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Astron. Astrophys. 349, 11-28 (1999)

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6. Discussion

The observations of Mkn 501 by the HEGRA IACT system during the long outburst in 1997 convincingly demonstrate for the first time that the energy spectrum of the source extends well beyond 10 TeV. We believe that this very fact, together with the discovery of a time- and flux- independent stable spectral shape of the TeV radiation will have considerable impact on our understanding of the nonthermal processes in AGN jets.

This discussion is not an attempt at detailed modeling of the result presented in the previous section; this will be done elsewhere. Neither shall we systematically invoke multi-wavelength observations. Our purpose is rather to point out the multiple facets of the [FORMULA]-ray phenomenon on its own. They stem from the fact that the emission probably originates from a population of accelerated particles within a spatially confined relativistic jet, specifically oriented towards the observer, and that subsequently the radiation must propagate through the diffuse extragalactic background radiation field (DEBRA) before it reaches us. Each of these circumstances can influence the characteristics of the emitted spectrum, and we shall address them in turn below.

6.1. Production and absorption of TeV photons in the jet

The enormous apparent VHE [FORMULA]-ray luminosity of the source, reaching [FORMULA] during the strongest flares which typically last [FORMULA] day or less, implies that the [FORMULA]-rays are most probably produced in a relativistic, small-scale (sub-parsec) jet which is directed along the observer's line of sight. The determining quantity is the so-called Doppler factor [FORMULA], where z is the redshift of a source moving with velocity [FORMULA] and Lorentz factor [FORMULA] along the jet axis that makes an angle [FORMULA] with the direction to the observer. Moreover, the assumption of relativistic bulk motion appears to be unavoidable in order to overcome the problem of severe [FORMULA] absorption by pair production on low-frequency photons inside the source (see e.g. Dermer & Schlickeiser 1994). Indeed, assuming that the [FORMULA]-radiation is emitted isotropically in the frame of a relativistically moving source, the optical depth at the observed [FORMULA]-ray energy is easily estimated as

[EQUATION]

Here [FORMULA], [FORMULA], [FORMULA], with E the energy of the gamma-ray in the laboratory frame, and [FORMULA] is the observed energy flux at [FORMULA] which for an order of magnitude estimate is assumed to be constant at the optical to UV wavelengths that predominantly contribute to the [FORMULA]-ray absorption. Furthermore [FORMULA] Mpc is the distance to the source with redshift [FORMULA], normalized to the value of the Hubble constant [FORMULA]. Assuming now that the observed optical/UV flux of Mkn 501, [FORMULA] (see e.g. Pian et al. 1998), is produced in the jet, the absorption of 20 TeV [FORMULA]-rays becomes negligible only when [FORMULA]; already for [FORMULA] internal absorption would be catastrophic for [FORMULA], with [FORMULA]. One may interpret the detected VHE spectrum of Mkn 501 given by Eq. 6 as a power-law production spectrum, modified by internal [FORMULA] extinction with optical depth [FORMULA]. This would give an accurate determination of the jet's Doppler factor taking into account the very weak dependence of [FORMULA] on all relevant parameters, namely [FORMULA]. Since there could be a number of other reasons for the steepening of the TeV spectrum, that estimate can only be considered as a robust lower limit on [FORMULA].

The strong steepening of the observed spectrum of Mkn 501 above several TeV could also be attributed, for example, to an exponential cutoff in the spectrum of accelerated particles. These could either be protons producing [FORMULA]-rays through inelastic [FORMULA] interactions and subsequent [FORMULA]-decay, or electrons producing [FORMULA]-rays via inverse Compton (IC) scattering. In IC models an additional steepening of the [FORMULA]-ray spectrum is naturally expected due to the production of TeV [FORMULA]-rays in the Klein-Nishina regime if the energy losses of electrons are dominated by synchrotron radiation in the jet's magnetic field. And finally, the exponential cutoff in the observed TeV spectrum could be caused by [FORMULA]-[FORMULA] absorption of TeV photons in a possible dust torus surrounding the AGN (Protheroe & Biermann 1996) and in the extragalactic diffuse background radiation (Nikishov 1962, Gould & Schreder 1965, Jelly 1965, Stecker et al. 1992).

Our current poor knowledge about the distortion of the source spectrum caused by internal and intergalactic absorption does not allow us to distinguish between hadronic and leptonic source models on the basis of their predictions concerning the TeV energy spectra. Fortunately the temporal characteristics are to a large extent free from these uncertainties. Thus we believe that real progress in this area can only be achieved by the analysis of both the spectral and temporal characteristics of X-ray and TeV [FORMULA]-ray emissions obtained during multiwavelength campaigns that investigate several X-ray selected BL Lac objects in different states of activity, and located at different distances within several 100 Mpc. Notwithstanding this belief, we show here that the high-quality HEGRA spectrum of Mkn 501 alone allows us to make quite a few interesting inferences about the [FORMULA]-ray production and absorption mechanisms.

6.1.1. IC models of the gamma ray emission

Currently it is believed (e.g. Ulrich et al. 1997) that the correlated X-ray/TeV flares discovered by multiwavelength observations of Mkn 421 (Takahashi et al. 1996, Buckley et al. 1996) and Mkn 501 (Catanese et al. 1997, Pian et al. 1998, Paper 1), support the hypothesis of both emission components originating in relativistic jets due to synchrotron/IC radiation of the same population of directly accelerated electrons (Ghisellini et al. 1996; Bloom & Marscher 1996; Inoue & Takahara 1996; Mastichiadis & Kirk 1997; Bednarek & Protheroe 1997). One of the distinctive features of leptonic models is that they allow significant temporal and spectral variations of TeV radiation. The stable shape of the TeV spectrum of Mkn 501 during the 1997 outburst does not contradict these models. It rather requires them to have two important features.

First of all the form of the spectrum of accelerated electrons should be essentially stable in time and be independent of the strength of the flare up to electron energies responsible for the production of the highest observed [FORMULA]-ray energies [FORMULA]. For a typical Doppler factor [FORMULA] this implies relatively modest electron energies [FORMULA] in the jet frame, assuming that the Compton scattering at the highest energies takes place in the Klein-Nishina limit. In this limit a significant fraction of the electron energy goes to the upscattered photon, i. e. [FORMULA].

For low values of the magnetic field in the emitting plasma, i.e. B [FORMULA] 0.01 G, the X-ray spectrum could be more sensitive to accelerated electrons with energies above 10 TeV. The typical observed energy of X-rays produced by electrons of energy [FORMULA] in the jet frame is

[EQUATION]

The BeppoSAX observations of Mkn 501 in April 1997 showed that the X-ray spectrum becomes very hard during strong flares. This is interpreted as a shift of the synchrotron peak to energies in excess of 100 keV (Pian et al. 1998). Formally this effect could be explained by a significant increase in each of the three parameters which determine the position of the synchrotron peak, i.e. the maximum electron energy [FORMULA], the magnetic field B, and the jet Doppler factor [FORMULA].

The rather stable energy spectrum of TeV radiation implies that the spectrum of the parent electrons does not significantly vary during the HEGRA observations, the latter performed with typical integration times between one and two hours. The condition of a constant acceleration spectrum does not yet guarantee a stable energy spectrum of TeV radiation. Therefore we need a second condition, namely to assume very effective radiative (synchrotron and IC) cooling of electrons, sufficiently fast to establish an equilibrium electron spectrum within [FORMULA]. The radiative cooling time is

[EQUATION]

[EQUATION]

where [FORMULA] is the total energy density of magnetic and photon fields. Thus, for a jet magnetic field of about [FORMULA] and a comparable low-frequency photon density ([FORMULA]) a radiative cooling time of less than 5 hours (in the jet frame) could be easily achieved.

6.1.2. [FORMULA] origin of gamma rays

The lack of correlation between spectral shape and absolute flux, as well as the very fact that [FORMULA]-rays with energy [FORMULA] are observed, could also be explained, perhaps even in a more natural way, by the assumption of a `[FORMULA]-decay' origin of the [FORMULA] -radiation. Yet the efficiency of this mechanism in the jet appears to be too low to explain the observed time variability and the high fluxes of the TeV radiation. This is due to the low density [FORMULA] of the thermal electron-proton plasma in the jet. The problem of variability could be at least in principle overcome by invoking adiabatic losses caused by relativistic expansion of the emitting "blob". However, this assumption implies very inefficient [FORMULA]-ray production with a luminosity [FORMULA], where [FORMULA] is the luminosity in relativistic protons, [FORMULA] is the adiabatic cooling time, and [FORMULA] denotes the characteristic emission time scale of [FORMULA]-decay [FORMULA]-rays. We shall assume here that the proton luminosity [FORMULA] should not exceed the total power of the central engine, roughly the Eddington luminosity [FORMULA] of a supermassive Black Hole of mass [FORMULA], or more empirically, the apparent ([FORMULA]) total luminosity of the source which is [FORMULA] erg/s, where [FORMULA] is the total observed radiative flux (see e.g. Pian et al. 1998). Then the observed TeV-flux of about [FORMULA] erg/s requires a lower limit [FORMULA] on the density of the thermal plasma in the jet. This makes the relativistically moving `blob' very heavy ([FORMULA]) with an unacceptably large kinetic energy [FORMULA] erg.

We would like to emphasize that these arguments hold against the [FORMULA]-origin of [FORMULA]-rays produced in a small-scale jet; they do not in general exclude hadronic models. In particular, scenarios like the one assuming [FORMULA]-radiation produced by gas clouds that move across the jet (Bednarek & Protheroe 1997), Dar & Laor 1997) remain an attractive possibility for hadronic models. They do not exclude either a "proton blazar" model (Mannheim 1993). It implies a secondary origin of the relativistic electrons that are the result of electromagnetic cascade, triggered by photo-meson processes involving extremely high energy protons in a hadronic jet (see Mannheim 1998and references therein).

6.2. Intergalactic extinction

The effect of intergalactic extinction of VHE [FORMULA]-rays in diffuse extragalactic background radiation fields became astrophysically significant (see e.g. Stanev & Franceschini 1998, Funk et al. 1998, Biller et al. 1998, Stecker & de Jager 1998, Biller 1998, Primack et al. 1998, Stecker 1998) after the discovery of TeV radiation from Mkn 421 and Mkn 501 up to energies of 10 TeV, as reported by the Whipple (Zweerink et al. 1997), and HEGRA (Aharonian et al. 1997a), CAT (Djannati-Atai et al. 1999), and Telescope Array (Hayashida et al. 1998) groups.

6.2.1. Gamma ray absorption

If we ignore the appearance of second generation [FORMULA]-rays (see Sect. 6.2.2), then extinction is reduced to a simple absorption effect which can be described by a single absorption optical depth [FORMULA].

The optical depth [FORMULA] of the intergalactic medium for a [FORMULA]-ray photon of energy E, emitted from a source at the distance [FORMULA], can be expressed for small redshifts z[FORMULA] 1 in a convenient approximate form using a quantity [FORMULA], where

[EQUATION]

[EQUATION]

with [FORMULA], [FORMULA], [FORMULA], and [FORMULA] being a correction factor which accounts for the specific form of the differential DEBRA photon number density [FORMULA]; the background photon energy is denoted by [FORMULA]. This expression is based on the narrowness of the [FORMULA] cross-section [FORMULA] as a function of [FORMULA] which peaks at [FORMULA] in an isotropic field of background photons (Herterich 1974). Thus for a large class of broad DEBRA spectra [FORMULA] the optical depth is essentially caused by background photons with energy centered around [FORMULA]. For (broad) power-law spectra, [FORMULA], the optical depth can be calculated analytically as [FORMULA], where [FORMULA] (Svensson 1987). Thus for relatively flat power-law spectra with [FORMULA] we obtain [FORMULA], i.e. approximately half of [FORMULA] is contributed by background photons with [FORMULA] in the interval given by [FORMULA].

We note that [FORMULA] for a power-law spectrum of DEBRA. For example, within the `valley' of the energy density at mid infrared wavelengths from several [FORMULA] to several tens of [FORMULA], where the energy density is expected to be more or less constant (i.e. [FORMULA]), the intergalactic extinction of [FORMULA]-rays is largest at the highest observed energies. In particular, according to Eq. 10, even at a very low and probably unrealistic level of the DEBRA intensity of [FORMULA] at [FORMULA], approximately [FORMULA] of the 25 TeV [FORMULA]-rays emitted by Mkn 501 are extinguished before they reach the observer. This implies that the observed TeV spectrum of Mkn 501 contains important information about the DEBRA, at least at wavelengths [FORMULA]. Moreover, it is quite possible that a non-negligible intergalactic extinction takes place also at low [FORMULA]-ray energies due to interactions with near infrared (NIR) background photons. Interpreting the power-law shape of the spectrum at [FORMULA] as an indication for weak extinction by, say, a factor less than or equal to 2, and ignoring the contributions from all other wavelengths beyond the interval [FORMULA] ([FORMULA]) from Eq. 10, one obtains [FORMULA], not far from the flux of DEBRA experimentally inferred (Dwek et al. 1998, de Jager & Dwek 1998) and theoretically expected (Malkan & Stecker 1998, Primack et al. 1998) at these wavelengths.

However, to the extent that the source spectrum of the [FORMULA]-rays is unknown, this cannot be considered as a model-independent upper limit (e.g. Weekes et al. 1997). Obviously for any quantitative estimate of the DEBRA one needs to know the intrinsic [FORMULA]-ray spectrum, and this can not be obtained from [FORMULA]-ray observations alone. As already emphasized above, a multi-wavelength approach is indispensable, for example in the form of detailed modeling of the entire or at least a large wave-length range of the nonthermal spectrum. To be specific, one avenue would be modeling the nonthermal X-ray and [FORMULA]-ray spectra in the framework of synchrotron-inverse Compton models, based on simultaneous multi-wavelength observations of X-ray selected BL Lac objects (Coppi & Aharonian 1999).

An illustration for the need of more than [FORMULA]-ray data alone is given by a DEBRA spectrum [FORMULA]. It does not change the spectral shape of [FORMULA]-rays at all (grey opacity), although formally the extinction could be arbitrarily large. The curves marked as "1" in Fig. 13 and Fig. 14 may serve as an example. In the case of Mkn 501 this ambiguity can be significantly reduced by rather general arguments regarding [FORMULA]-ray energetics . Requiring again that the [FORMULA]-ray luminosity should not exceed the total apparent luminosity of the source, [FORMULA], we must have [FORMULA]. For [FORMULA]-ray production in the jet with [FORMULA], this implies [FORMULA]. In fact, already a value of [FORMULA] (corresponding to curve 1 of Fig. 13) creates uncomfortable conditions for the majority of realistic models of high energy radiation from Mrk 501, assuming that the nonthermal emission is produced in the jet due to synchrotron and inverse Compton processes. Indeed, [FORMULA] implies that the [FORMULA]-ray luminosity of the source corresponding to the "reconstructed spectrum" (curve 1 in Fig. 14) exceeds the luminosity of the source in all other wavelengths by an order of magnitude which hardly could be accepted for any realistic combination of parameters characterizing the jet.

[FIGURE] Fig. 13. The energy fluxes of DEBRA for pure power-law differential spectra with [FORMULA] (curve 1), [FORMULA] (curve 2) and [FORMULA] (curve 3). Curve 4 is the sum of spectra 1 and 3. The absolute flux normalizations have been determined from the condition of [FORMULA] (for curve 1), and from the condition of the maximum possible flux of DEBRA which still "reproduce" reasonable [FORMULA]-ray source spectra shown in Fig. 14 (for curves 2 and 3). The horizontal bars correspond to the upper limits on DEBRA fluxes obtained using a method similar to the one suggested by Biller et al. (1998). The curve marked as "MBR" correspond to the density of the 2.7 K MBR. The tentative flux measurement at 3.5 µm is taken from Dwek & Arendt (1998), and the flux estimates based on the ISO survey at 6 and 15 µm are from Stanev & Franceschini (1998). The other measured fluxes and the upper/lower limit estimates of the DEBRA are taken from the recent compilation by Dwek et al. (1998), i.e., the upper limit at 2.2 µm is from Hauser et al. (1998), the lower limit at 2.2 µm is from Gardner et al. (1997), and the UV to optical detections are from Pozzetti et al. (1998).

[FIGURE] Fig. 14. The source spectra of Mkn 501 reconstructed for different models of DEBRA. The heavy dots correspond to the measured spectrum of HEGRA approximated by Eq. 6 with [FORMULA], and [FORMULA]. The curves 1, 2, and 3 correspond to the power-law DEBRA spectra shown in Fig. 13 by curves 1,2 and 3, respectively. The vertical line at 16 TeV indicates the edge of the [FORMULA]-ray spectrum of Mkn 501 measured by HEGRA.

Accepting the current lack of reliable knowledge of the [FORMULA]-ray source spectrum, it is nevertheless worthwhile to derive upper limits on DEBRA by formulating different a priori , but astrophysically meaningful requirements on the spectrum and the [FORMULA]-ray luminosity of the source. A possible criterion, for example, could be that within any reasonable source model the intrinsic spectrum of [FORMULA]-rays, [FORMULA], should not contain a feature which exponentially rises with energy at any observed [FORMULA]-ray energy. In practice this implies that the `reconstruction' factor [FORMULA] should not significantly exceed the exponential term of the observed [FORMULA]-ray spectrum from Eq. 6. This condition is most directly fulfilled by the power law-spectrum with [FORMULA] that has equal DEBRA power per unit logarithmic bandwidth in energy. It results in [FORMULA] and then yields an upper limit for the DEBRA density close to [FORMULA]. This limit corresponds to the borderline on which the source spectrum becomes a pure power-law Figs. 13 and 14, curves 2. A slight increase of the DEBRA density by as little as a factor of 1.5 leads to a dramatic (exponential) deviation of the reconstructed spectrum at the highest observed [FORMULA]-ray energies around 16 TeV from the [FORMULA] power-law extrapolation.

A power law with [FORMULA] would give similar and complementary results. The resulting source spectra (Fig. 14) and the upper limits on the DEBRA density (Fig. 13) obtained in this way assuming power-law spectra for DEBRA with [FORMULA] and 3, are given by the curves 2 and 3, respectively. The power law [FORMULA] corresponding to [FORMULA] complements Fig. 13.

It should be noted however that any realistically expected spectrum of the DEBRA in a broad range of wavelengths deviates from a simple power-law. In fact, all models of the DEBRA, independently of the details, predict two pronounced peaks in the spectrum at 1 µm and 100 µm contributed by the emission of the stars and of the interstellar dust, respectively, and a relatively flat `valley' at mid IR wavelengths around 10 µm (see e.g. Dwek et al. 1998). The strong impact of the DEBRA spectrum on calculations of the opacity of the intergalactic medium has been emphasized by Dwek & Slavin (1994) and Macminn and Primack (1996).

Note that the power law with [FORMULA] characterizes the shape of the spectrum of DEBRA at near IR wavelengths, typically between 1 and several microns, and the power-law with [FORMULA] characterizes the DEBRA between 10 and 100 microns. Interestingly, the sum of these two power law spectra "reproduces" a reasonable shape of the `valley'. This is seen in Fig. 13 and Fig. 15 where the measured fluxes or estimated upper and lower limits obtained directly at different wavelengths of the DEBRA are shown.

[FIGURE] Fig. 15. The same as in Fig. 13, but for different parameters of the power-law DEBRA. Curve 1: [FORMULA] with an absolute flux corresponding to [FORMULA]; curve 2: [FORMULA] with [FORMULA]; curve 3: [FORMULA]; curve 4: sum of spectra 1 and 3, truncated at 1 and 80 µm.

Finally, an interesting numerical criterion for the derivation of upper limits on the DEBRA was suggested by Biller et al. (1998). It relies on independent [FORMULA]-ray observations. A variant of this approach, where the additional restriction forbids a differential source spectrum harder than [FORMULA] within the observed energy range, is shown by the horizontal bars in Fig. 13.

The results described above could be "improved" assuming a more realistic, [FORMULA] optical depth for the [FORMULA] power law branch at short wavelengths, and a steeper, [FORMULA], power law branch at long wavelengths (see Fig. 15). The latter choice for [FORMULA] is due to the rapid rise of the data points towards far infrared wavelengths [FORMULA] in order to fit the recent measurements of the flux at [FORMULA] by DIRBE aboard the COBE satellite. For the far infrared (FIR) branch the absolute flux of the power-law with [FORMULA] is chosen again from the condition that the differential [FORMULA]-ray source spectrum should not exponentially rise at energies up to [FORMULA] (see Fig. 16). Note also that the criterion of [FORMULA] for the [FORMULA] branch at short wavelengths is pretty close to the level of the flux of the recent tentative detection of DEBRA at [FORMULA] (Dwek & Arendt 1998). The sum of the NIR and FIR power-law branches with the above indices and absolute fluxes results in a deeper mid IR "valley" and predicts a very steep spectrum of DEBRA between 30 and 100 [FORMULA]. In Figs. 15 and 16 this spectrum has been truncated at [FORMULA] that corresponds to the kinematic threshold of pair production at interactions with the maximum observed energy of [FORMULA]-rays of about 20 TeV. The spectrum is also truncated at [FORMULA] in order to avoid significant excess compared with the fluxes at optical/UV wavelengths recently derived from the Hubble Deep Field analysis (Pozzetti et al. 1998).

[FIGURE] Fig. 16. The same as Fig. 14 but for DEBRA fluxes shown in Fig. 15, with the addition of a reconstructed spectrum (curve 4) corresponding to curve 4 in Fig. 15.

The effect of "reconstruction" of the [FORMULA]-ray spectra of Mkn 501 corresponding to this "best estimate" of DEBRA between 1 and 80 µm is illustrated in Fig. 16. It shows, in the simple absorption picture (using two truncated power-laws) that even a conservative choice for the DEBRA field implies intergalactic extinction at all observed energies. Especially the reconstructed spectrum around 1 TeV could be considerably harder than the observed spectrum, with a maximum of [FORMULA] at 2 TeV (see Fig. 9). This also demonstrates that it would be dangerous to interpret the observed spectral slope at low energies in terms of a power law extending from still lower energies.

6.2.2. The effect of cascading in the DEBRA

The discussion of intergalactic [FORMULA] absorption effects is in principle incomplete without considering secondary radiations. Briefly, when a [FORMULA]-ray is absorbed by pair production, its energy is not lost. The secondary electron-positron pairs create new [FORMULA]-rays via inverse Compton scattering on the 2.7 K MBR. The new [FORMULA]-rays produce more pairs, and thus an electromagnetic cascade develops (Berezinsky et al. 1990, Protheroe & Stanev 1993, Aharonian et al. 1994). In fact, in our discussion of absorption we have neglected any secondary [FORMULA]-rays in the field of view of the detector and we finally turn now to these.

For a primary [FORMULA]-ray spectrum [FORMULA] harder than [FORMULA], extending to energies [FORMULA], the cascade spectrum at TeV energies could strongly dominate over the primary [FORMULA]-ray spectrum. In addition, the spectrum of the cascade [FORMULA]-rays that reach the observer has a standard form independent of the primary source spectrum with a characteristic photon index of [FORMULA] at energies between [FORMULA] and an exponential cutoff determined by the condition [FORMULA]. Thus, somewhat surprisingly, the measured time-averaged spectrum of Mkn 501 can in principle be fitted by a cascade [FORMULA]-ray spectrum for a reasonable DEBRA flux level of about [FORMULA], provided that the invisible source spectrum extends well beyond 25 TeV, say up to 100 TeV.

For an intergalactic magnetic field (IGMF) [FORMULA] the cascade [FORMULA]-rays could be observed in the form of an extended emission from a giant pair halo with a radius up to several degrees formed around the central source (Aharonian et al. 1994). Although the possible extinction of [FORMULA]-rays from Mkn 501, at least above 10 TeV, unavoidably implies the formation of a pair halo, the TeV radiation of Mkn 501 cannot be attributed to such a halo simply due to arguments based on the detected angular size and the time variability of the radiation.

However, the speculative assumption of an extremely low IGMF still allows an interpretation of the observed TeV [FORMULA]-rays of Mkn 501 within the hypothesis of a cascade origin. Instead of extended and persistent halo radiation, we expect in this case that the cascade [FORMULA]-rays penetrate from the source to the observer almost on a straight line. Yet at cosmological distances to the source even very small deflections of the cascade electrons by the IGMF lead to significant time delays of the arriving [FORMULA]-rays: [FORMULA] (Plaga 1995). This implies that in order to see more or less synchronous activity (within several days or less) of Mkn 501 at different wavelengths, as was observed during multiwavelength observations of the source (see e.g. Pian et al. 1998), we would have to require [FORMULA]. Although indeed quite speculative, such small magnetic fields on spatial scales large compared to 1 Mpc cannot be a priori excluded (e.g. Kronberg 1996).

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Online publication: August 25, 1999
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