Line-driven winds and the UV turnover in AGN accretion discs

Abstract

1 INTRODUCTION

The optical–UV emission in AGN is most likely the signature of accretion on to the central massive black hole through a thin accretion disc (AD; Shields 1978; Malkan 1983 and citations thereafter). Malkan & Sargent (1982) noted that the UV emission shows a turnover characteristic of a T_max ∼ 30 000 K blackbody. Following studies of larger samples showed that this is a general trend in AGN, where the SED shows a turnover from a spectral slope of α ∼ −0.5 (F_ν ∝ ν^α) at λ > 1000 Å to α ∼ −1.5 to −2 at λ < 1000 Å (Zheng et al. 1997; Telfer et al. 2002; Shang et al. 2005; Barger & Cowie 2010; Shull, Stevans & Danforth 2012; cf. Scott et al. 2004), which extends to ∼1 keV (Laor et al. 1997). The turnover at λ < 1000 Å, which corresponds to a blackbody with T_max ∼ 50 000 K, is in contradiction with the thin local blackbody AD models, which predict peak emission |$\nu _{\rm peak}\propto (\dot{m}/M)^{1/4}$|⁠, where M is the black hole mass and |$\dot{m}\equiv L/L_{\rm Edd}$| is the luminosity in Eddington units. Thus, ν_peak should range over more than an order of magnitude, as broad-line AGN extend over the range |$\dot{m}=0.01\hbox{--}1$| and M = 10⁶–10¹⁰ M_⊙, which is in contrast with the small range observed (e.g. Shang et al. 2005; Davis & Laor 2011, hereafter DL11). For example, some models predict a peak at ν_peak > 10¹⁶ (e.g. Hubeny et al. 2001; DL11), while objects with such SEDs appear to be extremely rare (e.g. Done et al. 2012). Furthermore, high-|$\dot{m}/M$| AD models predict significant soft X-ray thermal emission, which is also not observed (Laor et al. 1997), which again implies that the expected thermal emission from the inner hottest parts of the AD is missing.

The extreme UV (EUV) emission spectral shape can also be constrained based on various line ratios. The analysis of Bonning et al. (2013) of a sample of AGN reveals similar observed line ratios, again indicating similar EUV SEDs, and an absence of the dependence of the EUV emission on the predicted maximum thin AD temperature in each object.

In contrast, the SED of AD around stellar mass black holes, which peak in the X-ray regime, matches observations remarkably well, in particular near the peak emission which originates from the hottest innermost AD region (Davis et al. 2005; Davis, Done & Blaes 2006). The match is accurate enough that it can be used to determine the black hole spin (e.g. McClintock et al. 2011).

What prevents AD in AGN from generally reaching T_max ≫ 50 000 K? The universality of the observed ν_peak suggests that it is a local process in the AGN AD atmosphere, most likely related to an atomically driven process. This process should be effective at T ∼ 50 000 K, and absent at T ∼ 10⁷ K, relevant to AD around stellar mass black holes.

Interestingly, main-sequence stars show a similar maximum temperature. The hottest O stars also do not generally reach beyond ∼50 000 K (e.g. Howarth & Prinja 1989). Massive O stars produce a strong wind with a high mass-loss, |$\skew4\dot{M}_{\rm wind}$|⁠, which can reach 10⁻⁵ M_⊙ yr⁻¹ in the most luminous O stars with a luminosity L_* > 10⁶ L_⊙. Such stars have a mass of M_* ∼ 50-100 M_⊙, and thus lose a significant fraction of their mass on a time-scale |$t_{\rm wind}=M_*/\skew4\dot{M}_{\rm wind}\sim 5{\rm -}10$| Myr, which is comparable to their lifetime.

Is this |$\skew4\dot{M}_{\rm wind}$| regulation of the hottest and most massive O stars relevant to AGN? Can this mechanism explain the similar T_max observed in AGN AD and in O stars? The local structure of a stellar atmosphere is mostly set by the local flux, i.e. the effective temperature T_eff, and by the surface gravity g. AGN AD have T_eff ∼ 10⁴-10⁵ K at their inner regions, and g at the disc surface is set by the balance of radiation pressure and gravity. Thus, the local flux and g are always at the Eddington limit in the vertical direction. In O stars, the local flux and g also reach close to the Eddington limit. The radius of O stars is ∼10¹² cm. The radius of the UV emitting region in luminous AGN AD is ∼10¹⁵ cm, i.e. a 10⁶ larger surface area. Thus, if the mass-loss per unit surface area reaches similar values in O stars and in AGN, given their similar T_max and g, then the total |$\skew4\dot{M}_{\rm wind}$| from the AGN AD where T ∼ 50 000 K can reach 10 M_⊙ yr⁻¹, which can exceed the accretion rate |$\skew4\dot{M}$|⁠. The value of T_max in AGN AD may then be set by the radius at which the thin disc solution must break down as |$\skew4\dot{M}_{\rm wind}>\skew4\dot{M}$|⁠.

Below we explore this suggestion more quantitatively. In Section 2, we derive the mass-loss per unit surface area in stars as a function of the atmospheric properties. In Section 3, we provide a simple analytic estimate of the innermost disc radius, and the implied T_max in AGN AD, based on the O stars mass-loss. In Section 4, we derive a revised equation for the radial AD structure for a thin AD + wind. In Section 5, we provide numerical solutions for the revised thin AD radial structure. In Section 6, we derive the AD SED using various approximations. The results are discussed in Section 7, and the main conclusions are summarized in Section 8.

2 STELLAR MASS-LOSS

Below we use the stellar |$\skew4\dot{M}_{\rm wind}(L_*)$| relation to derive a relation between the mass-loss per unit area, |$\dot{\Sigma }$|⁠, and the locally emitted flux per unit area (i.e. T_eff). We assume that this relation applies to AGN AD, and use the AD expression for T_eff(R) to derive |$\dot{\Sigma }(R)$|⁠, where R is the radius. We then integrate over the AD surface area to derive the cumulative |$\skew4\dot{M}_{\rm wind}(R)$|⁠, and derive R_eq at which |$\skew4\dot{M}_{\rm wind}(R_{\rm eq})=\skew4\dot{M}$|⁠, where |$\skew4\dot{M}$| is the accretion rate coming in from infinity. This radius forms the effective inner thin AD boundary and sets the maximum thin disc temperature, T_max = T_eff(R_eq).

Figure 1.

Upper panel: the relation between the mass-loss per unit area |$\dot{\Sigma }$| and the local flux F, derived from the measured total mass-loss, luminosity and radius for 201 O stars tabulated by Howarth & Prinja (1989). The solid line marks the |$\dot{\Sigma }(F)$| relation used for AGN AD. Middle panel: the relation between |$\dot{\Sigma }$| and both F and g. This rather tight relation follows the predicted CAK relation between |$\skew4\dot{M}$| and both L_* and M_*, ignoring the 1 − L/L_Edd term in the CAK solution. The solid line marks the |$\dot{\Sigma }(F,g)$| relation used for AGN AD. Although this relation is significantly tighter than the |$\dot{\Sigma }(F)$| relation derived in the upper panel, this relation is expected to break in stars at the Eddington limit, which produces the same local conditions as in an AD atmosphere. Also, the value of g in the standard AD solution assumes only pure electron scattering opacity, an assumption which breaks down in AGN AD. Lower panel: the relation between the surface gravity and the local flux for the 201 O stars. The solid line shows the relation in an atmosphere supported by radiation pressure, with an electron scattering opacity, as expected in the inner part of an AGN AD. In O star atmospheres, the radiation force induced by electron scattering supports up to a half of the local gravity.

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Although the |$\dot{\Sigma }(F,g)$| relation (equation 3) is significantly tighter than the |$\dot{\Sigma }(F)$| relation (equation 2), its applicability to AGN AD is not clear. In the radiation pressure-dominated part of AD, relevant to AGN AD at small R, Γ = 1 at the disc surface, and the CAK expression for |$\skew4\dot{M}_{\rm wind}$| (equation 4) formally diverges. However, in AD the dynamics is different, as g ∝ z and F is constant for z ≪ R, in contrast with stars where both g and F are ∝ 1/R², so Γ > 1 does not lead to divergence as in the stellar case. We therefore use below both the |$\dot{\Sigma }(F,g)$| and the g averaged |$\dot{\Sigma }(F)$| relation, to get some indication of possible |$\skew4\dot{M}_{\rm wind}$| values.

We note in passing that additional |$\dot{\Sigma }(F,g)$| relations can be derived from various theoretical calculations presented by Vink, de Koter & Lamers (2000) and Lucy (2010). For the sake of simplicity, we use only the observationally derived relations given above.

3 ANALYTIC ESTIMATE OF T_max

3.1 Derivation for |$\dot{\Sigma }(F)$|

3.2 Derivation for |$\dot{\Sigma }(F,g)$|

Below we repeat the above derivation using the above expression for |$\dot{\Sigma }(F,g)$| (equation 3).

The wind flux is inversely correlated with g (equation 3). The value of g in hydrostatic equilibrium depends linearly on the gas opacity (equation 16). The electron scattering opacity used here is the minimal opacity for ionized gas. The additional contribution from line opacity increases g, and thus decreases |$\dot{\Sigma }$|⁠. As a result, the disc may extend further inwards, and thus reach a higher temperature than derived above (equations 22 and 23).

4 THE NAVIER–STOKES EQUATIONS FOR AN AD WITH MASS-LOSS

The above estimates suggest that line-driven winds from AGN AD prevent the formation of the hot inner AD regions with T > 10⁵ K, which may explain the uniformity of the far-UV (FUV) SED of AGN. However, these estimates are rather crude, as the expression used for F (equation 8) ignores the reduction in |$\skew4\dot{M}$| due to the wind mass-loss. Below we derive the AD structure, based on the mass, momentum and energy continuity equations, including a wind mass-loss term. We then calculate the revised AD SED, first using the local blackbody approximation, and then using the stellar atmospheric solution code tlusty (Hubeny & Lanz 1995; Hubeny et al. 2000).

The derivation below is for a viscous flow, described by the Navier–Stokes equations. We use cylindrical coordinates, R, z, ϕ, and assume axial symmetry (no ϕ dependence). We further assume that the R and z solutions are separable, which is likely valid in the thin disc approximation, and we solve for the radial dependence only. The solution below yields the radial dependence of the vertically integrated viscous torque W_Rϕ(R), which is required in order to get a steady-state solution. This quantity, together with Ω(R) – the angular velocity radial dependence, uniquely determines F(R). The physical origin of W_Rϕ(R) is an open question, heuristically addressed by the α-disc model (SS73). In ionized AD, it is now widely believed that angular momentum transport is provided by magnetorotational turbulence (Balbus & Hawley 1998), which when averaged over time and the vertical extent of the disc seems to be reasonably approximated by an α-disc solution (Balbus & Papaloizou 1999). If the disc is thin, only the surface density, Σ, and the vertical structure of the disc depend on the accretion stress mechanism. The expression for F(R) and the derived SED, in the local blackbody approximation, are independent of the nature of the angular momentum transport mechanism. A relation for the accretion stress is required to derive a detailed model of the AD vertical structure, which can then be used to derive the local |$\dot{\Sigma }$| from first principles, as done by CAK for O stars. Although the α-disc model allows us to solve the vertical disc structure, it is just a convenient way to parametrize our ignorance, and is far from being a first principles solution. Below we circumvent this difficulty by adopting the stellar |$\dot{\Sigma }$| as described above.

4.1 Derivation

The time-dependent AD equations can be derived by formulating the Navier–Stokes equations in cylindrical coordinates with vertical averaging (see e.g. Balbus & Papaloizou 1999). When the disc is sufficiently thin, the radial momentum equation is to lowest order simply a balance between rotational terms and gravity, with a slow radial inflow due to the stress. The gravitational potential then determines the rotation rate Ω(R), which is Keplerian for a point source. Balbus & Papaloizou (1999, see also Blaes 2004) show that with appropriate averaging, stresses arising from magnetorotational turbulence yield (to lowest order in H/R) essentially identical relations to the viscous relations when written in terms of the vertically integrated stress W_Rϕ.¹

These are identical to equations 26, 27 and 46 of Balbus & Papaloizou (1999) except for the appearance of fluxes of mass F_M, angular momentum and energy due to the outflow. The surface terms no longer vanish in the vertical integration, giving F_M ≃ ρ|v_z|, corresponding to a vertical momentum flux. This mass flux carries away an angular momentum flux proportional to the specific angular momentum of the material at its launching radius. Note that we have not attempted to account for an additional torque of the wind on the disc that might arise if e.g. the wind and disc are magnetically coupled. In some cases, the torque may be plausibly absorbed into W_Rϕ if there is associated dissipation that can be modelled as in equation (26). In general, this depends on the details of the torque mechanism (see e.g. Balbus & Papaloizou 1999).

The second term on the right-hand side of equation (26) accounts for possible work done by the radiation field in unbinding the outflow. Since the vertical component of gravity continues to increase (initially linearly), the radiation field will do work against gravity launching and accelerating any unbound material. We introduce a parameter ϵ, which corresponds to the fraction of the gravitational binding energy transferred from the radiation to the outflow, once the outflow leaves the thin disc. Hence, ϵ = 1 corresponds to a flow where all material removed from the thin disc reaches the local escape velocity.

If ϵ is treated as a constant, this prescription implies that the mass launched from an annulus at radius R is accelerated locally. Of course, this is generally not correct and more sophisticated calculations and numerical simulations (e.g. Murray et al. 1995; Proga, Stone & Kallman 2000) show that most of the outflow is accelerated above the surface of the disc, and predominately by radiation from regions interior to its launching radius. A realistic outflow model requires a global numerical simulation, and is beyond the scope of this paper. In the following, we simply adopt equation (26) as a useful heuristic which aids in the discussion of global energy conservation. We offer some discussion of the global aspects of the AD and outflow in Section 7.

4.2 The no wind solution

4.3 The disc + wind equations

For a Keplerian disc with mass-loss, equations (26)–(28) provide a system of coupled partial differential equations, since |$\dot{\Sigma }$| depends on F, which is computed using equation (26). For the form of |$\dot{\Sigma }$| given in equations (2) and (3), there is no simple analytical solution, and these equations must be integrated numerically.

4.3.1 An example for an analytic solution

For illustrative purposes, we derive below a simple analytic solution for F(R) for a Keplerian disc with a given analytic expression for |$\skew4\dot{M}_{\rm wind}(R)$| and ϵ = 0. This provides some insight on the effect of a wind on the AD SED.

5 RESULTS

Below we consider models which parametrize the mass flux with either |$\dot{\Sigma }(F)$| or |$\dot{\Sigma }(F,g)$|⁠. For each model, we consider the cases with both ϵ = 0 and ϵ = 1 in equation (26). The models with ϵ = 0 do not account for the energy lost in unbinding the flow while the set of models with ϵ ≥ 1 assumes that all |$\skew4\dot{M}_{\rm wind}$| becomes unbound and escapes the system. As we will see, models with lower M and high |$\dot{m}$| can lead to very large implied mass outflow rates with |$\skew4\dot{M}_{\rm wind} \simeq \skew4\dot{M}_{\rm out}$|⁠. In this regime, the fraction of |$\skew4\dot{M}_{\rm out}$| that is removed from the disc is very sensitive to our choice of ϵ, but we shall see that the derived radiative flux from the disc is rather insensitive to this assumption.

We present the derived T_eff(r) ≡ (F/σ_B)^1/4 for the disc+wind solution, and the associated SED (note that r = R/R_g). We show the significantly reduced dependence of ν_peak on |$\dot{m}/M$|⁠, as expected from the simplified analytic derivation above (equations 15 and 23). In the absence of a wind, the inner disc radius is often assumed to correspond to the innermost stable circular orbit r_ISCO, which in turn is set by the black hole spin a_*. In the absence of a wind, higher a_* AD spectra are harder. Below we show that since generally r_eq ≫ r_ISCO, the value of a_* has no effect on the observed SED, as the innermost thin disc region is gone with the wind. We also show below that sufficiently cold discs are not affected by the wind, and the SED of objects with low enough |$\dot{m}/M$| values should be well fitted by the standard disc solution.

5.1 The numerical solution

In Appendix A, we describe the general relativistic generalizations of equations (26) and (28), which now depend explicitly on a_*. Equations (A9) and (A10) still form a closed set of two coupled equations that need to be numerically integrated, with boundary conditions for both |$\skew4\dot{M}$| and W_Rϕ. If we set our boundary condition at r_in as in SS73, |$\skew4\dot{M}_{\rm in}$| is a parameter of the problem. We follow SS73 by setting W_Rϕ(r_in) = 0 and integrate outwards. Alternatively, we could start at a large radius where T_eff is low, |$\dot{\Sigma }=0$| and |$\skew4\dot{M}=\skew4\dot{M}_{\rm out}$| parametrizes the disc model. However, there is some ambiguity in the specification of W_Rϕ(r_out) in this case. This is problematic for models where |$\skew4\dot{M}_{\rm wind}$| becomes a substantial fraction of |$\skew4\dot{M}(r_{\rm out})$|⁠, as the integrated mass-loss and its radial profile can be quite sensitive to the precise choice of W_Rϕ(r_out). Therefore, we focus our attention on solutions that start from r_in and integrate outwards.

Since our model presumes that W_Rϕ is generated by local stresses within the disc, there is no reason to expect that W_Rϕ, set by local conditions at r ≫ r_in, happens to have the exact value required to match on to a stationary solution with W_Rϕ(r_in) = 0. In a disc with no outflow, matter should have time to diffusively adjust as it slowly spirals in, since the viscous time decreases inwards. Hence, it should generally follow the equilibrium solution specified by the inner torque assumption, unless an instability in the flow is present (see e.g. Shakura & Sunyaev 1976). However, it is less clear whether the same argument applies to the models considered here when there is significant mass outflow. It is possible that such flows show substantial time variability, but we only explore steady-state models here.

Fig. 2 shows a comparison of models with ϵ = 0, 1 and 2 for a 10⁷ M_⊙, a_* = 0 black hole with mass-loss parametrized by |$\dot{\Sigma }(F)$| and |$\dot{m}_{\rm out}=2$|⁠. The model with ϵ = 1 launches a wind where all material just reaches its escape velocity, while model with ϵ = 2 reaches infinity with a kinetic energy which equals the binding energy at its launching radius. The top panel shows that the overall mass-loss is larger for the ϵ = 0 model, and peaks at a somewhat larger radius than the ϵ ≥ 1 models. The ϵ = 1 and 2 profiles for |$\skew4\dot{M}_{\rm wind}(r)$| are very similar and the outflow is more broadly distributed in radius. The middle panel shows the corresponding F(r) for the models in the upper panel, evaluated using equation (A6). The outflow models all yield significantly lower values than the no mass-loss model. The strong sensitivity of |$\dot{\Sigma }$| on F (equations 2 and 3) serves as a thermostatic effect on the maximal possible F value in all models. In the ϵ ≥ 1 models, this happens primarily through the second term on right-hand side of equation (A6) so that the larger fraction of energy that is lost in unbinding the flow is offset by lower implied outflow rates. In the ϵ = 0 case, the cap on F only occurs through a reduction of W_rϕ, which depends less directly on |$\dot{\Sigma }$| through equation (A4). Due to the strong similarity of F(r) in the models with ϵ = 1 and 2, we only show the ϵ = 0 and 1 cases as representative examples in subsequent plots.

Figure 2.

Upper panel: the implied mass-loss rate versus R for models with ϵ = 0 (blue dotted), 1 (red dashed) and 2 (green dot–dashed). All models correspond to M/ M_⊙ = 10⁷, a_* = 0 and |$\dot{m}=2$| at a large radius. Mass-loss is modelled using the |$\dot{\Sigma }(F)$| prescription. The integrated mass-loss decreases as ϵ increases, with the largest outflow coming from the ϵ = 0 model, for which a larger fraction of the outflow occurs at a larger radius. Middle panel: the flux corresponding to the model shown in the upper panel and a model with no outflow (solid black). The larger fraction of energy that is lost in unbinding the flow for larger ϵ is offset by lower implied outflow rates, and the flux profiles for the models with mass-loss are all very similar. Bottom panel: the difference between the radiative flux and the energy required to unbind the implied outflow for the model with ϵ = 0. The thin dotted curve shows where this quantity is negative. The negative value implies that mass-loss is so great that there is insufficient energy to unbind it. Hence, most of the mass lost from the thin disk, if ϵ = 0, must ultimately accrete, possibly in the form of a much hotter and geometrically thick flow.

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The ϵ = 0 case assumes that no kinetic energy is taken by the outflow. Can the outflow still gain enough energy by intercepting enough of F(r), say in the form of a radiation pressure-driven wind, to produce a wind which escapes to infinity? The bottom panel explores this question by showing a modified F(r), for the ϵ = 0 model, when we subtract from the original value the binding energy of the mass lost. We compute this by subtracting off the flux of kinetic energy |$(1-E^\dagger )\dot{\Sigma }c^2$|⁠, which is required to (just) unbind the outflow. This value becomes negative at r < 90. Hence, there is insufficient energy in the radiation field to unbind the implied outflow. So, if there is no physical mechanism which can provide ϵ > 0, i.e. a mechanism which can convert directly some of the local dissipated energy in the disc into kinetic energy of the mass lost, then a large fraction of the implied |$\skew4\dot{M}_{\rm wind}$| must ultimately form a ‘failed wind’ and accrete. We discuss the observational implications of such failed winds in Section 7.2.

Fig. 3 presents numerical solutions for T_eff(r). The results are presented for the two |$\dot{\Sigma }$| relations and for different M values. The top panel presents the |$\dot{\Sigma }(F)$|⁠, ϵ = 0 AD model solution for T_eff(r), for M = 10⁷, 10⁸ and 10⁹ M_⊙. All models correspond to a_* = 0 and |$\dot{M_1}=0.94$| at r ≫ r_in, which corresponds to |$\dot{m}=2$|⁠, 0.2 and 0.02, respectively. For comparison, we also show the corresponding standard SS73 solution for T_eff(r) with no mass-loss, i.e. a constant |$\dot{m}$| (⁠|$=\dot{m}(r_{\rm out})$|⁠). Note the similar T_max in the M = 10⁷ and 10⁸ M_⊙ models, with a ratio of 1.4, in contrast with the ratio of 3.16 for the SS73 solution (equation 13). The simplified analytic solution ratio (equation 14) of 10^0.14 = 1.38 is remarkably close to the numerical solution ratio of 1.4. The absolute value of T_max in the analytic solution, 7.2 × 10⁴ K for M = 10⁷ M_⊙, is also very close to the numerical solution value of 7.5 × 10⁴ K. The M = 10⁹ M_⊙ model is cold enough to suppress |$\dot{\Sigma }(F)$|⁠, so that |$\dot{m}_{\rm wind}(r)/\dot{m}\ll 1$|⁠, and the solution overlaps the SS73 no wind solution.

Figure 3.

Upper panel: the T_eff versus R for models with M/ M_⊙ = 10⁷, 10⁸ and 10⁹. All models correspond to a_* = 0 and |$\dot{M_1}=0.94$| at a large radius, which corresponds to |$\dot{m}=2$|⁠, 0.2 and 0.02, respectively. The solid curves denote standard AD models with no mass-loss while the dashed curves show models with mass-loss given by |$\dot{\Sigma }(F)$| with ϵ = 0. Note the reduced difference in T_max between the M/ M_⊙ = 10⁷ and 10⁸ mass-loss models, compared to the standard solution. The AD in the M/ M_⊙ = 10⁹ model is too cold to produce significant mass-loss, and the T_eff profile remains the same. Middle panel: same as the upper panel but the dotted curves correspond to |$\dot{\Sigma }(F,g)$| with ϵ = 0. This mass-loss term yields a weaker R dependence of |$\skew4\dot{M}$|⁠, and thus a more gradual rise at smaller R in the deviation of T_eff from the standard solution. Bottom panel: same as the upper panel, but the dot–dashed curves correspond to |$\dot{\Sigma }(F)$| with ϵ = 1. In this case, the T_eff continues to increase very mildly as radius decreases, rather than decreasing slightly as in the ϵ = 0 case.

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The middle panel shows the numerical solution using |$\dot{\Sigma }(F,g)$| and ϵ = 0 for the same parameters as in the upper panels. The mass-loss is more pronounced, and its rise towards the centre is more gradual, as expected from the simplified analytic solution (equation 20 versus equation 11). The wind remains significant also for the M = 10⁹ M_⊙ models. The ratios of T_max from the M = 10⁷ and 10⁸ M_⊙ models are, 2.03, somewhat larger than those with the |$\dot{\Sigma }(F)$| relation.

The bottom panel shows a model with |$\dot{\Sigma }(F)$| and ϵ = 1. Although the implementation of the outflow differs significantly from the top panel, the thermostatic effect is still apparent. In this case, the ratio of T_max for the M = 10⁷ and 10⁸ M_⊙ models is 1.65, compared to 1.4 for the ϵ = 0 model above, and significantly less than the SS73 model. As above, there is very little mass lost for the M = 10⁹ M_⊙ model. Models with |$\dot{\Sigma }(F,g)$| and ϵ = 1 (not shown) have T_eff(r) profiles qualitatively similar to those with |$\dot{\Sigma }(F,g)$| and ϵ = 0.

Fig. 4 shows the implied |$\dot{m}(r)$| profiles for the models in Fig. 3. In the simplified analytic solution, the disc truncates at r_eq, where |$\dot{m}_{\rm wind}(r_{\rm eq})=\dot{m}(r_{\rm out})$|⁠. In the numerical solutions, the drop in |$\dot{m}(r)$| towards the centre suppresses the rise in T_eff(r), and thus reduces a further rise in |$\dot{m}_{\rm wind}(r)$| towards the centre. This negative feedback prevents |$\dot{m}_{\rm wind}(r)$| from ever reaching |$\dot{m}(r_{\rm out})$|⁠, and the disc always (nominally) extends down to r_ISCO. However, the implied total mass-loss can be very large. The |$\dot{\Sigma }(F,g)$| model with ϵ = 0 yields |$\dot{m}(r_{\rm in})/\dot{m}(r_{\rm out})=0.000\,92$| for M = 10⁷ M_⊙, and the |$\dot{\Sigma }(F)$| model with ϵ = 0 yields |$\dot{m}(r_{\rm in})/\dot{m}(r_{\rm out})=0.0024$|⁠. Thus, although formally the thin disc extends down to r_ISCO, it effectively terminates at a larger radius in these cases. For example, |$\dot{m}(r)/\dot{m}(r_{\rm out})=0.5$| at r = 62 for the |$\dot{\Sigma }(F)$|M = 10⁷ M_⊙ model and at r = 20 for the M = 10⁸ M_⊙ model (the simplified analytic solution, equation 12, gives truncation radii of r = 85 and 28, respectively). Inside these transition radii, the implied |$\dot{m}(r)$| profiles should be regarded with suspicion, given a possible feedback of the mass-loss on the thin disc solution. Obtaining reliable profiles in this region probably requires a more sophisticated disc model.

Figure 4.

The local |$\dot{m}$| versus R for the models shown in Fig. 3. The implied drop in |$\dot{m}$| towards R_ISCO becomes larger with decreasing M, as the AD gets hotter, and can reach a factor of 1000 in the most extreme model, corresponding to M = 10⁷ M_⊙, |$\dot{\Sigma }(F,g)$| and ϵ = 0. The models with ϵ = 1 generically produce less implied mass-loss, with most extreme case (M = 10⁷ M_⊙) having |$\skew4\dot{M}_{\rm out} \sim 6 \skew4\dot{M}_{\rm in}$|⁠. The M/ M_⊙ = 10⁹ and |$\dot{\Sigma }(F)$| models show no significant mass-loss for either value of ϵ.

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The coldest models with M = 10⁹ M_⊙ has no significant mass-loss for models employing the |$\dot{\Sigma }(F)$| relation. This is consistent with the T_eff(r) solution in the upper and lower panels of Fig. 3, which matches the no wind SS73 solution, as |$\dot{m}(r)$| remains effectively constant. The |$\dot{\Sigma }(F,g)$| relation yields a more gradual drop in |$\dot{m}(r)$| towards the centre for all M, but with a significantly larger amplitude, with significant wind also for M = 10⁹ M_⊙.

Fig. 5 explores the effect of the black hole spin a_* on T_eff(r), for the models assuming outflow rates of |$\dot{\Sigma }(F)$| and ϵ = 0 and 1. The solutions of T_eff(r) are presented for a_* = 0, 0.7 and 0.9. All models correspond to M = 10⁸ M_⊙ and |$\dot{M_1}=0.94$| at large r, which corresponds to |$\dot{m}=0.2$|⁠, 0.36 and 0.54, respectively. The SS73 solutions are also shown for comparison. In contrast with the SS73 solution, where T_max rises with a_*, as r_ISCO gets smaller, in the ϵ = 0 case all models reach a nearly identical T_max, which occurs at r ≃ 10, well outside the largest r_ISCO = 6. The value of T_max is well below the SS73 range of values, as shown above in Fig. 3. The AD extension to smaller r with increasing a_* just produces an extended inner region with T ≃ T_max, from r ≃ 20 down to r ≃ r_ISCO. The strong dependence of |$\dot{\Sigma }$| on the local F effectively serves as a local thermostat, which prevents F from rising above the limiting T_max value. Thus, the wind breaks the tight relation between T_max and a_*, which exists in models with no mass-loss.

Figure 5.

The T_eff(r) solution for models with a_* = 0, 0.7 and 0.9. All models correspond to M = 10⁸ M_⊙ and have |$\dot{M_1}=0.94$| at a large radius, which corresponds to |$\dot{m}=0.2$|⁠, 0.36 and 0.54, respectively. The solid curves denote standard AD (SS73) models with no mass-loss. The dashed and dot–dashed curves denote models with mass-loss given by |$\dot{\Sigma }(F)$| for ϵ = 0 and 1, respectively. In contrast with the SS73 models, all ϵ = 0 models reach the same T_max. This reflects the thermostatic effect of |$\dot{\Sigma }$|⁠, which sets a cap on T_max, and produces a nearly isothermal AD at r ∼ 2-20. For the ϵ = 1 models, T_eff continues to rise to small R, but much more weakly than in the SS73 models. Since the emitting area decreases, these innermost radii contribute little to the disc integrated luminosity. At larger R, which dominate the bolometric output, the profiles nearly overlap, as in the ϵ = 0 models.

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The ϵ = 1 models show a slight rise in T_eff(r) towards smaller r, but since the emitting area scales as r², we will see that this weak increase has almost no effect on the observed SED.

The results presented in Figs 3 and 5 clearly show that T_max remains well below 10⁵ K. There is some dependence of T_max on the AD parameters, but the dependence is significantly reduced compared to the solutions with no winds, particularly for the |$\dot{\Sigma }(F)$| relations. The numerical solutions are rather close to the simplified analytic estimate made in Section 3 for r_eq and T_max. The qualitative similarities between the models with ϵ = 0 and 1 suggest that the thermostatic effect of the wind could be quite robust. The difference between the ϵ = 0 and 1 models is their significantly different |$\dot{m}(r)$| in the hottest models. The models utilizing the |$\dot{\Sigma }(F,g)$| outflow prescription generally provide too much outflow, leading to discs considerably colder than those observed.

6 THE DERIVED SED

6.1 The local blackbody models

We now consider the SED predicted by the disc models in the presence of a mass outflow. The outflow can modify the spectrum in two primary ways: by modifying the underlying thin disc solution, as derived above, and also via its direct emission or reprocessing (absorption and scattering) of radiation from the underlying disc. In this work, we focus only on the effects on the underlying thin disc solution, which may produce the universal turnover at λ < 1000 Å. A more complete calculation requires modelling the outflow (Murray et al. 1995; Proga et al. 2000) to compute the effect of reprocessed emission (see e.g. Sim et al. 2010) on the SED. Such models require detailed numerical simulations that are beyond the scope of this work.

We first study the derived SED based on the simple local blackbody SED calculations, computed directly from the T_eff(r) profiles discussed above. We later present the results from a more detailed model which includes radiative transfer and the vertical structure of the atmosphere. The advantage of the simplified local blackbody calculation is that the results are insensitive to assumptions about the vertical dissipation distribution. We break the disc up into concentric annuli equally spaced in log r. At each radius, we compute a blackbody spectrum at T_eff(r), weighting by the emitting area of the annulus and summing over all radii yields full disc SEDs. For the sake of simplicity, the effects of relativity on photon propagation are neglected at this stage, and are included in the following more detailed atmospheric calculations.

Fig. 6 presents the SED for the outflow prescriptions described in Section 5.1. We use the T_eff(r) profiles presented in Fig. 3 for M = 10⁸ M_⊙, |$\dot{m}(r_{\rm out})=0.2$| and a_* = 0. The blackbody SED with no outflow peaks at ∼600 Å, too far in the UV to be consistent with the observed SEDs of 10⁸ M_⊙ black holes. The ϵ = 0 and 1 |$\dot{\Sigma }(F)$| models both yield peaks at ∼1000 Å, as typically observed. The |$\dot{\Sigma }(F, g)$| models, which predict more mass-loss, peak longwards of 2000 Å, too long to be consistent with observations of most luminous AGN.

Figure 6.

The specific luminosity λL_λ versus λ for blackbody emission corresponding to the models with different |$\dot{\Sigma }$| and ϵ, as noted in the plot. All models assume no torque at the inner boundary, a_* = 0, M = 10⁸ M_⊙ and |$\dot{m}=0.2$| at large R. The standard model with no mass-loss (solid) peaks at λ ∼ 600 Å, significantly shorter than typically observed. The |$\dot{\Sigma }(F)$| models peak at λ ∼ 1000 Å, as typically observed. The |$\dot{\Sigma }(F, g)$| models, characterized by a stronger and more extended mass-loss, peak at λ > 2000 Å, colder than typically observed.

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Fig. 7 presents the SEDs as a function of M. It compares the SEDs derived from the |$\dot{\Sigma }(F)$| model for ϵ = 0 and 1 with the standard SS73 SEDs. As expected from the T_eff(r) solutions (Fig. 2), the M = 10⁷ and 10⁸ M_⊙ wind solution SEDs show similar peak wavelengths. For ϵ = 0, the wavelength ratio is 1.4 (830 versus 1170 Å), as expected from their T_max ratio of 1.4. The M = 10⁹ M_⊙ SED is nearly identical to the solution with no wind, as expected from the negligible |$\skew4\dot{M}_{\rm wind}$|⁠. The factor of 10 in the ν_peak position (210 versus 2130 Å) for the local blackbody solution of M = 10⁷ and 10⁹ M_⊙ with a fixed |$\skew4\dot{M}$| is reduced to a factor of <3 (830 versus 2250 Å). At long enough wavelengths (λ > 3000 Å), the SED remains unchanged, as the emission originates from outer colder regions in the AD where the wind is negligible.

Figure 7.

The specific luminosity λL_λ versus λ for blackbody emission corresponding to the models presented in Fig. 3 using the |$\dot{\Sigma }(F)$| prescription with ϵ = 0 (dashed) and ϵ = 1 (dot–dashed). The peak emission wavelength decreases with M in all models, but in the models with outflow this sensitivity to M is significantly reduced. With these relations, the 10⁹ M_⊙ models are cold enough to have a negligible wind effect on the SED. The integrated luminosity drops with M when mass-loss is included in the models, as expected from the |$\dot{m}(r)$| solutions. There are only very modest difference between the ϵ = 0 and 1 models, suggesting that the thermostatic effect on the SED is fairly robust.

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Similar conclusions hold for the models assuming ϵ = 1. In this case, the M = 10⁷ and 10⁸ M_⊙ models have a peak wavelength ratio of 1.45, which is somewhat less than their T_max ratio of 1.65. This difference is a result of the shallow T_eff(r) profile in Fig. 3. Even though T_eff still rises as R declines, the rise is small, and the maximum occurs at smaller radii with lower emitting area, and thus contributes relatively little to the SED. Therefore, the ratio of the peak emission wavelengths is set by somewhat larger radii where the T_eff ratio between the two different mass models is smaller. Models with |$\dot{\Sigma }(F,g)$| (not shown) are too cold to correspond to the observed SEDs.

Fig. 8 presents the dependence of the SED on a_* based on the models shown in Fig. 5. The models with no mass-loss show a harder SED with increasing a_*, as r_ISCO gets smaller and the AD reaches a higher T_max. In sharp contrast, the dependence of the SED on a_* disappears completely once |$\dot{\Sigma }$| is included. Although the thin AD still extends down to r_ISCO (see Fig. 5), this innermost region is not hotter than the outer regions for the ϵ = 0 models, due to the thermostatic effect of |$\dot{\Sigma }$| on T_max mentioned above (Section 5.1). The models with ϵ = 1 have a shallow rise in T_eff towards the centre, but the small emitting area again means these hotter regions contribute very little to the overall emission. In both cases, the contribution to the SED from the r ∼ 2 region is negligible compared to the contribution of the r ∼ 20 region. The SED is thus blind to the inner extension of the disc, and therefore to the value of a_*, for the AD parameters explored in this figure.

Figure 8.

The specific luminosity λL_λ versus λ for blackbody emission corresponding to the models with different a_* presented in Fig. 5. Even though the SEDs for models without mass-loss (solid) vary significantly with a_*, the models with mass-loss are essentially identical. This reflects the thermostatic effect of |$\dot{\Sigma }$|⁠, which produces a nearly isothermal disc at r < 20 (see Fig. 5). The SED is thus blind to the value of a_*.

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6.2 The tlusty models

Due to atomic features and electron scattering, it is expected that the local SED may differ significantly from blackbody emission (SS73; Kolykhalov & Sunyaev 1984). Detailed modelling of the disc vertical structure is required to accurately model these departures from blackbody emission (see e.g. Hubeny et al. 2000, 2001). When mass-loss is an appreciable fraction of the mass accretion rate, a substantial portion of the disc surface layers can no longer be in hydrostatic equilibrium. In the CAK theory, these departures from hydrostatic equilibrium are due to the force multiplier from line opacity. In principle, these lines could be modelled directly by a stellar atmosphere code such as tlusty, but this would significantly increase the complexity and computational cost of such calculations (e.g. Kudritzki & Puls 2000). Therefore, we approximate the disc emission using hydrostatic models, as used in previous studies with no mass outflow (Hubeny et al. 2000). Although the characteristic peak energy of the SED should be reasonably insensitive to this assumption, the spectrum of emission at shorter wavelengths may be significantly modified.

Even with hydrostatic models, the mass-loss has a significant impact on the spectrum through its modification of F(r) and Σ(r), as non-blackbody models are generally sensitive to both. We now proceed using the same integration method described in Section 5.1, again assuming no torque at the inner boundary. We combine the resulting profiles of F(r) and Σ(r) with a radial profile of the vertical gravity, and use the interpolation methods described in Davis & Hubeny (2006) to construct full AD SEDs, accounting for relativistic effects on photon geodesics (Dexter & Agol 2009). We compute Σ(r) using an α relation for the stress and solving an algebraic equation that smoothly transitions between the gas and radiation pressure-dominated limits, as described in appendix B of Zhu et al. (2012). The only difference here is that we compute F₁ = W_Rϕ/α using our numerically integrated W_Rϕ rather than assuming that the last equality in their equation B3a holds.

Fig. 9 compares the tlusty-derived SEDs as a function of M for models with and without mass-loss. All models have |$\skew4\dot{M}(r_{\rm out})=0.94 \rm \,\mathrm{M}_{\odot }\,yr^{-1}$|⁠, a_* = 0, i = 40°. Here we only consider one mass-loss prescription, using the |$\dot{\Sigma }(F)$| relation with ϵ = 0 as an example. The tlusty-based models with no mass-loss peak at higher energies compared to the local blackbody models of the same parameters (Fig. 7), due to the larger fraction of the opacity dominated by electron scattering at short wavelengths, and the resulting modified blackbody emission. Absorption edge features are also present. All the models with mass-loss are significantly colder, as expected. An interesting new feature is that, in contrast with the local blackbody models, where there is a gradual shift to lower ν_peak with a rising M, here all models show a similar peak position near the Lyman edge, and a break in the spectral slope above and below the edge. This occurs because of the jump in absorption opacity across the edge. This spectral break is remarkably similar to the universal break observed at λ ∼ 1000 Å in the mean SEDs of AGN (Telfer et al. 2002; Shull et al. 2012). The slope shortwards of 1000 Å depends on M, and the lowest M = 10⁷ M_⊙ model shows that the spectral slope can remain close to −1 far into the EUV.

Figure 9.

Comparison of the specific luminosity λL_λ versus λ for tlusty-based models with (dashed) and without (solid) mass-loss for different M values. Mass-loss assumes |$\dot{\Sigma }(F)$| and ϵ = 0, with each model computed to yield |$\skew4\dot{M}(r_{\rm out})=0.94 \,\mathrm{M}_{\odot }\,{\rm yr^{-1}}$|⁠. All models account for relativistic effects on photon propagation, and assume an inclination of 40°. The tlusty-based models are much harder due to modified blackbody effects resulting from significant electron scattering opacity and the presence of strong edge features. All the models with mass-loss show a peak and a spectral break near 1000 Å, similar to the universal break at 1000 Å seen in the spectra of AGN. However, the spectral slope at λ < 1000  Å is predicted to be harder at lower M models.

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6.3 The derivation of |$\skew4\dot{M}$|

How does the modified AD SED derived here affect the derivation of |$\skew4\dot{M}$| from the optical luminosity in AGN? In an earlier work (DL11), we provide a useful expression which allows us to derive |$\skew4\dot{M}$| based on the optical luminosity, when M is known, assuming that the continuum is produced by a thin AD with no mass-loss. As shown above [excluding some implausible very cold AD models produced by the |$\dot{\Sigma }(F,g)$| relation], the wind becomes significant only in the hotter UV emitting regions, and thus it has a negligible effect on the optical emission. Thus, the AD-based |$\skew4\dot{M}$| derivation remains valid. This method was applied to the PG sample of quasars to derive the radiative efficiency |$\eta \equiv L_{\rm bol}/\skew4\dot{M}c^2$|⁠, which was found to show a clear trend with M of the form η ∝ M^0.5 (DL11). This trend can be interpreted as an indication for a trend of a rising a_* with M. However, as discussed in DL11, the η ∝ M^0.5 relation can be derived from the universal SED of AGN, and it was not clear whether the trend of η with M just happens to lead by coincidence to a universal SED, or whether the universal SED is a more fundamental property, which leads to an apparent η ∝ M^0.5 relation. Now that we have a physical mechanism which may lead to a rather universal SED, we can explore its effect on the η versus M relation, and in particular study whether the observed η versus M relation has any implication on an a_* versus M relation.

Fig. 10 presents the η versus M relation derived in DL11 for the PG sample of quasars. We now explore whether a similar relation can be derived if the SED of these objects is produced by an AD with mass-loss, using the |$\dot{\Sigma }(F)$| relation with ϵ = 0, but with a fixed value of a_*. For each object, we construct a local blackbody AD+wind model with the tabulated M in DL11, and a fixed value of a_* = 0.9 for all objects. We iterated over the boundary value of |$\skew4\dot{M}(r_{\rm in})$| in order to get the tabulated |$\skew4\dot{M}$| for that object in DL11. We then integrated over the AD luminosity to get the predicted L_bol, and from that derive the predicted radiative η, which is plotted in Fig. 10. The η derived here tends to be lower by a factor of 2–3 from the one measured in DL11; however, it shows a rather tight correlation with M, of a similar slope to the one derived in DL11. Since the η ∝ M^0.5 correlation can result from a universal SED, it is not surprising that the disc+wind model SED used here to derive L_bol leads to a similar relation, as this model produces similar SEDs over a wide range of M and |$\skew4\dot{M}$|⁠. This trend can also be understood from the simplified analytic solution (equation 12), which gives r_eq ∝ M^−0.48, and the fact that in a thin AD η is given by the innermost disc radius. Thus, the observed η versus M relation in DL11 does not necessarily imply an a_* versus M relation, as η may be set by r_eq, which is independent of a_*.

Figure 10.

The radiative efficiency η versus M for a sample of 80 PG quasars. The black filled squares denote the efficiencies inferred by DL11, assuming a standard disc model with no mass-loss to estimate |$\skew4\dot{M}$|⁠. The blue open circles show the implied efficiencies in models with mass-loss given by |$\dot{\Sigma }(F)$| for the same M and |$\skew4\dot{M}$| used in the DL11 analysis. They assume a_* = 0.9. The slope of the correlation implied by the models is qualitatively consistent with the DL11 results, but with a lower mean and reduced variance of η.

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7 DISCUSSION

The SS73 solution was constructed with stellar mass black hole systems in mind, which have a much hotter AD than in AGN. As a result, the dominant opacity in stellar systems is electron scattering and free–free absorption. In AGN AD, the maximum temperature drops from ∼10⁷ to ∼10⁵ K, and UV line opacity becomes the dominant photospheric opacity source. This likely explains why the observed SED in binary black hole systems is well matched by simple thin AD models (Davis et al. 2005, 2006), while in AGN there is generally a gross mismatch between the predicted and observed emission from the innermost part of the AD. Various AGN AD models did take bound-free opacity into account (e.g. Czerny & Elvis 1987; Laor & Netzer 1989; Ross, Fabian & Mineshige 1992; Storzer 1993; Sincell & Krolik 1997), and also included careful calculations of the vertical structure coupled to the radiative transfer (Hubeny et al. 2000, 2001). However, none of the models included line opacity. In O stars, the line opacity inevitably leads to a wind, and it may have a similar effect in AGN.

Winds are prevalent in AGN, as indicated by the broad and blueshifted resonance line absorption observed in broad absorption line quasars (e.g. Reichard et al. 2003). The winds most likely originate from the AD, and a likely driving mechanism is radiation pressure on resonance lines, as indicated by both analytic solutions (Murray et al. 1995) and numerical calculations (Proga et al. 2000). Here we find that radiation pressure-driven winds may modify significantly the disc structure. Applying the mass-loss per unit area measured in O stars, we find that the simple thin disc solution effectively terminates at a few tens of R_g. The steep dependence of the local mass-loss on T_eff, |$\dot{\Sigma }(F)\propto T_{\rm eff}^{7.6}$| (equation 2) sets a cap on the maximum T_eff, which is well below 10⁵ K. This can explain why the observed AGN SEDs do not show a rise towards the EUV, and may also explain the rather universal turnover observed at λ < 1000 Å.

The softening effect of an AD wind mass-loss on the SED of AGN was noted by Witt, Czerny & Zycki (1997) and Slone & Netzer (2012), and for cataclysmic variables (CV) by Knigge (1999). The new result here is the application of the stellar mass-loss to AGN AD, which yields a UV turnover similar to the one observed, with no free parameters. In contrast with Slone & Netzer (2012) and Knigge (1999), where general |$\skew4\dot{M}_{\rm wind}(r)$| relations were assumed, here we find that the derived |$\skew4\dot{M}_{\rm wind}(r)$| relation which matched the observed SED has a negligible effect on the derived |$\skew4\dot{M}(r_{\rm out})$| (DL11), as there is negligible wind at the region where the optical emission is produced. However, it is certainly correct that |$\skew4\dot{M}(r_{\rm in})$| may be significantly smaller than |$\skew4\dot{M}(r_{\rm out})$|⁠, as was pointed out by DL11 and Slone & Netzer (2012).

Lawrence (2012) proposed that obscuration near, but external to, the AD produces the nearly constant SED peak. Obscuring material off the plane of the disc but at low radius is assumed to be provided by outflows from (or instabilities in) the disc. This model assumes that mass lost to obscuring clouds is not large enough to modify the intrinsic disc emission, which is instead altered by transfer through the obscuring clouds. Although we have not computed the effects of reprocessing by the outflow, such reprocessing likely occurs and the effects described in Lawrence (2012) may be present at some level.

7.1 Are O stars winds applicable to AGN?

The CAK wind solution applies for spherically symmetric systems, where both gravity and the radiation field fall off as 1/r². In AD there are significant differences. Thin AD are rotationally supported, and in the local disc frame, close to the disc surface, gravity increases linearly with height, while the radiation field is independent of height. Thus, it is not clear that the CAK solution is relevant even locally close to the surface of the AD. Far enough above the disc, where h > r, both gravity and the radiation field become radial and fall off as 1/r². Thus, in contrast with the stellar case, in AD both the relative strength and directions of the radiative and gravitational forces change with position. Proga et al. (2000) find that as a result there is no steady-state solution, in contrast with the steady-state stellar wind solution.

A related issue is the difference in the velocity fields in AD and in O stars. AD are expected to have Keplerian velocity shear and be highly turbulent. Since line-driven wind models from O stars generally assume monotonically increasing velocity profiles, non-monotonic velocity distributions (due e.g. to the turbulence) may modify the acceleration of the outflow. Modelling such effects requires fairly sophisticated radiative transfer calculations beyond the scope of this work, which to the best of our knowledge have not been considered elsewhere.

In addition, the integrated SED in AGN is harder than in stars, and thus the wind is subject to overionization once it becomes exposed to the harder EUV – soft X-ray radiation, which may shut off line driving if the ionization is large enough. Previous studies (e.g. Murray et al. 1995) have generally assumed that such irradiation only becomes important after the matter has been lifted significantly above the disc surface. However, if the inner region of the disc is thick enough, it may directly irradiate the disc at larger radii, increase the ionization level and reduce the force multiplier at the disc surface. Such direct irradiation is not expected in the radiation-dominated regions of an SS73 solution because the AD scaleheight is nearly constant with radius. In any case, the irradiating flux falls off as r⁻³ and likely remains a small fraction of the locally dissipated flux (Blaes 2004). Significant flaring of the disc or scattering of radiation by the failed wind might increase the irradiation, enhancing the ionization level at the disc surface and potentially inhibiting the outflow. For the line-driven wind to be a viable explanation of the ‘universal’ UV break, the outflow clearly cannot be completely quenched.

The local force multiplier within an unirradiated disc atmosphere is likely similar to that in an atmosphere of a star with the same T_eff. The force multiplier is Γ ∼ 10³ (Lamers & Cassinelli 1999) at a column of ∼10¹⁸ cm⁻¹ from the disc surface, which inevitably leads to a modification of the disc vertical structure. The SS73 solution assumes only electron scattering, leading to a thin disc with a height h < r for |$\dot{m}<1$|⁠. An increase in the opacity, i.e. Γ > 1, will expand the disc atmosphere vertically, and can lead to a disc thickness h^′ ∼ Γh. For Γ > r/h, the disc atmosphere becomes geometrically thick. If h^′ > r, then g ∝ 1/r², a hydrostatic solution is not possible anymore and a wind is launched. Since r/h ∼ 10-100 typically in thin discs, Γ ∼ 10³ should lead to a wind. However, once the wind is exposed to the central ionizing continuum, it may get significantly ionized (depending on its density), leading to Γ ≲ 10 (Murray et al. 1995, fig. 9 there). An acceleration length of ∼r/Γ will bring the wind to the escape velocity, and allow it to escape, even if it gets overionized at the coasting phase. If the outflow does not attain the escape speed before being over ionized, the gas will fall back to the disc, forming a ‘failed wind’.

Proga et al. (2000) present a model for a UV line-driven wind from an AD with |$m_8=1, \dot{m}=0.5$|⁠. They find a time-averaged |$\skew4\dot{M}_{\rm wind}/\skew4\dot{M}=28$| per cent, for |$\skew4\dot{M}_{\rm wind}$| measured at r > 200. This wind is shielded by the failed wind produced close to the inner boundary of the simulation (at r = 150), which is overionized by the central X-ray source. The AGN AD wind simulation of Proga & Kallman (2004) shows a sharp rise in the local mass-loss at r < 100 (fig. 1 there), and thus the integrated |$\skew4\dot{M}_{\rm wind}$| lifted from the surface may reach |$\skew4\dot{M}_{\rm wind}/\skew4\dot{M}\sim 1$| closer to the centre, as derived by our simplistic use of the CAK solution. As noted by Proga (2005), this line driving can change the vertical structure of the disc and likely produce a ‘puffed-up’ disc.

7.2 What happens in the innermost disc?

In Section 5, we considered two sets of models which can have rather different implications for mass-loss from the disc. Models in which the work done in unbinding the outflow is negligible (ϵ = 0) can nominally drive almost all of the matter out of the thin disc. However, the energy required to drive all this material to escape velocity can exceed the energy released by the remaining disc material which accretes to the centre (e.g. the 10⁷ M_⊙ models with |$\dot{m}_{\rm out}= 2$|⁠, see Fig. 3). Hence, there must be a substantial failed wind in this scenario and most of the material in this failed wind must eventually accrete, although possibly not in the form of a standard, thin disc.

In the second set of models with ϵ = 1, we account for the energy lost by radiation in unbinding the outflow to compute F and use this to evaluate the |$\dot{\Sigma }$| relations. In this case, the mass-loss is limited by the corresponding reduction in F. The mass outflow rate can still be quite large (∼90 per cent of |$\skew4\dot{M}$| at a large radius), but the accreted mass is significantly larger relative to the ϵ = 0 models. All the energy needed to accelerate the outflow to escape velocity is accounted for, and there is no need for a failed wind to form a hotter, radiatively inefficient flow.

Although the latter models have the appeal of being explicitly energy conserving, the numerical simulations discussed above (e.g. Proga et al. 2000) suggest that winds may not operate in this fashion. They are predominately accelerated by the radiation from annuli interior to their launching radius, so the assumption of a constant ϵ is likely a poor approximation. More importantly, the winds behave (in certain respects) more like the ϵ = 0 models, in that they launch more matter from the thin disc than they accelerate to infinity and forming a ‘failed wind’. These simulations assume a constant |$\skew4\dot{M}$| AD as a boundary condition, and find outflow rates as high as 50 per cent of this prescribed |$\skew4\dot{M}$|⁠. In principle, even higher outflow rates could be obtained, but such models have not been considered due to the assumed lack of feedback on the boundary condition in the models used (Proga, private communication).

Our best guess is that a real system would behave like some combination of the above models: a significant fraction of the radiative flux will go into accelerating some fraction of the outflow that does exceed escape velocity and becomes a wind, but not all of the mass removed from the thin disc will become unbound. Much of it may form a failed wind that returns to the disc or accretes through a hotter, geometrically thick flow.

The distinction between the thin disc and the outflow probably breaks down when a non-negligible fraction of the disc is no longer in hydrostatic equilibrium. Since the calculations presented in Section 5 do not account for the detailed vertical structure, the radial distributions of T_eff and |$\skew4\dot{M}$| in regions where |$\skew4\dot{M}_{\rm wind}(r) \sim \skew4\dot{M}(r_{\rm out})$| should be interpreted with this in mind. If most of the material lifted from the disc falls back, say due to overionization, then it must eventually all accrete to the centre. In this case, observations would seem to require a hot, radiatively inefficient flow, since substantial thermal EUV emission is not seen. The bulk of ‘fallback’ flow must remain hotter and thicker than the standard solution, cooling primarily via inverse-Compton scattering and contributing significant radiation only in the X-ray band. In this scenario, line driving would result in a transition from a thin, radiatively efficient disc to a geometrically thick, radiatively inefficient flow.

A low radiative efficiency can occur in hot flows if the optical depth is very low or very high (see e.g. Abramowicz et al. 1988, 1995; Narayan & Yi 1994, 1995). When the optical depth is large, the radiation is advected inwards inside a thick disc, when the radiation diffusion time-scale is longer than the infall time-scale. This effect is expected to be present in discs with |$\dot{m}>1$|⁠, where the AD becomes slim, rather than thin (Abramowicz et al. 1988). Interestingly, all objects in DL11 with an observed radiative efficiency below 2 per cent have measured |$\skew4\dot{M}$| which corresponds to an AD with |$\dot{m}\sim 3\hbox{--}30$| for a 10 per cent efficiency. So, advection of radiation may indeed suppress the emission from the innermost disc. However, the SED of such a high |$\dot{m}$| disc typically peaks well shortwards of 1000 Å, so it is inconsistent with the observed λ ∼ 1000 Å turnover. Another process is responsible for the universal λ ∼ 1000 Å turnover.

The unbound material in the inner AD must remain hot enough to avoid emitting significantly in the observable UV band. What is the expected emission from this hot inner region? Can it form the thick and hot inner structure often invoked as a source for the X-ray emission? Let us assume that the X-ray emission is produced in the inner AD region through Comptonization of the incident thin disc emission which comes from r > r_eq, where the thin disc resides. The Compton cooling takes place only in a τ_es ∼ 1 surface layer of the illuminated hot and thick configuration at r < r_eq. So, only a small fraction of the inner disc volume will cool radiatively, while the rest of the dissipated energy should be advected inwards. The estimated X-ray luminosity of this surface layer can be derived based on the thermal energy stored in this layer divided by the cooling time. The likely projected surface area of the inner hot disc is A = cos(θ)2πr², where r ∼ r_eq (equation 12) and cos(θ) ∼ 0.1 is the illumination angle of the external radiation, which yields |$A=1.6\times 10^{29} m_8^{1.04}\dot{M_1}^{0.48}$| cm². The illuminated column density corresponds to τ_es ∼ 1. The electron temperature is T ∼ 10⁹ K (from the spectral slope of −1 for Comptonization; Rybicki & Lightman 1979, equation 7.45b). Thus, the total thermal energy of the electrons in the illuminated layer is |$E=A\Sigma kT=5\times 10^{46}m_8^{1.04}\dot{M_1}^{0.48}$| erg. The Compton cooling time of electrons embedded in a blackbody at T = 10⁵T₅ is |$t_{\rm C}=81.5 T_5^{-4}$| s (e.g. Laor & Behar 2008, equation 53). Approximating the disc emission as a blackbody at T_max (equation 14) gives a cooling rate of the illuminated layer of |$E/t_{\rm C}=L_{\rm X}\sim 10^{44}\dot{M_1}^{0.76}m_8^{0.48}$|⁠. How does it compare with the bolometric luminosity? The expected L_bol from the geometrically thin AD, which extends down to r_eq, is |${\sim } \skew4\dot{M}c^2/r_{\rm eq}$| or |$L_{\rm bol}\sim 10^{45}\dot{M_1}^{0.76}m_8^{0.48}$|⁠. We thus get L_X ∼ 0.1L_bol, which is close to the typical ratio observed in AGN.

To summarize, the radius of the inner thick disc, the assumption that it is maintained at T ∼ 10⁹ K by the viscous dissipation, the implied Compton cooling time based on the incident radiation from the thin AD and the thermal energy content of the cooling surface layer of the inner hot disc happen to combine together and give a constant L_X/L_bol, of the order of magnitude observed.

If the inner X-ray source is produced by a breakdown of the thin disc solution due to line opacity driving, while the X-ray source is significant enough to quench the disc wind, this may lead to an instability in the X-ray and EUV emission, and an anticorrelation between the two bands. The inner AD may switch back and forth from a thermal EUV emitting thin AD state, which develops a strong wind and turns into a thick and hot X-ray emitting configuration, which shuts off the wind, and turns back to the thin thermal EUV emission state, which again develops a strong wind.

7.3 What happens at lower masses?

With decreasing M the disc gets hotter, the wind starts at a larger r (equations 12 and 21) and the radiative efficiency is expected to get lower (see also η ∝ M^0.5; DL11). This may partly explain why accreting M ≤ 10⁶ M_⊙ systems are rare. A wind launched at a larger r may find it easier to escape, and low-M black holes may find it harder to grow by accretion. However, at a low enough M, r_eq will move out into the gas pressure-dominated regime of the disc. In this regime, the disc is vertically supported by the gas pressure, so an increase in radiation pressure due to Γ > 1 does not necessarily lead to a significant change in the vertical structure. A value of Γ > 10³ close to the disc surface will most likely overcome gravity and drive a wind. However, in contrast with the uniform vertical density profile of the radiation-dominated part of the disc, in the gas-dominated part the density drops steeply with height, and thus the density at the sonic point, which feeds the base of the wind, may be significantly lower, lowering |$\skew4\dot{M}_{\rm wind}$|⁠.

Another effect which comes in with decreasing M is that T_max increases, reaching >2 × 10⁵ K for M = 10 M_⊙ (equations 15 and 23). At this temperature, Γ likely decreases due to overionization of the primary atomic UV absorbers, which will also reduce |$\skew4\dot{M}_{\rm wind}$|⁠.

Observations of CV which harbour AD around white dwarfs yield winds with |$\skew4\dot{M}_{\rm wind}\ll \skew4\dot{M}$| (e.g. Feldmeier, Shlosman & Vitello 1999), despite the fact that the CV AD peaks in the UV. However, these systems are characterized by |$\dot{M_1}\sim 10^{-8}$|⁠, m₈ ∼ 10⁻⁸ and r_in ∼ 10⁴. Plugging these values into equation (11) gives |$\skew4\dot{M}_{\rm wind}/\skew4\dot{M}\sim 10^{-2}$|⁠, which is likely an overestimate as the AD in CV is gas pressure rather than radiation pressure dominated. Thus, CV AD have only weak winds, despite their peak UV emission and the associated high Γ values, due to their large r_in (see equation 11).

In X-ray binary (XRB) systems, the disc is much more compact, reaching T ∼ 10⁷ K. Therefore, the gas becomes fully ionized and line opacity is negligible. If the outer disc extends far out enough, it can reach the Γ ≫ 1 regime, this time from the other side of the opacity barrier, probably at T < 10⁶ K, where the line opacity starts to build up, which may also produce a wind, this time from the outer disc, rather than the inner disc. The wind is expected to move inwards in the low hard state, as the disc gets cooler, which may disrupt the thin disc formation down to the centre.

For an M < 10⁶ M_⊙ black hole radiating near Eddington, the inner regions of an SS73 disc reach T > 10⁶ K, and should emit predominantly in the soft X-rays (Done et al. 2012). At this temperature, the ionization may be high enough to reduce Λ and allow a thin disc solution with no wind. Further out the disc will be colder, and significant mass-loss is likely to occur, which may strongly suppress |$\skew4\dot{M}$| which can arrive to the centre. However, if a fraction of L_bol, which peaks at soft X-rays, is intercepted farther out in the disc, wind launching may be quenched further outside. Such a scenario may explain some low-mass (M < 10⁶ M_⊙) narrow-line Seyfert 1s with large soft X-ray ‘excesses’, such as RE J1034+396 (Done et al. 2012).

7.4 Can a_* be measured from the SED and the Fe Kα line?

Since the line-driven wind effectively terminates the thin disc well outside r_ISCO, the value of a_* does not affect the thin disc SED. Thus, in contrast with XRB, we generally do not expect the AGN SED to provide a useful constraint on a_*. However, for a sufficiently cold disc, with a turnover at λ > 1000 Å, the wind disappears. This may explain the remarkably good fit of a simple local BB AD model to SDSS J094533.99+100950.1 (Czerny et al. 2011; Laor & Davis 2011), which is a weak line quasar with a UV turnover at λ ∼ 2000 Å (Hryniewicz et al. 2010). We therefore expect that other cold AD quasars, i.e. quasars with a blue SED at λ > 3000 Å (which excludes dust reddening) and a UV turnover at λ > 1000 Å, can also be well fitted by a simple local BB AD model. One may therefore be able to constrain a_* based on the SED in these quasars, as commonly done now in XRB (e.g. Li et al. 2005; Shafee et al. 2006; McClintock et al. 2011).

Similar issues may apply to efforts to measure a_* via models of the Fe Kα line and ionized reflection. The numerical solutions indicate that an optically thick disc extends down to r_ISCO. However, unless the disc is cold, only a small fraction of |$\skew4\dot{M}$| extends down to r_ISCO. The numerical solution ignores the material which left the thin disc, which may form a thick configuration in which the thin disc is embedded, if it exists at all. A lack of a thin cold bare disc which extends down to r_ISCO will affect the expected profile of the fluorescence Fe Kα line, produced by X-ray reflection from the AD. The line may be dominated by emission outside r_eq, which will make the line narrower, and its profile independent of the value of a_*. Line emission inside r_eq will depend on the gas temperature, and the line profile may be modified by possible scattering effects at a thick and hot surface layer. In any case, the emission is not expected to originate from a thin bare AD, as commonly assumed, as a thin bare disc which extends down to r_ISCO is simply not observed. We do predict that the colder the disc is, the further it extends inwards, and the broader the Kα line can be.

7.5 What is the effect on the characteristic half-light radius?

The outflow changes the radial distribution of the emitted flux in different frequency bands. In particular, the reduction of the FUV emission relative to the SS73 solution leads to an increase in the half-light radius for the optical-to-UV bands. This is because the Rayleigh–Jeans tail of the emission from the UV peaked regions of the disc contributes a non-negligible fraction to the overall SED at longer wavelengths. Removing this emission or shifting it to X-ray wavelengths increases the fraction of the long-wavelength emission from larger radii.

This effect is interesting in light of claims based on microlensing analysis of lensed quasars that the optical-to-UV emission comes from radii which are factors of ∼3–10 larger than expected from the standard thin disc model (e.g. Mortonson, Schechter & Wambsganss 2005; Pooley et al. 2007; Morgan et al. 2010). However, the increases in the half-light radius (which the microlensing results are claimed to measure) that we infer are typically <30 per cent at 2000 Å and <10 per cent at 4000 Å relative to the models with no outflow, so this effect cannot account for the large discrepancies that are claimed. A caveat is that we have not considered the reprocessing of the thin disc emission by the outflow. For such large mass-loss rates, the wind will likely be optically thick to Thomson scattering in the radial direction (Sim et al. 2010). Some fraction of this will be scattered downwards and be reprocessed by the disc at larger radii. This reprocessed emission should increase the half-light radius at longer wavelengths, although a more detailed calculation is required to estimate the possible magnitude of the effect.

7.6 Predictions

The inward extent of the AD can be probed through the position of the UV turnover. The |$\dot{\Sigma }(F)$| wind relation generally predicts a turnover at λ_peak ∼ 1000 Å. However, there is still some residual dependence on |$\dot{m}$| and m₈, possibly in the form of |$T_{\rm max}\propto (\dot{m}/m_8)^{0.07}$| (equation 15), which should be observationally detectable. The quantity |$\dot{m}/m_8$| can be estimated based on the broad emission lines, which gives |$M=v_{\rm BLR}^2 R_{\rm BLR}/G$|⁠, where the Hβ full width at half-maximum is used to derive v_BLR. In addition, R_BLR ∝ L^0.5 based on reverberation mappings and the above relations imply |$\dot{m}/m_8\propto v_{\rm BLR}^4$|⁠. Thus, the |$\dot{\Sigma }(F)$| wind relation (equation 15) implies that |$T_{\rm max}\propto v_{\rm BLR}^{0.28}$| or |$\lambda _{\rm peak}\propto v_{\rm BLR}^{-0.28}$|⁠. A factor of 10 increase in v_BLR will therefore be associated with a decrease by factor of 2 in λ_peak. This is true as long as the disc is not too cold to become windless, i.e. when v_BLR is not too large ( ≲ 10 000 km s⁻¹; e.g. Laor & Davis 2011). In general, the detection of a correlation of λ_peak with |$\dot{m}$| and m₈ can provide a hint on the specific form of |$\dot{\Sigma }$| in AGN AD.

A clear prediction is a metallicity dependence of λ_peak. Radiation-driven wind models for hot stars predict a close-to-linear relation of Z and |$\skew4\dot{M}$| (e.g. Abbott 1982; Leitherer, Robert & Drissen 1992; Vink, de Koter & Lamers 2001; Kudritzki 2002). Since |$r_{\rm eq}\propto \skew4\dot{M}_{\rm wind}^{0.27}$| for the |$\dot{\Sigma }(F)$| wind relation, and in AD T_max ∝ r^−3/4, we expect that λ_peak ∝ Z^−0.2. Although the expected dependence of λ_peak on Z is weak, it is a robust prediction of line-driven winds, and its detection will provide strong support for the wind interpretation put forward in this paper.

8 CONCLUSIONS

We derive the Navier–Stokes equations for gas on circular orbits in a thin AD, including local mass-loss and angular momentum loss terms. We then solve these equations numerically for a Keplerian disc, with zero torque inner boundary condition. We apply the local mass-loss terms per unit area measured in hot stars, as a function of F and g, at the AD surface. We assume no angular momentum loss induced by the mass-loss. We calculate the derived SED based on the local blackbody approximation, and also using the tlusty stellar atmosphere code. We find the following.

• Line-driven winds put a cap on the AD T_max < 10⁵ K, with a weak dependence on M and |$\dot{m}$|⁠. This cap is consistent with observation of AGN SEDs, which generally show a spectral turnover near 1000 Å.

• In most cases, the thin AD is effectively truncated at a few tens of R_g, well outside r_ISCO. The derived SED is thus independent of the value of r_ISCO, and is therefore independent of the value of a_*.

• In a cold AD, defined by an SED with a turnover longwards of 1000 Å, the line-driven wind is negligible. It may therefore be possible to use the SED of objects predicted to have a cold AD, based on their measured M and |$\dot{m}$|⁠, to derive the black hole a_*.

• The tlusty-based models with winds tend to show a spectral break at λ ∼ 1000 Å, due to a combination of the Lyman edge and the truncation of the hot inner part of the AD due to the wind.

• Depending on the mass-loss prescription, M and |$\dot{m}$| of the model, the material removed from the thin disc either escapes as a wind or forms a failed wind that must accrete. In either case, the thin disc solution of SS73 cannot generally extend to r_ISCO, as the emission from the inner region of a thin disc is generally not observed.

• For a sufficiently large failed wind, the inner disc must be radiatively inefficient. It may form a geometrically thick hot inflow. If the electron temperature can be maintained at T ∼ 10⁹ K, then Compton cooling leads to L_X ∼ 0.1L_bol.

• The |$\dot{\Sigma }(F)$| wind relation implies a radiative efficiency which scales as M^0.5, which agrees with the measured relation (DL11). Thus, the low radiative efficiency of low-M AGN does not imply a low a_*, but may be induced by high mass-loss and the implied large truncation radius of the inner thin AD.

• If the UV turnover is indeed a line-driven mass-loss effect, then the effect is necessarily Z dependent. Higher Z objects should show a larger λ_peak, i.e. a softer ionizing SED at a given M and |$\skew4\dot{M}$|⁠.

Rather detailed wind simulations are available, and it will be interesting to explore the radiation transfer through the wind/failed wind, and its possible impact on the observed SED. However, a major uncertainty in wind models remains concerning the vertical structure of the disc, in particular the top layer from which the wind is launched. The structure depends on the exact nature of the turbulence and how it is dissipated. Thus, one cannot yet derive estimates for |$\skew4\dot{M}_{\rm wind}(r)$| from first principles. We went around this major uncertainty by adopting |$\skew4\dot{M}_{\rm wind}(r)$| derived based on |$\skew4\dot{M}_{\rm wind}(g, T_{\rm eff})$| of O stars. Although the models adopted here are rather simplified, they yield SEDs remarkably similar to those generally observed, without significant sensitivity to the details of the launching mechanism. However, the structure of the innermost disc and its possible feedback, in particular when the implied outflow becomes comparable to the accretion rate, need to be further explored.

One possible feedback is X-ray irradiation of the innermost disc which, if strong enough, may ionize the surface disc layer to a level which will quench the line-driven wind. If the hot inner disc is indeed formed by a line-driven failed wind, then quenching the wind may quench the X-ray source, which will allow the wind to form again.

Despite the associated uncertainty in the above analysis, a plausible statement one can make is that in AGN, as in O stars, ‘a static atmosphere is not possible’ (CAK, section II there). The true structure of the inner AD remains to be understood.

We thank the referee, Ramesh Narayan, for a critical and constructive review, which significantly improved the paper. We also thank Oren Slone for pointing out that applying |$\skew4\dot{M}(r)$| to the SS73 solution is invalid. We thank Norm Murray, Daniel Proga and Ramesh Narayan for helpful discussions. This research was supported by the Israel Science Foundation (grant No. 1561/13). SWD is grateful for financial support from the Beatrice D. Tremaine fellowship.

REFERENCES

APPENDIX A: RELATIVISTIC AD WITH OUTFLOW