Effects of wind on radiation spectra from magnetized accretion discs

Abstract

The effects of a wind on the emerging spectrum from an inefficiently radiating accretion flow in a global magnetic field are examined, based on the analytic solution obtained recently by one of the present authors. The results exhibit the steepening of the negative slope appearing in the intermediate frequency range of bremsstrahlung spectrum and the decrease in the luminosity ratio of thermal synchrotron to bremsstrahlung radiation, in accordance with the increasing wind strength. Both effects are caused by a suppressed mass accretion rate in the inner disc, caused by a mass loss in terms of wind.

In order to demonstrate the reliability of this model, Sagittarius A* (Sgr A*) and the nucleus of M31, both of which have been resolved in the X-ray band by Chandra, are taken up as the best candidates for the broad-band spectral fittings. Although the observed X-ray data are reproduced for these objects by both of the inverse-Compton and the bremsstrahlung fittings, some evidence of a preference for the latter is recognized. The wind effects are clearly seen in the latter fitting case, in which we can conclude that a widely extending accretion disc is present in each nucleus, with no or only weak wind in Sgr A* and with a considerably strong wind in the nuclear region of M31. Especially in Sgr A*, the inferred mass accretion rates are much smaller than the Bondi rate, the estimate of which has become reliable because of Chandra. This fact strongly suggests that the accretion in this object does not proceed in accordance with Bondi's prediction, though its extent almost reaches the Bondi radius.

accretion, accretion discs, MHD, galaxies: individual: M31, galaxies: individual: Sgr A*, galaxies: nuclei

1 Introduction

The broad-band spectrum of Sgr A*, from radio to hard X-ray bands, was reproduced fairly well for the first time by Narayan, Yi & Mahadevan (1995) based on an optically thin advection-dominated accretion flow (ADAF) model. In contrast with the standard accretion-disc model (Shakura & Sunyaev 1973), the optically thin ADAF model could reproduce the wide spread of the observed spectrum. Namely, the rather narrow peak in the radio band was explained by thermal synchrotron radiation from the inner part of a disc with relativistic temperature and X-ray luminosity by bremsstrahlung radiation from the whole disc. Another great advantage of this model is in its low efficiency in producing radiative fluxes. The latter feature could allow the model to explain the low luminosity of Sgr A* compared with that expected from a standard disc of approximately the Bondi accretion rate (see, e.g., Melia & Falcke 2001).

This type of model for optically thin ADAFs has been developed by many authors (e.g. Ichimaru 1977; Rees et al. 1982; Narayan & Yi 1994, 995a,b; Abramowicz et al. 1995; Blandford & Begelman 1999). Since, in this category, the viscosity of the accreting plasma plays a dominant role in both energy dissipation and angular-momentum extraction processes, we call it the viscous ADAF model. The model seemed to be successful in explaining the broad-band spectra not only of Sgr A* (Narayan et al. 1995; Manmoto, Mineshige & Kusunose 1997; Narayan et al. 1998b), but also of Galactic X-ray sources and of low-luminosity active galactic nuclei (LLAGNs; for a review, see Narayan, Mahadevan & Quataert 1998a).

Meanwhile, if the presence of ordered magnetic fields in the nuclear regions of galaxies is taken seriously, another type of optically thin ADAF model can be constructed (Kaburaki 2000; K00 hereafter). Since, in this model, the gravitational energy of accreting matter is liberated through the electric resistivity of the plasma it will be called the resistive ADAF model. Angular momentum, on the other hand, is extracted from the accreting matter by an ordered magnetic field that is penetrating the disc and is twisted, to a certain extent, by the rotational motion of the accreting matter. The observed spectrum of Sgr A* could also be reproduced by this model (Kino, Kaburaki & Yamazaki 2000, KKY hereafter) as satisfactorily as the viscous ADAF models could.

Recently, however, a fairly drastic change in the above stated situation has arised with the appearance of the results of the X-ray telescope, Chandra. Owing to its high resolution, the apparent nuclear sources of a few nearby galaxies have been further resolved into several point sources including a true nuclear source in each case (e.g. Garcia et al. 2000 for M31; Baganoff et al. 2001a for Sgr A*). The intrinsic X-ray luminosity of these true nuclear sources are, in fact, therefore considerably lower and moreover, their spectra have turned out to be much softer than previously believed.

In particular, this softness of the spectra requires a critical reconsideration of the broad-band spectral fittings of these objects by optically thin ADAF models, because this fact may exclude the hitherto accepted bremsstrahlung fitting to the X-ray band of the spectra. Furthermore, as has already been demonstrated by some authors (e.g. Quataert & Narayan 1999; Di Matteo et al. 2000), the presence of winds emanating from accretion discs can alter the ratio of X-ray luminosity to radio luminosity. Therefore, it is also necessary to include the presence of winds in the basic models, based on which the spectral fittings are performed. In this context, the resistive ADAF model has been developed to include winds emanating from the disc surfaces (Kaburaki 2001, K01 hereafter).

The present paper is devoted to describing general predictions on the broad-band spectra radiated from such accretion discs as described by the analytic model of K01, and to report the results of applications to Sgr A* and the nucleus of M31. Although Di Matteo et al. (2000) insist on the presence of winds in the nuclei of some nearby elliptical galaxies on the basis of their broad-band spectral fittings by a wind-version of the viscous ADAF model, the results are still uncertain because the X-ray fluxes they used are obtained by ASCA and hence the nuclear sources are not resolved (see also Quataert & Narayan 1999 for a Galactic X-ray transient and Sgr A*). We therefore restrict our applications only to objects that have been resolved by Chandra and the results of observations have been open to the public.

2 Resistive Adaf Model Including Winds

In this section, we first introduce the basic ideas and physics contained in the resistive ADAF model proposed in K01, in which the presence of winds from the disc surfaces is allowed for. A schematic drawing of the global configuration presumed in this model is given in Fig. 1. An asymptotically uniform magnetic field is vertically penetrating the accretion disc and is twisted by the rotational motion of the plasma. Owing to the Maxwell stress of this twisted magnetic field, a certain fraction of the angular momentum of the accreting plasma is carried out to infinity, and this fact ensures that the plasma gradually infalls towards the central black hole.

Figure 1

Conceptual drawing of the global configuration. The poloidal and toroidal components of the magnetic field lines are drawn as thin solid lines. The wind emanating from the disc surfaces and the poloidally circulating electric current are shown with white and black arrows, respectively. Also shown is the Lorentz force acting on the return current in the polar region, which has the components both for collimation and for radial acceleration of a jet. We distinguish jets from winds by their location and by the mechanisms for their launching.

In a stationary state, the deformation of magnetic field lines is determined by a balance between motional dragging and diffusive slippage of the field lines. Since the representative magnetic Reynolds number ℜ is large [i.e. ℜ²(r) ≫ 1] in the disc (except for the region close to its inner edge, where ℜ∼ 1), the deformations are also large. In this sense, the disc can be said to be as weakly resistive. Generally, the deformation in the toroidal direction is larger than that in the poloidal direction (i.e. b_ϕ/b_p∼ℜ, where b denotes the deformed part of the magnetic field).

The vertical structure of the disc is maintained by a pressure balance between the magnetic pressure of the toroidal field, which is dominant on the outside the disc, and the gas pressure in the disc; i.e. the accreting plasma is magnetically confined in a geometrically thin disc. Reflecting this fact, the gas pressure and density in the disc become quantities of the order of ℜ² (see equations 49 and 58 in K01). In general, this balance is not a static balance in its strict sense, and there may be a vertical flow from the upper and lower surfaces of the disc. We call such outflows (or inflows depending on the case) winds, and distinguish them from jets that may be formed within the inner edge of the accretion disc (also see below).

Since the wind velocity obtained in K01 is much smaller than the rotational velocity (by a factor of the order of ∼Δℜ⁻¹, where Δ≪ 1 is the half-opening angle of the disc), its inertial force can hardly affect the vertical force balance described above. However, the wind may be accelerated, by some mechanisms with which we do not concern ourselves in this paper, to a considerable speed after it has been injected from the accretion disc to the wind region outside the disc. Here, we only expect that the upward wind proceeds to infinity because its total energy per unit mass (i.e. the Bernoulli sum in K01) is positive and hence the flow is unbounded in the gravitational field.

The rotational velocity is a certain fraction of the Kepler velocity (i.e. a reduced Keplerian rotation) because the radial pressure-gradient force, together with the centrifugal force, sustains the gravitational pull on the plasma. In contrast to the rotational velocity, the infall velocity is a small quantity of the order of ℜ⁻¹ as long as the disc is weakly resistive (i.e. except for the region near the inner edge). The ratio does not depend on the disc thickness Δ as long as ℜ is regarded as a free parameter such as the viscosity parameter α in the viscous ADAF models.

It has been demonstrated in K01 that if the flow is completely adiabatic (i.e. if the liberated gravitational energy remains in each fluid element and is merely advected down the flow), the flow cannot drive winds. On the other hand, if some mechanism of energy transport (i.e. non-adiabaticity) allows the energy flow towards the accretion disc from the region within its inner edge, the disc can drive an upward wind. In K01, the presence of such mechanisms is treated implicitly in terms of a parameter n that specifies the strength of a wind. The essence of the results is in that the presence of such a non-adiabaticity affects solely the radial profiles of the quantities such as the density, pressure and magnetic field components (velocity components and temperature are not affected) but not on the vertical profiles. Thus, the presence of a wind appears unambiguously as a radius-dependent mass accretion rate.

The analytic solution obtained in K01 describes the physical quantities in the accretion disc (i.e. the solution is only valid between the inner and outer edges of the disc) with decreasing accuracy toward the upper and lower surfaces of the disc. The latter part of this statement means that the solution admits a few inconsistencies of the order of tan h²η, where η≡ (θ−π/2)/Δ, in its variable-separated forms. These are negligible, however, except near the disc surfaces. Furthermore, the solution is not dependent on the radiation loss. Nevertheless, since it has been confirmed retrospectively that the expected radiation flux from such a disc is negligibly small as long as the accretion rate is sufficiently smaller than the Eddington rate, the solution is consistent within this restriction. The disc spectrum can, therefore, be calculated with sufficient accuracy by using this solution, without any correction to the radiation losses.

If we extrapolate our physical understanding obtained within the disc even to the surrounding space, we are naturally led to the following picture of a galactic central engine. The engine is essentially a hydroelectric power station in which the ultimate energy source is the potential energy of the accreting plasma in the gravitational field. The accretion disc is a DC dynamo of a magnetohydrodynamic (MHD) type and drives a poloidally circulating current system, which is the cause of the toroidal magnetic component added to the originally vertical field (i.e. the twisting of the field lines). In the configuration shown in Fig. 1, the radial current is driven outward in the accretion disc, and part of the current closes its circuit through the near wind region while another part closes after circulating remote regions (probably reaching the boundary of a ‘cocoon’ enclosing hot winds). Anyway, these return currents concentrate within the polar regions in the upper and lower hemispheres, and finally return to the inner edge of the disc.

A bipolar jet may be formed from the plasma in the polar current regions, because the Lorentz force caused by the toroidal field has always both of the necessary components for collimation and for radial acceleration, as shown in the figure. The often suggested universality of the association of an accretion disc and a bipolar jet is thus understood naturally in terms of one physical entity, the poloidally circulating current system. It is very likely that only a small fraction of the infalling matter input in the disc can actually fall on to the central black hole, with the remaining part being expelled as a bipolar jet and a wind from disc surfaces.

3 Scaled Quantities

The radial and co-latitude dependences of all the relevant quantities in the resistive ADAF solution are given in K01. In rewriting these quantities in a scaled form convenient for the discussion of LLAGNs, the following normalizations are adopted:  

1

where and B₀ denote the radial distance, black hole mass, mass accretion rate at the outer edge of the disc, r =r_out, and strength of the external magnetic field at r_out, respectively. The radius is normalized to the gravitational radius of the central black hole (actually neglecting its rotation, by the Schwarzschild radius r_S) and the accretion rate, by the critical accretion rate defined in terms of the Eddington luminosity L_E as , taking into account a typical conversion efficiency of 0.1.

Furthermore, we eliminate ∣B₀∣ with the aid of the relation (see K01)  

2

or  

3

where G is the gravitational constant. This is because B₀ seems rather difficult to infer from observations compared with the outer edge radius r_out. Substituting the latter expression into the radial part of the variable-separated forms of the relevant quantities, we finally obtain  

4

 

5

 

6

 

7

 

8

and  

9

 

10

 

11

where v_ϕ, ρ, p and Δ are the rotational velocity, density, gas pressure and the half-opening angle of the disc, respectively, and the tildes above them denote their radial parts. The opacity is dominated by the electron scattering, and the resulting optical depth in the vertical direction τ_es (x) depends on the radius such as x⁻²ⁿ_outx²ⁿ = (r/r_out)²ⁿ. Therefore, it is much smaller than unity as far as and x_out≫ 1 since r/r_out < 1. We can confirm the explicit dependence of the accretion rate on the radius in equation (10).

Since the radii of inner and outer edges are mutually related by x_in =ℜ⁻²₀x_out (see K01), they are written in the present units as  

12

Although the numerical values appearing in the above scaled expressions are their formal values at the black hole horizon, a direct inward extension of the above solution beyond the inner edge is meaningless because the solution becomes invalid there. Finally, we note that the number of free parameters contained in our solution is five, i.e. and n.

4 Radiation Processes

The method of calculating the expected spectrum from an accretion disc, based on the analytic solution of the resistive ADAF model including winds, is essentially the same as stated in KKY for the no-wind version of that model. The main radiation mechanisms are thermal cyclo-synchrotron emission (most of which contributes to the flux in the radio band), thermal bremsstrahlung (that may contribute mainly in the X-ray band) and inverse Compton scattering of the synchrotron (and bremsung) photons, which may appear as one or two peaks in the middle frequency range (and the scattered bremsung flux that may be scarcely recognized above the exponentially decaying part of the spectrum).

In the present scheme there are, however, two main differences with respect to the former version. One is an improvement in the treatment of the Gaunt factor and the other is in the approximation used in evaluating the unscattered (the sum of synchrotron and bremsung processes) flux from a disc. They are described in the following subsections. Hereafter, we need only the radial parts of the relevant physical quantities, and hence we quote them without tildes for simplicity.

We first assume that an observed spectrum from a galactic nucleus is produced only by its accretion disc, and neglect the possible contributions from other components such as a jet or a wind. The validity of this assumption will be discussed in Section 5.3.

4.1 Unscattered flux

Since the Compton unscattered flux consists of the synchrotron emission and the bremsung (both of which are assumed to be isotropic), the absorption coefficient is calculated from Kirchhoff's law  

13

where χ_ν denotes the volume emissivity for both processes and B_ν is the Planck function. Assuming local thermodynamic equilibrium (LTE), we have for the specific intensity  

14

where τ_ν =τ*_ν/ cos θ, τ*_ν =α_νH and H is the height of the disc.

In this situation, we approximate the emerging flux by an interpolation formula as  

15

This formula has the merit of giving the correct answers πB_ν and (χ_ν,br+χ_ν,sy)H/2 in the optically thick and thin limits, respectively. The difference between this and the two-stream approximation adopted in KKY, however, remains small.

4.2 Bremsstrahlung

According to Narayan & Yi (1995b) and Manmoto et al. (1997), KKY used the non-relativistic version of the Gaunt factor g_ff (ν, T) not only for electron—ion collisions but also for electron—electron collisions, and a constant frequency-integrated Gaunt factor g_B (T). Compared with the stricter treatments of these quantities by Skibo et al. (1995), the old scheme evidently overestimates the contribution from the e—e process when hν < m_ec² (where h is Planck's constant and m_e is the electron mass), and underestimates it by approximately one order of magnitude when hν>m_ec² (for more details seeYamazaki 2002). Therefore, in this paper we adopt the formulae given by Skibo et al. (1995). The effect of this change on a spectrum is illustrated in Fig. 7 later (see Section 5.3; the thick and thin solid curves are calculated using the new and old versions, respectively, for the same fitting parameters).

Figure 7

Comparison of various fittings. As a reference curve, the best-fitting curve with no wind in Fig. 3 (thick solid curve) is adopted. The thin solid curve is the prediction of our old version of the calculation scheme, for the same values of the fitting parameters. Their difference results from the difference in the adopted Gaunt factors. The thin curve runs slightly above the thick curve until it falls rapidly at higher frequencies. The small dip above the peak B is artificial. The dashed and dotted curves are the predictions in the present scheme, when the central mass alone is replaced by the dynamical mass and when the accretion rate alone is replaced by the Bondi rate, respectively.

In order to grasp the basic ideas, we first assume that the Gaunt factor is nearly constant even in the relativistic temperature regime. Then, the luminosity caused by bremsstrahlung is written roughly as  

16

where k_B is the Boltzmann constant, dV = 4πΔr² d r is the volume element in the disc and  

17

Evaluating this integral f properly in each of the typical frequency ranges, we have the frequency dependence  

18

In fact, however, in addition to the above features, there may appear an increase in the luminosity in the frequency range ν∼k_BT_in/h, which is caused by the enhancement of the Gaunt factor at relativistic temperatures (see peak B′ in Fig. 3 in Section 5.1).

Figure 3

Bremsstrahlung fittings to Sgr A*. The Chandra X-ray spectrum is fitted by negative slopes appearing in the intermediate frequency range of thermal bremsstrahlung spectra. The solid curve represents the no-wind case (n = 0), while the dashed curve represents a case of weak wind (n = 0.1). As seen in each curve, the contribution from the bremsstrahlung has double peaks. The higher-frequency peak (B′) is caused by an enhanced Gaunt factor in the relativistic temperature range.

The predicted power-law index of the luminosity νL^bs_ν, i.e. on the frequency, in the intermediate frequency range (k_BT_out/h < ν≪k_BT_in/h) exhibits an explicit difference from the corresponding prediction by the viscous ADAF model including winds (Quataert & Narayan 1999), (where p = 2n). In the no-wind case (n =p = 0), the former predicts a negative slope while the latter predicts a positive slope in the ν—νL_ν diagram. This difference comes from the difference in the radial dependences of the density in the basic analytic solutions (K00 and Narayan & Yi 1994, respectively). In both models, the inclusion of a wind (n > 0) equally reduces these values depending on its strength. This is because the presence of a wind reduces the contribution from the inner parts of a disc to the bremsstrahlung (since ), resulting in a reduction of the higher-frequency side of the spectrum.

4.3 Synchrotron emission

The adopted formulae for the calculation of synchrotron emission and inverse Compton scattering is the same as in KKY. This synchrotron formula is known to be accurate when the electron temperature T_e is in the range 10⁸ < T_e < 3 × 10¹⁰. This temperature range corresponds in our solution to the radius range of 30 < x < 10⁴. Since, however, the contribution from the lower-temperature region (x > 10⁴) is negligibly small compared with that from the bremsung, we can even extend our calculation to such regions (until the effects of recombination on the bremsung becomes important), by safely truncating the cyclo-synchrotron emission.

The critical frequency under which the radiation becomes optically thick at a fixed r is given (Narayan & Yi 1995b; Mahadevan 1997) by  

19

Since ν_c increases towards the inner edge, the inner part of a disc is apt to be optically thick at a given frequency ν, with a critical radius  

20

Anyway, there is a radius x_p near x_in where the contribution to the self-absorbed radiation is the largest so that the peak frequency is determined as ν_p =ν_c (x_p).

Then, for the self-absorbed luminosity per frequency [ν_min < ν < ν_p, where ν_min =ν_c (x_out)], we have  

21

where the final expression is only valid in the frequency range ν≪ν_p in which x_c≫x_in. In the crude approximation in which x_c is regarded as a constant, the luminosity of the self-absorbed synchrotron emission is determined solely by the central mass m.

However, substituting the above estimation for x_c in equation (20), we have a more accurate dependence,  

22

The strongest dependence on the parameters is still on the central mass, and its power varies slightly from 1.80 to 1.85 for the variation of n in the allowed range, from 1/2 to −1/4. The dependence on the other parameters are rather weak, and this fact is very favourable to determine, at least within the framework of the present model, the central mass from the spectral fittings of this self-absorbed part, without much ambiguity (however, see also Section 5.3). The power of ν also varies only slightly from 1.6 to 1.7 over the same range of n.

The main effect of a wind on the synchrotron spectrum is a reduction of its peak intensity (when n > 0) according to the wind strength. This is because the synchrotron emission comes mainly from the innermost region of an accretion disc where the magnetic field is the strongest and the electrons are most energetic, so that the reduction of the accretion rate in this region caused by wind loss causes a decrease in the optically thin part of the emission.

5 Applications

As stated in the previous section, we can calculate the spectrum emanating from an accretion disc by specifying the values of five parameters, m and n. In other words, we can determine these values from the process of spectral fitting only if we have enough data points distributed over the full range of the spectrum of a specific object. In the applications of our model to Sgr A* and M31, we first proceed in this spirit, and regard all five parameters as free fitting parameters.

In fact, however, there is other information independent of the spectral data. For example, the central black hole masses are known for these objects fairly accurately by the observations based on the dynamics. In the case of Sgr A*, the Chandra observations provide us with a good estimate of the Bondi accretion rate. In principle, the results obtained from a spectral fitting should not conflict with this additional information. If, however, there are some discrepancies between them, it may offer some important information concerning the inadequacy of the present status of the basic model or of our general understanding. Such considerations will be given in Section 5.3.

5.1 Sgr A*

Chandra has resolved a weak source at the radio position of Sgr A* within an accuracy of 0.35 arcsec (Baganoff et al. 2001a). Its absorption-corrected luminosity in the 2–10 keV band is 2.4^+3.0_−0.6× 10³³ erg s⁻¹ and the power-law fitted photon index is 2.7^+1.3_−0.9. For the observational data aside from this X-ray band, we use those compiled by Narayan et al. (1998b). For the sake of comparison, the ROSAT value is also plotted in some figures (2, 3 and 7) as an empty circle. Among the radio observations, the 86-GHz point with the highest resolution obtained by very long baseline interferometry (VLBI) is taken most seriously in the following spectral fittings.

The most restrictive feature of the Chandra results to the spectral fittings is the softness of the X-ray spectrum, the most probable slope of which in the ν—νL_ν diagram is negative, in contrast to the ROSAT observation in which the slope in a similar X-ray range is positive. Although a rapid X-ray flaring is reported (Baganoff et al. 2001b), we do not discuss it here because the event is a transient phenomenon localized in a small region of the accreting plasma.

One possible way of reproducing this negative slope in our model is to use the decreasing side of the first-order Compton peak. Hereafter, we call this type of fitting the ‘Compton fitting’. In order for this negative slope to reach the Chandra X-ray band, the optically thin (i.e. the high-frequency) side of the synchrotron peak should be located in a high enough frequency range. Nevertheless, the location of its self-absorbed (i.e. the low-frequency) side is fixed by the 86-GHz point. Thus, the synchrotron peak in this fitting is required to be wide enough. Also, in order to keep the required X-ray luminosity, we require a sufficiently high luminosity for the synchrotron peak. However, these two requirements are apt to contradict with the upper limit data in the infrared (IR) band, and hence the Compton fitting becomes very tight. We do not include wind in this type of fitting, because its inclusion does not cause any change in the overall spectral shape but reduces the luminosity only similarly at every frequency.

Actually, the above stated requirement only allows a unique fitting, which is shown in Fig. 2 with the full curve. The contribution from bremsstrahlung is buried in the second Compton peak. This curve seems to be marginally fitted to the observational constraints. The curve traces the observational data points fairly well on the low-frequency side of the radio peak except around its top, but the X-ray slope is nearly at its steepest limit. If we try to improve the fit to the X-ray slope, it becomes impossible for the curve to go through the 86-GHz point.

Figure 2

Compton fitting to Sgr A*. The Chandra X-ray spectrum (Baganoff et al. 2001a) is fitted by the shoulder of the once Compton-scattered peak (C1) of the synchrotron photons (S). The thin dotted curve represents the corresponding spectrum in which the Compton-scattered components are suppressed. It can be seen that the contribution from bremsstrahlung (B) is buried in C2. The open circle just above the Chandra error box indicates the luminosity observed by ROSAT. Other observational data points are compiled by Narayan et al. (1998b). In this type of fitting the wind is suppressed (n = 0).

In this fitting, the accretion rate has to be reduced from the previously inferred value (KKY) to a considerably lower value in order to suppress the contribution from bremsstrahlung. The position of the low-frequency side of the synchrotron peak is fixed by adjusting mainly the black hole mass m (see equations 21 or 22). The most effective parameter to shift the Compton peak frequency is x_out, and a shift toward the high-frequency side means a reduction of x_out. In order to avoid the appearance of inner edge radii smaller than that of the marginally stable circular orbit, the magnetic Reynolds number at the outer boundary, ℜ₀, should stay at rather small values. Since the obtained ratio x_out/x_in from this fitting is not so large, the amplification of the seed magnetic field B₀ by the sweeping and twisting effects of the accretion flow remains rather small. In other words, this case of Compton fitting requires considerably large external magnetic fields, since b_ϕ(x_in) is almost fixed by the height of the synchrotron peak.

The resulting values of the fitting parameters are summarized in Table 1. When a figure is expressed as a three-digit number, it means that a change in the final digit causes a noticeable shift of the fitting curve. The dimension-recovered values and the related physical quantities of interest are:  

23

Table 1

Summary of model parameters.

Another type of possible fitting is called the ‘bremsstrahlung fitting’, in which the Chandra spectrum is reproduced by the intermediate frequency range of the bremsstrahlung (equation 18). In order for the Chandra frequency band to fall in this negative slope region, we have to take the outer edge of the disc x_out as so large that its temperature decreases below approximately 1 keV. This corresponds to the outer edges of x_out > 10⁵, and these values make a great contrast with the former value of 2.6 × 10³ that was obtained in KKY in reproducing the positive X-ray slope of the ROSAT result by the low-frequency range of the bremsstrahlung. The relatively faint X-ray luminosity of Chandra leads to a fairly small in this case, but not to such an extremely small volume as in the Compton fitting case.

Two good examples of such fittings are shown in Fig. 3. The full curve is for a no-wind case, and the dashed curve is for a weak-wind case. Although the fitting to the X-ray slope is much improved by the inclusion of a weak wind, the fitting to the low-frequency side of the synchrotron peak becomes rather worse around its top. At present, we cannot clearly point out which is the best fit to Sgr A*, because of a lack of data points that are effective in distinguishing them. High-resolution observations in the submillimetre to IR band would be very useful, not only for this purpose but also to decide between the Compton and bremsung fittings.

The values of the fitting parameters obtained for each case are given in Table 1, and the resulting values for the physical quantities are:  

24

for the solid curve, and  

25

for the dashed curve.

In order to maintain a relatively higher synchrotron luminosity compared with the X-ray luminosity in the spectrum of Sgr A*, large values are required for ℜ₀ (i.e. small values for Δ), and this fact results in reasonably small values for x_in in spite of the large value for x_out. The latter fact (i.e. x_in≪x_out), in turn, guarantees the appearance of a well-extended negative-slope region in the spectrum, and comfortably small values for the external magnetic field compared with the case of Compton fitting. The total bremsstrahlung hump in such a case shows a double-peaked structure in which the second peak (B′ in Fig. 3) on the higher-frequency side is emphasized by the enhancement of the Gaunt factor in the relativistic temperature regions located in the inner part of the disc (see also Fig. 7, later in Section 5.3).

The results in this case, therefore, predict a fairly wide extent for the ADAF state up to x_out > 10⁵ with a fairly low accretion rate of the order of a few ×10⁻⁶. For the obtained black hole mass of ∼ a few ×10⁵ M_⊙, the former value corresponds to a size of r_out>10¹⁶ cm, which is in good agreement with the marginally resolved radius of 0.02 pc for the nuclear X-ray source (Baganoff et al. 2001a). The corresponding values of the mass accretion rate at the outer radius, , is ∼ a few ×10⁻⁸ M_⊙ yr⁻¹. In the weak-wind case, this value is further reduced to at the inner edge. Therefore, these values are consistent with the limit ⩽10⁻⁸ M_⊙ yr⁻¹ recently imposed on (Agol 2000; Quataert & Gruzinov 2000) from the interpretation of polarization measurements in radio emission (Aitken et al. 2000, however, see also Bower et al. 2001).

5.2 M31

The nuclear source observed by ROSAT at the centre of M31 has also been resolved into five point sources by Chandra (Garcia et al. 2000). The true nuclear source is identified with that located within 1 arcsec of the supermassive black hole and has an anomalously soft spectrum. They report that its luminosity in the 0.3–7.0 keV band is 4.0⁺¹²_−2.8× 10³⁷ erg s⁻¹ and the power-law-fitted photon index is 4.5 ± 1.5 in the 0.3–1.8 keV range (we assume this value to hold up to 7.0 keV), after corrections for interstellar absorption. Although there seems to be some debate concerning these results, we tentatively fit our model to this observation.

Other observational data are obtained by VLA (Crane, Dickel & Cowan 1992; Crane et al. 1993), IRAS (Soifer et al. 1986;Neugebauer et al. 1984), KPNO (McQuade, Calzelli & Kinney 1995), Palomar 5-m telescope (Persson et al. 1980) and Hubble Space Telescope (HST) (Brown et al. 1998; King, Stanford & Crane 1995). The radio flux seems to have slight time variations. Among the HST data, the resolution of the 1.7 × 10¹⁵ Hz observation is much better (∼0.5 arcsec, King et al. 1995) than the other two at (1.1 and 1.5 ×10¹⁵ Hz, Brown et al. 1998).

Although the available data points are rather few, the Compton fitting is very tight. Namely, if we try to adjust the fitting parameters such that the predicted curve passes through the radio and X-ray points (satisfying both the luminosity and the spectral index), it also automatically fits the 1.7 × 10¹⁵ Hz point (Fig. 4). Other HST and IR data should then be regarded as an excess, the origin of which may be attributed to components other than the ADAF under consideration. The fitted slope in the Chandra band seems somewhat harder, but it is the best we can do.

Figure 4

Compton fitting to M31. The solid curve is the almost uniquely determined best-fitting curve, with no wind (n = 0). The thin dotted curve represents the corresponding spectrum in which the Compton-scattered components are suppressed. Also in this case, it can be seen that the contribution from bremsstrahlung (B) is buried in C2.

The determined fitting parameters are given in Table 1, and the physical quantities of our interest are calculated as  

26

On the other hand, if we adopt the bremsung fitting, the fit to the X-ray slope can be much improved. In Fig. 5, there are three curves that demonstrate the variations caused by the change in the wind parameter. Other parameters are adjusted in each curve to fit the radio, UV and X-ray luminosities simultaneously. As predicted in the previous section, we can clearly see the main effects of wind on the spectra, i.e. the steepening of the bremsung fall off toward the high-frequency side and the suppression of the synchrotron peak. For a more complete demonstration of such effects caused by winds, in Fig. 6 we show the variation of the spectrum according to the values of the wind parameter over its full range.

Figure 5

Bremsstrahlung fittings to M31. The dashed curve is the best-fitting curve with no wind (n = 0). It is clear that the negative slope in the Chandra band cannot be reproduced without wind. However, both the high-frequency shoulder of the bremsstrahlung and the height of the synchrotron peak become suppressed with increasing wind strength: n = 0.3 for the thin solid curve and n = 0.5 for the thick solid curve. Thus, the Chandra spectrum is well reproduced by these strong-wind cases.

Figure 6

Demonstration of the wind effects. The wind parameter is varied almost over its full range, as indicated on each curve. The base is the dashed curve in Fig. 5 (n = 0), and the parameters other than n are all fixed. Negative values of n denote downward winds toward the disc surfaces.

The values of the fitting parameters for each curve are also summarized in Table 1. The fit to the X-ray slope seems satisfactory when n is in between 0.3 and 0.5. However, it should be noted that the limiting case of n = 0.5 corresponds to a flat density profile (i.e. it is independent of x; see equation 5), which seems rather implausible. The mass accretion rate in this fitting case increases by approximately one order of magnitude compared with the Compton fitting case.

The resulting dimensional quantities in the case of the thin solid curve (n = 0.3), for example, are  

27

If we accept the bremsung fittings, we can conclude that M31 has a strong wind (0.3 < n < 0.5). However, it is again difficult to clearly insist on the superiority of the bremsung fitting with respect to the Compton fitting because of the shortage of observational data points. High-resolution observations at millimetre wavelengths are desirable for this purpose.

5.3 Discussion

In the previous subsections we could reproduce the observed broad-band spectra of Sgr A* and M31 fairly satisfactorily, by both of Compton and bremsung fittings. Although some superiority of the latter case may be recognized in the fitting to the X-ray slope, it is difficult to say clearly which is the better one from the goodness of the fit only. However, the preference becomes much clearer when we combine information other than just that from their spectra. In the course of this the main issue relating to our basic model will also become evident.

The central dark mass of Sgr A* is measured by the method of tracing stellar trajectories, and the result is 2.61 ± 0.35 × 10⁶ M_⊙ within the inner 0.015 pc (Eckert & Genzel 1997; Ghez et al. 2000). On the other hand, the point mass obtained from our spectral fittings is almost one order of magnitude smaller than the ‘dynamical’ mass. For M31, the central mass is estimated to be 3.3 × 10⁷ M_⊙ by using HST photometry (Kormendy & Bender 1999). Also in this case, our model largely underestimates it (apparently by approximately 30–40 times), even if the ambiguity caused by the shortage of data points is taken into account.

Since in our model the black hole mass is determined essentially by the fit to the luminosity of the self-absorbed part of the synchrotron emission, the above fact means that our model is overestimating this luminosity. This can be confirmed by the dashed curve in Fig. 7 which is plotted by adopting the dynamical mass (m = 2.6 × 10⁻²) without changing other parameters from the case of the solid curve in Fig. 3. The main reason for this overestimation may be attributed to the overestimation of the electron temperature near the inner edge of the disc, in spite of the rather large inner-edge radii that results (a few tens of r_S) compared with other models.

This overestimation is avoided in the viscous ADAF models by introducing a two-temperature scheme in which the electron temperature deviates from a virial-type ion temperature, especially in an inner region (x < 100), owing to the increasing synchrotron loss and inefficient energy supply from ions (e.g. Narayan & Yi 1995b; Narayan 2002). This may indeed be a plausible explanation, but it is still uncertain whether or not the effective collisions between ions and electrons can remain sufficiently small even in the presence of strong turbulence in the accretion flows (Balbus & Hawley 1998; Kaburaki, Yamazaki & Okuyama 2002).

In addition to increasing radiation cooling in the innermost accretion discs, there may be other boundary effects such as launching of a jet, which are not included in our idealized treatment of the accretion flow under the assumption of large magnetic Reynolds number. Another possibility may be the effect of the viewing angle of the self-absorbed part of the disc, although it has not been considered in the current ADAF fittings (except by Manmoto 2000). Indeed, we see only a smaller effective area compared with the face-on case, when the viewing angle becomes closer to the edge-on case. However, this effect can scarcely amount to more than an order of magnitude.

As for the mass accretion rate, it is widely believed that the Bondi accretion rate gives a plausible estimate of the actual accretion rate in remote (i.e. at the Bondi radius) regions (e.g. Melia & Falcke 2001). However, we have to emphasize here that it is nothing more than a fiducial value, such as the Eddington accretion rate. This is because there is no definite evidence that Bondi's spherical accretion processes are actually realized generally. Instead, the validity of that model has been criticized explicitly by Narayan (2002).

Anyway, Chandra provides a good estimate of the Bondi accretion rate for the case of Sgr A*. Baganoff et al. (2001a) report that there is ∼2-keV gas with a number density of n₀∼ 100 cm⁻³ spread over approximately an arcsec around the central source. This size is consistent with the Bondi radius, r_B = 2GM/V²_B, where V_B is the velocity of sound at r_B and the resulting Bondi accretion rate is .

Since this value corresponds to a normalized accretion rate of the order of for the observed dynamical mass, we show in Fig. 7 the prediction of our model in this case by also fixing the other parameters at the same values as in the case of the thick solid curve. It is evident that the result overestimates the observed luminosity by approximately three orders of magnitude. We can also recognize in this figure that the overall luminosity is much more sensitive to the mass accretion rate than to the central mass. Therefore, uncertainties in the mass determination affect the determination of the mass accretion rate less strongly than the other parameters.

Our best-fitting value for the normalized mass accretion rate is and similar values are also found for the resistive ADAF fittings (Narayan 2002). Since the outer edge of the ADAF in our bremsung fitting case (and also in the fittings by Narayan 2002) is of the order of the Bondi radius, the large discrepancy between the predicted mass accretion rates and the Bondi rate suggests that the accretion (at least in Sgr A*) does not proceed as in Bondi's prediction. Although there is no firm reasoning to relate our to the Bondi rate , a simple relation seems to account for the results of spectral fittings fairly well. The ratio represents the geometrical fraction of a disc accretion with a half-opening angle Δ compared with the spherical accretion. This relation should be compared with a similar one in the viscous ADAF models, , where α is the viscosity parameter (Narayan 2002).

As has already been pointed out in KKY, the low-frequency radio (ν < 86 GHz) excess seen above our fitting curves to Sgr A* is very likely to come from the inner jet that is located within the inner edge of the accretion disc and extend to the vertical direction along the polar axis. Although its existence in the Galactic Centre, in particular, has not been established observationally as yet, the universality of the disc—jet association seems plausible on both theoretical (see Section 2 of this paper) and observational (Ho 1999; Nagar, Wilson & Falcke 2001; Ulvestad & Ho 2001) grounds. There are already many such models in which the spectrum of Sgr A* is reproduced by the jet-only or jet-plus-disc model (e.g. Falcke & Markoff 2000; Yuan 2000; for a more comprehensive review see Melia & Falcke 2001). Therefore, we have to check the above idea by including the contribution from a jet in our calculation of the spectra. However, this is beyond the scope of the present paper.

The major concern with the case of the Compton fit would be the resulting large external magnetic fields. A typical value obtained is ∼1 G, and this value is uncomfortably large as the ambient values near the outer edge, even if a pre-amplification of the interstellar magnetic field caused by the sweeping effect of accreting flow is taken into account. On the other hand, this value decreases to ∼ a few ×10⁻³ G in the bremsung case.

The bremsung fit predicts that a widely spreading ADAF exists with a very small half-opening angle of Δ∼ 10⁻³—10⁻² in each of the objects under consideration. It is very favourable for this result that Chandra seems to have resolved the diameter of the central source in Sgr A* as 1 arcsec. This corresponds to a radius of 6 × 10¹⁶ cm and is in very good agreement with our result for the bremsung fit. Combined with the consideration in the previous paragraph, we can say that we have fairly good reasons for a preference for the bremsung fittings over the Compton fittings.

If such a geometrically thin disc as predicted by our fittings is surrounded by a tenuous wind plasma, the configuration is somewhat reminiscent of the disc—corona structure in the evaporation model (e.g. Meyer, Liu & Meyer-Hoffmeister 2000). However, the equatorial disc in our case is an ADAF, not an optically thick disc of standard type. The greatest concern with this situation would be the global stability of such a widely spreading twisted magnetic structure. At present, this is an open question. In this connection, we only quote a recent work by Tomimatsu, Matsuoka & Takahashi (2001) in which the stabilizing effect of rotating magnetic fields on the screw instability is reported.

6 Summary and Conclusion

We have examined the expected effects of a wind on the emerging spectrum from an ADAF in a global magnetic field, based on the recently proposed resistive ADAF model including winds (K01). The main effects are seen both in the spectral index (the power of ν) appearing in the intermediate frequency range of the thermal bremsstrahlung, and in the luminosity ratio of the thermal synchrotron emission to the bremsstrahlung. These two values decrease according to the strength of a wind. This fact can be explained by a suppressed mass accretion rate in the inner disc caused by wind loss.

In order to test the plausibility of the resistive ADAF model, we have fitted the observed broad-band spectra of Sgr A* and of the nucleus of M31 by this model. For each observed spectrum, there are two possible types of fitting. One is the Compton fitting in which the negative X-ray slopes in the spectral energy distribution, which are obtained by Chandra for both objects, are reproduced by the synchrotron self-Compton process, and the other is the bremsung fitting in which the negative slopes are reproduced by the intermediate frequency range of the thermal bremsstrahlung.

On the grounds of just the goodness of fit to the observational data points currently available, it is difficult to clearly distinguish the superiority of one type of fitting over the other, given the shortage of observational data. However, we prefer the bremsung fittings for both objects. The main reasons for this are the uncomfortably large values required for the strength of the seed magnetic field in the Compton fittings (0.5–3 G for both objects) and the very wide extension of accretion discs in the bremsung fittings, which seems favourable for the X-ray observations in the case of Sgr A*.

If the bremsung fittings are more plausible, we can conclude that the ADAFs extend so far as to reach the Bondi accretion radii [for both objects r > (1–5) × 10⁵r_S], with no or very weak wind in the Sgr A* case, and with a fairly strong wind in the M31 case. The resulting mass accretion rates for both objects are smaller than the Bondi rates by more than an order of magnitude, and this fact strongly suggests that the actual accretion processes in these objects are certainly different from Bondi's spherical accretion.

The major concern of our model in its present form is in the point that it largely underestimates the central masses. This fact seems to come from the overestimation of the electron temperature near the inner edge of the disc. Therefore, the improvement of the accuracy of the model especially near the inner boundary is strongly desired.

7 Note Added in Proof

After this manuscript had been accepted for publication, we were informed by Albert Kong that Garcia's group had changed their identification of the nucleus of M31 (Kong et al. 2002). The newly identified source is located ∼ 1 arcsec to the north of the former candidate (Garcia et al. 2000), and is about one order of magnitude underluminous, with a normal (not so extremely soft) spectrum. If we apply our fitting to this source, we may need no, or merely weak, wind in M31 and have a somewhat smaller accretion rate (probably a sub-Bondi rate, as in Sgr A*).

Acknowledgments

We are grateful to Michael Garcia for correspondence concerning his data, and to Makoto Hattori for helpful discussions on X-ray observations. We also appreciate the valuable comments made by the anonymous referee, which have deepened our understanding of the role of the Bondi accretion rate.