tokens: the numerical code. The second term is the source term, , which describes the injection of new power-law particles into the causal volume. In general, S ′ ( p e ) receives contributions from other processes, such as Compton scattering, pair-production and annihilation. The advection coefficient for the adiabatic cooling of particles is given by Here, we have used the scaling of particle energy with comoving volume due to adiabatic expansion, such that where is the adiabatic index for a relativistic (non-relativistic) particle distribution. In general, the advection coefficient includes contributions from other processes that change the energy of particles, e.g. Compton scattering, synchrotron cooling, Coulomb interactions, and particle heating (as described below). The adiabatic cooling timescale is obtained from the rate of volume expansion, where where ζ = 7/3 for r r s when Γ ∝ r 1/3 , and ζ = 2 for r ≥ r s when Γ = Γ ∞ . By expressing the LHS of equation ( A3 ) using the number density, we find where we have conveniently defined the comoving growth time of the causal volume, , through Next, we use equation ( A2 ) to express the rate of change of the comoving number density with comoving time in terms of and the causal volume , so that which in turn implies where the comoving particle injection time into the causal volume is defined by and is the particle injection rate per unit volume. Note that the causal comoving volume grows faster than the total comoving volume V ′ since it occupies an increasing fraction of it, , where . Now, using equations ( A3 ) and ( A6 ) lead to the equation for the number density evolution where . By integrating the above equation over p e , it can be shown that the adiabatic cooling term vanishes and , where so that the equation reads . The comoving rate of particle injection per unit volume per unit dimensionless momentum comprises of two terms, where a fraction ξ of the total is accelerated into a power-law (non-thermal) distribution and the remaining fraction (1 − ξ) forms a thermal distribution, where represents the normalized momentum distribution. The total energy density per unit rest mass energy of the injected distribution can be expressed as where 〈γ e 〉 th is the mean energy per rest mass energy of the thermal distribution and for the power-law distribution 〈γ e 〉 nth is given in equation ( 10 ). Here, we make the assumption that the injected thermal particles are cold, so that . The momentum distribution of power-law electrons is given by The normalization is obtained by equating the rate of energy injection per unit volume to that given to the electrons due to dissipation, such that which for γ M > γ m ≫ 1 gives where we take γ M = γ max , the maximum LF corresponding to the outer boundary of the particle momentum grid in the simulation. The efficiency of acceleration is controlled by the mean energy of the power-law electrons, which depends on ξ, and ultimately by the minimal energy of power-law
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