Astronomy & Celestial Mechanics
the sky, computed
Series

Quasars — the series

4 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    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.

    part 1 · galaxies
  2. A single-epoch mass, good to a factor of 2.6. The error budget of a black-hole mass obtained from one spectrum, as probability densities in the logarithm of the ratio of the estimate to the truth. Four independent widths go in. The radius comes from the luminosity through a relation with 0.19 dex of scatter about it. The line width has to be measured on a profile that is blended with narrow lines and with iron, and it enters the mass squared, so 0.12 dex. The luminosity was not measured at the epoch the relation was calibrated for and the object has varied since, which is 0.053 dex. And the term this figure exists for is orientation: the virial factor is one number fitted to a sample, every individual object has its own according to how its broad-line region is inclined, and the spread of that is 0.35 dex — the largest of the four by some margin. In quadrature the total is 0.42 dex — a factor of 2.6 — and the drawn total's half-width is checked against that combination rather than assumed. The bracket at the foot is a different kind of error and is not in the quadrature: the absolute calibration of the virial factor, uncertain by about a factor of 3, which moves every quasar mass ever published by the same 0.24 dex and widens nothing.

    The factor that multiplies every quasar mass

    A reverberation lag and a line width give a length times a velocity squared, which is a mass multiplied by an unknown number of order one. That number is fixed by making a few dozen nearby active galaxies lie on a correlation measured in quiescent ones — so every black-hole mass at redshift two rests on a fit performed at redshift zero.

    part 2 · galaxies
  3. Three arrangements of gas, three velocity–delay maps. The response of a broad emission line to a flash of the continuum, as the density of emitting gas in delay (up, days) and line-of-sight velocity (across, km/s; positive is receding), for gas between 10 and 30 light-days from a hundred-million-solar-mass black hole in three kinematic states, 6,000 clouds each. Left: a rotating disc inclined at 30° with orbits tilted up to 15° from its plane — an ellipse symmetric in velocity, with the fastest gas, nearest the hole, responding soonest. Middle: gas falling in at the free-fall speed — the near side is moving away from the observer and responds first, so the red side leads and the blue side lags, by 27 days on average beyond 1,500 km/s. Right: the same gas flowing out — the mirror image, the blue side leading and the red lagging by 27 days. The three maps are as different as maps can be; their projections onto either axis alone are not.

    A delay that tells which way the gas is moving

    A broad emission line echoes the continuum after a delay, and the delay is a radius. Measure it separately in the line's blue wing and its red wing and it becomes a map — symmetric for gas in orbit, lagging on the blue side for gas falling in, on the red side for gas blown out — and the one route to the geometry factor for an individual black hole.

    part 3 · galaxies
  4. A size–luminosity relation with a population below it. The lag of the broad Hβ line behind the continuum, in light-days, against the continuum luminosity at 5100 ångströms, for a synthetic population of active nuclei with accretion rates spread from slow to well above the Eddington rate. The line is the radius–luminosity relation calibrated on slowly accreting nuclei, log R = 1.527 + 0.533 log L₄₄. Nuclei accreting at dimensionless rates below 3 scatter about it by about 0.13 dex; those above (filled) lie systematically below it, by 0.41 dex on average in this model, in which the shortening grows as the accretion rate to the power −0.4. A mass computed from the line for one of them takes a radius that is too large.

    A lag too short for the light

    Most black-hole masses beyond the nearby universe come from one spectrum and one relation: the size of the broad-line region scales with the quasar's luminosity, calibrated on nuclei that accrete slowly. Nuclei accreting near or above the Eddington rate have lags shorter than the relation predicts, so their masses are overestimated — and since the accretion rate is computed from the mass, it is underestimated, which hides the very shortening that caused the error.

    part 4 · galaxies

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