We reexamine the outer-gap size by taking the geometry of the dipole magnetic field into account. Furthermore, we also consider that instead of taking the gap size at half of the light cylinder radius to represent the entire outer gap, it is more appropriate to average the entire outer-gap size over the distance. When these two factors are considered, the derived outer-gap size f ðP; B; hriðÞÞ is a function not only of the period P and magnetic field B of the neutron star but also of the average radial distance to the neutron star, hri, which depends on the magnetic inclination angle . We use this new outer-gap model to study the -ray luminosity of pulsars, which is given by L ¼ f 3 ðP; B; hriðÞÞL sd , where L sd is the pulsar spin-down power, and to study the death lines of -ray emission of the pulsars. Our model can predict the -ray luminosity of an individual pulsar if its P, B, and are known. Since different pulsars have different , this explains why some -ray pulsars have very similar P and B but very different -ray luminosities. In determining the death line of -ray pulsars, we have used a new criterion based on a concrete physical property, i.e., that the fractional size of the outer gap at the null-charge surface for a given pulsar cannot be larger than unity. In an estimate of the fractional size of the outer gap, two possible X-ray fields are considered: (1) X-rays produced by neutron star cooling and polar-cap heating, and (2) X-rays produced by the bombardment of relativistic particles from the outer gap onto the stellar surface (the outer gap is called a ''self-sustained outer gap''). Since it is very difficult to measure in general, we use a Monte Carlo method to simulate the properties of -ray pulsars in our Galaxy. We find that this new outer-gap model predicts many more weak -ray pulsars, which have a typical age between 0.3 and 3 Myr, than does the old model. For all simulated -ray pulsars with self-sustained outer gaps, the -ray luminosity L satisfies L / L sd , where the value of depends on the sensitivity of the -ray detector. For EGRET, $ 0:38, whereas $ 0:46 for GLAST. For -ray pulsars with L sd P L crit sd , $ 1, and L crit sd ¼ 1:5 Â 10 34 P 1=3 ergs s À1 is determined by f ðhri $ r L Þ ¼ 1. These results are roughly consistent with the observed luminosity of -ray pulsars. These predictions are very different from those of the previous outer-gap model, which predicts a very flat relation between L and L sd .

We study the -ray emission properties of pulsars by using a new self-consistent outer gap model. The outer gap can exist in pulsars older than 1 million yr if the effect of magnetic inclination angle, as well as the average properties of the outer gap, are considered. The mature -ray pulsars, whose ages are between 0.3 and 3 million yr, are able to move up to high Galactic latitude. Moreover, their -ray luminosities are weaker, and their spectra are significantly softer than those of younger -ray pulsars in the Galactic plane. We use a Monte Carlo method to simulate the statistical properties of -ray pulsars in the Galaxy as well as in the Gould Belt. We find that -ray pulsars located at j b j< 5 and j b j > 5 have very different properties. High Galactic latitude -ray pulsars are dominated by mature pulsars with longer periods, weaker fluxes, and softer spectra. If the pulsar birth rates in the Galaxy and the Gould Belt are $10 À2 and $2 ; 10 À5 yr À1 , respectively, there are 42 and 35 radio-quiet -ray pulsars for j b j< 5 and j b j > 5 , respectively. The number of radio-quiet -ray pulsars from the Gould Belt are 2 and 13 for j b j< 5 and j b j > 5 , respectively. We suggest that a good fraction of unidentified EGRET -ray sources may be these radio-quiet -ray pulsars. Furthermore, -ray pulsars located at j b j > 5 satisfy L / L sd , whereas L / L sd for -ray pulsars in the Galactic plane, where $ 0:6 and $ 0:3, respectively.

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