modes as distinct carriers whose relative abundances are determined by $P_{\mathrm{jump}}$: $$I^e_\nu=P_{\mathrm{jump}}(I^X_\nu+I^O_\nu)+(1-P_{\mathrm{jump}})(I^+_\nu+I^-_\nu).
\label{eq:inucarriers}$$ Note that, for the application to the Kes 79 CCO (Section 3), at fields $B\sim 10^{10}-10^{11}$G and energies $E\sim1-5$keV, $\rho_V$ is smaller than the photosphere densities, and therefore vacuum polarization does not play an important role. We therefore treat all photons as “jump” photons ($P_{\mathrm{jump}}=1$), and do not compute intensities for the + and – modes.

In general, to compute the individual specific intensities for each carrier, we integrate the standard radiative transfer equation, $dI^i_{\nu}/d\tau^i_{\nu}=B_\nu-I^i_\nu$. However, for the present purposes it is sufficient to use the Eddington-Barbier relation:

$$I^i_\nu(\textbf{k})\approx {1\over 2}B_\nu\left[T(\tau^i_\nu(\textbf{k})=2/3)\right].
\label{eq:eddington}$$

That is, the emergent intensity (toward $\textbf{k}$) due to carrier $i$ is equal to the Planck function evaluated at the depth at which the optical depth $\tau^i_\nu$ for that carrier (along $\textbf{k}$, measured from the outside in) is equal to $2/3$. The factor of $1/2$ arises because the photon number distribution is split among two carriers at all times. Thus, to calculate the total emergent specific intensity, only a few pieces of information are required. These are: the direction and magnitude of the magnetic field, the opacities for each carrier, and the atmospheric temperature and 