2. CORRELATIONS WITH MEASURABLE PARAMETERS

In the 1920's and 30's the systematics of the HR diagram and the "theory of the spectral type" paved the way for the modern theories of stellar atmospheres and interiors. It is likely that current efforts to discover the basic physical parameters behind the more successful classification systems will likewise advance our understanding of the nature and evolution of galaxies. In this sense the best classification system is that which is most closely correlated with the most basic physical parameters such as mass, angular momentum, gas fraction, etc. - or, perhaps, with the largest possible number of measurable derived parameters such as absolute luminosity, color, rotation velocity, hydrogen content, etc.

Principal component analysis as well as multiple correlation studies (see de Vaucouleurs 1974 for references) have already indicated that 2 or 3 parameters are dominant. For example, 2 variables contribute more than 80 percent of the total variance in a 6-variable analysis of 31 galaxies (Brosche 1973). Pending the results of new analyses of larger samples based on improved data, the dependence on T and L of various measured parameters has been investigated anew from RC2 data.

2.1 Concentration Indices

Let k(r*) be the fraction of the total luminosity L_T emitted within (equivalent) radius r*, k(r*) = L(r*) / L_T, with L(r*) = 2 ₀ I(r*) r* dr*. If r*₁, r*_e, r*₃ are the radii such that k(r*₁) = 1/4, k(r*_e) = 1/2, k(r*₃) = 3/4, then C₂₁ = r*_e / r*₁ and C₃₂ = r*₃ / r*_e are light concentration indices. Such indices have been derived from detailed surface photometry for some 120 galaxies (de Vaucouleurs 1962, 1977c; Fraser 1972, 1977; de Vaucouleurs and Agüero 1973). For greater simplicity consider only the combined concentration index C₃₁ = C₂₁C₃₂. The correlation between C₃₁ and morphological type T is shown in Fig. 5. The large scatter is caused by measuring errors, inclination effects and structural details (e.g. SB vs. SA). A more precise relationship emerges from a similar analysis of integrated magnitude-aperture relations m(A) derived in RC2 from photoelectric photometry for some 1000 galaxies (Fig. 5).

Figure 5. Correlations between concentration index C31 and morphological type t. Dots are for individual objects from detailed photographic photometry; dashed line is mean from photoelectric aperture photometry of ~ 1000 objects.

Light concentration is highest for ellipticals (t = -5) following the r^1/4 law (Section 3.1) and lowest for magellanic irregulars (t = +10) following the exponential law. The smooth, monotonic transition with morphological type t demonstrates the validity of the revised Hubble sequence. The concentration index is, of course, a measure of the varying bulge/disk ratio (Section 3.1) along the sequence, but it is more objective in that it does not presume any specific model for the light distribution nor its decomposition into 2 or more components.

Similar conclusions can be reached by considering the n index in Sérsic's (1968) formula

where m'' = m₀ - 1.086n, and m = -2.5 log [0.921 ^m₀ IS(m) dm] is the integrated magnitude within the isophote of area S; m₀ is the surface magnitude at the center. The exponent n varies with morphological type from n = 8 at type E (t = -5) to n = 2 at type Im (t = +10).

However, the scatter in the C(t) arid n(t) relations is too large to make either C or n acceptable quantitative substitutes for t. In particular inclination effects, possible differences between families (SA vs. SB) and varieties (r vs. s) at t = const. have not yet been investigated.

2.2 Color Indices

While concentration indices are structural parameters, color indices are population parameters. Total (or asymptotic) color indices (U - B)^o_T, (B - V)^o_T corrected for galactic extinction, internal absorption and redshift are listed in RC2 for, respectively, 583 and 953 galaxies.

a) Normal Colors

For each morphological type the average intrinsic colors of normal galaxies are defined by the mean corrected colors C^o_T(t) obtained after iterative rejection of aberrant residuals (generally negative and caused by line emission) until the residual distribution function becomes quasi-normal. Within statistical errors, the normal colors < C^o_T > vary smoothly and monotonically with t (Fig. 6a, b) as expected from earlier work (Holmberg 1958, de Vaucouleurs 1961, G. and A. de Vaucouleurs 1972); in particular, the broad-based index (U - V)^o_T varies from 1.36 at t = -5 (E) to 0.04 at t = +10 (Im) (Fig. 7a). The monotonic variation of C(t) demonstrates again the basic validity of the revised Hubble system, and in particular confirms the order of the stages along the sequence (analogy with color-spectral type relation).

Figure 6. Correlations between mean corrected color indices (B - V)oT, (U - B)oT and morphological type t. Open circles: before rejection of aberrant values; filled circles: after rejection.

Figure 7. Correlation between mean corrected color index (U - V)oT and morphological type t. (a) normal scale. (b) probability scale.

A simple, convenient analytical expression of the functional relation C^o_T(t) is by means of a Laplace-Gauss integral (Fig. 7b) with mean < (U - V)^o_T > = 0.70 at < t > = 3, total amplitude 1.40, and dispersion _t = 1.85. The observed mean colors are listed in Table 4a; the adopted values interpolated as in Fig. 7b are listed in Table 4b. The color-color relation is very nearly linear (Fig. 8); in the spirit of principal component analysis it is, therefore, convenient to replace the original variables x₁ = (B - V)^o_T, x₂ = (U - B)^o_T by the dominant variable ₁ = x₁ cos + [x₂ + K] sin and its associated rectangular variable ₂ = -x₁ sin + [x₂ + K] cos , where = 56°.85 and K = 0.8913. The values of < ₁ >, < ₂ > are also given in Table 4b; note that < ₂ > is everywhere less than 0.01 mag. The total range of < ₁ > is about 1.00 from 0.65 to 1.65; a normalized principal color parameter *₁ may then be defined by *₁ = ₁ - 0.65, varying from 0 to 1 when t varies from E to Im. Except for observational error and cosmic scatter (mainly due to line emission, see below) this normalized color parameter could, in principle, be used as an objective measure (among others) of Hubble stage t.

Figure 8. Mean color-color relation (a) mean points in usual variables, (b) interpolated values in principal variables 1, 2 as a function of t type.

Table 4A. Mean corrected colors by type t ¹

t	< (B - V)oT >	m.e.	1	n	< (U - B)oT >	m.e.	1	n
-5	0.894	0.003	0.033	134	0.471	0.006	0.057	98
-3	.863	.006	.051	67	.462	.007	.039	30
-2	.856	.005	.049	82	.419	.013	.092	51
-1	.808	.011	.071	42	.374	.018	.078	19
0	.804	.011	.053	23	.180	.067	.298	20
1	.704	.015	.097	41	.267	.016	.070	18
2	.737	.009	.057	40	.226	.022	.106	24
3	.657	.010	.080	70	.060	.017	.123	51
4	.564	.008	.066	70	- .042	.015	.104	47
5	.511	.008	.070	81	- .072	.016	.103	44
6	.441	.014	.080	34	- .190	.024	.110	21
7	.444	.020	.079	16	- .213	.039	.102	7
8	.418	.027	.098	13	- .218	.028	.085	9
9	.353	.019	.087	20	- .337	.028	.087	10
10	0.370	0.015	0.079	27	-0.331	0.025	0.108	19
1 After 8 cycles of 2 rejection of large residuals.

Table 4B. STANDARD FULLY CORRECTED COLORS BY TYPE

t	< (B - V)oT >	< (U - B)oT >	< 1 >	< 2 >	< *1 >	ð< *1 >
-6	0.892	0.486	1.641	+0.006	0.991	0.009
-5	.888	.478	1.632	+ .004	.982	.015
-4	.881	.465	1.617	+ .004	.967	.021
-3	.870	.447	1.596	+ .003	.946	.031
-2	.854	.420	1.565	+ .002	.915	.047
-1	.830	.380	1.518	.000	.868	.063
0	.797	.326	1.455	- .002	.805	.083
1	.753	.256	1.372	- .003	.722	.100
2	.699	.171	1.272	- .004	.622	.113
3	.637	.077	1.159	- .004	.509	.112
4	.575	- .017	1.047	- .003	.397	.104
5	.517	- .103	0.943	- .002	.293	.086
6	.469	- .174	0.857	.000	.207	.066
7	.431	- .228	0.791	+ .002	.141	.048
8	.404	- .268	0.743	+ .003	.093	.034
9	.386	- .296	0.709	+ .002	.059	.023
10	.373	- .315	0.686	+ .003	.036	0.014
11	0.365	-0.327	0.672	+0.003	0.022
< *1 > = < 1 > - 0.65

b) Color Residuals

Individual galaxies have corrected colors C^o_T which differ from the mean for their estimated type, that is, leave color residuals

Previous work (de Vaucouleurs 1961, G. and A. de Vaucouleurs 1972) has shown that the cosmic scatter is mainly along the color-color sequence, i.e., along the ₁ axis, while the scatter perpendicular to it, i.e., in ₂, is due mainly to observational errors. This is again demonstrated in Fig. 9 where the ₁, ₂ residuals are plotted for each type. The scatter in ₂ is < 0.15 mag (₂ 0.05), consistent with the observational m.e.'s of the original RC2 colors (0.03 in B - V, 0.04 in U - B). The ₁ residuals are generally < 0.25 mag (₁ 0.08), significantly in excess of measuring errors. Part of this scatter may be introduced by classification errors, especially near the middle of the sequence (0 < t < 6) where ð₁ / ðt 0.10 mag per stage unit and (t) 0.75 (Brosche 1973), but the large negative residuals are clearly caused by line emission, particularly among types earlier than Sc (t < 5) (for a more detailed discussion see G. and A. de Vaucouleurs 1972).

Figure 9. Color residuals in 1, 2 coordinates for main morphological types. Large negative residuals in are due mainly to line emission; scatter in 2 is due mainly to measuring errors. Effect of classification errors of ± 1 t unit is indicated for types where it is largest.

c) Luminosity Effects

For ellipticals and lenticulars (t < 0) there is a well-known correlation between integrated colors and absolute magnitude which can be used as a distance indicator (de Vaucouleurs 1961, see de Vaucouleurs 1974 for references). However, compact dwarfs (t = -6) seem to have normal colors (Faber 1971).

Curiously, there is no luminosity effect in the color residuals for spirals (t 0) as shown by Table 5 confirming previous work (G. A. de Vaucouleurs 1972); a weak apparent correlation between color and luminosity class is present only if the original Hubble types Sc and Irr are used (de Vaucouleurs 1961, Iye and Kodaira 1976).

Table 5. Mean color residuals as a function of luminosity class

L	1	2	3	4	5	6	7	8
< (U - V)oT >	+0.04	+0.01	+0.01	+0.04	-0.00	+0.01	+0.02	-0.02
1 1	0.03	0.06	0.03	0.04	0.05	0.05	0.10	0.11
N	24	18	49	20	17	6	6	5
1 1 is s.d. of individual values from mean.

Likewise, there is little indication of a systematic dependence of color residuals on RDDO class (S0, A or S). According to van den Bergh (1976) the mean corrected color C₀ (from RC1) decreases (at constant Hubble type) from S0 to A to S. There is only a slight tendency in this sense in the RC2 data (Table 6, left half), but no definite effect is indicated by the color residuals (B - V)^o_T at t = const. (Table 6, right half).

Table 6. Mean corrected colors and color residuals for rddo types¹

RDDO	S0	A	S	RDDO	< (B - V)oT >	n
a	0.83 (9)	0.82 (4)	0.845 (2)	S0a-b	-0.02	14
ab	0.805 (2)	0.825 (2)	0.75 (7)	Aa-ab	-0.02	22
b	0.84 (3)	0.74 (15)	0.715 (30)	Sa-bc	+0.06	46
bc	-	0.57: (3)	0.63 (8)	Sc	-0.035	32
c	-		0.495 (33)
1 < (B - V)oT > and (n) (left); < (B - V)oT > at t = const. and n (right).

2.3 Hydrogen Index

Let S_H be the total flux density (in W m^-2) in the 21 cm line of HI from a galaxy integrated over frequency and solid angle; the HI mass M_H is given by

(2)

if = distance of galaxy is in Mpc. Values of log S_H where S_H is in 10^-28 W m^-2 are listed in RC2 for 474 galaxies with an average m.e. 0.07.

By analogy with the optical case, define a "magnitude" corresponding to S_H, corrected for self-absorption,

(3)

where

(4)

The constant is chosen to give approximately equal optical and 21 cm magnitudes when S_H is in 10^-28 W m^-2 units; the 21 cm self-absorption is given by

(5)

(Heidmann, Heidmann and de Vaucouleurs 1971) for a galaxy whose spin axis is inclined i degrees to the line-of-sight, if the effective optical depth is

(6)

The inclination can be replaced by the apparent axis ratio R as explained in RC2 where the individual values of A₂₁ are given. The validity of equation (5) with the adopted value of *₀ has been verified by Balkowski (1973).

Then the hydrogen index, defined as HI = m^o₂₁ - B^o_T, is a measure of the ratio of the mass of neutral hydrogen to the corrected B-band optical luminosity of the galaxy. Values of HI are given in RC2 for 240 galaxies. If M_B() = +5.41, the normalized hydrogen/luminosity ratio is given by

(7)

where M^o_H and L^o_B are in solar units. The average value of this ratio varies with morphological type from ~ 0.07 for t 3 to ~ 0.5 for t 7 (Sd-Im) as shown by Fig. 10. Late-type spirals contain about 7 times as much hydrogen per unit of optical luminosity as do early-type spirals. However, at t = const., HI and g = M_H / L_B vary also with color and luminosity; that is, HI is a function of both T and L and is actually more closely correlated with the composite luminosity index than with either of its components. From 288 galaxies for which both HI and are known, a linear correlation is indicated (Fig. 11)

(8)

The residuals (HI) = HI - < HI > are weakly correlated with the color residuals ₁ [or (B - V)^o_T for a larger sample] in both the high and low luminosity groups ( < 1.0, 1.0) (Fig. 12a, b): in the mean

(9)

but with a large scatter, too large to permit using HI and (B - V)^o_T as a substitute for .

a) Hydrogen-Poor Galaxies

A small number of galaxies stand out as unusually hydrogen-deficient for their type and luminosity (HI > +1.2, log M^o_H / L^o_B < -0.5); examples at < 1.0 are NGC 613, 2613, 3627, 4321, 4501, 5247, 5676; two of them (NGC 4321, 4501) are members of the Virgo cluster which has led to the speculation that galaxies in clusters may be depleted of their interstellar gas through encounters (Davies and Lewis 1973, Bottinelli and Gouguenheim 1974); however, the others are members of pairs (NGC 3627, 5676) or are isolated. Two examples at > 1.0 are the LMC and the Pegasus dwarf = DDO 216; one might speculate that the LMC lost half or more of its hydrogen to the magellanic stream and/or to the LMC - SMC common envelope (but, perhaps, the integrated HI flux from the LMC should first be checked by new observations).

Figure 10. Mean hydrogen index HI and hydrogen/luminosity ratio g = MoH / LoB versus morphological type t. Circles are mean points. Dashed lines show trend of luminosity effects (schematic).

Figure 11. Mean hydrogen index HI versus luminosity index . Circles are mean points.

Figure 12. Correlations between residuals in hydrogen index HI and color index (B - V)oT (above) or principal color parameter 1 (below) in two intervals of luminosity index . Hydrogen rich galaxies tend to be bluer than the average for their type.

b) Hydrogen-Rich Galaxies

Conversely, a small number of galaxies have unusually high hydrogen/luminosity ratios (HI < -1.2, log M^o_H / L^o_B > +0.5). Examples at < 1.0 are NGC 628, 7640 and IC 342; in the latter two cases HI is uncertain; one might suspect that the internal extinction correction A(i) (N 7640) or the galactic extinction correction A_B (IC 342) were overestimated or, perhaps, that because of obscuration the luminosity classes are in error. An example at 1.0 is NGC 4244, an edge-on Scd whose luminosity classification and internal extinction correction are unavoidably uncertain.

2.4 Radio Continuum Indices

The radio continuum emission of most normal galaxies can be characterized by the integrated monochromatic flux density S_R at the standard frequency _R 1410 MHz and by 2 spectral gradients = -ð log S / ð log , _- at < _R, ₊ at > _R. Values of log S_R (in 0.01 Jy = 10^-28 W m^-2 Hz^-1) are given in RC2 for 279 galaxies with m.e.'s of 0.04 to 0.10 for sources stronger than 0.1 Jy. Spectral indices are given for 253 (_-) and 239 (₊) objects.

Following Hanbury Brown and Hazard (1961) a radio magnitude may be defined by

(10)

where the constant was chosen to roughly match radio and optical magnitudes. The radio magnitude corrected for redshift is

(11)

where _R = (a_- + a₊) / 2 is the average spectral index near = _R. For z < 0.04 (cz < 12,000 km s^-1) a close enough approximation is

The correction is < 0.05 mag for z < 0.04 and normal values of _R (0 < _R < 1). By analogy with the color index define the (corrected) radio index as

where B^o_T is the corrected total B magnitude. Values of RI are given in RC2 for 174 galaxies ranging from -4.2 for strong radio galaxies (3C 88, 3C 75, NGC 7236 - 37) to +5.0 for the weakest detectable normal galaxies (M 33, NGC 55, 3368).

The mean radio index < RI > = +2.65 is essentially constant and independent of morphological type for t 2 with a large scatter ( 0.5 to 1.5 mag) (Fig. 13a). In other words the emission ratio is independent of type for normal spirals, whose radio and optical emissions are dominated by the disk component. For t < 0 (E, L types) < RI > = -1.25, indicating for this selected sample (dominated by nuclear and/or double sources) a radio/optical emission ratio some 36 times stronger than for the average spiral ( = 2.5 mag).

The mean spectral index < _- > is almost completely independent of type (Fig. 13b) with < _- > = 0.70 for all types, but < ₊ > = 0.80 for t < 0 and t > 6, and 0.95 for 0 t 5.

Figure 13. (a) Mean radio index RI (above) and (b) mean spectral indices +, - (below) as a function of morphological type t.

Continuum radio emission and its spectral distribution are clearly not good indicators of morphological type. However, for spirals which have a strong nuclear radio source, a loose correlation between the monochromatic power at _R and Byurakan nuclear class has been noted by van der Kruit (1973) (Fig. 14). The most powerful sources (log P > 21) are generally found in Seyfert nuclei (BYU N = 5) (Fig. 15).

Figure 14. c Radio power at 1415 MHz and Byurakan nuclear class N, after van der Kruit (1973).

Figure 15. Correlation between radio power at 1415 MHz and infrared emission at 10µ, after van der Kruit (1973). Filled circles refer to Seyfert galaxies. Two values of radio/infrared spectral index are shown.

2.5 Ellipticities

In a large collection of galaxies a few of each type will be seen almost exactly edge-on; the maximum apparent flattening measured by log R = log a/b must be about equal to the true (maximum) flattening log R₀ = log a/c of a given type (Fig. 16). This maximum increases from log R(max) 0.3 (c/a = 0.5) for ellipticals to ~ 0.5 (c/a = 0.3) for lenticulars to ~ 0.5 to 0.7 (c/a 0.3 to 0.2) for spirals Sa to Sd, and drops back to ~ 0.5 at Sm-Im. More precisely, an analysis of the observed frequency function of apparent ellipticities of the isophotal diameters of over 2000 galaxies (de Vaucouleurs and Pence 1973; see also de Vaucouleurs 1974) leads to the following conclusions:

a) Elliptical galaxies have true ellipticities e = 1 - c/a = 1 - q₀ distributed normally about a mean = 0.36 (i.e., E 3.6) ⁽²⁾ with a small cosmic dispersion _e 0.1; the flattest bona fide ellipticals have e = 0.55, spherical galaxies (e 0) are rare or absent. ⁽³⁾

Figure 16. Correlation between maximum observed ellipticity log R25 and morphological type.

c) Spirals from S0/a to Sm have ellipticity distributions similar to the lenticulars with a major group (~ 70% of sample) at < e > = 0.7 to 0.8 and a minor one (~ 30%) at < e > 0.4. These two components probably correspond to two classes of spirals (and lenticulars) having respectively small and large bulge/disk ratios. The relative abundance of the latter group apparently increases along the Land S sequences (see Figs. 5 & 6 in de Vaucouleurs 1974).

d) Magellanic irregulars Im are distinctly less flat than Sc-Sd spirals and may also be divided into a major group (60%) at < e > 0.6 and a minor one (40%) at < e > 0.35. However, some barred irregulars lack an optically visible disk (e.g. NGC 4214, 4449, 6822) and treating bars (prolate spheroids) as oblate spheroids must vitiate the statistics.

This analysis suggests that the popular concept of the Hubble sequence as a single angular momentum sequence is oversimplified; at each stage t of the sequence there is a wide range of intrinsic ellipticities and, therefore, of bulge/disk ratios (Fig. 17). This conclusion is consistent with the large scatter observed in the correlation between type t and concentration index C₃₁ (Section 2.1).

Figure 17. Trend of fractional luminosity in spheroidal component KI = LT(I) / LT as a function of morphological type t for galaxies with different bulge/disk ratios (schematic). Compare FIG. 20.

2.6 Angular Velocities

The inner part of the rotation curve of a normal galaxy is often well approximated by a straight line (inflection tangent) (Fig. 18). Mayall (1960) and Mayall and Lindblad (1970) have discussed the correlation between the mean angular velocity < w > and radius R_l of this "linear" branch of the rotation curve as a function of morphological type. The lower angular velocities of barred spirals at constant R_l and t suggested by this work needs confirmation; if it is not an artifact due to preferential orientation of the spectrograph slit along the bars it may be an important clue to the dynamics of barred spirals.

Figure 18. Schematic rotation curve showing definition of RM, VM and "linear branch" Rl.

Because both R_l and w depend on some estimate of the distance of the galaxy neither is suitable as an empirical classification parameter. (See, however, 3.3.)

2.7 Maximum Rotational Velocities

Brosche (1971) found that the maximum rotational velocity V_M is loosely correlated ( -0.8) with morphological type t, and for spirals (t 2)

(12)

Iye and Kodaira (1976) report a similar correlation ( -0.83) between V_M and luminosity class L for late-type spirals (t 4).

Actually, V_M is better correlated with the luminosity index (Section 1.1) than with either of its components; this is not surprising since is correlated with the mass via the f = M / L ratio and M V²_M. A preliminary statistic from 20 systems (Table 7) gives

(13)

Table 7. Maximum rotational velocities¹

NGC	t		log VM	log 2 RM	M	M + M
0045	8	1.6	2.00	2.20	+0.30	2.30
0224	3	0.5	2.43	2.95	-0.24	2.19
0253	5	-	2.31	2.20	-0.09	2.22
0300	7	1.3	1.97	2.47	+0.16	2.13
0598	6	1.0	2.00	2.76	0.00	2.00
0925	7	1.1	(2.15)	(1.78)	(-0.19)	(1.94)
2403	6	1.1	2.11	2.37	+0.14	2.25
3031	2	0.4	2.40	2.08	-0.30	2.10
3109	9	1.7	1.73	2.51	+0.48	2.21
3115	-3	-	2.44	1.10	-0.76	[1.68]
3359	5	0.8	2.12	1.85	+0.05	2.17
3521	4	0.7	(2.41)	(1.48)	(-0.45)	(1.96)
4236	8	1.5	1.95	2.30	+0.11	2.06
4244	6	1.3	2.00	2.25	+0.22	2.22
4631	7	1.2	2.08	2.20	+0.17	2.25
5194	4	0.5	2.42	1.98	-0.04	2.38
5236	5	0.7	(2.25)	(1.90)	(-0.17)	(2.08)
5457	6	0.7	2.31	2.4:	-0.05:	2.26
6946	6	0.7	2.32	2.0:	-0.12:	2.20
1342	6	0.8	2.28	2.4:	+0.05:	2.33
1 Best observed systems from compilations by Brosche (1971), Huchtmeier (1975), Wakamatsu (1976).
M = log VM, M = log 2 RM / D0.

The radius R_M where V = V_M is poorly defined; nevertheless, recent data (Huchtmeier 1975, Wakamatsu 1976) confirming earlier conclusions (Brosche 1971) indicate that 2R_M / D₀, where D₀ is the corrected face-on isophotal diameter in the RC2 system, is a rapidly varying function of morphological type t (Fig. 19a) or possibly of , but the scatter is too large to make it a useful classification index.

is not (Fig. 19d) at least for spirals ( 0.4). More and better data will be needed to establish whether this is a small sample accident or a general property.

Figure 19. Correlations between M = log VM, M log 2RM / D0 and morphological type t or luminosity index .

Instead of V_M, the "quasi-total" width W of the 21 cm line in galaxies unresolved by the antenna beam can be used as an indicator of absolute luminosity and distance (Roberts 1969, 1975; Balkowski et al. 1974; Tully and Fisher 1975, 1977; Sandage and Tammann 1974) at least for disk systems of sufficient inclination (i > 20°). For face-on systems the line width is a measure of the velocity dispersion _z normal to the plane and it should not be used to calculate V_M or M_T.

Remark: Both _M = log V_M and _M log 2R_M / D₀ are correlated with (Fig. 19b, c), but within the accuracy of the data

is not (Fig. 19d) at least for spirals ( 0.4). More and better data will be needed to establish whether this is a small sample accident or a general property.

2.8 Velocity Dispersion

In spheroidal systems where most of the kinetic energy is in random motions rather than rotation and M_T ²_V R, relations of the form _V M_T corresponding to f = M_T / L_{T V} may be expected to hold. Such relations are in fact observed for elliptical galaxies and the spheroidal components of lenticulars and Sa-Sb spirals (Poveda 1961, Einasto 1972, Faber and Jackson 1976). The latter authors find L_T ⁴_V and f ^1/2_T, but both L_T, which is distance dependent, and _V are known for too few objects and too uncertain to use as a basis for quantitative classification at the present time.

² not E 6.5 as printed in IAU 58, p. 13. Back.

³ The large excess of E0 galaxies in catalogues based on POSS prints is an artifact. Back.