Astron. Astrophys. 338, 273-291 (1998)

4. Results

4.1. The reference case

We will begin our presentation of the results by following in some detail the evolution of the wind bow shock in case A. Fig. 6 shows, in a selection of snapshots, the most relevant phases in the evolution of the gas density in the computational plane, starting years after the "switch-on" of the stellar wind.

Fig. 6. Evolution of the gas density in case A. The time sequence proceeds from left to right, and from top to bottom. The first frame corresponds to years since the "switch-on" of the stellar wind, the second to years, the third to years, and the third to years. The star moves downward with a velocity km s-1. The intensity of grey is proportional to the logarithm of the density, with black corresponding to particles cm-3, and white to 4.2 particles cm-3. The size of the computational grid is 5.36 pc 2.68 pc ().

The formation stage of the bow shock takes place on a timescale of only a few years. Soon after the start of our simulations, the pressure inside the bubble of hot gas expanding around the star drops to values comparable to the ram pressure of the ambient medium, and its distortion is already well noticeable after only years. The decreasing pressure inside the bubble in turn increases the distance (measured from the star) at which the free-flowing stellar wind is shocked. The bubble expansion in these early stages causes the velocity with which the ambient gas is shocked ahead of the star to be well above . The resulting very high temperature of the shocked ambient gas makes its cooling time long enough so that the formation of a dense shell at the head of the bow shock is not possible.

The second frame in Fig. 6, showing the density distribution after years, corresponds to the stage in which the bubble has reached its maximum size. Its expansion velocity has dropped to zero now, and the ambient gas passes through the shock with a velocity in its reference frame. The decrease in shock strength, and thereby the reduction in post-shock temperature, shortens the cooling time of the shocked ambient gas, and a moderately dense layer begins to form. The higher density of this layer results in a decrease of its thickness, so that the outer boundary of the bow shock structure recedes towards the star, as can be seen by comparing the second and third frames of Fig. 6.

The fourth frame of Fig. 6 shows a structure like the one sketched in Fig. 1, corresponding to a situation of equilibrium. In this configuration, the distance of the shock limiting the extent of the freely flowing wind region is very close to . This is expected from the fact that the sum of thermal and ram pressures should be a constant across the different regions. The ambient gas flowing towards the star is first heated and compressed by the shock and, while it moves inwards into the bow shock structure, it cools down rapidly. The cool, dense layer of post-shock ambient gas formed in this way at the inner surface of the bow shock defines the boundary separating the stellar-wind material from the ambient gas. Nevertheless, the temperature of this gas is much lower than that of the shocked stellar wind closer to the star, causing a flow of energy from the hot interior of the bow shock toward the cool and dense shell due to thermal conduction. In turn, this produces an inward flow of matter from the dense layer into the region of shocked stellar wind. These different regions can be clearly identified in Fig. 7, where we have plotted the density and temperature distributions of the gas as one moves away from the star in the direction of the apsis of the bow shock.

Fig. 7. A section of the bow shock (the one including the star and the apsis) shows the temperature (top frame) and density (bottom frame) structure for the reference case (A, see Fig. 6). At this time, the structure has reached a stationary configuration. The free-flowing wind encounters a strong shock at a distance slightly in excess of , resulting in a steep increase in temperature to almost  K. However, thermal conduction and expansion of the gas in directions heading away from the apsis of the bow shock cause a reduction in temperature to  K just before the inner boundary of the bow shock, which is recognized by the strong and narrow peak in density. This thin and dense layer is in thermal equilibrium with a temperature of  K. A hotter layer of shocked ambient gas with a peak temperature of  K is present in front of the bow shock.

The velocity pattern close to the bow-shock apsis is shown in projection against a contour map of the density in Fig. 8. The interstellar gas crossing the outer shock front is progressively dragged downstream as it travels towards the inner surface of the bow shock. The velocities along the bow shock in the area covered by Fig. 8 are small, and the shear is only moderate.

Fig. 8. The gas flow inside the bow shock close to the apsis (positioned at the left edge of the diagram) shown for the reference case (A, see Fig. 6) at years (last panel Fig. 6). For clarity, the velocity vectors have been plotted in the regions where the volume density exceeds 3 times the ambient density. The area covered by the plot is 1.34 pc 0.94 pc.

4.1.1. Comparison with the semi-analytical approximation

The steady configuration reached in the reference case allows us to perform a comparison with the semi-analytical predictions described in Sect. 2. An obvious difference is that the numerical simulations show that the thickness of the layer of shocked stellar wind is not negligible. Therefore, the distance of the apsis of the bow shock to the star is actually larger than , as defined in Eq. (1). Nevertheless, we can still retain the definition of as the equilibrium distance between the bow-shock apsis and the star, which gives an appropriate lengthscale of the bow shock. By scaling the parameters to , we can compare basic observable parameters such as the shape of the bow shock, its surface density distribution, and the gas flow velocity along the bow shock. However, in doing so we should bear in mind that the value of is not linked anymore to the defining parameters of the problem by an expression as simple as Eq. (1).

To carry out the comparison, we adopt as a convenient definition of the bow shock position the locus of the computational cells where the density presents a maximum as we move along lines parallel to direction of motion of the star. Likewise, the direction tangent to the bow shock is taken as that defined by the gas velocity vector in the points of highest density. This definition works satisfactorily close to the apsis, where the layer of cool shocked gas is very thin, but becomes less meaningful as we move downstream where the dense layer flares out (Fig. 6). A comparison between the shapes of the bow shock close to the apsis is presented in Fig. 9. The shapes are very similar in both cases, despite of a factor of in scale, which is intentionally hidden in Fig. 9 by scaling the parameters to .

Fig. 9. A comparison of the bow-shock shape close to the apsis between the results obtained from numerical simulations (case A, solid line) and the semi-analytical approximation (dashed line). The lines illustrate the position of maximum density. As discussed in Sect. 2, the predicted shape of the bow shock should be the same in both the instantaneous cooling case and the case of a finite cooling time of the stellar wind, which seems to be confirmed by this figure. Note that, although the shapes are very similar, the scale factor differs by a factor , as pointed out in the text.

The prediction for the velocity pattern of the bow shock resulting from the semi-analytical case where the finite cooling of the stellar wind is taken into account is very different from that of the instantaneous cooling case (Fig. 10). The numerical simulations closely reproduce the behaviour described by the former, as expected from the lack of mixing of the shocked stellar wind with the dense bow shock. Interestingly, this difference in behaviour suggests that it should be possible to determine the degree of mixing between the interstellar gas and the stellar wind in the bow shock observationally by mapping velocities along the bow shock. This could be done by deprojecting measured radial velocities.

Fig. 10. A comparison between the velocity patterns along the bow shock, scaled to the spatial velocity of the star . The "solid" line corresponds to the numerical simulations, the dotted line to the semi-analytical case taking into account the finite cooling of the stellar wind, and the dashed line to the instantaneous cooling case.

The predictions of the surface density pattern also reveal a clear difference between the two scenarios discussed in Sect. 2. However, the comparison is not straightforward, as the structure of the bow shock is somewhat more complex than assumed in Sect. 2. The interstellar gas does not cool down immediately after crossing the outer shock front, but rather forms a hot layer of moderate density that is progressively dragged away by the stream of gas inside the bow shock, as shown in Fig. 8. The fact that the freshly incorporated gas is only gradually moving away from the apsis of the bow shock allows it to stay there for a longer time than predicted, thus increasing the surface density with respect to predictions of the semi-analytical calculations. This effect is illustrated in Fig. 11, in which the results of the simulations exceed by far the values predicted by the semi-analytical cases. The agreement becomes better for low values of if we take into account only the densest parts of the bow shock; here the gas flow closely follows the behaviour predicted by the semi-analytical approximation taking into account the finite cooling of the stellar wind (Fig. 10), but is still far away from the prediction of the instantaneous cooling case.

Fig. 11. A comparison between the density patterns along the bow shock, scaled to . The solid line corresponds to the numerical simulations, where we calculated the column density of the bow shock by adding the computational cells in which the density is at least 1.1 times that of the ambient interstellar gas. The dashed-dotted line is obtained in the same way, but now only taking into account the computational cells where the density exceeds 10 times that of the ambient gas. The dashed line describes the predicted behaviour in the semi-analytical case which takes the finite cooling of the stellar wind into account. The dotted line corresponds to the instantaneous cooling case.

The oscillations and jigsaw appearance of the curves representing the numerical simulations in Fig. 11 are due to two reasons: First, the thickness of the dense layer is only a few computational cells, and the effects of pixellation start to become noticeable. Secondly, the cooling of the gas just outside the dense layer, which results in accretion of the gas, takes place on a very short timescale. As a consequence, local fluctuations occur in the surface density of the dense layer. In general, we should expect the semi-analytical approximation to lie somewhere in between the two curves describing the results of the simulations, namely the one which considers all the material contributing to the surface density, and the one which only takes into account the densest parts. The surface density is actually expected to decrease and eventually drop to zero well away from the apsis of the bow shock (Fig. 11). The reason for this is the expansion of the bow shock in its downstream motion and the defined threshold. We should mention that the surface density of the bow shock is further reduced by the evaporation of dense gas due to thermal conduction. A more precise assessment on this effect is given in Sect. 4.6.

Again, the semi-analytical model taking into account the finite cooling of the stellar wind is found to provide a better reproduction of the numerical simulations, given the significantly lower surface density predicted by the instantaneous cooling case.

4.2. Effects of the stellar-wind strength: cases B and C

The main difference between cases A, B, and C, is the scale of the bow shock (characterized by ) due to the change in wind parameters. The post-shock properties of the stellar wind are also very different but, as the velocity of the star and the ambient density are identical, the strength of the shock exerted on the interstellar medium is the same for the three cases. The differences in the structure of the bow shock thus arise from the rate at which the shocked gas is able to travel a significant distance away from the apsis of the bow shock, considering the difference in the overall size of the structure.

The differences become obvious from Fig. 12. The "weak-wind" case (B) reaches a stable equilibrium configuration after about years; a dense bow shock is not formed. This is due to the rapid flow of the shocked ambient gas away from the apsis, which does not provide time to cool down significantly, as a consequence of the small value of and the strong curvature at the apsis. In the "strong-wind" case (C, lower panels), the time needed for the ambient shocked gas to move over a distance away from the apsis is much longer than the cooling time. As a result, a well-developed dense bow shock appears.

Fig. 12. Changing the stellar-wind strength: Greyscale maps showing the density distribution in the computational plane on a logarithmic scale. The upper panel corresponds to the "weak-wind" case (B), and the series in the lower panel to the "strong-wind" case (C). The areas covered by the simulations are very different, due to the change in the stellar wind parameters which set the characteristic lengthscale . The upper panel corresponds to a size of 0.85 pc 0.42 pc, and the lower ones to 20.74 pc 10.37 pc. Note the absence of a dense bow shock in the upper panel, as a result of the rapid flow of gas away from the apsis. The peak density in the shocked area is 0.45 cm-3 (4.5 times that of the ambient interstellar medium). The cut levels are (black) and  cm-3 (white). The whole structure is stable, with the equilibrium configuration shown here being reached less than years after the stellar wind has been switched-on. In the lower panels (case C), the shocked ambient gas has time to cool down before moving significantly away from the bow shock apsis, giving rise to a highly unstable and dense bow shock. The peak density at the bow shock in the frames shown, corresponding to the interval between and years after the wind switch-on, is approximately 20 cm-3 (cut levels and  cm-3).

In the "strong-wind" case (C) the much more massive dense layer comprised in the bow shock makes the equilibrium configuration as discussed in the reference case very unstable in the regions close to the apsis. Irregularities start to appear in this region approximately years after the stellar wind has been switched-on, and are already well developed at the time of the first frame in Fig. 12. The parameters used in this case are similar to those adopted by Raga et al. (1997), although they used a much higher mass-loss rate:   yr^-1, and a ten times higher density of the ambient medium ( and identical). These authors also find the bow shock to be highly unstable; in their case, the instability is very probably enhanced due to the higher ambient density, as our results in Sect. 4.4 suggest.

With reference to Sect. 2.1, the scenario in which the instabilities develop is similar to the one described by Mac Low & Norman (1993). When the part of the dense shell near the apsis of the bow shock is rippled, the gas flow along the bow shock is modified due to the conservation of the component parallel to the shock front. The flow close to the regions displaced outwards ("peaks") diverges, while it converges in the inwards-displaced regions ("valleys") (see Fig. 1 in Mac Low & Norman 1993). Assuming that, in a first approximation, the unperturbed shock front is perpendicular to the flow of ambient gas approaching it, the ripples introduce an obliquity in the shock that increases the component of the post-shock velocity in the direction of the impinging pre-shocked gas flow. This tends to move both peaks and valleys inwards towards the star, so that the instabilities do not lead to the formation of protrusions outwards from the bow shock.

At this point, it is interesting to note the difference between the overstability appearing in expanding shells around stars at rest and the situation described here. In both cases, the converging gas flow near the valleys causes an increase of the density inside the valleys. In an expanding, decelerating shell this implies that the valleys will experience a smaller deceleration exerted by the ram pressure of the ambient medium than their surroundings. Therefore, the valleys eventually overtake the rest of the shell and turn into peaks. However, bow shocks around runaway stars are stationary, and thus there is no restoring factor to turn the valleys into peaks. The valleys of the perturbed areas of the bow shock thus grow steadily inwards, until they reach regions where the rapid flow of shocked stellar wind disperses them away. On the other hand, as anticipated in Sect. 2.1, the flow of gas parallel to the bow shock tends to move the ripples away from the apsis, propagating them towards the tail of the bow shock. Both aspects are visible in the evolution of the on- and off-axis features appearing in the lower frames of Fig. 12.

4.3. Effects of the velocity of the star: cases D and F

Also the strength of the shock on the ambient gas, primarily determined by the velocity of the runaway star, has a large impact on the bow-shock structure. The consequences of reducing the velocity of the star to 50 km s^-1, as in case D, are demonstrated in the upper panel of Fig. 13. The compression factor is smaller, which should enhance the stability of the dense layer comprised in the bow shock. Indeed, instabilities do not develop, at least not during the timespan ( years after switch-on) covered by our simulations. A thin layer of hot gas is present at the outer edge of the bow shock, a consequence of a long cooling time due to the low post-shock density (rather than to high post-shock temperatures). The low velocity of the runaway star also results in a thicker layer of shocked stellar wind, which now becomes larger than , due to the smaller ram pressure acting on it.

Fig. 13. Changing the space velocity: Greyscale maps showing the density distribution in the computational plane (logarithmic scale). The upper panel corresponds to case D (low space velocity), and the lower panel to case F (high space velocity). The areas covered by the simulation are 10.71 pc  5.36 pc (upper panel) and 3.57 pc 1.78 pc (lower panel). In both cases the structure is stable: in case D, the compression exerted on the bow shock by the external ram pressure is now too small for the instabilities to develop. In case F, the combination of a high post-shock temperature and a small value of prevents the formation of a bow shock. The cut levels are (black), (white) and ,  cm-3 for case D and F respectively.

An increase in space velocity to 150 km s^-1 (case F), as illustrated in the bottom panel of Fig. 13, also leads to stability, although for different reasons. In this case, the absence of a dense bow shock has two causes: the first one is, as in case B ("weak wind"), the decrease in size of the whole structure with respect to the reference case, because of the increase in the confining ram pressure. The second one is the increase in post-shock temperature induced by the greater strength of the shock, and the subsequently longer cooling time. The result is a structure very similar to that of case B, although the shocked stellar wind is much hotter now due to the higher terminal wind velocity: just after the shock on the stellar wind, the temperature reaches a maximum of  K, as compared to the peak temperature of  K in case B.

4.4. Effects of increased ambient density: case E

The tenfold increase of the ambient density (case E) with respect to the reference case decreases the post-shock cooling time by two orders of magnitude; is reduced by only a factor . The cold layer of the bow shock is much denser and thicker now, and instabilities appear very quickly. Fig. 14 shows a well developed bow shock showing the first hints of instability already years after the wind switch-on. The resulting structures quickly propagate downstream, due to the small size of the bow shock (note that in both of the previously studied cases where was smaller compared to the reference case, namely cases B and F, a dense bow shock did not form). Clumpy and sheet-like structures are produced which appear as filaments in the computational plane displayed in the figures. In our two-dimensional simulations, the dense sheets create "bags" of hot gas that cannot flow downstream, causing the pressure inside them to increase and a further distortion of the bow shock. This is most likely an artifact of the imposed cylindrical symmetry: in reality, the sheets are not expected to continuously extend as a function of azimuthal angle, but may rather resemble filaments through which the flow of hot gas could penetrate. A full three-dimensional treatment would probably result in a more regular overall shape of the bow shock than displayed here, although its detailed structure should still include abundant and quickly evolving irregularities similar to those shown in Fig. 14.

Fig. 14. Increasing the ambient density: Greyscale maps showing the evolution of the density distribution in the computational plane (logarithmic scale) in the high ambient density case (E). The area covered by the simulations is 1.69 pc 0.85 pc. The first frame corresponds to years after the onset of the stellar wind; the subsequent frames are spaced by years. The high post-shock density makes the cooling time short and the bow shock becomes very unstable. The evolution of the instability towards the tail of the structure is very fast due to the small size of the bow shock. The cut levels are: (black) and  cm-3 (white).

4.5. Numerical simulations with instantaneous cooling: case G

As anticipated in Sect. 2.1, the instantaneous cooling of both the shocked stellar wind and the shocked ambient medium leads to large distortions of the bow shock shape. These distortions are initially produced by shear inside the bow shock, which in turn is amplified by the distortions themselves.

The evolutionary sequence shown in Fig. 15 illustrates one of the dramatic consequences that these instabilities can have. The first panel depicts a situation in which the distorted bow shock has reversed its curvature near the apsis. In this situation, a larger fraction of the momentum injected by the wind goes into the flow of gas parallel to the shock front, rather than into opposing the momentum transferred to the bow shock by the ambient interstellar gas. The bow shock becomes more "aerodynamic" with respect to the stellar wind, and the ram pressure of the interstellar medium can push it closer to the star. However, as the head of the reversed bow shock moves inwards, the outwards ram pressure exerted by the wind increases as , and eventually halts and reverses the bow shock's motion. Away from the apsis, though, the wings of the bow shock continue to move downstream. The head and the wings of the reversed bow shock thus travel in opposite directions, and finally detach from each other. This can be most clearly seen in the fourth panel of Fig. 15. When this happens, the branches of the bow shock move quickly backwards with respect to the star, with the parts closest to it being pushed away, as seen in the fifth panel in Fig. 15. The "old" bow shock is thus swept back by the ambient gas, and a new bow shock begins to form around the apsis. The process vaguely resembles the disconnection events taking place in cometary tails, although the physics involved in both phenomena is very different. The last two panels simultaneously show the "old" and "new" bow shocks. Instabilities in the new bow shock become obvious a few tens of thousand years later, and a new cycle, comparable to the one described here, starts.

Fig. 15. Greyscale maps showing the density distribution in the computational plane for the instantaneous cooling case (G) (on logarithmic scale). The imposition of a constant temperature across the computational plane suppresses the thick layer of shocked stellar wind, and puts the wind in direct contact with the bow shock. The incorporation of the stellar wind into the bow shock makes the bow shock very unstable and produces important large scale departures from the usual shape. The time sequence shown here covers years, and starts years after the onset of the stellar wind. The area covered by this simulation is 5.36 pc  2.68 pc, like in Fig. 6 (case A). The cut levels are: (black) and  cm-3 (white).

The increased importance of shear with respect to the reference case (A) can be appreciated by comparing Figs. 16 and 8. The vector diagram superimposed on the isodensity contours corresponds to the third frame in Fig. 15. Although the flow pattern is in general similar to that in Fig. 8 (with the exception of a slight motion of the freshly shocked ambient gas towards the apsis as a result of the reversed curvature of the bow shock), the velocities in the star-facing side of the bow shock are more than twice as large as obtained in the reference case. Moreover, the gradient of the transverse velocity across the bow shock is greatly enhanced by the decrease in thickness due to the instantaneous cooling of the ambient gas.

Fig. 16. The pattern of velocities in the bow shock at an age of years for case G (instantaneous cooling) is superimposed on a contour map of the density (on a logarithmic scale). The area plotted here is an enlargement of the third panel from the top in Fig. 15, covering 1.00 pc 0.67 pc. For clarity, arrows have been plotted only on those areas where the density is higher than three times the ambient gas density. The shear between the material coming from the ambient medium and the stellar wind is clearly seen within the bow shock.

4.6. Numerical simulations without thermal conduction: case H

Our last simulation differs from the reference case in that thermal conduction is not considered here. The physics included in this model should be comparable to that considered by Raga et al. (1997). Some immediate consequences are: the disappearance of the layer of intermediate density and temperature separating the bow shock from the shocked stellar wind; the temperature increase of the shocked wind region; and a faster downstream flow of shocked stellar wind. The impact of the suppression of thermal conduction can be demonstrated by comparing the behaviour of the density and temperature as a function of distance from the star for the case with (A, drawn lines) and without thermal conduction (H, dashed lines) (Fig. 17). As will be shown later, the bow shock is unstable in case H. However, the development of the first instabilities is rather slow, and it is possible to find a nearly stationary configuration in which the bow shock has already fully developed, but instabilities still have not appeared. The dashed curves in Fig. 17 correspond to an age of the bow shock of years after the wind has been switched-on. The now disturbed near-equilibrium configuration was already present at an age of years.

Fig. 17. Behaviour of the temperature (top panel) and density (bottom panel) in a section of the bow shock containing the star and the apsis. To demonstrate the importance of the inclusion of thermal conduction in the simulations, we compare the results obtained for the reference case (A) and for case H where thermal conduction is not included. The solid curves are the same as in Fig. 7 (case A), and the dashed curves correspond to a stage in the evolution of case H previous to the development of instabilities in the bow shock.

The differences between the solid and dashed lines in Fig. 17 can be explained in qualitative terms. The most obvious difference is the smaller extent of the whole structure in case H. The bow shock is positioned at a distance of 1.5  from the star, instead of 2  as in the case with thermal conduction (in both cases, the definition of given by Eq. (1) is used). Fig. 17 clearly shows that, when thermal conduction is taken into account, the additional 0.5  in thickness corresponds to the thickness of the intermediate density layer. The velocities of the gas in this layer directed away from the apsis are much lower than close to the region where the freely flowing stellar wind shocks. Most of the mass in the layer of intermediate density and temperature originates from the evaporation of the inner surface of the bow shock due to thermal conduction. This is supported by the fact that the dense layer comprised in the bow shock becomes significantly thinner when conduction is taken into account. Since the rate at which the hot gas flows away from the bow shock axis is roughly given by the pressure gradient divided by the density, the gas in this intermediate density region moves downstream much more slowly than the gas closer to the star. This maintains a thick layer between the shocked stellar wind and the bow shock even though the evaporation rate of the bow shock is moderate. An additional consequence of the inclusion of thermal conduction is the flow of internal energy from the inner, low-density regions towards the intermediate density layer where the rate of energy loss by radiation (proportional to the density squared) is much higher. This produces a permanent pressure imbalance between the inner and the outer parts of the region interior to the bow shock. As a consequence, the shock on the stellar wind is allowed to move away from the star up to a distance exceeding . When this flow of energy by thermal conduction is suppressed, however, the stellar wind finds the shock at a distance of almost exactly .

The higher surface density of the bow shock due to the absence of evaporation facilitates the appearance of instabilities. Moreover, when thermal conduction is taken into account, the layer of intermediate density, temperature, and velocity provides a fairly smooth transition zone between the physical conditions in the bow shock and those in the regions lying just outside the shock on the stellar wind. With the suppression of this layer, the strong discontinuity in density and velocity at the inner surface of the bow shock causes the amplification of irregularities in the dense layer. This leads to the development of structures very similar to those appearing in some of the cases discussed so far. The evolution of the first of such major instabilities is displayed in Fig. 18.

Fig. 18. Evolution of the density in the computational plane for case H. All the parameters are the same as in the reference case (A), but thermal conduction has been suppressed. The scale is the same as in Fig. 6. The sequence shows the development of the first important instability, starting years after the wind has been switched-on. The panels are spaced by time intervals of years. The cut levels are: (black) and  cm-3 (white).

© European Southern Observatory (ESO) 1998

Online publication: September 8, 1998
helpdesk.link@springer.de