10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 N E 0 B$_{p,12}$ =1000 B$_{p,12}$ =100 B$_{p,12}$ =10 B$_{p,12}$ =1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 N E 0 P$_{0}$ =10 P$_{0}$ =1 P$_{0}$ =0.1 $_{c}$=R Pair cascades of strongly-magnetized neutron stars 19 Figure 10. Secondary electron/positron multiplicities N$_{E}$ as a function of the initial Lorentz factor of the primary electron γ$_{0}$ , for P$_{0}$ = 1 and a dipole geometry at several different surface field strengths (left panel), and for B$_{p,}$$_{12}$ = 100 at several different periods/field geometries (right panel). tude r , the ratio of the number of synchrotron photons to the number of curvature photons at low energies is largest for large γ$_{0}$ , small P , or small R $_{c}$. The ratio of synchrotron photons to curvature photons is also largest for B$_{p}$ close to B$_{crit}$ , such that the point of maximum synchrotron radia- tion, B ≃ B$_{crit}$ (see Fig. 11), occurs at a low altitude. Note that for ϵ < ∼ 0 . 1 MeV the cumulative and final photon spec- tra all have power-law shapes with index Γ = 2 / 3 (where dN/dϵ = ϵ − $^{Γ}$), regardless of whether synchrotron radiation from the secondary pairs or curvature radiation from the primary electron dominates; this is because both processes have spectra that depend on F ( x ) /ϵ ∝ ϵ − 2 / 3 at low energies [i.e., for x ≪ 1; see Eqs. (15) and (35)]. will pair produce through the ( jk ) = (00) channel and ⊥ photons will split into two ‖ photons before pair producing (see Section 3.2). For B < 0 . 5 B$_{Q}$ , pair production dominates over photon splitting, such that a few synchrotron photons are emitted from electrons and positrons in the n = 1 Lan- dau level. For B < B$_{crit}$ , electrons and positrons are created in higher Landau levels, such that many synchrotron pho- tons are produced by each electron or positron ( ∼ 10; see Appendix C1). The minimum photon energy for pair pro- duction ϵ$_{min}$ ∝ B − 1 [Eq. (48)] grows with radius while the energy of the typical curvature photon ϵ$_{CR}$ ∝ γ 3 [Eq. (10)] and synchrotron photon ϵ$_{SR}$ ∝ ϵ$_{CR}$ (Appendix C1) fall with radius; therefore the number of electron-positron pairs cre- ated per photon decreases rapidly with radius. Figures 12 and 13 show the final spectra (i.e., the spec- tra as measured at the light cylinder) of photons and pairs, as well as the spectra of curvature photons emitted by the primary electron, for a variety of magnetic field strengths, spin periods, and cascade energies. Figure 14 shows cumu- lative photon and pair spectra at various magnetosphere radii. These spectra are generated by recording the energy of any photon, electron, or positron which reaches various radii (such as r/R = 1 . 2 , 2 , 5 , 20). Figure 15 shows the effect of resonant inverse Comp- ton scattering on the final photon and pair spectra. Both a “warm” hot spot ( T$_{6}$ = 1, θ$_{spot}$ = 0 . 3), and a “cool” spot ( T$_{6}$ = 0 . 3, θ$_{spot}$ = π/ 2) are considered, as well as the case where ICS is inactive. For the cascade parameters adopted in our simulations, the results for “hot” hot spots ( T$_{6}$ = 3, θ$_{spot}$ = 0 . 1) are nearly the same as for warm spots, and so are not shown. This is because the increased cascade activ- ity due to the larger T almost exactly cancels the decreased activity due to the smaller maximum photon-electron inter- section angle ψ$_{crit}$ ). We find due to resonant ICS, a major- ity of the electrons and positrons with energies in the range E$_{RICS}$ ∼ 20–2000 B$_{p,}$$_{12}$T − 1 6 MeV [Eq. (53)] radiatively cool to below that range. This has a strong effect on the pair spectra if the average electron/positron energy ¯ E lies in this range (which occurs, e.g., for B$_{p,}$$_{12}$ < ∼ 44 . 14 when P$_{0}$ ≥ 0 . 1 and γ$_{0}$ ≤ 4; see Table 2). In general, we find that including RICS tends to make the cascade pair energy distribution narrower. On the other hand, ICS has only a minor effect on the photon spectra; its effect is moderate when ¯ E ∼ 20– 2000 B$_{p,}$$_{12}$T − 1 6 MeV and E tot > ε$_{tot}$ (which only occurs when R c = R and B$_{p,}$$_{12}$ = 10; see Table 2). Regardless of the cas- cade parameters, resonant ICS by the secondary particles has very little effect on the multiplicity of photons N$_{ϵ}$ or Figures 12–14 show many of the same trends as are seen in Table 2 and Fig. 11. For example, the multiplicity N$_{E}$ in- creases with increasing B$_{p}$ , γ$_{0}$ , P − $^{1}$, or R − 1 c . But there are also several new features. First, increasing B$_{p}$ or γ$_{0}$ , or de- creasing P or R $_{c}$, tends to increase the maximum energy and decrease the minimum energy of the cascade pairs, so that the pair spectrum becomes broader in energy (see Sec- tion C2). Second, the cumulative photon and pair spectra at a given radius r (Fig. 14) are nearly independent of B$_{p}$ , as long as the local field strength at that radius is larger than B$_{crit}$ (e.g., the spectra at r = 2 R are nearly the same for B$_{p,}$$_{12}$ = 44 . 14 and 1000). Third, under certain condi- tions, synchrotron radiation dominates the cumulative/final photon spectrum at low energies ϵ < ∼ 1 MeV (e.g., in the top middle and top right panels of Fig. 13): For a given alti-