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Astron. Astrophys. 333, 970-976 (1998)

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3. The energy distribution of relativistic electrons

The energy distribution of the relativistic electrons responsible for the observed emission changes with time due to different types of losses, that depend on the environment where the emission takes place. Electrons are affected by synchrotron losses in the presence of the magnetic field, and undergo Coulomb collisions with the particles of the background thermal plasma.

A detailed study of the influence of the different types of loss mechanisms on the electron distribution have been developed for the analysis of the decaying phase of flares (see Franciosini & Chiuderi Drago 1995 and references therein). Generally speaking, Coulomb collisions have a typical time scale [FORMULA] hours, where [FORMULA] is the thermal electron density, while for synchrotron losses [FORMULA] hours. In the case of UX Arietis the quoted values for the magnetic field range from [FORMULA] 10 G in the corona to [FORMULA] G at photospheric level, while an almost constant thermal density with [FORMULA] cm-3 can be assumed (see Massi & Chiuderi Drago 1992 for details). Hence, collisional losses will influence the emission in the same way everywhere in the loop and on time scales of the order of [FORMULA] day, while synchrotron losses will strongly suppress the emission from the compact region near the photosphere in a few minutes after the injection onset, but will affect the radiation from the extended coronal region only after some hours. Therefore, the consequences of radiative losses will be seen first at higher frequencies, while the low frequency emission coming from the diffuse region remains unaffected. The above considerations imply that radiative losses must be taken into account even during the rising phase of the flare, i.e., during the phase in which a large amount of new accelerated particles is injected in the loop.

In order to quantify how synchrotron and collision losses influence the electron distribution, it is necessary to define a specific model of the radio source. The geometry of the region where the flare energy is produced is not known, although the emission properties, related to the presence of the magnetic field, let us imagine loop-like structures analogous to those present in the solar corona. This picture is consistent with the one suggested in the previous section by the spectral shape and can also account for different radiative losses, as outlined above. In fact, in a magnetic loop anchored to the photosphere the compact region, whose emission peaks at higher frequencies, is the one which suffers from stronger radiative losses and the one which is more easily obscured by the stellar rotation. The extended coronal part of the loop emits longer and constitutes almost an halo around the star.

We assume here that the loop structure is the same of the model by Franciosini & Chiuderi Drago (1995), i.e. a dipolar magnetic field generated by a dipole buried under the stellar surface, and connecting two starspots of radius [FORMULA] and angular separation [FORMULA]. The strength of the dipole is determined by the value [FORMULA] of the magnetic field in the center of the spots. As shown in Fig. 2, the magnetic field is described in terms of polar coordinates ([FORMULA]) on the plane perpendicular to the line of sight, where [FORMULA] corresponds to the loop top and r is the distance from the dipole, and is assumed to be constant along the line of sight.

[FIGURE] Fig. 2. A sketch of the magnetic field structure: the dashed curves identify the loop and the bold segments the starspots

We suppose that electrons are injected at the top of the loop ([FORMULA]), uniformly, isotropically and at a constant rate. The dynamical time for relativistic electrons is very short, [FORMULA] sec; hence, the injected electrons spread out rapidly from the acceleration site into the whole magnetic loop and, being reflected in the appropriate mirror points, reach in a very short time a definite spatial distribution. From our point of view, the attainment of this spatial distribution can be considered instantaneous since, for the moment, we have no way of investigating this short transient phase.

In order to derive the electron energy distribution as a function of space and time, we must solve the full kinetic equation. In our case, however, it is not possible to have any spatial resolution of the observed fluxes, so it has no sense to solve the equation exactly for the spatial dependence. Moreover, the time scales over which we are interested to investigate the effects of radiative and collision losses and of the continuous particle acceleration are much longer ([FORMULA]) than the dynamical time needed to attain a stationary distribution in absence of losses ([FORMULA]). For this reason we decided to analyse the problem in two separate steps: we first derived the spatial density distribution just after the short transient phase neglecting energy losses. We then used this distribution as source function to study the time evolution including all losses (see Eq. (3)).

The solution of the kinetic equation along the magnetic field lines has been studied in the context of solar flares (see e.g. McTiernan & Petrosian 1990). For the first step we are interested in the stationary solution in the case of a non-uniform magnetic field and in the absence of losses. In the simplest case of an isotropically injected distribution at the loop top, as the one we are considering here, Leach & Petrosian (1981) show that the electron number density remains unaltered while electrons stream along the magnetic field lines and are reflected at the mirror points. This happens only if the thermal density scale height is larger than the magnetic one and if the adimensional column depth, [FORMULA] ([FORMULA] is the coordinate along the loop axis), is smaller than one. This is the case for the emitting region in this binary system since an almost constant thermal density [FORMULA] cm-3 is quoted (see before) and since the maximum loop length can not be larger than the binary separation [FORMULA] cm. Therefore, for times longer than the dynamical time the electron density is independent of the spatial coordinate. Thus, we can assume that relativistic electrons are injected uniformly and isotropically in the whole loop at a constant rate [FORMULA] el. cm-3 sec-1 and we can study how this uniform distribution evolves in time due to losses.

The equation describing the time evolution of the electron energy distribution, [FORMULA], in the case of injection plus energy losses is:

[EQUATION]

where

[EQUATION]

represents the energy variation due to synchrotron and collision losses, respectively, and [FORMULA] represents the electrons that escape from the emitting region into the loss-cone. The loss-cone term has been evaluated by Franciosini & Chiuderi Drago (1995) as:

[EQUATION]

where [FORMULA] is the loss-cone solid angle, with [FORMULA], ([FORMULA] local magnetic field), and [FORMULA] sec ([FORMULA]) is the "deflection time" (Spitzer 1962), i.e. the characteristic time scale over which collisions with the background protons isotropize again the electron distribution.

The general solution of Eq. (3) is given by Melrose & Brown (1976) as

[EQUATION]

where

[EQUATION]

[FORMULA] is the electron energy at time [FORMULA] and the relation between [FORMULA] and [FORMULA] can be derived by integrating Eq. (4) (see Chiuderi Drago & Franciosini 1993 for detailed calculations). The complete expression of the energy distribution at time t is obtained by carrying out the integral in Eq. (6) and substituting the relation [FORMULA]. We have assumed for the injected energy distribution [FORMULA] and for the initial distribution [FORMULA] the same functional dependence:

[EQUATION]

and

[EQUATION]

In the above expressions [FORMULA] and [FORMULA] is the total electron number density at [FORMULA]. The result of the integral in Eq. (6) depends on the value of [FORMULA]: since at a given time t the distribution [FORMULA] is defined for [FORMULA], while [FORMULA] is defined only for [FORMULA] (and is zero elsewhere), the effective integration limits are:

[EQUATION]

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© European Southern Observatory (ESO) 1998

Online publication: April 28, 1998

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