3. H I radial surface density distribution
So far the H I surface density distributions have only been used to define the sizes of the H I discs (e.g. and ; see Table 1), but these distributions themselves are of interest as well. For our sample they show a considerable diversity. When normalized by a certain radius scale (e.g. optical radius as is usually done) and averaged per morphological type, there seems to exist a characteristic profile for each morphological type (see Figs. 7 and 8 in Paper I; see also Cayatte et al. 1994), although there is a significant scatter about the mean profile. The Sb-Sc galaxies tend to have very similar H I density profiles: more or less flat up to around , and a steep drop beyond that point (Broeils & van Woerden 1994; Sancisi 1983). The Sd or later type galaxies mostly have density profiles that keep increasing towards the central parts (see e.g. Paper I). Early type galaxies (S0, Sa, Sab) are not well represented in the present sample; however, it is generally believed that these galaxies have central H I depressions (e.g. van Driel 1987; Sancisi 1983) and that the hydrogen gas might have an external origin.
In this paper we will take the analysis one step further: Principal Component Analysis (PCA) is used to investigate systematics of the H I radial surface density distributions. PCA is a statistical method which focuses on inter-object correlations, reduces their (parameter-space) dimensionality, and consequently finds the least number of the new dimensions (principal components) (see Murtagh & Heck 1987). Traditionally, the PCA method has often been used to find the most significant components among various global parameters (e.g. Whitmore 1984). Recently, Han (1995) has applied PCA to find the most important components in the surface brightness distributions of the spiral galaxies. He found that about 94 % of the variation in the surface brightness distribution of galaxies can be accounted for by just two principal components. In this paper we follow Han's line of approach to study the H I surface density distributions.
In constructing the input dataset, we start with scaling the radii of the galaxies by . In order to avoid that all data points converge into one point ( at ), we choose for the scaling radius instead of . We choose 15 sampling radii, from 0 to 1.5 , with equal intervals in linear scale. This sampling roughly corresponds to one beam size on the major axes of typical galaxies in our sample. The H I surface densities of each galaxy (in linear scale) are then sampled at these radii. The 15 points represent the H I surface density distributions of the sample galaxies at the common scaled radii and PCA is expected to find the most important components in the H I distribution of galaxies. We have tested with different numbers, ranges and intervals of sampling points without finding significant differences. Neither does the use of logarithmic scaling of the H I surface densities change the results significantly.
In deriving the H I surface density distribution from the H I strip integrals, we have used an iterative deprojection method developed by Warmels (1986), with the assumptions of azimuthally symmetric distribution of H I, and constant inclination throughout the gas and stellar discs. Any violation of these assumptions, among other things, would increase the uncertainty in the H I density profile, and consequently, increase the dimensionality of the dataset studied in this paper. The possible existence of non-linear relations between the parameters could therefore lead to overestimation of some weak components.
Applying PCA to our input dataset produces a series of principal components (eigen vectors): The mean H I surface density profile and the first four principal components are presented in Fig. 6. First of all, the mean H I surface density profile shows a shoulder typical of Sb-Sc galaxies (Sancisi 1983) extending to about , which coincides with the optical radius . Note that our data sample is heavily biased towards Sc galaxies. Fig. 6b further shows that the first principal component has small variation in amplitude with radius and no change in sign. This implies that the most important physical process for the H I surface density profile works in a similar way at all positions within the galaxies. The first principal component is found to carry about 65 % of the total variance as listed in Table 2. The second principal component accounts for about 16 % of the total variance. It shows different behaviours for the inner and outer parts: Its sign changes around 0.5 from positive to negative and beyond around 0.5 it changes slowly with radius, staying negative. This component mainly determines the degree of depression (or boost) in the central part (see Fig. 7). The third component is found to explain about 7 % of the total variance. This component might be responsible for the wiggles and bumps in the H I surface density profile. It is possible that this component is related to the existence of the H I rings, bars or spiral-like structures. The fourth component accounts for 5 % of the total variance. It is not clear whether this component reflects weak but true features of the galaxies or just show random noise.
Table 2. Results of the Principal Component Analysis: eigenvalues () for the first ten principal components.
Therefore, at least three principal components are needed to account for about 90 % of the total variance, meaning that three (see Table 2) physical parameters mainly determine the shape of the H I radial surface density profiles. In Fig. 7, the variations of the H I radial surface density profiles are presented in the and planes. The principal component parameters , , and indicate the strength of the first, second, third and fourth principal components, respectively. The role of each principal component is better visible in these diagrams. Fig. 7 also shows that the shape of the HI surface density profiles can be parameterized by the principal components. Therefore, potentially, certain combinations of the principal component parameters could yield a good objective classification system for the H I surface density profiles. Furthermore, it could offer a better tool for the comparison of H I surface density profiles between field and cluster galaxies (cf. Cayatte et al. 1994). An asymmetry parameter can be defined by measuring the principal component parameters separately for the receding and approaching sides of a galaxy.
The question naturally arises whether the principal components are correlated with other parameters. The morphological dependence of the mean shape of the H I surface density profiles (see above discussions) is confirmed in Fig. 8 except for the Sa, Sab galaxies (the filled circle).
We have also examined the principal component parameters of the galaxies grouped by the presence of optical bar and ring structures. Mean differences in the principal component parameters are marginal for the galaxies with and without bars. In the plane ringed galaxies tend to shift toward the upper-left direction from the origin, where non-ringed galaxies are located. It would be interesting to investigate whether the principal component parameters show more prominent differences between the galaxies with and without H I bar and ring structures, but two-dimensional information on the distribution of H I is needed to do this.
We also compared the principal component parameters and their simple combinations (e.g. , , and so on) of the sample galaxies with any of the global galaxian parameters listed in Table 1 and others from LEDA (IRAS far-infrared fluxes and colours), and with their combinations. In general, the principal component parameters do not show tight correlations with global parameters. The first principal component parameter shows some trend, though weak, with the maximum rotation velocity and H I diameter. It seems to correlate better with the (optical, far infrared and H I) mean surface brightness parameters and colours (U-B and B-V), but the correlations are weak at best. The second principal component does not show any trend with other global parameters. The third principal component shows a very weak trend with the optical (B band) mean surface brightness parameter. Combinations of the principal component parameters also tend to correlate better with the mean surface brightness parameters or colours.
© European Southern Observatory (ESO) 1997
Online publication: May 5, 1998