SPIDER. IV. OPTICAL AND NEAR-INFRARED COLOR GRADIENTS IN EARLY-TYPE GALAXIES: NEW INSIGHT INTO CORRELATIONS WITH GALAXY PROPERTIES

Galaxy stellar masses can be estimated by fitting their observed spectral energy distributions (SEDs) with stellar population synthesis models. Brinchmann & Ellis (2000) demonstrated that such techniques, combining optical and near-IR fluxes with spectroscopic redshifts, could be used to derive masses with a factor of two uncertainty to z ∼ 1. Local studies by Bell et al. (2003) and Cole et al. (2001) examined the stellar mass function using similar methods. Salim et al. (2005, 2007) have shown that galaxy properties derived from SED fitting with broadband colors are consistent with those derived from spectroscopic data. Thus, we use SED fitting to derive galaxy stellar masses.

We employ the LePhare code (S. Arnouts & O. Ilbert) used by the COSMOS survey (Ilbert et al. 2009) to fit a suite of Bruzual & Charlot (2003, hereafter BC03) model SEDs to our observed 8 band photometry and the known spectroscopic redshift. LePhare performs a χ² minimization between the observed fluxes (and their associated errors) and those synthesized from the models. Because early types are unlikely to have significant ongoing star formation, we use a simple set of BC03 SEDs, with six different star formation e-folding times τ, five internal reddenings E(B − V), and four metallicities Z/Z_☉, for a total of 120 possible models. The values of these parameters are given in Table 1. We assume a Calzetti et al. (2000) extinction law and Chabrier (2003) initial mass function (IMF). The reddening E(B−V) is limited to 0.5 mag to avoid incorrect fitting of observationally red galaxies as highly reddened blue galaxies; such objects should be absent from our sample. Because each BC03 template is normalized to one solar mass, we use the observed K-band flux to determine the normalization of the best-fit template which then gives the stellar mass. We use the K band for the mass scaling since it is only weakly affected by dust extinction and is quite insensitive to the presence of young, luminous stars (Kauffmann & Charlot 1998; Bundy et al. 2005).

Galaxy fluxes are estimated in an adaptive aperture of radius 3 × r_K,i in all wavebands, where r_K,i is the i-band Kron radius (see Paper I). We use these aperture fluxes as they have a better accuracy than those obtained from (2DPHOT) total magnitudes. In order to derive total stellar masses, M_⋆, we apply an aperture correction. We calculate the fraction of total to 3 × r_K,i aperture flux in the K band, and then multiply the estimated stellar mass by this fraction. This procedure assumes that stellar mass in a galaxy is distributed in the same way as the K-band light.

Rather than use an individual best-fitting SED (based on the χ²), we use a mass computed from the model corresponding to the median $e^{-\chi ^{2}/2}$. This reduces stochastic mass errors due to individual models which happen to have very low χ² values, since some degeneracy remains in the SED fits even with eight filters. A small number of galaxies (1465 of 39,993 with optical data only, and 286 of 4546 with NIR data) are not well fit by any of the SEDs, resulting in $e^{-\chi ^{2}/2} = 0$ and thus no stellar mass estimate. These objects, comprising 3.6% and 6.3% of the optical and optical+NIR samples, respectively, are excluded from analyses that require the stellar masses.

We tested our stellar masses by comparing to those derived by a group at Max Plank using SDSS photometry alone.⁷ The overall agreement is excellent. We find a modest offset in $M_{*,\rm{SPIDER}} - M_{*,\rm{MPI}}$ of 0.14 dex; this is because we use Kron magnitudes rather than fiber magnitudes; Kron magnitudes are ∼0.5 mag brighter, which translates into a difference ∼0.2 in log(M). The rms scatter between the two mass estimators is only 0.1 dex, a factor of ∼2 smaller than most systematic errors and biases introduced by different choices of IMF, extinction law, etc. More details of the SED fitting and potential sources of errors and biases in the stellar mass estimates will be presented in an upcoming paper in this series devoted to this issue alone (R. Swindle et al. 2010, in preparation).

We note that both the SED fitting and the analysis of color gradients (Section 7) are performed with a different set of models (i.e., BC03) than that used for the spectral analysis (i.e., MILES, see Section 3.3). The BC03 models provide a high resolution of 1 Å in the wavelength range 3322 Å–9300 Å (griz), with a median resolution of 10 Å in zYJHK. We do not use the MILES library here as we do for other tests in this paper due to its limited wavelength coverage (3525 Å to 7500 Å). With 7500 Å lying near the central wavelength of the i band, ignoring the izYJHK data inevitably yields failures in SED identification, provided the range of model parameters given in Table 1. This uncertainty, coupled with the benefit of the K-band flux for mass scaling, motivates this choice of models for the SED fitting. The technical justification for using the MILES library in analyzing spectra and color gradients is also detailed in Section 3.3.

Dynamical masses, M_dyn, are obtained from galaxy effective radii and velocity dispersions. Applying the virial theorem, we write

$$\begin{equation} M_{\rm dyn} = A \times \sigma _0^2 r_{e,K}, \end{equation} \tag{ 1 }$$

where r_e,K is the effective radius in K band, and A is a scale factor that accounts for the internal structure of ETGs. Most previous works estimated A by assuming a single universal model for the light distribution in ETGs, such as the King model (Jørgensen et al. 1996). However, this does not account for the variety of ETG light profile shapes (e.g., Caon et al. 1993), as parameterized by the Sersic index. To account for this, following a similar procedure to that of D'Onofrio et al. (2008), we estimate A using de-projected Sersic galaxy models. The models and the de-projection procedure are detailed in Coppola et al. (2009). The models describe spherical, non-rotating, isotropic galactic systems. We also assume that the spatial distribution of dark matter follows that of the stellar component. Under these assumptions, each model consists of a single Sersic component characterized by three parameters: the half-mass radius, the total mass, and the Sersic index (see Coppola et al. 2009 for details). The use of more complex models (i.e., those with two components) is well beyond the scope of this paper, and we postpone it to a forthcoming contribution. For each ETG, we construct the corresponding Sersic model, using the effective radius and Sersic index of the galaxy in K band. We estimate A by computing the ratio of the quantities M_dyn and σ²₀ × r_e,K for the given model. We compute σ₀ by projecting the model in two dimensions, within a circular aperture of radius 15, i.e., the same aperture as the SDSS fibers. Since we do not have K-band structural parameters for the optical sample of ETGs, M_dyn is only estimated for galaxies in the optical+NIR sample. Figure 1 shows the dynamical masses versus the Petrosian magnitudes, from which we see that our limiting magnitude corresponds to a lower mass limit of ∼3 × 10¹⁰ M_☉.

Zoom In Zoom Out Reset image size

As described in Paper I, we have re-measured velocity dispersions for ETGs in the SPIDER sample using STARLIGHT (Cid Fernandes et al. 2005). This software determines the linear combination of simple stellar population (SSP) models which, broadened with a given σ, best matches the observed galaxy spectrum. Hereafter, we refer to the set of SSP models provided as input to STARLIGHT as the SSP basis, and to the output linear combination of SSPs as the synthetic spectrum. For the purpose of remeasuring the σ₀'s, we run STARLIGHT with a basis of SSPs from the MILES galaxy spectral library (Vazdekis et al. 2010), covering a wide range of ages and metallicities, but with a fixed solar abundance ratio ([α/Fe] = 0).

Here we also use STARLIGHT to estimate the Age, [Z/H], and [α/Fe] of each ETG. In contrast to the more common practice of using index–index diagrams to estimate SSP-equivalent stellar population parameters,⁸ STARLIGHT uses all of the information present in the data, by fitting a set of SSP models to the entire galaxy spectrum. A detailed comparison of stellar population parameter estimates from different techniques for the SPIDER sample is currently under way, and will be the subject of a forthcoming paper in this series. Here, we provide only the essential information on how we use STARLIGHT to derive Age, [Z/H], and [α/Fe]. For each galaxy, we run STARLIGHT using the α-enhanced MILES (hereafter α-MILES) spectral library (Cervantes et al. 2007). As detailed in Appendix A, the α-MILES library extends the MILES galaxy spectral library to SSP models with non-solar abundance ratios, in the [α/Fe] range of −0.2 to +0.6, allowing the spectra of ETGs to be modeled with better accuracy than that achieved with solar-abundance MILES models. The α-MILES SEDs cover the same spectral range (3525–7500 Å), with the same spectral resolution (2.3 Å), as the (solar abundance) MILES library. Hence, they are well suited to analyzing SDSS spectra, whose spectral resolution is ∼2.36 Å FWHM, in the wavelength interval 4800–5350 Å. To run STARLIGHT, we select a basis of 176 α-MILES SSPs, with ages ranging from 1 to ∼14 Gyr in 11 steps, metallicities of [Z/H] = −0.68, − 0.38, 0.0, 0.2, and enhancements of [α/Fe] = +0.0, + 0.20, + 0.40, + 0.60. This grid in Age, [Z/H], and [α/Fe] allows us to cover a wide range of stellar population properties without overly increasing the execution time required to run STARLIGHT.⁹ For each galaxy, we first smooth the basis models to match the wavelength-dependent resolution of the SDSS spectrum (see Paper I for details). STARLIGHT outputs the light percentage, X, of each SSP basis model in the output synthetic spectrum. Hence, for a given property Y (e.g., Age), one can estimate its luminosity-weighted value, Y_L, as

$$\begin{equation} Y_L = \frac{\sum Y \times X}{\sum X}, \end{equation} \tag{ 2 }$$

where the summation is performed over all the SSP basis models. The Age is obtained directly from Equation (2). Since metallicity and enhancement are logarithmic quantities, we first apply Equation (2) by setting Y_L = 10^[Z/H] and Y_L = 10^[α/Fe], and then compute the logarithm of Y_L in both cases. The uncertainties on Age, [Z/H], and [α/Fe] are computed by comparing the estimates of these quantities for 2313 galaxies with repeated observations in SDSS, following the same procedure described in Paper I to obtain the errors on σ₀. The average uncertainties are ∼20% on Age, and ∼0.05 for [Z/H] and [α/Fe].

An important potential bias when using the spectra from SDSS is the effect of a fixed spectroscopic fiber aperture on the Age, [Z/H], and [α/Fe] estimates. The fraction of galaxy light inside the fiber will vary with galaxy size and redshift. Because we select the SPIDER sample based on the eClass parameter, the galaxies are selected to have passive spectra in their centers. This means that, as discussed in Paper I, our putative early-type sample could be contaminated by early spiral systems, i.e., spiral galaxies with a prominent bulge component. Indeed, in Paper I we show that this contamination is about 15%, falling to 5% for a higher quality sample defined on the basis of visual image classification. As we discuss at the end of Section 8.3 (see Figure 16), the trends with stellar population properties remain unchanged when restricting the SPIDER sample to the lower contamination subsample. The reverse might also hold, i.e., we might be missing ETGs with strong emission features localized to their cores. However, very few ETGs are expected to have spectra exhibiting such strong central emission.

In Figure 2, we display how Age, [Z/H], and [α/Fe] vary with the fraction of light inside the fiber with respect to the total light, L(r <r_fiber)/L_tot, along with the distribution of this fraction, showing that ∼65% of the galaxies have a ratio between 0.2 and 0.6. The median trends, negligible compared to the scatter in Age, [Z/H], and [α/Fe], show that the derived stellar population parameters are not affected by the aperture effect, reinforcing what we have presented in Appendix B, i.e., that the average r_e does not change much with respect to the stellar population parameters, implying that the correlation of color gradients with stellar population properties has no dependence on r_e.

Zoom In Zoom Out Reset image size

The values of Age, [Z/H], and [α/Fe] used here are not SSP-equivalent parameters, as commonly used in the literature, but are luminosity-weighted quantities inferred from the galaxy spectra. As mentioned above, a detailed comparison of SSP-equivalent and luminosity-weighted parameters will be performed in a forthcoming contribution.

We consider two stellar populations characterized by different physical parameters, such as age, metallicity, α-enhancement, etc. In general, the problem is how one can use the difference between the color indices of the two stellar populations to infer the corresponding difference(s) in physical parameters. The color index difference may be (1) radial—the internal color gradient, (2) between galaxies with different masses—the Fundamental Plane coefficients in different wavebands (see La Barbera et al. 2010b; hereafter Paper II), (3) in luminosity—the slope of the color–magnitude relation, or (4) in the residuals in luminosity about a given scaling relation. We first describe the general approach we adopt, and then, in Section 7, we focus on its application to point (1).

We indicate as Δ_i the variation of a given physical property between the two stellar populations, where the index i runs over the considered set of properties. The two stellar populations have differences in colors δ_j, where j spans the set of colors defined by the available wavebands. One can introduce the following approximations:

$$\begin{eqnarray} \delta _j & = & \sum _i A_{ij} \cdot \Delta _i, \end{eqnarray} \tag{ 3 }$$

where A_ij is the partial derivative of the jth color index with respect to the ith stellar population parameter. The above equation holds if (a) the color indices are well-behaved functions of the stellar population parameters and either (b1) the Δ_i are small enough to assume a linear approximation or (b2) the color indices are linear functions of the stellar population parameters. The quantities A_ij have to be evaluated at the average values of the stellar population parameters of the two stellar populations. Equations (3) can be solved in a χ² sense, by minimizing the expression:

$$\begin{eqnarray} \chi ^2 & = & \sum _i \left(\delta _i - \sum _k A_{ki} \cdot \Delta _k \right)^2. \end{eqnarray} \tag{ 4 }$$

Requiring that the derivatives of χ² with respect to Δ_j vanish, one obtains the following system of equations:

$$\begin{eqnarray} \sum _k \left(\sum _i A_{ki} A_{ji} \right) \Delta _k & = & \sum _i A_{ji} \delta _i. \end{eqnarray} \tag{ 5 }$$

Setting M_jk = (∑_iA_kiA_ji) · (∑_iA_jiδ_i)⁻¹, we obtain a new system of N linear equations in the N unknown stellar population quantities Δ_j:

$$\begin{eqnarray} &&\sum _j M_{ij} \cdot \Delta _j = 1, \end{eqnarray} \tag{ 6 }$$

where the quantities M_ij are completely defined by the partial derivatives A_ij and the measured color differences δ_i. If the matrix M_ij is non-degenerate, we can invert the linear system of Equations (6) and derive a unique solution for the unknown quantities Δ_i (e.g., differences in age, metallicity) from the measured color differences δ_j.

In this section, we investigate how the color indices in SPIDER can be used to constrain variations of stellar population properties in ETGs, specifically the effects of age and metallicity. Since we analyze color gradients in the form ∇_g−X (Section 4), we consider here color indices of the form g − X, where X is one of rizYJHK. The g-band includes the 4000 Å break at the median redshift of the SPIDER samples, making the g − X colors most sensitive to the effects of stellar population variations.

For the optical sample, each ETG is characterized by three independent colors, g − r, g − i, and g − z. Assuming that the relevant stellar population parameters are the age t and metallicity Z, Equations (3) can be rewritten as

$$\begin{eqnarray} &&\delta _{g-r} = \frac{\partial (g-r)}{\partial {\log t}} \cdot \Delta _{t} + \frac{\partial (g-r)}{\partial {\log Z}} \cdot \Delta _{Z} \end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray} \delta _{g-i} & = & \frac{\partial (g-i)}{\partial {\log t}} \cdot \Delta _{t} + \frac{\partial (g-i)}{\partial {\log Z}} \cdot \Delta _{Z} \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \delta _{g-z} & = & \frac{\partial (g-z)}{\partial {\log t}} \cdot \Delta _{t} + \frac{\partial (g-z)}{\partial {\log Z}} \cdot \Delta _{Z}, \end{eqnarray} \tag{ 9 }$$

where Δ_t and Δ_Z are the logarithmic differences of age and metallicity between two stellar populations. As noted above (see beginning of Section 6.1), the quantities δ_g−r, δ_g−i, and δ_g−z could be the differences of color indices between galaxies having different parameters (e.g., mass), although here we focus on the case where δ_g−X is the difference in color index per radial decade in a galaxy, i.e., the internal color gradient (δ_g−X ≡ ∇_g−X). In this case, the quantities Δ_t and Δ_z provide the difference in age and metallicity per radial decade (i.e., Δ_t ≡ ∇_t and Δ_z ≡ ∇_z, see Section 7.1). Using the formalism of Section 6.1, Δ_t and Δ_Z are derived by solving the linear system:

$$\begin{eqnarray} &&M_{t,t} \Delta _{t} + M_{t,Z} \Delta _{Z} = 1 \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} &&M_{Z,t} \Delta _{t} + M_{Z,Z} \Delta _{Z} = 1, \end{eqnarray} \tag{ 11 }$$

where the matrix M is computed directly from the partial derivative of color indices with respect to log t and log Z, and the measured color differences δ_g−r, δ_g−i, and δ_g−z. For the optical+NIR sample of ETGs, the system of Equations (7)–(9) includes four extra equations, corresponding to the g − Y, g − J, g − H, and g − K color indices. The quantities Δ_t and Δ_Z are still obtained by solving the system of Equations (10) and (11), where the matrix M is defined by using the color differences in all available wavebands.

Since the NIR light is far less sensitive to metallicity (through line-blanketing) than the optical, and is also less sensitive to age, we expect that including the NIR wavebands helps to (partly) break the age–metallicity degeneracy, i.e., the degeneracy of the matrix M is reduced when including all the g − X color indices. To examine this, we set δ_i = 0.01 as a typical uncertainty in color differences. We adopt this value of δ_i as it is an upper bound for the typical uncertainty on the mean value of color gradients. In fact, the typical error on the color gradient of a given galaxy is ∼0.2–0.3 (Table 2). For the optical+NIR sample of ETGs, this implies a typical uncertainty on the mean color gradient of ∼0.003–0.005.¹⁰ In order to estimate the derivatives of color indices with respect to age and metallicity, we adopt a polynomial approximation of the magnitude of a given stellar population model with respect to t and Z. Details of this approximation are provided in Appendix C. Here, we consider SSP models from the Bruzual & Charlot (2003) synthesis code (hereafter BC03) with a Scalo IMF. We compute the matrix $\bf M$ for different values of t and Z of these SSP models, in the range where the polynomial approximation holds, i.e., $5\!<\! t (\rm Gyr)\!<\!13.5$ and 0.2 < Z/Z_☉ < 2.5. Figure 4 plots the determinant of the matrix M, $q = \mbox{det}(\bf M)$, as a function of log t for different metallicities. The case where only optical colors are used is shown by the dashed curves. q is close to one for non-solar metallicity, while it is always smaller than one (by almost an order of magnitude) in the case of Z = Z_☉. This implies that for solar metallicity the matrix $\bf M$ is nearly degenerate and hence, as expected, optical colors alone are not able to effectively constrain age–metallicity variations. The solid curves show the cases where all the optical+NIR colors are used. q is then always greater than ∼10. In particular, for solar metallicity, q is two orders of magnitude greater than with optical colors alone, implying that using all of the grizYJHK wavebands places much better constraints on age and metallicity variations.

Zoom In Zoom Out Reset image size

To illustrate this point more quantitatively, we perform Monte Carlo simulations, setting Δ_t = 0.05 and Δ_Z = −0.3, which correspond to two stellar populations with a difference in age of ∼11% and a factor of two difference in metallicity. These values of Δ_t and Δ_Z are close to those of the mean age and metallicity internal gradient of ETGs measured by LdC09. Using Equations (3), we compute the color differences, δ_i, between the two stellar populations for all the seven g − X colors. The quantities A_ij are computed for $t = 10\, \rm Gyr$ and Z = Z_☉. For each iteration, the color indices are shifted according to a normal distribution with a width of σ_δ, and then we estimate Δ_t and Δ_Z with the procedure outlined above. The procedure is repeated twice, adopting (1) σ_δ = 0.002 and using only the griz colors, and (2) with σ_δ = 0.005 and all the grizYJHK bands. The above σ_δ's represent the typical uncertainties on the mean internal color gradients in the optical and optical+NIR SPIDER samples of ETGs (see the errors on μ reported in Tables 3 and 2, respectively). Figure 5 shows the distributions of inferred Δ_t and Δ_Z. In both cases, there is a strong correlation between Δ_t and Δ_Z, implying that even with the full grizYJHK filter set, one cannot completely break the age–metallicity degeneracy. However, the amplitude of the correlated variation is much smaller in the case where the optical and NIR bands are adopted. In fact, although for case (2) σ_δ is larger than in case (1), the corresponding error bars on Δ_t and Δ_Z are much smaller, only ∼0.01 for both t and Z. In other terms, optical+NIR colors constrain the size of the age and metallicity differences more effectively. From Figure 5, we conclude that even a small age gradient of ∼11% can be detected at ∼2.5σ with the optical+NIR data set.

Zoom In Zoom Out Reset image size

We adopt the procedure described in Section 6.2 to infer the radial gradients in age (∇_t) and metallicity (∇_Z). Setting δ_g−X = ∇_g−X in Equations (7)–(9), the quantities Δ_t and Δ_Z become the logarithmic differences in age and metallicity per radial decade inside a galaxy, i.e., Δ_t ≡ ∇_t and Δ_Z ≡ ∇_Z. Equations (7)–(9) are then rewritten as follows:

$$\begin{eqnarray} &&\nabla _{g-X} = \frac{\partial (g-X)}{\partial {\log t}} \cdot \nabla _{t} + \frac{\partial (g-X)}{\partial {\log Z}} \cdot \nabla _{Z}, \end{eqnarray} \tag{ 12 }$$

with X = rizYJHK. As shown in Section 6.2, in order to derive ∇_t and ∇_Z, one has to solve the linear system:

$$\begin{eqnarray} M_{t,t} \nabla _{t} + M_{t,Z} \nabla _{Z} & = & 1 \end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray} M_{Z,t} \nabla _{t} + M_{Z,Z} \nabla _{Z} & = & 1, \end{eqnarray} \tag{ 14 }$$

where the matrix M is defined as in Section 6.1, setting δ_j = ∇_g−X. The computation of M is performed using different stellar population models. For each model, we use Equations (13) and (14) to derive the corresponding ∇_t and ∇_Z. This procedure, as presented in Section 6, minimizes the age–metallicity degeneracy depending on how many color terms are available and how they track the most conspicuous spectral features (e.g., g − r probes the 4000  Å break, yielding important information on the age of the underlying stellar population).

We consider SSP models from three different sources: BC03, Maraston (2005, M05), and S. Charlot & G. Bruzual (2010, in preparation; CB10). These models are based on different synthesis techniques and have different IMFs. The M05 model uses the fuel consumption approach instead of the isochrone synthesis of BC03 and CB10. The CB10 code implements a new asymptotic giant branch phase treatment (Marigo & Girardi 2007). The IMFs are Scalo (BC03), Chabrier (M05), and Salpeter (CB10). Moreover, we also use a composite stellar population model from BC03 having an exponential star formation rate (SFR) with an e-folding time of τ = 1 Gyr (hereafter $\mbox{BC}03_{\tau = 1\rm Gyr}$). The models are convolved with the grizYJHK throughput curves, and the polynomial approximation described in Appendix C is used to calculate the partial derivatives entering in M. The derivatives are computed assuming an age of t = 9.27 Gyr, corresponding to a formation redshift of z = 2.5 at the median redshift of the SPIDER sample (〈z〉_median ∼ 0.07), and solar metallicity (Z = Z_☉). For each stellar population model and each galaxy, we perform N = 1000 iterations where in each of them we perturb the observed color gradients according to the corresponding uncertainties. The errors on ∇_t and ∇_Z are then the widths of their distributions resulting from all iterations. Note that in the above procedure all of the models are computed at the median redshift of our sample (z = 0.0725), neglecting the (small) redshift range of the sample. In Appendix D, we show that this approximation does not affect our results.

As discussed in Section 6.2, even using the entire grizYJHK set of wavebands we cannot break the age–metallicity degeneracy. This is further shown in Figure 6, where we plot the age versus metallicity gradients obtained with the above procedure, for the optical+NIR sample, with the BC03 SSP models. The ∇_Z and ∇_t estimates are strongly correlated. As shown from the blue dashed line in Figure 6, obtained from an orthogonal least-squares fit of ∇_Z versus ∇_t, the correlation is well described by a linear relation:

$$\begin{equation} \nabla _Z \propto \alpha \cdot \nabla _t. \end{equation} \tag{ 15 }$$

The correlation is very similar to that between Δ_Z and Δ_t in Figure 5. This is shown by the red dotted line in Figure 6, which is obtained by a linear fit of ∇_Z versus ∇_t with slope fixed to that of the red dot-dashed line in Figure 5. The slopes of the blue dashed and red dotted lines differ by only 4%, implying that the strong correlation of ∇_t and ∇_Z is due mainly to correlated errors on differences in age and metallicity, such as the one between Δ_Z and Δ_t shown in Figure 5. In other terms, individual ∇_Z and ∇_t estimates are affected by large, correlated statistical uncertainties because of the age–metallicity degeneracy (see the error ellipses in Figure 6). However, as shown in Section 6.2, using optical+NIR data allows unbiased estimates of the mean ∇_Z and ∇_t to be obtained, provided that the number of galaxies is large enough to make statistical uncertainties on the means small enough relative to the absolute values of the differences in age and metallicity (i.e., ∇_t and ∇_Z). This allows us to bin galaxies according to different properties and perform a meaningful comparison of the mean ∇_t and ∇_Z among different bins (see Section 9).

Zoom In Zoom Out Reset image size

In order to avoid the correlated uncertainties on individual ∇_t and ∇_Z estimates, we can define an effective color gradient:

$$\begin{equation} \nabla _{\star } = \nabla _Z - \alpha \cdot \nabla _t. \end{equation} \tag{ 16 }$$

From a geometrical viewpoint, ∇_⋆ measures the distance of a given point in the (∇_Z, ∇_t) plane along the ∇_Z direction to the line defining the age–metallicity degeneracy. By definition, the correlated variation of ∇_Z and ∇_t, which is described by Equation (15), does not change the ∇_⋆. Hence, the effective color gradient is not affected by the age–metallicity degeneracy. For galaxies having no age gradients, ∇_⋆ reduces to the galaxy metallicity gradient. Moreover, since α is negative (see below), Equation (16) can be rewritten as

$$\begin{equation} \nabla _{\star } = \nabla _Z + |\alpha | \cdot \nabla _t. \end{equation} \tag{ 17 }$$

This shows that when ∇_Z=0, ∇_⋆ is directly proportional to the age gradient. More generally, Equation (17) shows that ∇_⋆ behaves like a color gradient. A bluer outer stellar population relative to the inner one in a galaxy (because of either a metallicity or age gradient) yields a lower ∇_⋆. This is further shown in Figure 7, where we plot ∇_⋆ as a function of all the color gradients (see Section 4) for the optical+NIR sample of ETGs. On average, ∇_⋆ monotonically increases as a function of each ∇_g−X. In essence, ∇_⋆ can be seen as a color gradient that takes into account all of the colors.

Zoom In Zoom Out Reset image size

Using ∇_⋆ provides three important advantages over traditional color gradients. First, because ∇_⋆ combines all of the color gradient, its statistical uncertainty is significantly smaller than that on any color gradient using one pair of passbands. Second, different samples (bins) of galaxies can be compared using just one quantity (∇_⋆), rather than separately for each color gradient. Finally, as we prove below (Section 7.3), the effective color gradient is a completely model-independent quantity (in the same way as the observed color gradients), which is not the case for ∇_Z and ∇_t.

For each stellar population model, we derive α via an orthogonal least-square fit of ∇_Z versus ∇_t for all galaxies. Then we calculate ∇_⋆ for each galaxy, using Equation (16). The value of α changes between different models, as well as for a given model when using a different set of wavebands (i.e., optical versus optical+NIR). For the optical+NIR wavebands, we obtain α = −0.943 (BC03), −0.663 (CB10), −1.341 ($\mbox{BC}03_{\tau = 1\rm Gyr}$), and −0.713 (M05). For the optical wavebands, we obtain α = −1.280, − 1.186, − 1.756, and −1.007, respectively.

We examine how sensitive ∇_⋆ is to varying the set of utilized wavebands. This is shown in Figure 8, where we compare the ∇_⋆ measured with optical+NIR and only optical bands. These two values of ∇_⋆ are linearly related, indicating that optical data alone can still provide a reliable estimate of the effective color gradient. This is consistent with the fact that ∇_⋆ is not affected by the age–metallicity degeneracy (influencing the optical more than the optical+NIR gradients) and is also proportional to each of the color gradients. On the other hand, the slope of the correlation between ∇_⋆ (optical) and ∇_⋆ (optical+NIR) is significantly greater than 1 at 2.00 ± 0.03. This reflects the reduced statistical uncertainty on ∇_⋆ when incorporating the NIR information. This is further shown in Figure 9 where we compare the histograms of ∇_⋆ obtained using optical and optical+NIR data. The width of the ∇_⋆ distribution is significantly larger when using optical with respect to optical+NIR data. Also, a small offset in the peak of the two distributions is present. These results are quantified in Table 4 where we report the statistics, i.e., the peak (μ) and width (σ) as estimated using the bi-weight estimator (Beers et al. 1990), for different samples and stellar population models. In the same table, we also report the median uncertainty, $\sigma _{\star }^{\rm{err}}$, as well as the intrinsic width, σⁱ_⋆, defined in the same way as the quantity σⁱ in Section 5. From these data we conclude that for the optical+NIR sample, the peaks of the ∇_⋆ distributions are consistent for different stellar population models, i.e., the ∇_⋆ estimated from optical+NIR is model independent. In contrast, when using only optical data, a significant systematic different is present between the peak values of different models, though this difference is quite small (<0.02) with respect to the absolute peak value of ∇_⋆. Furthermore, in agreement with Figure 9, we see that σ is significantly larger when using optical data (σ ∼ 0.4) as opposed to using optical+NIR data (σ ∼ 0.25), i.e., ∇_⋆ is affected by significantly larger uncertainties (by a factor of ∼1.6) when using only optical data. Finally, the peak values of the optical and optical+NIR ∇_⋆ are slightly different, in the sense that the effective color gradients are on average more negative by ∼0.06 for the optical+NIR than for the optical samples. The differences in μ might be due to the different shapes of the optical and optical+NIR ∇_⋆ distributions, rather than some intrinsic difference. In fact, as seen in Figure 9, the distribution of ∇_⋆(optical) is clearly asymmetric.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 4. Statistics of the Effective Color Gradients

Model	Optical+NIR Sample				Optical Sample
	μ	σ	$\sigma _{\star }^{\rm{err}}$	σi⋆	μ	σ	$\sigma _{\star }^{\rm{err}}$	σi⋆
BC03	−0.283 ± 0.007	0.246 ± 0.005	0.163	0.184	−0.219 ± 0.004	0.407 ± 0.002	0.377	0.152
M05	−0.286 ± 0.007	0.246 ± 0.005	0.164	0.182	−0.209 ± 0.003	0.388 ± 0.002	0.355	0.157
CB10	−0.296 ± 0.007	0.254 ± 0.005	0.181	0.178	−0.235 ± 0.004	0.436 ± 0.003	0.406	0.160
$\mbox{BC}03_{\tau = 1\rm Gyr}$	−0.291 ± 0.008	0.257 ± 0.005	0.162	0.200	−0.225 ± 0.004	0.417 ± 0.002	0.390	0.147

Download table as:  ASCIITypeset image

The robustness of ∇_⋆, ∇_Z, and ∇_twith respect to the adopted stellar population model is illustrated in Figure 10, where we compare the ∇'s obtained using the optical+NIR sample of ETGs for BC03 and M05. We see that the scatter in ∇_⋆ is significantly reduced in comparison with that for ∇_Z or ∇_t. As noted above, ∇_⋆ is model independent, while this is obviously not the case for ∇_Z and ∇_t . This gives extra support to using ∇_⋆ as a useful and robust representation of the radial gradients of stellar population properties in galaxies.

Zoom In Zoom Out Reset image size

We bin ∇_⋆ with respect to the following photometric parameters: r- and K-band effective half-light radii (R_e,r and R_e,K); r- and K-band total magnitudes (^0.07M_r and ^0.07M_K), obtained through a point-spread function (PSF)-convolved Sersic modeling of the galaxy image; logarithmic median Sersic index (〈log n〉); median axis ratio(〈b/a〉); the a₄ parameter, which characterizes the galaxy isophotal shape (boxiness: a₄< 0 and diskyness: a₄> 0, see Paper I); the optical-NIR color index (r − K) and the median PSF fitting $\rm \chi ^2$ ($\rm \chi ^2_{PSF}$, Figure 11). For each galaxy, median values are computed using data at the available wavebands, i.e., griz and grizYJHK. We adopted the r- and K-band radii and magnitudes as representative of the sizes and luminosities of galaxies in the optical and NIR spectral regions, respectively. The r − K color index is estimated using 2DPHOT total galaxy magnitudes. Each bin includes the same number of galaxies, i.e., N = 200 ETGs for the optical+NIR sample and N = 2000 ETGs for the optical sample.

Zoom In Zoom Out Reset image size

Figure 11 shows the relations between ∇_⋆ and the photometric quantities listed above. Panels (a) and (c) show that ∇_⋆ changes significantly with the r-band effective radius and r-band total absolute magnitude in the sense that larger and more optically luminous ETGs have stronger (more negative) effective color gradients. This result is somewhat expected as a more negative ∇_⋆ implies that bluer (either less metal rich or younger) stars in the galaxy are preferentially distributed toward the galaxy periphery. We note that the trend with ^0.07M_r is weaker than that with R_e,r. When we restrict ourselves to only optical data the trends with radius and magnitude are less pronounced (see below). Moreover, ∇_⋆ has no dependence on radius or luminosity when we use K band (panels (b) and (d)). This finding is of particular interest, as the NIR light follows more closely the stellar mass distribution than the optical light. It implies that for bright ETGs the effective color gradient does not vary significantly along the stellar mass sequence. A mild correlation exists between ∇_⋆ and Sersic index, as shown in panel (e). For $\log n \,{\widetilde{>}}\, 0.75$ ($n \,{\widetilde{>}}\, 5.6$) ETGs with higher n tend to have stronger effective color gradients, likely because galaxies with higher n have higher luminosity due to the (optical) luminosity–Sersic index relation of ETGs (Caon et al. 1993). Panel (f) shows no trend whatsoever of ∇_⋆ with axis ratio while with the boxy/disky parameter a₄, shown in panel (g), a weak correlation is present especially when optical+NIR data is used. In panel (h), we show that a strong correlation exists between ∇_⋆ and total galaxy colors. For r − K < 2.9, galaxies with bluer colors also tend to have stronger gradients. This is not a spurious result arising from the correlation of galaxy colors and magnitudes. In fact, the color–magnitude relation implies that brighter galaxies should have redder colors. This, together with the trend of ∇_⋆ with ^0.07M_r, would produce a trend opposite to that seen in panel (h). Moreover, as shown in Paper I, no significant correlation is found between total magnitudes and total (rather than aperture) galaxy colors. Finally, panel (i) probes possible systematics in color gradient estimates related to the PSF modeling. It shows that ∇_⋆ is not correlated with the χ² from fitting the galaxy image with a PSF convolved Sersic light distribution. Hence, our PSF models are accurate enough to provide, on average, unbiased ∇_⋆ estimates.

In summary, we find that the effective color gradient in (bright) ETGs do not correlate with (NIR) luminosities and galaxy radii, while a correlation exists with galaxy colors. The former finding is further investigated in the next section, where we analyze the trend of ∇_⋆ with galaxy mass. The correlation of ∇_⋆ with colors is further studied in Section 8.3, where we bin ∇_⋆ with respect to the stellar population parameters of ETGs.

The lack of correlation between ∇_⋆ and R_e,K (panel (b) of Figure 11) deserves further comments. In Paper II we show that the ratio of optical (g-band) to NIR (K-band) effective radii, R_e,g/R_e,K decreases as a function of R_e,K, with the largest galaxies having R_e,g/R_e,K ∼ 1 (see Figure 7 of Paper II). This variation implies that the slope of the Kormendy relation for ETGs increases from g through K. A similar result, at optical wavebands (g − r), has been recently reported by Roche et el. (2010). Using the ratio of optical radii as a proxy for the color gradient, the decrease of R_e,g/R_e,K with R_e,K implies that larger galaxies should have shallower gradients (see also Tortora et al. 2010), in contrast with the lack of correlation in panel (b) of Figure 11. However, color gradients are determined not only from the ratio of effective radii but also from that of the Sersic indices. In fact, the variation of the surface brightness profile between two wavebands is determined from both R_e and n. Figure 12 plots the logarithmic binned ratio of g to K band Sersic indices, log n_g/n_K, as a function of log R_e,K. We find that n_g/n_K decreases as a function of R_e,K, i.e., larger galaxies are more concentrated in the NIR than lower-R_e,K systems. This compensates for the trend of R_e,g/R_e,K with R_e,K, making ∇_⋆ nearly constant with NIR radius. Thus there is no inconsistency between the lack of correlation in panel (b) of Figure 11 and the correlation shown in Figure 7 of Paper II.

Zoom In Zoom Out Reset image size

As mentioned above, we find that color gradients exhibit a complex behavior with galaxy luminosity. When estimated from optical+NIR color gradients, ∇_⋆ becomes more negative as the optical magnitude becomes brighter. On the contrary, for the optical sample of ETGs, i.e., when ∇_⋆ is estimated from ∇_g−r, ∇_g−i, and ∇_g−z, this steepening disappears, and ∇_⋆ does not vary with M_r (panel (c) of Figure 11). The reason for the different behavior of ∇_⋆, when estimated using optical versus optical+NIR data is illustrated in Figure 13, where we plot the median color gradient, ∇_g−X, as a function of the optical magnitude for all the available wavebands (X = rizYJHK). The trends of ∇_g−X with ^0.07M_r are modeled with second-order polynomials (dashed curves in the figure). The most remarkable feature is that the color gradient variation with magnitude depends on the waveband. The ∇_g−r gradient exhibits a double-valued behavior, becoming flatter in both more and less luminous galaxies. For ∇_g−z, the double-valued behavior disappears and the ∇ decreases monotonically as a function of ^0.07M_r. At redder wavebands, the trend reverses, and $\bf \nabla _{g-X}$ becomes a monotonically increasing function of ^0.07M_r. As shown in Figure 7, the effective color gradient is proportional to each of the individual color gradients from which it is estimated. When using optical data alone, ∇_⋆ mainly reflects the behavior of ∇_g−r, ∇_g−i, and ∇_g−z. As a result, ∇_⋆ exhibits no strong variation with luminosity, similar to the double-valued behavior of ∇_g−r. When ∇_⋆ is estimated from optical+NIR data, the optical–NIR gradients dominate the trend, with ∇_⋆ an increasing function of ^0.07M_r. The main conclusion is that the behavior of color gradients with luminosity depends on the waveband where the luminosity is estimated (panels (c) and (d) of Figure 11), as well as the wavebands used to estimate the gradient itself. This complex dependence may explain, at least partly, the discrepant results found in the literature about the relation between color gradients and luminosity (mass). For instance, Peletier et al. (1990) first reported a surprising lack of correlation between color gradients and galaxy luminosity, noting that brightest ellipticals do not exhibit less steep gradients, as expected if they were the debris of repeated mergers of lower mass systems. de Propris et al. (2005) also reported no correlation between the size of the color gradients and galaxy luminosity. Tamura & Ohta (2003) found that color gradients in cluster ETGs become steeper (more negative) in brighter galaxies. The same result was found by Balcells & Peletier (1994) for the bulges of spiral galaxies. On the other hand, La Barbera et al. (2005) found that color gradients do not depend on galaxy luminosity, and Choi et al. (2007) found that optical color gradients are essentially constant over a wide range of galaxy luminosity, with a weak trend of becoming flatter at both fainter and brighter magnitudes. Recently, Roche et el. (2010, hereafter RBH10), using the ratio of g- to r-band effective radii from the SDSS as a proxy for the color gradient, also found that ETGs with intermediate luminosity have stronger gradients than those at low and high luminosities, in agreement with the double-valued behavior we observe for ∇_g−r.

Zoom In Zoom Out Reset image size

Figure 14 plots the median ∇_⋆ as a function of galaxy stellar (M_⋆) and dynamical (M_dyn) mass. The binning is performed as in Figure 11, with each bin having the same number of galaxies. To quantify the dependence of ∇_⋆ on mass, we perform linear least-squares fits of ∇_⋆ with respect to M_⋆ and M_dyn, with the slopes reported in the figure. We find no trend of the effective color gradient with mass. In particular, the slope of ∇_⋆ versus M_dyn is consistent with zero at the 1.5σ significance level, while the slope of ∇_⋆ versus M_⋆ is consistent with zero at ∼0.5σ. For M_dyn < 2 × 10¹¹ M_☉, the ∇_⋆ becomes more negative than at higher mass, but this effect might not be real, as in this mass regime, the SPIDER sample is incomplete due to the r-band magnitude selection (see Figure 1). We verified that the trend with stellar mass remains unchanged when using different stellar population models (e.g., CB10) to perform the SED fitting (Section 3.1).

Zoom In Zoom Out Reset image size

Figure 15 plots the median effective color gradient, ∇_⋆ , as a function of central velocity dispersion, age, metallicity, and α-enhancement, for both the optical and optical+NIR samples. For the optical, the plot of ∇_⋆ versus log σ₀ seems to suggest a double-valued behavior, with ∇_⋆ reaching a minimum around log σ₀ ∼ 2.2. However, this is not confirmed by the optical+NIR data, where no trend appears. This result supports the findings of Sections 8.1 and 8.2, that ∇_⋆ is not driven by galaxy mass. On the contrary, we find a strong dependence of ∇_⋆ on stellar population properties, i.e., the age, [Z/H], and [α/Fe]. In order to quantify these trends, we perform a robust linear fit of ∇_⋆ as a function of each of the three parameters, minimizing the sum of absolute residuals in ∇_⋆. The uncertainties on fitting coefficients are estimated by the width of their distribution among 1000 iterations, where the median ∇_⋆ in each bin are shifted according to their error bars, and the fitting process is repeated. The slope and offset of the best-fit lines are reported in Table 5. From panel (b), we see that the effective color gradient is flat for galaxies with older stellar populations. The trend is statistically significant for both the optical and optical+NIR samples, with the slopes of ∇_⋆ versus Age relations greater than zero at ∼3σ and ∼9σ, respectively. We note that galaxies with ages younger than ∼4 Gyr seems not to follow the same trend as galaxies with older ages. As discussed below, this inversion in the trend of ∇_⋆ with Age is caused by galaxies with faint spiral-like morphological features and less accurate structural parameters. The behavior of ∇_⋆ with metallicity is less clear. For the optical sample, the slope of the ∇_⋆ versus [Z/H] best-fit relation is larger than zero at over 7σ. However, the trend is only weakly detected in the optical+NIR sample, where the slope is only marginally larger than zero, at ∼2.2 σ. We find that ∇_⋆ strongly depends on the α-enhancement parameter, in the sense that galaxies with larger [α/Fe] also have shallower effective color gradients. The slope of the best-fitting line is significantly different from zero, at more than 5σ, for both the optical and optical+NIR samples.

Zoom In Zoom Out Reset image size

Table 5. Coefficients of the Linear Fit of ∇_⋆ Versus Stellar Population Properties

Property	Optical+NIR Sample		Optical Sample
	Offset	Slope	Offset	Slope
$\log Age \, (\rm Gyr)$	−0.529 ± 0.058	0.227 ± 0.063	−0.714 ± 0.050	0.498 ± 0.055
[Z/H]	−0.321 ± 0.007	0.225 ± 0.100	−0.263 ± 0.004	0.430 ± 0.057
[α/Fe]	−0.239 ± 0.016	−0.733 ± 0.125	−0.174 ± 0.010	−0.785 ± 0.086

Download table as:  ASCIITypeset image

As discussed in Paper I, the SPIDER sample of ETGs is affected by a small contamination from late-type galaxies with a prominent bulge component. To analyze the impact of this contamination, we define a purer sample of ETGs, where contamination is reduced to at most 5%. To analyze if the trends of ∇_⋆ with stellar population properties are affected by the presence of late-type contaminants, we define two subsamples of ETGs, by matching the optical and optical+NIR samples with the lower contamination sample of Paper I. We also select only galaxies with better quality structural parameters, by removing objects whose two-dimensional fitting χ² in any of the available wavebands is larger than 1.25 (see Paper I). The optical and optical+NIR subsamples include 3928 and 31, 523 ETGs, respectively. Figure 16 plots the same trends as in Figure 15 but for the better quality subsamples. The coefficients of the best-fitting lines of ∇_⋆ versus age, metallicity, and enhancement are reported in Table 6. The better quality subsamples exhibit trends fully consistent with those obtained for the full samples. In the case of ∇_⋆ versus age, we do not find that galaxies with younger ages (<4 Gyr) have flatter gradients, as found for the full samples.

Zoom In Zoom Out Reset image size

Table 6. The Same as Table 5 for the Samples with Lower Contamination and Better Quality of the Structural Parameters

Property	Optical+NIR Sample		Optical Sample
	Offset	Slope	Offset	Slope
$\log Age \, (\rm Gyr)$	−0.465 ± 0.062	0.159 ± 0.067	−0.539 ± 0.046	0.325 ± 0.050
[Z/H]	−0.319 ± 0.008	0.121 ± 0.101	−0.246 ± 0.005	0.269 ± 0.059
[α/Fe]	−0.244 ± 0.017	−0.626 ± 0.130	−0.174 ± 0.012	−0.621 ± 0.094

Download table as:  ASCIITypeset image

A further issue is that of the fixed aperture size (15 SDSS fiber radius) where the stellar population properties are measured. Because of radial gradients in stellar population properties, this fixed aperture size might produce a correlation of the estimated Age, [Z/H], and [α/Fe], parameters with galaxy radius. Since the ∇_⋆'s correlate with the (optical) R_e (see panel (a) of Figure 11), the aperture effect might also bias the correlation of the ∇'s with Age, [Z/H], and [α/Fe]. However, as shown in Appendix B, the variation of R_e along σ₀, Age, [Z/H], and [α/Fe], is negligible with respect to the full range of R_e values (panel (a) of Figure 11), implying that the trends in Figure 15 are not affected at all by the aperture effect.

In Table 7, we report the peak of the distributions of age ($\mu _{_{\nabla _t}}$) and metallicity ($\mu _{_{\nabla _Z}}$) gradients for all the stellar population models described in Section 7. In agreement with previous works (e.g., Peletier et al. 1990; Tamura & Ohta 2003, 2004; La Barbera et al. 2003), we find that color gradients imply the presence of a negative metallicity gradient in ETGs (∇_Z < 0), with the outer stellar populations being less metal-rich than the inner ones. The peak value of ∇_Z spans the range of −0.45 to −0.3, depending on the stellar population model. In agreement with our previous work (LdC09), we find that the peak ∇_t is significantly greater than zero, i.e., ETGs have on average younger stars in the center than the outskirts. ∇_t ranges from 0.05 to 0.2, being significantly larger than zero for all models. Hence, as discussed in LdC09, the existence of a small, but significantly positive age gradient is a robust (model-independent) result. From the values of ∇_Z and ∇_t we see that, for all models, metallicity is the main driver of color gradients.

Table 7. Peak Values of the Distributions of Radial Gradients in Age (∇_t) and Metallicity (∇_Z) of ETGs

Model	$\mu _{_{\nabla _Z}}$	$\mu _{_{\nabla _t}}$
BC03	−0.401 ± 0.021	0.130 ± 0.023
M05	−0.417 ± 0.017	0.198 ± 0.025
CB10	−0.309 ± 0.010	0.052 ± 0.012
$\mbox{BC}03_{\tau = 1\rm Gyr}$	−0.456 ± 0.023	0.134 ± 0.019

Download table as:  ASCIITypeset image

Figure 17 plots the median ∇_Z (red circles) and ∇_t (blue squares) in different bins of stellar population parameters (age, metallicity, and enhancement), velocity dispersion, stellar, and dynamical mass. For each bin of a given quantity, we also show the median ∇_⋆ in that bin (black triangles). ∇_⋆, ∇_Z, and ∇_t are estimated using BC03 SSP models. Age gradients are multiplied by the absolute value of the α parameter (see Section 7), so that, for each bin, the sum of age and metallicity gradients gives, approximately,¹¹ ∇_⋆ in that bin (see Equation (16)). This allows us to analyze the relative contribution of age and metallicity to the trends of color gradients with respect to the different quantities.

Zoom In Zoom Out Reset image size

9.2.1. Trends with Mass Proxies

9.2.2. Trends with Age

9.2.3. Trends with Metallicity

9.2.4. Trends With the Enhancement

Since the trend of age and metallicity gradients with respect to different mass proxies depends on the range of galaxy mass, we bin the sample of ETGs according to M_dyn, and analyze the trends of ∇_Z and ∇_t in each bin. We define two bins, including low- (M_d1 < M_dyn < M_d2), and high-mass (M_dyn ⩾ M_d2) galaxies, where M_d1 and M_d2 are defined above. Although the sample of ETGs becomes incomplete at $M_{\rm dyn} \,\widetilde{<}\, 3 \times 10^{10}\, M_\odot$, we adopt a slightly lower mass limit of M_dyn ∼ 2.5 × 10¹⁰ M_☉, as this is approximately the M_dyn at which the trend of ∇_t and ∇_Z with mass changes behavior (see panel (f) of Figure 17).

Figures 18 and 19 are the same as Figure 17, but plot only galaxies in the low- and high-mass bins, respectively. In order to quantify the trends of ∇_t, ∇_Z, and ∇_⋆ with respect to a given parameter, we fit each trend with a linear relation, minimizing the sum of absolute residuals of median gradients (see Section 8.1). The slopes and their uncertainties are reported in Tables 8 and 9 for the low- and high-mass bins, respectively. From Figures 18 and 19, as well as Tables 8 and 9, we see that

• 1.  
  Low mass. The effective color gradient, ∇_⋆, does not show any significant correlation with either σ₀ or M_dyn, while it tends to marginally decrease with stellar mass, as the ∇_⋆–M_⋆ slope in Table 8 is less than zero (−0.12 ± 0.05) at the 2.2σ confidence level. No significant trend of ∇_Z and ∇_t is detected with respect to any mass proxy (M_dyn, M_⋆, and σ₀). On the other hand, for stellar population parameters, we find that ∇_⋆ strongly correlates with Age and [α/Fe]. The effective color gradient increases for larger Age, and becomes more negative as the [α/Fe] increases. The slopes of the ∇_⋆–Age and ∇_⋆–[α/Fe] linear fits differ from zero at more than 4σ, proving that the trends are highly significant. The ∇_⋆ also tends to increase with metallicity, but the slope of the ∇_⋆–[Z/H] fit is larger than zero at only 2.2σ. As noted above, the trend of ∇_⋆ with Age is mainly due to ∇_t, while that with [α/Fe] is driven by ∇_Z.
• 2.  
  High mass. For high-mass galaxies, the effective color gradient does not vary significantly with Age, metallicity nor with any mass proxy. In fact, the slopes of the correlations involving ∇_⋆ in Table 9 are all consistent with zero at less than 2σ. For the [α/Fe], there is a significant trend of ∇_⋆ decreasing as the enhancement increases. The slope of the ∇_⋆–[α/Fe] linear fit (−0.84 ± 0.2) differs from zero at over 4σ, and is consistent with that obtained for low-mass ETGs (−0.68 ± 0.17, see Table 8) and for the entire sample (−0.73 ± 0.13, see Table 5). For all quantities, we find anti-correlated variations of ∇_Z and ∇_t, which cancel each other in the trends of ∇_⋆. For [α/Fe], the metallicity gradient dominates, resulting in the strong correlation of ∇_⋆ and [α/Fe].

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Table 8. Slopes of the Linear Fits of Median ∇_t (Column 2), ∇_⋆ (Column 3), and ∇_Z (Column 4), as a Function of Different Galaxy Parameters

Parameter	∇t	∇⋆	∇Z
log σ0	0.18 ± 0.25	0.050 ± 0.104	−0.16 ± 0.24
${\it Age}\, (\rm Gyr)$	0.59 ± 0.25	0.406 ± 0.091	−0.10 ± 0.21
[Z/H]	−0.02 ± 0.30	0.338 ± 0.151	0.33 ± 0.27
[α/Fe]	0.16 ± 0.37	−0.681 ± 0.165	−0.48 ± 0.38
M⋆[1011 M☉]	−0.01 ± 0.17	−0.111 ± 0.055	−0.05 ± 0.15
Mdyn[1011 M☉]	0.19 ± 0.16	0.009 ± 0.068	−0.13 ± 0.16

Notes. Each line in the table corresponds to a different parameter, as reported in Column 1. The slopes refer to the sample of ETGs in the low-mass bin.

Download table as:  ASCIITypeset image

Table 9. The Same as Table 8 but for Galaxies in the High-mass Bin

Parameter	∇t	∇⋆	∇Z
log σ0	1.51 ± 0.24	−0.193 ± 0.129	−1.53 ± 0.31
${\it Age}\, (\rm Gyr)$	1.54 ± 0.33	0.229 ± 0.132	−1.05 ± 0.30
[Z/H]	1.26 ± 0.42	0.006 ± 0.165	−0.98 ± 0.37
[α/Fe]	1.00 ± 0.54	−0.756 ± 0.190	−1.36 ± 0.45
M⋆[1011 M☉]	0.26 ± 0.13	−0.020 ± 0.052	−0.26 ± 0.13
Mdyn[1011 M☉]	0.56 ± 0.15	−0.082 ± 0.060	−0.55 ± 0.15

Download table as:  ASCIITypeset image

From Figures 17–19, we see that positive age gradients in ETGs are more associated with high rather than low-mass galaxies. At high mass, the age gradient strongly increases as a function σ₀, M_dyn, and to a lesser extent with stellar mass, while these trends are not observed for low-mass galaxies. However, for both low and high masses, ∇_t becomes significantly positive for galaxies older than ∼7 Gyr. This implies that age gradients in ETGs are mainly driven by mass, and only secondarily by Age. Panel (b) of Figure 17 also shows that for Age < 5 Gyr, the trend of ∇_t with Age reverses: the younger the age, the more positive the age gradient becomes. Consistent with the downsizing picture (Cowie et al. 1996), galaxies at Age < 5 Gyr are low-mass systems, as shown by the inversion in the ∇_t–Age trend being observed only for the low-mass bin (panel (b) of Figure 18).

Stellar population parameters, such as Age, metallicity, and enhancement, are known to correlate with proxies of galaxy mass, such as velocity dispersion and stellar mass (see, e.g., Thomas et al. 2005; Nelan et al. 2005; Gallazzi et al. 2006). Since for high-mass galaxies we find that internal gradients correlate with both mass and stellar population properties, a natural question is if these trends are just because of the correlation among stellar population properties and mass. In order to address this issue, we apply a correction procedure that removes the correlations of each ∇ quantity with mass from all the trends in Figures 18 and 19. The procedure is illustrated in Figure 20, where we show how the trends of the ∇'s with respect to to [Z/H] are corrected, for galaxies in the high-mass bin.

• 1.  
  First, we model the ∇'s-M_dyn trends with third-order polynomials in M_dyn (dashed curves in the top panel).
• 2.  
  We compute the median M_dyn in each bin of [Z/H] (mid panel). As expected from the [Z/H]–mass correlation (e.g., Thomas et al. 2005), the mass increases as [Z/H] increases.
• 3.  
  In each bin of [Z/H], using the corresponding median M_dyn, we calculate the expected values of ∇_⋆, ∇_Z, and ∇_t from the best-fit polynomials.
• 4.  
  The expected ∇'s in each [Z/H] bin are subtracted from the measured ∇ in that bin; the subtracted values are re-normalized, by suitable additive terms, to the medians of the ∇_⋆, ∇_Z, and ∇_t distributions (bottom panel in Figure 20). Comparing top and bottom panels in Figure 20 illustrates how the correction procedure (partly) removes the correlations of ∇_⋆, ∇_Z, and ∇_t with [Z/H].

Zoom In Zoom Out Reset image size

We apply this correction scheme for both low- and high-mass ETGs, modeling the ∇'s–M_dyn trends (point (1) above) independently for each case. We obtain the following results:

• 1.  
  High mass. Figure 21 exhibits the correlations of ∇_⋆, ∇_Z, and ∇_t, with different quantities, for high-mass galaxies (M_dyn ⩾ M_d2), after removing the ∇'s–M_dyn trends. By construction, the procedure removes the correlations of the ∇'s with M_dyn (panel (f)). However, it does not impact the trends involving stellar population properties. This is because such trends are due to the ∇–σ₀ correlations (rather than ∇'s–M_dyn). This is shown in Figure 22, where we repeat the entire procedure by removing the ∇'s–σ₀ trends. The ∇'s–σ₀ corrections remove all the trends, with the remarkable exception of the correlations among ∇_⋆ (∇_Z) and the enhancement. A linear fit to the ∇_⋆–[α/Fe] trend in Figure 22 gives a slope of −0.8 ± 0.2, still fully consistent with that of the uncorrected trend (−0.84 ± 0.19, see Table 9).
• 2.  
  Low mass. In the mass range of M_d1 to M_d2, there is no variation of ∇_⋆ with velocity dispersion or dynamical mass. Hence, the correlations between gradients and stellar population parameters cannot arise from age, metallicity, and enhancement varying with mass. However, at a given σ₀, the stellar population parameters are correlated with each other, with age being anti-correlated with both metallicity and enhancement (Graves et al. 2009). To account for this, we re-derive the trends between the ∇'s and other parameters by removing the correlations with the Age. The procedure is the same as that adopted to remove the ∇'s–M_dyn and ∇'s–σ₀ correlations for high-mass galaxies (see above). Figure 23 shows the same plots as Figure 18, but after the ∇'s–Age trends are removed. We see that this correction does not affect significantly the correlations of the ∇'s with respect to metallicity and enhancement. In particular, after the correction is applied, a linear fit of ∇_⋆ with respect to Z/H gives a slope of 0.29 ± 0.12, fully consistent with that of 0.33 ± 0.15 reported in Table 8. For ∇_⋆ versus [α/Fe], the slope is −0.59 ± 0.15, still consistent with that in Table 8 (−0.84 ± 0.19). It follows that, for low-mass systems, all the correlations of internal gradients with stellar population parameters are independent of each other, i.e., the gradients exhibit a genuine correlation with each single parameter (in particular age and enhancement).

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size