2 The electron distribution shapes

The Maxwellian distribution, generally considered for the electron distribution in astrophysical plasmas, $N_{\rm e}(E)$, is defined as:

 

$$% {\rm d}N_{\rm e}(E) n_{\rm e}\ f^{\rm M}_{\rm E}(E)\ {\rm d}E$$ = (1)

$$f^{\rm M}_{\rm E}(E) \frac{2}{\sqrt{\pi}}\ (kT)^{-3/2}\ E^{1/2}\ {\rm e}^{-\frac{E}{kT}}$$ = (2)

where E is the energy of the electron, T is the electronic temperature and $n_{\rm e}$ the total electronic density. In this expression the Maxwellian function $f^{\rm M}_{\rm E}(E)$ is normalised so that $\int^{\infty}_{0}~f^{\rm M}_{\rm E}(E)~{\rm d}E=1$.

It is convenient to express this distribution in term of the reduced energy x=E/kT:

  

$$% {\rm d}N_{\rm e}(x) n_{\rm e}\ f^{\rm M}(x)\ {\rm d}x$$ = (3)

$$f^{\rm M}(x) \frac{2}{\sqrt{\pi}}\ \ x^{\frac{1}{2}}\ {\rm e}^{-x}.$$ = (4)

The corresponding scaled (non-dimensional) distribution $f^{\rm M}(x)$ is an universal function, of fixed shape.

Non-Maxwellian electron distributions expected in the vicinity of shock waves, as in young supernova remnants, seem to be reasonably described by a Maxwellian distribution at low energy up to a break energy $E_{\rm b}$, and by a power-law distribution at higher energy (e.g., Berezhko & Ellison 1999; Bykov & Uvarov 1999). We call hereafter this "Maxwellian/Power-law'' type of electron distribution the Hybrid electron distribution ($f^{\rm H}$). It is defined, in reduced energy coordinates, as:

$$% {\rm d}N_{\rm e}(x) = n_{\rm e}f^{\rm H}(x) {\rm d}x$$ (5)

 

$$% f^{\rm H}(x) C(x_{\rm b},\alpha)\ \frac{2}{\sqrt{\pi}}\ x^{1/2}\ {\rm e}^{-x} ~~~~~~~~~~~~~~~ x \leq x_{\rm b}$$ (6)

$$f^{\rm H}(x) C(x_{\rm b},\alpha)\ \frac{2}{\sqrt{\pi}}\ x_{\rm b}^{1/2}\ {\rm e}^{-x_{\rm b}}\ \left(\frac{x}{x_{\rm b}}\right)^{-\alpha}~~x \geq x_{\rm b},$$

where $x_{\rm b}=E_{\rm b}/kT$ is the reduced break energy, and $\alpha$ is the energy index of the power-law ($\alpha>1$). Note that for $\alpha\leq$ 2  the energy diverges (in practice a cutoff at very high energy occurs). Since for the calculations of the ionization and recombination rates the very high energies ( ${\geq}20~kT$) have negligible effect, for simplicity, we use here a power-law defined from $x_{\rm b}$ to infinity.

The normalisation factor of the power-law distribution is defined so that the electron distribution is continous at $x_{\rm b}$. The factor $C(x_{\rm b},\alpha)$ is a normalisation constant, so that $\int^{\infty}_{0}~f^{\rm H}(x)~{\rm d}x=1$:

$$% C(x_{\rm b},\alpha) = \frac{\sqrt{\pi}}{2}\ \frac{1}{\gamma... ...}) + (\alpha-1)^{-1} \ x_{\rm b}^{3/2}\ {\rm e}^{-x_{\rm b}}}$$ (7)

where $\gamma(a,x)$ is the gamma function defined as $\gamma(a,x)=\int^{x}_{0}t^{a-1}\ {\rm e}^{-t}\,{\rm d}t$. For $x \leq x_{\rm b}$, the Hybrid distribution only differs from a Maxwellian distribution by this multiplicative factor. The scaled distribution $f^{\rm H}(x)$ only depends on the two non-dimensional parameters, $x_{\rm b}$ and $\alpha$. The dependence on kT of the corresponding physical electron distribution is $f^{\rm H}_{\rm E}(E)=(kT)^{-1} f^{\rm H}(E/kT)$.

The Hybrid distributions $f^{\rm H}(x)$, obtained for several values of the energy break $x_{\rm b}$, are compared to the Maxwellian distribution in Fig. 1. The slope has been fixed to $\alpha =2$, a typical value found in the models referenced above. The variation of the reduced median energy of the distribution with $x_{\rm b}$, for $\alpha=1.5,2.,3.$, is plotted in Fig. 2, as well as the variation of the normalisation factor $C(x_{\rm b},\alpha)$.

Figure 1: The Hybrid (Maxwell/Power-law) electron distribution for different values of the break parameter $x_{\rm b}$, compared to a Maxwellian distribution ( $x_{\rm b}=\infty$). The slope of the power-law has been fixed to $\alpha =2$. Reduced energy coordinates, x=E/kT, are used.

Figure 2: Variation of the median energy (top panel) and of the normalisation constant (bottom panel) with the break parameter $x_{\rm b}$. The lines are labeled by the value of the slope parameter $\alpha$. The bold part of the curves corresponds to $x_{\rm b}\geq \alpha +1/2$. The black horizontal lines correspond to the values obtained for a Maxwellian distribution.

As apparent in the figures, there is a critical value of $x_{\rm b}$, for each $\alpha$ value, corresponding to a qualitative change in the behavior of the Hybrid distribution. This can be understood by looking at the distribution at the break energy $x_{\rm b}$. Whereas the distribution is continuous, its slope changes. The logarithmic slope is $1/2-x_{\rm b}$ on the Maxwellian side and $-\alpha$ on the power-law side. There is no break in the shape of the Hybrid distribution (full line in Fig. 1), only for the critical value of $x_{\rm b}=\alpha +1/2$. For $x_{\rm b}>\alpha+1/2$, the power-law always decreases less rapidly with energy than a Maxwellian distribution and does correspond to an enhanced high energy tail. The contribution of this tail increases with decreasing $x_{\rm b}$ (and $\alpha$). Thus the median energy increases and the normalisation parameter, which scales the Maxwellian part, decreases (see Fig. 2). On the other hand, when $x_{\rm b}<\alpha+1/2$, there is an intermediate region above the energy break where the power-law decreases more steeply than a Maxwellian (see dotted line in Fig. 1). This results in a deficit of electrons at these energies as compared to a Maxwellian distribution, more and more pronounced as $x_{\rm b}$ is small. The median energy thus starts to decrease with decreasing $x_{\rm b}$ and can be even lower than the median energy of a Maxwellian distribution (Fig. 2). In this paper we only consider the regime where $x_{\rm b}\geq \alpha +1/2$. It corresponds to clear cases where the high energy part of the distribution is indeed increased, as expected when electron are accelerated in shocks. Furthermore, the distribution used here is only an approximation, valid when the hard tail can be considered as a perturbation of the original Maxwellian distribution. The simulations of Bykov & Uvarov (1999, see their Fig. 2) clearly show that the low energy part of the distribution is less and less well approximated by a Maxwellian distribution, as the "enhanced'' high energy tail extends to lower and lower energy (lower "break''). Although we cannot rigorously define a corresponding quantitative lower limit on $x_{\rm b}$, the cases presented by Bykov & Uvarov (1999, see their Fig. 2) suggest a limit similar to the one considered here, i.e. a few times the Maxwellian peak energy.

The distribution considered here differs from the so-called "kappa-distribution'' or the "power distribution'', relevant for other physical conditions (see e.g. Dzifcáková 2000 and references therein). These two distributions have been used to model deviations from a Maxwellian distribution caused by strong plasma inhomogeneities, as in the solar corona, and their impact on the ionization balance has been extensively studied (e.g. Roussel-Dupré 1980; Owocki & Scudder 1983; Dzifcáková 1992; Dzifcáková 1998). Although the effect of the Hybrid distribution is expected to be qualitatively similar, it has never been quantitatively studied. In the next section we discuss how the ionization and recombination rates are modified, as compared to a pure Maxwellian distribution, depending on the parameters $x_{\rm b}$ and $\alpha$.