Annu. Rev. Astron. Astrophys. 2015. 53:115-154 Copyright © 2015 by Annual Reviews. All rights reserved

7. MOMENTUM, ENERGY OR RADIATION?

The study of AGN outflows and their effects on the host galaxy has two main aims. A viable picture must explain both the scaling relations, and simultaneously the fact that galaxy spheroids appear ultimately to be largely swept clear of gas by high–speed molecular outflows which have significantly greater scalar momenta v than the radiative value L / c of the central AGN (the clearout problem). The discussion given above offers plausible physical grounds that the shock interaction characterising the black hole wind feedback changes from momentum–driven, acting on small spatial scales near the black hole, to energy–driven, instead acting globally on the whole galaxy bulge and producing a high–energy clearout of its gas. The M − σ relation marks the point where this transition occurs in a given galaxy. We will argue below (Section 8) that observations support this picture of local–global transition in several ways, but before accepting this conclusion we should consider other possibilities.

First, the switch from momentum to energy–driving depends on the details of gas cooling. It is sometimes argued (e.g. Silk & Nusser 2010) that strong cooling of the ambient interstellar medium enforces momentum–driving by the central SMBH throughout. In fact cooling the ambient gas is not relevant to the question of energy or momentum driving: as we have seen, it is the cooling of the shocked wind gas which decides this. But as we emphasised in Section 4, at least some of the physics of the suggested momentum–energy switch is still beyond a full numerical treatment with realistic parameters. It is sensible then to check our treatment above by considering the momentum–driven and energy–driven cases in isolation, and then the effect of direct radiation pressure.

7.1. Wind Momentum Driving

We first simply assume that a black hole wind acts on its surroundings by pure momentum–driving alone, at all radii. For generality we let the pre–shock wind have speed v_w and take its mechanical luminosity _w v_w² / 2 as a fixed fraction a of L_Edd, i.e. we do not explicitly assume that the wind has the Eddington momentum, as seems to hold for UFOs. Then the momentum feedback first becomes important at a critical black hole mass M_crit roughly given by equating the wind thrust _w v_w = 2a L_Edd / v to the weight

(73)

of the overlying gas in an isothermal potential (cf equation 37). With L_Edd = 4GM_crit c / κ we get

(74)

By definition a = _w v_w² / 2L_Edd and L_Edd = η_Edd c², so

(75)

(cf Fabian 1999). We see that for general wind parameters the critical mass differs from M_σ. We find M_crit = M_σ only if v_w = η c _Edd / _w, which is equation (22). This immediately implies that the wind momentum is Eddington, i.e. _w v_w = η_Edd c = L_Edd / c. In other words, assuming pure momentum driving gives the critical mass as M_σ if and only if the driving wind has the Eddington momentum, i.e. has the properties observed for UFOs.

But pure momentum driving is unable to drive off the bulge gas without a significant increase of the black hole mass above M_crit. Several authors have reached this conclusion (cf Silk & Nusser 2010, McQuillin & McLaughlin 2012). Moreover, if galaxy bulges accrete at the rates suggested by cosmological simulations it seems unlikely that any hypothetical momentum–driven outflows would have enough thrust to prevent infall and so could not turn off star formation definitively (cf Costa et al. 2014). We conclude that pure momentum–driving, even given the lack of a likely shock cooling process, probably does not give a realistic picture of the interaction between SMBH and their hosts.

7.2. Wind Energy Driving

The direct opposite case from that considered in the last subsection is pure energy–driving by winds, where radiative cooling is assumed negligible throughout. This was often the implicit assumption in early treatments (e.g. Silk & Rees 1998, Haehnelt et al. 1998). The equation of motion for this case is (46). This shows that gas is always driven out at constant speed for any SMBH mass, however small: setting R = v_e t and using the definition of L_Edd (cf equation 17) gives the speed v_e as

(76)

This expresses the fact that the adiabatically expanding shocked gas pushes the interstellar gas away as a hot atmosphere for any SMBH mass. One can easily find the corresponding mass outflow rate by setting _out v_e² / 2 ∼ L_w, since we know that the outflow mechanical luminosity is a significant fraction of the driving wind mechanical luminosity L_w = η lL_Edd / 2.

To define a critical SMBH mass for energy–driven outflow one has to impose a further condition. This is usually taken as v_e ∼ σ, defining some kind of escape velocity. But it is not obvious that this is appropriate: the outflow is driven by pressure, so the ballistic escape velocity is not relevant. Even if the AGN switches off, the residual gas pressure still drives outflow for a long time (cf Fig. 9). If we nevertheless impose this condition we find a critical mass

(77)

which is a factor 3σ / η c ∼ 1/50 too small in comparison with observations.

Silk & Rees (1998) considered the growth of a protogalaxy (i.e. gas with f_g ∼ 1) around a supermassive seed black hole which formed earlier, but their argument applies to the coevolution of the SMBH and host also, provided we take f_g ∼ 0.1. They assume the wind sweeps mass into a shell with speed v_s, and implicitly neglect pressure work, and the fact that energy is shared between the shocked wind and the swept–up outflow. This would imply a relation

(78)

as each new shell of mass 4π r²ρ(r) v_s now simply acquires kinetic energy v_s² / 2 as it is swept up. Using the isothermal relation (34) and requiring v_s ∼ σ gives

(79)

where f_w = L_w / L_Edd. The neglect of pressure work overestimates the wind–driving efficiency, so this mass is even smaller than (77). It is clear that wind energy–driving of this type does not correctly reproduce the M − σ relation, giving a critical mass too low by factors 50 – 100.

A more promising approach has recently been outlined by Nayakshin (2014), Zubovas & Nayakshin (2014) and Bourne et al. (2014), who consider the effects of strong inhomogeneity of the bulge gas. They assume first that inverse Compton shock cooling may not be effective because of two–temperature effects (but see Section 4.2 above). Second, they suggest that sufficiently dense clouds of interstellar gas would feel a net outward force ∼ ρ v² per unit area when overtaken by a free–streaming UFO wind of preshock density ρ and speed v, thus mimicking a momentum–driven case. The density of these clouds is a factor 1/f_g ∼ 6 below the star–formation threshold. If most of the ISM gas mass is in the form of such clouds, SMBH feeding must stop when the outward force overcomes gravity, which gives a relation like (39) up to some numerical factor. This idea throws up several gas–dynamical problems. First, a cogent treatment must explain the origin and survival of clouds at densities close to but just below the star formation threshold, which must contain most of the interstellar gas. The clouds must be completely immersed in the wind, so the net outward force on them is a surface drag, which is dimensionally also ∼ ρ v² per unit area. Estimating this surface drag requires knowledge of how the cloud–wind interfaces evolve on very small scales. Since these are formidable theoretical tasks, we should ask for observational tests. The main difference from the quasispherical momentum–driven case is that instead of being radiated away, most of the energy of the UFO wind now continuously drives the tenuous intercloud part of the ISM out of the galaxy at high speed. If this tenuous gas is a fraction f_t of the total gas content, equations (50, 57) show that for SMBH masses not too far from M_σ this outflow should have speed v_out ∼ 1230 f_t^−1/3 km s⁻¹ and mass–loss rate M_out ∼ 4000f_t M_⊙ yr⁻¹, and so be potentially observable for many AGN spheroids. From the work of Section 5.4 one might also expect a continuously elevated star formation rate in the central parts of their galaxy discs also, which is not in general observed.

7.3. Cosmological Simulations

Cosmological simulations often produce an empirical M − σ relation as part of much larger structure formation calculations. Limits on numerical resolution inevitably require a much more broad–brush approach then adopted here. The effect of the SMBH on its surroundings is usually modelled by distributing energy over the gas of the numerical ‘galaxy’ at a certain rate. This injected mechanical luminosity is then iterated until the right relation appears. This empirical approach (e.g. di Matteo et al. 2005) seems always to require a mechanical luminosity 0.05L_Edd to produce the observed M − σ relation. This is precisely what we expect (cf equation 23) for a black hole wind with the Eddington momentum _out v = L_Edd / c.

But the success of this procedure is puzzling. If the ambient gas absorbed the full numerically injected mechanical luminosity 0.05L_Edd the resulting outflows would give the energy–driven (32) or Silk–Rees mass (79) above, which are too small compared with observations. The fact that cosmological simulations instead actually iterate roughly to the observed M − σ value (42) suggests that they somehow arrange that the injected energy only couples to the gas at the very inefficient rate which occurs in momentum driving, or possibly that the numerical gas distribution is highly inhomogenous. The real physics determining this in both cases operates at lengthscales far below the resolution of any conceivable cosmological simulation, so the inefficiency must be implicit in some of the ‘sub–grid’ physics which all such simulations have to assume (cf Costa et al. 2014, Appendix B).

7.4. Radiation Driving

7.4.1. Electron scattering opacity

We remarked in the Introduction that in principle direct radiation pressure is the strongest perturbation that a black hole makes on its surroundings, but its effects are more limited than this suggests. As we already suggested, the reason is that in many situations radiation decouples from matter before it has transferred significant energy or momentum. This is particularly likely for radiation emitted by an AGN in the center of a galaxy bulge. The gas density (cf equation 34) is sharply peaked towards the center, and so is its tendency to absorb or scatter the radiation from the accreting black hole. The electron scattering optical depth from gas outside a radius R for example is

(80)

which is mostly concentrated near the inner radius R. This means that gas initially close to the AGN is probably swept into a thin shell by its radiation, and so at radius R has optical depth

(81)

very similar to the undisturbed gas outside R (cf equation 80). Gas distributed in this way has large optical depth near the black hole when its inner edge R is small (i.e. less than the value R_tr specified in equation 82 below). Then the accumulating accretion luminosity L of the AGN is initially largely trapped and isotropized by electron scattering, producing a blackbody radiation field whose pressure grows as the central AGN radiates. This growing pressure pushes against the weight W = 4f_gσ⁴ / G (equation 73) of the swept–up gas shell at radius R. This is exactly like the material energy–driving we discussed in Section 5.3, except here the photon ‘gas’ has γ = 4/3 rather than γ = 5/3 there. Clearly the effectiveness of this radiation driving is eventually limited because the shell’s optical depth falls off like 1/R as it expands. The force exerted by the radiation drops as it begins to leak out of the cavity, until for some value τ_tot(R) ∼ 1 it cannot drive the shell further.

This shows that the sweeping up of gas by radiation pressure must stop at a ‘transparency radius’

(82)

where (up to a logarithmic factor) the optical depth τ_tot is of order 1, so that the radiation just escapes, acting as a safety valve for the otherwise growing radiation pressure. This process is discussed in in detail by King & Pounds (2014), who suggest that the stalled gas at R_tr may be the origin of the ‘warm absorber’ phenomenon (cf Tombesi et al. 2013). The radius R_tr is so small that very little accretion energy is needed to blow interstellar gas to establish this structure, and to adjust it as the galaxy grows and changes σ.

7.4.2. Dust

At larger radii much of the cold diffuse matter in the galaxy bulge may be in the form of dust. The absorption coefficient of dust depends strongly on wavelength and is far higher than electron scattering in the ultraviolet, but decreases sharply in the infrared (e.g. Draine & Lee, 1984). The energy of an ultraviolet photon absorbed by a dust grain may be re–emitted almost isotropically as many infrared photons, which then escape freely. The net effect is that dusty gas feels only the initial momentum of the incident UV photon, while most of the incident energy escapes. Then a spherical shell around an AGN would experience a radial force ≃ L/c, where L is the ultraviolet luminosity, as long as it remained optically thick to this kind of dust absorption. This is dynamically similar to wind–powered flows in the momentum–driven limit, and this type of radiation–powered flow is often also called ‘momentum–driven’, even though the physical mechanism is very different.

An important distinction between the wind and radiation–powered cases is that ambient gas in the path of a wind cannot avoid feeling its effects, whereas this is not true for radiation, as the gas may be optically thin. Galaxies are generally optically thin to photons in various wavelength ranges, and a radiation–driven shell may stall at finite radius because its optical depth τ becomes so small that the radiation force decouples, as we saw in the electron–scattering case. Ishibashi & Fabian (2012, 2013, 2014) appeal to this property to suggest that star formation in massive galaxies proceeds from inside to outside as radiation–momentum driven shells of dusty gas are driven out and then stall at the dust transparency radius R_dust ≃ (κ_d/κ) R_tr. For large dust opacities κ_d ∼ 1000κ this can give R_dust ∼ 50 kpc. In contrast galaxies are probably never ‘optically thin’ to winds, and the density of a black hole wind is always diluted as 1/R², so it inevitably shocks against a swept–up shell of interstellar gas at large R.

The mathematical similarity (cf eq 37) between wind–powered and radiation–powered momentum driving allows an empirical estimate of an M − σ relation for the latter if we assume that observed AGN define the relation, and that their observed luminosities correspond to L / L_Edd ∼ 0.1 − 1. This gives M_crit = (L_Edd / L)M_σ ∼ 1 − 10 M_σ (Murray et al. 2005). Optical depth effects might narrow this range closer to the observed one (Debuhr et al. 2012). This suggests that radiation driving might be compatible with the M − σ relation, but a momentum–driven outflow like this can never simultaneously reproduce the high–velocity molecular outflows characterising the clearout phase. In particular its momentum is L / c < L_Edd / c, considerably smaller than the observed ∼ 20L_Edd / c of such flows (see Section 5.3). In other words, we have the usual difficulty that momentum–driving can accommodate the M − σ relation, but not simultaneously solve the clearout problem.

One way of possibly overcoming this (e.g. Faucher-Giguère et al. 2012; Faucher-Giguère & Quataert, 2012) is to assume (cf Roth et al. 2012) that instead of degrading incident high–energy photons to lower–energy ones that escape freely, the effect of dust absorption is to retain much of the incident radiant energy. Then if the dust is distributed spherically and is in a steady state the radiation force on it is τ L / c, where τ is the radial optical depth of the dust (cf Roth et al. 2012). This form of radiation driving of optically thick dust can in principle produce outflows whose scalar momenta are boosted above that of the driving luminosity L / c by a factor ∼ τ because photons may be reabsorbed several times. For τ ≫ 1 the radiation field is effectively trapped and presumably approaches a blackbody form (cf the discussion of the electron scattering case above), limiting the boost.

Evidently for radiation driving of dust to solve simultaneously both the SMBH scaling and clearout problems requires a sharp transition in the properties of the dust opacity at the critical M − σ mass. This must change the outflows from effectively momentum–driven (incident photons are absorbed but their energy escapes as softer photons) to energy–driven (incident photons trapped) at this point, in a switch analogous to the turnoff of Compton cooling in the wind–driven case. There have so far been no suggestions of how this might happen, but the physics of dust opacity is sufficiently complex that this is perhaps not surprising.