Figure 5.9. A BPT diagram for emission-line galaxies from the Sloan Digital Sky Survey. The points to the left are star-forming galaxies where the line emission arises in HII regions associated with hot stars. The upper-right points are Seyfert galaxies and the lower-right points are LINERs; these are actually two distinct populations, although with some overlap in their emission-line flux ratios. The two curves are delimiters from Kauffmann et al. (2003) (solid line) and Kewley et al. (2006) (dashed line). Courtesy of B. Groves, based on an original from Groves et al. (2006).

of low-ionization relative to high-ionization lines and accounting for the total energy in the emission lines.

In principle, many of the ambiguities can be resolved by using the technique of emissionline reverberation mapping (Blandford & McKee 1982; Peterson 1993), which makes use of the natural continuum variability of AGNs to probe the BLR. For reasons still not understood, the continuum emission from AGNs varies with time, and the broad emission-line fluxes change in response to these variations, but with a delay due to the time taken for light to travel across the BLR. We thus begin this section with a brief phenomenological discussion of AGN continuum-variability characteristics.

5.7.1 The nature of continuum variability The continuum emission from AGNs is variable at every wavelength at which it has been observed, from gamma rays to long-wavelength radio. Blazars exhibit the most extreme variations and, unlike radio-quiet quasars and Seyfert galaxies, significant variations in the radio and long-wavelength infrared. Blazar variability is fundamentally tied to jet physics and relativistic beaming, and, since the jets are too well-collimated to have much influence on the line-emitting regions, we will restrict our discussion of variability to the continuum emission that arises in the accretion disk and its immediate vicinity.

In all cases, the observed continuum-flux variations in AGNs are aperiodic and of unpredictable amplitude. It has been found that a useful way to characterize AGN variability is in terms of the power-density spectrum (PDS), which is the product of the Fourier transform of the light curve and its complex conjugate. The common parameterization of the PDS for AGNs is as a power law in frequency f of the form P(f) <x f-a. Depending on the observed frequency and on the particular AGN, the power-law index is generally in the range 1 < a < 2.5, with X-ray variations tending towards the lower values and UV/optical variations tending towards the larger values. On shorter timescales (days, for Seyfert galaxies), X-ray variations are poorly correlated with UV/optical variations, but on longer timescales (months to years) the X-ray variations appear to be well corrrelated with the UV/optical variations (e.g. Uttley et al. 2003). This is not true in a few cases where the phenomenology, let alone the physics, remains baffling; in the case of NGC 3516, for example, the X-ray and optical variations seem oddly disconnected (Maoz et al. 2002).

On the basis of early spectral-monitoring campaigns that revealed the correlation between continuum and emission-line variations, we can draw some basic conclusions that will simplify mapping the BLR.

(i) The rapid response of the broad emission lines to continuum variations tells us that the BLR is small (because the light travel-time is short) and that the BLR gas is dense (so the recombination time is short).

(ii) The BLR gas is optically thick in the H-ionizing continuum, i.e. at A < 912 A. If the gas were optically thin, the emission lines would not respond strongly to continuum variations.

(iii) The variations of the observable continuum in the UV/optical must trace those of the unobservable H-ionizing continuum. Specifically, any time delay between the variations in the ionizing continuum and the observable continuum must be small.

On the basis of these very basic conclusions, we can make some further assumptions that will aid us in developing a theoretical framework for reverberation mapping. These assumptions are justifiable ex post facto by the following observations.

(i) The continuum is postulated to originate in a single central source that is small compared with the BLR. As noted earlier, the size of the UV/optical continuum-emitting region in an accretion disk around a black hole of mass 108M0 is ~ 10Rg, or about 1.5 x 1014 cm or 0.06lt-day, much smaller than the typical radius of the BLR in such a system, i.e. ~ 1016 cm. We can thus treat the continuum as a point source.*

(ii) The light-travel time across the BLR, tlt = RBLR/c, is the most-important timescale. Other potentially important timescales include the following.

(a) The dynamical time (introduced earlier), Tdyn, which is the timescale over which the structure of the BLR could change due to bulk motions of the line-emitting gas. Since Tdyn/TLT = c/FWHM « 100, the BLR structure is stable over the time it takes to map the BLR by reverberation, which is typically at least a few times tlt .

(b) The recombination time Trec = (neaB)-1, where ne is the particle density, which we previously estimated to be greater than ~ 1010 cm-3, and aB is the case-B recombination coefficient, is the timescale over which photoioniza-tion equilibrium is re-established when the incident ionizing flux, i.e. Q;on(H), changes. For the high densities of the BLR, Trec < 400 s, which means that the BLR gas responds virtually instantaneously for the purposes of reverberation mapping.

(iii) There is a simple, though not necessarily linear, relationship between the observable UV/optical continuum and the ionizing continuum that drives the emissionline variations. There is evidence that the continuum variations at shorter wavelengths precede those at longer wavelengths and the variations are more rapid and have more structure, but these are comparatively small effects.

* In contrast, this simplifying assumption cannot be made in reverberation mapping of the X-ray FeKa line (Reynolds et al. 1999).

5.7.3 The transfer equation The continuum variations are described by the light curve C(t) and the emission-line behavior as a function of time t and line-of-sight (LOS), or Doppler, velocity VLOS is L(t,VLOS). Over the duration of a reverberation experiment, the continuum and emission-line variations are generally small, so it is useful to write both light curves in terms of deviations from mean values (C} and (L(Vlos)}, i.e. C(t) = (C} + AC(t) and L(t, VLOS) = (L(Vlos)} + AL(t, VLOS). Thus, even if the emission-line response or the relationship between the ionizing continuum and the observable continuum is non-linear, we can use a linearized-response model to describe the relationship between the continuum and emission-line variations,

-to where ^(t, VLOS) is the "transfer function" and Equation (5.16) is the "transfer equation." It is apparent by inspection of Equation (5.16) that ^(t, VLOS) is the response of the emission line as a function of time and Doppler velocity to a ¿-function continuum outburst. As we will see below, it is the projection of the BLR responsivityt in the LOS velocity-time-delay plane; for this reason, we often refer to ^(t, VLOS) by the more descriptive name "velocity-delay map."

5.7.4 Velocity-delay maps At this point, we can construct a velocity-delay map from first principles by considering how an extended BLR would respond to a ¿-function continuum outburst, as seen by a distant observer. For illustrative purposes and simplicity, we consider the response of an edge-on (inclination i = 90°) ring of radius R that is in Keplerian rotation around a central black hole of mass Mbh. Photons produced in the continuum outburst would stream outward and, after some time R/c, some of these photons encounter the gas in the orbiting ring, are absorbed, and are quickly reprocessed into emission-line photons. The emission-line photons would be emitted in some pattern (we can assume isotropy for simplicity, but this is not necessarily the case), and some of them would return to the central course after another interval R/c. A hypothetical observer at the center of the system would observe the continuum flare go off, and after a time delay 2R/c would observe the response of the gas in the ring. From this privileged location, the observer sees the entire ring respond simultaneously because the total light-travel time is the same in each direction. At any other location in the ring plane, the total light-travel time from the continuum source to the ring to the observer differs for different parts of the ring; the locus of points in space for which the light-travel time from the source to the observer is constant is obviously an ellipse, with the source and the observer at the respective foci. Given that the observer is essentially at infinity, this locus of constant time delay, or "isodelay surface," is a parabola, as illustrated in Figure 5.10, in which the ring-shaped BLR is intersected by several isodelay surfaces corresponding to different time delays as seen by the distant observer.

The velocity-delay map we wish to construct transforms the BLR from configuration space into the observable space of LOS velocity and time delay. The upper panel of Figure 5.11 shows our ring-shaped BLR intersected by a single, arbitrary isodelay surface that intersects the BLR at polar coordinates (R, 0) centered on the continuum source.

t By "responsivity," we mean the marginal change in line emissivity SL as the continuum changes by SC.

Figure 5.10. The circle represents a ring of gas orbiting around a central continuum source at a distance R. Following a continuum outburst, at any given time the observer far to the left sees the response of clouds along a surface of constant time delay, or isodelay surface. Here we show five isodelay surfaces, each one labeled with the time delay (in units of the shell radius R) we would observe relative to the continuum source. Points along the line of sight to the observer are seen to respond with zero time delay. The farthest point on the shell responds with a time delay 2R/c.

Figure 5.10. The circle represents a ring of gas orbiting around a central continuum source at a distance R. Following a continuum outburst, at any given time the observer far to the left sees the response of clouds along a surface of constant time delay, or isodelay surface. Here we show five isodelay surfaces, each one labeled with the time delay (in units of the shell radius R) we would observe relative to the continuum source. Points along the line of sight to the observer are seen to respond with zero time delay. The farthest point on the shell responds with a time delay 2R/c.

Relative to continuum photons traveling from the central source to the distant observer, the time delay for this surface is shown by the dotted path whose two components have distances R and R cos 0, so the time delay for this isodelay surface is t =(1 + cos 0)R/c, (5.17)

which, as we anticipated earlier, is the equation for a parabola in polar coordinates. To transform the orbital velocity as a function of position into the observable line-of-sight velocity, we see by inspection that

where worb = (GMBH/R)1/2 is the orbital speed. In the lower frame of Figure 5.11, we show this transformation into velocity-delay space. A circular Keplerian orbit in configuration space transforms into an ellipse of semiaxes worb and R/c in velocity-time-delay space.

It is also instructive to "collapse" the velocity-delay map of Figure 5.11 in velocity to obtain the delay as a function of time for the entire emission-line flux (sometimes called the "one-dimensional transfer function" or the "delay map") and in time delay to obtain the emission-line profile. This case can be treated analytically in a straightforward fashion. We can assume that the azimuthal response of the ring is uniform, so ^(0) = e, where e is a constant. To express the response as a function of time delay, we use the transformation d0

and from Equation (5.17) we have dT R — =--sin d0 c

Isodelay surface

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