 1e-20  1e-18  1e-16  1e-14  1e-12  1e-10 10 -2 10 -1 10 0 10 1 10 2 10 3 P ·  (s s -1 ) P (s)$^{Dipole}$ 0.95R offset 100 cap polar cap B p,12 =1 10 100  1e-20  1e-18  1e-16  1e-14  1e-12  1e-10 10 -2 10 -1 10 0 10 1 10 2 10 3 P ·  (s s -1 ) P (s)$^{ℜ}$$^{$_{c}$=R}$ B p,12 =1 10 100 4 Zach Medin and Dong Lai Figure 1. Pulsar death lines. Death lines are shown for pulsars with dipole magnetic fields, dipole fields offset from the center of the star by Δ r = 0 . 95 R , and magnetic fields with extended polar caps 100 times larger than the dipole value θ$_{cap}$ = √ Ω R/c (left panel); and for pulsars with magnetic field curvatures R$_{c}$ = R at the surface (right panel). Note that these death lines do not apply for the millisecond pul- sar population in the lower left corner of the diagram, as their short periods and low magnetic field strengths cause the primary electron to be radiation reaction limited. In each panel, rotation-powered pulsars (ATNF catalog, http://www.atnf.csiro.au/research/pulsar/psrcat) are labeled by crosses, while magnetars (McGill catalog, http://www.physics.mcgill.ca/ ∼ pulsar/magnetar/main.html) are labeled by solid circles and the two radio magnetars are labeled by solid triangles. ( R c = R ; cf. Ruderman & Sutherland 1975) at the surface and a dipole curvature at higher altitudes ( r > 2 R ); the im- plementation of this non-dipole model into our simulation is discussed in more detail in Section 3. The death line in this approximation is shown on the right panel of Fig. 1. Because of the increased efficiency of the cascades in these highly-curved magnetic fields, the Lorentz factor of the pri- mary electron upon emerging from the gap is a factor of ∼ 10 lower than in the dipole case, in the range γ$_{0}$ ≃ (2–4) × 10 6 (Hibschman & Arons 2001a; Medin & Lai 2007). or more of the assumptions made above. In some mod- els the magnetosphere geometry is not a centered dipole, but instead is an offset dipole (Arons 1998), or a twisted dipole (e.g., Thompson, Lyutikov, & Kulkarni 2002), or con- tains quadrupole or higher multipole components (e.g., Pavan et al. 2009). Numerical simulations suggest that due to twisting of the field lines near the light cylinder the po- lar cap is slightly larger than in the pure dipole case (e.g., Contopoulos et al. 1999; Spitkovsky 2006); the polar cap could be significantly larger if, as the pulsar spins down, field line reconnection (conversion of open field lines into closed field lines) is too slow to keep pace with the expand- ing light cylinder (Contopoulos 2005). All of these models increase the size of the theoretically allowed P - ˙ P space for pulsars by decreasing the radius of curvature of the mag- netic field lines R c in the region where the pair cascade occurs, i.e., above the polar cap. Since pair creation is more efficient along tightly-curved field lines, a smaller radius of curvature allows the cascade to occur at a lower γ$_{0}$ . How- ever, the magnetosphere must be highly non-dipolar near the polar cap in order for the models to include all pul- sars on the active side of the death line. For example, the offset dipole model can fully match observation only if the dipole is offset by 0 . 95 R or more, while an expanded po- lar cap (due to a twisted field or delayed field reconnec- tion) must be hundreds of times larger than the pure dipole cap; see Fig. 1. It is unclear whether such a strongly non- dipolar field is stable (cf. Beloborodov & Thompson 2007, where a field twist Δ φ > ∼ 1 radian is unstable even in mag- netars). In addition, observations of pulsar radio emission suggest that this emission is coming from a purely dipole region of the magnetosphere (e.g., Rankin & Wright 2003), so any non-dipole structure at the surface must give way to a dipole configuration at the altitude where radio pulses are generated (typically about 1% of the light cylinder radius; e.g., Gangadhara & Gupta 2001; Dyks, Rudak, & Harding 2004). In our simulation, we model the effects of a non-dipole magnetosphere by giving the magnetic field a large curvature This equation is nearly the same as Eq. (3) for young pul- sars, where Φ ≪ Φ$_{cap}$. However, for pulsars near death, with Φ < ∼ Φ$_{cap}$, the spindown power is much lower for a given po- lar cap voltage. Conversely, for a given P and ˙ P [which de- termines the observed spindown power, Eq. (2)] the polar cap voltage is much larger than would be assumed by us- In some models the spindown power of the neutron star differs from that of an orthogonal magnetic dipole, Eq. (3). For example, Contopoulos & Spitkovsky (2006) show that for nearly-aligned pulsars (with the angle between the mag- netic field and rotation axes α < ∼ 30 $^{◦}$), the spindown power can be approximated by In some models the pair cascade occurs primarily within the gap, due to efficient inverse Compton scattering by the primary electron, rather than above the gap. These cas- cades occur at much lower energies of the primary electron ( γ ∼ 10 $^{3}$–10$^{4}$), such that all observed pulsars can provide the voltage necessary to initiate this type of cascade. How- ever, ICS cascades are generally very weak, producing < ∼ 10 electron-positron pairs per primary electron, while most ra- dio emission models assume a secondary particle density 100-1000 times that of the primary electron beam (e.g., for the development of a two-stream instability; see Usov 2002). The inclusion of this type of cascade into our simulations is discussed below, in Section 2.2. ˙ E ≃ 2Φ 2 $_{cap}$c 3 ( 1 − Φ Φ$_{cap}$ ) . (5)