Astron. Astrophys. 344, 421-432 (1999)

5. The relations between the measured parameters

It is well known that the observational parameters describing spiral galaxies are correlated, in such a way that two quantities can describe most of the variance in the parameter space. In his pioneering analysis, Brosche (1973) used the method of the Principal Component Analysis (PCA) to study the data (morphological type, optical size, color, absolute luminosity, maximum rotation velocity and HI mass) of 31 spiral galaxies. He found that the parameter space has two significant dimensions. Bujarrabal et al. (1981) also used the PCA to study a sample of 100 objects with optical and radio data and also conclude that 2 parameters could suffice to describe the data: size (given equivalently by the optical size, the HI mass or the luminosity) and aspect (given by the morphological type or the color). Whitmore (1984) obtained the same conclusion for a sample of 60 spirals with about 30 observed parameters, and assigned the two dimensions to scale (optical size, luminosity) and form (color, bulge to total light ratio). More recently, the PCA of I-images of some 1600 spirals led Han (1995) to find also two principal dimensions. Magri (1995), with a sample of 492 spirals with compiled and new data (Magri 1994), and a refined version of the PCA to allow for the inclusion of non-detections, also concluded in the same vein.

Even if the PCA has some limitations related with the bias in the results that could be introduced if the number of parameters involved in the analysis is not big enough (Magri 1995), or the fact that it assumes that the correlations among the parameters are strictly linear, it appears as a well suited method to find the minimum number of variables describing the data set we have constructed for isolated spiral galaxies. In our case, given that all possible effects of the interaction are in principle excluded, what could be expected from that kind of analysis is to obtain the bare, intrinsic structural correlations between the parameters. We have already discussed how the inclusion of interacting galaxies could increase the scatter in the relations between parameters. In the same sense, Folkes et al. (1996) have shown that the inclusion of perturbed or peculiar galaxies could produce misleading results, in the sense of smoothing the morphology-spectrum relationship found for normal galaxies.

The main result that emerges from the PCA of our data is that the parameter space has essentially 2 dimensions, in agreement with all the previous results. The new aspects of our analysis are that till 95 of the variance of the sample is explained, and the identification of a better suited form parameter. Indeed, the first eigenvector , corresponding to the scale parameter, is given by either the total luminosity, L_B, the total mass, M, or the size, as usual. For the second, the form parameter, we find that the best choice is the inner gradient, G, or, equivalently, the bulge to disk ratio, B/D. In Fig. 5 we show the projection of the vector constructed with the luminosity, the bulge to disk ratio, the total mass, the G parameter and a color index onto the plane described by the two eigenvectors. We note that we find no correlation between the color and the luminosity, whereas color is correlated with the bulge-to-disk ratio, in the expected sense of redder galaxies having a greater relative contribution from the bulge component.

Fig. 5. Results of the PCA. The vectors are 14=G, 19=LB, 20= B/D=, 21=(B-V), 23=Mass.

It is the first time that the G-parameter is revealed as equivalent to the form eigenvector . Some other were proposed, like some color index, or the morphological type, but what our data on isolated galaxies show is that the scatter is minimum when G, or the B/D ratio, is used. The general trends between the morphology, the colors and the B/D ratios are well known, and at the base of the Hubble classification scheme. In that sense, Whitmore (1984) pointed out that the inner gradient of the rotation curves could be a good discriminant of the central density, i.e., the importance of the bulge component. On the other hand, Baiesi-Pillastrini (1987) already noticed that the G-parameter is not directly correlated with the morphological type, so the relation between the B/D ratio and the morphology would present an important scatter, as it is. What we find here is that the G-parameter is a much more useful property to classify a spiral galaxy, in the sense that it is essentially coincident with one of the 2 eigenvectors of the parameter space, i.e., normal to the size parameter (see Fig. 5).

Indeed, a number of strong linear correlations between parameters does exist. Regarding the photometric data, we have already pointed out that the optical size and the luminosity in each band are well correlated. The disk and bulge parameters, that have been discussed in the previous section, also present good correlation, the tightest being between the surface brightness and the size, both for the bulge (the Kormendy relation), and for the disk. For the bulge parameters, we find , i.e., somewhat smaller slope but compatible with that of the Kormendy relation for other spiral bulges (Andredakis et al. 1995; Hunt et al. 1998) and for ellipticals (Bender et al. 1992).

Among the relations involving kinematical parameters, we find a correlation between V_M and the size, in agreement with the linear relationship found by Zasov & Osipova (1987). V_M (and V_max) are also related with the B luminosity (Rubin et al. 1982; Persic & Salucci 1991). We also find a good correlation between the angular momentum, J = M V R, and the total mass, log(J) = -4.27 ( 0.24) + 1.70 ( 0.02) log(M), in good agreement with Campos-Aguilar et al. (1993). The scatter is smaller in our case (see Fig. 6). V_max and R_max are correlated with the total magnitude with the same overall tendencies found by Courteau & Rix (1997) (they use R magnitudes from Courteau 1996) but, at variance with them, we do not find any clear trend between total colors and total luminosity, what could be due to the limited luminosity and subtype range covered by our sample. Total absolute luminosity and total mass are well correlated, as shown in Fig. 7. We obtain M = 4.80 ( 0.15) L_B with a correlation coefficient r = 0.9. This implies a mass to luminosity ratio which is within the range of (M/L) values (from 3.9 to 6.6 in solar units, with H₀ = 75 km s^-1 Mpc^-1) measured for Sc galaxies by Rubin et al. (1985). It is also compatible with (M/L_R) ratios found for field galaxies by Forbes (1992) (from 3 to 6 in solar units, with H₀ = 75 km s^-1 Mpc^-1). M/L ratio tends to be higher for more massive (or more luminous) galaxies, as found by Broeils & Courteau (1996).

Fig. 6. Correlation between the angular momentum, J (see text) and the total mass, M. The dashed lines represent the average and range found in Campos-Aguilar et al. (1993). The solid line corresponds to the fit to our data, log(J) = -4.27 + 1.70 log(M).

Fig. 7. Correlation between the mass inside , M25 (see text) and the total luminosity in B, LB.

The new result is the tight correlation between the inner gradient, G, and the bulge to disk ratio, B/D (Fig. 8) we have found. The slope is (6.80 0.16)10^-4 and the scatter amounts only to 0.02 in the determination of B/D. As already mentioned, either of those two parameters builds up the second eigenvector of the parameter space of the isolated spiral galaxies. The existing correlation reflects what could be intuitively expected, that is, that galaxies with bigger bulges have higher inner gradients. Baiesi-Pillastrini (1988) found a somewhat similar correlation but much more scattered. This could be due to the inclusion of non-isolated spirals in his sample since, as we show in the next Paper III of the series, the correlation is affected by the interaction, as it is much weaker for spirals in pairs.

Fig. 8. Correlation between the inner gradient of the rotation curve, G (km s-1 Kpc-1, and the bulge to disk luminosity ratio, B/D.

The correlation between G and the B/D ratio can in principle be used for distance determination. We notice however that, as just mentioned, the scatter of the relation increases when non isolated galaxies are considered, so only strictly isolated galaxies should be used.

© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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