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Astron. Astrophys. 331, 949-958 (1998)

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4. Analysis and results

4.1. Galactic rotation

First, we check for differential Galactic angular rotation by plotting separately the Galactic longitudinal and latitudinal components of proper motion after removal of the basic Solar motion, versus longitude (see Fig. 1). Except for the Solar correction (especially for nearby stars), the positions of the data-points are only weakly dependent on distance. This is useful, since the bulk of the distance estimates are photometric, with relatively large errors (typically [FORMULA] 30%). In Fig. 1, we have superposed the expected Galactic angular rotation (see previous section) in l for [FORMULA] (the dependence on b for typical small values of b is very weak) and for three distances, which span the distance range for the bulk of the data. Note again the weak dependence on the distance. The rotation effects in b also depend on b, but these are negligeably small and thus not shown.

[FIGURE] Fig. 1. Net proper motion [FORMULA] versus Galactic longitude for each of the l and b components. Circles refer to O stars, triangles to WR. Filled symbols are for [FORMULA], open symbols [FORMULA]. The zero line is drawn in for the b component. Flat rotation curves ([FORMULA]) are shown for the l component: solid, dotted, and dashed curves are for [FORMULA], respectively. Stars that deviate by at least 10 [FORMULA] in either l or b are identified.

Fig. 1 shows that the O stars and the WR stars are fairly well mixed, with two effects emerging: (1) A global double-wave trend in [FORMULA] versus l, with minima (zero actually) in absolute value at [FORMULA] and [FORMULA], as expected from the rotation model. The perfect double sine-wave for zero distance becomes increasingly distorted with larger distance, with maxima in absolute value spreading out gradually below [FORMULA] and above [FORMULA]. (2) The weak dependence of the model curves on distance allows one to easily locate stars that deviate significantly from the trend of Galactic rotation (but conversely makes proper motions essentially useless for rotation parallaxes, given the observational uncertainties). In Fig. 1 and Table 2 we identify those stars for which at least one of the componenets of proper motion deviates from the curve, appropriate for the distance, by more than [FORMULA]. These stars have very significant peculiar motion relative to the general Galactic rotation curve. More modestly deviating stars can be easily identified in Table 1.


Table 2. Program stars with the most significant deviation ([FORMULA])in [FORMULA] or [FORMULA] from Galactic rotation

Fig. 1 also reveals ranges in Galactic longitude, where coherent deviations appear to occur, in particular in [FORMULA] for [FORMULA] and in [FORMULA] for [FORMULA]. These may be due to large-scale deviations from the assumed circular rotation, towards the Carina region and the anticenter, respectively.

4.2. Peculiar tangential motions

Significant peculiar proper motion does not necessarily imply fast absolute tangential motion in [FORMULA]. To obtain tangential velocities, we have no choice but to take published photometric distances, since geometric distances are still not precise enough for most OB and WR stars. The peculiar (total) tangential motions (see Table 1 and Fig. 2) were calculated using

[FIGURE] Fig. 2. Peculiar tangential motion of all O and WR stars from Table 1, versus Galactic longitude. Symbols are as in Fig. 1. The dashed horizontal line refers to the base limit of 42 km/s for selecting runaways based on peculiar tangential velocities. Stars with [FORMULA] are identified.


Errors in [FORMULA], [FORMULA], are calculated using error propagation in all the observed quantities, assumed to be independent, and assumimg a 30% uncertainty in the distances, corresponding to a 0.7 mag rms error in distance modulus, quite reasonable given the cosmic scatter in [FORMULA] for single OB and WR stars. This technique of error calculation works well for relatively small errors; for large errors - as prevail in some cases - this will tend to overestimate the lower error bound and underestimate the upper error bound. However, for such non-linear functions, there is no standard way to correct for this. In any case, this effect will be partly compensated for by the increase in [FORMULA] introduced by the same error bias. By not correcting for either of these, we tend to err on the conservative side.

In order to make a meaningful selection of runaways based on tangential motion alone, we adopt the criterion


The base value of 42 km s-1 is consistent with a selection in [FORMULA] alone above 30 [FORMULA] (e.g. Cruz-Gonzalez et al. 1974), i.e. allowing for a factor [FORMULA] for the two components of velocity that go into [FORMULA].

This criterion leads to a selection of 19 stars, 6 WR and 13 O-type (see Table 3 for a detailed list). Note that only eight of these have very significant peculiar proper motion according to Table 2; because of large distances, 11 new stars have appeared in Table 3. Within the small numbers, there is no reason to assume that WR and O stars differ at all in their frequency of runaways. In any case, the pre-selection of O stars with high [FORMULA] may account for the slightly higher number of O runaways. Four stars in the Carina region have large [FORMULA] ; some of these may be spurious, being part of a global velocity perturbation in this localized region of the Galaxy. The average runaway frequency in our overall sample is 14%. This seems low, but is compatible with the observed runaway frequency of O stars from RVs, allowing for our somewhat more severe selection here to allow for a one- [FORMULA] error. Following Stone (1991), extrapolating this distribution tail to a Gaussian would lead to significantly higher frequency for the so-called high-velocity massive stars.


Table 3. Stars with [FORMULA], in order of HIP number.

Some stars in Table 3 deserve special mention. While several O type RV runaways appear in our peculiar tangential motion study, some others are missing (e.g. µ Col); this is not surprising, however, since high peculiar motion need not always appear in both the radial and tangential components simultaneously (see below). Among the WR stars, neglecting the Carina region, there is a preference for WN8 stars to be runaway, based on peculiar motion. This is reinforced by the known very high [FORMULA] of the runaway WN8 star WR 124 (Moffat et al. 1982), whose [FORMULA] is not well determined here, and the avoidance of clusters by the WN8 subclass (Moffat 1989). Other WN8 stars could have high [FORMULA], without having been observed as such. Even the strange SB1 binary WR 148 with WN7 subclass (WN8h according to the revised classification of Smith et al. 1996) may be related to these objects. The only other non-Carina WR runaway in Table 3 is WR71 (WN6, a previously suspected runaway: Isserstedt et al. 1983).

We now compare the peculiar tangential and radial motions of the O stars in Fig. 3, in which we have reduced the total tangential component from two to one dimension through division by [FORMULA]. In preparation for this figure, we have updated many of the Cruz-Gonzalez et al.(1974) RVs and recalculated the Solar correction and Galactic rotation in a way that is consistent with our above treatment of the tangential data. In particular, we write the observed component of heliocentric radial velocity:


in which


We have adopted [FORMULA] uncertainty level in the observed RVs for calculating the error bars in Fig. 3. All the O-star RV data are summarized in Table 1.

[FIGURE] Fig. 3. Absolute value of peculiar radial velocity versus peculiar tangential velocity normalized to one component, for the program OB stars. Symbols are as in Fig. 1, with additional crosses indicating MXRBs. The horizontal dashed line indicates the Cruz-Gonzalez et al. (1974) cutoff in [FORMULA]. The dashed arcs refer to lines of constant [FORMULA]. Some of the most extreme stars are identified.

Fig. 3 shows that some dozen O-stars now have peculiar RVs that are below the Cruz-Gonzalez et al. (1974) limit to be classified as RV runaways (30 km s-1). Also, the majority (but not all) of the MXRBs lie below the limit, since they were not selected for high [FORMULA]. However, as expected, many stars with low [FORMULA] have high [FORMULA], and vice versa, within the prescribed RV cutoff of the Cruz-Gonzalez et al. (1974) RV data. The overall impression here is that, within the errors, both radial and tangential data give consistent results, leading to a more-or-less circular distribution centred on (0,0) in the [FORMULA] plane. Note that some runaways have both components large. This situation reassures us that obligatorally taking only tangential peculiar motions for the WR stars is a statistically sound procedure for recognizing some WR runaways.

4.3. Kinematic ages

Kinematic ages ([FORMULA]) are important to constrain the origin of the runaways. They can be obtained in principle for individual stars from the distance to their origin divided by the runaway (i.e. peculiar space) velocity. For the SN binary scenario, one should have [FORMULA], where [FORMULA] is the current nuclear age of the observed star (Stone 1991). For the cluster ejection scenario, [FORMULA]. Two methods can be used to trace the place of origin: from the natal cluster/association or from the Galactic plane. If one cannot locate unambiguously a natal cluster or association in the Galactic plane, this technique must be abandoned. As noted in the Introduction, we will not deal with this method here; we will concentrate on the latter method.

Assuming all massive stars are born in or near the Galactic plane (and in clusters/associations), one can obtain [FORMULA] simply from the current separation from the plane ([FORMULA]) and the current velocity perpendicular to and away from the plane (massive runaways are too young to have peaked and be returning to the plane: Gies & Bolton 1986). The latter can be written


For the majority of our objects, b is small, so Z depends mainly on [FORMULA]. In any case, systemic RVs are generally not known for WR stars; hence we will simply take as adequate approximation for all stars in this context


Following Scheffler & Elsässer (1987), one has the equation of motion perpendicular to the Galactic plane for any given star, valid out to [FORMULA] 0.5 kpc:


with [FORMULA], in which [FORMULA] is the restoring force per unit mass towards the plane. Scheffler & Elsässer give [FORMULA]. More recent studies (Kuijken & Gilmore 1989a,b; cf. also Bhattacharya et al. 1992) show that such a linear acceleration-distance law is limited to [FORMULA] 0.1 kpc only (where the disk potential dominates), beyond which the slope gradually flattens to a value dominated by the halo for [FORMULA] 0.7 kpc. However, as we shall see below, our analysis will be limited to stars below [FORMULA] 0.35 kpc, for which a value of [FORMULA] 0.06 km s-1 pc [FORMULA] s-1 is most representative. The scatter in the data does not justify going beyond a linear force law; thus we adopt this value from here on. Integration leads to


The sinusoidal oscillation about the plane occurs with period [FORMULA] 100 Myr and maximum separation [FORMULA], where [FORMULA] is the initial perpendicular velocity in the plane at [FORMULA]. Since we do not know [FORMULA] for each star, we eliminate a by taking the ratio


In order to apply the above scenario to our sample, it is not useful to look at individual runaway stars, in view of the large errors both in tangential velocity and in distance. However, grouping also poses a problem, since different stars will have different nuclear ages. O stars have lifetimes [FORMULA] 2-10 Myr, compared to [FORMULA] 2-7 Myr for WR stars at [FORMULA] (Maeder & Conti 1994). The upper limit for the latter decreases for lower metallicity. In the Solar environment of the Galaxy, allowing for the strong bias of the IMF towards lower-mass stars (as reflected by the preponderance of late-type O stars in Table 1, which are also the progenitors of WR stars) a typical mean value lies in the range [FORMULA] 8-9 Myr for the average O star and [FORMULA] 5-6 Myr for the average WR star. In a random sample, [FORMULA] for O stars and [FORMULA] for WR stars (since the WR He-burning phase is short compared to the progenitor main sequence phase). Thus, a global value [FORMULA] 5 Myr probably reflects a good overall average for the (statistically more viable) combined O and WR stars in our sample.

We therefore adopt [FORMULA] in the above equation, and make a linear fit [FORMULA]. [We also assume equal weights; otherwise the result will be fortuitously dominated by a small number of data points of very high weight.] Avoiding the 11 stars beyond [FORMULA] pc ([FORMULA] 5 scale heights from the Galactic plane), where errors in distance are large, we find [FORMULA] at the 95% level (this fit is shown in Fig. 4). The positive value of [FORMULA] is compatible with stars leaving the Galactic plane, as expected. Taken at face value, this leads to [FORMULA] Myr. This appears to be compatible with the cluster ejection scenario, i.e. [FORMULA] at the [FORMULA] level.

[FIGURE] Fig. 4. Velocity perpendicular to the Galactic plane [FORMULA] versus separation from the Galactic plane [FORMULA], in normal and zoomed scales. Symbols are as in Fig. 1. A linear unweighted fit is shown by the straight line [FORMULA].

4.4. Correlation with bow shocks

If [FORMULA] is supersonic with respect to the ISM (e.g. [FORMULA], [FORMULA]: Scheffler & Elsässer 1987), a bow shock (region of enhanced emission) will occur in the direction of motion of the (windy O or WR) star (e.g. Van Buren 1993), assuming a relatively uniform surrounding ISM and neglecting complexities of multi-wind interactions around WR stars. In order to best explore whether the motion of the star is compatible with the formation of such bow shocks, we use only the tangential velocity vector, which is a projection of the space velocity on the sky. The radial component of peculiar velocity will not lead to a clear, unambiguous bow shock arc.

We have thus scanned the literature for arc-shaped bow shocks (enhanced emission in the direction of peculiar tangential motion) around all of our program O and WR stars. About 85% of the OB stars from our list (Table 1) were previously searched for the presence of bow shock-like emissivity on 60µm IRAS maps by Van Buren et al. (1995). The fraction of OB stars from our list that manifest bow shock-related phenomena is [FORMULA] (WR stars have not yet been searched systemetically for bow-shocks), in complete agreement with the estimation of Van Buren et al. Only 5 of them (HD 34078,41997,66811,210839 and HDE329905) can be recognized as well-established runaways (cf. Tables 2 and 3). Interesting cases (see Table 4) among the OB and WR stars are:

  1. HD 189957 is moving toward a bright knot (unresolved bow shock structure?) seen on [FORMULA] IRAS image (Van Buren et al. 1995).
  2. The vector of the peculiar motion of HD 210839 is directed [FORMULA] along the axis of symmetry of the clearly seen (Van Buren et al. 1995) bow shock structure.
  3. HD 77581 (Vela X-1). This is an especially interesting case (Kaper et al. 1997). With the new Hipparcos data (see Fig. 5a), we now find that the star is heading right towards the emissivity maximum of the bow shock.
  4. WR 6, moving toward the brightest part of the surrounding ring nebula (cf. the maps from Arnal & Cappa 1996; Van Buren et al. 1995; also Fig. 5b here). One is inclined to suggest that this relative brightening is directly related to the bow-shock phenomenon.
  5. The same brightening of the surrounding IS media in the direction of the stellar tangential motion is seen for WR 16 and WR 40 (Marston 1995), WR 23 and WR 55 (Chu et al. 1983; Marston et al. 1994), WR 133 (Marston 1996), WR 134 and WR 136 (Miller & Chu 1993; note that for WR 136 the IR maximum is somewhat displaced from the optical maximum - cf. the map from Marston 1996)
  6. A spectacular system of shells (Heckathorn et al. 1982; Miller & Chu 1993) is stretched to one side of the tangential motion vector of WR 128.


Table 4. OB and WR stars moving towards recognized/potential bow shock structures.

[FIGURE] Fig. 5. Proper motion vectors superposed on ISM emission maps of the surroundings of two sample stars a  HD 77581 (Vela X-1) and b  HD 50896 (EZ CMa = WR6). The maps are from Kaper et al. (1997) and Van Buren et al. (1995), respectively.

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Online publication: March 3, 1998