largely independent of field strength. When ICS is inactive, almost all of the cascade energy resides in the pairs; i.e., E tot ≃ ϵ$_{0}$ . The pair cascade will be very weak regardless of photon energy, with n$_{E}$ < 10 and n$_{ϵ}$ = 0 or 1 (i.e., at most one photon escapes the magnetosphere without pair produc- tion). This is because the e ± pairs are produced exclusively through the ( jk ) = (00) or (01) channel (see Section 3.2), so that at most one synchrotron photon is emitted per pair. For B$_{p,}$$_{12}$ > 20, photon splitting causes all pairs to be produced We find significant differences in the behavior of the cascades at magnetic field strengths below and above Our results for photon-initiated cascades are presented in Figs. 6–9. We consider primary photons with energies in the range of 10 $^{3}$–105 MeV; for B$_{p,}$$_{12}$ = 1–1000, the primary electron should emit very few photons (via either resonant ICS or curvature radiation) above this energy range (see Section 2). Unless otherwise stated, the primary photon is emitted from near the surface, in the direction tangent to the last open field line. Thus the radius of curvature near the point of emission is R c ≃ 9 × 10 $^{7}$P 1 / 2 0 cm for dipole fields [Eq. (14)]. We first present our results for photon-initiated cas- cades (see Section 3.4), as they are simpler and aid us in our discussion of the results for the full cascade (initiated by a primary electron). where N$_{0}$ is the number of photons produced by the primary electron (see Section 2.2). From our numerical results we in- fer various empirical relations for each cascade; quantitative arguments for the validity of several of these relations are given in Appendix C. are strongly affected. Thus hot surface spots with higher T tend to lower E tot more. As expected, we find that photon splitting does not affect the cascade at these field strengths (see Section 3.2). The photon and pair cascade spectra for B$_{p,}$$_{12}$ = 1 are shown in Fig. 6, both when ICS is inactive and when ICS is active from a “warm spot” ( T$_{6}$ = 1, θ$_{spot}$ = 0 . 3). At high fields ( B > ∼ B$_{crit}$ ), a primary photon injected from the surface will pair produce when When ICS is active from a hot spot (Section 3.3), the num- ber of pairs produced does not change, since the photons produced through ICS at these field strengths have energies ∼ B 2 p, $_{12}$T − 1 6 MeV [Eq. (8)] and can not pair produce. The total pair energy E tot decreases, however, since the ICS pro- cess transfers energy from the pairs to photons. Although resonant ICS is most important for electrons and positrons at γ$_{crit}$ ≃ ϵ$_{c}$/kT [Eq. (7); see Section 2.2], we find in these cascades that all electrons and positrons with energies in the range of In this section we present the results of our simulations of photon- and electron-initiated cascades (Sections 4.1 and 4.2, respectively), for a variety of different surface field strengths, rotation periods, field geometries, and initial en- ergies of the primary particle. For each type of cascades, we present the “final” spectra of the cascade photons and pairs as they cross the light cylinder and escape from the magneto- sphere. For the electron-initiated cascades we also show the spectra at several intermediate stages (i.e., the spectra of all photons and pairs that cross the height r = 1 . 2 R, 2 R, 5 R , etc.). The photon spectra are plotted over the energy range 10 keV-1 TeV, since for energies < ∼ 1 keV the thermal pho- tons dominate the spectra while above ∼ 1 TeV fewer than one photon is produced per primary electron. We are par- ticularly interested in the pair multiplicities, i.e., the total number of cascade electrons + positrons produced per pri- mary particle. We use n$_{E}$ to denote the number of electrons and positrons per “primary” photon and N$_{E}$ to denote the number per primary electron; the two multiplicities are re- lated by respectively. These results are (largely) independent of the hot spot model used. When ICS is inactive, the cascade elec- tron/positron has final energy (after it has finished radiating synchrotron photons) extending from [Eq. (C3)] (for the first pair produced) down to ∼ 0 . 1 B$_{p,}$$_{12}$ϵ$_{min}$ for the lowest-energy pairs, and the total energy of the pairs is [Eq. (C4)] where R 8 is the radius of curvature R c in units of 10 8 cm, evaluated at the surface along the last open field line. Strong cascades, where more than one electron-positron pair is pro- duced, typically occur at energies ∼ 10 times ϵ$_{min}$ . For ϵ$_{0}$ in the range from ϵ$_{min}$ to ∼ 10 5 MeV, we find that the mul- tiplicities of photons and e $^{+}$e − particles produced in the cascade are and tra to some extent, we modify the synchrotron emission pro- cedure of Section 3.3 for secondary particles in high Landau levels ( n ≥ 3). In every step 10 photons are emitted, each with a weighting factor of 0 . 1Δ N$_{ϵ}$ (rather than one photon with a weighting factor of Δ N$_{ϵ}$ , as before). Each photon has a different energy ϵ$_{⊥}$$_{,i}$ [chosen randomly according to Eq. (35)], so that the total energy lost by the secondary particle becomes γ$_{⊥}$m$_{e}$c 2 = 0 . 1 ∑ 10 i =1 ϵ$_{⊥}$$_{,i}$ . We do not apply this procedure to the synchrotron emission from secondary particles in Landau levels n = 1 or n = 2, as it would not gain anything; each of the 10 photons emitted would have the same value of ϵ$_{⊥}$$_{,i}$ . B$_{crit}$ ≃ 3 × 10 12 G [Eq. (B17)]. At low fields B < ∼ B$_{crit}$ , the primary photon can pair produce if [Eq. (C2); see also Hibschman & Arons 2001a] Pair cascades of strongly-magnetized neutron stars 13 ϵ$_{0}$ > ϵ$_{min}$ ∼ 3000 B − 1 p, 12 R 8 MeV , (48) n$_{ϵ}$ ∼ ϵ$_{0}$ 500 MeV R − 1 8 (49) n$_{E}$ ∼ ϵ$_{0}$ 10 4 MeV B$_{p,}$$_{12}$ R − 1 8 , (50) E$_{max}$ ∼ 0 . 1 B$_{p,}$$_{12}$ϵ$_{0}$ (51) 4 RESULTS E tot ∼ 2 E$_{max}$ + 0 . 1 B$_{p,}$$_{12}$ϵ$_{min}$n$_{E}$ ln ( 0 . 075 ϵ$_{0}$ ϵ$_{min}$ ) . (52) N$_{E}$ = N$_{0}$ × n$_{E}$ , (47) E$_{RICS}$ ∼ (0 . 3 − 30) γ$_{crit}$m$_{e}$c 2 ≃ (20 − 2000) B$_{p,}$$_{12}$T − 1 6 MeV (53) ϵ$_{0}$ > ϵ$_{min}$ ∼ 200 R 8 MeV , (54) 4.1 Results: photon-initiated cascades