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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 $γ$-ray observations. We apply outer gap model for the $γ$-ray emission process, and study the radiation characteristics as a function of the magnetic inclination angle $(α)$ and the Earth viewing angle ($ζ$). In our model, the $γ$-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 $ζ\rightarrow 90^{\circ}$ and $p\sim 1.2-1.3$ for $ζ\ll 90^{\circ}$. Our simulation predicts that the pulsars with larger inclination angles ($α=70-90^{\circ}$) and larger viewing angles ($ζ=70-90^{\circ}$) have been preferentially detected by the $Fermi$ $γ$-ray telescope, and hence the observed pulse profiles of the $γ$-ray pulsars have the double peak structure rather than single peak. In the simulation, most $γ$-ray millisecond pulsars are categorized as the radio-quiet $γ$-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 $γ$-ray-selected millisecond pulsars, indicating the radio-quiet millisecond pulsars must contribute to the $Fermi$ unidentified sources and/or the $γ$-ray background radiations. The Unique radiation properties of the low-efficient $γ$-ray pulsar, PSR J0659+1414, can be explained by the present our gap model with a viewing geometry of $α\sim ζ=40^{\circ}-50^{\circ}$.