## 4. The differential equation for acceleration and damping of an electronWe are now able to give a closed expression in form of a differential equation of the acceleration and damping processes an electron undergoes within a neutron star magnetosphere. For the acceleration we use the model of Goldreich & Julian (1969) and the conditions of Michel (1974). In this model a linear acceleration for the electrons is assumed within an acceleration length equal to the polar cap radius and up to a maximum value of With this we obtain a natural scale length for an electron to receive kinetic energy in the order of its rest energy Therefore, the acceleration can be described by the differential equation with and given in Eq. (23) and (14) respectively. Additionally, we include energy loss by curvature radiation in our calculations which is given by (e.g. Jackson, 1998, p. 667): For our calculations we set the curvature radius , typical for the magnetic field line curvature at the rim of the thermal cap with (see also Sturner 1995). Finally, using the expressions we derived in Sect. 2 for the damping processes by nonresonant and resonant inverse Compton scattering (ICS and RICS, respectively) we come out with the differential equation for the discussed model: The here used quantities are given in Eqs. (24), (4), (13), with the use of (17), and (25). © European Southern Observatory (ESO) 2000 Online publication: May 3, 2000 |