A&A 402, 1013-1019 (2003)
DOI: 10.1051/0004-6361:20030320
E. Meyer-Hofmeister - F. Meyer
Max-Planck-Institut für Astrophysik, Karl- Schwarzschildstr. 1, 85740 Garching, Germany
Received 23 December 2002 / Accepted 21 February 2003
Abstract
We develop a new method to describe the accretion flow in the corona
above a thin disk around a black hole in vertical and radial extent.
The model is based on the same physics as the earlier one-zone
model, but now modified including inflow and outflow of mass,
energy and angular momentum from and towards neighboring zones.
We determine the radially extended coronal flow for different mass
flow rates in the cool disk resulting in the truncation of the thin
disk at different distance from the black hole. Our computations show
how the accretion flow gradually changes to a pure vertically extended
coronal or advection-dominated accretion flow (ADAF). Different
regimes of solutions are discussed. For some cases wind loss causes
an essential reduction of the mass flow.
Key words: accretion, accretion disks - black hole physics - X-rays: stars - galaxies: nuclei
The new fascinating results from XMM-Newton and the Chandra X-ray Observatory allow a deeper insight into the physical processes in many astrophysical objects, on scales ranging from binary stars to galaxies. One of these topics is the accretion onto black holes. This can happen in form of an advection-dominated accretion flow (ADAF) or via a standard geometrically thin Shakura-Sunyaev accretion disk. Since an ADAF is only possible in the inner region around the central accretor one has in many objects an advection-dominated accretion flow in the inner region, fed by the mass flow through a standard disk at larger distances from the black hole (for a recent review on ADAF models see Narayan 2002). The situation is the same for accretion onto black holes in galactic X-ray binaries and onto supermassive black holes in active galactic nuclei (AGN).
The spectra arising from these two different ways of accretion are very different, a soft multi-temperature spectrum from the black-body like accretion disk and a hard power law spectrum from the Comptonizing much hotter gas in an ADAF (one observes the combination of both originating at different distances from the black hole). The observed change of the spectral type, the soft/hard spectral transitions in X-ray binaries, were successfully modeled as related to a changing mass flow rate by Esin et al. (1997, 1998), thereby supporting this picture of an ADAF in the inner region surrounded by a standard disk. This means that at a certain distance from the black hole the accretion mode changes from the flow via a thin disk to a hot gas flow. The distance where this happens depends on the mass flow rate from outside.
Meyer et al. (2000) developed a model for a corona above a geometrically thin standard disk around a black hole (basically the same physics as already discussed earlier for disks of dwarf nova systems, where the compact object is a white dwarf (Meyer & Meyer-Hofmeister 1994)). The corona is fed by gas which evaporates from the cool thin disk underneath. An equilibrium establishes between the cool accretion stream and the hot flow. The efficiency of this process increases towards the black hole. This means that at a certain distance all matter is evaporated and the disk is truncated. From this distance on inward all gas is in the hot flow and proceeds towards the black hole as an advection-dominated flow. This original model is a simplifying description of the two-dimensional accretion flow (of vertical and radial extent with a free boundary condition on a radially extended surface). The resulting evaporation efficiency as a function of distance (and of the central black hole mass) allows to determine the location at which the thin disk is truncated.
For some X-ray binaries in quiescence the location of the inner edge of the disk can be deduced from the orbital velocity there, which is inferred from the observed line profiles. This requires that the disk temperature at the innere edge does not exceed the ionization temperature of hydrogen. Using the ADAF model for the fit to the spectra of soft X-ray transients as e.g. A620-00, V404 Cyg and Nova Mus 1991 the mass accretion rates were derived (Narayan et al. 1996, 1997). If one compares these results with the disk truncation radius from the evaporation model the agreement is reasonable (Liu et al. 1999; Meyer-Hofmeister & Meyer 1999). The change from disk accretion to an ADAF was also investigated for several low luminosity AGN and elliptical galaxies (Quataert et al. 1999; Gammie et al. 1999; Di Matteo et al. 1999, 2000), also the theoretically expected disk truncation (Liu & Meyer-Hofmeister 2001). The results confirm that the truncation of the thin disk is located at smaller distance from the black hole for higher mass flow rates in the thin disk. Only for low luminosity AGN the observed spectra seem to demand a disk truncation at radii too small for the accretion rate (Quataert et al. 1999). But this apparent discrepancy might be resolved by the effect of magnetic fields from a dynamo in the underlying disk on the coronal gas flow (Meyer & Meyer-Hofmeister 2002). We now present work which is an essential step beyond the one-zone model. We develop a new method to describe the accretion flow in the corona above a thin disk in its vertical and radial extent. The model is based on the same physics as in the one-zone model, but now modified including inflow and outflow of mass, energy and angular momentum from and towards neighboring zones. In the earlier model inflow from outward regions was neglected. But this is necessary if one considers regions inside the evaporation maximum (in the one-zone model at about 300 Schwarzschild radii, compare Fig. 3 in Meyer et al. 2000). For the inner regions it is also important to take into account different ion and electron temperature and, in the case of a high mass flow rate in the thin disk, the effect of Compton cooling of coronal electrons by disk photons. Both were included in the one-zone model description by Liu et al. (2002), where emphasis was put on evaluation of the coronal flow in the case of high accretion rates as in narrow-line Seyfert 1 galaxies.
The co-existence of hot and cold gas around galactic black holes and in AGN was also investigated by Rózanska & Czerny (2000a,b). Different ion and electron temperatures were already included. In their work the physical picture is basically the same as in ours, but the results differ in detail. The investigation focuses on the innermost region near the black hole. In earlier work (Meyer-Hofmeister & Meyer 2001) we discussed the difference of the results. Processes of evaporation very close to the black hole were studied by Spruit & Deufel (2001).
In Sect. 2 we discuss the physics of interaction between the hot corona and a cool disk underneath. We introduce the modifications necessary for a consistent treatment of the mass, energy and angular momentum flow. We describe the procedure how one finds the solution for the two-dimensional accretion flow, put together from the separately computed structures for different radii, and we present the new computational results (Sect. 3). This multi-zone model describes the increase of the mass flow in the hot corona towards the black hole. A part of the matter is lost in a wind from the hot corona. In Sect. 4 we discuss the different regimes of solutions, wind loss from the coronal flow and the consequences of a varying mass flow rate in the thin disk on spectral transitions.
To describe the structure of the corona above the cool disk we take the standard equations of viscous hydrodynamics: conservation of mass, the equations of motion and the first law of thermodynamics. We want to determine the vertical structure of the corona at different distances r from the black hole with emphasis on the inner accretion regions. For the earlier "one-zone model'' only the zone at the edge of the thin disk was considered. Since the evaporation efficiency increases steeply in radial direction inward this approach gives already good results (but only for distances from the black hole where the evaporation efficiency indeed increases inward). We now want to investigate the coronal structure in its full extension in two dimensions. We determine the vertically extended coronal structure for a series of successive radial zones. In this multi-zone model the divergence in radial direction is replaced by inflow/outflow of mass and angular momentum in the zones (in the former one-zone model this could only be taken into account approximately).
We use cylindrical coordinates with the z-axis perpendicular to the disk midplane. We consider stationary and azimuthally symmetric flows. We use basically the same equations as for the one-zone model, also used in Liu et al. (2002), but modified to take mass, energy and angular momentum inflow and outflow in its radial dependence into account.
In the following we list these five ordinary differential equations. The dependent variables are the vertical mass flow ( density, v_{z} vertical flow velocity), the vertical component of the heat flux , the pressure P and the ion and electron temperature and , the independent variables are r and z.
We use the following equation of state
(1) |
From the conservation of mass we get our first equation
(2) |
The second equation is the z-component of the equation of motion,
(3) |
The vertical conductive heat flux provides our third equation
(4) |
The two remaining equations, the energy equations for ions and for
electrons, as derived by Liu et al. (2002),
include the cooling and heating processes in the hot
corona and the equations are now modified for the multi-zone model.
For ions the energy balance is determined by viscous
heating, cooling by collision with electrons and radial and vertical
advection. The friction is taken proportional to the pressure with a
standard -prescription.
(5) |
(6) |
(7) |
For the multi-zone model there are two changes compared to the one-zone model, (1) the mass and energy modification factors and (2) we now take into account the radial dependence of in the formula for the specific energy (see Sect. 2.3.3).
For electrons the energy balance is determined by the processes of heating by collisions with ions, cooling by bremsstrahlung (free-bound and free-free transitions), Compton cooling, and vertical thermal conduction.We neglect here the radial thermal conduction in a first approximation. In the main part of the vertical structure solution it is not a dominant term (see Meyer et al. 2000).
The second energy equation is
= | (8) |
In the energy balance for the electrons one could omit the terms multiplied with due to the low ratio of electron to ion mass. We keep the equation for ions, Eq. (5), and replace the equation for electrons by a joint equation for electrons and ions (with u the specific energy of electrons and ions).
We first derive the formula for the radial diffusive velocity and then the mass and energy advection modification terms. For simplicity these modifications are taken in a height averaged way.
(9) |
(10) |
(11) |
(12) |
(13) |
(14) |
(15) |
= | |||
= | (16) |
(17) |
(18) |
(19) |
(20) |
(21) |
As the lower boundary of our calculations at z=z_{0}, we take the level where electron temperature and ion temperature are already about the same and have the value K. Below an analytic solution suffices to cover the small remaining extent before the disk chromosphere is reaching it and yields a relation between pressure and downward heat flux (for a more detailed discussion see Meyer et al. 2000).
At the upper boundary with no pressure confinement at infinity we require sound transition at some height z=z_{1} (free boundary), . Further with no influx of heat from infinity there, we require neglecting a small remaining outward heat flow.
This constitutes 5 boundary conditions for the 5 ordinary differential equations in z.
For the determination of the coronal mass flow in a radially extended region the difference of inflow and outflow in a succession of radial zones is now properly accounted for. This means that the coronal mass flow at a certain radius depends on the coronal structure in the neighboring zones and must be determined iteratively.
We first compute the coronal structure for
successive distances and a series of values for the modification terms
to construct a grid of curves
.
The derivative of f only appears in and in the modification terms. We therefore take these terms together
as
(22) |
Figure 1: Values of the viscosity integral for different parameters, basic curves for the evaluation of the coronal flow. solid lines: , (0.5, 0.5), (1, 1); long dashes: (0.5, 1), (1, 1.5) For (0, 0.2) the line is approximately the same as for (0, 0). Thick line: example of a consistent curve log f, including the mass, energy and angular momentum flow consistently (in our computations we investigate the accretion flow around a black hole). | |
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What we finally need is a series of coronal structures
at each distance determined for given
values
(
indicates the dependence on both
and
)
so that the curve
log
has the slope
that fits to these values taken for
.
From Eqs. (19), (21), and (22) and the definition of
s we obtain
(23) |
We use the following procedure. We start at a chosen distance where the thin disk is truncated. There the boundary condition is which means that all angular momentum carried inward into the inner disk free region is returned by friction (see discussion in Meyer et al. 2000). Now we construct a consistent log f curve step after step. Step 1: we choose an initial value of log (from our computed grid of curves); from Eq. (23) we see which value has to be chosen for to give the values and that belong to the chosen initial value of f(the derivative is approximated by a difference quotient , we used ). This determines the slope s with which to proceed to the next log f value at the next radius. Step 2: the slope determined in step 1 yields the next value for log f; again we determine the further slope so that relation (23) is fulfilled. Each step thus determines the further outward slope sof the log f curve so that the values and are consistent.
For a given truncation radius and for each initial value of log fwe get with this procedure one consistent log curve. A second boundary condition at an outer boundary of the interval for which we determine the coronal flow allows to discriminate between the various curves that belong to one truncation radius. The slope at the outer distance is a possible boundary condition. For s= 1/2 the radial diffusive velocity becomes zero, no mass flow inward or outward. Then closer to the black hole the mass flows inward, at larger distance the mass flows outward. In different disks the outer boundary condition might be different. In the case of X-ray binaries it seems plausible that tidal forces at the disk boundary prevent outer coronal mass outflow, that is becomes -1/2 there.
Figure 2: Consistent solutions for log f for truncation radii log r=8.1 to 9.3. Diamonds on two curves mark where . The coronal flow at the truncation radius is proportional to f. Dashed parts indicate uncertain extrapolation outside of the computed basic grid curves. The dotted curve is identical with the uppermost curve in Fig. 1 (accretion flow around a black hole). | |
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Comparing the f curves of the solutions with the field which our basic grid curves cover in the log f-log r diagram we see that the solution f- curves can easily lead into areas outside of the grid, either to higher or lower values of log f. If they stay within the range of computed basic curves necessarily the slope becomes less than -1/2. No consistent solutions have been constructed which have at a very large distance. The too steeply decreasing curves correspond to the situation that the coronal flow is inward only near the inner boundary but turns into an outflow already a short distance away from it. To continue these curves correctly we need the coronal structures for high values, which means a high net radial mass outflow. The flatter curves also can leave the field of standard solutions, that is . The corona would then have a negative radial net outflow, which means less mass radially leaves the zone at the inner radius than enters into it at the outer radius.
Figure 3: Upper panel: solid lines: coronal mass flow rate above the thin disk around a black hole, truncated at log , mass and energy flow modification values and . evaporation rate (see text). Dashed lines: and for disk truncation at log . Lower panel: mass flow rate in the thin disk (gray area) and in the corona (dotted area) as function of distance r from the black hole, area remaining above indicates the rate of gas loss in the wind from the corona. | |
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Figure 4: Thick line: series of solutions for the coronal mass flow rate , for disk truncations at log r from 8.5 to 9.7; thin lines: ion and electron temperature at coronal height z=r together with the virial temperature . | |
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