18 Zach Medin and Dong Lai Table 2. Energies and multiplicities for a cascade initiated by a single electron. Listed are the initial and final energy of the primary electron, γ$_{0}$m$_{e}$c 2 and γ$_{f}$m$_{e}$c $^{2}$, respectively; the total energy of the cascade photons, ε$_{tot}$ ; and the multiplicity and total and average energies of the secondary electrons and positrons, N$_{E}$ , E$_{tot}$ , ¯ E = E$_{tot}$ /N$_{E}$ , respectively. Each cascade is specified by the magnetic field strength, field geometry (curvature radius R$_{c}$ or spin period in the case of dipole fields), and cascade energy ( γ$_{0}$m$_{e}$c $^{2}$). P B$_{p}$ γ$_{0}$m$_{e}$c 2 γ$_{f}$m$_{e}$c 2 ε$_{tot}$ E$_{tot}$ N$_{E}$ ¯ E (1 s) (10 12 G) (MeV) (MeV) (MeV) (MeV) (MeV) 10 10 1.022e7 8.1e6 2.1e6 7.4e3 1.3e1 5.7e2 100 2.0e6 8.1e4 1.7e2 4.8e2 1000 1.9e6 1.8e5 5.1e2 3.5e2 1 1 4.8e6 5.4e6 1.9e4 5.3e1 3.6e2 10 4.9e6 5.8e5 9.6e2 6.0e2 100 4.0e6 1.4e6 3.0e3 4.7e2 1000 3.4e6 2.0e6 5.9e3 3.4e2 0.1 1 2.4e6 7.4e6 4.0e5 1.5e3 2.7e2 10 4.4e6 3.4e6 6.0e3 5.7e2 100 2.8e6 5.0e6 1.1e4 4.5e2 1000 2.1e6 5.7e6 1.7e4 3.4e2 1 1 2.044e7 4.9e6 1.5e7 6.3e5 8.2e2 7.7e2 10 9.5e6 5.9e6 3.8e3 1.6e3 100 6.3e6 9.2e6 6.9e3 1.3e3 1000 4.8e6 1.07e7 1.1e4 1.0e3 R$_{c}$ = R 1 1.022e6 5.8e5 3.9e5 5.2e4 6.5e3 8.0e0 10 8.0e4 3.7e5 2.0e4 1.9e1 100 1.3e4 4.3e5 1.9e4 2.3e1 1000 1.5e4 4.3e5 1.8e4 2.4e1 mately by Φ$_{cap}$ = Φ$_{death}$, or i.e., The value of γ$_{death}$ changes very little if we alter the critical value of N$_{E}$ in Eq. (56) by a factor of ∼ 10, because of the steep dependence of N$_{E}$ on γ$_{0}$ in this region. Therefore, although it is unknown exactly what value N$_{E}$ must have for a pulsar to be active, γ$_{0}$ = γ$_{death}$ is a good predictor of pulsar death regardless. Using Fig. 10 we find empirically that for B$_{p,}$$_{12}$ = 1–1000 this is nearly the same as the death line depicted in Fig. 1 ( P$_{0}$ = 0 . 6 B 8 / 15 p, 12 ). P$_{0}$ = B 1 / 2 p, 12 ; (60) N$_{E}$ ( γ$_{0}$ = γ$_{death}$ ) = 1 . (56) γ$_{death}$ ≃ 1 . 5 × 10 $^{7}$B − 1 / 6 p, 12 R 2 / 3 8 . (57) γ$_{death}$ ≃ 1 . 4 × 10 $^{7}$B − 1 / 6 p, 12 P 1 / 3 0 = ( Φ$_{death}$ Φ$_{cap}$ ) 1 / $^{6}$( e Φ$_{death}$ m$_{e}$c 2 ) , (58) Φ$_{death}$ = 7 × 10 12 V . (59) For dipole fields, R 8 = 0 . 9 P 1 / 2 0 , and we can write where Φ$_{cap}$ is given by Eq. (1) and Therefore, for dipole fields, the death line is given approxi- An example of how the cascades develop spatially is pre- sented in Fig. 11, which shows the number of photons and secondary pairs as a function of the radius at which each par- ticle is created, for surface fields strengths B$_{p,}$$_{12}$ = 1, 44 . 14 (i.e., B$_{p}$ = B$_{Q}$ ), and 1000. The right two panels show the difference in the behavior of the cascade when the local mag- netic field strength B is above or below B$_{crit}$ ≃ 3 × 10 12 G [see Eq. (27)], i.e., below or above r ≃ 2 . 5 R for B$_{p}$ = B$_{Q}$ and r ≃ 7 R for B$_{p,}$$_{12}$ = 1000 [see Eq. (11)]. For B > ∼ B$_{crit}$ , pair production by curvature photons is very efficient; for the conditions depicted in Fig. 11 ( P$_{0}$ = 1 and γ$_{0}$ = 2 × 10 $^{7}$) approximately one e ± pair is created for every two photons emitted by the primary electron. This is because about half of all the photons created at low altitudes lie above the limit ϵ$_{min}$ ∼ 200 MeV of Eq. (54) (see Fig. 14). Photon splitting has very little effect on the curvature photons, since the ma- jority (7 / 8) of these photons are ‖ -polarized. There is no synchrotron radiation for B > 0 . 5 B$_{Q}$ , since the ‖ photons