Shock acceleration of electrons in the presence of synchrotron losses - I. Test-particle theory

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We discuss a semi-analytical solution of the transport equation for electrons at a non-relativistic shock in the presence of synchrotron energy losses. We calculate the spectrum of accelerated (test) particles at any point upstream and downstream of the shock for an arbitrary diffusion coefficient and we specialize the results to three cases: 1) diffusion constant in momentum ($D(p)=D_{0}$), 2) Bohm diffusion ($D(p)\propto p$), and 3) Kolmogorov diffusion ($D(p)\propto p^{1/3}$). Of special importance is the determination of the shape of the cutoff in the electron spectrum which depends on the diffusion properties felt by particles in the shock region. The formalism can be generalized to the case of a shock with an upstream precursor induced by the dynamical reaction of accelerated particles.