Pair cascades of strongly-magnetized neutron stars 3 the range Φ ∼ (1–2) × 10 13 V, regardless of the accelera- tion model (e.g., Hibschman & Arons 2001a; Medin & Lai 2007, hereafter ML07). For the surface field strengths we are considering, B ≥ 10 12 G, the primary electrons are not radiation-reaction limited within these gaps (ML07; cf. the millisecond pulsar models of Harding, Usov, & Muslimov 2005), so we can set γ$_{0}$ = e Φ /m$_{e}$c $^{2}$. We therefore restrict γ$_{0}$ to the range (2–4) × 10 7 for dipole fields. Note that these large voltage drops do not occur in pulsars where the gap electric field is fully screened due to inverse Compton scat- tering by the primary electron. We discuss this case in Sec- tion 2.2. The voltage drop across the gap can be no larger than the voltage drop across the entire polar cap of the neutron star (e.g., Ruderman & Sutherland 1975): 1999; Gruzinov 2005; Spitkovsky 2006; Timokhin 2006; Komissarov 2006; Kalapotharakos & Contopoulos 2009), but they do not yet include any particle acceleration, or pair creation self-consistently. Therefore, in this paper we decouple particle acceleration from the rest of the cascade and focus on the cascade produced by a primary electron 1 injected into the magnetosphere with a given initial Lorentz factor γ$_{0}$ (cf. Daugherty & Harding 1982). We also consider the cascade produced by a single “primary” photon emitted by the primary electron, in the case where photon emission within the acceleration gap is important (i.e., for cascades where the dominant mechanism for high-energy photon pro- duction is inverse Compton scattering). The outline of the paper is as follows. In Section 2 we summarize our method for estimating the initial parameters (e.g., γ$_{0}$ ) of the primary cascade particles, for use in our sim- ulations. In Section 3 we describe the details of the numer- ical simulations, both for cascades with photon production dominated by curvature radiation and by resonant inverse Compton scattering (resonant ICS, or RICS). In Section 4 we present our results (e.g., photon and pair plasma spec- tra) for a wide range of parameters: surface magnetic fields B = 10 $^{12}$–1015 G, rotation periods 0 . 1–10 s, surface tem- peratures T = (0 . 3–3) × 10 6 K, and pure dipole and more complex field geometries. In Section 5 we summarize our findings and discuss their implications for the radio emis- sion and high-energy (hard X-ray and gamma-ray) emission from pulsars and magnetars. Some technical details (on our treatment of inverse Compton scattering, on our treatment of attenuation coefficients and e $^{+}$e − energy levels for pair production, and on deriving semi-analytic fits to our nu- merical results) are given in the appendix. where R is the radius of the star (assumed in this pa- per to be 10 km), P$_{0}$ is the spin period in units of 1 s, and B$_{p}$ = 10 $^{12}$B$_{p,}$$_{12}$ G is the polar surface magnetic field strength. If the voltage drop, Φ, required to initiate pair cascades is not available, i.e., Φ > Φ$_{cap}$, the magnetosphere should not produce pulsed radio emission; the locus of points where Φ = Φ$_{cap}$ defines the pulsar death line. 2 A typical death line for an inner gap model, plotted in P - ˙ P space, is shown on the left panel of Fig. 1. The line was made using three assumptions: (i) The magnetosphere field geometry is dipolar. (ii) The pair cascade occurs primarily above the gap, (through curvature radiation) once the primary elec- tron has reached a large Lorentz factor γ$_{0}$ ∼ 10 $^{7}$. (iii) The spindown power of the pulsar, given by is approximately equal to the spindown power of a magnetic dipole with its magnetic field and rotational axes orthogonal to each other: In our cascade simulation (described in Section 3) we do not include an actual acceleration region, since we wish in this work to remain as model-independent as possible. Instead, we model the effect of this region on the cascade by giving the primary electron an initial energy γ$_{0}$m$_{e}$c 2 equivalent to the energy it would reach upon traversing the entire gap, and injecting it into the magnetosphere at the neutron star surface (cf. Daugherty & Harding 1982). Obviously, this ap- proximation excludes a proper treatment of the slot gap and outer gap acceleration models. However, in most parts of the polar cap region (i.e., except for the boundary region adja- cent to the open field lines), the main voltage drop occurs near the stellar surface, regardless of the nature of the “gap” (vacuum gap or space charge limited flow). Such inner gap models and other global models with near-surface accelera- tion are allowed in our analysis. For a dipole magnetic field geometry, most active pul- sars with inner gap accelerators have gap voltage drops in A well-known problem with the death line made using these assumptions is that it cuts right through the middle of the main group of pulsars (Ruderman & Sutherland 1975; Hibschman & Arons 2001a; Harding & Muslimov 2002; Medin & Lai 2007); i.e., the model incorrectly predicts that there will be no radio emission from many neutron stars that are observed to be active pulsars. where ˙ P$_{−}$$_{15}$ is the period derivative in units of 10 − 15 s/s and I = 10 45 g-cm 2 is assumed. Several authors have proposed models of the neu- tron star magnetosphere that shift the theoretical death line closer to the observed death line by altering one 1 Although the primary particle could also be a positron (or even an ion), we assume here for simplicity that the pulsar is oriented such that electrons are accelerated away from the star. 2 This applies to the vacuum gap model. In the space-charge- limited-flow model the condition is Φ = κ$_{g}$ Φ$_{cap}$, with κ$_{g}$ ≃ 0 . 15 (e.g., Hibschman & Arons 2001a). The polar magnetic field strength inferred from this frequently-used approximation is ˙ E ≃ B 2 $_{p}$Ω $^{4}$R6 6 c 3 = 2Φ 2 $_{cap}$c 3 . (3) ˙ E = − I Ω ˙ Ω = 4 π $^{2}$I ˙ P P 3 , (2) 2.1 Primary electrons 2 ESTIMATING THE INITIAL PARAMETERS FOR THE PRIMARY PARTICLES Φ$_{cap}$ ≃ Ω B$_{p}$ 2 c R $^{3}$Ω c = 7 × 10 $^{12}$B$_{p,}$$_{12}$ P − 2 0 V , (1)