10 1 10 2 10 3 10 4 dN/dln E, dN/dln  e $^{+}$e - pairs Photons e $^{+}$e - pairs Photons 10 0 10 1 10 2 10 3 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 10 5 E,   (MeV) e $^{+}$e - pairs Photons 14 Zach Medin and Dong Lai Figure 6. The final photon and pair spectra of photon-initiated cascades for surface magnetic fields B$_{p,}$$_{12}$ = 1. The NS spin period is P$_{0}$ = 1 and a dipole field geometry is adopted. In the upper panels, ICS is assumed to be inactive, while in the lower panels, ICS from a hot spot with T$_{6}$ = 1, θ$_{spot}$ = 0 . 3 is included in the simulation. The primary photon is injected from the surface and has an energy of 10 4 MeV (left panels) or 10 5 MeV (right panels); for photons with energy 10 3 MeV, no cascade is initiated. The spike in the pair spectra of each panel represents the electron-positron pair produced by the primary photon. The spectra in the top panels (where ICS is inactive) are nearly identical to the spectra generated, e.g., by a photon injected at r$_{0}$$_{,}$$_{ph}$ = 3 R above a star with surface field B$_{p,}$$_{12}$ = 3 3 = 27, such that the local field strength at the injection point is B = 10 12 G [Eq. (11)]. to ϵ$_{RICS}$ , the energy of a typical ICS photon upscattered by the primary electron at high fields [Eq. (8)]; therefore, at high fields a typical ICS photon is able to initiate a weak cascade. We find that for B$_{p}$ < ∼ 0 . 5 B$_{Q}$ a significant fraction ( ∼ 20%–60%) of the total cascade energy ϵ$_{0}$ resides in the photons; as in the low field case [Eq. (52)], this fraction de- creases as either ϵ$_{0}$ or B$_{p,}$$_{12}$ increases. For B$_{p}$ > ∼ 0 . 5 B$_{Q}$ , even ignoring photon splitting, the total photon energy fraction is very low, < 10%; but with photon splitting included, it is almost negligible, < 1%. The photon and pair cascade spec- tra for B$_{p,}$$_{12}$ = 10 and 100 when ICS is active from a “warm spot” ( T$_{6}$ = 1, θ$_{spot}$ = 0 . 3) are shown in Fig. 7. with ( jk ) = (00), such that the cascades are even weaker: n$_{E}$ ≤ 4 and n$_{ϵ}$ = 0. When ICS is active, both n$_{ϵ}$ and n$_{E}$ can be larger, though not as large as would be predicted by an extrapolation of Eqs. (49) and (50) to high fields. In order for ICS to affect the cascade, however, the primary photon must have an energy the energy of the electron/positron produced by this photon, E$_{max}$ ∼ (0 . 1–0 . 5) ϵ$_{0}$ [Eq. (C9)], must be larger than the min- imum energy at which ICS is effective, ∼ 0 . 3 γ$_{crit}$m$_{e}$c 2 (see Appendix C1). Note that this energy is approximately equal