We performed Monte Carlo simulations of curvature-radiation-induced electromagnetic cascades developing above a polar cap. The cascade development due to magnetic photon absorption accompanied by $e^\pm$ -pair creation and then synchrotron emission was followed in a 3D space in order to analyse pulse properties. As an example we choose a model with basic parameters of the Vela pulsar: $B_{\rm pc} = 6.8$ TG, P=0.0893 s. In order to meet observational restrictions for the Vela, both spectral and temporal, the following general requirements within polar-cap scenarios had to be satisfied: 1) a polar-cap accelerator should be placed a few stellar radii above pulsar's surface (Dyks et al. 2001); 2) an inclination angle $\alpha$ of the magnetic dipole with respect to the spin axis must not be large, and the pulsar has to be a nearly-aligned rotator (Daugherty & Harding 1994). Recently Harding & Muslimov (1998) proposed a physical mechanism for lifting the polar cap accelerator up to $0.5{-}1\ R_{\rm NS}$ above the surface. However, this altitude is still too low to explain the 10 GeV radiation emerging the Vela magnetosphere unattenuated. Therefore, we placed the polar-cap accelerator at the altitude of $h_0 = 4\ R_{\rm NS}$ to ensure that the magnetosphere is not entirely opaque to curvature photons of energy $\la$ 10 GeV (see Dyks et al. 2001 for the detailed model spectral fitting for the Vela pulsar). Similarly, Miyazaki & Takahara (1997) achieved the best agreement between the observed and their modelled pulse profiles of the Crab pulsar placing the accelerator at $h_0 = 4\ R_{\rm ns}$ . To reproduce the observed peak-to-peak separation $\Delta^{\rm peak} \simeq 0.42$ (Kanbach et al. 1994) we assumed (after Dyks & Rudak 2000) for the inclination angle $\alpha$ and the observer's angle $\zeta _{\rm obs}$ (an angle between the line-of-sight and the spin axis) that $\alpha = \zeta_{\rm obs} = 7.6^\circ$ .

$\begin{figure} \par\includegraphics[width=14.4cm,clip]{MS2183f5.eps}\end{figure}$

Figure 5: Pulse profile integrated for $\varepsilon > 100$ MeV (left panel) and the phase averaged energy spectrum (right panel) are shown for the case of initial energy of primary electrons $E_0 = 2\times 10^7$ MeV. Note the relatively weaker wings outside the peaks and much softer spectrum in comparison with the previous case of E₀ = 10⁷ MeV (Fig. 3a, third row from the top). The data for the Vela pulsar are laid over the model spectrum: boxes are the data from the first COMPTEL source catalogue (see Table 3 in Schönfelder et al. 2000), and dots (plus an upper limit above 10 GeV) are the EGRET data (Thompson et al. 1997). The flux level $\varepsilon F_{\varepsilon }$ is in MeV cm^-2 s^-1 units.

Our numerical results are presented in Fig. 3 (a + b). The three columns of Fig. 3 show (from left to right): 1) mapping onto the parameter space $\zeta _{\rm obs}$ vs. $\phi$ of outgoing photons with energy $\varepsilon > \varepsilon _{\rm limit}$ (where $\phi$ denotes a phase of rotation), 2) double-peak pulse profile due to these photons when $\zeta _{\rm obs}= 7.6^\circ$ , and 3) phase-integrated energy spectrum of these photons, with the position of $\varepsilon _{\rm limit}$ indicated with dotted vertical line. The eight rows correspond to 8 different values of $\varepsilon _{\rm limit}$ : 1, 10, 10², 10³, $4\times 10^3$ , $6\times 10^3$ , $8\times 10^3$ , and 10⁴ MeV (top to bottom). An asymmetry in the double-peak profiles is noticable even though the rotator is nearly aligned: at the highest energies, above $\sim$ 6 GeV, the leading peak LP is less intense than the trailing peak TP (three lowermost panels in the middle column in Fig. 3b). This is a direct result of stronger magnetic absorption of the LP photons comparing to the TP photons. The distribution of these photons in the corresponding panels of $\zeta _{\rm obs}$ vs. $\phi$ (the left column) shows that at viewing angles $\zeta _{\rm obs}$ larger than $7.6^\circ$ (not allowed due to the fixed peak-to-peak separation of 0.42) the asymmetry in pulse profile would be even stronger. This demonstrates an increasing role of rotational effects as the distance from the spin axis increases. In the course of magnetic absorption high-energy curvature photons are converted into electron-positron pairs which in turn emit low-energy synchrotron photons. Asymmetry in the absorption rate as discussed above means, therefore, an identical asymmetry in the $e^\pm$ pair production rate. Consequently, higher number of low-energy synchrotron photons emerges at the LP than at the TP. This is the reason for a dominance of the LP over the TP below $\sim$ 100 MeV, noticable in Fig. 3a. Combining the results from both energy domains, a characteristic inversion in the relative strentgh of the LP and the TP occurs across the gamma-ray energy space. A qualitatively similar inversion of peak intensities takes place in the gamma-ray double-pulse of the Vela pulsar (Thompson 2001).

The beam of synchrotron radiation in our cascades occupies a very narrow range of magnetic colatitudes; in other words - it is highly anisotropic. The reasons for this include a very limited range of altitudes at which the $e^\pm$ pairs are created and the effects of relativistic beaming. By comparison, curvature radiation below $\sim$ 100 MeV is much less anisotropic. Therefore, the prominent peaks visible at $\varepsilon < 100$ MeV (two uppermost panels of Fig. 3a) consist almost entirely of synchrotron radiation (SR) photons, whereas the apparently flat wings outside the peaks (i.e. within the "offpulse" region corresponding to high altitudes) are composed of curvature radiation (CR) photons. A close-up view of the double-peak pulse profile for $\varepsilon > 1$ MeV shown in Fig. 4 reveals that the CR wings are not flat - in fact their intensity decreases with increasing phase $\vert\phi\vert$ ; moreover, their shapes can be reproduced with analytical means: spectral power of curvature radiation $\frac{{\rm d} P_{\rm cr}}{{\rm d} \varepsilon }$ well below a characteristic photon energy $\varepsilon_{\rm crit}\propto \frac{\gamma^3}{\rho_{\rm cr}}$ does not depend on the energy of radiating particles $\gamma$ but on the curvature radius $\rho_{\rm cr}$ of magnetic field lines solely. Since primary electrons reside within a pulsar magnetosphere for a limited period of time $\varepsilon_{\rm crit}$ has a lower limit which equals roughly $\la$ 100 MeV (see Rudak & Dyks 1999 for details). Therefore, the wings in the pulse profiles below 100 MeV fall off due exclusively to an increase in the curvature radius $\rho_{\rm cr}$ of magnetic field lines: this proceeds according to the following relation

As noted by Daugherty & Harding (1996) the wings within the offpulse region must not be too strong within the entire energy range of EGRET if the theoretical pulse profiles are to resemble those of the Vela pulsar. We find that the intensity of wings relative to the intensity of peaks depends sensitively on the richness of the cascades, i.e. on the multiplicity ${\cal M}$ (the number of created pairs per primary electron). The results discussed above and presented in Fig. 3 had been obtained for the initial energy of primary electrons E₀ = 10⁷ MeV which yielded ${\cal M} = 73$ . By increasing the initial energy E₀ up to $2\times 10^7$ MeV the multiplicity reaches ${\cal M} = 830$ and the corresponding pulse profile at 100 MeV (left panel of Fig. 5) changes notably with respect to its counterpart of Fig. 3a. It reveals now a much lower level of wings outside the peaks. Equally important is the change in the shape of the phase-averaged energy spectrum which becomes much softer by gaining more power in the low-energy range (right panel in Fig. 5). Both new features are in rough agreement with the data for the Vela pulsar, contrary to the case with E₀ = 10⁷ MeV. It is interesting to note that the association of the broad peaks at 100 MeV with the relatively hard spectrum (Fig. 3a) on one hand, and of the narrow peaks with the soft spectrum (Fig. 5) on the other hand do resemble qualitatively the observed characteristics of Geminga and the Vela pulsar, respectively.

We may now test our model of the double-peak asymmetry by comparing the numerical results obtained for specific pulsar parameters with the data for real objects. Since the effect is induced by magnetic absorption the expected weakening of the leading peak with respect to the trailing peak occurs only in the vicinity of the high-energy spectral cutoff. Therefore, it is essential to have good photon statistics also at the highest energy bins, i.e. above 1 GeV. As far as the EGRET data are concerned this requirement is barely satisfied even for Vela. With these limitation in mind, we consider Vela as the only appropriate case to provide the test. We used the EGRET data for Vela to calculate the ratio (denoted as P2/P1) of the photon counts in the LP and the TP (denoted as P1 and P2, respectively). For each energy bin (the energy bins cover the range between $\sim$ 30 MeV and $\sim$ 10 GeV) we calculated P1 (P2) by summing all photons within the range $\phi_{\rm lp}\pm 0.05$ ( $\phi_{\rm tp}\pm 0.05$ ) in phase, where $\phi_{\rm lp}$ ( $\phi_{\rm tp}$ ) is the phase of maximum in the LP (the TP) at 100 MeV. Figure 6 shows the observational points as well as their estimated errors along with the results of model calculations performed for three different altitudes: $h_0 = 2 R_{\rm NS}$ , $3 R_{\rm NS}$ , and $4 R_{\rm NS}$ . The overall qualitative and quantitative behaviour of P2/P1 for the EGRET data is very similar to the dependence presented by Kanbach et al. 1980 for the COS-B data. The data points certainly can acommodate our model. However, to answer the question of whether it would be necessary to invoke any additional processes to reproduce the increase in P2/P1 inferred from the data requires better photon statistics at the spectral cut-off and careful statistical analysis.

4 Numerical modelling of the gamma-ray data