2. Semi-analytical approach
The 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-1, i.e. , in a diffuse warm ionized medium. The characteristic scale length of the bow shock is given by the standoff distance , defined as the distance from the star where the ram pressures of the ambient gas and of the stellar wind balance:
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 instantaneous cooling approximation, used by e.g. Wilkin (1996) and Dgani et al. (1996b), and justified by Mac Low et al. (1991) in their treatment of ultracompact H II regions. However, if cooling of the shocked stellar wind is inefficient, its high temperature sets a limit on the density compression factor which, for the limiting case of no cooling, would be equal to 4 (adiabatic shock). In such a case, a thick layer of very hot, low density gas (with an accordingly long cooling timescale, except in the case of a very dense ambient medium, like in the models of Mac Low et al. 1991) can be expected to lie in between the freely flowing stellar wind and the bow shock, forcing the latter to recede away from the star to a distance somewhat greater than . The numerical simulations performed by Raga et al. (1997) confirm these expectations.
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 (z, ) are centered on the star with z parallel to its space velocity. R and are, respectively, the radial and polar spherical coordinates, also centered on the star so that the velocity of unshocked stellar wind has only a component along R. The star moves in the direction. The unit vector normal to the shock front is , and forms an angle with the z direction. The vector parallel to the surface in the direction of growing is .
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 R from the star,
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
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 z axis (see Fig. 3). The tangential momentum injected by the stellar wind in the area of the shocked layer immediately inside the bow shock has a modulus
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 z and directions are thus:
where we have defined in Eq. (12a). The shape of the bow shock is found using
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 v along the bow shock (measured in the stellar rest frame) are given by:
Note that, due to the proportionality of both and to , , . Fig. 4 and 5 compare our predictions of respectively and v with those obtained for the instantaneous cooling case, adopting a ratio in the latter case.
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:
The factor stems from the assumption that hydrogen is totally ionized, and a helium abundance . The proton mass is , k is the Boltzmann constant, and is the temperature of the bow shock. Its thickness is , and the number of recombinations (per unit area of the bow shock) is , where is the recombination rate to all the levels except the ground level, and the proton number density: . The flux of ionizing photons required to balance the rate of recombinations per unit area at the head of the bow shock 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-1 (Schaerer & de Koter 1997).
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 shock
A 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 transverse acceleration instability , has been presented by Dgani et al. (1996a,1996b). This instability occurs as a consequence of the acceleration in the flow normal to the shock due to its curved surface. Another instability appearing under the same conditions, the non-linear thin-shell instability , arises from the shear in shock-bounded slabs produced by deviations from equilibrium (Vishniac 1994, Blondin & Marks 1996). Dgani et al. (1996b) have discussed the regions of the bow shock where either instability is likely to dominate, and have pointed out the caveat that another instability, not considered in those studies, may in fact dominate the dynamics of the gas in the bow shock. This instability is due to the zero-order shear between the gas coming from the stellar wind and the gas from the ambient medium, which triggers Kelvin-Helmholtz instabilities all along the surface of the bow shock. As a result, real bow shocks may be expected to have severely distorted shapes in the instantaneous cooling case. The development of this zero-order shear instability is clearly seen in numerical simulations of colliding unequal winds (Stevens et al. 1992), and is confirmed by the results of our numerical simulations (Sect. 4.5). In fact, the shear is also present when the outer shock front is isothermal, as in general the velocity of the gas flowing along the bow shock differs from the instantaneous tangential velocity of the shocked gas locally incorporating to it (, see Fig. 1). However, given the large ratios in the cases of interest here, the direct incorporation of the shocked stellar wind into the bow shock that occurs in the isothermal case has a much larger impact.
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 v along the bow shock is smaller than the local sound speed . The first-order development of Eq. (17) gives
and therefore v keeps subsonic as long as
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