1ST-ORDER FERMI ACCELERATION IN SPHERICALLY SYMMETRICAL FLOWS - SOLUTIONS INCLUDING QUADRATIC LOSSES

We obtain a new, exact solution for the Green's function describing the transport of relativistic particles in steady, spherically symmetric background flows, including the effects of first-order Fermi acceleration, spatial diffusion, bulk advection, and losses proportional to the square of the particle momentum. The flow velocity of the background (scattering) plasma and the spatial diffusion coefficient are assumed to vary as upsilon(r) is-proportional-to r(-alpha) and kappa(p, r) is-proportional-to r(beta)K(p), respectively, where r is the radius and K(p) is an arbitrary function of the particle momentum p. The momentum loss rate is assumed to vary quadratically with the particle momentum as [p]loss = Ap2-upsilon/r, where A = constant. Examples of quadratic loss mechanisms include synchrotron and inverse Compton; in the synchrotron case, the implied magnetic field variation is B2 is-proportional-to upsilon/r, which may be satisfied in galaxy cluster cooling flows. Losses exactly balance first-order Fermi acceleration at the critical momentum p(c) = (2 - alpha)/(3A), which is a single constant for the entire flow if A = constant. Finite losses cause a compression of the dynamic range of the momentum variable relative to the lossless case (A = 0), since particles injected with momentum p0 are confined to the interval (p0, p(c)). The sign of upsilon is unrestricted, and therefore our model can be used to study the transport of relativistic particles in both winds and accretion flows. Previous studies of particle transport in radio sources have considered the effects of spatial diffusion and synchrotron losses on relativistic electrons propagating through static background plasmas. The inclusion of bulk motions of the plasma affects both the energization of the electrons (via Fermi acceleration) and the spatial transport (via advection). In accretion flows, advection tends to drag the electrons inward, reducing the number escaping from the flow, while Fermi acceleration tends to increase the energy of the electrons that do escape. We find that the spectrum of electrons at large radii (and hence the associated synchrotron emission) is significantly harder in the presence of bulk motions, and that extended, power-law radio spectra are a natural consequence of either monoenergetic or power-law electron injection. In particular, we conclude that Fermi acceleration may help to explain the production of core-halo radio emission in cooling flows. Therefore core halos may provide independent evidence for dynamical cooling processes in clusters of galaxies. It has been previously pointed out that the flux of relativistic particles emerging from an accretion flow vanishes for certain values of the parameters (alpha, beta, gamma) when K(p) is-proportional-to p(gamma). This region of the parameter space is consequently unobservable if the relativistic particles are photons. However, if the relativistic particles are electrons, we show that observable power-law radio emission can be produced even when the emergent flux of electrons vanishes.