Mass and spin coevolution during the alignment of a black hole in a warped accretion disc
Abstract
In this paper, we explore the gravitomagnetic interaction of a black hole (BH) with a misaligned accretion disc to study BH spin precession and alignment jointly with BH mass and spin parameter evolution, under the assumption that the disc is continually fed, in its outer region, by matter with angular momentum fixed on a given direction We develop an iterative scheme based on the adiabatic approximation to study the BHdisc coevolution: in this approach, the accretion disc transits through a sequence of quasisteady warped states (BardeenPetterson effect) and interacts with the BH until the spin aligns with For a BH aligning with a corotating disc, the fractional increase in mass is typically less than a few percent, while the spin modulus can increase up to a few tens of percent. The alignment timescale is of yr for a maximally rotating BH accreting at the Eddington rate. BHdisc alignment from an initially counterrotating disc tends to be more efficient compared to the specular corotating case due to the asymmetry seeded in the Kerr metric: counterrotating matter carries a larger and opposite angular momentum when crossing the innermost stable orbit, so that the spin modulus decreases faster and so the relative inclination angle.
keywords:
accretion, accretion disc – black hole physics – galaxies: active, evolution – quasars: general1 Introduction
Astrophysical black holes (BHs) are Kerr black holes fully characterized by their mass and spin , customarily expressed in terms of the dimensionless spin parameter (), and unit vector :
(1) 
The spin and mass of BHs residing in galaxy nuclei do not remain constant, close to their birth values, but change sizeably through cosmic time, in response to major accretion events. In current cosmological scenarios for the evolution of galaxies, repeated interactions among gasrich halos play a key role not only in shaping galaxies, but also in triggering quasar activity (White & Rees, 1978; Di Matteo et al., 2005). Massive gaseous nuclear discs that form in the aftermath of major galaxy mergers (Mihos & Hernquist, 1996; Mayer et al., 2007) may provide enough fuel to feed, on subparsec scales, the BH through a Keplerian accretion disc (Dotti et al., 2007; Dotti et al., 2009). If these episodes repeat recursively and/or at random phases (King & Pringle, 2006) the BH spin is expected, initially, to be misaligned relative to the direction of the angular momentum of the disc at its unperturbed, outer edge . In this configuration, the gas elements inside the disc undergo LenseThirring precession (see, e.g. Wilkins, 1972). In the fluid, the action of viscosity onto the differentially precessing disc ensures that the inner portion of the accretion disc aligns (or antialigns) its orbital angular momentum with the BH spin , out to a transition radius beyond which the disc remains aligned to the outer disc, as first shown by Bardeen & Petterson (1975)(see also Armitage & Natarajan, 1999; Nelson & Papaloizou, 2000; Fragile & Anninos, 2005; Fragile et al., 2007). Warping of the inner disc at distance from the BH is communicated through the fluid elements on a timescale related to the vertical shear viscosity of the accretion disc. Therefore, the inner regions of the disc align (or counteralign if the disc is counterrotating) with the BH spin on the scale when the viscous time for vertical propagation of disturbances equals the LenseThirring precession time. On a longer timescale, the joint evolution of BH+disc system restores full axisymmetry, with the BH spin direction aligned relative to the total angular momentum of the composite system (Rees, 1978; Thorne et al., 1986; King et al., 2005). The change in is a consequence of angular momentum conservation: since the BH acts on the disc with a torque that warps the disc, then an equal and opposite gravitomagnetic torque acts on the BH that modifies its direction only.
BH spin alignment has been studied in two main contexts. In the first, explored by King et al. (2005) and Lodato & Pringle (2006), the focus is on an closed system where the accretion disc has a finite mass and radial extent. Here, the total angular momentum is well defined vector, and the BH eventually aligns it spin vector to the direction of . In the second, explored by Scheuer & Feiler (1996), Natarajan & Armitage (1999) and Martin et al. (2007), the focus is on an open system, where the accretion disc has infinite extension and it is continually fed at its outer edge by matter whose angular momentum has constant direction . In this second case, the BH aligns its spin to the outer disc direction on a timescale that exceeds by a few orders of magnitude (Scheuer & Feiler, 1996; Martin et al., 2007).
In this paper, we progress on the study of BH alignment including the contemporary change in mass and spin modulus due to accretion of matter, neglected in previous works. During BH precession and alignment, matter flows inward and accretes carrying the energy and the specific angular momentum of the innermost stable circular orbit (ISO). This study thus provides estimates of the fractional increase of mass and spin during BH alignment (the subscript refers to initial conditions), together with a sensible expression for the alignment time . In our context, we assume a continuous and coherent feeding of the accretion disc around the BH, at least for a time as long as the alignment timescale . Thus, we consider an open system and we fix the orbital angular momentum direction at the outer edge of the disc.
In Section 2, we introduce key parameters and highlight our model assumptions. Section 3 surveys properties of steady state warped discs and key scales associated to the BardeenPetterson effect; disc models with constant and powerlaw viscosity profile are explored for completeness. In Section 4, we describe the equations for the BH mass and spin evolution, and introduce the adiabatic approximation to solve these equations. In the same section we also revisit the expression for the BH alignment time. Results are illustrated in Section 5; there we explore also the tendency to alignment in initially counterrotating warped discs. Section 6 contains the discussion of the results and our conclusions.
2 Initial assumptions and main parameters
We consider a BH with spin , surrounded by a geometrically thin, standard ShakuraSunyaev disc (e.g. Shakura & Syunyaev, 1973; Frank et al., 2002). The disc is initially misaligned relative to , i.e. the angular momentum unit vector of the disc at the outer edge is ; the relative inclination angle between the two unit vectors is .
Following Pringle (1992), we assume that the accretion disc has a high viscosity (, where is the disc vertical scale height) so that perturbations propagate diffusively. We introduce two viscosity parameters, and : is the standard radial shear viscosity while is the vertical share viscosity associated to the diffusion of vertical warps through the disc, due to LenseThirring precession. For we adopt the prescription
(2) 
where is the sound speed inside the accretion disc. It is still poorly understood which is the relation between the radial and the vertical viscosity; in particular, if or . In order to simplify our discussion, we refer to the recent analysis of Lodato & Pringle (2007), and for we take:
(3) 
where (given in Table 1) is a coefficient determined in numerical simulations that accounts for nonlinear effects.
The disc model is defined after specifying five free parameters (subscript 0 will be introduced to indicate initial values when mass and spin evolution is considered):
(1) The BH mass, ; we explore a mass range between . For the BH mass we introduce the dimensionless parameter as
(2) The spin modulus, in terms of the dimensionless spin parameter , which varies between . We do not use the theoretical limit because, if accretion is driven by magnetorotational instabilities in a relativistic MHD disc, the final equilibrium spin due to continuous accretion is (Gammie et al., 2004);
(3) The relative inclination angle between the spin versor and the orbital angular momentum versor at the external edge of the accretion disc, . This angle varies isotropically from 0 to . In the following, however, we will confine this interval to () in order to satisfy the used approximations.
(4) The viscosity parameter which is assumed to vary between to bracket uncertainties (King et al., 2007). For our purposes we selected values of according to Lodato & Pringle (2007), as in Table 1. In this study, is considered as a constant inside the disc.
0.18  1.00 
0.15  0.85 
0.09  0.60 
0.05  0.38 
(5) The accretion rate onto the BH, is expressed in terms of the Eddington ratio and of the accretion efficiency (where is the Eddington luminosity): . We consider values of in the interval and compute as a function of the BH spin modulus.
If the disc, warped in its innermost parts, is described to first order by the ShakuraSunyaev model, both and follow a powerlaw. If viscosities satisfy relation (3) and is assumed to be constant, their exponent are equal:
(4) 
Following standard ShakuraSunyaev disc solutions for external regions of an accretion disc (Frank et al., 2002), we have and
(5)  
In equation (5), and are the coefficient and the BH radiative efficiency in unit of . is tabulated in Table 1 (Lodato & Pringle 2007).
3 Warped accretion disc
3.1 The angular momentum content of discs: extended versus truncated discs
The dynamics of a fluid element in a misaligned disc around a spinning BH is given by the combination of three different motions: the Keplerian rotation around the BH; the radial drift, due to radial shear viscosity, and finally the LenseThirring precession, due to the gravitomagnetic field generated by (see, e.g., Weinberg, 1972; Thorne et al., 1986). In response to LenseThirring induced precession, viscous stresses in the disc acts rapidly to produce in the vicinity of the BH an axisymmetric configuration whereby adjacent fluid elements rotates in the equatorial plane of the spinning BH. The disc thus warps and the warp disturbance propagates diffusely (Papaloizou & Pringle, 1983) in the disc.
As the BardeenPetterson effect modifies the inclination of the orbital plane of consecutive infinitesimal rings, then the warped profile of the accretion disc can be described by the specific angular momentum density, , expressed as
(6) 
where is a unit vector indicating the local direction of the orbital angular momentum, is the modulus, is the surface density of the disc and the local Keplerian angular velocity. The angle describing the tilted disc is defined as
(7) 
so that carries information of the warped structure of the accretion disc. The angular momentum of the accretion disc within radius is given by
(8) 
where the integration domain extends from the innermost stable orbit out to . In order to calculate the total disc angular momentum we define an outermost radius, . For an extended disc with , the disc angular momentum always dominates over
Real discs are likely to be truncated by their own selfgravity that becomes important at distances where the disc mass (see, e.g., Pringle, 1981; Frank et al., 2002; Lodato, 2007). Outside the truncation radius, gas can be either turned into stars or expelled by winds from stars which do form (Levin, 2007; King & Pringle, 2007). Thus, we are led to define a disc outer edge as the distance where the Toomre parameter for stability, (where ), becomes less than unity, and the cooling timescale of the clumping gas is less than its dynamical timescale. When the Toomre parameter drops toward unity, the disc becomes unstable on a lenghtscale (Polyachenko et al., 1997; Levine et al., 2008); for a nearly Keplerian, ShakuraSunyaev disc, this scale is much smaller than the disc radial dimension, and the cooling time of the associated perturbation is less or of the same order of its orbital period. Then, as long as the accretion disc can be described as a ShakuraSunyaev disc^{1}^{1}1This condition is fullfilled only for . If this condition is not satisfied the gas temperature drops below K, in the external region of the disc where is still greater than unity. The change in the opacity likely modifies the structure of the outer disc, and we can not explicitly use (9). In this paper we assume that the outer region is sufficiently extended to provide matter and angular momentum to the inner regions and use selfconsistently the ShakuraSunyaev model to describe the disc in regions where the gravitomagnetic interaction takes place. , the external radius can be defined from the condition , so that
(9) 
where is the Schwarschild radius. At the outer edge of the disc, and
Definitions (6) and (8) for and hold for any disc profile. At first order, we can neglect details about the warped disc structure around assuming , and estimate the modulus of the orbital angular momentum within radius , , using ShakuraSunyaev solutions for a flat disc. In this approximation, the surface density is (see, e.g., Pringle, 1981; Frank et al., 2002) and
(10) 
Using equations (8) and (10), and espression (4) for in the case of , the modulus of the disc angular momentum within reads:
(11) 
If espression (11) is estimated at the outer radius (9), the resulting dimensionless ratio between the disc and BH angular momenta is
(12) 
3.2 Timescales and warp radius
The timedependent evolution of the disc is described by the continuity equation
(13) 
where is the radial component of the velocity vector, and by the equation of conservation of angular momentum. In presence of a gravitomagnetic field, for a geometrically thin disc characterized by the two viscosities and , the equation reads (Pringle, 1992):
(14)  
The last term is the LenseThirring precession term and the associated angular velocity is
(15) 
The timedependent equation (14) describes the radial
drift of matter and the diffusion of warping disturbances across the
highviscosity disc.
This equation introduces several key scales:

The viscous/accretion timescale for radial drift, related to the angular momentum transport parallel to , . It can be seen as the time it takes for a fluid element at to accrete onto the BH (see, e.g., Pringle, 1981) Considering equation (14), the balance between the advection term and the viscous term proportional to (both on the right side of equation 14) leads to an estimate of the accretion time:
(16) According to equation (16), we can introduce the disc consumption timescale , a concept useful when considering transient, truncated discs, as the accretion timescale at the outer radius:
(17) 
The timescale for warp propagation, related to the radial diffusion of gravitomagnetic perturbations that transport the component of the disc angular momentum lying in the plane of the disc; this scale is inferred from equation (14) considering the term proportional to ,
(18) The physical interpretation of this timescale has been recently investigated by solving numerically equation (14) for a thin disc (Lodato & Pringle, 2006): starting at with a flat disc misaligned relative to the fixed BH spin, indicates the time it takes for the radial diffusion of the warp to reach radius ; on longer timescale, the disc approaches a steady warped state.

The characteristic extension of the warp , defined as the distance at which the BardeenPetterson timescale equals the LenseThirring precession timescale :
(19) For powerlaw viscosity model, equations (1), (5) and (19) give
(20) The warp radius represents the dividing between the outer region for , where the disc keeps its original inclination, given by , and the inner region for , where the disc aligns (or antialigns) its orbital angular momentum with the BH spin, . The warp radius fixes also the magnitude of the relevant BardeenPetterson timescale, which reads
(21) If we define the function
(22) and the radius where the disc is maximally deformed
(23) we expect that:
(24) with of order unity. has two important properties: first, if it is the radius where the disc is maximally warped, i.e. where the diffusive propagation of vertical perturbations is more significant; second, it provides a reliable estimate of the distance from the BH where the gravitomagnetic interaction is stronger. From equation (14) this interaction is proportional to : this term vanishes in the inner part of the disc () since the BardeenPetterson effect aligns with , and also in the outer regions (), due to the rapid decline with . Accordingly, the region near (or equivalently ) is the only one significantly misaligned with .
3.3 Analytical solutions
In this Section we summarise the properties of the steady warped disc structure used to compute the joint evolution of the disc and the BH.
Following previous studies we assume that the viscosity profiles are
powerlaws with exponent , as in equation (4), and explore two possible cases.
In the first, we formally extend the ShakuraSunyaev solution
everywhere in the disc, i.e. with
, and given by equation (3)
(Martin
et al., 2007).
In the second case, we assume the viscosities to remain approximately
constant everywhere in the disc (Scheuer &
Feiler, 1996).
In order to compare the two models (Martin
et al., 2007, cfr.), we impose
the continuity of the viscosities at where the gravitomagnetic torque
is most important.
Before solving equations (13) and (14)
we introduce two appropriate reference frames.
The first is the inertial reference frame referred to the outer disc; we can always rotate it so that its axis is parallel to the direction of .
The second reference frame is the notinertial frame referred to the BH
spin, which is always centered to the BH and whose axis is parallel to the black hole time varying spin . If we use the adiabatic approximation, then frame can be approximated, with time , as a
sequence of frames, one for every quasistationary state of the system.
The shape of the warped accretion disc is studied in the frames and
the cartesian components of any vector are there indicated
as ; is the natural frame to study the temporal
evolution of the BH spin and here the Cartesian components of the previous
vector are denoted as .
For a stationary state, continuity equation (13) can be
easily intergrated introducing the accretion rate as
constant of integration:
(25) 
while the projection of equation (14) along reads:
(26) 
In the small deformation approximation (Scheuer & Feiler, 1996) the warp is gradual and we can neglect the nonlinear term, proportional to . Using the boundary condition , the integral of (26) is
(27) 
This means that, in this approximation scheme, the modulus of the angular momentum density for a warped accretion disc far from the horizon is the same as for a flat disc, equation (10).
Following Scheuer & Feiler (1996) we study the disc profile of the steady disc introducing the complex variable and considering the case . Using powerlaw viscosities according to (4), analytic solutions of equation (14) in the small deformation approximation have been found by Martin et al. (2007):
(28)  
where is a complex constant of integration, depending on the boundary condition at the external edge, the subscript ”PL” is a reminder of the powerlaw viscosities and is the modified Bessel function of order . In the particular case where we consider constant viscosities, i.e. , the solution can be written as
(29) 
where is a complex constant of integration and the subscript ”C” is a reminder of the constant viscosity model (Scheuer & Feiler, 1996). In this latter case, is calculated selfconsistently, using the definition (24) and the prescription for constant viscosity evaluation at ; we find that, for every possible parameters set, . We notice that (and so also ) depends on the radius through the dimentionless ratio (Martin et al., 2007).
In Figure 2 we plot the modulus of the gradient of , , which is a local measure of the deformation degree of the disc, for a particular set of parameters and for three different angles, . The shape of is similar for the two different disc profiles and for all the angles: there is a well defined maximum near , where we expect the disc to be more deformed. At radii smaller than and far from the disc is almost flat (note that the graph is logarithmic in both axes). For the constant viscosity (powerlaw) profile the peak is at (). In Figure 1 we also see that a constant viscosity disc is less warped (since the maximum deformation is the smaller) than the power law viscosity disc. The ratio between the maximum deformations in the power law vs constant viscosity is roughly a factor 2, and it does not depend on the inclination angle (except for a scale factor, approximately equal to the ratio between the corresponding angles).
3.4 Validity of the approximation
We calculated the warped disc profile under the small deformation approximation. We neglected second order terms in equation (26) and found an analytic solution for ; in order to verify the consistence of this approximation, we define as the ratio between the neglected term and the first term into the round brackets of (26), assuming to have a Keplerian disc with powerlaw viscosity profile with exponent , like in equation (4). Considering equations (27) for , we have and then reads
(30) 
Once we know the explicit solutions, the consistence of this approximation can be tested a posteriori calculating : the approximation is well satisfied if From equation (30), can be expressed also as a function of and :
(31) 
Figure 2 shows the function for constant viscosity
profiles (dashed lines) and the function
for powerlaw viscosity profiles (solid lines), for the
same parameters as in Figure 2.
The function exhibits a maximum, ,
around . Far from the accuracy of the approximation increases, albeit slowly.
The function is most sensitive to the inclination angle, as expected (notice that Figure 2 uses logarithmic axes).
In Figure 3, we test the validity of the
small deformation approximation plotting,
in the BH mass versus plane, the color coded
values of , for different values of the viscosity parameter
(Table 1), using the constant
viscosity profile model (we fix and ).
White zones represent the regions where , i.e.
where the small deformation approximation becomes invalid.
shows mainly a strong dependence on inclination angle
, but also a weaker dependence on the BH mass which reveals
that the small deformation approximation is less accurate for
and increasing BH mass.
Comparing different values, the approximation is better satisfied
for large viscosities parameters (i.e. ).
We repeated the analysis for the powerlaw viscosity model
that shows no significant differences in the parameters dependence.
In Figure 4, using the same colours conventions, we explored
in the versus (left panels), and versus (right panels) planes, once we have fixed the viscosity parameter (), the BH mass (), and for the left panels and for the right panels. For both constant () and powerlaw () models the relative inclination angle is again the leading parameter gauging
the goodness of the fit as the approximation depends very weakly on
and .
4 Black hole evolution
4.1 Basic equations
In this section we explore the equations for the BH evolution. The BH is accreting and its mass increases, from an initial value according to
(32) 
where is the energy per unit mass of a test particle at the innermost stable orbit. is related to the efficiency that depends only on the spin parameter (Bardeen, 1970; Bardeen et al., 1972). Equation (32) introduces a natural timescale for BH mass growth, known as Salpeter time :
(33) 
As argued by Rees (1978) and shown by Thorne et al. (1986), there is a coupling between the BH spin and the angular momentum of the disc. Even though the disc is much less massive that the BH, the moving fluid elements perturb the Kerr metric and interact with the BH spin, causing spin precession, and if viscous dissipation is present, alignment. For an infinitesimal ring of inviscid matter with total angular momentum , the BH spin precesses, following the equation
(34) 
with a precession frequency
(35) 
Equation (34) can be extended to the case of an accretion disc to yield:
(36)  
The first contribution is due to accretion of matter at where indicates the orbital angular momentum per unit mass carried by matter at ISO; the BardeenPetterson effect ensures that the direction of is parallel or antiparallel to , so that the accretion modifies only the spin modulus. As shown by Bardeen (1970), a variation of mass is necessary to pass from a Schwarzschild BH () to an extreme Kerr BH (), while spin flip of , due only to accretion on an initially extreme Kerr BH, needs . So, the spin accretion timescale for the spin modulus is of the same order of the mass accretion timescale . The second term in equation (36) describes the gravitomagnetic interaction between the rotating viscous disc and the BH spin vector. This term modifies only the spin direction of the BH in order to conserve the total angular momentum of the system. Under the working hypothesis that the disc is continually fed by matter carrying the same angular momentum (see Section 6 for a critical discussion), the BH aligns its spin in the direction of . Alignment implies that goes to with time. Figure 5 shows the function defined as the modulus of the integral kernel of equation (36)
(37) 
as a function of for different value of , where is computed along the profile of the steady warped disc of equation (28) and (29). The function , similarly to (defined in eq. [22]), peaks near . Contrary to , power law viscosity profiles have lower peaks, compared with constant viscosity profiles. This figure indicates also that the BHdisc gravitomagnetic interaction is spread over a relatively small region of the disc around the warp radius; the characteristic spreading length, which is slightly larger for constant viscosity profiles, is usually of a few warp radii.
4.2 Alignment time
In this Paragraph we want to give simple estimations for the alignment and the precession timescales, starting from equation (36).
Assuming BH mass and spin modulus variations due to accretion to be small compared with gravitomagnetic effects during the alignment, we neglect the term proportional to in (36); if BH spin aligns and precess, left hand side of (36) can be estimated introducing a characteristic gravitomagnetic timescale as
and the integral on the right hand side as
since the bulk of the gravitomagnetic interaction occurs around Equating these two expressions and using equation (27) for the specific angular momentum density modulus, we obtain
(38) 
Using equation (19) and (11) which imply , the gravitomagnetic scale can be written in terms of the BardeenPetterson warp timescale (eq. [18]):
(39) 
and also in term of the accretion timescale (eq. [16])
(40) 
where is the disc angular momentum modulus within the warp radius, estimated by (11). Finally, considering equation (11) and (16) for together with the expression for the spin modulus and Schwarzschild radius, of expression (40) can be rearranged as
(41) 
Since the disc carries very little angular momentum at the warp radius, from equation (39) always. The gravitomagnetic BHdisc interaction causes BH spin precession and alignment at the same time, and then introduces two scales related with , the precession and the alignment timescales, and respectively. We separate their relative importance following Martin et al. (2007) results, and define the parameter , so that
(42) 
The exact value of depends on the viscosity profile, and can be estimated either analytically (Martin et al., 2007), or numerically as in this paper. Initially, we assume alignment and precession to have the same timescale, according to Scheuer & Feiler (1996). Substituting expressions (5) for the viscosities, (20) for the warp radius, and (38) for in (42), the alignment time reads
(43) 
The timescale increases with , indicating that a rapidly rotating Kerr BH offers some resistance before changing its direction. Interestingly, the alignment timescale does not depend on the initial inclination since a more inclined configuration implies more pronunced disc deformations and stronger mutual gravitomagnetic interactions (as also shown in Figure 2 and 5). has a weak dependence on the BH mass and scales nearly as : a higher accretion rate implies a higher angular momentum density and thus a stronger gravitomagnetic coupling. We notice also that, apart from numerical factors of order unity, this timescale is consistent with the alignment scales found by by Scheuer & Feiler (1996); Natarajan & Pringle (1998); Natarajan & Armitage (1999); Martin et al. (2007).
4.3 The adiabatic approximation
In Section 3.2 and 4.1 we described the equations governing the evolution of a warped accretion disc around a fixed BH, and the evolution of an accreting Kerr BH in gravitomagnetic interaction with its accretion disc. The BH and the accretion disc evolve contemporary and their evolution is coupled, so that we can solve simultaneously equations (13) and (14) for a Keplerian disc and (32) and (36) for the accreting and precessing BH.
In this paper, we solve these coupled equations using the adiabatic
approximation that
separates the rapid temporal evolution of the warped disc from the
longer temporal evolution of the BH.
Equations are integrated starting from given initial conditions:
at the BH spin is inclined with respect
to by an angle
and the warped disc profile is described by a quasystationary profile
; and are the initial
BH mass and spin.
In order to justify this approximation scheme,
we survey the BH and disc timescales, as functions of and , for
two selected values of the viscosity and spin parameter: and
In Figure (7) and in Figure (7), we draw in the
 plane lines of constant
and ratios.
The comparison between the different timescales lead to the following
hierarchy of timescales:
(44) 
Then, in the adiabatic approximation, the disc transits through a sequence of warped states over the shortest timescale while, on the longer timescale the BH aligns its spin to , and modifies a little its spin modulus and mass due to accretion. Considering one of these disc quasisteady states, initially at time , after a time gap the BH mass and spin are updated according to
(45) 
and these variations produce a new quasistationary warped state at
, .
For the BH mass variation , we integrate equation (32)
from to :
(46) 
where is the last innermost stable orbit associated with the
current value of .
For the spin variation, we need to integrate equation (36)
that includes
the two different and coupled contributions due to accretion and gravitomagnetic interaction; if and are small on the timescale , to first order the two contributions decouple and they can be integrated
separately:
(47) 
(48)  
where is due to accretion and changes only the spin modulus while is due to gravitomagnetic interaction and changes only the spin direction. After the interval the angular momentum of (45) are updated according to this rule
(49)  
This procedure can be repeated iteratively on a timescale to study the coupled evolution of , and during the alignment process.
5 Spin alignment
5.1 Set up
In this Section we study the coupled evolution of the BH and warped accretion
disc using the approximation scheme described in the previous Section,
in order to infer
the evolution of and as a function of time, in response
to the gravitomagnetic interaction and matter accretion.
VP  

()  (in units of )  (in units of )  
0.15  2.2  0.67  1.57  
1  0.30 