## 3. The energy distribution of relativistic electronsThe 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
hours, where is the thermal electron density,
while for synchrotron losses hours. In the case
of UX Arietis the quoted values for the magnetic field range from
10 G in the corona to G
at photospheric level, while an almost constant thermal density with
cm 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 and angular separation
. The strength of the dipole is determined by
the value 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 () on the plane
perpendicular to the line of sight, where
corresponds to the loop top and
We suppose that electrons are injected at the top of the loop (), uniformly, isotropically and at a constant rate. The dynamical time for relativistic electrons is very short, 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 () than the dynamical time needed to attain a stationary distribution in absence of losses (). 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,
( 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
cm The equation describing the time evolution of the electron energy distribution, , in the case of injection plus energy losses is: represents the energy variation due to synchrotron and collision losses, respectively, and 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: where is the loss-cone solid angle, with , ( local magnetic field), and sec () 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 is the electron energy at time
and the relation between
and can be derived by integrating Eq. (4) (see
Chiuderi Drago & Franciosini 1993 for detailed calculations). The
complete expression of the energy distribution at time and In the above expressions and
is the total electron number density at
. The result of the integral in Eq. (6) depends
on the value of : since at a given time © European Southern Observatory (ESO) 1998 Online publication: April 28, 1998 |