Sporadically Torqued Accretion Disks Around Black Holes

Abstract

The assumption that black hole accretion disks possess an untorqued inner boundary, the so-called zero torque boundary condition, has been employed by models of black hole disks for many years. However, recent theoretical and observational work suggests that magnetic forces may appreciably torque the inner disk. This raises the question of the effect that a time-changing magnetic torque may have on the evolution of such a disk. In particular, we explore the suggestion that the ``Deep Minimum State'' of the Seyfert galaxy MCG--6-30-15 can be identified as a sporadic inner disk torquing event. This suggestion is motivated by detailed analyses of changes in the profile of the broad fluorescence iron line in XMM-Newton spectra. We find that the response of such a disk to a torquing event has two phases; an initial damming of the accretion flow together with a partial draining of the disk interior to the torque location, followed by a replenishment of the inner disk as the system achieves a new (torqued) steady-state. If the Deep Minimum State of MCG--6-30-15 is indeed due to a sporadic torquing event, we show that the fraction of the dissipated energy going into X-rays must be smaller in the torqued state. We propose one such scenario in which Compton cooling of the disk corona by ``returning radiation'' accompanying a central-torquing event suppresses the 0.5-10 keV X-ray flux coming from all but the innermost regions of the disk. Comment: 12 pages, 24 figures, ApJ accepted

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arXiv:astro-ph/0501455v2 21 Jan 2005
DRAFT VERSION FEBRUARY 2, 2008
Preprint typeset using LATEX style emulateapj
SPORADICALLY TORQUED ACCRETION DISKS AROUND BLACK HOLES
DAVID GAROFALO1AND CHRISTOPHER S. REYNOLDS2
Draft version February 2, 2008
ABSTRACT
The assumption that black hole accretion disks possess an untorqued inner boundary, the so-called zero torque
boundarycondition,has beenemployedbymodelsofblackholedisks formanyyears. However,recenttheoretical
and observational work suggests that magnetic forces may appreciably torque the inner disk. This raises the
question of the effect that a time-changingmagnetic torquemay have on the evolutionof such a disk. In particular,
we explore the suggestion that the “Deep Minimum State” of the Seyfert galaxy MCG–6-30-15 can be identified
as a sporadic inner disk torquingevent. This suggestion is motivated by detailed analyses of changes in the profile
of the broad fluorescence iron line in XMM-Newton spectra. We find that the response of such a disk to a torquing
event has two phases; an initial damming of the accretion flow together with a partial draining of the disk interior
to thetorquelocation,followedbya replenishmentoftheinnerdiskas thesystem achievesa new(torqued)steady-
state. If the Deep Minimum State of MCG–6-30-15 is indeed due to a sporadic torquing event, we show that the
fraction of the dissipated energy going into X-rays must be smaller in the torqued state. We propose one such
scenario in which Compton cooling of the disk corona by “returning radiation” accompanying a central-torquing
event suppresses the 0.5–10keV X-ray flux coming from all but the innermost regions of the disk.
Subject headings: accretion, accretion disks – black hole physics – magnetohydrodynamics– galaxies:active –
galaxies:individual(MCG–6-30-15)–X-rays:galaxies
1. INTRODUCTION
There is direct evidence that active galactic nuclei (AGN) are powered by disk accretion onto supermassive black holes. For
example, the central regions of the giant elliptical galaxy M87 were spatially resolved by the Hubble Space Telescope sufficiently
to actually see a ∼ 100pc-scale disk of ionized gas in orbit about an unseen mass of 3 × 109M⊙, presumed to be a supermassive
black hole (Ford et al, 1994; Harms et al, 1994). The fact that this gas disk is approximately normal to the famous relativistic jet
displayed by this object is circumstantial evidence that it is, indeed, the outer regions of the accretion disk that powers this AGN.
In another important case, radio observations of megamasers in the spiral galaxy NGC 4258 (M106) reveal an almost perfectly
Keplerian ∼ 0.5pc gas disk orbiting a compact object with mass 3.5 × 107M⊙(Miyoshi et al. 1995; Greenhill et al. 1995).
Even before the observational evidence for black hole accretion disks became compelling, the basic theory of such disks had been
extensively developed. Building upon the non-relativistic theory of Shakura & Sunyaev (1973), Novikov & Thorne (1974) and Page
& Thorne (1974) developed the “standard” model of a geometrically-thin, radiatively-efficient, steady-state, viscous accretion disk
around an isolated Kerr black hole. In addition to the assumptions already listed, it is assumed that the viscous torque operating
within the disk becomes zero at the radius of marginal stability, r = rms. Physically, this was justified by assuming that the accretion
flow would pass through a sonic point close to r = rmsand hence flow ballistically (i.e., “plunge”) into the black hole.
Even while setting up this boundary condition, Page & Thorne (1974) noted that magnetic fields may allow this zero-torque
boundarycondition(ZTBC) to be violated. Giventhe modernviewpointof accretiondisks, that the very“viscosity” drivingaccretion
is due to magnetohydrodynamic (MHD) turbulence, the idea that the ZTBC can be violated has been revived by recent theoretical
work, starting with Gammie (1999) and Krolik (1999a). In independent treatments, these authors show that significant energy and
angular momentum can be extracted from matter within the radius of marginal stability via magnetic connections with the main
body of the accretion disk. Agol & Krolik (2000) have performed the formal extension of the standard model to include a torque at
r = rmsand show that the extra dissipation associated with this torque produces a very centrally concentrated dissipation profile. As
shown by Gammie (1999), Agol & Krolik (2000), and Li (2002), this process can lead to an extraction (and subsequent dissipation)
of spin energy and angular momentum from the rotating black hole by the accretion disk. In these cases, the magnetic forces might
be capable of placing the innermost part of the flow on negative energy orbits, allowing a Penrose process to be realized (we note
that Williams [2003] has also argued for the importance of a non-magnetic, particle-particle and particle-photonscattering mediated
Penrose process). A second mechanism by which the central accretion disk can be torqued is via a direct magnetic connection
between the inner accretion disk and the (rotating) event horizon of the black hole. In this case, as long as the angular velocity of
the event horizon exceeds that of the inner disk, energy and angular momentum of the spinning black hole can be extracted via the
Blandford-Znajek mechanism (Blandford & Znajek 1977). We note that field lines that directly connect the rotating event horizon
with the body of the accretion disk through the plunging region are seen in recent General Relativistic MHD simulations of black
hole accretion (e.g., Hirose et al. 2004).
Interest in these torqued relativistic disks has received a boost from recent X-ray observations. The X-ray spectra of many AGN
(and, indeed, Galactic Black Hole Candidates) reveal the signatures of “X-ray reflection” from optically-thick matter. In some cases,
examination of these features allows us to detect strong Doppler and gravitational shifts indicative of circular motion close to a black
1Dept.of Physics, University of Maryland, College Park, MD20742
2Dept.of Astronomy, University of Maryland, College Park, MD20742
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hole (Fabian et al. 1989; Tanaka et al. 1995; Fabian et al. 2000; Reynolds & Nowak 2003). In these cases, the observed range
of Doppler and gravitational shifts can be used to map the X-ray emission/irradiation across the surface of the accretion disk. The
XMM-Newton satellite is particularly well suited to the study of these features due to its combination of good spectral resolution and
high throughput. Using XMM-Newton, Wilms et al. (2001)and Reynolds et al. (2004)studied the Seyfert-1 galaxyMCG–6-30-15in
its peculiar “DeepMinimumState” first discoveredby ASCA (Iwasawa et al. 1996). Confirmingthe principalresult of Iwasawa et al.
(1996), the X-ray reflection features were found to be extremely broad. The degree of gravitational redshifting required the majority
of the X-ray emission to emerge within a radius of r ∼ 2GM/c2from a near-extremal Kerr black hole (i.e., a black hole with a
spin parameter of a = 0.998). As explicitly shown in Reynolds et al. (2004), it is very problematic to explain these data within the
framework of the standard accretion disk model. Fabian & Vaughan (2003) and Miniutti & Fabian (2004) suggest that gravitational
focusing of the primarycontinuumX-rays might producesuch a centrally-concentratedemissivity profile. Alternatively,Reynolds et
al. (2004) has shown that a torqued disk can readily explain the Deep Minimum spectrum provided the source is assumed to be in a
torque-dominatedstate (or, in the terminology of Agol & Krolik [2000], an “infinite-efficiency” state) whereby the power associated
with the innermost torque is instantaneously dominating the accretion power. In other words, the X-ray data suggest that during this
Deep Minimum state of MCG–6-30-15 the power derived from the black hole spin greatly exceeds that derived from accretion.
Of course, this state of affairs cannot last forever or else the central black hole in MCG–6-30-15would spin down to a point where
it could no longer provide this power. At some point in its history, the system must be in an accretion-dominatedphase in which the
black hole is spun up. However, even in its spin-dominated state, the spin-down timescale of the central black hole is of the order of
100 million years or more. Thus we could envisage a situation in which the system shines via a quasi-steady-state, spin-dominated
accretion disk. There are hints, though, that accretion disks may switch between spin-dominated and accretion-dominated on much
shorter timescales. In its normal spectral state, the X-ray reflection features in MCG–6-30-15 are much less centrally concentrated
than in the Deep Minimum State, suggesting that the normal state might be accretion-dominated. It is also important to note that
this system can switch between its normal state and the Deep Minimum State in as little as 5–10ksec (Iwasawa et al. 1996), which
corresponds to only a few dynamical timescales of the inner accretion disk. Thus it is of interest to consider the physics of an
accretion disk that undergoes a rapid torquing event. That is the prime motivation for this paper.
In Section 2 we will begin our study of sporadically torqued accretion disks by investigating an analytic solution for a torqued
Newtonian disk. In Section 3, we generalize to the fully relativistic equations and obtain numerical solutions. In Section 4, we relate
our results with the observed properties of the “Deep Minimum State” of MCG–6-30-15, and consequently discuss the effect that
a torquing event may have on the physics of the X-ray emitting disk corona. In particular, we suggest that the enhanced Returning
Radiation associated with a torquing event might suppress 0.5–10keV coronal emission in all but the inner portion of the disk.
Section 5 summarizes our main conclusions.
2. AN ANALYTIC “TOY” MODEL OF A TORQUED DISK
We beginour investigationof sporadicallytorqueddisks via the study of a simple case that lends itself to a straightforwardanalytic
solution.
We shall constructamodelofa radiatively-efficientaccretiondiskfollowingthe usualapproachofPringle(1981). We shall assume
that the accretion disk is axisymmetric, geometrically-thin and in Keplerian motion about a point-mass M. Using a cylindrical polar
coordinatesystem(r,z,φ) withthe axispassingthroughthe centralmass normaltothe diskplane,we shall denotethe surfacedensity
of the disk by Σ(r,t), the angular velocity of the disk about M as Ω(r) and the radial velocity of the disk material as vr(r,t). The
equations that determine the structure of the thin disk assuming radiative efficiency are mass and angular momentum conservation,
r∂Σ
∂t+∂(rvrΣ)
∂r
= 0,
(1)
r∂(Σr2Ω)
∂t
+∂(rΣvrr2Ω)
∂r
=
1
2π
∂G
∂r,
(2)
respectively, where G(r,t) is the torque exerted by the disk outside of radius r on the disk inside of that radius.
In standard disk models, the torque G is the integrated value of the only stress tensor component (Srφ) that survives the condition
of axisymmetry and geometric-thinness. From Krolik (1999b) we have,
G =
?
rSrφdz
?
rdφ = 2πr3νΣ∂Ω
∂r,
(3)
where we have introduced an ”effective kinematic viscosity”, ν. To generalize these models to the case of an externally imposed
torque, we set
G = 2πr3νΣ∂Ω
∂r+ GT.
(4)
Combining eqns. 1, 2 and 4, and specializing to a Keplerian rotation curve, we get the usual diffusion equation for surface density
modified for the effects of the external torque,
∂Σ
∂t=3
r
∂
∂r
?
r1/2∂(νΣr1/2)
∂r
?
−
1
rπ(GM)1/2
∂
∂r
?
r1/2∂GT
∂r
?
.
(5)

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For the rest of this paper, we shall work in units where GM = 1. Changing variables to x = r1/2and ψ = νΣx and assuming that
ν has no explicit time dependence, we get
∂ψ
∂t=
4x2
∂x2
We now consider a particular torquingevent. Suppose that the disk suffers no external torques for the periodt < 0. Then, at t = 0,
we engage an external torque (possibly resulting from a magnetic connection to the plunging region or spinning event horizon) that
deposits angular momentum into a narrow annulus at r = r0. If the rate at which angular momentum is being deposited is β, we
have
∂GT
∂r
giving
GT(r,t) = βΘ(r − r0)Θ(t),
where Θ is the Heavyside step function. At this point, we specialize to a particular viscosity law. We set ν = kr for mathematical
convenience, although it will not make qualitative difference. We can now rewrite eqn. 6 as,
3ν
∂2
?
ψ −GT
3π
?
.
(6)
= βδ(r − r0)Θ(t),
(7)
(8)
∂ξ
∂t=3k
4
?∂2ξ
∂x2
?
−
β
3πΘ(x − x0)δ(t),
(9)
where,
ξ=ψ −β
3πΘ(x − x0)Θ(t),
(10)
and the delta-function in time results from the time-derivative of the Θ(t) term.
Suppose that the disk is in the untorqued steady state at t < 0. From eqn.9, one can easily see that such a steady state is given by,
ξss= ψ = A(x − xi)(t < 0),
(11)
where A is a normalization constant and ri≡ x2
torques vanish. Examination of eqn. 9 shows that the time-dependent behavior of the torqued disk at times t > 0 is given by the
simple diffusion equation,
∂ξ
∂t=3k
4
with an initial condition set by integrating through the delta-function in time, ξ(x,t = 0) = ξss− βΘ(x − x0)/3π. The appropriate
boundary condition is ξ → ξssas t → ∞. Standard methods (i.e., separation of variables) give the following solution:
iis the inner edge of the untorqued disk defined as the location where the “viscous”
∂2ξ
∂x2,
(12)
ξ(x,t) = ξss(x) +
β
3π2
?∞
0
1
λ[sinλx0cosλx − (cosλx0 + 1)sinλx]exp?−3kλ2t/4?dλ.
(13)
With this solution, we can compute the surface density of the disk at any given radius and time. Armed with the surface density,
we can then compute all other quantities of interest including the viscous dissipation rate per unit surface area of the disk:
D(r) =νΣr2
2
Ω′2,
(14)
where Ω′= dΩ/dr. Plots of the radial dependence of Σ(r) and D(r) for various times are shown in Figures 1–2. Also shown is the
time dependence of the total viscous dissipation obtained by integrating D(r) across the whole disk (i.e. luminosity). We can see
that the response of the disk to the onset of an external torque can be separated into two phases. In the first phase, the accretion flow
is “dammed” at r = r0due to the inability of the accretion flow to transport the angular momentum deposited by the external torque.
This leads to a build-up of mass (i.e., an increase in the surface density) in the region r > r0. Concurrently, matter in the region
r < r0continues to accrete thereby partially draining away the surface density. The inevitable result is a growing discontinuity
in the surface density at r = r0. The angular momentum transport associated with this discontinuity grows until mass can, once
again, flow inwards across this radius. One then enters the second phase of evolution, whereby the surface density in the region
r < r0is replenishedback to its original level while the surface density discontinuityis maintainedat approximatelya constant level.
Eventually, one achieves the torqued steady-state solution (e.g., Agol & Krolik 2000). The two sets of figures are for an external
torque at r = 4 and one closer to the inner edge at r = 2. Note how, for a given injection rate of angular momentum, the effect on
the disk structure is much more dramatic for smaller radius.
Since our disks are assumed to be radiatively-efficient, the instantaneous total luminosity of the accretion disk can be formally
decomposed into two components, one due to the decrease in gravitational potential energy of the accreting gas, and a second due to
the work done by the external torque, i.e.,
L = 2
?
2πrD(r)dr =1
2
?
GM˙M
r2
dr +
?
Ω∂GT
∂r
dr.
(15)

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FIG. 1.— Evolution of our “toy” model disk with a torque acting at r = 4. Panel (a) shows the initial state of the surface density profile for the non torqued disk
(solid line) and the resulting torqued steady-state (dashed line). Panel (b) shows four times in the early evolution of the surface density profile (solid-line:t = 0 the
untorqued steady-state, dashed-line:t = 10−3, dot-dashed-line:t = 10−2, dotted-line:t = 10−1; we use units such that k = 1 which corresponds to scaling with
respect to the viscous timescale of the inner disk). Notice how the initial evolution is such that density drops inward of the torque location and increases outward
of it due to the “damming” of the accretion flow. Panel (c) shows the subsequent late evolution towards the torqued steady-state (solid-line:t = 0, or untorqued
steady-state, dashed-line:t = 1, dot-dashed-line:t = 10, dotted-line:t = 100). Panels (d) and (e) show the dissipation profiles D(r) in the early and late stages,
respectively, of the evolution with the type of line and time corresponding to those of figures (b) and (c). In this case, the effect on D(r) is subtle. In panel (f), we
show the luminosity profile obtained by integrating the dissipation profile over the disk surface. The final steady-state torqued luminosity profile is enhanced with
respect to the non-torqued steady-state profile due to work done by the torque.

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FIG. 2.— Evolution of our “toy” model disk with a torque acting at r = 2. Panel (a) shows the initial state of the surface density profile for the non torqued disk
(solid line) and the resulting torqued steady-state (dashed line). Panel (b) shows four times in the early evolution of the surface density profile (solid-line:t = 0,
dashed-line:t = 10−3, dot-dashed-line:t = 10−2, dotted-line:t = 10−1). Panel (c) shows the subsequent late evolution towards the torqued steady-state (dashed-
line:t = 1, dot-dashed-line:t = 10, dotted-line:t = 100). For reference, the lower solid-line shows the untorqued steady-state disk. Panels (d) and (e) show the
dissipation profiles D(r) in the early and late stages, respectively, of the evolution with the same line-type/time arrangement of figure 1. In other words, the line
types of figure (b) match those of figure (d) and those of figure (c) match those of figure (e). In panel (f), we show the luminosity profile obtained by integrating the
dissipation profile over the disk surface for both torques at r=2 (dashed line) and r=4 (solid line).