trajectories in the outer gap. Once the radial distances of the emission regions are determined, the spectrum of photon emission can be calculated for a given radial distance $r$. Figure 4 shows the comparison between the observed phase-resolved spectra of the Crab pulsar and the calculated spectra in the energy range of 10 MeV to 10 GeV (Zhang et al. 2000). The more detailed phase-resolved spectra only in the energy range of EGRET can be found in CRZ. Figure 5 show the model energy dependent light curves of X-rays in four different energy channels (Zhang and Cheng, 2001b). Figure 6 shows the phase dependent spectral indices of X-rays in four different energy ranges.

[Fig. 3. The variation of radial distance with pulse phase for the Crab. The inclination angle is $65^\circ $. Five regions for different pulse phase are indicated.\[fig:f3\]]{}

![image](f4){width="176mm"}

[Fig. 4. Phase-resolved $\gamma $-ray spectra from $10$MeV to $10$GeV for peak 1, bridge, peak 2 and phase-average of the Crab pulsar. Observed data are taken from Ulmer et al. (1995).\[fig:f4\]]{}

![image](f5_3){width="110mm"}

[Fig. 5. Expected X-ray pulse profiles of Crab pulsar for four different energy bands. The first model light curve in the energy band from $0.3-4.2$ keV corresponds 