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3.1 Luminosity Distribution Laws: Photometric Parameters

Detailed surface photometry confirms the validity of one of the two basic Hubble classification criteria, namely that the luminosity of normal galaxies can be resolved into two principal components:

(I) a spheroidal component characteristic of elliptical galaxies (de Vaucouleurs 1948, 1953, 1959, 1962; Fish 1964; King 1962, 1966), of the bulge of lenticulars (de Vaucouleurs 1959-1975; Tsikoudi 1977) and early spirals Sa, Sb (de Vaucouleurs 1958b, van Houten 1961), and even of other Population II samples such as the system of globular clusters (de Vaucouleurs 1977d),

(II) a flat component characteristic of the disks of spirals Sb-Sd (de Vaucouleurs 1958b, 1959, 1974; Freeman 1970; Schweizer 1976) and of magellanic irregulars Sm-Im (Ables 1971; de Vaucouleurs and Freeman 1972).

In reduced units (rho* = r* / r*e, J = I/Ie) the mean surface brightness distribution in near face-on systems is well represented by

Equation 14   (14)


Equation 15   (15)

or by a linear combination for intermediate types. In a first approximation the relative integrated luminosity curves

Equation 16   (16)

form a one-parameter family as a function of type t (see Fig. 5 in RC2).

The total luminosity of the spheroidal component is

Equation 17   (17)

and that of the flat component

Equation 18   (18)

The fraction KI = LT(I) / LT of the total luminosity LT = LT(I) + LT(II) which is contributed by the spheroidal component is, therefore, also a function of t and a quantitative measure of the first Hubble classification criterion.

From a study of available photometric data for 24 galaxies, Yoshizawa and Wakamatsu (1975) found that Upsilon ident LT(I) / LT(II) = KI / (1 - KI) varies with t as shown in Fig. 20 and that the absolute magnitude of the spheroidal component is loosely correlated with Upsilon while the absolute magnitude of the disk component is essentially independent of t [but varies with M°T at t = const. as noted by Freeman (1970)].

Figure 20

Figure 20. Correlation between mean bulge/disk luminosity ratio Upsilon = LT(I) / LT(II) and morphological type t, after Yoshizawa and Wakamatsu (1975). Note KI = Upsilon / (1 + Upsilon) is the bulge fractional luminosity LT(I) / LT.

This leads to a first quantitative 2-parameter classification system using KI (or Upsilon) as a type indicator of the bulge/disk ratio and LT = LT(I) / KI as luminosity indicator. (4)

A second, more refined 3-parameter scheme rests on the fact that for LT = 1, the two-component model has 2 free parameters alpha = r*e(I) / r*e(II) and beta = Ie(I) / Ie(II), the ratios of the effective radii and specific intensities of the spheroidal and flat components which determines as before the bulge/disk ratio Upsilon = 1.91 alpha2 beta. The 3rd parameter may be LT or a scale factor, say r*e for the whole galaxy. This scheme allows also for large vs. small, as well as bright vs. faint bulges and disks, all of which are observed in various combinations, although observational selection effects may conspire to restrict to a small range the bulge or disk parameters currently available; this in turn may produce deceptively tight correlations.

Work is in progress to apply this photometric classification scheme to several hundred galaxies with adequate photographic and photoelectric data.

Remark: For galaxies with a nuclear source, e.g. Seyfert galaxies, the same scheme is applicable after subtraction of the (often variable) contribution of the quasi-stellar source (see de Vaucouleurs 1973, de Vaucouleurs et al. 1973), that is

Equation 18a

3.2 Color Distribution Laws: Colorimetric Parameters

Integrated color indices C(rho*) = B(rho*) - V(rho*) are available for ~ 1000 galaxies (and for several hundreds in U - B); the average color-aperture relations C(xi, t) where xi = log A / Ae is the normalized aperture (Ae = aperture transmitting half the total B flux) have been derived for all the major morphological types t (see Fig. 6 in RC2). In principle, the total color indices CoT (Section 2.2) and the color amplitude DeltaCT = CN - CT, where CN = C(xi << -1) is the nuclear color, could define an equivalent type t. However, the variable effects of line emission, inclination (absorption) and the intrinsic (cosmic) color range at t = const. does not make this approach a promising one.

Also, attempts at establishing a color-magnitude diagram for galaxies have been frustrated by the lack of correlation between color and luminosity at t = const.; as noted in Section 2.2 a weak correlation appears only when all late-types (Sc to Im) are dumped into the original Sc Hubble class.

At present broad-band colors do not appear promising as a basis for a quantitative classification scheme, unless supplemented by detailed spectral data. Emission-free colors measured through narrow-band filters are more promising, but are not yet precise and numerous enough to be a practical alternative at this time.

3.3 Kinematical Parameters

The empirical observations that (1) VM is a function of t and L, i.e., of Lambda (Fig. 19b), and (2) 2RM / D0 is a function of t, i.e., of the bulge/disk ratio (Fig. 19c) could be used to set up a kinematical classification scheme with tauM = log 2RM / D0 serving as a type indicator and upsilonM = log VM as luminosity indicator; however, RM is too poorly defined and measured to make this scheme practical.

A variant uses the centrifugal force at RM, KM = V2M / RM as a type indicator (Fig. 21a, b) (Wakamatsu 1976). This has the disadvantage that its 2 parameters KM and tauM are not independent, leading to a deceptively tight correlation which is mainly a result of systematic errors in RM.

Figure 21

Figure 21. Correlations between centrifugal force KM at RM, and morphological type t (a) and normalized radius tauM (b), after Wakamatsu (1976).

A better combination of parameters could be upsilonM and taul, = log 2Rl / D0 which is more precisely defined and is available for a greater sample of galaxies.

Both schemes suffer from the serious drawback that reliable values of VM, and especially RM, are currently limited to a small number of nearby galaxies having inclinations i > 20°.

3.4 Dynamical Parameters

Basic dynamical parameters such as total mass

Equation 19   (19)

and total angular momentum

Equation 20   (20)

where m(r) = projected surface density, have been proposed as elements of a more fundamental classification scheme (Wakamatsu 1976). However, the two parameters are not independent and are poorly defined by the observations because a large fraction of the angular momentum is in the outer regions of disk galaxies (at r >> where RM) where V(r) and m(r) are essentially unknown (Nordsieck 1973).

In particular, the tight correlations reported between MT and AT (Wakamatsu 1976, see de Vaucouleurs 1974 for earlier references) are largely trivial (Freeman 1970, Nordsieck 1973). The only significant difference that seems well established is that ellipticals and lenticulars have smaller specific angular momenta (i.e., per unit mass) than do spirals (Bertola and Capaccioli 1975, 1977). However, this marginally detectable difference is not a practical classification criterion.

A better scheme involves MT and a mass distribution parameter Upsilon' which depends on the bulge/disk ratio (Wakamatsu 1976), assuming that bulge and disk are each characterized by a constant M / L = f ratio; thus if MI = Upsilon' MII, Upsilon' = fI LI / fII LII, and MII = 2pi S0 / alpha2 is the mass of the exponential disk (Freeman 1970)

Equation 21   (21)


Equation 22   (22)

where f(Upsilon'), g(Upsilon') are numerical factors depending on the mass distribution within the bulge. Upsilon' = (fI / fII)Upsilon is essentially a morphological type indicator strongly correlated with KM = V2M / RM. This scheme provides a dynamical interpretation of the (T, L) empirical system, but again is hardly a practical classification system in view of the current scarcity of reliable photometric and dynamical data. (5)

3.5 Physical Parameters

Another quantitative classification system could be based on fundamental physical parameters such as total mass MT and the gas/stars ratio measured by the neutral hydrogen to total mass ratio h = MH / MT, since principal component analysis shows that MH and MT are correlated, but h and MT are not (Brosche 1971). However, MT requires dynamical information and, as noted in Section 3.4, is severely restricted; an acceptable substitute is the total luminosity LT since f = MT / LT is highly correlated with color index CT, at least for the spheroidal component (Fig. 22) (for details and references see de Vaucouleurs 1974). Similarly h = MH / MT which requires MT and is distance dependent, could be replaced by g = MH / LT which is independent of distance; this ratio is measured by the hydrogen index HI as discussed in Section 2.3.

Figure 22

Figure 22. Correlation between mass/luminosity ratio fB = log MT / LB and (U - V)oT color index for spheroidal systems.

Other parameters such as the average surface density of hydrogen, the effective radius of the HI distribution and its ratio to the optical effective radius, etc. have not yet been measured consistently for a sufficient number of galaxies.

It is hoped that a workable multivariate classification system based on a combination of optical, kinematical and physical parameters will emerge from studies now in progress at several institutions.

Some of the statistical studies presented in this report were supported in part by grants from the National Science Foundation.

4 A simplified version of this scheme which makes no assumption on the forms of the luminosity laws would be to use the concentration index C31 as a rough measure of the bulge/disk ratio. Back.

5 The same remark applies to the interpretation of luminosity classification and arm/disk ratios (Schweizer 1976) within the framework of the density wave theory (Roberts et al. 1975). Back.

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