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 16 Zach Medin and Dong Lai Figure 8. The final photon and pair spectra of photon-initiated cascades with active ICS, for surface magnetic fields B$_{p,}$$_{12}$ = 10 (left panel), 100 (middle panel), and 1000 (right panel). The NS period is P$_{0}$ = 1, a dipole field geometry is adopted, and thermal photons are emitted from a hot spot with T$_{6}$ = 3, θ$_{spot}$ = 0 . 1. The primary photon has ϵ$_{0}$ = 10 5 MeV. The spikes in the pair spectra of the left and right panels represent, from shortest to tallest, one, two, or four electrons/positrons. space-charge-limited flow; inner vacuum gap accelerators, for example, would yield N$_{0}$ about 20–100 times lower. Also note that, for a given acceleration model, an accurate de- termination of N$_{E}$ requires that rather than just setting ϵ$_{0}$ = ϵ$_{RICS}$ , a distribution of energies be used which takes into account resonant (and possibly non-resonant) scattering away from the thermal peak (i.e., γ = γ$_{crit}$ ): although fewer photons are upscattered at energies greater than ϵ$_{RICS}$ , these photons can have an important effect on the total multiplic- ity of pairs produced, since n$_{E}$ grows approximately linearly with photon energy ϵ$_{0}$ ; the ϵ$_{0}$ > ϵ$_{RICS}$ photons are especially important for cascades where n$_{E}$ ( ϵ$_{RICS}$ ) = 0 (e.g., cascades with B$_{p,}$$_{12}$ = 10 and P$_{0}$ = 1). than in dipole magnetospheres. At high fields, we find that this dependence on curvature radius is much weaker: n$_{E}$ ∝ R − 1 / 2 c at B$_{p}$ ≃ 0 . 5 B$_{Q}$ and n$_{E}$ ∝ R − 1 / 4 c power at larger fields. Thus for high fields, the cascades in non-dipole and dipole magnetospheres are of similar sizes. The photon and pair cascade spectra for B$_{p,}$$_{12}$ = 1, 10, and 100 and a R c = R non-dipole magnetosphere are shown in Fig. 9. Two aspects of the cascade depend strongly on the al- titude ( r$_{0}$$_{,}$$_{ph}$ ) at which the primary photon is injected: the local magnetic field strength, B ∝ r − $^{3}$, and the effectiveness of ICS, which is completely negligible for r$_{0}$$_{,}$$_{ph}$ ≥ 3 R (regard- less of the temperature and size of the hot spot). The other cascade parameters have a much weaker dependence on al- titude (e.g., radius of curvature R c ∝ √ r$_{0}$$_{,}$$_{ph}$ ). We find that the photon and pair spectra for primary photons injected at r$_{0}$$_{,}$$_{ph}$ > R are very similar to the spectra for photons injected at the surface, as long as ICS is inactive and the local magnetic field strengths are the same in both cases. For example, the spectra for r$_{0}$$_{,}$$_{ph}$ = 3 R and B$_{p,}$$_{12}$ = 27 [such that the local field strength at the point of injection is B$_{12}$ = 1; Eq. (11)] is nearly identical to the spectra given in the first row of Fig. 6, where r$_{0}$$_{,}$$_{ph}$ = R , B$_{p,}$$_{12}$ = 1, and ICS is inactive. We can use the pair multiplicity per primary photon, n$_{E}$ ( ϵ$_{0}$ ), obtained from our numerical simulations to estimate the pair multiplicity per primary electron, N$_{E}$ = N$_{0}$n$_{E}$ , when ICS is the dominant cascade emission process. For ϵ$_{0}$ and N$_{0}$ , we use the expressions for the typical energy ϵ$_{RICS}$ and total number ≃ 10 B − 1 p, $_{12}$P 3 / 4 0 T 5 / 2 6 of resonant ICS photons upscattered by the primary electron [Eqs. (8) and (9); see Section 2.2]. The results are shown in Table 1. Note that although ϵ$_{RICS}$ is independent of the acceleration model used, the number of upscattered photons, N$_{0}$ , as given by Eq. (9) is applicable for only inner gap accelerators with In Table 2 we list some key quantitative results of our simulations: For each cascade (characterized by the spin pe- Our results for electron-initiated cascades are presented in Table 2 and Figs. 10–15. In our simulation, the primary electron is emitted from the surface along the last open field line ( θ$_{0}$ = θ$_{cap}$ ). We consider the cases of γ$_{0}$ = 2 × 10 7 and 4 × 10 7 in dipole magnetospheres, as well as the case of γ$_{0}$ = 2 10 6 in non-dipole R c = R magnetospheres, as discussed in Section 2.1. Although a larger initial primary energy γ$_{0}$ gives rise to more cascade particles with a larger total energy, the behavior of the cascade at γ$_{0}$ > 4 × 10 7 is qualitatively similar to that at γ$_{0}$ = (2–4) × 10 7 (or γ$_{0}$ = 2 × 10 6 and R c = R ) and key quantities such as cascade multiplicities and energies can be extrapolated from our results. For most of this section we “turn off” ICS in our simulation and allow photon splitting only through the ⊥→‖ ‖ mode; at the end of this section we discuss how changing these simulation parameters affects the cascade. 4.2 Results: electron-initiated cascades