Pair cascades of strongly-magnetized neutron stars 15 10 1 10 2 10 3 10 4 dN/dln E, dN/dln  e $^{+}$e - pairs Photons 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 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 Figure 7. The final photon and pair spectra of photon-initiated cascades with active ICS, for surface magnetic fields B$_{p,}$$_{12}$ = 10 (upper panels) and 100 (lower panels). The pulsar spin period is P$_{0}$ = 1, a dipole field geometry is adopted, photon splitting ⊥→‖ ‖ is active, and thermal photons are emitted from a hot surface spot with T$_{6}$ = 1, θ$_{spot}$ = 0 . 3. The primary photon has an energy of 10 3 MeV (left column of panels), 10 4 MeV (middle column), or 10 5 MeV (right column). The spikes in the pair spectra of several panels represent the electron-positron pair produced by the primary photon. Note that there are actually two such pairs in the bottom left panel ( B$_{p,}$$_{12}$ = 100 , ϵ$_{0}$ = 10 3 MeV), as the primary photon has split in that case; while in the upper right panel ( B$_{p,}$$_{12}$ = 10 , ϵ$_{0}$ = 10 5 MeV) the electron and positron have significantly different energies from each other and so are represented by two shorter spikes. Also note that photon spectra do not appear in the lower panels. This is due to a combination of weak synchrotron emission and efficient pair production near the surface of a B$_{p,}$$_{12}$ ≥ 100 neutron star; very few secondary photons are created, and none of them survive to escape the magnetosphere. times larger than warm spots; see Fig. 8 [from Eq. (55), at B$_{p,}$$_{12}$ = 1000 a strong cascade requires T$_{6}$ ≥ 3]. Cool spots give much smaller multiplicities (factors of > 10 smaller) than warm or hot spots regardless of the primary photon energy. At low fields, the multiplicities of photons and e $^{+}$e − particles produced per primary photon, n$_{ϵ}$ and n$_{E}$ , depend on R c to the ( − 1) power [Eqs. (49) and (50)], such that cascades in non-dipole magnetospheres can be much larger As discussed in Section 3.3, we consider three hot sur- face spot models for active ICS: a “cool” T$_{6}$ = 0 . 3, θ$_{spot}$ = π/ 2 spot; a “warm” T$_{6}$ = 1, θ$_{spot}$ = 0 . 3 spot; and “hot” T$_{6}$ = 3, θ$_{spot}$ = 0 . 1 spot. At low fields we find that the only effect the various hot spot models have is to lower E tot relative to the total cascade energy ϵ$_{0}$ (see above). At high fields, the cascades due to warm and hot spots are similar in multiplicities n$_{ϵ}$ and n$_{E}$ for energies ϵ$_{0}$ < ∼ 10 4 MeV; but for energies ϵ$_{0}$ > ∼ 10 5 MeV, the hot spots give multiplicities ∼ 3