that the spectral indices change as a function of phase.

[Fig. 7a. The variation of radial distance with the pulse phase for different outer gap surfaces of the Geminga pulsar. The inclination angle is $50^\circ $. Five regions for different pulse phases which are the same as observed those are indicated.\[fig:f7a\]]{}

![image](f7b){width="160mm"}

[Fig. 7b. Phase-resolved $\gamma $-ray spectra for different phases (peak one, trailing wing one, bridge, leading wing two, peak 2 and phase-average) of the Geminga pulsar. Observed data are taken from Fierro et al. (1998).\[fig:f7b\]]{}

Now we apply to this model to explain the pulse profile and phase-resolved spectra of the Geminga pulsar. The key differences between the Geminga pulsar and the Crab pulsar are: (1) According to the model described in previous section, we can find that the fractional height of the outer gap for Geminga is $f_0\sim 0.7$,which is a very thick gap. Certainly we cannot use a single surface to represent the emission regions. We have used five layers to approximate the emission regions (cf. Figure 7a). (2) The mean free path of the primary photons from the gap is longer than the light cylinder. So the observed $\gamma$-rays are curvature photons from primary charged 