Method for Temperature Profile Computation
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As an all-purpose tool, the computational procedure described in Sections 2.1-2.2 requires the knowledge of the atmospheric temperature profile $T(\tau)$. In theory, for a given combination of effective temperature, magnetic field strength and direction, $T(\tau)$ is obtained by self-consistent atmosphere modeling to achieve radiative equilibrium. However, this modeling must be done for every single patch of the NS surface, and is not practical to implement. Here we propose the “Temperature Template with Full Transport” (TTFT) method for constructing temperature profiles. While this method is not accurate for predicting the observed synthetic radiation spectrum, we show that it is reliable insofar as light curve shape (and therefore pulse fraction) is concerned.

The TTFT method consists of the following steps: (i) We write the temperature profile in the form $T(\tau)=T_0 F(\tau)$, characterized by the dimensionless template function $F(\tau)$ and the adjustable constant $T_0$. The template function may be obtained from a specific NS atmosphere model. (ii) For each $T(\tau)$, we carry out full radiative transfer calculation (Section 2.2) to obtain the total radiation flux from a single surface patch, =dd\^2[**k**]{} I\_\^e([**k**]{}) [**k**]{}, \[eq:calF\]where the integration $\int d^2{\bf k}$ covers the outgoing direction relative to the surface 