The first equation describes the time evolution of f(p), the momentum distribution of the non-thermal particles, from here on supposed to be relativistic electrons, since we are considering a leptonic model:
(1)$$\begin{eqnarray*}
\frac{\partial f}{\partial t} = \frac{1}{p^2} \frac{\partial }{\partial p} \left[ p^2 D_p \frac{\partial f}{\partial p} + p^2 \left(\frac{\partial p}{\partial t} \right)_{\text{rad}} f \right] + \frac{f}{t_\text{esc}} +I_f ,
\end{eqnarray*}$$where p is the particle momentum, Dp the momentum diffusion coefficient, (∂p/∂t)rad the cooling coefficient due to radiative emission, and If is the particle injection rate. Starting from momentum distribution, it is possible to calculate the electron energy density, in fact n(γ) = 4πp2f(p)mec, where me is the electron mass. Supposing only parallel and antiparallel propagating magnetohydrodynamic (MHD) waves interacting with electrons, the acceleration time can be estimated as (Miller & Roberts 1995; Kakuwa 2016):
(2)$$\begin{eqnarray*}
t_\text{acc} = \left[ \frac{2 \beta _w^2 c}{U_B r_g}\int _{k\gt k_\text{res}} \frac{W_B(k)}{k} dk \right]^{-1} ,
\end{eqnarray*}$$where βw = va/c is the wave velocity in unit of c, UB = B2/8π is the magnetic energy density, rg = γmec2/eB is the Larmor radius, WB(k) ≈ W(k)/2 is the magnetic component of the energy density of turbulence fields per unit of k [indicated with W(k)] and kres = 2π/rg is the resonant wavenumber. Note that an electron with a fixed γ can interact only with Alfvén waves whose wavelength is smaller than its Larmor radius. From the time acceleration, the momentum diffusion coefficient can be obtained using Dp = p2/2tacc. Neglecting inverse Compton and adiabatic cooling for simplicity, the cooling time is equal to:
(3)$$\begin{eqnarray*}
t_\text{cool} = \frac{6 \pi m_e c}{\sigma _T B^2 \gamma } ,
\end{eqnarray*}$$where B is the mean magnetic field and σT is the Thomson cross-section. From the cooling time, it is possible to calculate the cooling coefficient, in fact $t_\text{cool} = p/\dot{p}$. Another effect influencing the electron spectra is the spatial escape from the acceleration site:
(4)$$\begin{eqnarray*}
t_\text{esc} = \frac{R}{c} + \frac{R^2}{k_\parallel } ,
\end{eqnarray*}$$where k∥ = crg/9ζ(kres) is the spatial diffusion coefficient along the mean magnetic field, while ζ(k) = kWB(k)/UB is the relative amplitude of the turbulent magnetic field energy density for a given k. Note that if the mean free path is large (which implies a small diffusion coefficient), the escape time is simply equal to the geometric escape time, that is the ratio between the dimension of the acceleration region and the particles velocity. Finally, the last term of equation (1) describes the particle injection, identified here as the electrons accelerated at the recollimation shock and advected downstream. Supposing a strong shock, the injected electron density per unit time is equal to:
(5)$$\begin{eqnarray*}
I_n(\gamma) = I_{n,0} \, \gamma ^{-2}\, e^{-\frac{\gamma }{\gamma _\text{cut}}} \quad \text{with} \quad \gamma _\text{inj}\lt \gamma \lt 100\, \gamma _\text{cut} .
\end{eqnarray*}$$The extremes of the injection range are fixed and they are estimated thanks to recent simulations of diffusive shock acceleration: γinj = 103 and γcut = 105 (Zech & Lemoine 2021). From the injected electron density per unit time, it is possible to obtain the injection distribution in the momentum space, in fact In(γ) = 4πp2mecIf(p).
Citation: Kakuwa 2016
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