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Astron. Astrophys. 342, 687-703 (1999)

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5. Discussion

The simultaneous observations of NGC 4051 in the IR-optical and X-ray wavebands by Done et al. (1990) have given very strong constraints on the spatial distribution of the emitting regions. Effectively, in this object, the limits on the amount of rapid variability in the optical/IR were below 1 and 5 per cent while the X-ray flux continually flickered by up to a factor 2. It clearly rules out models in which the IR/optical and X-ray continuum emission are produced in the same region. Nonetheless, the IR/optical continuum could be the sum of two different components. The first one could originate in the outflows observed in most Seyfert galaxies (Wilson 1993, Colbert et al. 1996), through synchrotron process on large scale magnetic field. Due to the large sizes of the flows, we expect no rapid variabilities from this emission. On the contrary, a second component, whose flux is noted [FORMULA], could be associated with the synchrotron emission of the non-thermal distribution of relativistic electrons producing X-rays, and thus concentrated in a much smaller region. Since rapid X-ray variability is a common features in such objects (Mc Hardy et al. 1985, Mushotzky et al. 1993, Grandi et al. 1992) and is likely associated with instabilities in the source of particles, we expect flickering from this second component too. We assume that its variability amplitude is of the order of the flux, that is [FORMULA] which seems reasonable since it is the case in the X-ray range (Mushotzky et al. 1993, Ulrich et al. 1997). The treatment allows thus to estimate an upper limit of this variable component by measuring [FORMULA] and therefore to constrain the intrinsic properties of the local environment of the emission region. Our assumptions are presented in the following.

5.1. Basic hypotheses

We suppose the non-thermal plasma region to be spherical, with radius R. As explained above, the particles emit synchrotron radiation in a magnetic field of strength B. We also assume the electrons density distribution follows a power law with spectral index s, i.e. [FORMULA], with [FORMULA]. If we assume the magnetic field to be uniform throughout the emitting region and with a random direction in the line of sight, the spectral density of the synchrotron flux received by an observer at a distance D away, can be approximated by (Blumenthal & Gould 1970):

[EQUATION]

In this equation, [FORMULA] is a function of s solely, [FORMULA] the cut-off frequency of the radiation which depends on the maximum Lorentz factor of the electrons (Blumenthal & Gould 1970, Rybicki & Lightman 1989):

[EQUATION]

and [FORMULA] the synchrotron self-absorption frequency separating the optically thin and optically thick regimes of synchrotron emission (see Pacholczyk 1970).

On the other hand, the same electron population produces X-ray radiation by Inverse Compton (IC) process on UV photons, generally supposed to be produced by an accretion disk. We assume that the UV source is roughly at a distance Z from the non-thermal plasma. Finally we suppose that the UV photons density can be approximate by a delta function, and thus, at the location of the hot source, this density can be expressed as follows:

[EQUATION]

where [FORMULA] is the observed UV flux. We can then deduced the X-ray flux received by an observer at a distance D away (Blumenthal & Gould 1970, Rybicki & Lightman 1989):

[EQUATION]

where [FORMULA] is solely a function of s. This expression is representative of the common spectrum of Seyfert galaxies between 2-10 keV which is well fitted by a power with mean spectral index [FORMULA] (Mushotzky et al. 1993).

5.2. Constraint deduced on R and Z

First of all, it seems likely that [FORMULA], where [FORMULA] is the Schwarzschild radius of the black hole supposed to power the AGN. We obtain a lower limit for [FORMULA] through the Eddington limit. Assuming [FORMULA] as roughly equal to the bolometric luminosity, it gives:

[EQUATION]

On the contrary, the smaller X-ray time variability [FORMULA] (if known) gives an upper limit for the size of the non-thermal source:

[EQUATION]

Finally, we must have at least:

[EQUATION]

On the other hand, it appears from Eq. (6) that, to observe no synchrotron emission at the I band frequency [FORMULA], a sufficient (but not necessary) condition is [FORMULA], that is the upper cut-off of the spectrum lies below our observed frequency. It gives thus a possible upper limit for the strength of the magnetic field:

[EQUATION]

We can also constraint [FORMULA] since we know that the X-ray spectrum of Seyfert galaxies can be fitted by a power law from [FORMULA] to [FORMULA], where an exponential cut-off is observed (Jourdain et al. 1992; Maisack et al. 1993; Dermer & Gehrels 1995). Since the mean frequency of the soft UV photons is roughly in the range [FORMULA] (Walter et al. 1994), the maximum Lorentz factor [FORMULA] of the particles must be in the range 50-300.

Besides, limits on [FORMULA] resulting from our data analysis (see Sect. 3) give upper limits on the flux of the variable component [FORMULA] for each galaxy. Consequently, combining Eqs. (6) and (9) we obtain another possible upper limit for the magnetic field:

[EQUATION]

In this equation [FORMULA] is the mean X-ray frequency depending on the X-ray data for each objects, and [FORMULA] is the associated mean flux. Thus, no microvariability detection in any galaxy of our sample, means that:

[EQUATION]

We have studied these differents constraints for only seven galaxies of our sample whose UV and X-ray luminosity and spectral index are reported in Walter & Fink (1992). These data are gathered together in Table 4, with the corresponding values of [FORMULA], [FORMULA], [FORMULA] and [FORMULA] for each of the galaxies. The galaxy NGC 4051 is the only one for which a variability in the X-ray is known, down to 100 s. As a conservative value to estimate the maximum X-ray size for this galaxy, we use [FORMULA] = 300 s.


[TABLE]

Table 4. Characteristics of 7 galaxies of the sample. The flux density are given in [FORMULA], lengths in centimeter and magnetic fields in gauss units. Data are taken from Walter & Fink 1992. The maximum of [FORMULA] gives an absolute upper limit on the magnetic field in the AGN in order not to detect variability.


Further constraint come from equipartition between particles and magnetic field. Effectively, non-thermal particles need to be accelerated to compensate synchrotron and Inverse Compton losses and magnetic field is generally invoked in the acceleration process (Fermi processes in a shock for example). In this case the magnetic energy density must be equal or larger than the particles energy density. Defining the equipartition value [FORMULA] for the magnetic field:

[EQUATION]

and deducing [FORMULA] from Eq. (9), we must have finally:

[EQUATION]

Inequalities (15) and (17) reduce finally to inequalities between Z and R:

[EQUATION]

or

[EQUATION]

Plots Z vs. R of Fig. 6 compiled the constraints described above. We have plotted the curves (type I) corresponding to constraint (18) for each galaxy in dashed line. The second inequality (19) gives a set of limiting curves (type II) on the assumed value of [FORMULA]. Since these curves represent the equipartition [FORMULA], they can also be considered as isocontours of [FORMULA]. We have plotted type II curves corresponding, from left to right, to [FORMULA], which correspond to [FORMULA] and [FORMULA]. The diagrams must be read as follows:

  1. For each galaxy, the allowed region is constrained by Eqs. (10), (11), (12), (18) and (19). It is colored in grey in each plot for [FORMULA]. Other [FORMULA] values would correspond to another curve of type II. The hashed regions are forbidden by Eqs. (10) and (11).

  2. At a given point inside the allowed region, a lower limit of B is given by [FORMULA], represented by the type II curve passing through this point. An upper limit is given by [FORMULA] if Eq. (18) applies or by [FORMULA] if Eq. (19) applies. [FORMULA], [FORMULA] and [FORMULA] are plotted on the right of each graphic. The equality [FORMULA] is realized, for a given assumed value of [FORMULA], when type I and type II curves intersect. An absolute maximum of the magnetic field is obtained for the smaller value of Z in the allowed region. This value is also reported in Table 4.

[FIGURE] Fig. 6a-g. The left part of each plot gives limits on R and Z for 7 galaxies of our sample whose parameters are reported in Table 4. The dash lines represent the equipartition [FORMULA] (Eq. (18)) whereas the set of dot-dash lines represents the equipartition [FORMULA] for different values of [FORMULA] (Eq. (19)). From left to right, [FORMULA]= 50, 100, 200 and 300, corresponding to [FORMULA] 32000 G, 8000 G, 2000 G and 1000 G. Finally, the solid line refers to the [FORMULA] (Eq. (12)). For each galaxy, the allowed region is constrained by Eqs. (10), (11), (12), (18) and (19). It is colored in grey in each plot for [FORMULA]. Other [FORMULA] values would correspond to another dot-dash curve (called type II in the text). The hashed regions are forbidden by Eqs. (10) and (11). On the right part of each graphic, we have plotted [FORMULA] in solid line. The dot line and three dots-dash line correspond respectively to [FORMULA] and [FORMULA]. Thus [FORMULA] when Eq. (18) applied and [FORMULA] when Eq. (19) applied

An allowed region exists for each galaxy, with a critical case for NGC 4051, where the space parameter is strongly constrained. However our results for this galaxy disagree with those of Celotti et al. (1991), since if we assume, like them, that the size of the X-ray region is strictly equal to [FORMULA], we are intside the allowed region for non-thermal models. But these results need to be used with care, in the case of this galaxy, since it seems unlikely for R and Z to be so fine tuned. These different results are obviously affected by the lack of simultaneous X-ray and Optical-UV data and constraints could be tightened if rapid X-ray variability were detected for most of these objects. It appears however that non-thermal model can not be ruled out by our data and can still explain the high energy spectra of Seyfert galaxies.

[FIGURE] Fig. 7a-u. Light curves of the different galaxies and associated comparison stars of the sample.

[FIGURE] Fig. 7a-u. (continued)

[FIGURE] Fig. 7a-u. (continued)

[FIGURE] Fig. 7a-u. (continued)

[FIGURE] Fig. 7a-u. (continued)

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999
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