## 2. Semi-analytical approachThe parameters defining the scenario of a runaway star moving
supersonically in a diffuse medium are the stellar wind's mass-loss
rate, ; the terminal wind velocity,
; the velocity of the star in the rest frame of
the gas, ; and the density of the ambient medium,
. The supersonic motion implies that the thermal
pressure of the ambient gas is smaller than its ram pressure in the
direction of motion of the star by a factor ,
where is the Mach number; for large values of
, therefore, the thermal pressure can be
neglected. The space velocities of OB runaway stars range from a few
tens to about a hundred km s If the interaction between the stellar wind and the ambient medium
were to take place in an infinitely thin layer, where the gas coming
from both the wind and the ambient medium were instantaneously cooled
and compressed to much higher densities, would
directly give the distance to the star of the apsis of the bow shock.
This is the At the outer side of the wind bow shock, the structure depends on the rate at which the ambient gas cools down and the velocity with which the shocked gas flows downstream along the bow shock (for a sketch, see Fig. 1). In general, a layer of moderately hot gas constitutes the outer boundary. Cooling of this gas, if it takes place before it has moved downstream significantly, can then create a denser, cooler zone further inside. If cooling of the shocked ambient gas is very fast, the outer shock is practically isothermal, and the outer hot layer vanishes. The actual structure is thus rather sensitive to the parameters which determine the cooling time and the gas flow rate along the bow shock. This property can be used to indirectly constrain the values of some of the basic parameters of the problem, provided that other parameters are known (cf. Sect. 4). We note that, due to the long cooling timescale of the shocked stellar wind in all the cases considered here, the dense shell defining the bow shock is composed of swept-up ambient gas only, with no contribution from the stellar wind, contrary to the case of instantaneous cooling. The situation studied throughout this paper is in some respects the inverse of that described by Borkowski et al. (1992), in which a slow stellar wind is shocked and quickly cooled by the interaction with a hot, low density medium with a much longer cooling time. In the latter case, the bow shock is composed of shocked stellar wind only, with no contribution from the ambient gas.
The equations determining the shape of the wind bow shock can be derived using geometry and some considerations on momentum conservation, plus a number of simplifying assumptions. The main one adopted here is that the hot layer separating the freely flowing stellar wind from the shocked ambient medium has a thickness which is small compared to , i.e. the inner surface of the layer of shocked ambient gas is parallel to the surface of the shocked stellar wind. Later on, the numerical simulations will show to what extent this approximation is realistic. The geometry of the wind bow shock is depicted in Fig. 2. We use
the same notation as Wilkin (1996), and follow closely his approach to
the problem in the remainder of this section. The axial and radial
cylindrical coordinates (
The shock front makes an angle with the direction of the unshocked wind. The shocked stellar wind produces a pressure normal to the layer of shocked ambient gas equal to the sum of the ram pressure of the shocked wind, plus its thermal pressure. By conservation of the momentum across the shock front, the total normal pressure in the shocked wind equals the normal component of the ram pressure of the unshocked wind perpendicular to it, so that where is the density of the stellar wind at
a distance The momentum flux injected in an area element of the bow shock by the ram and normal pressures of the shocked wind perpendicular to its surface is thus with Using , one can show that the contribution of the stellar wind to the momentum flux normal to the bow shock surface (measured within a cap of the bow shock included by the angle ) is: At this point we should note a difference with Wilkin's development. If the cooling is instantaneous, the shocked stellar wind reaches a high density and both its mass and its total momentum (not only the component perpendicular to the surface) are incorporated into the bow shock. The vector sum of and the tangential component of the momentum yielded by the stellar wind then provides the total wind contribution , which depends only on by virtue of momentum conservation arguments. Subsequently, can be easily integrated, providing the analytical solution found by Wilkin. The basic difference with our approach is that we assume, as a starting point, that the shocked stellar wind and the shocked ambient medium do not mix together in the bow shock. As a consequence, the tangential momentum contained in the shocked stellar wind inside a solid angle is not incorporated immediately onto the area of the bow shock included by , contrary to the instantaneous cooling case. In our case, a fraction of the tangential momentum must indeed be
injected into the bow shock inside the cap included by
: only the projection of this momentum parallel
to the bow shock at angle will not contribute
to the global momentum of the cap. Therefore, we need to calculate the
sum of the infinitesimal tangential momenta inside the cusp along a
direction forming an angle with respect to the
The projection in the direction is Therefore, the contribution of this component to the momentum injected by the stellar wind in the bow shock is or, using again (3), The last contribution to the momentum of the cap we have to consider, is that of the ambient medium: The summed components of the momentum in the where we have defined in Eq. (12a). The shape of the bow shock is found using Then, Using Eqs. (14), it is straightforward to numerically integrate Eqs. (12) and determine the function describing the shape of the bow shock. However, it is possible to anticipate that the solution must indeed be the same as found by Wilkin (1996): the balance of momenta determining the shape is essentially the same as described here, regardless of whether the momentum of the shocked wind is entirely incorporated into the bow shock, as in the instantaneous cooling case, or whether a part of it flows in a hot layer beneath it, as discussed here. The analytical solution found by Wilkin, also applicable to our case, is and explicit expressions as a function of
can then be easily found for Eqs. (14). Although their ratio is the
same, the values of and
are in general different from the instantaneous cooling case, as well
as the distribution of the surface density and the tangential velocity
in the bow shock. The surface density of the bow shock only contains
the contribution from the ambient gas and, as this gas and the stellar
wind do not mix, the mass accumulated by the bow shock moves away from
the apsis more slowly, since it is not dragged by the shocked wind.
The surface density and the velocity Note that, due to the proportionality of both
and to
, ,
. Fig. 4 and 5 compare our predictions of
respectively and
Under some conditions, the bow shock can be dense and thick enough so that it blocks the ionizing flux of the star. The combination of parameters for which this happens can be easily estimated in the neighbourhood of the apsis, where the shocked gas reaches its largest compression. For , the surface density can be approximated by The volume density at that point is determined by the jump conditions across the shock: with The factor stems from the assumption that
hydrogen is totally ionized, and a helium abundance
. The proton mass is ,
where is the number density of hydrogen
nuclei in the unshocked ambient gas. Eq. (21) neglects the absorption
of ionizing photons by dust inside the bow shock, and assumes an
arbitrarily large ionization cross section of hydrogen atoms;
therefore, it actually provides a lower limit to
, which nevertheless should be useful in giving
an order of magnitude of the required ionizing flux. As a reference,
for a B0 V star the predicted ionizing flux is
s This makes the shocked layer sensitive to a kind of instabilities discussed by Giuliani (1982) and García-Segura & Franco (1996): in a slightly denser region of the shocked layer, the ionization front moves closer to the star, and a lateral pressure against it is provided by the less dense, but hotter surrounding areas which are still within reach of the ionizing flux. This pressure makes the dense region to grow even denser, further distorting the ionization front and the shape of the bow shock. We will not consider this here, and instead assume that the value of is always large enough to maintain the bow shock ionized. ## 2.1. Some considerations on the stability of the bow shockA number of instabilities may affect the flow of gas inside the bow
shock, depending on the cooling efficiency of the gas and its position
with respect to the apsis. A detailed analysis of an instability
characteristic to bow shocks with instantaneous cooling of the stellar
wind, the so-called When the shocked stellar wind cools rather inefficiently, the above mentioned instabilities are not expected to have a great relevance. Instead, another kind of instabilities, similar to the blast wave overstability described by Ryu & Vishniac (1987), may be present. This overstability appears in shocked layers bounded by thermal pressure on one side, and by ram pressure on the other. It is a result of the fact that the ram pressure acts only in the direction of motion, while thermal pressure acts perpendicular to the surface of the shock. A clear qualitative description and numerical simulations of this overstability are presented by Mac Low & Norman (1993). The blast wave overstability, in the form studied by Ryu & Vishniac (1987) and Mac Low & Norman (1993), applies to expanding bubbles in a medium at rest, or to planar shocks. The basic physical scenario applies to runaway stars as well in the proximities of the apsis of the bow shock. Away from it, shocked gas rapidly flows along the bow shock surface, and the results presented by these authors are not likely to apply. From a qualitative point of view, one can expect that the supersonic flow of gas along the surface will quickly smear out any ripples distorting the bow shock as a consequence of this instability. On the other hand, the fact that the bow shock remains at a constant distance from the star, rather than expanding at a decreasing velocity, introduces important differences in the development of the overstability that actually turn it into an instability. We defer a detailed discussion of these differences to Sect. 4, where we follow the development and evolution of the instabilities numerically. Nevertheless, the results on expanding shells are still useful to establish some basic features of the instabilities that can be expected to appear in bow shocks, and we will use them in the remainder of this section. We can use lengthscale arguments to show that bow shocks with
non-instantaneous cooling are indeed expected to be unstable near the
apsis. Ryu & Vishniac (1987) present figures displaying the growth
rate of the overstability on a spherical surface as a function of its
angular wavelength, showing that for an infinite compression of the
gas at the shock, modes with angular wavelengths smaller than
are overstable. For finite compression factors,
a range of overstable wavelengths is found, unless the compression
factor is less than . The numerical simulations
by Mac Low & Norman (1993) show a good qualitative agreement with
these results. The smaller the wavelength of the perturbation, the
faster the growth rate, but the amplitude of saturation is expected to
be smaller too (Blondin & Marks 1996). Therefore, one expects the
longest overstable wavelengths to dominate the overall distortion of
the shell. In our case, as explained above, the longest overstable
wavelength that can be sustained is in addition limited by the
condition that the zero-order velocity must be small, so that the
dynamics of the gas in the bow shock is dominated by the
overstability. In this way, we may estimate the extent of the unstable
region around the apsis by using the condition that the velocity
and therefore The size of the unstable area around the apsis is thus . For the Mach numbers that we will consider here (of the order of ), this is much smaller than the size of the longest overstable wavelength, , where is the radius of curvature of the bow shock at the apsis. Nevertheless, we expect the effects of the overstability to propagate well beyond the area delimited by , as the distortions produced in this region are carried away from the apsis by the gas flow along the bow shock. © European Southern Observatory (ESO) 1998 Online publication: September 8, 1998 |