Gamma-ray Luminosity and Death Lines of Pulsars with Outer Gaps

We re-examine 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 is not only the function of period P and magnetic field B of the neutron star, but also the function of the average radial distance to the neutron star <r>. We use this new outer gap model to study $\gamma$-ray luminosity of pulsars, and to study the death lines of $\gamma$-ray emission of the pulsars. Our model can predict the $\gamma$-ray luminosity of individual pulsar if its $P, B$ and $\alpha$ are known. Since different pulsars have different $\alpha$, this explains why some $\gamma$-ray pulsars have very similar $P$ and $B$ but have very different $\gamma$-ray luminosities. In determining the death line of $\gamma$-ray pulsars, we have used a new criterion based on concrete physical reason. In estimate of the fractional size of the outer gap, two possible X-ray fields are considered: (i) X-rays are produced by the neutron star cooling and polar cap heating, and (ii)X-rays are produced by the bombardment of the relativistic particles from the outer gap on the stellar surface. Since it is very difficult to measure $\alpha$ in general, we use a Monte Carlo method to simulate the properties of $\gamma$-ray pulsars in our galaxy. We find that this new outer gap model predicts many more weak $\gamma$-ray pulsars. For all simulated $\gamma$-ray pulsars with self-sustained outer gaps, $\gamma$-ray luminosity $L_{\gamma}$ satisfies $L_{\gamma}\propto L^{\delta}_{sd}$; where the value of $\delta$ depends on the ensitivity of the $\gamma$-ray detector.