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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 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- 

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mately by Φ$_{cap}$ = Φ$_{death}$, or 

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 ). 

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 

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