The input parameters for our simulation are the initial energy of the electron ( γ$_{0}$m$_{e}$c $^{2}$) or photon ( ϵ$_{0}$ ), its initial po- sition (in most cases, the intersection of the last open field line with the stellar surface), the general pulsar parameters (surface magnetic field strength B$_{p}$ = 10 $^{12}$–1015 G, rotation period P = 0 . 33–5 s, and surface temperature T = 10 6 K or 5 × 10 6 K), and the geometry of the magnetic field. In each run of the simulation, the magnetic field structure is given by one of two topologies: (i) a pure dipole field ge- ometry; or (ii) a more complex field geometry near the stel- lar surface which gradually reverts to dipole at higher alti- tudes (a non-dipole, or “multipole” field geometry). Mod- eling the dipole field geometry is straightforward (see, e.g., von Hoensbroech, Lesch, & Kunzl 1998), but there is no ob- viously correct way to model the geometry for the multipole field case (see Section 2.1). Two features of a multipole field geometry have a strong effect on the pair cascade dynam- ics and must be incorporated into our model: First, the ra- dius of curvature R c is much smaller than dipole (we choose R c = R , the stellar radius) near the surface of the star. This leads to a much larger number and peak energy of photons emitted through curvature radiation than in the dipole field case. Second, as a photon propagates through the magne- tosphere the angle between the photon and the field, which scales like ΔΘ$_{ph}$ ∼ s$_{ph}$/ R $_{c}$, where s$_{ph}$ is the distance trav- eled by the photon from the point of emission, grows much faster than dipole. This leads to a much more rapid decay of photons into pairs than in the dipole case. The integration of these two features into our model is discussed in the relevant subsections below (Section 3.1 and Section 3.2, respectively). Note that Arendt & Eilek (2002) consider the first aspect of a multipole field geometry in their model (that R c = R ) but ignore the second. In all of the simulation runs we assume that the local magnetic field strength varies as in the dipole case, where ( r, θ, φ ) are the spherical coordinates (with the mag- netic north pole at r = R and θ = 0). Our approximation therefore ignores any amplification of the field strength near the surface caused by the complex topology. For simplicity we consider a “two-dimensional” cascade model in which all photons are emitted and travel in the plane defined by the local magnetic field line. Both the pho- tons and the electrons/positrons are tracked in the “coro- tating” frame (the frame rotating with the star), and any bending of the photon path due to rotation is ignored – this is expected to be valid since the cascade takes place far in- side the light cylinder. Thus we shall also call this corotating frame the “lab” frame for the remainder of the paper. With this approximation the particle positions and trajectories are defined only in terms of r and θ in our simulation. We justify this approximation below (Sections 3.1 and 3.2). As an additional simplification we ignore any effects of general relativity on the photon/particle trajectory. The cascade simulation can naturally be divided into three parts: (i) the propagation and photon emission of the primary electron; (ii) photon propagation, pair production, and splitting; and (iii) the propagation and photon emission as they propagate from the point of emission through the magnetosphere, until they decay into electron-positron pairs through magnetic pair production or escape to infinity. In the superstrong field regime, the photon (if it has the cor- rect polarization; see Section 3.2) also has a finite proba- bility of splitting into two photons before pair production, in which case we follow the two child photons in a similar way. The electrons and positrons created by these photons are tracked as they radiate away their transverse momenta through synchrotron radiation and then gradually lose their forward momenta through inverse Compton scattering. Sub- sequent generations of photons and electrons/positrons are also tracked, in a recursive manner, and the total numbers and energies of photons and electrons + positrons that es- cape the magnetosphere are recorded. We track each parti- cle until it is destroyed or reaches a height comparable to the light cylinder radius. Tracking particles out to the light cylinder is most important for the primary electron, since although in general there is no discernible pair production above the radius r ∼ 10 R , curvature radiation continues up to very high altitudes (albeit very weakly, with photon energies ϵ < 10 MeV near the light cylinder). ∼ In the second version of our cascade simulation, a pho- ton with energy ϵ$_{0}$ ∼ 10 $^{3}$–105 MeV (see Section 2.2) travels outward from the last open field line at the stellar surface, in the direction tangent to the magnetic field at that point. While in theory the photon should be emitted at an an- gle to the field line of ΔΘ$_{ph}$ < ∼ 1 /γ$_{e}$ , where γ$_{e}$ is the final Lorentz factor of the electron after emitting the photon, in practice this does not matter, as there is no change in the final products of the cascade whether we use ΔΘ$_{ph}$ = 1 /γ$_{e}$ or ΔΘ$_{ph}$ = 0. (This is true even for photons upscattered by resonant ICS at B > ∼ B$_{Q}$ ; see Section 3.3.2). The primary photon and subsequent generations of electrons/positrons and photons are tracked in the same way as in the main cascade simulation. The cascade as modeled by the second simulation is sketched in the inset of Fig. 2. Figure 2. A schematic diagram showing the magnetosphere pair cascade, from initiation by a high-energy electron to completion. Photon splitting is also shown. The inset shows the beginning of a cascade initiated by a photon upscattered through the inverse Compton process. B ( r, θ, φ ) = B$_{p}$ ( R r ) 3 √$_{3 cos}$ 2 θ + 1 2 , (11) 6 Zach Medin and Dong Lai