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Unformatted text preview: 3. Coulomb and ionization losses 3.1 Non-relativistic treatment Any charged particle, electron or ion, suffers an electrostatic interaction when it passes through the Coulomb field of another charged particle. In high-energy astrophysics we are usually interested in interactions between a very fast particle and a slow, cold particle, that is the encounter of an energetic particles with ambient quasithermal matter. The energetic particle is then treated as a test particle, i.e. we assume very few energetic particles compared with the cold background medium, so one can neglect interaction between two high-energy particles and considers only interactions with the cold background gas. This basic electrostatic interaction can have a number of effects, each of which has its own name. The energetic particle is usually decelerated, it suffers energy losses called Coulomb losses or ionization losses depending on the interaction partner, usually an electron, being free or bound. The particle will also (slowly) change direction, so we are dealing with scattering. Finally, the energy loss and the scattering are the results of acceleration, to which are charges react by radiating. This radiation process is called bremsstrahlung or free-free radiation. The treatment of the problem is somewhat different depending on whether the energetic particle is relativistic or not. Let us first consider the non-relativistic case. Suppose a particle with charge q 1 and mass m 1 passes by a second particle with characteristics q 2 and m 2 . I will assume that particle 1 is so fast that during the encounter particle 2 can be considered at rest. Note that orbital electrons have a kinetic energy that equals the binding energy, so the inner electrons of heavier atoms from Oxygen to Iron have a couple of keV in kinetic energy, so the approximation is not necessarily good for those electrons. Particle 1 may have an unperturbed trajectory that would carry it with a minimum separation b , the impact parameter, past particle 2. For opposite charges we know the perturbed trajectory to be a Kepler orbit. However, for fast incoming particles the interaction time is short and the deflection is small. It is therefore sufficient to calculate the effects of the interaction using the unperturbed trajecory, i.e. a straight line. Indeed, Rutherfords scattering formula shows that the differential cross section falls of rapidly with the scattering angle . d d 1 sin 4 parenleftBig 2 parenrightBig The force is defined as the total time-derivative of the momentum, therefore the change in momentum can be calculated as the time integral of the force. The component of the force in direction of trajectory, F bardbl , is antisymmetric and the time-integral vanishes. So we only need to consider the force perpendicular to the trajectory, F . Let x be the coordinate along trajectory with x = 0 at the point of closest approach. Then F = q 1 q 2 4 r 2 sin r 2 = b 2 + x 2 sin = b r (3 . 1 . 1) 1 The change in momentum of the energetic particle is most easily calculated by a change of...
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