ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Model I-Model O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O
tokens: does not need to precompute and store anything for these surfaces. To solve equation ( A4 ), we need only the current position and direction, and the θ boundaries of the current cell. A1.3 Constant phi surfaces The constant φ surfaces are planes. Determine the Cartesian coordinates for the four edgepoints of the surface. Then, define, for the k plane, the vectors Take the cross-product and normalize to get the normal vector to the plane . Note that an intersection point on the plane will obey . For the k + 1 plane use instead and then corresponding operations for and d . The intersection point ( x i , y i , z i ) and the distance to this point, d s , are then found from solving the linear equation system No solution means the packet is moving parallel to the plane and will not impact it. If d s 0, the packet is moving in the wrong direction and will not impact the plane. The four numbers are computed once and stored for all surfaces. A1.4 Entry surface flagging If the surface through which the packet entered the cell is allowed also an as exit surface, small numeric errors could lead to the packet getting stuck at the boundary. To avoid this the entry surface is initially flagged to be disallowed as solution for which next boundary to encounter. This flagging is removed if an interaction occurs inside the cell. For two surfaces, the ‘convex out’ boundaries at the constant r outer radius and constant θ conical boundary closer to the equatorial plane, it is possible for the packet to enter and exit the same boundary on a straight path. However, this occurrence is very rare, in particular for a high-resolution grid as those used here. We therefore kept the flaggings also for these cases, and was removed only after a small distance into the cell had been travelled. This treatment implies a small distortion to the flight path for photons gracing these surfaces but the effect of this should be very small, and this treatment avoids an more expensive algorithm for doing the transfer exactly. A1.5 Positional corrections Once the shortest distance has been calculated, the new Cartesian coordinates of the packet are calculated from . The spherical coordinates are then calculated from these. It may sometimes happen that in these transformations small numeric errors lead to the new spherical coordinates being (slightly) outside the limits of the new cell. If that happens severe errors can occur in calculating the next movement. Therefore, at each new cell entry, this is checked and if a discrepancy has arisen, the coordinates are adjusted to be just inside (by a multiplicative factor 1 ± 10 −5 ) the surface in question. A1.6 Next interaction point For continuum opacity, standard Monte Carlo procedures are followed by drawing a random optical from τ = −ln z where z is a random number between 0 and 1. Then As the distances are computed in the stellar rest-frame one should ideally use the direction-dependent opacity in this frame. We did not make the