10 -4 10 0 10 4 10 8 10 12  1  10 R’ (cm -1 ) x= ’/(2m$_{e}$c$^{2}$) B=B$_{Q}$ Pair production Photon splitting $_{⊥ →}$  || + ||  e $^{+}$e - || e $^{+}$e - Pair cascades of strongly-magnetized neutron stars 9 and the sum is taken over all possible states for the electron- positron pair. Since pair production is symmetric with re- spect to the electron and the positron, R ′ jk = R ′ $_{kj}$; for sim- plicity we hereafter use R ′ jk to represent the combined prob- ability of creating the pair in either the state ( jk ) or ( kj ) (i.e., R ′ new jk = R ′ old jk + R ′ old kj ). For a given channel ( jk ), the threshold condition for pair production is where ϵ ′ = ϵ sin ψ is the photon energy in the perpendicular frame and E$_{n}$ = m$_{e}$c $^{2}$√ 1 + 2 β$_{Q}$n is the minimum energy of an electron/positron in Landau level n (the energy of an electron/positron with the momentum along the magnetic field p$_{‖}$ = 0). In dimensionless form, the condition [Eq. (23)] can be written as ϵ ′ > E$_{j}$ + E$_{K}$, (23) x$_{00}$ < x$_{01}$ < x$_{02}$ < x$_{11}$ < x$_{03}$ < · · · , β$_{Q}$ < 4 ; x$_{03}$ < x$_{11}$ < , β$_{Q}$ > 4 . (25) Figure 5. Attenuation coefficients in the perpendicular frame (the frame where the photon is traveling perpendicular to the magnetic field), for both photon splitting, labeled by ⊥→‖ + ‖ , and pair production, labeled by ‖→ e $^{+}$e − and ⊥→ e $^{+}$e $^{−}$. The local magnetic field strength is B = B$_{Q}$ ≡ 4 . 414 × 10 13 G. created in intermediate Landau levels. Therefore, in our sim- ulation we only consider the first three attenuation coeffi- cients for ‖ -polarized photons ( R ′ ‖ , $_{00}$, R ′ ‖ , $_{01}$, R ′ ‖ , $_{02}$) and the first two non-zero attenuation coefficients for ⊥ -polarized photons ( R ′ ⊥ , $_{01}$, R ′ ⊥ , $_{02}$). If the photon reaches the threshold for the (03) or (11) channel [whichever is reached first; see Eq. (25)], we use the asymptotic formula, Eq. (26). The to- tal attenuation coefficient for pair production (as given by this approximation) is plotted in Fig. 5 for both ‖ and ⊥ polarizations at β$_{Q}$ = 1. The first three attenuation coefficients (corresponding to the three lowest threshold levels x$_{00}$, x$_{01}$, x$_{02}$ ) for both ‖ and ⊥ polarizations are given in Appendix B, Eqs. (B6)-(B10); see also Daugherty & Harding (1983). Note that R ′ ⊥ , 00 = 0, and thus the first non-zero attenuation coefficient for ⊥ polarized photons is actually R ′ ⊥ , $_{01}$, not R ′ ⊥ , $_{00}$. We include photon splitting in our simulations. Based on the kinetic selection rule (Adler 1971; Usov 2002, but see Baring & Harding 2001), only the process ⊥→‖ ‖ is allowed. Therefore, for ‖ -polarized photons, the attenuation coeffi- cient for photon splitting is zero ( R ′ sp ‖ = 0). For ⊥ -polarized photons we use the following formula, adapted from the nu- merical calculation of Baring & Harding (1997): In our simulation a photon is typically created with x below the first threshold ( x$_{00}$ or x$_{01}$ , depending on the photon polarization). As long as x remains below the first threshold, R ′ = 0 and the optical depth to pair production remains zero. As the photon propagates into the magneto- sphere and crosses the first threshold, R ′ > 0, Δ τ > 0, and τ begins to grow. As it continues to travel outward, both τ and the number of Landau levels available for pair production j$_{max}$ and k$_{max}$ increase. Depending on the local magnetic field strength [Eq. (11)], the photon may reach a large enough optical depth ( τ ∼ 1) for pair production after crossing only a few thresholds (so that j$_{max}$ and k$_{max}$ are small) or after crossing many thresholds (so that j$_{max}$ and k$_{max}$ are very large). For “weak” magnetic fields ( β$_{Q}$ < ∼ 0 . 1) the optical depth increases slowly with s$_{ph}$ and it is valid to use the j$_{max}$, k$_{max}$ ≫ 1 asymptotic attenuation coefficient for pair production (e.g., Erber 1966), where g ( β$_{Q}$, x ) = β 3 Q exp( − 0 . 6 x $^{3}$). For x ≤ 1, this expression reproduces the results of Baring & Harding to better than 10% at both β$_{Q}$ ≤ 0 . 5 and β$_{Q}$ ≫ 1, while underestimat- ing the results at β$_{Q}$ = 1 by less than 30%. The ⊥→‖ ‖ attenuation coefficient for photon splitting is plotted in Fig. 5 at B = B$_{Q}$ . Because the attenuation coefficient R ′ sp drops rapidly with field strength for β$_{Q}$ < 1, photon split- ting is unimportant for β$_{Q}$ < ∼ 0 . 5 (e.g., Baring & Harding 2001). However, for ⊥ -polarized photons propagating in su- perstrong fields β$_{Q}$ > ∼ 0 . 5, photon splitting is the dominant attenuation process: even though above the first threshold ( x ≥ x$_{01}$ for ⊥ photons) the attenuation coefficient for pho- ton splitting is much smaller than that for pair production, in superstrong fields the photon splits before reaching the first threshold (see Fig. 5). R ′ sp ⊥→‖ ‖ ≃ α 2 f 60 π $^{2}$a$_{0}$ ( 26 315 ) $^{2}$(2 x ) $^{5}$β6 Q [ g ( β$_{Q}$, x ) + 0 . 05] [0 . 25 g ( β$_{Q}$, x ) + 20] , (28) R ′ ‖ , ⊥ ≃ 0 . 23 a$_{0}$ β$_{Q}$ exp ( − 4 3 xβ$_{Q}$ ) , (26) which applies for both polarizations. For stronger fields, however, pairs are produced in low Landau levels, and the more accurate coefficients of Daugherty & Harding must be used. In Appendix B2 we find that the critical magnetic field strength separating these two regimes is [see Eq. (B17)]. We also find that the boundary between the two regimes is very sharp: pairs are either created at the first few Landau levels ( n ≤ 2) for B > ∼ B$_{crit}$ or in very high Lan- dau levels for B < B$_{crit}$ , with very few electrons/positrons In the simulation, whenever τ ≥ 1 or τ$_{sp}$ ≥ 1 the pho- ton is destroyed (i.e., turned into a pair or two photons). More precisely, the photon should only be destroyed with probability 1 − exp( − τ ). But in practice we find that such B$_{crit}$ ≃ 3 × 10 12 G (27)