6 Discussion

We have shown that pulsar rotation induces an asymmetry in the magnetic absorption rate with respect to the magnetic dipole axis. Its consequences are potentially interesting in constraining the phase-space of parameters in the polar cap models of high-energy radiation, provided that very high quality gamma-ray data (e.g. as expected from GLAST) are at hand. Its magnitude depends mainly on the linear velocity $\beta$ of the magnetosphere at sites of particle acceleration and magnetic photon absorption. When the region of electron acceleration is placed just above the neutron star surface rotation does not produce any detectable effects even for relatively fast rotating young gamma-ray pulsars. However, it has been argued that at least in the case of the Vela pulsar, such a situation is difficult to reconcile with the spectral high-energy cutoff at about 10 GeV (e.g. Dyks et al. 2001). We find then that raising the accelerator up to $\sim$4 neutron star radii (in the spirit of Harding & Muslimov 1998) above its polar cap produces asymmetric gamma-ray pulse profiles even in the case of nearly aligned rotators with a spin period of $P\sim 0.1$ s. The resulting features - softer spectrum of the leading peak and the dominance of the trailing peak above $\sim$5 GeV - do agree qualitatively with the EGRET data of the bright gamma-ray pulsars (Thompson 2001).

We are far from concluding that the rotation effects alone can account for the observed asymmetry in the double peaks of the bright EGRET pulsars. On the contrary - some axial asymmetry intrinsic to the region of electron acceleration is inevitable in order to explain the double-peak properties at $\> 100$ MeV of Geminga and B1706-44, where the leading peak is weaker than the trailing peak. Strong deviations of the actual magnetic field structure from the pure dipole at the stellar surface (e.g. Gil et al. 2002) might be responsible for maintaining axial asymmetry at the site of electron acceleration (unlike the symmetric initial conditions introduced in Sect. 2). This in turn would lead to electromagnetic cascades whose properties vary with magnetic azimuth. It is important, however, that the propagation effects due to rotation work in the right direction, i.e. they explain qualitatively the observed weakening of the leading peak with respect to the trailing peak. We emphasize that this weakening occurs only in the vicinity of the (phase-averaged) high-energy spectral cutoff, where the flux level decreases significantly.

Another consequence of the magnetic absorption of high energy photons is a noticeable change in the separation $\Delta^{\rm peak}$ between the two peaks in the pulse, taking place near the high-energy spectral cutoff (Dyks & Rudak 2000). In the model discussed above, with electrons ejected only from a rim of the polar cap, the higher energy of photons requires higher emission altitudes to avoid absorption. Therefore, a slight increase in $\Delta^{\rm peak}$ is visible in the three lowermost pulse profiles in Fig. 3b. However, if the emission from the interior of the polar cap were included, just the opposite behaviour would occur: $\Delta^{\rm peak}$ would decrease near the high-energy cutoff in the spectrum. This is because in this case of a "filled polar cap tube", the highest energy non-absorbed photons are emitted closer to the magnetic dipole axis (see Fig. 2 in Dyks & Rudak 2000). The latter case agrees qualitatively with the marginal decrease in peak separation found in the EGRET data for Vela (Kanbach 1999).

Stimulated by high-quality observations of gamma-ray pulsars anticipated with GLAST we analysed in Sect. 5 the importance of rotation-driven asymmetry in magnetic absorption for a broad range of pulsar parameters. A decline in gamma-ray flux at high-energy spectral cutoff should inevitably be accompanied by strong changes in pulse profiles: whereas at lower photon energies the profile is determined by the density distribution of primary electrons over the polar cap and the efficiency of photon emission mechanism, in the vicinity of the cutoff it becomes additionally constrained by likely high values of the asymmetry parameter ${\cal R}_{\rm esc}(\theta_{\rm pc})$ - the situation anticipated for fast rotating (P < 0.01 s), and highly inclined ( $\alpha \ga 45^\circ$) pulsars.

Acknowledgements

We thank V.S. Beskin and A.K. Harding for useful comments on the issue of magnetospheric distortions. We are grateful to Gottfried Kanbach for providing us with the EGRET data on Vela, and to Aga Wozna for calculating the P2/P1 ratios used in Fig. 6. We acknowledge comments and stimulating suggestions made by the anonymous referee. JD appreciates Young Researcher Scholarship of Foundation for Polish Science. This work was supported by KBN (grants 2P03D02117 and 5P03D02420) and NCU (grant 405A).