Pair cascades of strongly-magnetized neutron stars 5 The primary electron is most efficient at scattering photons when ing Eq. (3). Depending on alignment and how close the gap potential drop Φ is to Φ$_{cap}$, many pulsars which were pre- dicted to be dead may actually have a potential drop large enough to generate pair cascades (Φ ≃ 10 7 V). According to Contopoulos & Spitkovsky, the standard death line shown in Fig. 1 is consistent with the observed P - ˙ P values for all pulsars if the magnetic inclination angles α of nearly-dead pulsars are weighted towards α = 0. If this is the case, we can use γ$_{0}$ ≃ (2–4) × 10 7 for all pulsars and do not need to invoke a strongly-curved magnetic field geometry ( R c ≃ R ) or efficient inverse Compton scattering by the primary elec- tron in order to reproduce the observed pulsar death line. where T is the surface temperature of the star and ϵ$_{c}$ = ¯ heB/m$_{e}$c is the electron cyclotron energy. 3 Therefore, for photons scattered from near the surface, where B = B$_{p}$ , the typical energy of a scattered photon is where T$_{6}$ is the surface temperature in units of 10 6 K and f ( β$_{Q}$ ) = 1 − 1 / √ 1 + 2 β$_{Q}$ is evaluated at the surface. Setting ϵ$_{0}$ = ϵ$_{RICS}$ and assuming a T$_{6}$ range of 0 . 3–3, we obtain ϵ$_{0}$ in the range 1–10 MeV at B$_{p,}$$_{12}$ = 1 up to (0 . 4–4) × 10 5 MeV at B$_{p,}$$_{12}$ = 1000. The number of resonant ICS photons scattered by the primary electron is more difficult to estimate, since it de- pends on the acceleration model. Inner gaps with space- charge-limited flows have (e.g., Hibschman & Arons 2001a; Medin 2008) As the primary electron traverses the acceleration re- gion it can pass through two distinct regimes (see, e.g., Hibschman & Arons 2001a). First, for Lorentz factors γ ∼ 10 $^{2}$–10$^{4}$, the electron efficiently upscatters photons through the inverse Compton process. Second, for Lorentz factors γ > ∼ 10 $^{6}$, the electron efficiently emits curvature radiation. The above treatment of the acceleration zone (Section 2.1) is best suited for pulsars where the electrons reach the sec- ond regime. In that case we can safely ignore the contribu- tions to the cascade made by photons emitted before the primary electron reaches full energy ( γ$_{0}$ ), since the number and energy of photons emitted through curvature radiation increase strongly with γ (i.e., ˙ N$_{CR}$ ∝ γ , E$_{CR}$ ∝ γ $^{3}$). The ap- proximation is poor, however, if inverse Compton scattering and subsequent pair production within the gap is efficient enough to screen the accelerating potential before the elec- trons can reach the second regime. In that case the photons produced in the gap are critical to the cascade, while the photons produced above the gap have a negligible effect on the cascade (the upscattered photons must travel a finite distance before pair production in order to screen the gap; in that distance the primary electron is accelerated to above resonance and exits the gap with a Lorentz factor between the first and second regimes of efficient photon production). Because ICS is strongly peaked at resonance, primary electrons traveling through this second type of gap will emit a large number of photons at a characteristic “resonance” energy and very few at other energies. The effect of this type of gap on the cascade is better modeled by N$_{0}$ photons of en- ergy ϵ$_{0}$ emitted from the surface (cf. Arendt & Eilek 2002), rather than one electron of energy γ$_{0}$m$_{e}$c $^{2}$. We therefore run a second version of the simulation, this time tracking the cascade initiated by a “primary” photon. The quantitative results of this simulation can be multiplied by N$_{0}$ to obtain the full cascade results (e.g., the number of electron-positron pairs produced per primary electron). For a primary electron resonantly upscattering primary photons, we estimate the value of ϵ$_{0}$ as follows. When the primary electron reaches a Lorentz factor γ , it upscatters photons to a mean energy (e.g., Beloborodov & Thompson 2007) 3 Note that the actual resonance condition is ϵ$_{i}$γ (1 − β cos ψ ) = ϵ$_{c}$ , where ϵ$_{i}$ ∼ kT is the initial (before scattering) photon energy, β = √ 1 − 1 /γ 2 is the ratio of the electron speed to the speed of light and ψ is the incident angle of the photon with respect to the electron’s trajectory. However, because the scattering rate depends inversely on γ (see Appendix A), photons with cos ψ ≪ 1 are far more likely to scatter off the electron than photons with cos ψ < ∼ 1. The general picture of the pair cascade as modeled by our numerical simulation is sketched in Fig. 2. At the start of the simulation, an electron with initial Lorentz factor γ$_{0}$ ∼ 10 6 . $^{5}$–107 . 5 (Section 2.1) travels outward from the stel- lar surface along the last open field line. As it travels it emits high-energy photons through curvature radiation or inverse Compton upscattering. The simulation tracks these photons 3 NUMERICAL SIMULATION OF PAIR CASCADES: PHYSICS INGREDIENTS AND METHODS For dipole fields the typical curvature photon has an energy ϵ$_{CR}$ < ∼ 10 $^{3}$–104 MeV (for γ ≤ γ$_{0}$ ∼ 10 7 . $^{5}$), while for fields with R c = R we have ϵ$_{CR}$ < ∼ 10 $^{2}$–103 MeV (for γ ≤ γ$_{0}$ ∼ 10 6 . $^{5}$). Note that we can also use this second cascade simula- tion as a diagnostic tool for the main simulation. For exam- ple, we can study the partial cascade initiated by a single curvature radiation photon emitted at some altitude in the magnetosphere to understand how the strength of the local magnetic field affects the cascade. The characteristic energy of curvature photons is Inner vacuum gaps, with accelerating electric fields on the order of 6 times larger, have N$_{0}$ values at least 20-100 times smaller (the primary electron is more rapidly accelerated out of resonance; see, e.g., ML07). 2.2 Primary photons N$_{0}$ ≃ 10 B − 1 p, $_{12}$P 3 / 4 0 T 5 / 2 6 . (9) γ = γ$_{crit}$ ≃ ϵ$_{c}$/kT , (7)