10 0 10 1 10 2 10 3 10 4 10 -2 10 -1 10 0 10 1 10 2 10 3 10 4 dN/dln E, dN/dln  E,   (MeV) e $^{+}$e - pairs Photons 10 -1 10 0 10 1 10 2 10 3 10 4 E,   (MeV) e $^{+}$e - pairs Photons 10 -1 10 0 10 1 10 2 10 3 10 4 10 5 E,   (MeV) e $^{+}$e - pairs Photons Pair cascades of strongly-magnetized neutron stars 17 Figure 9. The final photon and pair spectra of photon-initiated cascades for a non-dipole magnetosphere with local radius of curvature R$_{c}$ = R , for surface magnetic fields B$_{p,}$$_{12}$ = 1 (left panel), 10 (middle panel), and 100 (right panel). Here, the primary photon has ϵ$_{0}$ = 10 4 MeV and ICS is active from a hot spot with T$_{6}$ = 1, θ$_{spot}$ = 0 . 3. Table 1. Pair multiplicity N$_{E}$ when resonant ICS is the dominant photon emission process of the primary electron. For simplicity, all photons are assumed to be upscattered from the thermal peak ( γ$_{crit}$ ; see text). Each entry in the table gives N$_{E}$ = N$_{0}$n$_{E}$ for a different surface magnetic field strength B$_{p}$ , radius of curvature (either R$_{c}$ = R or a dipole field curvature with the pulsar spin period P$_{0}$ is specified), and hot spot temperature T and size θ$_{spot}$ . The pair multiplicity is zero for B$_{p,}$$_{12}$ = 1 and the hot spot models used here, even when R$_{c}$ = R ; we therefore omitted these entries from the table. T$_{6}$ = 1, θ$_{spot}$ = 0 . 3 T$_{6}$ = 3, θ$_{spot}$ = 0 . 1 P$_{0}$ = 10 P$_{0}$ = 1 R$_{c}$ = R P$_{0}$ = 10 P$_{0}$ = 1 R$_{c}$ = R B$_{p,}$$_{12}$ = 10 0 0 14 0 0 220 B$_{p,}$$_{12}$ = 100 2.2 1.2 1.2 35 6.2 31 B$_{p,}$$_{12}$ = 1000 0.7 0.1 0.1 11 1.9 4.4 of a secondary electron/positron, ¯ E = E $_{tot}$/N$_{E}$ , is directly proportional to the total cascade energy (i.e., ¯ E ∝ γ$_{0}$ ), but depends only weakly on B$_{p}$ , P , and R c (e.g., ¯ E is approx- imately the same for R c = R and γ$_{0}$ = 2 × 10 7 as in the dipole case for γ$_{0}$ = 2 × 10 $^{7}$). riod for dipole fields or the curvature radius for multipole fields, the surface field strength B$_{p}$ and the primary electron energy γ$_{0}$m$_{e}$c $^{2}$), we give γ$_{f}$m$_{e}$c 2 (the final energy of the pri- mary electron when it escapes the light cylinder), ε$_{tot}$ (the total energy of the cascade photons), E tot (the total energy of the secondary e $^{+}$e − pairs), and N$_{E}$ (the multiplicity of e $^{+}$e − pairs produced per primary electron). Note that the total cascade energy must satisfy γ$_{0}$m$_{e}$c 2 = γ$_{f}$m$_{e}$c 2 + E tot + ε$_{tot}$ . We find that N$_{E}$ is largest for cascades with strong sur- face fields, short rotation periods, multipole geometries, or large initial energies for the primary electron, but that re- gardless of cascade parameters the particle multiplicity sat- urates at N$_{E}$ ∼ 10 $^{4}$. Increasing B$_{p}$ or γ$_{0}$ , or decreasing P or R $_{c}$, tends to increase the ratio f$_{E}$ = E $_{tot}$/ ( ε$_{tot}$ + E $_{tot}$); i.e., under these conditions a larger fraction of the “secondary” energy, γ$_{0}$m$_{e}$c 2 − γ$_{f}$m$_{e}$c $^{2}$, is transfered from the photons to the pairs. At low fields only a small fraction of the secondary energy is held by the secondary pairs (e.g., for B$_{p,}$$_{12}$ = 1, f$_{E}$ < ∼ 0 . 05), but at high fields this fraction is typically above 50% (e.g,, for B$_{p,}$$_{12}$ = 1000, f$_{E}$ > ∼ 0 . 8). The average energy Figure 10 shows the secondary pair multiplicities N$_{E}$ as a function of γ$_{0}$ , for various field strengths and periods/field geometries. We find that for dipole geometries N$_{E}$ is a strong function of both γ$_{0}$ and the ratio B$_{p}$P − 2 (i.e., the polar cap voltage), but depends very weakly on either P or B$_{p}$ alone. For example, the N$_{E}$ versus γ$_{0}$ curves for B$_{p,}$$_{12}$ = 1 , P$_{0}$ = 1 and B$_{p,}$$_{12}$ = 100 , P$_{0}$ = 10 are nearly the same, as is shown in Fig. 10. Assuming that a dense secondary pair plasma is a necessary ingredient in pulsed radio emission from neutron stars (see Section 1), we can use our pair multiplicity results to estimate the conditions for pulsar “death” — the condi- tions under which a NS is no longer active as a radio pulsar: For a particular set of cascade parameters B$_{p}$ , P , and R $_{c}$, we define γ$_{death}$ to be the value of γ$_{0}$ at which on average only one electron/positron is created per primary electron;