A&A 407, 403-421 (2003)
DOI: 10.1051/0004-6361:20030868
V. I. Pariev^{12} - E. G. Blackman^{1} - S. A. Boldyrev^{3}
1 - Department of Physics and Astronomy, University of Rochester,
Rochester, NY 14627, USA
2 -
P.N. Lebedev Physical Institute, Leninsky Prospect 53, Moscow
117924, Russia
3 -
Department of Astronomy and Astrophysics, University of Chicago,
5640 South Ellis Avenue, Chicago, IL 60637, USA
Received 6 March 2003 / Accepted 3 June 2003
Abstract
We develop a model of thin turbulent accretion discs supported
by magnetic pressure of turbulent magnetic fields.
This applies when the turbulent kinetic and magnetic energy
densities are greater than the thermal energy density
in the disc. Whether such discs survive in nature or not
remains to be determined, but here we simply
demonstrate that self-consistent solutions
exist when the -prescription for the viscous stress,
similar to that of the original
Shakura-Sunyaev model, is used.
We show that
for the strongly
magnetized case and we calculate the radial structure
and emission spectra from the disc in the regime when it is optically
thick.
Strongly magnetized optically thick discs can
apply to the full range of disc radii for objects 10^{-2} of the
Eddington luminosity or for the outer parts of discs in higher luminosity
sources.
In the limit that the magnetic pressure is equal to the thermal
or radiation pressure, our strongly magnetized disc
model transforms into the Shakura-Sunyaev model with .
Our model produces spectra quite similar to those of standard
Shakura-Sunyaev models. In our comparative study, we also
discovered a small discrepancy in the spectral calculations
of Shakura & Sunyaev (1973).
Key words: accretion: accretion disks - turbulence - magnetohydrodynamics (MHD) - plasmas
The well known and most widely used model of the accretion disc was proposed and elaborated by Shakura (1972) and Shakura & Sunyaev (1973). In this model the disc is vertically supported by the thermal pressure (Shakura & Sunyaev 1973). Turbulent viscosity is invoked in the Shakura-Sunyaev model to explain the angular momentum transfer required by the accretion flow. As originally pointed out in Lynden-Bell (1969) and Shakura & Sunyaev (1973) a magnetic field can also contribute to the angular momentum transport. A robust mechanism of the excitation of magnetohydrodynamical (MHD) turbulence was shown to operate in accretion discs due to the magneto-rotational (MRI) instability (Balbus & Hawley 1998). The growth of the MRI leads to the excitation of turbulent magnetic fields and self-sustained MHD turbulence. The contribution of Maxwell stresses to the transport of angular momentum is usually larger than Reynolds stresses. However, the magnetic energy observed in many numerical experiments was smaller than the thermal energy of the gas in the disc (Brandenburg 1998). Simulations of the non-linear stage of MRI are typically local simulations in a shearing box of an initially uniform small part of the disc. Attempts to expand the computational domain to include a wider area of radii and azimuthal angle (Armitage et al. 2001; Hawley & Krolik 2001; Hawley 2001) are underway. However, even before the recent focus on the MRI Shibata et al. (1990) observed the formation of transient low state in a shearing box simulations of the non-linear Parker instability in an accretion disc.
Vertical stratification has been considered in the shearing box approximation (Miller & Stone 2000; Brandenburg et al. 1995). In particular, Miller & Stone (2000) investigated discs with initial Gaussian density profiles supported by thermal pressure. The initial seed magnetic field grows and starts to contribute to the vertical pressure balance. The computational domain extends over enough vertical scale heights to enable Miller & Stone (2000) to simulate the development of a magnetically dominated corona above the disc surface. In the case of an initial axial magnetic field, Miller & Stone (2000) observed that the saturated magnetic pressure dominates thermal pressure not only in the corona but everywhere in the disc. As a consequence, the thickness of the disc increases until it reaches the axial boundaries of the computational box. The formation of low filaments in magnetized tori was also observed in global MHD simulations by Machida et al. (2000). Although further global MHD simulations of vertically stratified accretion discs are needed, this numerical evidence suggests that magnetically dominated thin discs may exist.
Previously, analytic models of thin accretion discs with angular momentum transfer due to magnetic stresses were considered by Eardley & Lightman (1975) and Field & Rogers (1993a,b). Both these works included magnetic loops with size of the order of the disc thickness. In Eardley & Lightman (1975), the magnetic loops were confined to the disc. Loop stretching by differential rotation was balanced by reconnection. The reconnection speed was a fraction of the Alfvén speed. Radial magnetic flux was considered as a free function of the radius. Vertical equilibrium and heat transfer were treated as in Shakura & Sunyaev (1973), with the addition of the magnetic pressure in the vertical support. No self-consistent magnetically dominated solutions were found in model of Eardley & Lightman (1975).
In contrast, dominance of the magnetic pressure over the thermal and radiation pressure was postulated from the beginning by Field & Rogers (1993a,b) and verified at the end of their work. These authors assumed that the ordered magnetic field in the disc, amplified by differential rotation, emerges as loops above the surface of the disc due to Parker instability. Because the radial magnetic field in the disc has an intially sectorial structure, the loops above the disc come to close contact and reconnect. All dissipation of magnetic field occurs in the corona in the model of Field & Rogers (1993a,b). Such a corona was assumed to be consisting of electrons and some fraction of positrons and no outflow from the disc is present. Electrons and positrons are accelerated to relativistic energies at the reconnection sites in the disc corona and subsequently emit synchrotron and inverse Compton photons. Because reconnection was assumed to occur at loop tops, Field & Rogers (1993a,b) found that up to 70 per cent of the energy released in reconnection events in the corona will be deposited back to the surface of the disc in the form of relativistic particles and radiation. Only thin surface of optically thick disc is heated and cools by the thermal emission, which is the primary source of soft photons for the inverse Compton scattering by relativistic particles in the corona.
Since the characteristic velocity of rise of the loops of the buoyant magnetic field is of the order of the Alfvén speed, it takes about the time of a Keplerian revolution for the loop of the magnetic field to rise (e.g., Beloborodov 1999). This is also about the characteristic dissipation time of the magnetic field in shocks inside the disc (see Sect. 2). The model we explore here differs from that of Field & Rogers (1993a,b) in that the dissipation of the magnetic energy occurs essentially inside the disc and the heat produced is transported to the disc surface and radiated away. Observations of hard X-ray flux indicate the presence of hot coronae where a significant fraction of the total accretion power is dissipated. For example, the X-ray band carries a significant fraction of the total luminosity of Seyfert nuclei: the flux in the 1-10 keV band is about 1/6 of the total flux from IR to X-rays, and the flux in 1-500 keV band is about 30-40 per cent of the total energy output (Mushotzky et al. 1993). Another example is the low/hard state of galactic black hole sources, where the borad band spectrum is completely dominated by a hard X-ray power law, rolling over at energies of (Done 2002; Nowak 1995). Also, in the so called very high state, some of galactic black hole X-ray sources show both thermal and non-thermal (power law) components, with the ratio of non-thermal to total luminosity of 20-40 per cent (Nowak 1995). Reconnection events and particle acceleration should also happen in rarefied strongly magnetized corona of the disc in our model and could cause observed X-ray flaring events. However, we do not consider the coronal dynamics here, and instead just focus on the structure and the emission spectrum of the disc itself.
Models of magnetized accretion discs with externally imposed large scale vertical magnetic field and anomalous magnetic field diffusion due to enhanced turbulent diffusion have also been considered (Ogilvie & Livio 2001; Shalybkov & Rüdiger 2000; Campbell 2000). The magnetic field in these models was strong enough to be dynamically important. But those models are limited to the subsonic turbulence in the disc and the viscosity and magnetic diffusivity are due to hydrodynamic turbulence. Angular momentum transport in those models are due to the large scale global magnetic fields. Both small scale and large scale magnetic fields should be present in real accretion discs. Here we consider the possibility that the magnetic field has dominant small scale component, that is magnetic field inside the disc consists mostly of loops with size less than or comparable to the thickness of the disc.
We consider vertically integrated equations describing the radial structure of the magnetically dominated turbulent accretion disc and provide the solutions for the radial dependences of the averaged quantities in Sect. 2. In Sect. 3 we analyse the conditions for a magnetically supported disc to be self-consistent. In Sect. 4 we calculate thermal emission spectra of magnetically supported disc taking into account scattering by free electrons.
Here we neglect effects of general relativity and do not consider the behaviour
of the material closer than the radius of the
innermost circular stable orbit
.
We assume a non-rotating black hole with
,
where the gravitational radius of the black hole of mass
is
.
We assume that accretion occurs
in the form of a geometrically thin accretion disc and verify this assumption
in Sect. 3.
We consider a disc of half-thickness H, surface density ,
averaged over the disc thickness volume density
,
accretion rate ,
and radial
inflow velocity v_{r}, v_{r}>0 for the accretion. We take
to be the angular Keplerian frequency at the radial distance r from the black
hole. Then, equation of mass conservation reads
Equation (4) can be solved to give H as
The free parameters are
and .
Also, we need to specify one more physical condition,
since the dependence of B on radius
in Eq. (7) is undetermined.
Such a condition should come from the modelling of supersonic turbulence.
Lacking a detailed model, we assume that the radial dependence of
the vertically averaged magnetic field in the disc is the power law
Let us now summarize the similarities and differences between our model and the standard Shakura-Sunyaev model. If we replace the thermal pressure in the standard model by the sum of the magnetic and turbulent pressures, the equations for mass conservation (1), angular momentum conservation (2), the viscosity prescription (5) and vertical pressure support (4) are the same as in the standard model. The pressure in the standard model is determined by the rate of the cooling of the disc, while the coefficient can be an arbitrary function of r, . In our model we have the magnetic pressure unspecified in its radial dependence as soon as it exceeds the thermal pressure, but for all r. The latter results from the much faster dissipation of supersonic turbulence than subsonic turbulence assumed in the standard model. Both our model and the standard model have only one undetermined function of radius, ( in the standard model and B(r) in our model). The determination of this free function would eventually come from detailed modelling of the MHD turbulence.
Now let us use the solution for the disc structure provided by Eqs. (6),
(7) and (15) and obtain constraints on free parameters of
the model, such that our model of thin magnetized accretion disc is self-consistent.
Using Eqs. (7) and (6) to substitute for
in
Eq. (1) we can express the radial inflow velocity
as
A necessary condition for the existence of a magnetically dominated disc is that
the vertical escape time for radiation must be shorter than
,
so that the energy density of
radiation,
,
remains smaller than the energy density of the
magnetic field
.
For optically thin,
geometrically thin discs this condition is always satisfied since the inverse of
escape time
.
As we will see, Thomson scattering
is the dominant source of opacity in most cases of optically thick discs.
The average time it takes for a photon to escape out of optically thick disc with
optical depth
is .
For Thomson scattering
,
where
is the number density of free electrons,
is the Thomson cross section
and
is the Thomson scattering opacity. For simplicity
we assume the composition of the disc to be completely ionized hydrogen.
Then,
is equal to the number density of protons in the disc, n. The necessary
condition now becomes
The dominance of the magnetic and turbulent energy compared to the energy density
of radiation is expressed as
.
One can substitute here for
from Eqs. (28) and (26). Heating rate Q is given by
Eq. (11). After using
Eqs. (18-20) to manipulate with ,
B and H,
one can reduce the condition of magnetic pressure dominance over the
radiation pressure to be
Figure 1: Plots of conditions when our model is valid. Plots are for and . On each plot values of radius extend from to . The panels differ by values of . On each panel, any given model of the disc is represented with a horizontal line . Filled areas indicate regions where a magnetically dominated geometrically thin and optically thick disc can exist. The difference in filling represents different types of spectra emitted locally from the surface of the disc: the regions with black body spectra are filled with lines in top-left to bottom-right direction and the regions with modified black body spectra are filled with lines in bottom-left to top-right direction. There are seven types of lines on each panel: (1) upper solid line curved upward on each plot bounds the region where radiation pressure is small compared to magnetic pressure and turbulent stress (below the curve); (2) lower solid line curved downward on each plot bounds the region where the disc is thin, i.e. H< r (above the curve); (3) long-dashed line bounds the region where thermal gas pressure is small compared to magnetic pressure and turbulent stress (below the curve); (4) short-dashed line separates the regions where the disc Thomson optical depth (above the line) and (below the line); (5) long-dashed and dotted line separates the region where the disc free-free optical depth (above the line) and (below the line); (6) short-dashed and dotted line separates the regions where the disc effective optical depth (above the line) and (below the line); (7) dotted line separates the regions where the disc free-free optical depth exceeds Thomson optical depth, , (above the line) from where , (below the line). | |
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Figure 2: Plots of conditions when our model is valid. Plots are done in the plane of B_{10} and for and . All notations are the same as in Fig. 1. | |
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Finally, the ratio of the disc semi-thickness H to the
radius r using (18) is
Depending on the power in the dependence of magnetic field , optically thick magnetically dominated accretion discs can exist only at a limited interval of radii. For (as on our plots for ) a thin magnetically dominated disc (shadowed regions on the plots) is possible for . The window for the strength of the magnetic field is not very wide: about one order of magnitude or less. This window is narrower for low masses of the central black hole and is wider for higher masses. The value corresponds to about the Eddington accretion rate for , and because of in our model (see Sect. 2). Higher values of correspond to higher accretion rates. Allowed values of the magnetic field are higher for higher accretion rates. The magnetic fields in the discs around higher mass black holes are smaller than in the discs around lower masses black holes as is temperature of the disc ( ) and the surface radiation flux. For large luminosities ( ) the inner disc cannot be optically thick for true absorption but can be optically thick to free electron scattering. Comptonisation becomes significant for in the inner regions at high luminosities (see Sect. 4). We leave consideration of Comptonised regimes for future work.
Physically, the limitations on the magnetic field strength can be understood as follows: suppose one decreases B_{10} while keeping constant. Then, is increasing; , and , both decreasing (Eqs. (18-20)). Scattering opacity through the disc strongly decreases, so the heat is transported to the surface faster and decreases (Eq. (28)); thermal and radiation pressures decrease as and respectively; plasma parameter decreases, so the disc becomes more magnetically dominated; and both decrease since their decrease due to lower overcomes the increase of the absorptive opacity from the drop of the temperature. Therefore, there exists only a limited interval of B_{10} such that still the disc is optically thick to true absorption.
Free-free, bound-free and cyclotron emission could contribute to the radiation spectrum. In Appendix A we show that, because of the self-absorption in the dense disc, the total flux of cyclotron emission from the disc surface is negligibly small compared to the total radiated power Q. This power is entirely due to free-free and bound-free radiative transitions. Cyclotron and synchrotron emission can be important in the rarefied and strongly magnetized disc corona (Ikhsanov 1989; Field & Rogers 1993a; Di Matteo et al. 1997), but our focus here is on the disc.
We perform a simplified calculation of the emergent spectrum. We assume local
thermodynamic equilibrium and do not consider effects of the temperature change with
depth. This is justified when the spectrum is formed in thin layer near
the disc surface. For simplicity we do not include the bound-free
contribution to the opacity. Free-free opacities
are the dominant source of thermal absorption for
,
so
our simplified spectrum is most relevant for smaller masses
of the central black hole, for which the inner disc is hotter. Our goal here is to
capture the effect of the magnetic field on the shape of the spectrum.
We consider only
optically thick disc models with both
and
.
The electron scattering opacity
does not depend on frequency in
the non-relativistic limit, whereas free-free absorption opacity
is a function of frequency:
The energy transfer due to repeated scatterings (Comptonisation process) is
characterized by Compton
parameter (Rybicki & Lightman 1979)
In the case of coherent scattering the approximate expression of the radiative
flux per unit surface of an optically thick medium is given by
(Rybicki & Lightman 1979)
When a significant interval of radii exists where the emitted spectrum is a modified
black body, e.g. ,
it is possible to get an approximate analytic
expression for .
For
we use expression (44)
for ,
which becomes
In summary, we see that magnetically dominated accretion discs have power law spectra with the spectral index depending on the radial distribution of magnetic field strength such that, . This contrasts the standard weakly magnetized -disc which shows a declining modified black body formed from the inner radiation dominated disc with .
As a side remark we note that the value for the spectral index close to 0 found for the latter regime of accretion disc by Shakura & Sunyaev (1973) (text on page 349 after Eq. (3.11) of that work) is different from ours . It is easy to follow the exact prescription of Shakura & Sunyaev (1973), namely calculate integral [3.10] in their work for spectrum [3.2] and temperature [3.7]. As a result we obtain rather than given in Shakura & Sunyaev (1973). We need to point out this discrepancy because it is widely stated in many textbooks on accretion discs (Shapiro & Teukolsky 1983; Krolik 1999) with the reference to Shakura & Sunyaev (1973) that high luminosity accretion discs have almost flat plateau in its spectrum before the exponential cut off corresponding to . However, the flat spectrum is produced by part (b) of the standard disc model, where gas pressure dominates over radiation pressure. The spectral index is close to the given in Shakura & Sunyaev (1973) but the radial dependence of the surface temperature in zone (b) is rather than given by their formula [3.7]. Thus, the standard -disc possessing both (b) and (a) zones should have spectrum steepening from plateau to and then exponentially cutting off at the temperature of the inner edge. Because the intervals of r, where approximate analytic expressions for emitted spectrum are valid, do not typically exceed one order of magnitude (the same is true for our disc model as well), one does not see "pure'' extended power laws when calculating spectra numerically by using general expressions (43) and (45). For example, Wandel & Petrosian (1988) found slope in the narrow interval between 1000 Å and 1450 Å by numerically integrating disc spectra.
Figure 3: Dependencies of the half-thickness of the disc H on radius for magnetically dominated disc (solid line) and Shakura-Sunyaev disc with the same and M_{8} parameters and viscosity parameter (short-dashed line) and (long-dashed line). The breaks in the curves for the Shakura-Sunyaev disc occur at the interface of zones (a) and (b) and are the results of using approximate analytic expressions in zone (a) and zone (b). For zone (b) extends down to the inner edge of the disc. | |
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We present results of the simplified analytical integration of the spectrum using Eq. (48) as well as more exact numerical integration using Eq. (43), solving for T from Eq. (46) and integrating Eq. (47). Function was kept in numerical calculations, so the results are applicable to the innermost parts of the disc, where the most of energy is radiated. For a given M_{8} and , an optically thick magnetically dominated discs exist within only for in the interval of about 1 to 1.4. In Figs. 3-11 we illustrate models for the four choices of parameter sets:
Figure 4: Dependencies of the surface density on radius for magnetically dominated disc (solid line) and Shakura-Sunyaev disc with the same and M_{8} parameters and viscosity parameter (short-dashed line) and (long-dashed line). The breaks in the curves for the Shakura-Sunyaev disc occur at the interface of zones (a) and (b) and are the results of using approximate analytic expressions in zone (a) and zone (b). For zone (b) extends down to the inner edge of the disc. | |
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In Fig. 4 we plot the dependencies of the column thickness through the disc (Eq. (19)) on the radius and also compare to Shakura-Sunyaev standard disc. The magnetically dominated disc is much less massive than the standard disc. Both and are smaller for magnetically dominated discs, and only in the inner are the densities comparable.
Figure 5: Dependencies of temperatures on radius: for magnetically dominated disc (solid line); for Shakura-Sunyaev disc with the same and M_{8} parameters and viscosity parameter (short-dashed line) and (long-dashed line); (dashed-dotted line). The breaks in the curves for the Shakura-Sunyaev disc occur at the interface of zones (a) and (b) and are the results of using approximate analytic expressions in zone (a) and zone (b). For zone (b) extends down to the inner edge of the disc. | |
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The dependencies of mid-plane temperature on radius given by Eq. (28) are shown in Fig. 5. On the same figure we also plot in the Shakura-Sunyaev model and given by Eq. (26), which is the same for magnetically dominated and standard discs. Because of the lower column density of the magnetically dominated disc, is less than for standard -discs.
Figure 6: Dependencies of pressures on radius for parameters of disc considered in Sect. 4. Magnetic plus turbulent pressure is plotted with a solid line, is plotted with a short-dashed line, and is plotted with a long-dashed line. | |
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The dependencies of magnetic plus turbulent pressure, , radiation pressure in the disc mid-plane , and thermal pressure in the disc mid-plane are presented in Fig. 6. We see that the assumption of magnetic pressure dominance is well satisfied for our models except in the innermost regions, , for higher accretion rates , where radiation pressure becomes comparable to the magnetic pressure. The latter fact limits the existence of magnetically dominated regime in the innermost parts of accretion discs for higher luminosities. Plasma parameter defined as decreases with radius and varies from 1 to 10^{-2} in our models.
Figure 7: Comparison of limit of magnetically dominated disc model with , , and M_{8}=1 and Shakura-Sunyaev model. Top-left plot shows the radial profiles of magnetic plus turbulent pressure (solid line), (short-dashed line), and (long-dashed line) in the magnetically dominated model. Bottom-left, top-right, and bottom-right plots show the radial profiles of , , and Hin the magnetically dominated model (solid lines) and the Shakura-Sunyaev model with (dashed lines). The parameters and M_{8} are the same for magnetically dominated model and Shakura-Sunyaev model. | |
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In the limit magnetic pressure is comparable to the largest of radiation or thermal pressures and our strongly magnetized disc model transforms into Shakura-Sunyaev model with . If in our model, then the radial scaling of the magnetic and turbulent pressures, , is the same as that of the radiation pressure inside the disc, , aside from the factor . If , then the radial scaling of is the same as that of the thermal pressure inside the disc, . Therefore, by choosing and adjusting the magnitude of B_{10} one can construct the model with approximately constant in the zone where radiation pressure exceeds thermal pressure. By choosing one can construct constant model in the zone where thermal pressure dominates radiation pressure. We illustrate this in Fig. 7, where we show the dependencies of pressures, , H, and on r for our model with , , and M_{8}=1, and for the Shakura-Sunyaev model with for the same accretion rate and M_{8}. The transition from zone (a) to zone (b) in this Shakura-Sunyaev model occurs at . The breaks on the curves corresponding to the Shakura-Sunyaev model occur at in Fig 7. We adjusted B_{10} such that the magnetic pressure will be in equipartition with the radiation pressure in our model. Then, as it is seen from the top-left plot in Fig. 7, thermal pressure is less than magnetic and radiation pressures for r less than some and exceeds magnetic and radiation pressures for , so our model is not applicable for . The subsequent three plots show that . Two right plots and bottom-left plot in Fig. 7 show that , H, and in our model for are very close to , H, and in Shakura-Sunyaev model in the radiation pressure dominated zone . A similar conclusion holds for the transition of our model with to a Shakura-Sunyaev zone (b) model for and . The radiation spectra of our model in the limit also approach that of the Shakura-Sunyaev model, as shown by direct numerical calculations. The power law modified black body spectrum derived in Sect. 4.1 becomes for and for , which is coincident with the modified black body power laws for the Shakura-Sunyaev zone (a) and zone (b) spectra (see Sect. 4.2).
The dependencies of optical depths through the half disc thickness on radius are shown in Fig. 8 for four parameter sets. The three curves plotted are: given by Eq. (36), , and the effective optical thickness . For , the effective optical thickness is almost constant throughout the disc, but when deviates from 5/4, starts to approach 1 either at the inner or at the outer edge of the disc and so our model breaks down at those radii. With the decrease of the accretion rate, the disc becomes cooler and denser so the absorbing opacity rises and becomes larger than the scattering opacity in the outer parts of the disc.
Figure 8: Dependencies of optical depth through the half disc thickness on radius for parameters of disc considered in Sect. 4. Dashed line is , dashed-dotted line is , and solid line is . | |
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Figure 9: Dependencies of temperatures on radius for parameters of disc considered in Sect. 4. Short dashed line is , long dashed line is , dotted line is , and solid line is T. The latter two temperatures are the surface temperatures of the disc calculated in Sect. 4. | |
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Figure 10: Dependencies of Comptonisation parameter y on radius for parameters of disc considered in Sect. 4. Solid line is for y calculated using temperature T, exact expression (42) for , and . Dashed line is for y calculated using , assuming that , and . | |
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Figure 11: Spectral energy distribution for the total flux from the disc. Frequency is plotted in units of . Exact values of calculated using temperature T are plotted with solid lines; values of calculated from analytical expression (50) are plotted with short-dashed lines. The latter are shown only for frequencies at which is larger than the minimal value of , which is achieved at about to . Spectra of Shakura-Sunyaev discs are plotted for viscosity parameter with long-dashed lines. | |
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In Fig. 9 we show: given by Eq. (26), given by Eq. (28), given by Eq. (49), and T(r) by solving Eq. (46) numerically. Note that and surface temperature T are always smaller than the for an optically thick disc. For low accretion rate, , and . In this case, is ill defined and values of are unphysical on the plot for and also for on the plot for , . The temperature becomes a good approximation for T when (scattering dominates over absorption in the surface layer). Unlike the values and slope of , which substantially increases with increasing , the value of T is less sensitive to : only inner parts of the disc becomes slightly hotter for larger values of . Both T and are changed significantly when the accretion rate or mass M are changed.
Figure 10 shows the results of calculating Comptonisation y parameter according to Eqs. (41) and (42). We conservatively set x=5 for the calculation of y. Then, y is the function of radius alone. On the same figure we also show y in the regime of modified black body spectrum, using and writing the simplified version of Eq. (41) as . We see that Comptonisation is not important for our models even in the inner disc.
Energy spectra
are presented in Fig. 11.
We normalized frequency to the characteristic frequency of an effective
black body from the inner disc, namely, we plot
versus
,
where
We have found self-consistent solutions for thin, magnetically supported turbulent accretion discs assuming the tangential stress . When compared to the standard -disc models (Shakura & Sunyaev 1973) magnetically dominated discs have lower surface and volume densities at the same accretion rate. This is due to the more efficient angular momentum transport by supersonic turbulence and strong magnetic fields than the subsonic thermal turbulence of the standard model. As a result, magnetically dominated discs are lighter and are not subject to self-gravity instability. In the limit of plasma , magnetic pressure is comparable to the largest of radiation or thermal pressures and our strongly magnetized disc model transforms into the Shakura-Sunyaev model with .
When we derived the disc structure, we made no explicit distinction between turbulent and magnetic pressure support and angular momentum transfer. As such, our model would be valid in any situation in which the magnetic and turbulent kinetic energies are comparable to, or greater than the thermal energy density. The assumption that the kinetic and magnetic energies are nearly comparable is natural because turbulence should result in the amplification of small scale magnetic fields in highly conducting medium due to dynamo action. Typically, in a sheared system, the magnetic energy can be even slightly larger than turbulent kinetic energy since the magnetic energy gains from the additional shear. We find that the thermal spectrum from the surface of the magnetically dominated disc in the optically thick regime is close to the spectrum of the standard Shakura-Sunyaev disc.
The issue arises as to how the magnetic field could reach sonic or supersonic energy densities. To obtain sonic turbulence and produce a disc, the MRI might be sufficient. To obtain a supersonic turbulence may require something else. One possibility in AGN appeals to the high density of stars in the central stellar cluster surrounding AGN accretion discs. Passages of stars through the disc might be an external source of supersonic turbulence analogous to the supernovae explosions being the source of supersonic turbulence in the Galaxy. Stars pass through the disc with the velocities of order of Keplerian velocity, which is much larger than the sound speed in the disc. We consider the support of turbulence by star-disc collisions in Appendix B and find that statistically speaking, star-disc collisions are unlikely to provide enough energy to sustain supersonic turbulence in most AGN accretion discs, however the possibility remains that a small number out of a large population could become magnetically dominated.
Indeed whether a disc could ever really attain a magnetically dominated state is important to understand. The present answer from simulations is not encouraging, but not completely ruled out. Further global MHD simulations of turbulence in vertically stratified accretion discs with realistic physical boundary conditions are needed along with more interpretation and analysis. Magnetic helicity conservation for example, has not been fully analyzed in global accretion disc simulations to date, and yet the large scale magnetic helicity can act as a sink for magnetic energy since magnetic helicity inverse cascades.
As an intermediate step in assessing the viability of low discs, it may be interesting to assess whether they are stable. One can take, as an initial condition, the stationary model of the magnetically dominated accretion disc given by expressions (15), (18-20) with the initial magnetic field satisfying all constraints of our model and falling into the shadowed regions on plots in Figs. 1-2.
One point of note is that magnetically dominated discs may be helpful (though perhaps not essential, if large scale magnetic fields can be produced, Blackman 2002; Blackman & Pariev 2003) in explaining AGN sources in which 40% of the bolometric luminosity comes from hot X-ray coronae. If the non-thermal component in galactic black hole sources is attributed to the magnetized corona above the disc (e.g., Di Matteo et al. 1999, also Beloborodov 1999 discusses possible alternatives), then magnetically dominated discs can naturally explain large fractions, up to 80% (Di Matteo et al. 1999), of the accretion power being transported into coronae by magnetic field buoyancy (although disc solutions are also possible, Merloni 2003). Though coronae can form in systems with high interiors, the percentage of the dissipation that goes on in the interior vs. the coronae could be dependent.
The main purpose of our study was simply to explore the consequences of making a magnetically dominated analogy to Shakura and Sunyaev, and filling in the parmeter regime which they did not consider. In the same way that we cannot provide proof that a disc can be magnetically dominated, they did not present proof that a disc must be turbulent, but investigated the consequences of their assumption. We also realize that the naive disc formalism itself can be questioned and its ultimate validity in capturing the real physics is limited. Nevertheless it still has an appeal of simplicity.
Finally, we emphasize that our model does not describe dissipation in the corona and interaction of the corona with the disc. Further work would be necessary to address relativistic particle acceleration and emission, illumination of the disc surface by X-rays produced in the corona and subsequent heating of top layers of the disc, and emergence of magnetized outflows.
Acknowledgements
V.I. Shishov is thanked for the remarks that improved presentation of the results. VP and EB want to acknowledge their stay at the Institute of Theoretical Physics in Santa Barbara, where part of this work was done. SB acknowledges his stay at the University of Rochester when this work was initiated. This research was supported in part by the National Science Foundation Under Grant No. PHY99-07949. VP and EB were also supported by DOE grant DE-FG02-00ER54600.
Since the characteristic temperature inside the disc (
given by Eq. (28)) is non-relativistic, cyclotron emission of an
electron occurs at frequencies close to multiples of the gyrofrequency
,
where
,
and
Cyclotron self-absorption also occurs in narrow lines centred on multiples of
.
At some frequency ,
emission and absorption occurs
only in spatially narrow resonant layers inside the disc,
where the magnetic field strength matches
the frequency, i.e.
is small. The width of emission and absorption
frequency intervals is determined mainly by the thermal Doppler shifts
(Zhelezniakov 1996).
The width of such resonant layers can be estimated as
.
Zhelezniakov (1996) (chapter 6) gives the expression for the optical thickness through
such gyroresonance layers on the second harmonic
,
which
for our disc is
When a star passes through a disc, it creates strong cylindrical shock propagating in the surrounding gas in the disc. The aftershock gas is heated to temperatures exceeding the equilibrium temperature in the accretion disc. As the shock weakens, this heating decreases until at some distance from the impact point the incremental heating becomes comparable to the equilibrium heat content. The scale substantially affected by a star passage is , much larger than the radius of the star . The shock front can become unstable and turbulence can occur in the aftershock gas. The heated gas becomes buoyant, rises above the disc and falls back because of gravity. Fall-back occurs with supersonic velocities and can further excite turbulence. Turbulence will derive energy from both heating by star passages and shear of the flow. The energy, which can be derived from shear, is equal to Q given by expression (12). It is possible that star-disc collisions might mainly be a trigger for the available shear energy to be converted into supersonic turbulence, and additional energy deposited into the disc by star-disc collisions is negligible. However, it seems unreasonable that the star-disc collisions can influence the global structure of the accretion disc unless the energy deposition from them is some fraction of the energy necessary to sustain turbulence level Q in the disc.
The energy deposition rate by stars per unit surface of the disc is
The resolution of observations is only enough to estimate the
number density of stars at about 1 pc for M32 and M31 and about 10 pc for
nearest ellipticals. In line with these observations we assume a star density
at 1 pc distance
from the central massive black hole (Lauer et al. 1995).
To estimate
for
we need to rely on the theory of central
star cluster evolution.
The gravitational potential inside the central 1 pc will be always dominated
by the black hole. Bahcall & Wolf (1976) showed that, if the evolution of
a star cluster is dominated by relaxation,
the effect of a central Newtonian point mass on an isotropic cluster would
be to create a density profile
.
However, for small
radii (0.1-
)
the physical collisions of stars
dominate two-body relaxations. Also, regions
near the black hole will be devoid of stars due to tidal disruption and
capture by the black hole. Numerical simulations
of the evolution of the central star cluster, taking into account
star-star collisions, star-star gravitational interactions,
tidal disruptions and relativistic effects were recently
performed by Rauch (1999). Rauch (1999) showed that star-star
collisions lead to the formation of a plateau in stellar density for small rbecause of the large rates of destruction by collisions.
We adopt the results of model 4 from Rauch (1999) as our fiducial model.
This model was calculated for all stars having initially one solar mass.
The collisional evolution is close to a stationary state,
when the combined losses of stars due to collisions, ejection, tidal
disruptions and capture by the black hole are balanced by the replenishment
of stars as a result of two-body relaxation in the outer region
with a
density profile. Taking into account the
order of magnitude uncertainty in the observed star density at 1 pc,
the fact that model 4 has not quite reached
a stationary state can be accepted for order of magnitude estimates.
For
we approximate the
density profile of model 4 as
We see that star-disc collisions cannot excite turbulence and strong magnetic
fields in the very inner part of the accretion disc, for
,
and such excitation
should be weak for
.
The relative width of the star depleted region,
,
decreases with increasing M.
For
and star-disc collisions happen all over the disc. For
and for
and star-disc collisions are unimportant for the structure of
the accretion disc. Let us assume that it should be
where the fraction
f is less than unity but not much less than unity. Further, we use
expression (12) for Q, expression (19) for ,
and
the value for n in the constant density core of the star cluster, second raw
in expression (B.2), to substitute into
.
Since the relation
should be satisfied
for all values of r, the value of
is determined and turns out to be
.
Solving the rest of the equation for the magnitude of magnetic field B_{10} at
we obtain
We explored all feasible range of parameters M_{8}, , f>10^{-3}, n_{5} < 10^{3} and found that the magnetic field calculated from expression (B.3) is always too strong to fall in the allowed range of parameters discussed at the end of Sect. 3. In particular, the constraint that magnetic and turbulent pressure dominate thermal and radiation pressure is violated. The minimum number density of stars necessary to satisfy this constraint at the most favourable values of other parameters still plausible for some AGN (M_{8} =40, , f=10^{-3}), turns out to be . Such a high number density of stars would imply total mass in stars of order of inside the central parsec from the central black hole. This mass exceeds observational and theoretical limitations.