2. Neutral hydrogen in the region of the Cepheus Bubble
The HI data were taken from the Leiden/Dwingeloo HI survey (Hartmann & Burton 1997). The angular and velocity resolution of the spectra are and 1.03 kms-1, respectively, covering the velocity range kms-1. The observed positions are distributed on a regular grid with steps of both in l and b. This grid provides a spatial resolution of up to 3 times higher than obtained by SVSG, although the Leiden/Dwingeloo sampling is somewhat coarser than that of the HI data set of Patel et al. (1998).
2.1. HI distribution in the channel maps
In the region to , to HI emission is dominated by a narrow galactic plane layer of at any radial velocity between -110 kms-1 and kms-1. Although HI emission is detectable over a large range in both b and , at the most prominent emission features appear in two discrete velocity intervals at  kms-1 and kms-1. The HI structure at kms-1 belongs to the extended Outer Arm high velocity cloud (Wakker and van Woerden 1991), whose study is beyond the scope of this paper.
We focus on the kms-1 velocity range, and search for hydrogen structures possibly associated with the infrared ring found in the IRAS maps. We display in Fig. 1 a series of HI maps by integrating the spectra over 4 kms-1 velocity intervals between -38 kms-1 and +10 kms-1. The maps show that the bulk of HI emission arises from the  kms-1 velocity range. Since the infrared ring is expected to correlate with the projection of the most prominent HI features on the sky, we plotted in Fig. 2 the hydrogen emission integrated between -14 kms-1 and +2 kms-1. The figure shows a well-defined closed ring around the low emission region to to , a result also published by SVSG and by Patel et al. (1998). A comparison of Fig. 2 with the IRAS map of the Cepheus Bubble will be the topic of Sect. 3.
An inspection of the HI maps of Fig. 1 reveals loop structures in several velocity regimes. The most prominent ring structure, with sharp inner edge in the direction of the Cepheus Bubble, appears in the kms-1 range. A similar ring-like pattern is clearly recognizable at more negative velocities as well. Between -26 kms-1 and -14 kms-1, Fig. 1 shows a low emission area (`hole'), bounded by stronger HI emission regions, and the whole structure extends over about parallel to the galactic plane. Although the centre of this higher negative velocity hole () is at slightly higher galactic latitude than that of the ring in Fig. 2, the transition between these two loop structures is continuous in the velocity space (Fig. 1), providing a strong evidence for their physical link. At even higher negative velocities ( kms-1) the upper boundary of the hole is fragmented, and the loop structure is no longer visible. The fragments are, however, still recognizable at more negative velocities, roughly following the trend that fragments of higher negative radial velocities appear closer to the centre of the former ring. At kms-1 even these fragments disappear. The interpretation of these results in terms of an expanding shell is given in Sect. 2.3.
So far we identified the significant cloud complexes related to the Cepheus Bubble by visual inspection of the HI maps. This method, however, is not automatic, can be somewhat subjective, and works less efficiently in regions where the resolution of the kinematic distances, resulting from the differential rotation of the Galaxy, is poor (like in the Cepheus region which is close to the tangent point). Visual inspection may also fail to identify structures which extend over very large radial velocity ranges due to internal and/or peculiar motions. In the next subsection we use a multivariate statistical method for identifying the main structures in the data cube representing the Cepheus Bubble, free from subjective bias.
2.2. Multivariate analysis of the HI channel maps
The positional and velocity data of the neutral hydrogen form a data cube . We assume that the HI emission is optically thin, and the observed channel maps are weighted superpositions of k components which represent the main hydrogen cloud complexes, i.e.:
where , and are the measured intensities in the channel maps, the weighting coefficients, and the contributions of the components, respectively, and n is the number of the channel maps. Normally, we may assume that . Description of the observed variables by linear combination of hidden variables (factors) is a standard procedure of multivariate statistics, called factor analysis.
We make the assumption that the correlation between the components is negligible. This assertion enables us to apply the principal components analysis (PCA) for finding the number of significant components (factors) and their numerical values, using standard techniques implemented in statistical software packages. The PCA represents the observed variables ( values in our case) as linear combinations of non-correlated background variables (principal components). We note that although PCA is often used for finding the factors, there are many other techniques for obtaining a factor model. PCA and factor analysis represent two different procedures, strongly related but not identical.
PCA obtains the factors by solving the eigenvalue equation of a matrix built up from the correlations of the observed quantities. The components of the obtained eigenvectors serve as coefficients of the factors (the significant principal components in this technique) in the equation given above. The eigenvalues give some hint for the `importance' of the corresponding components. The and ratios indicate what percentage of the variance of the observed variables can be explained by the i-th principal component (explained percentage) and by the linear combination of the first i principal components (cumulative percentage). For further details of this technique see Murtagh and Heck (1987). PCA is a standard procedure of many statistical software packages. Balázs, Tóth and Kun (1989) used this technique to separate the galactic background from the zodiacal light.
We analyzed a matrix built up from the mutual correlations between the HI channel values. We used altogether 43 channels in the [-38,+6] kms-1 region, corresponding to a sampling frequency of about 1 kms-1. Table 1 shows the eigenvalues and the explained percentages of the principal components, as well as their cumulative percentages. We found that the 6 major principal components having eigenvalues larger than 1.0 can describe 95.4% of the variance of the observed HI channel maps. We kept these principal components for getting the factors describing the observed HI distribution. The results demonstrate that the 43 channel maps can be represented by only 6 maps created by the PCA, while the remaining 37 maps carry mainly redundant information, and may be dropped from the further data analysis. However, the PCA does not guarantee that a factor map contains only physically related objects: if two independent HI clouds appear exactly in the same velocity range, they would be included in the same factor.
Fig. 3 presents maps of the 6 factor values, and Fig. 4 shows the weighting coefficients for these 6 factors as a function of radial velocity. Fig. 4 reveals that each factor has a well-defined radial velocity interval where it is dominant and where the contributions of the others are almost negligible (these velocity ranges are also given in Table 1).
In the following we compare the results of the PCA with those derived in the previous subsection. The well defined loop structure in the  kms-1 velocity range (Fig. 1) can easily be identified with Factor 2, by both their patterns and their velocity ranges. Factor 1, although dominated by a very strong feature at , contains also the loop visible in the maps of Fig. 1 between -26 kms-1 and -14 kms-1. Fragments of this loop towards the centre can be associated with Factor 3. On the more positive velocity side, Factor 5 is dominated by a concentration towards the interior of the ring, although at this velocity significant foreground contamination due to local HI can be expected. Factor 4, which is important only at more positive velocities, and the weak Factor 6 apparently do not carry substantial information on the bubble.
We found that all prominent emission structures, recognized in the HI maps of Fig. 1 (Sect. 2.1), were identified by the PCA as well, and the results of the multivariate analysis could be converted into useful physical information. This approach offers an objective way to get an unbiased estimate of the characteristic radial velocities of the most significant structures, which is not given by the visual inspection. The method also shows how to reduce the size of our data cube without losing too much information, and therefore it could be used for automatic analysis of larger data sets, too.
2.3. HI distribution in the position-velocity space
The existence of an extended depression in the hydrogen emission at high negative velocities is also evident from Fig. 5, a position-velocity diagram taken perpendicularly to the galactic plane at . The figure shows a large hole between and , 37 and -4 kms-1, and suggests that most HI features in this velocity range (including the fragments at kms-1) belong to a large interstellar structure, forming a closed loop around the hole. Similar cuts at and reveal similar structures but with a somewhat smaller diameter, justifying our choice of as the main cross section of the bubble.
We propose to interpret the observed spatial-velocity distribution as radial expansion of a 3-dimensional shell. According to this interpretation, the regular ring patterns in the kms-1 velocity range (Fig. 1) correspond to different cross sections of the shell, while the blueshifted fragments at kms-1 represent its approaching part. The shift of the fragments toward the hole's centre at larger negative radial velocities is consistent with the expansion model, which predicts that at blueshifted velocities an expanding shell appears in the form of concentric rings of apparent radii decreasing with radial velocity. The receding wall of the expanding shell, however, is not easy to identify. Fig. 5 suggests that the receding side is seen at kms-1, but this emission could be seriously contaminated by HI emission from the solar neighbourhood expected at kms-1. The map of Factor 5 (Fig. 3), however, which contains emission having central velocity of kms-1, reveals a mass concentration towards the interior of the bubble instead of the picture of a homogeneous foreground emission. This result may indicate that the shell is closed at kms-1.
The apparent centre of the shell lies at kms-1 and (Fig. 5). This velocity is significantly more negative than the kms-1 derived in the previous subsection as the characteristic velocity of the best defined cross section of the bubble. Fig. 5, however, shows that the emission of the shell along the velocity axis is asymmetric, concentrating towards more positive radial velocities (the measured brightness temperatures are approximately 15 K and 60 K in the directions of the approaching and receding sides of the shell, respectively). This asymmetric mass distribution may indicate that the Bubble was formed at the near side of a large cloud complex. Assuming optically thin emission, the 1:4 brightness temperature ratio between the approaching and receding sides is transformed into the same ratio for the corresponding column densities. In order to determine the true central velocity of the expansion, we weighted the radial velocities of the approaching ( kms-1) and receding ( kms-1) sides by their corresponding column densities. The result is kms-1, close to the kms-1 yielded by the PCA in the previous subsection, but in clear contradiction with the value of -2 kms-1 given by Patel et al. (1998), who assumed that the systemic radial velocity of the Bubble is identical to that of the ionized gas of IC1396. This difference in the systemic velocities may explain also the discrepancy between the kinetic energies obtained by them and those calculated below. Following the same procedure, i.e. weighting the velocities with the corresponding HI densities, we also calculated an `effective expansion velocity' of kms-1. These velocities will be used to model the possible physical origin of the bubble.
© European Southern Observatory (ESO) 2000
Online publication: February 9, 2000