## Abstract

Orientation of parsec-scale accretion discs in active galactic nuclei (AGNs) is likely to be nearly random for different black hole feeding episodes. Because AGN accretion discs are unstable to self-gravity on parsec scales, star formation in these discs will create young stellar discs, similar to those recently discovered in our Galactic Centre. The discs blend into the quasi-spherical star cluster enveloping the AGN on time-scales much longer than a likely AGN lifetime. Therefore, the gravitational potential within the radius of the black hole influence is at best axisymmetric rather than spherically symmetric. Here we show that, as a result, a newly formed accretion disc will be warped. For the simplest case of a potential other mechanisms of accretion disc warping, have a direct relevance to the problem of AGN obscuration, masing warped accretion discs, narrow Fe Kα lines, etc.

## Introduction

Observations show that a significant fraction, perhaps a majority, of active galactic nuclei (AGNs) of different types are obscured by a screen of a cold dusty matter, thought to be a molecular torus-like structure with a scale between 1 and 100 pc (Antonucci & Miller 1985; Antonucci 1993; Maiolino & Rieke 1995; Risaliti, Maiolino & Salvati 1999; Jaffe et al. 2004; Sazonov & Revnivtsev 2004). Moreover, many lines of observational evidence suggest that the unobscured AGNs have similar torii as well, but we are viewing these AGNs at an angle allowing a direct view of their central engines. The orientation-dependent obscuration is thus said to unify the different AGN classes (Antonucci & Miller 1985; Antonucci 1993).

Despite the observational importance, the theory of molecular torii is still in an exploratory state, which is indicative of the difficulty of the problem (Krolik & Begelman 1986, 1988; Pier & Krolik 1992; Yi, Field & Blackman 1994; Vollmer, Beckert & Duschl 2004). Krolik & Begelman (1988) have shown that the large geometrical thickness of the torus and yet its apparently low temperature are consistent only if the torus is made of many molecular clouds moving with very high random velocities (Mach number ≳30). Given that stellar feedback processes appear inefficient to explain these large random speeds, the authors were ‘forced to seek a much more speculative solution: viscous heating of the cloud system due to partially elastic cloud–cloud collisions.’ However, until now there has been no definitive answer (via numerical simulations) on whether clouds colliding at large Mach numbers would behave even partially elastic, or would rather share and dump the random velocity component and collapse to a disc configuration.

Wada & Norman (2002) took issue with the assertion of Krolik & Begelman (1988) of the weakness of the stellar processes. Considering ‘large’ (e.g. 100 pc) torii and using numerical simulations of supernova explosions resulting from star formation in a self-gravitating, massive disc, they have shown that large random cold gas velocities result from the interaction of the gas with the supernova shells. However, as such, about 99 per cent of the gaseous disc in the simulations is consumed in the star formation episode rather than being accreted by the black hole. Unless most of the stars formed in the disc are later somehow accreted on to the supermassive black hole (SMBH), this mechanism can only work for torii on scales much larger than the SMBH gravitational sphere of influence, *R*_{h}, i.e. where the stellar mass is much greater than *M*_{BH}. It is also not clear whether this mechanism would work for smaller accretion/star formation rates, because then supernova explosions become too rare. Finally, the time variability of obscuring column depths argues for much smaller radial sizes of the ‘torus’, e.g. in the range 0.1–10 pc (Risaliti, Elvis & Nicastro 2002).

Here we would like to emphasize that accretion discs in AGNs are very unlikely to be planar. There are several mechanisms that are capable of producing strong warps. We believe that such warps have to be an integral part of the AGN obscuration puzzle. In particular, we describe and calculate a disc warping mechanism driven by an axisymmetric gravitational potential.

The sequence of events that takes place in an AGN feeding cycle in our model is as follows. First of all, a merger with another galaxy or a satellite, or another source of cold gas, fills the inner part of the galaxy with plenty of gas. This gas will in general have a significant angular momentum oriented practically randomly. The gas will start settling in a disc that is too massive and too cold to be stable against self-gravity (e.g. Paczynski 1978; Shlosman & Begelman 1989; Collin & Zahn 1999; Goodman 2003). Stars are formed inside the accretion disc, producing a flat stellar system. This long-lived axisymmetric (or perhaps also warped) structure will torque any orbit which is not exactly co-planar with it, resulting in a precession around the symmetry axis. Now, as time goes on, the orientation of the angular momentum vector of the incoming gas changes, and the newly built disc is exposed to the torque from the stellar disc remnant. Different rings of the disc precess at different rates, and thus the disc becomes warped.

We emphasize that the existence of these stellar disc remnants is hardly a question based on the severe self-gravity problems for the standard accretion discs at large radii. Recent observational evidence also supports this conclusion. The best-known example of such flat stellar systems is in our Galactic Centre (Genzel et al. 2003) where two young stellar rings are orbiting the SMBH only a tenth of a parsec away. The orbital planes of these rings are oriented at very large angles to the Galactic plane. Such flat stellar systems are also being found elsewhere (e.g. Krajnovic & Jaffe 2004).

In this paper, we concentrate on the linear gravitational warping effect to investigate its main features. Taking the simplest case of a warp produced by a massive ring, we calculate the gravitational warping torque. Starting from a thin test accretion disc, we calculate its time-dependent shape. Under quite realistic assumptions, the disc becomes strongly warped in some 10^{2}–10^{3} orbital times, i.e. in 10^{5}–10^{7} yr. Because the warping is gravitational in nature, gaseous and clumpy molecular discs alike are subject to such deformations. We speculate that in the non-linear stage of the effect, a clumpy warped disc will form a torus-like structure.

## Torques in a Linear Regime

The basic physics of the effect is very simple and has been explained by, for example, Binney (1992) in his consideration of the galactic warps (most galactic discs are somewhat warped). Consider a nearly circular orbit of radius *R* for a test particle in an axisymmetric potential (e.g. a central point mass plus a flat disc). Let the axis *z* be perpendicular to the plane of the disc. The particle makes radial and vertical oscillations with slightly different frequencies, and the difference is the precession frequency, ω_{p}. Due to the symmetry, the *z*-projection of the angular momentum of the particle is exactly conserved (see, for example, section 3.2 in Binney & Tremaine 1987). Thus, the angle between the angular momentum of the particle and the *z*-axis, β, is a constant. However, the line of the nodes (the line over which the two planes intersect) precesses.

Now consider a ‘test particle’ disc, initially in the same plane as the circular orbit. The disc is a collection of rings, i.e. circular orbits. Because precession rates ω_{p} are different for different *R*, rings turn around the *z*-axis on unequal angles. The initially flat disc will be warped with time.

### Gravitational torques between two rings

Let us calculate the gravitational torque between two rings with radii *R*_{1} and *R*, inclined at angle β with respect to each other. We work in two rigid coordinate systems, *x*, *y*, *z* and *x*′, *y*′, *z*′, the centres of which are at the SMBH. For convenience, we place the first ring in the *z* = 0 plane, whereas the second ring is in the *z*′= 0 plane. The angle β is obviously the angle between the axes *z* and *z*′. Further, the axes *x* and *x*′ are chosen to coincide with the line of the nodes.

The total torque exerted by the second ring on the first is

where σ_{1}=

*M*

_{1}/2π

*R*

_{1}and σ=

*M*/2π

*R*, with

*M*

_{1}and

*M*being the masses of the two rings, respectively. The integration goes over angles φ

_{1}and φ, which are azimuthal angles in the respective frames of the rings. From this expression, it is immediately clear that co-planar rings do not exert any torque on each other, [

*r*_{1}

**×**

**]= 0. In addition, if β=π/2, the integral vanishes as well because for each**

*r*

*r*_{1}the opposites sides of the second ring (e.g. φ and φ+π) make equal but opposing contributions.

It is also possible to show that due to symmetry the torque's *z*-projection vanishes. Thus, we only have the τ_{21,x} =−τ_{12,x} component, meaning that the angular momentum vector of the ring will rotate without changing its magnitude. Also, if *M*_{1}≫*M*, then we can neglect warping of the first ring.

The absolute distance between two ring elements with the respective angles φ_{1} and φ is

*I*(δ, β) with and

The total angular momentum of the second ring is *L* =*M*Ω_{K}*R*^{2}. Recall that *L _{z}* =

*L*cos β= const, whereas the component of

*in the plane of the first ring (*

**L***z*= 0) precesses. We have chosen it to be initially in the

*y*-direction, so that

*L*(

_{y}*t*= 0) =

*L*sin β (cf. equation 12). The precession frequency for the second ring is then

In general, the integral in equation (5) is calculated numerically, but for δ≪ 1 we can decompose (1 −δ cos λ)^{−3/2}≈ 1 + 3/2 δ cos λ and obtain *I*(δ, β) ≈ (3/8) δ sin 2β. When this approximation holds, that is when *R*≫*R*_{1} or *R*≪*R*_{1}, the precession frequency for the second ring is

A few estimates can now be made. First of all, ω_{p} reaches a maximum at radius *R*_{m} at which . The maximum growth rate of the warp is thus

*M*

_{1}/

*M*

_{BH}may be expected to be of the order of a per cent or so because this is when the accretion discs become self-gravitating (e.g. Nayakshin & Cuadra 2005), and the resulting stellar mass would probably be of that order too. In this case we notice that to produce a sufficiently large warp, we have to wait for at least ∼10 ×

*M*

_{BH}/

*M*

_{1}∼ 1000 Ω

^{−1}

_{K}or ∼150 orbital times at the radius of the maximum warp

*R*

_{m}. While this is a fairly long time, it is still much shorter than the corresponding disc viscous time (cf. for example, fig. 2 in Nayakshin & Cuadra 2005).

The asymptotic dependence of precession frequency on radius *R* is

### Test disc warping

We now want to calculate the shape of a light non-self-gravitating disc warped by a massive ring *R*_{1}. The disc is treated as a collection of rings with different radii *R* and negligible mass. We assume that the mass of the whole disc, *M*, is negligible in comparison with the mass of the ring, *M*≪*M*_{1}, and that therefore the massive ring's orientation (angular momentum) does not change with time.

In application to the real discs, it should be remembered that any attempt to warp a disc will be resisted by disc viscous forces (e.g. Bardeen & Petterson 1975) that transfer the angular momentum through the disc. The viscous time-scale for such a disc is of the order of *t*_{visc}∼Ω^{−1}_{K}α^{−1}(*R*/*H*)^{2} (Shakura & Sunyaev 1973). We can then neglect viscous redistribution of the angular momentum in the disc if ω_{p}≫*t*^{−1}_{visc}, which is quite possible for thin accretion discs, for which *H*/*R*∼ 0.001–0.01 ≪ 1 for a realistic parameter range (see, for example, fig. 1 in Nayakshin & Cuadra 2005). The anonymous referee of this paper noted that the viscous stress can be significantly different in clumpy accretion discs. These discs are supported by random cloud motions rather than by gas thermal pressure so *H*/*R* is far larger. Viscosity is established mainly via cloud–cloud collisions (e.g. Krolik & Begelman 1986; Kumar 1999; Vollmer et al. 2004). However, here we are interested in the thin discs which, when warped, may completely eliminate the need for geometrically thick clumpy discs. Nevertheless, even if disc α-viscosity is unimportant for the time-scales considered, the restoring force from self-gravity of the disc may be important. We defer the study of this and other non-linear effects to a future paper.

It is convenient to work with angle β, already introduced, and one additional angle, γ. Recall that angle β is the local angle of the ring's tilt to the *z*-axis and it remains constant in the test particle regime. Angle γ is needed to introduce the projections of the unit tilt vector normal to the ring, *l*(*R*, *t*), on the *x*- and *y*-axis (Pringle 1996):

*z*-axis. With the chosen coordinate system, at time

*t*= 0, γ=π/2 (and we obtain, in accordance with the conventions of Section 2.1,

*L*(0) =

_{y}*L*sin β,

*L*(0) = 0). As each of the disc rings precesses

_{x}It is useful to write the expression for the unit tilt vector in the (*x*′, *y*′, *z*′) coordinate system rigidly bound to the initial disc plane:

*l*′

_{x}=

*l*′

_{y}= 0, e.g. the rings are never tilted with respect to the

*z*=

*z*′ axis, as it should. Also, when γ=π/2,

**′ indeed coincides with the**

*l**z*′-axis, i.e. the disc is flat.

Using equations (14)–(16), we can now calculate the shape of the warped disc in that system given the function γ(*R*, *t*). To accomplish this, we first introduce the azimuthal angle φ on the surface of the ring. The coordinates of the points on the ring, * r*, are then given by equations (2.2) and (2.3) in Pringle (1996). The corresponding coordinates in the primed system of reference are easily obtained by

*x*′= (

*r***′), etc., where**

*e*_{x}**′ and so on are the unit coordinate vectors of the primed system. The result is**

*e*_{x}### An example

To illustrate the results, we calculate the precession rates ω_{p}(*R*) for the following case: *R*_{1} = 2, *M*_{1} = 0.01 *M*_{BH}, β=π/4. The resulting shape of the disc in the original unwarped disc plane is plotted in Fig. 1. The time is in units of Ω^{−1}_{K} at *R* = 1, i.e. *t* = 1 corresponds to time *t* = 1400 *r*^{3/2}_{pc}*M*_{8} yr, where *r*_{pc} is the distance in units of pc and *M*_{8} =*M*_{BH}/10^{8} M_{⊙}. Note that the warp is strongest around radius *R*∼*R*_{1} = 2, as expected. The inner disc is hardly tilted, which is not surprising given that ω_{p}(*R*) → 0 for small *R* (equation 10). The same is true for the outer radii, where the small tilt of the original plane can be seen on the edges of the disc (we do not calculate the tilt beyond *R* = 5.2, and hence the original disc plane is still seen at the edges of the figure).

## Discussion

### Obscuration of the central engine in active galactic nuclei

We believe that the gravitational disc warping due to non-spherical mass distribution within the SMBH sphere of influence is a common occurrence in real AGNs. It is hard to see why AGN discs should always form in one plane, and even why they should be planar when they are born (Phinney 1989). The time-scales for the development of strong warps through the mechanism discussed here can be estimated as follows. Accretion discs become gravitationally unstable and may form stars when the disc mass is of the order of *M*_{min} ≳ (*H*/*R*) *M*_{BH}. We can then assume that the mass of a stellar ring or a disc will be comparable to *M*_{min}. For thin discs, this translates into 0.1–1 per cent of the SMBH mass. The number can be higher for thicker discs, or long-lived star-forming discs. From equation (9), the warp will then grow in hundreds to thousands of orbital times, which, depending on *M*_{BH} and the disc radius, will be in the range of 10^{5}–10^{8} yr, comparable to characteristic AGN lifetimes (which are thought to be in the range of 10^{7}–10^{8} yr). We thus expect that the outer edges of the disc will obscure a significant fraction of the sky as seen from the central source, and hence are directly relevant to the unification schemes of AGNs (Antonucci 1993).

Fig. 2 shows the warped disc surface from Fig. 1 at different times as seen from the origin of coordinates. The vertical axis shows and the horizontal axis shows φ defined previously. The shape of the disc at different times is shown with different shades. At time *t* = 0, the disc is flat and its projection is simply cos θ= 0 for all φ. At later times the rings of the disc near *R* =*R*_{1} = 2 precess and bulge out of the initial disc plane. With time, the fraction of the sky obscured from the central engine becomes greater than half. In reality, non-linear effects and the interaction of the disc with AGN winds and radiation, etc., will become important in shaping the disc for significant warps.

### Non-linear evolution

In general, the non-linear stage of the evolution of the system of discs or rings of stars and molecular gas is far from a thin disc as long as collisions are of minor importance. We have already explored the non-linear evolution of collisionless systems with *N*-body codes and the results will be reported elsewhere. When two massive discs warp each other, mixing occurs when γ−π/2 becomes large, and the resulting configuration is similar to that of a torus.

Now, inelastic dissipative collisions between gas clumps will eventually become important. Relaxation of a collisionless *N*-body system leads to non-circular orbits, and hence different disc rings will start to overlap. Collisions should then tend to destroy the random motions and should establish a common disc ‘plane’. However, such a disc will normally be warped itself. Therefore, inelastic dissipative collisions do not necessarily turn off the obscuration in our model. Further, if molecular gas clumps are constantly supplied from the outside and come with fluctuating angular momentum, the disc may never arrive in a flat thin configuration (see also Phinney 1989).

Nenkova, Ivezić & Elitzur (2002) and Risaliti et al. (2002) convincingly argued that the AGN obscuring medium must not be uniform in density. This does not contradict our model at all, because AGN accretion discs on large (e.g. 0.1 pc and beyond) scales are self-gravitating unless the accretion rates are tiny (e.g. Shlosman & Begelman 1989).

The source of warping potential does not have to be a thin stellar ring or a disc. It may be any non-spherical distribution of stars in the SMBH vicinity that retains a non-zero quadrupole moment; it can also be a second (smaller) SMBH during a merger of two galaxies.

Both the collision-dominated (Krolik & Begelman 1988) and the stellar-feedback inflated torii (Wada & Norman 2002) share a common starting point: the torus is the result of some internal disc physics. If our interpretation applies, AGN torii lend their existence to the way in which the cold gas arrives in the central part of the galaxy. Compared with the model of Krolik & Begelman (1988), large random speeds for the cold gas are not required in our model. Although the gas may be quite high up away from the original plane of the disc, the disc is locally coherent and thin. High-speed elastic collisions between molecular clouds are thus not needed to explain obscuration of AGNs in our model.

There are also two other mechanisms potentially able to produce warped discs at parsec scales, and beyond, around AGNs. As mentioned in the X-ray binaries context, accretion discs develop twists and warps due to instabilities driven by X-ray heated wind off the disc surface (Schandl & Meyer 1994). The same can be achieved by the radiation pressure force from the central source (Pringle 1996). However, it seems that the majority of discs in X-ray binaries are either not warped or warped not strongly enough (Ogilvie & Dubus 2001) to provide the large obscuration needed for the AGN unification schemes. In contrast, the warping mechanism discussed here is not applicable to X-ray binary systems, and hence it may be natural that AGN discs are more strongly warped than X-ray binary discs on appropriately scaled distances from the centre.

### Stellar discs in Sgr A*

The two young stellar discs discovered recently in Sgr A* present a challenge to the usual star formation modes because the gas densities required to avoid tidal shearing are many orders of magnitude larger than the highest densities observed anywhere in the galactic molecular clouds. On the other hand, star formation inside a massive accretion disc is a long-expected outcome of the self-gravitational instability of such discs (e.g. Paczynski 1978; Shlosman & Begelman 1989; Collin & Zahn 1999; Goodman 2003). As such, the young stars in the Sgr A* star cluster are therefore a first example of star formation in this extreme environment. It is very likely that the star formation efficiency and the initial mass function (IMF) are quite different in the immediate AGN vicinity and elsewhere in a galaxy. The ‘astro-archeology’ of Sgr A* can be used to study these issues.

The gravitational warping effect discussed in this paper constrains the time-averaged mass of the outer ring, *M*_{outer}, since the moment of its creation, assumed to be *t* = 2 × 10^{6} yr. The inner stellar ring is rather well defined (Genzel et al. 2003), and we thus estimate ω_{p}*t* for the inner ring to be smaller than π/4. Taking the radius of the inner stellar ring to be *R* = 3 arcsec ≃ 4 × 10^{4}*R*_{S} for the GC black hole, and for the outer, *R*_{1} = 5 arcsec, and using equation (8) with cos β= 1/4, we obtain *M*_{outer} < 10^{5} M_{⊙}. Preliminary numerical *N*-body simulations show that this limit may be even smaller.

### Other implications for active galactic nuclei

There are clearly other observational implications of gravitationally warped discs. For example, narrow Fe Kα lines, observed in many Seyfert galaxies, can be explained with X-ray reflection off such warped cold discs (as earlier suggested by, for example, Krolik, Madau & Zycki 1994; Hartnoll & Blackman 2000). In addition, warped discs will yield different coherent paths for maser amplification than flat discs do, which may be part of the explanation for the complexity of the observed AGN maser emission.

## Conclusions

We have argued that clumpy self-gravitating accretion discs in AGNs are generically strongly warped. We believe such warping should be an integral part of the explanation for the AGN unification schemes. Our model provides arguably the easiest way to obscure the central engine without the need to lift cold gas clouds off the disc plane high up via elastic collisions, supernova explosions or winds.

### Acknowledgements

The author thanks Jorge Cuadra and Walter Dehnen for their help with numerical simulations which motivated this semi-analytical paper, and for very useful discussions. In addition, the author benefited from discussions with Andrew King and Friedrich Meyer. Finally, the anonymous referee is thanked for comments that improved the paper.