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Astron. Astrophys. 363, 455-475 (2000)

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8. Wind solutions: a summary of results

8.1. General considerations on energetics and momentum deposition

We have already discussed the necessity of taking into account a parameterized heating rate to be able to obtain complete and sensible solutions for the stationary wind-type outflow problem analysed in the AGN context.

Inward resolution of wind equations, starting from the sonic point, is particularly critical from the energetics point of view, since we want to keep our model wind in the regime of well-behaved solutions, fulfilling the criteria we have identified in Sect. 2, that is that of tenuous, hot winds (see Sect. 5), representing a background connection for other denser and clumpy (line emitting or absorbing) AGN components. In fact, we expect temperatures in the inner region of the wind that can be easily much higher than the Compton temperature [FORMULA] for a typical AGN spectrum, that can be estimated around [FORMULA]K (Krolik 1999). This leads to a very strong energy loss rate through Inverse Compton process for the wind plasma in favor of the radiation field photons. Thus, we have found that, to obtain a solution, a heating function (of unspecified origin at this level) is required essentially to avoid sort of an IC "catastrophe" that would stop a regular integration in the inner region. In the resolution of our problem, in particular, we have chosen a representative value [FORMULA]K for a broad-band spectrum, following Krolik 1988 (see also Mathews & Ferland 1987, Mathews & Doane 1990). We also tested different values of [FORMULA], to allow for possible different spectral distributions; however the wind models we present for the value mentioned above are quite representative of the typical results.

It seems here appropriate to stress the following issue. One of our wind requirements (Sect. 2) concerns the value of the total optical depth to scattering, that must be [FORMULA], to be able to consistently neglect the consideration and the analysis of non-linear effects on the radiation field, that can be thus assumed to be substantially unaffected by the interaction with the nuclear wind through Compton processes. However, at the same time, from the point of view of the tenuous wind plasma, these same Compton interactions do have an absolutely significant role in the energetics of the plasma itself. This is not at all contradictory, and it is testified by the need for an heating source distributed along the wind way.

We have tried to compare the properties of our regular wind solutions with well known and simple wind models. An easy treatment is that of polytropic winds, in which case the energy equation is substituted by the assumption of a given simple relation between pressure and density of the wind gas, namely [FORMULA]. In our energetic language, it can be shown that the conditions defined by such an assumption correspond to a situation in which the energy gain and loss mechanisms at work are such that the corresponding rates satisfy [FORMULA] (see Eq. (31) for the definition of [FORMULA]). In other words, this condition would be fulfilled for a model wind "mimicking" a simple polytropic wind, whose solution is guaranteed and behaviour well known. (Moreover, for a polytropic wind, it turns out that, whenever the wind is accelerating, [FORMULA], which is of the order of the expansion cooling rate.) To our purposes this issue only represents an example, since our wind energetics is more complex, but also it again reinforces our choice of introducing an additional parameterized heating rate for the wind plasma with the aim of obtaining regular solutions.

For further qualitative discussion of wind energetics, we refer to Appendix A2, where more reasons supporting the necessity of an additional heating source distributed along the wind way for the construction of our AGN wind model are discussed. This heating source is represented by the parameterized heating rate described in Sect. 4.4 (with possible modifications as explained in Sect. 7). Our parameterization of this function could suggest a seemingly large freedom in the choice of the various parameters; this is however really only apparent, since, as a matter of fact, reasonable and complete solutions turn out to exist only when parameters are chosen within pretty narrow ranges of values. For the sake of completeness for the present considerations, we anticipate here part of the discussion of our choice of parameters, to which next section is devoted, namely that part regarding parameters that define [FORMULA]. It turns out that the two parameters defining the spectral indices of the two power-laws (see Sect. 4.4) are restricted to the following values: [FORMULA], and [FORMULA]. As for the coefficients [FORMULA] and [FORMULA], they suffer strong limitations as well, [FORMULA] especially, to perform inward integration of the equations; in fact, this last parameter must be appropriately chosen, since, once a solution is obtained for a certain value of [FORMULA], variations of order [FORMULA] of this same value can prevent integration.

Fig. 1 shows three different examples of solutions of our problem, obtained for the case of constant mass flux, and identifying three distinct physical behaviours for the wind model; number density, temperature and velocity curves for the wind plasma are depicted in the three panels. The distinction among the three different classes of solutions that we exemplify in Fig. 1 is based essentially on the possible presence, value and relevance of a momentum deposition contribution [FORMULA] due to the heating mechanism. Indeed, the three solutions are obtained starting from the same set of parameters, except for the value of the heating rate coefficient [FORMULA]. Solid curves represent the physical quantities referring to a solution for which the energy deposition mechanism does require an associated momentum contribution to momentum deposition and the heating rate coefficient [FORMULA] is within a rather narrow range, as mentioned in the previous paragraph. This type of solutions is characterized by rather low density and high temperature in the inner region of the wind (although significantly lower than [FORMULA], as defined by Eq. (9) in Sect. 5, by at least one order of magnitude); the optical depth turns out to be [FORMULA] and the total power exchanged in heating and cooling processes of the wind plasma is rather "low" (see below for a more quantitative characterization) with respect to the central radiation field luminosity [FORMULA]. These solutions, therefore, identify a regime of "consistency" with respect to the requirements of Sect. 2 for the AGN wind we intend to model, and in these cases the heating rate coefficient [FORMULA] is such that the total "luminosity" [FORMULA] (where [FORMULA] is the outer integration radius) supplied to the wind by the heating source is

[EQUATION]

[FIGURE] Fig. 1. Comparison between three different solutions, exemplifying three distinct classes of wind "models", as explained in detail in the text, and respectively represented by continuous, dotted, and dashed curves. Number density, temperature and outflow velocity behaviour are shown respectively in the three panels; parameters at the critical point are as specified in the figure; for all the three solutions [FORMULA] and [FORMULA]. The only difference in parameters is in the respective values of [FORMULA], that correspond to a ratio [FORMULA], and [FORMULA].

Still for energy deposition mechanisms implying an associated momentum deposition contribution, increasing [FORMULA] by a minimum factor that can be as low as 5 (and increases with decreasing central source luminosity, to become as large as [FORMULA] for [FORMULA] erg s-1) leads again to complete integration of the wind equations, but it produces the transition of the obtained solution to a markedly different regime for the model solutions; these are characterized by much lower inner temperatures and strong acceleration of the wind plasma, but strong density gradients and large inner region densities. This class of solutions is exemplified in Fig. 1 by the solution represented by the dotted curves and it corresponds to the regime we have qualitatively anticipated in Sect. 5. Once the solution is within this second regime, further increase of [FORMULA] (i.e. of the heating rate and its associated momentum deposition rate), even by a very large factor, still produces a complete solution, with internal temperature that gets lower and lower; however, the resulting total optical depth values are very large, due to the large densities, and energy exchange rates for the energy deposition and loss processes turn out to be huge, confirming the qualitative considerations of Sect. 5. Our estimate of [FORMULA] for this example solution shown in Fig. 1 with the dotted curves gives, in fact, [FORMULA] erg s-1, several orders of magnitude larger than the central source radiation field luminosity, which is a clearly unphysical value. Indeed, for these solutions, due to the correspondingly strong energy deposition rate in the wind plasma, especially close to its origin, momentum deposition from the [FORMULA] term is the dominant agent of the wind starting up and acceleration; since energy deposition is distributed in distance, acceleration of the wind goes on with r increasing, inducing a positive gradient of the flow velocity and a correspondingly strong negative gradient in the wind density, approaching the sonic point; in this first phase, the plasma temperature smoothly decreases, mostly due to expansion, and the additional heating tends to compensate radiative losses that (being [FORMULA]) are the dominant energy loss mechanism. When the steep decrease in density brings radiative losses to be negligible with respect to heating, the wind temperature undergoes a rapid increase, just before the sonic point is reached. In the following supersonic region of the wind, because of the low density values reached, all the energy exchange processes are less relevant, and the physical quantities show a smooth behaviour.

On the contrary, the first solution discussed above, plotted with continuous curves, belongs to a regime in which thermal expansion still has a relevant role in starting up the wind, although momentum deposition helps to maintain the plasma temperature at the wind origin below the limiting value mentioned in Sect. 5, characteristic of pure "coronal expansion", and guarantees, at least close to the wind origin, an acceleration contribution for the wind plasma. In this case, the flow velocity profile is the result of a complex equilibrium between regions in which expansion slows down and others in which it accelerates, without anyway showing very large gradients.

The third solution shown in Fig. 1 and plotted with dashed curves is again obtained starting from the same set of physical parameters as the two previous ones, but refers to the the case in which there is no momentum deposition contribution due to the heating process. This is just the case in which, since the only momentum deposition is due to radiation force, the start up of the wind is dominated by thermal pressure driven expansion (see Sect. 5), and a temperature (much) closer to the limiting value [FORMULA] discussed in Sect. 5 is expected; in fact, the value of the temperature at the base of the wind for the solution shown is [FORMULA] K. As a matter of fact, except for the behaviour of the temperature in the inner region (i.e. within [FORMULA]) of the wind, showing significantly larger values, this third model solution is quite similar to the one represented by continuous curves (and actually the density and velocity curves, as well as the temperature curve for [FORMULA], are almost superimposed).

These two example solutions represent two opposite limit conditions as for the contribution to momentum deposition due to the heating process represented by [FORMULA] (see Sect. 3).

As already discussed, solutions in the "inconsistency" regime exemplified by the dotted curve solution are unacceptable following our criteria (see Sects. 2 and 5); on the other hand, the last type of solutions mentioned (exemplified by the dashed curve solution and characterized by no momentum deposition contribution due to the heating process) is also not very much promising, since the inner temperatures attained are really very high and imply very large energetic requirements.

Therefore, in the following sections we choose to examine in detail only the results pertaining to solutions in the regime exemplified by the solution represented with continuous curves in Fig. 1, in which a contribution to wind plasma acceleration is expected as a consequence of the heating process and for which the "consistency" and physical criteria of Sect. 2 are fulfilled.

Finally, we would like to just mention that in a first analysis we had taken thermal conduction as well into account; however, we have chosen to neglect it in our treatment, as it is apparent from the governing equations we have described in the previous sections, since thermal conduction turns out to be an efficient mechanism essentially when large gradients are present.

8.2. Physical parameters

We have analysed the physically interesting solutions for a sub-relativistic thermal wind-type flow originating in the very central regions of radio-quiet AGNs with a central luminosity source characterized by [FORMULA] erg s-1. We thus span the range of typical luminosities of Seyfert I galaxies. The present solutions are obtained by integrating the hydrodynamical equations as described in Sects. 3 to 7.

We have already mentioned the range of values explored as far as the luminosity of a central isotropic source is concerned. There are several other physical parameters that have to be set to solve for the wind-type outflow. As we have mentioned already, we have set the value of the radial distance from which the wind starts, i.e. the origin of the outflow as [FORMULA] (motivations of this choice are related to the limits for the central source dimensions derived from variability time-scales; see Koratkar & Blaes 1999).

We can actually divide model parameters into three groups. A first group refers to the AGN source parameters and it comprises the luminosity of the central source, [FORMULA], whose values we have mentioned above and the gravitational parameters; as for these last, we have chosen different values of the central black hole mass ([FORMULA]) depending on the assumed value for the central source luminosity: increasing [FORMULA] is associated with a larger value of [FORMULA], as it is physically reasonable, and consequently with a larger value of [FORMULA] and of [FORMULA] (since the ratio [FORMULA] is taken as constant). The values we have used for the solutions we present and discuss in the following are those appearing in Table 1. In this table we report the value of the Eddington luminosity, [FORMULA], corresponding to each adopted central mass: it is clear from the table that we did not assume a constant ratio [FORMULA]. The present choice looks like reasonable, since central mass-luminosity relation in AGNs is still matter of debate, but at present it seems somewhat established that the ratio of luminosity to mass ([FORMULA]) is increasing with luminosity, and our values do follow this trend (see Wandel 1998).

A second group of parameters specifies the physical conditions at the sonic point, namely the values of [FORMULA], temperature of the wind plasma at [FORMULA], and [FORMULA], that is the number density of the wind at the critical point, [FORMULA], when no mass input along the wind is accounted for, whereas it gives [FORMULA] through the relation [FORMULA] (see Sect. 6) when the wind is loaded by external mass along its way. The position of the sonic point is actually directly determined essentially by the chosen value of the temperature at the sonic point itself (see Eq. (17)): the higher is the temperature, the closer to the wind origin the outflow becomes supersonic.

Once defined these values, solutions depend on the choice of the parameters of a third group, those characterizing the additional parameterized heating rate function as defined in Sect. 4.4, and we have found that physically acceptable transonic solutions extending from the origin of the wind to the external asymptotic region of supersonic flow do exist only for a limited range of values of these last parameters, for a given choice of the ones previous described (see Sect. 8.1).

We have analysed different solutions, by changing the physical parameters, so as to meet the conditions imentioned in Sect. 2 as best as possible, and, consequently, to be able to identify the most favourable choices of the physical properties for the wind solutions themselves. The strongest restriction (as a matter of fact very significant for the effective existence of physically sensible solutions as well) turns out to be on the plasma density (characterized by the value of the parameter [FORMULA], the plasma number density at the sonic point); the wind density must be rather low, with typical values around a few [FORMULA] cm-3 at the wind base [FORMULA]. We have tried to analyze conditions under which the power input and/or exchange in the wind could be maintained much smaller than the total luminosity of the AGN, by varying the relevant parameters (especially those referring to the energetics of the problem), but it seems that to obtain a well behaved solution and meet the requirements above, we just need to keep the plasma density low.

8.3. Solutions without mass input along the wind

We started looking for the existence of solutions in which the wind mass flux is constant along the wind itself, meaning that no external mass is entrained by the outflow. In our formalism (see Sect. 3), this implies we require that the mass flux per steradiant is [FORMULA] and the mass function [FORMULA].

Fig. 2 and Fig. 3 show four different solutions of this type, obtained starting from a different set of parameters and substantially representative of the behaviour of wind solutions of this type. In Fig. 2, we present two distinct solutions, shown in panels (A) and (B), referring to the same value of the central source luminosity, namely [FORMULA] erg s-1. In each of the two panels, we plot the wind temperature and density, and the outflow radial velocity, as well as the sound speed as functions of the distance r from the central black hole, normalized to the gravitational radius [FORMULA]; notice that this last quantity has the same value for the two solutions shown in Fig. 2, since they refer to the same central source luminosity and, thus, following our choice as shown in Table 1, to the same value of the central black hole mass, [FORMULA]. The small circle around the point in which the two curves for outflow velocity and sound speed cross each other indicates of course the critical point position for the outflow, becoming supersonic at larger distances. It is easily seen that for the wind solution (B), the one shown in the lower panel, the sonic point position is located much farther from the central black hole with respect to the sonic point characteristic of solution (A), in the upper panel. In fact, the main and more relevant difference between the parameters of the two solutions is the critical point temperature value, [FORMULA], which is chosen to be [FORMULA]K for solution (A), whereas it is [FORMULA]K for solution (B); as we have mentioned already in Sect. 6 and it can be directly seen from relation (17), sonic point temperature is the most significant parameter determining the sonic point of the wind, resulting in a position closer and closer to the wind origin for larger values of [FORMULA] itself.

[FIGURE] Fig. 2a and b. Panels a and b show the physical quantities for two wind models characterized by the same central luminosity and different values of the temperature at the critical point, as specified in plots; outflow velocity, v, and sound speed [FORMULA], are expressed in km s-1; the circles drawn around the crossing point of outflow velocity and sound speed curves indicate the position of the sonic point. Solution in panel a is obtained with [FORMULA] cm-3 and [FORMULA], while for the solution in panel a it is [FORMULA] cm-3 and [FORMULA]; for both the solutions shown [FORMULA].

[FIGURE] Fig. 3a and b. Panels a and b show the physical quantities for two wind models characterized by the same temperature at the critical point and different values of the central source luminosity, as specified in plots; outflow velocity, v, and sound speed [FORMULA], are expressed in km s-1; the circles drawn around the crossing point of outflow velocity and sound speed curves indicate the position of the sonic point. Solution in panel a is obtained with [FORMULA] cm-3, [FORMULA] and [FORMULA], while while for the solution in panel b it is [FORMULA] cm-3, [FORMULA] and [FORMULA].

Fig. 3 shows two more solutions, again identified as (A) and (B), respectively in the upper and lower panel, but this time the upper panel refers to a central luminosity [FORMULA] erg s-1, whereas the lower one shows a solution for a much smaller luminosity, [FORMULA] erg s-1, meaning that the central black hole mass is different as well (see Table 1). In this figure, we have chosen to plot solutions characterized by the same value of the critical point temperature, [FORMULA]K, and the normalized values of the sonic point distance, [FORMULA], are indeed not so different, although they do not coincide; this is, of course, due to the fact that "illumination" conditions for the wind plasma are very different, implying that heating and cooling conditions required to obtain the complete resolution of the wind problem can be different as well (see Eq. (17)), thus rendering somehow significant, to the sonic point determination, parameters that have in general only a secondary effect with respect to critical point temperature, [FORMULA].

All the solutions shown meet our selection criteria for consistency and physical significance. They are characterized by a small total optical depth to scattering, typically [FORMULA], except for the one wind solution for [FORMULA] erg s-1, for which we have a somewhat larger value [FORMULA], that is anyway still well within the "thin" regime. Power exchanges (gain or losses) for the wind plasma are also acceptable being at least one full order of magnitude (or even more) smaller than the luminosity of the central source.

A very general feature of the wind solution behaviour in the framework we have built up is the smoothness of the plasma temperature all along the explored extension of the outflow; wind temperatures are very high and, allowing for a very low and slowly decreasing heating rate component in the farthest, supersonic region (well beyond the BLR distance) of the wind, where its gas is very tenuous, the temperature can be easily kept around [FORMULA]K or higher. This of course is consistent with our description of the wind gas as an essentially completely ionized plasma and with our rather schematic representation of radiation losses, since for such high temperatures radiation losses are substantially due to bremsstrahlung process.

Outflow velocity curves can be rather different depending on the solution, as can be seen comparing the solutions shown, although the outflow velocity values do not undergo strong variations. Decreasing critical temperature for given [FORMULA], or increasing luminosity for a given critical temperature result in inducing the presence of a dip in the subsonic portion of the velocity curve, right before getting to the sonic point. This follows from the complex interplay of the various processes, mentioned in Sect. 8.1. Also, it influences directly the density behaviour, although this is not so immediate from the figures, due to the different scales chosen, since n and v are directly related by continuity equations (mass flux conservation equation) in the present case, in which no mass input along the wind is allowed for.

We have already discussed the necessity of maintaining the density parameters of the wind at low values; indeed, the wind density tends to decrease rather quickly, especially in the external regions, where the wind is accelerating again, or at best, the outflow velocity is setting around its asymptotic value. The wind is therefore getting more and more tenuous with increasing distance from the inner region, so that it basically ends up to be almost physically "irrelevant"; unfortunately this may happen at distances that are comparable to those at which we deduce the presence of interesting phenomena, such as UV-X-ray absorption or even BLR. To circumvent this problem, we have examined wind models in which we allow for externally originated mass to be engulfed by the wind along the outflow, so as to try to maintain wind density around reasonable values even at large distances from the wind origin.

8.4. Solutions with mass input along the wind

Allowing for the mass flux per steradiant [FORMULA] to be effectively a function of distance and appropriately choosing the radial dependence of this function, we can build up wind models including a deposition of externally originated mass along the wind way. This has a twofold relevance. First, it allows to at least partially overcome the very low density problem we have just mentioned for constant mass flux solutions, when we consider the wind at large distances from the central source, since we allow for a mass input along the wind from a certain distance on, and this ends up in an increase of the wind density in the external regions of the outflow. Second, although we do not specify, here at least, the precise mechanism, it is pretty reasonable that a nuclear wind outflow, in which various other AGN components that are phenomenologically inferred to be at a certain distance from the central black hole are embedded, should have some sort of interaction with these components, and as a consequence, for example, entrain some mass to their expenses. This is in particular interesting with respect to the possible interaction/connection of this wind-type outflow with the physical component that gives origin to the well known broad emission lines, thus constituting the BLR. It is not the purpose of the present paper to model the relation between the nuclear wind we are studying and the BLR and to explore the details of their physical connection. This would, in particular, require also a specific model for the BLR itself, what is still matter of debate (see Korista 1999). We postpone this analysis to a subsequent paper (Torricelli-Ciamponi & Pietrini 2000), and in the present work we just start to study the characteristics of wind solutions with a mass deposition which is distributed along the wind essentially in a region more or less centered around the typical estimated distance for the BLR of an AGN of given luminosity [FORMULA]. The results we present all refer to the case with central source luminosity [FORMULA] erg s-1 and we estimate the BLR characteristic distance following the relation given by Netzer & Peterson (1997) and widely accepted, namely [FORMULA] pc, where [FORMULA] is the central source luminosity in units of [FORMULA] erg s-1. We have built up a parameterized mass deposition function [FORMULA], so as to mimic the desired behaviour of mass input along the wind. Its explicit form is the following:

[EQUATION]

where [FORMULA], [FORMULA] ([FORMULA] for the presently chosen value for [FORMULA] erg s-1), and w,q, and [FORMULA] are three parameters whose adjustment allows us to obtain the required mass deposition. The mass flux difference between the wind origin and the asymptotic region in which mass deposition comes to be negligible is therefore

[EQUATION]

mass function parameters must therefore be chosen so as to both appropriately "center" the mass deposition and maximize the value of [FORMULA], compatibly with other requirements for the AGN nuclear wind and with the supposed source of external mass, to obtain a non-negligible wind plasma density in the farthest wind regions. Fig. 4 shows the two different choices for [FORMULA] that we have used to build the two exemplifying solutions with mass deposition that we present in Fig. 5; the labels (A) and (B) indicate that the corresponding mass function refers respectively to solution (A) or (B) in Fig. 5.

[FIGURE] Fig. 4. Two examples of the mass input function [FORMULA] as defined by Eq. (38) corresponding respectively to the two different choices of the set of parameters identified by the labels (A) and (B); notice that these two functions are respectively those chosen to build the wind models whose solutions are shown in the following Fig. 5 and labeled with the same (A) and (B) notation. The dot on the distance axis identifies the position of the reference BLR radius in units of [FORMULA], [FORMULA], as estimated in the text, for [FORMULA] erg s-1.

[FIGURE] Fig. 5a and b. Two examples of mass-loaded wind models characterized by the same central luminosity and different values of the temperature at the critical point, as specified in the plots; outflow velocity, v, and sound speed [FORMULA], are expressed in km s-1; the circles drawn around the crossing point of outflow velocity and sound speed curves indicate the position of the sonic point. Solution in panel a is obtained with [FORMULA] cm-3, [FORMULA], and [FORMULA], while for the solution in panel b it is [FORMULA] cm-3, [FORMULA] and [FORMULA].

As for energy exchanges with entrained mass, since at this level we do not have any characterization of the thermal condition of this externally originated material, we have chosen not to model explicitly the energetics of the mass deposition in the wind; instead we simplify the problem by supposing that possible energy exchanges between the wind plasma and the deposed mass can be accounted for by our parameterized heating rate function [FORMULA].

A general consideration on this type of solutions is that increasing the wind plasma number density leads to an enhancement of the energetic requirements; in fact, ranges of parameters defining the energy deposition function that correspond to wind models whose properties are in the desired regime get even narrower than in the constant mass flux case.

Comparing the mass-loaded wind solutions in Fig. 5 with those for constant mass-flux in the analogous Fig. 2, the main difference lies of course in the number density curves (dotted lines); although starting substantially around the same values at the wind origin, for the solutions with mass-input we get rather flattened density curves in the regions of mass entrainment (whose extension can be identified by an inspection of Fig. 4). Indeed, for the solution in panel (B) there is actually a region around [FORMULA] in which number density turns out to have a locally positive gradient as an effect of mass deposition. Beyond the distance at which the chosen mass deposition function [FORMULA] reaches more or less its asymptotic value, i.e., the position from where on mass input is negligible, the density curves return to their steadily decreasing behaviour, but the global effect is that the density values in the external regions of the wind are significantly larger than those of the corresponding constant-mass-flux solutions (in Fig. 2), specifically by more than one order of magnitude in the examples shown. This is obtained still maintaining the solutions within the regime fulfilling our consistency criteria; in fact, for both the models shown, the resulting total optical depth to scattering is [FORMULA], and the total power exchanged by the wind plasma (for energy gains or losses) is still lower than [FORMULA] by a good order of magnitude at least.

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Online publication: December 11, 2000
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