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Astron. Astrophys. 357, 301-307 (2000)

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4. The differential equation for acceleration and damping of an electron

We 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

[EQUATION]

With this we obtain a natural scale length for an electron to receive kinetic energy in the order of its rest energy [FORMULA]

[EQUATION]

Therefore, the acceleration can be described by the differential equation

[EQUATION]

with [FORMULA] and [FORMULA] 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):

[EQUATION]

For our calculations we set the curvature radius [FORMULA], typical for the magnetic field line curvature at the rim of the thermal cap with [FORMULA] (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:

[EQUATION]

The here used quantities are given in Eqs. (24), (4), (13), with the use of (17), and (25).

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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