has been measured for an object that subtends 3.5·10⁻⁵ arcseconds at a hundred megaparsecs — some thirty times finer than the best optical interferometry has ever resolved anything, and reached here with a photometer and a clock. The irregularity of the continuum is what makes this work. A periodic source would give a cross-correlation with many equal peaks and no way to choose; a random one gives a single peak, and the whole method rests on active nuclei being erratic.
Fig. 10 The same twenty-day lag from the same pair of curves, sampled every fourteen days instead of every two. The cross-correlation still finds a peak, and it finds it in the wrong place, because the sampling grid is comparable to the quantity being measured and the peak can only land where a grid point is. A cadence is not a detail of scheduling; it is the resolution of the instrument.

The rule of thumb that comes out of that is that the cadence must be several times finer than the lag and the baseline several times longer, which brackets the measurable lags into little more than a decade of range for any one campaign. It is why the field has proceeded by building separate campaigns for separate luminosity classes rather than one survey covering all of them, and why the objects at the two ends of the radius–luminosity relation have never been measured by the same programme with the same systematics.

And the geometry is assumed rather than measured in almost every case. The virial factor that converts a width and a radius into a mass is a single number standing in for the inclination, the anisotropy of the line-emitting gas and whatever bulk motion it carries, and it is calibrated by requiring that the whole reverberation sample reproduce the local relation between black-hole mass and bulge velocity dispersion. That is a calibration against a different measurement of the same quantity, which makes the masses internally consistent and leaves them tied to whatever the local relation’s own zero point is worth.

And the continuum being monitored is not the continuum doing the ionising. The optical continuum is a proxy for the extreme ultraviolet, which is unobservable at any redshift because of intervening absorption, and the two are assumed to vary together with no delay. The disc itself has a light-crossing time, so that assumption is known to be imperfect at the level of a day or two — small against a twenty-day lag, and not small for the shortest ones.

Where this ladder goes next

This rung has taken an unresolvable object and got a length out of it, by using the one instrument that does not care about angular size: a clock, and a source that varies irregularly enough to be recognised when it arrives late.

The rung above is the transfer function rather than its centroid. Recovering the delay distribution rather than its mean would give the geometry of the region, and with it the virial factor for that object — which is the only route to removing the field’s dominant systematic, and which needs monitoring an order of magnitude better than what a lag requires.

Beside it lies the same technique at a completely different scale: the same interferometric instruments that resolve a nearby galaxy’s centre have now resolved a broad-line region directly, and the radius they measure agrees with the reverberation radius — which is the first external check the method has ever had.

And below it, the habit: when an angle cannot be measured, measure a time. A size is an angle times a distance, and it is also a delay times a speed. The second route needs no resolution, no distance, and no telescope larger than the one already in use.

Where this goes next

S2's orbit, and the mass it implies. The orbit of S2 about the centre of the Milky Way, drawn from its measured elements: a semi-major axis of 0.1251 arcseconds, which at 8.28 kpc is 1036 AU, an eccentricity of 0.8843, and a period of 16.05 years watched round twice. Kepler's third law in solar units — a³/P² — gives 4.31 million solar masses, by exactly the calculation that weighs a planetary system. At periapsis the star is 120 AU from the focus, 1407 Schwarzschild radii, and the mean density inside that radius is 5.3e+15 M☉ per cubic parsec — a million times the densest star cluster known. That density, not the mass, is the argument: there is no configuration of stars that would fit. The mass at the centre that is not stars One star at the centre of the Milky Way has been watched round a complete orbit. Its semi-major axis and its period give four million solar masses by Kepler's third law — and the volume that mass occupies is small enough to rule out every alternative to a black hole. The temperature of a disc around a stellar black hole. Effective temperature against radius, in units of the inner edge, for a stellar black hole of 10 solar masses accreting 10⁻⁸ solar masses a year. Two features are structural. The profile turns over rather than rising all the way in: the factor (1 − √(r_in/r)) is the statement that no torque acts across the inner edge, so nothing is dissipated there and the peak sits at 49/36 of it, measured here at 1.361. And outside a few inner radii the run is exactly r^−3/4, drawn as the dashed line, which is what makes a disc's spectrum broad: every decade of radius contributes at a temperature a factor of 5.6 lower. The peak is 3.46·10⁶ K here, so the disc radiates in X-rays, and integrating the whole profile gives 4.72·10³⁰ W — which is GMṀ/2r_in to a per cent, half the binding energy released and no more, because the other half is still going round. The disc that has to throw angular momentum away Matter cannot simply fall onto a compact object. At the energy it arrives with, it has far too much angular momentum, and the only way in is for some of it to be carried outwards — which is what a disc is for. An average that arrives, and a snapshot that never does. 2T/|U| against time for Burrau's problem — three masses released from rest, and never repeating, in two forms: the instantaneous ratio, and the ratio of the running time-averages. The instantaneous one runs between 0.03 and 1.97 — the system is a long way from equilibrium at almost every moment, because the bodies are alternately falling together and flying apart. The averaged one settles: after 50 time units it is 1.0142, against the exact 1 the virial theorem requires of any bound system. This system is not periodic and is not even permanently bound — Burrau's problem ejects its lightest body — so the average is drifting rather than converged, and that is the theorem's own condition made visible: it is a statement about bound systems and says nothing about anything that is leaving. What the figure cannot show is the error: a cluster observed once gives 2T/|U| with a scatter of this size, and what makes its mass believable is not the measurement but the relaxation. An average that weighs what cannot be watched A cluster's mass can be had from its speeds alone — no orbit followed, no period observed, no distance to any single star. The theorem that allows it is an average over time, which is exactly what a photograph is not.

What this makes readable

Essays that name this one as a prerequisite.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Accretion discBroad line regionContinuum variabilityCross-correlationDamped random walkEddington ratioLight-travel timeRadius luminosity relationReverberation mappingSingle-epoch massTransfer functionVirial factor