  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.
