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. , for Fig. 1 and for Fig. 2, respectively. In both cases is used for the radius of the hot thermal cap, whereas for the height of the acceleration zone is used as given in formula (14). From relation (22) follows and , 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.
Fig. 3 compares results of the calculations including (solid lines) and excluding (dashed lines) the non-resonant part of the ICS ( in Eq. (26)). For each situation the upper line shows the electron end energies with a pulsar rotation period , the lower lines represents the results for . For all calculations and . As can be seen just from this figure, neglecting the non-resonant part of the ICS may change the results significantly.
For a complete overview we present in Figs. 4 to 6 the developement of the electron energy as a function of height d above the polar cap and magnetic field strength B for three thermal temperatures 100 eV, 300 eV, and 1000 eV, respectively. For all three cases, the pulsar rotation period is set to and the radius of the hot thermal cap to .
In the situation with (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 . The end energy above the acceleration zone remains nearly uneffected.
In the case of (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 and above 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.
For (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).
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 100 eV, 300 eV, and 1000 eV, respectively. For all three cases, the radius of the hot thermal cap is set to . 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 , not shown in these figures).
© European Southern Observatory (ESO) 2000
Online publication: May 3, 2000