In more than four years of observation the Large Area Telescope on board the Fermi satellite has identified pulsed $gamma$-ray emission from more than 80 young pulsars, providing light curves with high statistics. Fitting the observations with geometrical models can provide estimates of the magnetic obliquity $alpha$ and aspect angle $zeta$, yielding estimates of the radiation beaming factor and luminosity. Using $gamma$-ray emission geometries (Polar Cap, Slot Gap, Outer Gap, One Pole Caustic) and radio emission geometry, we fit $gamma$-ray light curves for 76 young pulsars and we jointly fit their $gamma$-ray plus radio light curves when possible. We find that a joint radio plus $gamma$-ray fit strategy is important to obtain ($alpha$, $zeta$) estimates that can explain simultaneous radio and $gamma$-ray emission. The intermediate-to-high altitude magnetosphere models, Slot Gap, Outer Gap, and One pole Caustic, are favoured in explaining the observations. We find no evolution of $alpha$ on a time scale of a million years. For all emission geometries our derived $gamma$-ray beaming factors are generally less than one and do not significantly evolve with the spin-down power. A more pronounced beaming factor vs. spin-down power correlation is observed for Slot Gap model and radio-quiet pulsars and for the Outer Gap model and radio-loud pulsars. For all models, the correlation between $gamma$-ray luminosity and spin-down power is consistent with a square root dependence. The $gamma$-ray luminosities obtained by using our beaming factors not exceed the spin-down power. This suggests that assuming a beaming factor of one for all objects, as done in other studies, likely overestimates the real values. The data show a relation between the pulsar spectral characteristics and the width of the accelerator gap that is consistent with the theoretical prediction for the Slot Gap model.
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