1 Introduction

Pulsars moving supersonically with respect to the ambient medium are expected to give rise to a bow shock. In the case of interaction with the interstellar medium (ISM) the H$\alpha$  emission from the nebula may be detected. Although pulsars have high typical velocities, and there are more than one thousand known radio pulsars, only four bow shocks have been discovered so far: PSR 1957+20 (Kulkarni & Hester 1988), PSR 2224+65 (Cordes et al. 1993), PSR J0437+4715 (Bell 1995), and PSR 0740-28 (Jones et al. 2001); while it is not clear whether the cometary-like nebula associated with the isolated neutron star RX J185635-3754 (van Kerkwijk & Kulkarni 2001) is a bow shock or results from photo-ionization. In this kind of object the relativistic particles in the pulsar wind produce a synchrotron emission too weak to be detectable in radio wavelengths (Gaensler et al. 2000). These nebulae may be detected instead in optical Balmer lines (mostly H$\alpha$) as a signature of a non-radiative shock moving through a partially neutral medium (Chevalier & Raymond 1980). The Balmer lines are due to the de-excitations of neutral H atoms, following collisional excitations or exciting charge-exchange processes. When recombination times are long these are the dominant processes giving emission in the optical band, as it has been widely studied on various supernova remnants (Smith et al. 1991).

A puzzling point is that two out of the four known nebulae have a shape closely matching that of a classical bow shock, while the others show a more peculiar shape with a conical tail: the nebula associated with PSR 2224+65 in fact shows a remarkably conical tip followed by a bubble (which justifies why it has been named the "Guitar Nebula''); (Cordes 1996). Various hypothesis have been put forward to justify its shape: a peculiar ISM density distribution in the surroundings of the PSR 2224+65 (Cordes et al. 1993); effects like mass loading due to neutral atoms which might penetrate in the external layers of shocked material (Bucciantini & Bandiera 2001, hereafter Paper I). The fourth nebula, which has recently been discovered (Jones et al. 2001) near PSR 0740-28, seems to represent an intermediate case, with a "standard'' head and a conical tail.

A pulsar wind interacting with a homogeneous ISM is in principle a much easier problem to model than that with a surrounding supernova remnant, since in the latter case the ambient medium might be quite inhomogeneous. Thus our analysis will be restricted to the former case. The ISM is seen by the pulsar as a plane-parallel flow, likely with a constant density (at least on the typical length scale of the bow shock), and lasting long enough to produce a steady-state regime. However, modeling is made more complicated by the fact that the pulsar wind is relativistic and magnetized. Moreover the ISM is typically partly neutral and the neutral atoms may have collisional mean free paths comparable to the scale length of the system, then invalidating a purely fluid treatment. However as demonstrated in a previous paper (Paper I), if we suppose that the H atoms can be ionized only via collisions with electrons and protons of the shocked plasma, for a large number of pulsars the presence of a neutral component in the ISM can be neglected as far as the fluid dynamics is concerned, or at least it can be taken into account as a small perturbative effect.

The interaction of the relativistic magnetized pulsar wind with the ISM ionized component is not direct, because the relativistic particles themselves have very small cross sections, but is mediated by the magnetic field advected by the pulsar wind and compressed on the head of the nebula. The effective mean free path of particles is then their gyroradius, which is typically much smaller than the typical dimension of the nebula, scaling with the distance of the bow shock stagnation point from the pulsar:

$$d_{\rm o}=\sqrt{{\cal L}/4\pi c \rho_{\rm o}V_{\rm o}^{2}},$$ (1)

where ${\cal L}$ and $V_{\rm o}$ are the pulsar luminosity and peculiar velocity, and $\rho_{\rm o}$ the density of the dragged component of ambient medium (Paper I). For this reason we have assumed little mixing between the two media, which allows the use of a fluid approach to investigate the structure and dynamics of these objects.

In Sect. 2 we present the numerical code and the parameters of the various simulations we have performed; Sect. 3 discusses the results of these simulations and compares them to what is expected from an analytic "two thin layer'' model (Comeron & Kaper 1998); in Sect. 4 we evaluate the penetration thickness of the external layer for a neutral hydrogen atom of the ISM.