3. The alignment effect
3.1. Alignment of galactic nebulae with the galactic plane
Fig. 1 illustrates the distribution of the nebular position angles . It is immediately clear that the distribution is apparently not uniform and has a strong peak near , indicating an alignment with the galactic plane. Considering the two bins centred on 90o, the expected mean is 12/9 and the observed excess a 4 deviation. Another smaller peak is seen around () indicating that some objects are oriented perpendicular to the galactic plane 2. These tendencies are more clearly seen in Fig. 2 which illustrates the distribution of , the position angle relative to the plane of the Galaxy. The median of , noted , is equal to ; it is significantly different from 45o, the value expected from a uniform distribution.
For evaluating the significance of a possible alignment effect, it is important to recall that the observed position angle refers to the object long axis projected onto the plane of the sky. If nebular axes are randomly oriented in the three dimensional (3D) space (which is equivalent to uniformly distributing points on the surface of a sphere), all axes characterized by a given are located on a great circle, such that the projected position angles are equally likely 3. Hence, randomly oriented nebular axes have uniform distributions in and .
Statistical tests may then be used to see if the position angles are drawn from a uniform distribution, or not. Note that statistics for circular data (e.g. Fisher 1993) are needed when analyzing the angles , while usual linear statistics are adequate for . We therefore apply the well-known Kolmogorov-Smirnov (K-S) test to the distribution of . The K-S statistic is computed to be D = 0.489 for n = 12 objects. This corresponds to a probability that the observed distribution of is drawn from a uniform distribution. This suggests a significant deviation from uniformity.
However, such a test does not identify the nature of the deviation, and does not provide an estimate of the significance of the possible alignment. A rather simple measure of the angle concentration near is the median of the distribution. We therefore adopt this quantity as a statistic and evaluate the probability to obtain by chance only a value of the median , assuming the angles uniformly distributed. For this, we randomly generate 105 sets of 12 angles . The median is computed for each set, and its distribution built from the 105 realizations. From this distribution, we derive a probability that the observed value of the median is due to chance only, which indicates a significant alignment effect.
In order to see if this result is stable against uncertainties, we now randomly choose for each object its position angle in the interval assuming uniform deviates. For the 12 objects of the sample, a new median is then computed. By repeating this process 104 times, we derive the distribution of , which appears to be roughly normal with a mean = and a standard deviation . The median of the observed sample therefore lies within the 3 confidence interval [, ] and probability that it is due to chance only is computed to be between and . These values still indicates a significant deviation from uniformity.
Finally, if we simply discard from the sample the four nebulae with more uncertain measurements, i.e. M1-67, G79.29+0.46, S308 and WRA751 (cf. Sect. 2), the median is computed to be = 15.0 for n = 8 objects, a value which has a probability to be due to chance only.
We may therefore safely conclude that the major axes of the nebulae around LBV and WR stars are not randomly oriented, and that a significant tendency to alignment with the galactic plane is detected, even within our rather small sample.
Given this overall alignment, it is interesting to note that all but one misaligned nebulae have their axis nearly perpendicular to the galactic plane (Fig. 2). Although this weak tendency could be real, one should remark that two of these objects, He3-519 and RCW58, are both elliptical nebulae older and fainter than e.g. the AG Car nebula and therefore possibly affected by errors on their true morphological type (cf. Sect. 2), while the classification as a LBV-type nebula of the third object, G25.5+0.2, is still hypothetical (Subrahmanyan et al. 1993, Hutsemékers 1997).
3.2. Alignment of nebulae with the interstellar polarization in the Galaxy
Many distant stars are polarized in the visible due to dichroic absorption by aligned interstellar dust grains. The direction of this interstellar polarization is thought to follow the direction of the galactic magnetic field (e.g. Mathewson & Ford 1970, Axon & Ellis 1976). Since at the low latitudes where our objects lie, the magnetic field is essentially parallel to the galactic plane, the correlation found in the previous section will necessarily repeat itself when comparing the nebular position angles to the polarization position angles of neighbouring objects. However, there is some variation in the orientation of the interstellar polarization, and the nebular axes might be better aligned with the local interstellar polarization than with the galactic plane. This is particularly interesting to investigate for those misaligned objects.
The position angle of each nebula is therefore compared to the polarization position angles of the nearest neighbouring stars on the celestial sphere. The polarization data are taken from the Axon & Ellis (1976) compilation, the data related to the nebula central stars themselves being discarded. Since the considered LBV and WR nebulae are rather distant objects (Table 1), only distant stars ( pc) are accounted for, i.e. those stars lying beyond the local volume where most interstellar polarization is imprinted. Also, only sufficiently polarized objects are considered i.e. those with a polarization degree . Then if refers to the interstellar polarization position angle of a given star, we evaluate the angle difference
for each nebula and its ten polarized nearest neighbours. In general, the ten neighbouring stars are within a few degrees from the objects.
Fig. 3 illustrates the distribution of . It appears bimodal and indicates an overall alignment, as expected from the distribution of the nebular position angles. Also illustrated is the distribution of computed after rotating by 90o the axes of the three nebulae which are nearly perpendicular to the galactic plane. In this case the peak near = 0o is definitely stronger. This indicates that misaligned objects are not better aligned with the interstellar polarization than with the galactic plane. This is also illustrated in Fig. 4 where nebulae with various orientations are represented. Only the nebula around WRA751, the axis of which is not parallel nor perpendicular to the galactic plane, is possibly better aligned with the neighbouring interstellar polarization vectors. However, it should be clear that we cannot conclude from these results that the nebulae which are perpendicular to the interstellar polarization vectors are necessarily perpendicular to the direction of the interstellar magnetic field in their direct vicinity. Indeed, the interstellar polarization from the compilation of Axon & Ellis (1976) is measured from stars which are generally closer to us or closer to the galactic plane than the considered nebulae.
3.3. Alignment of nebulae with the magnetic field in the LMC
A completely different, external, point of view is provided from the LMC which is seen at medium inclination. In this case, the nebular axes may be directly compared to the magnetic field lines.
Using radio polarization measurements, Klein et al. (1993) have obtained a well-sampled map of the LMC magnetic field (which they found in overall good agreement with the optical polarization). The data were kindly provided by the authors, and are illustrated in Fig. 5 together with the long axes of the four considered nebulae (Table 2). We can see that the nebular axes closely follow the LMC magnetic field. We then evaluate the mean direction of the LMC magnetic field in the vicinity of each nebula, , by vectorially averaging the five nearest data points (i.e. roughly within half a degree around the objects). The difference is then computed using
and given in Table 2. All the values are small, clearly indicating that LBV-type nebulae, as well as the nebula around SN1987A, are aligned with the LMC magnetic field.
The sample of LBV-type nebulae from the LMC is unfortunately too small to derive a useful statistical significance for the observed alignment. However, in light of our previous results obtained for galactic nebulae (Sect. 3.1), this alignment with the LMC magnetic field clearly provides additional evidence for a correlation between LBV-type nebula orientations and galactic magnetic fields.
Note that from our results in the Galaxy, we would also expect the nebular axes to be aligned with the plane of the LMC. This may be verified for the SN1987A nebula since its inclination with respect to the line of sight has been estimated: (Burrows et al. 1995, Meaburn et al. 1995). Assuming the LMC inclined at with a line of nodes at a position angle (Westerlund 1990), we find that the long axis of the SN1987A nebula is tilted by from the LMC plane. In this case, the east side of the LMC being closer to us implies that the southern part of the SN1987A nebula is nearer us, as observed (Burrows et al. 1995). This result indicates that the long axis of the SN1987A nebula is reasonably aligned with the LMC plane, though the uncertainties of the involved quantities are large.
© European Southern Observatory (ESO) 1999
Online publication: March 10, 1999