E_1^2 B_{14}^2 f_B^{-2}~\mathrm{g~cm^{-3}},
\label{eq:rhov}$$ where $Y_e$ is the electron fraction, $E_1=E/(1~\mathrm{keV})$, $B_{14}=B/(10^{14}~\mathrm{G})$, and $f_B$ is a slowly varying function of $B$, of order unity. At resonance, the X and O polarization modes become degenerate, and therefore the photon may be converted from one mode to another when it traverses the resonance. We define two more effective polarizations modes: + and –. These modes act as X (O) polarization modes before the resonance and as O (X) modes after. Lai & Ho (2002) showed that photons satisfying the adiabatic condition, $E \gtrsim E_{\mathrm{ad}}$, act as + and – modes. Here $$E_{\mathrm{ad}}=2.52\Bigl[f_B \tan\theta_{kB}\left|1-(E_{Bi}/E)^2\right|
\Bigr]^{2/3}\left({1~\mathrm{cm}\over H_\rho}\right)^{1/3}\mathrm{keV},
\label{eq:ead}$$ where $\theta_{kB}\equiv\cos^{-1}\textbf{k}\cdot\hat{\textbf{B}}$, $E_{Bi}=0.63Y_e B_{14}~\mathrm{keV}$ is the ion cyclotron energy, and $H_\rho\equiv|ds/d\ln \rho|$ is the density scale height along the ray. More generally, a photon with energy $E\sim E_{\mathrm{ad}}$ will have a nonadiabatic ‘jump’ probability of $$P_{\mathrm{jump}}=\exp\left[-{\pi\over 2}\left(E/E_{\mathrm{ad}}\right)^3\right].
\label{eq:pjump}$$ If the ‘jump’ probability is low, most outward diffusing photons will evolve through the resonance adiabatically as + or – modes, undergoing mode conversion. On the other hand, if the ‘jump’ probability is high, X-mode (O-mode) photons will remain X-mode (O-mode) photons after traversing the resonance. We therefore elect to treat the X, O, +, and – photon 