M/L_B AND COLOR EVOLUTION FOR A DEEP SAMPLE OF M^⋆ CLUSTER GALAXIES AT z ∼ 1: THE FORMATION EPOCH AND THE TILT OF THE FUNDAMENTAL PLANE^*,^†,^‡

We based our sample on a magnitude-limited survey of galaxies in the field of MS 1054-03. The original selection was done using a limiting magnitude of J = 21.2 mag (Vega) from a catalog of galaxies observed with the CFH12K at CFHT and the WIRC camera on the du Pont. Each galaxy in this sample had either a spectroscopic redshift, or a photometric redshift from a catalog of BVRIJH imaging data. Galaxies with spectroscopic redshifts were preferred in our selection. These photometric redshifts, the catalog construction, and the spectroscopic subsample will be discussed in a later paper.

Beyond meeting the J magnitude limit, the main criterion for selection was that the galaxy lies in the ACS field of view. We selected a DEIMOS mask center and position angle that maximized the number of galaxies with ACS imaging and with J < 21.2. A total of 44 galaxies met these criteria. Beyond our main sample, 12 supplemental galaxies were selected to have 21.2 < J < 21.8 and an additional 4 were targeted outside of the HST field of view. Also, we re-observed three galaxies that were in the sample of Wuyts et al. (2004, hereafter W04). Therefore, the total number of galaxies targeted for dispersions was 62.

We also combine our sample with that from W04. This earlier sample has a brighter I selection, see W04 for details, and used LRIS (Oke et al. 1995), with a smaller field of view than DEIMOS, for spectroscopic observations. The result is that the combined catalog of our sample and that of W04 will contain more galaxies in the inner regions of the cluster and will have a higher completeness at brighter magnitudes.

Our sample of galaxies is listed in Table A1 in Appendix A. We list our observed and rest-frame quantities along with the velocity dispersions. This table includes the sample of W04, which are identified by a "w" in front of the identification number.

The ACS imaging used in this paper comes from two separate programs. The first, discussed in B06, used the F606W, F775W, and F850LP filters to image the center of the cluster. We will refer to the filters as V₆₀₆, i₇₇₅, and z₈₅₀ for the rest of the paper. The second program imaged the outer regions of MS 1054-03 with the V₆₀₆ and the F814W, or I₈₁₄ filter. Example images can be seen in Appendix A, Figure A1.

We fit a model to each galaxy to determine the effective radius r_e and surface brightness 〈I_e〉 in the filter that most closely matches the rest-frame B. For the inner regions, this is the i₇₇₅ filter while for those galaxies at larger radii, we used I₈₁₄. For each galaxy, we fit a model with a free Sérsic parameter constrained to fall within 1 ⩽ n ⩽ 4 using GALFIT (Peng et al. 2002). For this paper, we use the best-fitting Sérsic parameters for r_e and 〈I_e〉, as this was shown to yield the same resulting FP as fixing n = 4 (Kelson et al. 2000a). When the best-fitting Sérsic n was at the limit of the allowed range, i.e., n = 1 or n = 4, we refit the image fixing n to that value.

The total magnitude we use is the normalization of the model fit. To measure colors, we used the circularized half-light or effective radius, $r_e = a_{\rm hlr} \sqrt{q}$, where q is the ratio of the minor to major axis, or 1 − , and a_hlr is the half-light radius along the major axis of the best-fitting elliptical model as determined by GALFIT. This is the radius used both in our M/L analysis and the radius of the aperture for which we measured the galaxy's color, see B06 for details. From this circularized r_e, we compute the average surface brightness within that radius, or 〈I_e〉.

Using the simulation results of Holden et al. (2009), which placed real galaxy images in the ACS frames, we find that our measurements of the total magnitudes are too bright by −0.08 mag in the i₇₇₅ data or −0.04 mag in the I₈₁₄ data. We find a scatter of σ = 0.07 mag around the input magnitude for the simulations, after removing the offset, in good agreement with the expectations from total magnitude measurements in ACS imaging (Holden et al. 2005a). We apply this offset to our data in later sections.

In addition to fitting a bounded Sérsic model, we fit a fixed de Vaucouleur's model to our data so we can compare the two resulting fits. As expected, for galaxies classified as E or S0 the median best-fitting n value was n = 3.6 while, for those galaxies classified as spirals the median n was n = 1.04 (we discuss the classifications in the next section.) The n values showed a larger scatter for the spirals than for E and S0 systems as well. In cases where the best fitting n > 3, the values for r_e are good agreement with the n = 4 values, with differences of 11% ± 6% on average. When best-fitting bounded Sérsic n < 3, the differences in the r_e are much larger, as expected (Kelson et al. 2000a).

2.2.1. Morphologies

2.2.2. Redshifted Magnitudes

The spectral data were acquired over two observing runs in 2008 January and March, in excellent seeing and generally photometric conditions. The exposures were 20 minutes long using the 600 line mm⁻¹ grating on the DEIMOS spectrograph. The slits were 1'' wide, yielding a resolution of ∼3.8 Å or an instrumental σ = 60 km s⁻¹. The slits were at least 8'' long, and between each exposure we offset the telescope by 2'' along the slit. Therefore, the object does not land on the same detector pixels in sequential exposures, but is instead offset by 2''. Each mask was observed for six exposures, or 2 hr. The final exposure was 16 hr for the deepest galaxies, 10 hr for the brighter objects.

2.3.1. Spectral Data Reductions

2.3.2. Measuring the Velocity Dispersions

Our low-redshift comparison sample is a subset of the Coma sample from Holden et al. (2007). Each galaxy has surface brightness profiles fit to the Sloan Digital Sky Survey (SDSS) imaging data in the same manner as we have for MS 1054-03. Each galaxy is a member of the JFK96 E and S0 sample used in that FP study. We use the dispersions from JFK96 and the colors from Eisenhardt et al. (2007). The colors from Coma match the Bower et al. (1992) colors well, but cover a larger number of galaxies. We note here that we use only the half-light radius colors from Table 8 of Eisenhardt et al. (2007). For the total magnitudes, we use our values from the surface brightness profile fits to the SDSS images. The larger sample of sizes and colors allows us to expand on the sample of JFK96 which had B data for only a subset of all of the Coma galaxies with dispersions. We list in Table A2 the magnitudes, rest-frame colors and dispersions we use. Not all of the sample of JFK96 have colors listed in Eisenhardt et al. (2007) as noted in Table A2.

We use the same procedure and τ model population templates from BC03 as we used for the MS 1054-03, see Section 2.2.2, to transform the z = 0.023 observed U and V into z = 0 values U_z and V_z. We find that there is some curvature to the relation, and we require a quadratic fit of

$$\begin{equation*} (U-V)_z = (U-V) -0.067 (U-V)^2+ 0.220 (U-V) -0.220 \end{equation*}$$

to transform the observed magnitudes into the redshifted ones. For galaxies on the red sequence of Coma, we find (U − V)_z = (U − V) − 0.04 mag. Using the Coleman et al. (1980) elliptical model, we find (U − V)_z = (U − V) − 0.04 mag, in good agreement with our quadratic relation. We note that these values are significantly different from the (U − V)_z = (U − V) − 0.00 as found by Bower et al. (1992). For the rest of the analysis, we will use the second order equation above relating the observed U − V and the z = 0 (U − V)_z color.

The colors of the galaxy population in MS 1054-03 have been studied extensively in B06. We show in Figure 1 the color–magnitude diagram of the population of galaxies that we targeted for dispersions in this study and the larger sample of galaxies that could have been targeted, a superset of the B06 sample. We note here that we remeasured the colors for all galaxies in our sample, instead of using the values from B06 and we will use these colors for our later measurements of the evolution of the cluster population.

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Because two different programs observed MS 1054-03 which used two different sets of filters, we convert the colors and magnitudes of galaxies observed in I₈₁₄ to i₇₇₅ using the following relations

$$\begin{equation*} i_{775} = I_{841} + 0.248(V_{606} - I_{814}) - 0.211, \end{equation*}$$

$$\begin{equation*} V_{606} - i_{775} = 0.752(V_{606} - I_{814}) + 0.211, \end{equation*}$$

which are only valid between 1.2 < V₆₀₆ − I₈₁₄ < 2.4. In Figure 1, we also removed galaxies from our sample that do not have spectroscopic redshifts.

Our colors significant a strong color–magnitude relation with a small scatter. The scatter for the whole sample of E and S0 galaxies with redshifts is σ = 0.077 ± 0.008, larger than the scatter measured in B06. This larger scatter comes from the subsample of galaxies with V₆₀₆ − I₈₁₄ colors that have been transformed into V₆₀₆ − i₇₇₅ colors, removing them brings the scatter down to the size as measured in B06. This is consistent with the ∼0.01–0.02 mag error typically found in these color transformations. We find we recover the slope and intercept from B06 with our larger sample, showing consistency between the two sets of measurements.

Evaluating the degree of evolution in the galaxy population requires calculating the completeness of our sample. Part of this is estimating how successful we are in measuring dispersions as a function of galaxy property. We targeted a total of 62 galaxies, with 44 above the magnitude limit of J < 21.2. Thirty-six have measured dispersions, with 31 above J < 21.2.

The simplest way of examining the completeness is to plot the distribution of galaxies with measured dispersions as a function size, color, and magnitude. First, we show the color–magnitude plot in Figure 2, followed by the size–magnitude relation in Figure 3. In both cases, the magnitude is the J, the selection passband.

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3.2.1. The Success Rate of Measuring Dispersions for E and S0 Galaxies

3.2.2. The Success Rate of Dispersions for Early-type Spiral Galaxies

Thirty percent of the galaxies with J < 21.2 that could have been targeted had slits placed on them and 26% of galaxies with J < 21.2 have measured dispersions. These numbers, of course, depend mildly on the apparent magnitude of the galaxy.

The second sample we would like to consider is the published work of W04. W04 had a selection limit of I = 22 but preferentially targeted brighter galaxies, with more than half of the sample above I ⩽ 21 (or J ⩽ 19.8). This results in a sample with a much brighter effective magnitude limit. As can be seen in Figure 2, most of the galaxies that could have been targeted, and the ones that had been targeted, are on the red sequence.

Considering only E and S0 galaxies, our spectroscopic sample, combined with that of W04, targeted 38% of the galaxy population with J < 21.2, with 36% of E and S0 galaxies above J = 21.2 having dispersions. This combined sample of velocity dispersions represents a significant fraction of the E and S0 population in MS 1054-03. For the rest of the paper, we will use the completeness fraction of galaxies as a function of J magnitude as weights. The inclusion of the W04 sample makes these weights a strong function of magnitude, as can be seen in Figure 2. These completeness fractions range from 100% at the brightest magnitudes to 20% at our completeness limit of J = 21.2.

4.1.1. Measuring the Tilt of the Fundamental Plane

We compute the parameters of the FP by fitting the J < 21.2 magnitude limited subset of our dispersions (σ), effective radii (r_e), and our B_z surface brightnesses. To fit the FP, log r_e = αlog σ + βlog 〈I_e〉 + γ, where 〈I_e〉 is −0.4μ_B, we minimized the mean absolute orthogonal deviation, as was done in JFK96:

$$\begin{equation*} \Delta = \frac{\log r_e - \alpha \log \sigma - \beta \log \langle I_e \rangle - \gamma }{\sqrt{(}1+\alpha ^2 + \beta ^2)} \end{equation*}$$

around the FP in all three projections, r_e, σ, and 〈I_e〉. We computed the average values of each of the coefficients for our final estimates of α, β, and γ. We estimated the errors on all of our fits to the FP by bootstrapping the data.

To compute the best-fitting plane, we need include our incompleteness. We did this two different ways. First, we use our weights we computed above in Section 3.3. Thus, galaxies at the fainter end of the luminosity function, closer to our limiting magnitude, get a higher weight. Second, we trim the data at a fixed value of σ. Because galaxies become intrinsically fainter at lower masses, by trimming at a fixed σ, we remove the very lowest M/L galaxies where our sample will not have the corresponding high M/L galaxies at the same dispersion. We find that at J = 21.2, the typical dispersion is log₁₀σ = 2.10 ± 0.10, and thus we select limit of log₁₀σ = 2.20. We will examine how our best-fitting plane varies depending on whether or not we remove the lowest σ galaxies or use our weights.

Our resulting best-fit values are summarized in Table 2 and, for comparison, we give the relations from JFK96 for the B selected sample. We verified that we recover the FP values from JFK96 when we fit to the values listed in Table A2.

Table 2. Summary of FP Parameters

Cluster	Limiting log σ	Morphology	α	β
	(km s-1)
JFK96a		E and S0	1.20 ± 0.07	−0.83 ± 0.02
Comab	2.2	E and S0	1.18 ± 0.08	−0.78 ± 0.04
MS 1054-03	2.2	All	1.18 ± 0.16	−0.76 ± 0.06
MS 1054-03	2.2	E and S0	1.25 ± 0.11	−0.76 ± 0.03
MS 1054-03	2.1	All	1.19 ± 0.13	−0.78 ± 0.05

Notes. ^aThe best-fitting slopes for the B from the r selected Coma sample from Table 5 of JFK96. ^bThe best-fitting slopes as determined by for the Coma sample from Table A2 after selecting galaxies above the log σ limit and above the equivalent r magnitude limit that matches our J magnitude limit at z = 0.83.

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When fitting the FP in MS 1054-03, at first we restricted the sample to only those galaxies that are E and S0 galaxies, a similar selection as JFK96, but using the log₁₀σ>2.2 and using the weights from above, we find α = 1.25 ± 0.11 and β = −0.76 ±  0.03. The value of α is in good agreement with that of JFK96 but the value of β appears different at the level of two standard deviations. Hyde & Bernardi (2009) show that sample selection can cause large changes in the best-fitting slopes of the FP, and specifically trimming at a fixed σ. The JFK96 sample is complete to a fainter r limit than the equivalent J limit for our sample and did not have a limiting σ as we did. So, we restricted the sample of Coma galaxies to those galaxies that would match our high-redshift selection criteria, namely, E and S0 galaxies with an equivalent r selection corresponding to our J < 21.2 at z = 0.83 and the same limiting σ. We refit the FP for the Coma sample, and we find that value of β for the Coma galaxies decreases to β = −0.78 ± 0.04 with α unchanged. Thus, we conclude that the mild difference in β we find in comparison to JFK96 is likely a result of sample selection, and that there is no evolution in the tilt of the FP.

Including early-type spirals in our sample does not change the result significantly, as was seen in a z = 0.33 cluster by Kelson et al. (2000c). Fitting the FP to all galaxies with log₁₀σ>2.2 and using the weights from above, we find α = 1.18 ± 0.16 and β = −0.76 ± 0.06. Lowering the limiting σ to log₁₀σ>2.1 and using the weights from above, we find α = 1.19 ± 0.13 and β = −0.78 ± 0.05. In general, these numbers are in good agreement with the values for a more restrictive subsample of E and S0 galaxies with log₁₀σ>2.2. This shows that our results are robust to different limiting velocity dispersions and uncertainties in the sample selections.

From these tests, we conclude that the observations are consistent with an unchanging tilt in the FP. In Figure 4, we plot the FP for our z = 0.83 sample. We use the values of α and β from JFK96 to plot both the low- and high-redshift samples.

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4.1.2. Residuals around the Fundamental Plane

vv07 found that there was a strong correlation between the residuals around log r_e and other quantities related to the mass of the galaxy, such as σ or the dynamical mass. Such a deviation could be caused by curvature in the FP and is more thoroughly explored in van der Marel & van Dokkum (2007a, 2007b). We assume the FP relation from JFK96 and we plot, in Figure 5, the deviation between the measured r_e and the predicted r_e from the FP as a function of r_e, σ, and the dynamical mass M_dyn. We computed M_dyn is in M_☉ using the relation M_dyn = 5σ²r_e/G or

$$\begin{equation*} \log _{10} M = 2 \log _{10} \sigma + \log _{10} r_e + 6.07 \end{equation*}$$

with σ is in km s⁻¹ and r_e is kpc. We find galaxies with dynamical masses M_dyn < 10¹¹ M_☉ that lie close to the z = 0 relation after adjusting for passive M/L evolution (see below). This result is expected given the lack of evolution we find in the tilt of the FP.

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4.1.3. M/L Evolution from the Offset of the Fundamental Plane

We can combine our measurements of the total B light with the M to estimate a M/L_B value or

$$\begin{equation*} M/L_B = 5 \sigma ^2 /(2\pi G r_e \langle I_e \rangle). \end{equation*}$$

We show in Figure 6 the relation between M/L_B and σ for Coma and our sample of galaxies in MS 1054-03 from the virial M/L estimator. We fit the results at both low and high redshifts using a robust linear fitting technique that minimizes the median absolute deviation and estimate our errors with bootstrapping. The bootstrapping is performed in a weighted manner, based on the weights determined in Section 3.3. It appears that the slope has not evolved between the low- and high-redshift samples. In our sample of galaxies in MS 1054-03, we find for the Coma sample, M/L_B ∝ σ^0.98±0.10, in excellent agreement with the low-redshift result of van der Marel & van Dokkum (2007b), while for our sample in MS 1054-03 we find M/L_B ∝ σ^1.12±0.21. If we assume no change in the slope, we find Δlog₁₀M/L_B = −0.50 ± 0.02 at a fixed σ for the whole of the galaxy population with J < 21.2. For the red sequence, E and S0 population, we find Δlog₁₀M/L_B = −0.50 ± 0.03, or no measurable difference.

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Our measured M/L_B evolution using the virial estimator is higher than the Δlog₁₀M/L_B = −0.44 ± 0.03 we find in Section 4.1. In principle, these results should be the same if the evolution in the FP is caused entirely by M/L evolution. van der Marel & van Dokkum (2007b) showed that structural evolution, such as caused by higher rotation rates or size evolution, would cause the disagreement we observe between the two estimators of M/L_B evolution. van der Marel & van Dokkum (2007b) find, for their sample, that the virial estimator yields a result closer to the M/L as estimate by the dynamical modeling done by van der Marel & van Dokkum (2007b). For the rest of this paper, we will use the virial estimator as a consequence, but will note how using the FP results changes our results. This should avoid some of the bias found in Saglia et al. (2010). Finally, we add that van der Wel & van der Marel (2008) found little evidence for higher rotation rates in field E and S0 galaxies, thus evolution in rotation alone does not explain the results of T05 or vdW05.

In Figure 7, we show that, as expected, the rest-frame (U − V)_z colors of the MS 1054-03 galaxies are bluer than the colors of the Coma galaxies. This shift can be seen both at fixed luminosity and at fixed σ. Since σ is expected to evolve little or not at all as compared to luminosity, we adopt the color evolution at fixed σ as the quantify of interest. For Coma, we find the relation (U − V)_z = 0.653 ± 0.093 log₁₀(σ/200 km s^-1) + 1.475 ± 0.015.

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We find no evolution in the slope, as expected from the larger survey of Mei et al. (2009) which found no change in the slope of the color–magnitude relation for eight z > 0.8 clusters. We measure the offset in the average color for all red-sequence E and S0 galaxies in our sample with log₁₀(σ)>2.2 and we find Δ(U − V)_z = −0.24 ± 0.02 mag between z = 0.831 and z = 0.023. As above, we use bootstrapping to estimate the errors on the color–σ relation in Coma and on the amount of color evolution.

The evolution of M/L for passively evolving systems depends critically on the IMF, while the evolution in optical colors depends less so (Tinsley 1972). This comes about because both the colors and luminosity of a passively evolving stellar population are determined by the properties of the stars at the main-sequence turnoff. The color evolution depends mostly on the effective temperature of those stars, and thus is most strongly influenced by the age and typical metallicity. The luminosity evolution, however, depends on not just luminosity of the individual stars, but the relative numbers of stars as a function of mass. Thus, our measurements of the ratio of the pace of color and luminosity evolution can be used to constrain the IMF of stars at the main-sequence turnoff, around 0.8–1.0 M_☉ for old stellar populations in our target galaxies.

vD08 found that, for cluster galaxies with masses >10¹¹ M_☉, the color and M/L evolution favored an IMF very different from a Salpeter slope in the mass range of 0.8–1.0 M_☉. The assumptions we made in the previous section use a Salpeter-like IMF to estimate the formation epoch from the M/L evolution. vD08, however, shows that the IMF that most closely agrees with the sample in that paper yields a formation epoch of z_⋆ = 3.7^+2.3_−0.8 for cluster galaxies with >10¹¹ M_☉. In light of the results of vD08, we have examined our data removing the assumption of a fixed IMF.

vD08 provides a useful relation for constraining the IMF with the evolution in the color and M/L_B. vD08 finds that for solar metal abundances with the M05 models with ages >1 Gyr

$$\begin{equation*} 2.5 \frac{\Delta \log (M/L_B)}{\Delta (U-V)} = 6.93 - 1.81 x, \end{equation*}$$

where the terms Δlog₁₀(M/L_B) and Δ(U − V) represent the amount of evolution in the M/L_B and color, respectively. The term x is the slope of the IMF, where the Salpeter (1955) value is x = 1.35.

We plot Δlog₁₀M/L_B and Δ(U − V) along with the expected evolution in Δlog₁₀M/L_B and Δ(U − V) in Figure 8. We find that our data are in mild disagreement with the expected evolution from a Salpeter IMF, with a slope of IMF better described by x = 0.9 ± 0.2, shown in Figure 8 with a red line. This is a difference of 2.25 standard deviations from the Salpeter value of x = −1.35. This value modestly changes the best-fitting formation epoch from z_⋆ = 1.8^+0.2_−0.2 to z_⋆ = 2.0 ± 0.3. As we discuss above, such a shift is in line with the other systematic uncertainties.

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5.1.1. Contrasting Our Results with vD08

We find no evidence for any evolution in the tilt of the FP, nor do we find evidence for a change in slope of the color–σ relation. Therefore, our data would rule out the scenario where the median star formation epoch depends strongly on galaxy mass. We quantified this by fitting a model to the color and M/L data in both Coma and MS 1054-03. This model assumes that the galaxies in Coma and MS 1054-03 are coeval and have the same metal abundance, and leaves the scatter as a free parameter at each redshift. The model parameterizes the formation epoch as power-law function of the σ, t_f = t_o(σ/160 km s^-1)^α. Fitting this model to those galaxies above our magnitude and σ limit, we find α = 0.2^+0.15_−0.1 and recover a t_o corresponding to z_⋆ = 1.5. The low value for α means that, in our best-fitting model, the most massive galaxies (log₁₀σ = 2.5) have a z_⋆ = 2.3^+1.3_−0.3. Our resulting value for α is on the small side of those reported in the literature, which we will discuss in later, in Section 5.4.

There is an important caveat to this analysis. We assume that the whole population in both clusters is coeval. Because the cluster population grows by accretion of galaxies from outside of the cluster, however, the population of galaxies in MS 1054-03 represents only a subset of the z = 0 cluster galaxy population. Thus, we would overestimate the epoch of cluster galaxy formation as we would only include those galaxies that are already E, S0 or early-type spirals at z = 0.83. If this process is dramatic, we would expect to find rapid evolution in the fraction E and S0 galaxies in clusters. However, for mass-selected samples, the amount of observed evolution in the fraction of E and S0 galaxies is minimal (Holden et al. 2006, 2007). van Dokkum & Franx (2001) provide a method for calculating the impact of this bias. We find if we assume that 90% of the galaxies were in our sample at z = 1, there is no shift in the epoch of formation. Assuming that fraction is only 50%, the effective epoch of formation shifts downward by Δz = 0.2, z_⋆ = 1.4 for galaxies with log σ = 2.2. It is likely that this fraction actually varies with velocity dispersion, so, in effect, the actual slope of the relation between formation epoch and dispersion is mildly steeper than we measure. If we assume that 50% of the log σ = 2.2 galaxies were not present in the sample while 90% of the log σ = 2.5 were present, this shifts the resulting α from α = 0.2^+0.15_−0.1 to α = 0.25^+0.15_−0.1, a very modest change.

5.2.1. Implications for the IMF

A number of color-based measurements of the formation epoch of cluster galaxies have been made. B06 found that the scatter in the colors in MS 1054-03 and another z = 0.83 cluster, RX J0152-13, implied a star formation epoch of z_⋆ ≃ 2.2, in good agreement with our results. Mei et al. (2009) found a similar scatter in the colors of the E and S0 population in a sample of eight clusters, including the two in B06, indicating a similar formation epoch for the z ∼ 1 cluster population. Holden et al. (2004) measured z_⋆ = 3⁺²₋₁ by fitting the evolution of the zero point of the color–magnitude relation of E and S0 galaxies, as opposed to the scatter as was done in B06, for a sample of 24 clusters spanning a redshift range of 0.3 < z < 1.3. Eisenhardt et al. (2008) and Mancone et al. (2010) find a similar formation epoch for the galaxies on the red sequence in a much larger sample of z > 1 clusters than in Holden et al. (2004). All of these results all point to z = 2–3 being the important epoch for the star formation of cluster early-type systems. Interestingly, Eisenhardt et al. (2008) find evidence that the average formation epoch of the cluster red sequence increases for the higher redshift cluster galaxies, the expected result if more galaxies are joining the red sequence over time.

At z ∼ 2–3, Brammer & van Dokkum (2007), Zirm et al. (2008), Kriek et al. (2008), and Brammer et al. (2009) all find a well-defined red sequence in both clusters and the field environment. These early-type systems have finished forming their stars at z ∼ 3, a time higher than the mean age as measured by colors or by the FP. However, these galaxies represent only the oldest 10% of the early-type population, and so are consistent with a more typical epoch of z_⋆ ∼ 2 for the majority of the population. Therefore, color measurements show that ∼L^⋆ galaxies have stellar populations with formation times of z_⋆ ∼ 2–3, an epoch when a sizable fraction of the stars that exist today were formed and entirely consistent with our FP measurements.

Trager et al. (2000a, 2000b) found that age and dispersion were part of a hyper-plane also including metallicity and α enhancement. In general, higher dispersion galaxies have older population ages at a fixed metallicity and galaxies in denser environments show older ages. A number of papers have built on these results, finding a strong correlation between the properties of stellar populations and velocity dispersions of galaxies for very large samples (Nelan et al. 2005; Gallazzi et al. 2006; Graves et al. 2009a, 2009b; Smith et al. 2009). Interestingly dynamical mass is not the important variable, rather Graves et al. (2009a, 2009b) and Smith et al. (2009) both found that σ is the driving parameter for determining the stellar populations of z = 0 early-type galaxies.

The ages derived from these measurements are strikingly young for lower dispersion galaxies, roughly 30%–50% of galaxies at log₁₀σ = 2.2 having formation ages corresponding to z ∼ 0.8. Trager et al. (2008) found similar results for E and S0 galaxies in the Coma cluster, a typical age of 5–8 Gyr, or roughly a z_⋆ ∼ 0.5–1, regardless of galaxy mass. Correspondingly, many authors find a steep relation between the age of the stellar population and velocity dispersion, with logarithmic slopes of 0.4–0.6 common (Trager et al. 2000a; Nelan et al. 2005; Smith et al. 2009) and values up to ∼1.1 found (Bernardi et al. 2003b). Kelson et al. (2006) found a strong relation between metallicity and σ, but, in contrast, that after accounting for the selection limits the population of CL 1358+62 showed no age variation with σ.

Part of the reason for the difference in our results, a shallow age–dispersion relation, and the steeper ones found by others come about from our use of broadband photometric data as compared to fitting models of stellar populations to spectra. Tortora et al. (2009) found that the slope of the age–σ relation is much shallower when stellar populations are fit to only broadband photometry. La Barbera et al. (2010a) finds, when analyzing a sample of 0.05 < z < 0.95 early-types, that there is little change in the ages of stellar populations with mass when the ages are determined by the FP in different passbands. Further, Cooper et al. (2009) found that galaxies in high-density environments with the same broadband color and magnitude as galaxies in low-density environments, had both larger ages and larger metallicities using models fit to spectra from Gallazzi et al. (2006). This shows that the population parameters derived from spectra are not the same as those derived from color measurements alone. Trager & Somerville (2009) provide a potential framework to interpret these seemingly contradictory results. Trager & Somerville (2009) find that the ages that result from fitting simple stellar population models to spectral features are likely when the last 5%–10% of the stars formed. Thus, these young apparent ages are not when the median star was formed, but rather the epoch when the last star formation occurred.

Our sample consists entirely of cluster galaxies, so part of the reason we find a different result from field samples such as T05 or vdW05 could simply be environment. Bernardi et al. (2003a), Bernardi et al. (2006), Cooper et al. (2009), La Barbera et al. (2010b), and Saglia et al. (2010) all find differences in the FP or the stellar populations of galaxies such that galaxies in clusters appear older than the same mass field galaxies.

Interestingly, T05 show, in Figure 14 of their paper, that most massive (>10¹¹ M_☉) galaxies have a formation epoch of z ∼ 2–3. vv07 perform a separate analysis and find a similar result. The main differences between our results and T05 are at <10¹¹ M_☉. We find that ∼L^⋆ cluster galaxies have a typical formation epoch of z_⋆ = 1.7 to z_⋆ = 2, whereas T05 find z_⋆ ∼ 1.2 for similar mass galaxies in their sample. In our MS 1054-03 sample, we have galaxies that show rapid enough M/L evolution to be consistent with a formation epoch of z_⋆ ∼ 1.2, but the typical low σ galaxy has a much higher M/L (see Figure 6). In the context of our model presented in the previous section where t_⋆ ∝ (σ)^α, the results of T05 prefer an α ∼ 0.4–0.45 as opposed to our α ∼ 0.2. There are two possible explanations for why our results differ from T05. First, the selection limits of T05 z band filter galaxy selection, as opposed to our redder J, biased the resulting sample toward bluer galaxies than our sample,. The second possibility is that the results of T05 are different for lower mass field galaxies because lower mass field galaxies have a different star formation history as compared with lower mass cluster galaxies. Patel et al. (2009b) and Patel et al. (2009a) do find that field and cluster galaxies have different star formation rates and color distributions at z = 0.83, possibly explaining the difference between our results and T05.

T05 explain the apparently young stellar populations seen in lower mass galaxies because of small recent episodes of star formation and not because the galaxy population is, as a whole, young, akin to the model of Trager & Somerville (2009). This picture is given more support when compared with the recent results of Thomas et al. (2010), which find that a small fraction of galaxies in the local universe with an E and S0 morphology appear to have had recent, small episodes of star formation. Labeling these young galaxies as "rejuvenated," Thomas et al. (2010) found the fraction of these galaxies increases at lower velocity dispersions, mimicking the steep age–dispersion relation found in T05 and in other work. Rogers et al. (2010) also found a higher fraction of galaxies with recent star formation in lower mass dark matter halos. All of this is consistent with the Trager & Somerville (2009) picture. To explain the results of, say, Cooper et al. (2009) or Rogers et al. (2010), these events would be more common in field-like environments. These are likely the handful of galaxies we observe with bluer colors and lower M/L_B values at lower values of σ. This picture provides a natural explanation for the buildup of the red sequence from z ∼ 1 until today, see, for example, Harker et al. (2006) or Ruhland et al. (2009), and could provide the progenitors for the apparently young Coma galaxies in Trager et al. (2008).

For each galaxy, we fit the surface brightness profiles in multiple steps. A Sérsic model was fit first, using values from a catalog derived by SExtractor as initial guesses but with a restriction on n of 1 ⩽ n ⩽ 4. The output of the Sérsic model was used an initial guess for the de Vaucouleur's model fit, and the Sérsic model was refit using the previous iteration as an initial guess. If the best-fitting Sérsic index is n = 1, we refit freezing n = 1. This was done to ensure convergence.

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Galaxies in crowded environment, generally those near the center of the cluster, were fit simultaneously along with all of those galaxies' neighbors. We use the sizes from the SExtractor catalog to find galaxies with close neighbors. All galaxies within a radius twice the size of a target galaxy were fit along with the target galaxy. In addition, galaxies outside of that radius but with sizes such that the target galaxy lies within twice their size were also included in the joint fit. This later criterion was chosen because particularly large galaxies, like the brightest galaxy of the cluster, often have extended surface brightness profiles. After the fitting process, if the radii of the target galaxy were significantly larger than the initial SExtractor guess, we refit including more neighboring galaxies. In all cases, neighboring galaxies were fit with a Sérsic model constrained to lie in 1 ⩽ n ⩽ 4.

We note here that the ACS magnitudes and surface brightnesses have been adjusted by the offsets as we computed from simulations that are listed in Section 2.2.

A number of galaxies in our sample were morphologically typed multiple times. All of the galaxies in our sample were classified by Marc Postman twice, once for Postman et al. (2005) and once for the parent sample that the galaxies in this paper are drawn from. In addition, many of these galaxies were classified in WFPC2 imaging (Tran et al. 2007). For the purposes of this paper, the main classification is the distinction between a late-type or an early-type system, where the former are spiral and irregular galaxies while the later are ellipticals and S0 galaxies. The comparison in Postman et al. (2005) finds that 10% of the galaxies in the ACS sample are identified as spirals in the WFPC2 or vice versa. When comparing that two separate classifications done by Marc Postman, we find that 92% of galaxies are grouped in the same two broad bins.

We list all of the galaxies targeted in Table A1. For each galaxy, we list the redshift and dispersion, if available. For a summary of why some galaxies do not have dispersions, see Section 2.3.2. The galaxies in the sample of W04 are listed with a "w" in front of the identification number. The galaxies with redshifts are those we re-observed as part of our program. All of the other galaxies have spectroscopic information from W04 which does not list redshifts, though those can be found in Tran et al. (2007).