density profiles \[$T(\tau)$ and $\rho(\tau)$, where $\tau=\kappa_T y$ (with $\kappa_T=0.4$ cm$^2$g$^{-1}$) is the (zero-field) Thomson depth, and $y$ is the column density\].

We use free-free absorption opacities for each carrier, including the contribution of electron-ion Coulomb collisions, as described by Lai & Ho (2003). This opacity depends strongly on $\textbf{k}\cdot\hat{\textbf{B}}$, as well as on density and temperature.

In addition, at the magnetic field strengths we are considering, the opacity is further complicated by oscillations in the magnetic Gaunt factor (Potekhin 2010). However, the energies we consider have values of $\sim 1.5-5$ times the electron cyclotron energy and therefore these oscillations are not very strong. Furthermore, while the oscillations may affect the specific flux emitted at certain frequencies, we do not expect them to affect pulse shape. We therefore do not include this complication in our calculation, instead choosing to use the non-magnetic Gaunt factor, which captures the trend about which the full magnetic Gaunt factor oscillates.

The temperature profile $T(\tau)$ and density profile $\rho(\tau)$ are related by the equation of hydrostatic equilibrium. For all practical purposes, an ideal, non-degenerate gas equation of state can be used with better than a few percent accuracy (e.g., Ho & Lai 2001).

TTFT: A 