4.2. The Lyman Edge

To explain the dearth of Lyman edges in AGNs, the new accretion disk models have considered a number of theoretical approaches and improvements. The new models now include complex stellar atmosphere effects, and modern state-of-the-art stellar atmosphere codes. Preliminary steps have been taken toward investigating two-phase accretion disk models and including relativistic transfer functions in the codes.

The original calculations of accretion disks by Kolykhalov & Sunyaev (1984) relied on existing stellar atmosphere libraries and thus predicted large Lyman edge features. These libraries completely neglected non-LTE effects. These can be very important in the Lyman limit region because the large hydrogen photoionization opacity generally means that the Lyman continuum originates high in the atmosphere, where densities are low and radiative transition rates can dominate collisional transition rates. Even more important, the atmospheres that exist in those libraries generally have high densities compared to what is possible in high-luminosity accretion disks. This limited the range of parameter space that Kolykhalov & Sunyaev could explore to low- parameter ( < 10^-2), high photospheric density models. Because the neutral fraction of hydrogen becomes larger at high densities, this resulted in very large Lyman edge absorption opacity, thereby producing very large absorption edges in these disk models.

Low densities in the disk photosphere will reduce the ratio of Lyman continuum absorption opacity to scattering opacity and thereby decrease the flux difference at an absorption edge or even drive it into emission (Czerny & Pojmanski 1990). Ab initio radiative transfer calculations confirm this behavior (Coleman & Shields 1993; Shields & Coleman 1994; Hubeny & Hubeny 1997; Sincell & Krolik 1998), which shows that as the maximum effective temperature and/or the ratio of luminosity to Eddington luminosity increase, Lyman absorption edges become weaker, then disappear, become moderate emission edges, and finally become weak emission edges as the hydrogen becomes fully ionized and recombination rates are low. Non-LTE effects produce similar results (Sun & Malkan 1989; Shields & Coleman 1994; Störzer, Hauschildt, & Allard 1994; Hubeny & Hubeny 1997; Collin & Dumont 1997; Czerny & Dumont 1998). Ironically, non-LTE effects can also enhance the He II Lyman continuum (Hubeny & Hubeny 1997), which would have important implications for photoionization of the broad-line region clouds in AGNs. To get rid of the Lyman edge completely in a given atmosphere at a given radius in the disk using these effects would require fine tuning, but it is clear that the overall edge produced from an integration of diverse atmospheres at different disk radii, some with absorption edges and some with emission edges, might generically produce a small net effect (see Fig. 18). Note, however, that the requirement that the disk be hot enough to drive the edge into emission in the innermost radii requires high accretion rates compared to Eddington, which drives the optical/UV SED closer to the classic F ^1/3 limit (Sincell & Krolik 1998). The relatively cool disks discussed in the previous subsection to explain the SED would probably have observable Lyman absorption edges (see Fig. 22). There is as yet no accretion disk model that solves both problems (the Lyman edge and the optical/UV continuum slope) simultaneously.

Figure 18. Predicted SED of the accretion disk model of Fig. 20, after folding through the relativistic transfer function for three different observer inclination angles i (µ cos i). Dashed curves show the non-LTE hydrogen/helium spectra, while solid curves show the spectra as modified by LTE metal line emission and absorption but neglecting metals in the atmosphere structure calculation. The Lyman edge is virtually undetectable except for the near face-on model (µ = 0.98), which for clarity is blown up on a linear scale in the inset. Here the Lyman limit is shown by a vertical dashed line. In this model, a change in continuum slope in the Lyman limit region is still evident for a substantial parameter space (courtesy E. Agol).

To explain the X-ray spectra observed in AGNs, a number of models invoke thermal Comptonization of optical/UV photons by an energetically powerful, magnetized corona above the disk (see, e.g., Haardt & Maraschi 1991). Strong support for such models has been provided by the observation of Compton reflection features in hard X-rays (Nandra & Pounds 1994), which require that cold matter (presumably the disk) cover approximately half of the sky as seen from the X-ray source. Until recently, most accretion disk model calculations explored the parameter space for a "bare" accretion disk and neglected the effects of external heating of the disk atmosphere. Sincell & Krolik (1997) have taken the first steps in calculating the spectrum from an X-ray-illuminated accretion disk. In their simple two-phase model, all the accretion power is assumed to be dissipated in a corona above the disk. X-rays from the corona irradiate the accretion disk and the UV continuum is produced by reradiation of absorbed energy. There is no internal disk heating. Their calculations showed large Lyman edge features in emission, which are not present in the observations. Figure 19 shows the emission edges produced for three observer viewing angles by the Sincell & Krolik (1997) models. The sharpest edges occur for face-on disks, but substantial edges are still produced for edge-on disks. Physically, these large emission edges arise because taking accretion power out of the disk and putting it in the corona robs the disk of pressure support, producing larger photospheric densities and therefore larger hydrogen recombination rates. Although the simple two-phase model disagrees with the observations in the Lyman limit region, the parameter space for this model has not been thoroughly examined, and many sophisticated radiative transfer effects, particularly non-LTE effects on the level populations of the irradiated hydrogen gas, have not been included.

Figure 19. Predicted spectra for X-ray irradiated accretion disk with accretion luminosity equal to 3% of Eddington, around a nonrotating black hole of mass 2.7 × 108 M. Each of the three sets of points corresponds to a different observer inclination angle i, where µ cos i. The top set of points is more face-on, while the bottom set is more edge-on. The large Lyman edges predicted by these models are not observed (courtesy M. Sincell).

The accretion disk is embedded in the deep potential well of the supermassive black hole. It is then natural that many authors have considered the effects of relativistic Doppler shifts and gravitational redshifts to explain the dearth of Lyman edge features in AGNs (Sun & Malkan 1989; Laor & Netzer 1989; Lee, Kriss, & Davidsen 1992; Laor 1992; Shields & Coleman 1994; Wehrse 1997). The amount of Lyman edge smearing is inclination angle-dependent, with the largest effect seen when the object is viewed edge-on. In the unified schemes for AGNs, the objects for which we have observational data (Seyfert 1 galaxies and QSOs) are most likely to be viewed face-on, where the effect of smearing is predicted to be the smallest. The smearing also depends on the black hole spin. Kerr holes are more effective than Schwarzschild holes because the innermost stable orbit can be much closer to the horizon. However, to test effectively the accretion disk models that include all these effects is nearly impossible. Ideally, the observations need to be of an AGN with a "reasonably clean" line of sight and over a large wavelength region, and, as discussed in Section 3.2, such observations are rare!

One of the many ways to account for the few observed Lyman edge features in AGNs is to invoke a Compton scattering atmosphere (Czerny & Zbyszewska 1991; Lee et al. 1992). In these models, the spectral shape in the Lyman limit region is a function of the electron temperature of the Comptonizing corona, the optical depth to Compton scattering, and the inclination angle of the accretion disk. The Lyman edge is smeared out because Lyman limit photons are scattered across the Lyman edge, which makes the Lyman limit feature more difficult to detect. The models predict a definite break in the shape of the continuum at the Lyman limit (Lee et al. 1992). This is qualitatively similar to observations of the few candidate "partial" Lyman edge objects. As discussed in the next section, Compton scattering may impart polarization to the radiation higher than observed, but if the corona is magnetized, Faraday rotation can alleviate this problem.

Metal line opacity is likely to be important in the Lyman limit region, and no models exist as yet that properly take this into account. Hubeny & Hubeny (1998) have included this in a preliminary way by including the line opacity and emissivity in radiative transfer calculations through atmospheres whose structure was calculated by neglecting these lines. While not fully self-consistent, the results suggest rather strongly that the Lyman edge will be swamped by metal line features (see Fig. 20). Figure 18 shows the corresponding total spectra after integrating over all disk radii and folding through the relativistic transfer function (Agol, Hubeny, & Blaes 1998b). Even in the pure hydrogen/helium models, the Lyman edge is greatly reduced. A change in slope is still evident in the µ = 0.5 case, although it occurs shortward of 912 Å. A bump is present redward of 912 Å in the near face-on case. This is produced because the Lyman edge is in emission in the (more redshifted) inner parts of the disk and in absorption in the (less redshifted) outer parts of the disk (see Fig. 20). When metal lines are included, the change in slope near this bump is reduced but is still detectable conceivably. In addition, the dramatic reduction in flux shortward of 850 Å caused by the line-blanketing in this model could be detected. However, this is the face-on case in which relativistic smearing is most reduced. Self-consistent calculations need to be performed to make firmer predictions, but this work is encouraging.

Figure 20. Predicted flux emerging locally from various annuli at different radii in an accretion disk model around a Kerr hole of mass 2 × 109 M and spin a / M = 0.998. The accretion rate through the disk is 1 M yr-1. Only the wavelength range near the Lyman limit is shown. Solid curves show the full spectra, including a metal line source function in LTE, but with a fixed atmosphere structure calculated by neglecting metals. The solid curves are therefore not self-consistent but rather indicate that metal lines are likely to be important. Dashed curves show the self-consistent non-LTE continuum spectra computed by neglecting metals (data courtesy I. Hubeny).

At long wavelengths the spectra shown in Figure 18 obey F ^-7/3, in agreement with the standard F ^1/3 accretion disk prediction. However, flux is lost by the line-blanketing effects of metals, which causes the SED to drop at short wavelengths. In a self-consistent calculation, this absorbed radiation will have to be reemitted somewhere in the SED, and it is therefore conceivable that the overall SED continuum shape will change, likely becoming redder and therefore in better agreement with observations. Once again, this needs to be checked by computing self-consistent line-blanketed, non-LTE models in a broad range of parameter space.

An important fact about all the new "bare" accretion disk calculations is that the strength of the Lyman edge feature is not as strong as thought in the earlier models, but getting rid of the Lyman edge is difficult. Self-consistent calculations of Lyman edge features in disk-corona or multiphase accretion flow models are badly needed.