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

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5. Computation and results

To solve Eq. (26) we used various numerical methods. For the differential equation itself a Runge-Kutta algorithm fourth order was used. All calculations were repeated several times with different step sizes to check for the stability of the solution. For the integrations a recursive Simpson method was performed and all results were proofed by substituting the Simpson method for a Romberg algorithm with adaptive step size.

The two diagrams in Fig. 1 and Fig. 2 show our results for parameter settings used by Sturner (1995) in his Monte Carlo simulations, i.e. [FORMULA], [FORMULA] for Fig. 1 and [FORMULA] for Fig. 2, respectively. In both cases [FORMULA] is used for the radius of the hot thermal cap, whereas for the height of the acceleration zone [FORMULA] is used as given in formula (14). From relation (22) follows [FORMULA] and [FORMULA], respectively, for the maximum Lorentz factor of an electron being fully accelerated without any damping. From this and both figures it is obvious that damping caused by inverse Compton scattering of thermal photons does not have any significant effect for temperatures below 100 eV, in agreement with the results from Sturner (1995). For temperatures between 100 eV and 200 eV we find a stronger damping effect during acceleration than Sturner did (for a discussion see Sect. 6). Nevertheless, this is not relevant for most pulsar magnetosphere models because they only make use of the maximum Lorentz factor above the acceleration zone.

[FIGURE] Fig. 1. Electron Lorentz factor [FORMULA] as a function of height d (above the polar cap surface) and the thermal temperature kT of the hot polar cap. Here, for a comparison we used the same pulsar parameter settings like Sturner & Dermer (1995): [FORMULA], [FORMULA], and [FORMULA] for the thermal cap radius.

[FIGURE] Fig. 2. The same like Fig. 1 but for [FORMULA].

Fig. 3 compares results of the calculations including (solid lines) and excluding (dashed lines) the non-resonant part of the ICS ([FORMULA] in Eq. (26)). For each situation the upper line shows the electron end energies [FORMULA] with a pulsar rotation period [FORMULA], the lower lines represents the results for [FORMULA]. For all calculations [FORMULA] and [FORMULA]. As can be seen just from this figure, neglecting the non-resonant part of the ICS may change the results significantly.

[FIGURE] Fig. 3. Electron end energies [FORMULA] as a function of magnetic field strength for two different pulsar rotation periods. The dashed lines show the influence of resonant ICS whereas the solid lines are for the total, resonant and non-resonant ICS. In each case the upper line results from calculations with [FORMULA] the lower from [FORMULA]. The other parameter settings are: [FORMULA] and [FORMULA].

For a complete overview we present in Figs. 4 to 6 the developement of the electron energy [FORMULA] as a function of height d above the polar cap and magnetic field strength B for three thermal temperatures [FORMULA] 100 eV, 300 eV, and 1000 eV, respectively. For all three cases, the pulsar rotation period is set to [FORMULA] and the radius of the hot thermal cap to [FORMULA].

In the situation with [FORMULA] (Fig. 4) damping by ICS has only a slight influence on the electron energy within a part of the acceleration zone and for magnetic field strengths around [FORMULA]. The end energy above the acceleration zone remains nearly uneffected.

[FIGURE] Fig. 4. Electron Lorentz factor [FORMULA] as a function of height d above the polar cap surface and magnetic field strength B for [FORMULA]. The pulsar rotation period is set to [FORMULA] and the radius of the hot thermal cap is set to [FORMULA].

In the case of [FORMULA] (Fig. 5) and for a wide range of magnetic field strengths damping by ICS prevents the electrons from being fully accelerated within the acceleration zone. Only below [FORMULA] and above [FORMULA] the electrons recover to end energies approximately half an order below the undamped situation (compare to Fig. 4). Between these magnetic field strengths, the electrons completely lose their energy.

[FIGURE] Fig. 5. The same like Fig. 4 but for [FORMULA].

For [FORMULA] (Fig. 6) the electrons are not only braked within the acceleration zone but also above. As a result, only low energy electrons can be found within the magnetosphere above the acceleration zone (for the considered magnetic field strengths).

[FIGURE] Fig. 6. The same like Fig. 4 but for [FORMULA].

[FIGURE] Fig. 7. End electron Lorentz factor [FORMULA] as a function of pulsar rotation period P and magnetic field strength B for [FORMULA]. The radius of the hot thermal cap is set to [FORMULA].

[FIGURE] Fig. 8. The same like Fig. 7 but for [FORMULA].

[FIGURE] Fig. 9. The same like Fig. 7 but for [FORMULA].

Finally, Figs. 7 to 9 show the end energies the electrons can reach above the acceleration zone as a function of pulsar rotation period P and magnetic field strength B, again for the three thermal temperatures [FORMULA] 100 eV, 300 eV, and 1000 eV, respectively. For all three cases, the radius of the hot thermal cap is set to [FORMULA]. With increasing temperature the plateau of damped electrons enlarges to smaller rotation periods and to a wider range of magnetic field strengths. Nevertheless, as can clearly be seen, millisecond pulsars are not effected at all (even for magnetic field strengths around [FORMULA], not shown in these figures).

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000
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