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

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6. Discussion

All calculations base on Eq. (26), which neglects energy loss due to triplet pair production. Sturner (1995) showed that ICS dominates over all other energy loss effects when [FORMULA]. This is exactly the energy range we are interested in here. Electron bunching (to produce coherent radio emission by curvature radiation) could decrease the range of validity of our calculations, because energy losses by curvature radiation could then become much more important. On the other hand, the latter depends strongly on the assumed curvature radius of the magnetic field lines (see Eq. (25)). The value of [FORMULA] for the curvature radius used in the calculations corresponds to a typical curvature of the magnetic field lines at the rim of the thermal cap. Because this radius increases (and therefore energy loss by CR decreases) from the rim to the center of the polar cap, the mean value of the CR power is less by at least one order which turns out our calculations as being conservative. The effect of the energy loss by CR can be seen in Figs. 7 to 9, where it clearly restricts the maximum end energy of the electrons to a few [FORMULA] (in the edge of the diagrams with high values for B and low values for P).

In Sect. 5 we mentioned a difference in Lorentz factors within the acceleration zone for the situation with kT between 100 eV and 200 eV when compared to the values of Sturner (1995), whereas the final Lorentz Factor above the acceleration zone is comparable to their results. In Sect. 2 we mentioned that our computations do not include photon polarization effects and also no changes of the resonant cross section due to different incident angles. Both neglects result in a small overestimation of the damping by inverse Compton scattering. Therefore, it is understandable, that our results show a stronger damping in the acceleration zone compared to the results of Sturner. This difference vanishes when the electrons approach the top of the acceleration zone because RICS plays a minor rule there (for the considered magnetic field strength). Another point is, that we consider the magnetic field to be proportional to [FORMULA] and therefore do not include its dependence on the azimute angle. As a consequence, we regard the electrons as moving radially outward. This is surely a good approximation, because the dominant damping effect takes place up to a height of [FORMULA] of the neutron star radius, where the magnetic field lines have a negligible curvature.

On the other hand, the radius of the hot thermal cap [FORMULA] has a large influence on the results. All calculations shown in the figures of this paper refer to the case of a large hot thermal cap (case (b) in Sect. 3, [FORMULA]). If we use case (a) and set [FORMULA], at the top of the acceleration zone the maximum angle for the infalling photons decreases from [FORMULA] to [FORMULA], which results in a weaker damping. Additionally, the area of the polar cap is smaller by a factor of [FORMULA]. Both effects together lead to higher values for the end energy of the accelerated electrons. In the case of low magnetic field strengths ([FORMULA]) the electrons lose their energies at lower heights, where a different size of the hot thermal cap is of negligible influence. But in the situation of high magnetic field strengths ([FORMULA]) and high temperatures ([FORMULA]) this effect can change the end energy of the electrons by three orders. On the other hand, increasing [FORMULA] higher than [FORMULA] has no further influence, because the rim of the thermal cap is then beyond the horizon. Therefore, we chose [FORMULA] for our calculations as a reasonable value.

In Sect. 5 we already stressed the important effect of the non-resonant part of the ICS ([FORMULA] in Eq. (26)) on the electron end energies (see Fig. 3). The non-resonant ICS is often characterized as being negligible in comparison with the resonant part (e.g. Sturner 1995, Zhang & Qiao 1996, and others). This is fairly true for a wide range of parameters but it is obviously wrong for parameter settings used in Fig. 3, where for magnetic field strengths arround [FORMULA] and pulsar rotation periods of [FORMULA] the electron end energies are clearly affected by non-resonant ICS. These parameter values are absolutely typical for neutron stars.

Figs. 7 to 9 show the (expected) fact, that ICS has no influence on the end energies of electrons within the magnetosphere of millisecond pulsars. For standard pulsars ([FORMULA]) the situation strongly depends on temperature and magnetic field strength. Here, for any model of pulsar magnetospheres ICS has to be taken into account.

Our results have been derived assuming Michel's acceleration model. In the following we want to discuss briefly to what extent they are relevant for other polar cap acceleration models which use higher acceleration potentials resulting from either stronger electric fields or larger acceleration heights, or both. E.g. Arons & Scharlemann's (1979) acceleration scheme considers the effect of the divergence of the magnetic field lines which produces changes in the local Goldreich-Julian density and a consequent acceleration over a much larger length of the open field lines. In addition to this effect is the general relativistic frame dragging effect of Muslimov & Tsygan (1992) which also produces an additional acceleration.

In order to check on the effects of electron braking under such conditions we (1) increased the acceleration field in steps up to one hundred times over the value used within the model of Michel and we (2) increased the height of the acceleration zone by factors up to ten. In the first case we found curvature radiation as becoming increasingly important. It limits the end energies [FORMULA] of the electrons to a few times [FORMULA]. The energy loss due to ICS did not increase. In the second case we found the electrons being accelerated within the additional upper part of the acceleration zone without any significant damping. This is because at these heights the flux density of the thermal photons is already too low and also the angles of the infalling photons are too small to cause a significant ICS braking effect. The pair cascades occuring in the models of Arons & Scharlemann (1979) and Muslimov & Tsygan (1992) start at a height of about one neutron star radius. As pointed out above in that region the braking by thermal photons is already small for particles travelling along the field lines. We have not investigated the additional braking occuring due to the angular divergence of pair electrons. Concerning this point, e.g. see Zhang & Harding (2000). We want to point out, within the inner magnetosphere up to one neutron star radius in height our results concerning the energy loss by ICS remain valid, more or less independent of the choosen acceleration model.

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© European Southern Observatory (ESO) 2000

Online publication: May 3, 2000