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1979). We therefore randomly assign the photon a polariza- tion in the ratio of one ⊥ (perpendicular to the field) to every seven ‖ (parallel to the field) photons, corresponding to 75% averaged parallel polarization. 

## 3.2 Photon propagation, pair production, and splitting 

In our simulation, the photon is emitted/scattered from the point ( r$_{0}$$_{,}$$_{ph}$, θ$_{0}$$_{,}$$_{ph}$ ) with energy ϵ , polarization ‖ or ⊥ , and weighting factor Δ N$_{ϵ}$ (to represent multiple photons; Sec- tion 3.1). It has an optical depth to pair production, τ , and to photon splitting, τ$_{sp}$ , both of which are set to zero at the moment of the photon’s creation. The photon propagates in a straight line from the point of emission, at an angle Θ$_{ph}$ with respect to the magnetic dipole axis. Note that in the corotating frame (which is the frame we are working in for most of our simulation; but see Section 3.3) the path of the photon is in reality curved, with the angular deviation from a straight line growing approximately as s$_{ph}$ Ω /c = s$_{ph}$/r$_{LC}$ (cf. Harding, Tademaru, & Esposito 1978). Like the beam- ing angle (Section 3.1), this curved path modifies the growth of the photon-magnetic field intersection angle and the loca- tion of photon decay in our simulation. However, the total intersection angle grows much faster with photon distance s$_{ph}$ than the deviation does ( ∼ s$_{ph}$/ R c versus s$_{ph}$/r$_{LC}$ , or a factor of r$_{LC}$/ R c ≃ 100 P 1 / 2 0 larger for dipole fields), so we can safely ignore this deviation. 

In each step the photon travels a short distance through the magnetosphere, Δ s$_{ph}$ < 0 . 05 r$_{ph}$ , where ( r$_{ph}$, θ$_{ph}$ ) refers to the current position of the photon; our method for choos- ing the value of Δ s$_{ph}$ for a given photon is discussed at 

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the end of this section. At the new position the change in the optical depth for pair production, Δ τ , and for photon splitting, Δ τ$_{sp}$ , are calculated: 

where R$_{‖}$$_{,}$$_{⊥}$ = R ′ ‖ , $_{⊥}$sin ψ is the attenuation coefficient for the ‖ or ⊥ polarized photons, ψ is the angle of intersection between the photon and the local magnetic field, and R ′ is the attenuation coefficient in the “perpendicular” frame (i.e., the frame where the photon propagates perpendicular to the local magnetic field). For a dipole field geometry the intersection angle is given by 

where Θ$_{ph}$ is given by Eq. (17) and χ ( θ$_{ph}$ ) is the angle be- tween the magnetic axis and the magnetic field at the cur- rent location of the photon [Eq. (16)]; see Fig. 4 for a sketch. For the near-surface multipole field geometry we set 

This approximation has the advantage of accounting for the effect of a strong field curvature on the photon propagation without requiring knowledge of the actual field topology. 

The total attenuation coefficient (in the perpendicular frame) for pair production is given by (suppressing the sub- scripts ‖ , ⊥ ) R ′ = Σ$_{jk}$ R ′ $_{jk}$, where R ′ jk is the attenuation coefficient for the channel in which the photon produces an electron in Landau level j and a positron in Landau level k , 

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