Population study for gamma-ray pulsars: (III) Radiation characteristics and viewing geometry

We have performed a Monte-Calro simulation for Galactic population of pulsars and for the $\gamma$-ray observations. We apply outer gap model for the $\gamma$-ray emission process, and study the radiation characteristics as a function of the magnetic inclination angle $(\alpha)$ and the Earth viewing angle ($\zeta$). In our model, the $\gamma$-ray flux and the spectral cut-off energy tend to decrease as the inclination and viewing angles deviate from $90^{\circ}$. The emerging spectrum above 100 MeV becomes soft with a photon index $p\sim 1.8-2$ for $\zeta \rightarrow 90^{\circ}$ and $p\sim 1.2-1.3$ for $\zeta \ll 90^{\circ}$. Our simulation predicts that the pulsars with larger inclination angles ($\alpha=70-90^{\circ}$) and larger viewing angles ($\zeta=70-90^{\circ}$) have been preferentially detected by the $Fermi$ $\gamma$-ray telescope, and hence the observed pulse profiles of the $\gamma$-ray pulsars have the double peak structure rather than single peak. In the simulation, most $\gamma$-ray millisecond pulsars are categorized as the radio-quiet $\gamma$-ray pulsars, because its radio fluxes are under the sensitivities of the major radio surveys. Even we drastically increase the radio sensitivity by a factor of ten, the number of radio-selected millisecond pulsars detected by the $Fermi$ ten years observations is still much less than the expected $\gamma$-ray-selected millisecond pulsars, indicating the radio-quiet millisecond pulsars must contribute to the $Fermi$ unidentified sources and/or the $\gamma$-ray background radiations. The Unique radiation properties of the low-efficient $\gamma$-ray pulsar, PSR J0659+1414, can be explained by the present our gap model with a viewing geometry of $\alpha\sim \zeta=40^{\circ}-50^{\circ}$.