We analytically investigate the condition for a particle accelerator to be active in the outer magnetosphere of a rotation-powered pulsar. Within the accelerator (or the gap), magnetic-field-aligned electric field accelerates electrons and positrons, which emit copious gamma-rays via curvature process. If one of the gamma-rays emitted by a single pair materializes as a new pair on average, the gap is self-sustained. However, if the neutron-star spin-down rate decreases below a certain limit, the gap becomes no longer self-sustained and the gamma-ray emission ceases. We explicitly compute the multiplicity of cascading pairs and find that the obtained limit corresponds to a modification of previously derived outer-gap death line. In addition to this traditional death line, we find another death line, which becomes important for millisecond pulsars, by separately considering the threshold of photon-photon pair production. Combining these traditional and new death lines, we give predictions on the detectability of gamma-ray pulsars with Fermi and AGILE. An implication on the X-ray observations of heated polar-cap emission is also discussed.
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