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Astron. Astrophys. 351, 869-882 (1999)

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5. Statistics of NIR and optical-to-NIR colors

5.1. [FORMULA] and [FORMULA] colors

The galaxies have been distributed in eight broad generic types according to the morphological type T taken from the RC3 (see Table 2).


[TABLE]

Table 2. Correspondence between the generic types used in this paper and the RC3 types.


Because the uncertainties computed for individual [FORMULA] and [FORMULA] are overestimates, we simply give in Tables 3 and 4 the median colors for each type and compute their uncertainties by a bootstrap with replacement. The total and effective colors show almost no difference. But for a small reddening of both colors in Sa-Sc, [FORMULA] is nearly independent of the type, whereas [FORMULA] is clearly bluer in irregular galaxies than in E-Sd, indicating that the emission of Im in the J-band is contaminated by young stars. The redder colors in early-type spirals are more difficult to understand with current models of stellar evolution and atmospheres. As this effect is observed in both the total and effective colors and the extrapolation to compute the latter is small, this may not come from a problem with the growth curve. The differential internal extinction, negligible in the NIR, is also unlikely to cause this phenomenon. A possible explanation is that we begin to see in H and K the tail of the dust infrared emission at short wavelengths. We note in particular that the J and K samples contain a non-negligible fraction of galaxies with nuclear activity ([FORMULA] are also recorded in the Catalogue of Seyfert Galaxies of Lipovetsky et al. (1988)) of which central colors are in some cases as red as [FORMULA].


[TABLE]

Table 3. Median total and effective [FORMULA] colors (with their [FORMULA]-uncertainties) as a function of type. The values in parentheses are the mean RC3 types in each generic type. N is the number of galaxies.



[TABLE]

Table 4. The same as Table 3 for [FORMULA].


5.2. Optical-to-NIR colors: statistical estimators

The catalog is large enough to perform a statistical analysis of the colors as a function of the type, the luminosity, and the inclination or the shape ([FORMULA]) of the galaxy. An interesting quantity is also the intrinsic scatter in the colors at a given type, luminosity and [FORMULA], for it is a measure of the variations of the star formation history and of the effects of dust (e.g. Peletier & de Grijs 1998; Shioya & Bekki 1998). Note however that the intrinsic scatter may depend on the type-binning, especially if the colors evolve rapidly from type to type.

The [FORMULA] sample is by far the largest (more than 900 galaxies in [FORMULA] against 600 in [FORMULA] and [FORMULA]). It is also the most accurate (the uncertainty is typically 0.16 mag in [FORMULA] and 0.09 mag in [FORMULA]) and the most complete in type, whereas the [FORMULA] and [FORMULA] sample are strongly deficient in the latest types. For these various reasons, we will mainly focus our discussion in the following on the analysis of the [FORMULA] data.

We adopt here the formalism proposed by Akritas (1999). Let us assume that the true color [FORMULA] of the [FORMULA] galaxy depends linearly on p quantities [FORMULA] (the [FORMULA] being, e.g., [FORMULA] or the absolute magnitude):

[EQUATION]

where [FORMULA], the median of [FORMULA], is introduced merely to identify [FORMULA] with the typical color of the sample, and [FORMULA] is the deviation from the relation due to the intrinsic scatter. We assume that [FORMULA] independently of the other parameters. Let us write [FORMULA] to remind us that the scatter is also to be determined.

The observed variables, [FORMULA] and [FORMULA], are related to the true variables by [FORMULA], where [FORMULA] and [FORMULA]. We obtain

[EQUATION]

Assuming that [FORMULA] is not correlated with the [FORMULA] and [FORMULA], and that the covariances between [FORMULA] and [FORMULA] and between [FORMULA] and [FORMULA] are [FORMULA] and [FORMULA], respectively, we obtain that [FORMULA] with

[EQUATION]

We have tested two procedures to estimate the [FORMULA] from our sample. The first one estimates the [FORMULA] by maximizing the logarithm of the likelihood [FORMULA] with respect to the [FORMULA], where

[EQUATION]

according to the formulae given above. The covariance matrix of the [FORMULA] is computed by inverting the curvature matrix [FORMULA], [FORMULA], where

[EQUATION]

Maximum likelihood (ML) estimators are often biased and we therefore wish to compare to another method.

The second procedure (MCES estimators) has been developed by Akritas & Bershady (1996) and Akritas (1999). According to the authors, it yields unbiased estimators. However, the [FORMULA] are not used in the expression of the [FORMULA], which means that they have all the same weight. This is especially not satisfying if the intrinsic scatter is small or comparable to the uncertainties in the colors - which happens in particular for elliptical galaxies - and may give an excessive importance to outliers. The authors also do not provide the intrinsic scatter [FORMULA] and we estimate it from

[EQUATION]

but when the intrinsic scatter is smaller or of the same order than the uncertainties in the observables, it is underestimated 1 and we may even obtain a negative (and meaningless) value. We then assume [FORMULA]. Uncertainties on the [FORMULA] have been computed by a bootstrap with replacement on the [FORMULA] rather than by using the cumbersome formulae proposed by Akritas & Bershady (1996) and Akritas (1999), which anyway are not available for [FORMULA]. Note that the uncertainties determined either from the curvature matrix or by bootstrap become themselves unreliable when the number of galaxies is less than about 20 to 30.

In the following, values determined by the ML and MCES estimators are written in boldface and italic characters, respectively.

Because we are mainly interested in "normal" galaxies, we have to reject a few galaxies with unusual colors (cf. Buta et al. 1994). Most of them are extremely blue and, according to their optical colors (when available), are presumably starbursting galaxies. Some blue compact dwarfs may also be misclassified from their morphology as ellipticals. In the opposite, our sample contains some galaxies with very red colors, especially in the K-band and their NIR emission is dominated by an active nucleus. To reject these galaxies, we apply an iterative method. Rather than using the standard deviation around the best fit, which is very sensitive to the outliers, we estimate the observational scatter from the median of the absolute deviations. At each step, we compute

[EQUATION]

from the sample (including the "abnormal" galaxies), the factor 1.48 ensuring that in the case of a perfectly Gaussian distribution, the standard deviation is [FORMULA] (Wong 1997, p. 240); we define "normal" galaxies as those having [FORMULA] and then determine the [FORMULA] from them only by either of the methods discussed above. We iterate this procedure 4 times, but the convergence is usually achieved at the third one.

5.3. Optical-to-NIR colors: the color-type relation

The total [FORMULA] color has been plotted as a function of the morphological type in Fig. 2. A very impressive trend to bluer colors with advancing type is observed. The mean [FORMULA] of irregular galaxies is 1.35 mag bluer than the one of ellipticals (Table 5), to be compared to 0.68 in [FORMULA], 0.47 in [FORMULA], 0.46 in [FORMULA] and only 0.23 in [FORMULA] (Fioc & Rocca-Volmerange, in preparation). A similar gap between the [FORMULA] of ellipticals and irregulars was obtained in [FORMULA] by Buta (1995) from a sample of 225 galaxies.

[FIGURE] Fig. 2. Total [FORMULA] color as a function of the type. A random component in [FORMULA] has been added to the type to make the graph clearer. The diamonds are the standard colors determined by ML. The vertical segments show the intrinsic scatter and the horizontal ones the width of the type-bin.


[TABLE]

Table 5. Total and effective [FORMULA] colors per type: [FORMULA]. The values computed with the ML and MCES estimators are written respectively in boldface and italic characters. [FORMULA] is the intrinsic scatter. The numerical values in the first column are the mean RC3 types in each generic type.


The [FORMULA] colors determined by the ML estimator are almost always systematically redder by [FORMULA] mag than those computed using the MCES estimator. This bias comes from the fact that the uncertainties are smaller for brighter galaxies in the NIR, which are also usually redder. The ML estimator giving more weight to the data with smaller uncertainties, it is biased to the red.

A significant scatter is also observed within each type. Part of it is due to the observational uncertainties, but it comes mainly from the intrinsic scatter in the colors. The intrinsic scatter increases from [FORMULA], depending on the estimator, for ellipticals to [FORMULA] for Sb and remains nearly constant at later types.

5.4. Optical-to-NIR colors: the color-[FORMULA] relation

The color-inclination relation is potentially a very powerful constraint on the amount of dust and its distribution relatively to stars. As a disk becomes more and more inclined, its optical depth increases and the colors redden. This must be especially striking when one of the band (e.g. B) is heavily extinguished whereas the dust is almost transparent in the other one (e.g. H). For these reasons, various studies have tried to determine a color-inclination relation, regrouping the galaxies either as a function of type (e.g. Boselli & Gavazzi 1994) or as a function of their NIR absolute magnitude (Tully et al. 1998). The samples used in these studies where however small (about 100 galaxies) and it is worth to look at the relation once again with our much larger catalog.

For an oblate ellipsoid, which is a standard model for a galaxy disk (Hubble 1926), the inclination i (face-on corresponds to [FORMULA]) is related to the true ratio [FORMULA] of the minor to the major axis and to the apparent ratio [FORMULA] by [FORMULA].

Practically, we prefer to establish a relation between the colors and [FORMULA], both because [FORMULA] is the directly observed quantity and because the linear estimators used here provide a better fit when [FORMULA] is used rather than i or [FORMULA].

The relations are plotted in Fig. 3 and statistical estimators of the slope and the color are given in Table 6 for [FORMULA].

[FIGURE] Fig. 3. [FORMULA]-[FORMULA] relation per type. The solid and dashed lines correspond respectively to the ML and MCES estimators.


[TABLE]

Table 6. [FORMULA] color per type as a function of [FORMULA]: [FORMULA]. See Table 5 for other notations and conventions.


The slopes are represented as a function of type in Fig. 4. Also shown on this graph are the slopes derived from the much larger [FORMULA] (3063 galaxies) and [FORMULA] (2305 galaxies) samples built from the RC3.

[FIGURE] Fig. 4. Slopes of the [FORMULA]-, [FORMULA]-, and [FORMULA]-[FORMULA] relations as a function of type. The black and white squares correspond respectively to the ML and MCES estimators. The vertical segments are the [FORMULA]-uncertainties.

The typical slope of the [FORMULA]-[FORMULA] relation is about 1 for spiral galaxies, to be compared to 0.2 or 0.3 in [FORMULA] and [FORMULA]. The value of the slope is surprisingly high for Sa galaxies. However, the slope of the [FORMULA]-inclination relation would be shallower for Sa since [FORMULA] is a decreasing function of type (Bottinelli et al. 1983), i.e., Sa galaxies are thicker than later types, as also indicated by their smaller median [FORMULA].

A dip is observed for Sbc galaxies, which seems strange since the maximum extinction is expected precisely for this type (Fioc & Rocca-Volmerange 1997). Though consistent with a constant slope for all spirals, given the uncertainties, a similar dip also appears in the slopes of the [FORMULA]- and [FORMULA]-[FORMULA] relations, suggesting that this might be a real phenomenon. One should first remember that the slope is not a measure of the total extinction, but of the reddening relative to face-on . With a simple radiative transfer model where the stars and the dust are distributed homogeneously in an infinite slab, we find that when the face-on optical depth [FORMULA] in the V-band increases, the slope begins first to steepen (Fig. 5b), but when [FORMULA] becomes larger than 1 or 2, the slope flattens while the colors go on reddening (Fig. 5a). Although this modeling is simplistic and we cannot infer from it the value of the optical depth, this suggests that the extinction may indeed be higher for Sbc and that the slope has begun to decrease. This suggestion is reinforced by the fact that the relation for Sbc is flatter when we consider effective [FORMULA] colors ([FORMULA]), which are probably more extinguished than total colors. Now, the existence of the dip in [FORMULA] depends on a handful of galaxies, so definitive conclusions may not be drawn yet.

[FIGURE] Fig. 5. [FORMULA]-[FORMULA] relations computed with the radiative transfer code (see text) for different face-on V-band optical depths. We assume [FORMULA], a typical value for an Sb. Top: reddening relative to the dust-free case. Bottom: reddening relative to face-on.

We observe a positive slope for irregular galaxies in [FORMULA], contrary to [FORMULA] and [FORMULA] where none is detected. However, the uncertainties are large and the null slope is within 2[FORMULA]. Moreover, our "Im" type regroups in fact not only Im galaxies ([FORMULA]) but also Sm ([FORMULA]), which are more elongated and redder. The small size of the "Im" sample makes it impossible to decide whether this reddening is due to the dust or to intrinsically redder populations in Sm galaxies.

No slope is observed for lenticular galaxies, which may indicate that no significant amount of dust is present in S0 (see also Sandage & Visvanathan 1978). Another possibility is that the disk is overwhelmed by the bulge. No inclination dependence of the colors would then be expected if the bulge is spherically symmetric or if it is devoid of dust.

The most striking result is the negative slope observed for ellipticals. This certainly does not come from the dust because the slope would be positive. For ellipticals, [FORMULA] is not directly related to the inclination because their true shape is unknown. In the mean, rounder ellipticals seem redder and are probably more metal-rich (Terlevich et al. 1980). They also tend to be brighter (Tremblay & Merritt 1996) which may link the color-[FORMULA] relation of ellipticals to their color-magnitude relation. No such relation is observed in [FORMULA], which is anyway a poor indicator of the metallicity. The [FORMULA]-[FORMULA] is not very conclusive: whereas the ML estimator provides a negative slope, as in [FORMULA], the MCES estimator does not detect any. Since no obvious bias between the two estimators is observed for the other types and the intrinsic scatter is very small compared to the uncertainties on the colors for ellipticals - which is the worst case for the MCES estimators -, we tend to trust here more the result given by ML.

5.5. Optical-to-NIR colors: the color-magnitude relation

5.5.1. Global relation

The color-magnitude (CM) relation (Fig. 6) is an important constraint on the dynamical models of galaxy formation because the absolute magnitude may be related to the mass of the galaxy via a mass-to-luminosity ratio. The CM relation has also been proposed as a distance indicator (e.g. Sandage 1972; Visvanathan 1981; Tully et al. 1982). An important question is the choice of the reference band, optical or NIR, used for the absolute magnitude. Usually, the NIR has been adopted because it suffers little extinction. Another reason is that it is produced mainly by old giants and is therefore expected to be a better indicator of the mass than the optical, which is more sensitive to the recent star formation. The estimators of the CM relation with either the optical or the NIR as reference band are given in Table 7. The [FORMULA]-[FORMULA] relation is much steeper than the [FORMULA]-[FORMULA] one and its intrinsic scatter is smaller. If there were no intrinsic scatter, the slopes should be nearly the same. For example, if [FORMULA], then we should have [FORMULA], which has almost the same slope since [FORMULA]. The difference is due to the mixing of types with different slopes (see 5.5.2) and to the internal extinction which reddens the galaxies but makes also them fainter in B. The slope of the [FORMULA] relation is even positive in some spiral types! To detect a color-magnitude relation, the choice of the NIR as reference band is thus clearly favored.

[FIGURE] Fig. 6. Global [FORMULA]-[FORMULA] relation.


[TABLE]

Table 7. [FORMULA] [[FORMULA], H or K] vs. [FORMULA] or [FORMULA] color-magnitude relations: [FORMULA] or [FORMULA]. See Table 5 for other notations and conventions.


Though the overall agreement is very good, we note that the ML estimator of the slope is slightly biased when compared to the a priori unbiased MCES estimator. Its slope is "more positive" in the [FORMULA]-[FORMULA] relation because of the positive correlation of the errors in [FORMULA] and [FORMULA], and "more negative" in the [FORMULA]-[FORMULA] relation because of the negative correlation of the errors in [FORMULA] and [FORMULA].

The slope of the [FORMULA]-absolute magnitude relation increases with the wavelength of the NIR band for there is also a CM relation in the NIR colors. This effect is less apparent in [FORMULA] because the [FORMULA] sample contains many late-type galaxies which have a flatter slope: a linear fit for all the galaxies together is hence too simplistic.

5.5.2. Relation per type

The global color-magnitude relation discussed above is actually dominated by star-forming galaxies ([FORMULA] for Sa-Im against [FORMULA] for E-S0). These values are close to the slopes of the [FORMULA]-[FORMULA] relations ([FORMULA] for Sa-Sdm and [FORMULA] for E-S0) determined by Mobasher et al. (1986). The slope for spiral galaxies is much steeper than the relations obtained by Bershady (1995) for its spectral types bk, bm, am and fm. This is however not surprising, because using spectral types rather than photometric types bins galaxies as a function of their colors and tends to suppress any color-magnitude relation.

An especially interesting question is whether the mass is the main driving force of galaxy evolution and star formation, or if some other quantity possibly related to the morphological type - e.g. the bulge-to-disk ratio (Simien & de Vaucouleurs 1986) or the angular momentum (Caimmi & Secco 1986; Mao & Mo 1998) -, has also to be considered. To answer this question, it is worth to look at the color-magnitude relation per type more thoroughly.

We give in Table 8 the [FORMULA]-[FORMULA] and [FORMULA]-[FORMULA] relations as a function of type, where [FORMULA] is the color corrected to face-on using the [FORMULA] determined in 5.4. We assume that the extinction is negligible in the NIR and do not correct [FORMULA]. No correction is also applied to the colors of ellipticals because their color-[FORMULA] relation is not due to the extinction. We note that the intrinsic scatter in the corrected colors is significantly reduced. A more picturesque comparison of the relations for the different types is plotted in Fig. 7. Though the agreement between the estimators is not as good as previously and the uncertainties are large, the slope obviously depends on the type. We tentatively summarize the following results:

[FIGURE] Fig. 7. Analytical [FORMULA]-[FORMULA] relations. The segments correspond to the mean of the ML and MCES fits and extend from [FORMULA] to [FORMULA] for each type.


[TABLE]

Table 8. [FORMULA] and [FORMULA] per type as a function of [FORMULA]: [FORMULA] or [FORMULA]. See Table 5 for other notations and conventions.
Notes:
[FORMULA]) [FORMULA] is not available for a few (4) S0 galaxies and their [FORMULA] cannot be computed. The first value of N or [FORMULA] is for the fit of [FORMULA] and the second for [FORMULA].


Star-forming galaxies. The slope is much flatter for Sd-Im than for Sa-Sc galaxies. Mobasher et al. (1986) probably did not detect this because the few Sd-Im galaxies in their "late-spirals" sample were lost among Sbc and Sc galaxies. At a given NIR absolute magnitude, the mean colors of the types are different, indicating that the CM relation is not simply a type-mass relation: the NIR intrinsic luminosity must be completed by the type to characterize the colors (and the star formation history) of star-forming galaxies. A similar conclusion was drawn by Gavazzi (1993) but was later challenged by Gavazzi et al. (1996b).

E-S0. The color-magnitude relation of early-type galaxies is usually explained by the galactic winds produced by supernovae in a starbursting environment, which expel the gas and quench the star formation (Matthews & Baker 1971). Massive (and NIR bright) galaxies having a deeper gravitational well, they are able to retain the gas longer and to prolong the star-forming phase. This results in a higher mean stellar metallicity and thus in redder colors (Faber 1973).

We find a different slope for ellipticals and lenticulars. The slope of S0 seems closer to that of early and intermediate spirals than to that of ellipticals, which may give some clues on their evolution.

The MCES estimation does not detect any significant slope for ellipticals. Because of the small intrinsic scatter, it is maybe not the best estimator 2, but even the slope determined by ML is smaller than that of the comparable [FORMULA]-[FORMULA] relation established by Bower et al. (1992) in the Virgo and Coma clusters. We note however that their relation is based on aperture colors. Their aperture is comparable to the mean effective aperture of ellipticals in Virgo and samples only about half their luminosity (cf. also Kodama et al. 1997). Since the effective aperture of bright ellipticals is larger, only their inner regions are observed in the Bower et al.'s aperture. Because of the blue-outwards color gradient, their aperture color is redder and the slope is higher than the one derived from total colors. We therefore encourage the modelists of elliptical galaxies, either to compare their relations to total observed colors, or to predict aperture colors.

Our intrinsic scatter is also larger than that found by Bower et al. (1992). This might be due to an underestimation of the uncertainty on the aperture magnitudes [FORMULA] computed in Sect. 3: [FORMULA] was indeed evaluated only from the galaxies used in the construction of the growth curves, but the other ones might suffer from higher uncertainties. This would result in an underestimation of the uncertainties on the total magnitudes and colors and thus in an overestimation of the intrinsic scatter. Another possible explanation of this discrepancy is that our sample contains not only cluster but also field galaxies, which may have a different star formation history and increase the scatter (Larson et al. 1980; Kauffmann & Charlot 1998; Baugh et al. 1996; see yet Bernardi et al. 1998 and Schade et al. 1996). Note, however, that errors on the distances of field galaxies due, e.g., to large-scale motions might also artificially enhance the scatter. Such phenomenon does not affect the CM relation of a given cluster.

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Online publication: November 16, 1999
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