A. Appendix

Energy spectrum of energetic particles accelerated in a plasma by a stochastic type-Fermi acceleration process ( αβW) while losing energy simultaneously by collisional losses according to the general expression of [10], operative throughout all the range from suprathermal to ultrarelativistic energies, given in Eq. (2.1) in Section II. In this case, the equation to be solved when only collisional losses are competing with acceleration is

$$\frac{dW}{dt} = a\beta W - \frac{k}{\beta} \ln\left(k\_1 \beta^2\right) \left[R\_4 H(\chi\_\epsilon) + R\_5 H\left(\chi\_\rho\right)\right] \qquad \left(\frac{MeV}{\text{seg}}\right) \tag{A.1}$$

$$
\beta = \frac{\sqrt{\gamma^2 - 1}}{\mathcal{Y}} \tag{A.2}
$$

$$
\alpha \beta \mathcal{W} = \alpha \frac{\sqrt{\gamma^2 - 1}}{\gamma} M c^2 \gamma = M c^2 \alpha \sqrt{\gamma^2 - 1} \tag{A.3}
$$

$$\frac{d\boldsymbol{\eta}}{dt} = a\sqrt{\boldsymbol{\gamma}^2 - 1} - \frac{\boldsymbol{\kappa}}{Mc^2} \frac{\boldsymbol{\gamma}}{\sqrt{\boldsymbol{\gamma}^2 - 1}} \ln\left(\frac{\kappa\_1(\boldsymbol{\gamma}^2 - 1)}{\boldsymbol{\gamma}^2}\right) \left[R\_4 H(\mathbf{x}\_\varepsilon) + R\_5 H(\mathbf{x}\_\boldsymbol{\eta})\right] \tag{A.4}$$

$$dt = \frac{d\boldsymbol{\gamma}}{\alpha \sqrt{\boldsymbol{\gamma}^2 - 1} - \frac{\kappa}{\mu c^2} \frac{\boldsymbol{\gamma}}{\sqrt{\boldsymbol{\gamma}^2 - 1}} \ln \left(\frac{\kappa \boldsymbol{\gamma} (\boldsymbol{\gamma}^2 - 1)}{\boldsymbol{\gamma}^2} \right) \left[R\_4 H(\mathbf{x}\_\varepsilon) + R\_5 H(\mathbf{x}\_p) \right]} \tag{A.5}$$

$$t = \frac{1}{\sqrt{b^2 - 4ac}} \left[ \ln \left| \frac{2a(\sqrt{\gamma^2 - 1}/\gamma) + b - \sqrt{b^2 - 4ac}}{2a(\sqrt{\gamma^2 - 1}/\gamma) + b + \sqrt{b^2 - 4ac}} \right| \frac{2a(\sqrt{\gamma\_c^2 - 1}/\gamma\_c) + b - \sqrt{b^2 - 4ac}}{2a(\sqrt{\gamma\_c^2 - 1}/\gamma\_c) + b + \sqrt{b^2 - 4ac}} \right| \tag{A.6}$$

$$\left(\text{were}\right)a = -a; b = -f'(\gamma\_T); c = a - f(\gamma\_T) + \frac{\sqrt{\gamma\_T^2}}{\gamma\_T}f'(\gamma\_T);$$

$$f(\gamma) = \frac{1}{\gamma^3 - \gamma^-} \frac{\kappa}{mc^2} \ln\left(\frac{k\_1(\gamma^2 - 1)}{\gamma^2}\right) \left[R\_4 H(\mathbf{x}\_\epsilon) + R\_5 H(\mathbf{x}\_p)\right] \text{ (and)}$$

$$f'(\gamma) = \frac{\kappa}{Mc^2} \frac{\left[R\_4 H(\mathbf{x}\_\epsilon) + R\_5 H(\mathbf{x}\_p)\right]}{\sqrt{\gamma^2 - 1}} \left[\left(-3 - \frac{2}{\gamma^2 - 1}\right) \text{Im}\left(\frac{k\_1(\gamma^2 - 1)}{\gamma^2}\right) + \frac{2}{\gamma^2 - 1}\right]$$

$$+ \frac{\kappa}{Mc^2} \frac{1}{\gamma(\gamma^2 - 1)} \ln\left(\frac{k\_1(\gamma^2 - 1)}{\gamma^2}\right) \left\{R\_4 R\_2 e^{-\overline{x}\_p^2} \left[1 - c\left(1 - 2x\_\epsilon^2\right)\right] + R\_5 R\_3 e^{-\overline{x}\_p^2} \left[1 - c\left(1 - 2x\_\epsilon^2\right)\right]\right\}$$

$$N(\boldsymbol{\gamma})d\boldsymbol{\gamma} = \frac{N\_0}{\pi \mathbf{M}c^2} e^{-t/\tau} dt \tag{\text{A.7}}$$

$$e^{-t/\tau} = \left[ \frac{|2a(\sqrt{\gamma^2 - 1}/\gamma) + b - \sqrt{b^2 - 4ac}|}{|2a(\sqrt{\gamma^2 - 1}/\gamma) + b + \sqrt{b^2 - 4ac}|} \middle| \frac{2a(\sqrt{\gamma\_c^2 - 1}/\gamma\_c) + b - \sqrt{b^2 - 4ac}|}{|2a(\sqrt{\gamma\_c^2 - 1}/\gamma\_c) + b + \sqrt{b^2 - 4ac}|} \right]^{\frac{1}{\tau\sqrt{b^2 - 4ac}}} \tag{A.8}$$

$$N(\gamma)d\gamma = \frac{N\_0}{\tau Mc^2} \frac{\left[\frac{2a\left(\sqrt{\gamma^2 - 1}/\gamma\right) + b - \sqrt{b^2 - 4ac}}{2a\left(\sqrt{\gamma^2 - 1}/\gamma\right) + b + \sqrt{b^2 - 4ac}}\right] \left[\frac{2a\left(\sqrt{\gamma\_c^2 - 1}/\gamma\_c\right) + b - \sqrt{b^2 - 4ac}}{2a\left(\sqrt{\gamma\_c^2 - 1}/\gamma\_c\right) + b + \sqrt{b^2 - 4ac}}\right] \left[\frac{\frac{1}{\sqrt{b^2 - 4ac}}}{\sqrt{b^2 - 4ac}}d\gamma}{a\sqrt{\gamma^2 - 1} - \frac{\chi}{\mu c^2}\sqrt{\gamma^2 - 1}\ln\left(\frac{\kappa\_1(\gamma^2 - 1)}{\gamma^2}\right)\left[R\_4H(\mathbf{x}\_\varepsilon) + R\_5H(\mathbf{x}\_\mu)\right]}\right] \tag{A.9}$$

$$J(>\gamma) = \int\_{\gamma}^{\gamma\_m} N(\gamma)d\gamma = \frac{N\_0}{Mc^2} e^{t\{\gamma\_c\}/\pi} \left[ e^{-t\langle\gamma\rangle/\pi} - e^{-t\{\gamma\_m\}/\pi} \right] \tag{A.10}$$

$$I(>\mathcal{\gamma}) = \frac{N\_0}{Mc^2} \left| \frac{2a\left(\sqrt{\mathcal{\gamma}\_c^2 - 1}/\gamma\_c\right) + b - \sqrt{b^2 - 4ac}}{2a\left(\sqrt{\mathcal{\gamma}\_c^2 - 1}/\gamma\_c\right) + b + \sqrt{b^2 - 4ac}} \right|^{1/\pi\sqrt{b^2 - 4ac}}$$

$$\left[ \frac{2a\left(\sqrt{\mathcal{\gamma}^2 - 1}/\gamma\right) + b - \sqrt{b^2 - 4ac}}{2a\left(\sqrt{\mathcal{\gamma}^2 - 1}/\gamma\right) + b + \sqrt{b^2 - 4ac}} \right]^{1/\pi\sqrt{b^2 - 4ac}} - \begin{vmatrix} \frac{2a\left(\sqrt{\mathcal{\gamma}\_M^2 - 1}/\gamma\_M\right) + b - \sqrt{b^2 - 4ac}}{2a\left(\sqrt{\mathcal{\gamma}\_M^2 - 1}/\gamma\_M\right) + b + \sqrt{b^2 - 4ac}} \end{vmatrix} \tag{A.11}$$
