processes at play, we can start evolving the evolution equations once the particles exit the magnetar’s atmosphere, which is the only region where plasma effects are non-negligible. Therefore, our initial amplitudes at the magnetar’s surface correspond more precisely to the amplitudes outside the magnetar’s atmosphere. However, since the magnetar radius is of the order of kilometers while the atmosphere’s thickness is of the order of centimeters, we can choose $x=1$ as our initial point to solve the evolution equations without consequences. From this point on, the plasma is completely negligible and the analysis presented before is valid.

However, the dipolar approximation holds only several radii away from the magnetar. Hence, to assume a dipole magnetic field, it is necessary to redefine the initial amplitudes at the magnetar’s surface by the amplitudes several radii away from the magnetar where the dipolar approximation is valid. The evolution equations are thus evolved numerically from there and the average in is performed on the phase difference at that position. Since the conversion radius, where most of the ALP-parallel photon conversion occurs, is a few hundred radii away from the magnetar for our benchmark point, the dipolar approximation is clearly justified.

The particles thus evolve 