Suleimanov et al. 2010). This is currently not practical for exploration of a wide range of model parameters, since quantitative calculation of the light curve demands adding up many different surface patches (rather than one or two hot spots; cf. Ho 2007). Therefore we develop an approximate method (TTFT) (see Section 2.3) that allows us to obtain $I^e_\nu(\textbf{k})$ efficiently. Here, we describe our full radiative transfer calculation in NS atmospheres.

In the atmospheres of magnetized NSs, photon polarization plays an important role in determining $I^e_\nu(\textbf{k})$. For example, at photon frequencies smaller than the electron cyclotron frequency, the mean free path for ordinary (O) mode photons is much smaller than that of the extraordinary (X) mode photons, and therefore, the X-mode photons that contribute to the emergent flux are typically produced in the deeper, hotter layers of the atmosphere. At higher field strengths ($\gtrsim8\times10^{13}$G), vacuum polarization will also have a strong effect on the radiation spectrum (Lai & Ho 2002, 2003). While rigorous treatment of this effect is available (van Adelsberg & Lai 2006), we describe an approximate method below.

For a photon of a given energy $E$ propagating along $\textbf{k}$ in the NS atmosphere, vacuum resonance occurs at $$\rho_V=0.96 Y_e^{-1} 