4 Penetration thickness for neutral hydrogen

As we have demonstrated in a previous paper (Paper I) the penetration thickness $\tau$ for neutral hydrogen in the external layer, depends on the ambient density and the pulsar velocity. For a pulsar moving in a partially neutral medium (the ISM ionization fraction is of order 0.1; Taylor & Cordes 1993; Dickey & Lockman 1990), this parameter plays an essential role in determining the true size of the nebula and the H$\alpha$  photon flux. The penetration thickness depends on the process that could drag the atoms: essentially by charge-exchange and by ionization via the interaction with the protons confined in the external layer. The cross sections of such processes are strongly dependent on velocity (McClure 1966; Newman et al. 1982; Peek 1966; Ptak & Stoner 1973), and variations of the surface density in the external layer give rise to changes of the thickness moving away from the stagnation point. In our previous paper we have evaluated the thickness using the two thin layer model limited to a region near the axis. Our aim is to verify if such an approximation is acceptable and how far in the tail can it be extended.

The thickness of the external layer has been evaluated integrating the ionization and charge-exchange probabilities along the path of a neutral test hydrogen atom. So we have taken into account variations of density, relative velocity and temperature. We have verified that, for typical pulsar velocities, collisional ionization can be neglected, even in non equilibrium cases.

Figure 6: Penetration thickness of the external layer for neutral hydrogen coming from the ambient medium. $n_{\rm o}$ is the ambient particle density, $a_{\rm o}$ is the Bohr Radius. The four cases represent different velocity of the pulsar: A 150 km s-1; B 200 km s-1; C 300 km s-1; D 400 km s-1.

Figure 6 shows the penetration thickness, normalized to the external density. The differences in the values for the various simulations are due to variations of the cross section with velocity, while the effect of temperature is less sensitive. The thickness decreases at higher velocities as expected: in fact it follows the behaviour of the cross section for charge-exchange, which is the dominant process. Concerning the variation of $\tau$ with the distance from the stagnation point, we can see that it has a homogeneous trend for different velocity values. Moving away from the stagnation point, the surface density increases and the same holds for the cross section of charge-exchange, due to the decrease of relative velocity between the neutral hydrogen atoms coming from ISM and the protons flowing in the external layer. This effect, that tends to increase the penetration thickness, is however reduced because the interaction rate, which depends on the relative velocity, decreases even if the cross section is higher, so that the mean free path is longer. The thickness, at least for the head, ranges only over a factor 2. Thus we can say that the model restricted to the stagnation point, which we have proposed (Paper I), can reproduce within the same factor the behaviour of the whole head. This can be quite important, because, if the nebula is thin to the penetration of neutral hydrogen, it can remain thin in the tail, even far from the pulsar.