STELLAR KINEMATICS OF z ∼ 2 GALAXIES AND THE INSIDE-OUT GROWTH OF QUIESCENT GALAXIES^*,†

The galaxies in this paper are drawn from the NMBS-I (Whitaker et al. 2010) and the UKIDSS-UDS (UDS, Williams et al. 2009). They were selected to be bright in the H band, and to have z > 1.4, in order to obtain sufficient S/N. The SED from the broadband and medium-band photometry was required to indicate that they have quiescent stellar populations, and the rest-frame optical imaging could not show signs of large disturbance due to, e.g., mergers. We note that NMBS-COS7447 was presented in van de Sande et al. (2011), and UDS-19627 was presented in Toft et al. (2012). All data for both galaxies have been re-analyzed according to the following procedure for consistency. Our selection had no priors on mass or size, but could be biased in either one of these parameters. Full information on the photometric properties of the targets is listed in Table 1.

To investigate possible biases, we compare our targets to a sample of galaxies with mass >10^10.5 M_☉ at 1.4 < z < 2.1 from the NMBS-I and the UDS. Rest-frame U − V and V − J colors are commonly used to distinguish between star-forming and quiescent galaxies at this redshift (e.g., Williams et al. 2009). Figure 1(a) shows the UVJ diagram for all galaxies at redshifts between 1.4 < z < 2.1 with mass >10^10.5M_☉, together with the sample presented here. The sizes of the squares indicate the density of galaxies. For our targets, the rest-frame colors have been measured from the spectra, while for the full sample rest-frame colors are based on the broadband and medium-band data. As demonstrated by Williams et al. (2009), non-star-forming galaxies can be identified using a color selection indicated by the black lines. Within this selection region, our targets fall in the area occupied by young, quiescent galaxies (Whitaker et al. 2012). The median specific star formation rates (sSFRs), as indicated by the different colors, are in good agreement with the full high-redshift sample at the same place in the UVJ diagram. For their mass, however, NMBS-COS7447 and UDS-19627 have slightly bluer colors as compared to the full sample (Figure 1(b)). At fixed mass, the targets are among the brightest galaxies, except for NMBS-I-7865 (Figure 1(c)). This may not come as a surprise as they are among the youngest quiescent galaxies, and thus have relatively low M/L.

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Four different imaging data sets are used to measure the surface brightness profiles of our galaxies, as summarized below. (1) All our targets in the NMBS-I COSMOS field were observed with HST-WFC3 H₁₆₀ as part of the program HST-GO-12167 (PI: Franx). Each target was observed for one orbit (2611 s), using a four-point dither pattern, with half-pixel offsets. Reduction of the data was done in a similar way to the reduction described in Bouwens et al. (2010), but without sigma clipping in order to avoid masking the centers of stars. The drizzled images have a pixel scale of 006, with a full width at half-maximum (FWHM) of the point-spread function (PSF) of ∼016. (2) Our NMBS-I targets are complemented with HST-ACS I₈₁₄ imaging from COSMOS (ver. 2.0; Koekemoer et al. 2007; Massey et al. 2010), which has a 003 pixel scale and PSF-FWHM of ∼011. (3) For UDS-29410, we make use of the HST-ACS F814W, HST-WFC3 J₁₂₅, and H₁₆₀ from UDS-CANDELS (Grogin et al. 2011; Koekemoer et al. 2011). These data have the same properties as the data described in (1) and (2). (4) For UDS-19627, we use ground-based data from UKIDSS-UDS (Lawrence et al. 2007; Warren et al. 2007) Data Release 8 in the J, H, and K bands, as no HST data are available. Imaging in all three bands was drizzled to a pixel scale 0134, and the FWHM of the PSF is 07 in the K band. Color images of our galaxies are shown in Figure 2.

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Observations were performed with X-Shooter on the VLT UT2 (D'Odorico et al. 2006; Vernet et al. 2011). X-Shooter is a second-generation instrument on the VLT that consists of three arms: UVB, VIS, and NIR. The wavelength coverage ranges from 3000 to 24800 Å in one single exposure. The galaxies were observed in both visitor and service modes, and the observations were carried out between 2010 January and 2011 March (programs: Fynbo 084.A-0303(D), Van de Sande 084.A-1082(A), Franx 085.A-0962(A), and Toft 086.B-0955(A)). Full information on the targets and observations is listed in Table 2. All observations had clear sky conditions and an average seeing of 08. A 09 slit was used in the NIR, except for the first hour of UDS-19627 where a 06 slit was used. For the 09 slit, this resulted in a spectral resolution of 5100 at 1.4 μm. Observing blocks were split into exposures of 10–15 minutes each with an ABA'B' on-source dither pattern. For most targets, a telluric standard of type B8V–B9V was observed before and after our primary target, in order to create a telluric absorption spectrum at the same airmass as the observation of our target.

Table 2. Targets and Observations

Catalog	ID	R.A.	Decl.	Exp. Time	Slit Size NIR	S/N J	S/N H	S/N4020 < λÅ < 7000	Telluric Standard Star
				(minutes)	(arcsec)	(Å−1)	(Å−1)	(Å−1)	HipparcosID
NMBS-COS	7447	10:00: 6.96	2:17:33.77	120	09	4.98	8.48	6.31	050307, 000349
NMBS-COS	18265	10:00:40.83	2:28:52.15	90	09	3.37	6.99	4.18	050684, 000349
NMBS-COS	7865	10:00:17.73	2:17:52.75	434	09	1.64	5.86	4.12	049704, 057126, 046054,
									040217, 059987
UDS	19627	2:18:17.06	−5:21:38.83	300	06, 09	3.80	7.94	5.90	012377, 114656, 008352,
									000328, 015389
UDS	29410	2:17:51.22	−5:16:21.84	120	09	3.75	7.09	4.35	012377

Notes. R.A. and Decl. are given in the J2000 coordinate system. Exposure times are given for the NIR arm, the UVB and VIS arms had slightly shorter exposure times due to the longer read-out. Except for UDS-19627, all targets were observed with a 09 slit in the NIR. S/N ratios have been determined from comparing the residual of the velocity dispersion fit to the flux and are given for the J and H bands as well as for the region in which we determine the velocity dispersion. Last column gives the telluric standard stars from the Hipparcos catalog that were observed before and after each target.

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Data from the three arms of X-Shooter must be analyzed separately and then combined to cover the full range from the UV to NIR. In the NIR, we identified bad pixels in the following way. The data were corrected for dark current, flat-fielded, and sky-subtracted using the average of the preceding and subsequent frames. The ESO pipeline (ver. 1.3.7; Goldoni et al. 2006) was used to derive a wavelength solution for all orders. The orders were then straightened using integer pixel shifts to retain the pixels affected by cosmic rays and bad pixels. Additional sky subtraction was done on the rectified orders by subtracting the median in the spatial direction. Cosmic rays and bad pixels were identified by LA-Cosmic (van Dokkum 2001), and a bad pixel mask was created.

Further 3σ clipping was done on the different exposures, corrected for dithers, to identify any remaining outliers. The bad pixel masks of different orders were combined into a single file and then transformed back to the raw frame for each exposure. Masks will follow the same rectification and wavelength calibration steps as the science frames.

Next, the flat-fielded and sky-subtracted observations were rectified and wavelength-calibrated; only this time we used interpolation when rectifying the different orders. Again, additional sky subtraction was done. Per order, all exposures were combined, with exclusion of bad pixels and those contaminated with cosmic rays present in the mask file.

The telluric spectra were reduced in the same way as the science frames. We constructed a response spectrum from the telluric stars in combination with a stellar model for a B8V/B9V star from a blackbody curve and models from Munari et al. (2005). Residuals from Balmer absorption features in the spectrum of the tellurics were removed by interpolation. All the orders of the science observations were corrected for instrumental response and atmospheric absorption by dividing by the response spectrum.

The different orders were then combined and in regions of overlap weighted using the S/N of the galaxy spectrum. A noise spectrum was created by measuring the noise in the spatial direction below and above the galaxy. If the regions exceeded an acceptable noise limit, from contamination by OH lines or due to low atmospheric transmission, this spatial region was discarded for further use. The two-dimensional (2D) spectra were visually inspected for emission lines, but none were found. A 1D spectrum was extracted by adding all lines (along the wavelength direction), with flux greater than 0.1 times the flux in the central row, using optimal weighting.

Absolute flux calibration was performed by scaling the spectrum to the available photometric data. The scaling was derived for each individual filter that fully covered the spectrum. For our targets in NMBS-I, we used the following filters: J, J2, J3, H, H1, H2, and Ks, while for the targets in the UDS we only used J and H. We then used an error-weighted average obtained from the broadband magnitudes and scaled the whole spectrum using this single value. After scaling, no color residuals were found, and no further flux corrections were applied to the spectrum.

A low-resolution spectrum was constructed by binning the 2D spectrum in wavelength direction. Using a bi-weight mean, 20 good pixels, i.e., not affected by skylines or strong atmospheric absorption, were combined. The 1D spectrum was extracted from this binned 2D spectrum in a similar fashion to the high-resolution spectrum (see Figures 3–5).

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For the UVB and VIS arms, we used the ESO reduction pipeline (ver. 1.3.7; Goldoni et al. 2006) to correct for the bias, flat field, and dark current, and to derive the wavelength solution. The science frames were also rectified using the pipeline, but thereafter, treated in exactly the same way as the rectified 2D spectra of the NIR arm, as described above.

We estimate the stellar population properties by fitting the low-resolution (∼10 Å in the observed frame) spectrum in the visual and NIR in combination with the broadband and medium-band photometry with SPS models. We exclude the UVB part of the spectrum due to the lower S/N and the extensive high S/N broadband photometry in this wavelength region. Stellar templates from Bruzual & Charlot (2003, BC03) are used, with an exponentially declining star formation history (SFH) with timescale τ, together with a Chabrier (2003) initial mass function (IMF), and the Calzetti et al. (2000) reddening law.

Using the FAST code (Kriek et al. 2009), we fit a full grid in age, dust content, star formation timescale, and metallicity. We adopt a grid for τ between 10 Myr and 1 Gyr in steps of 0.1 dex. The age range can vary between 0.1 Gyr and 10 Gyr, but the maximum age is constrained to the age of the universe at that particular redshift. We note, however, that this constraint has no impact on our results as the galaxies are young. Step size in age is set as high as the BC03 templates allow, typically 0.01 dex. Metallicity can vary between Z = 0.004 (subsolar), Z = 0.08, Z = 0.02 (solar), and Z = 0.05 (supersolar). Furthermore, we allow dust attenuation to range from 0 to 2 mag with step size of 0.05. The redshift used here is from the best-fitting velocity dispersion (see Section 3.2). Results are summarized in Table 3.

Due to our discrete grid and the high-quality data, and also because metallicity and age are limited by the BC03 models, our 68% confidence levels are all within one grid point. Our formal errors are therefore mostly zero, and not shown in Table 3. This does not reflect the true uncertainties, which are dominated by the choice of SPS models, IMF, SFH, and extinction law (see, e.g., Conroy et al. 2009; Muzzin et al. 2009).

The low sSFR confirms the quiescent nature of the galaxies in our sample, and they match well with the sSFR of the general population in the same region of the UVJ diagram (Figure 1(a)). We find a range of metallicities, with the oldest galaxy having the lowest metallicity. However, due to the strong degeneracy between age and metallicity, we do not believe this result to be significant. Overall, the dust content in our galaxies is low.

We find very similar results for NMBS-C7447 as compared to van de Sande et al. (2011), and the small differences can be explained by the newer reduction. For UDS-19627, we find a slightly lower mass as compared to Toft et al. (2012), which is likely due to the lower dust fraction that we find, i.e., A_v = 0.2 versus A_v = 0.77 from Toft et al. (2012).

The galaxies in our sample are not detected at 24 μm, leading to a 3σ upper limit of 18 μJy for the galaxies in NMBS-COSMOS, and 30 μJy for UDS-19627 (see Whitaker et al. 2012; Toft et al. 2012). UDS-29410 has a strong detection at 24 μm of 232 ± 15 μJy. From these upper limits we calculate the dust-enshrouded SFRs that are listed in Table 2. We find a high SFR for UDS-29410, but we find no other signs for this high SFR. That is, we find no emission lines and the best-fitting SPS model indicates a low SFR. Therefore, we think that the strong 24 μm detection is likely due to an obscured AGN.

The clear detection of absorption lines in our spectra, together with the medium resolution of X-Shooter, enables the measurement of accurate stellar velocity dispersions. We use the unbinned spectra in combination with the Penalized Pixel-Fitting (pPXF) method by Cappellari & Emsellem (2004) and our best-fitting BC03 models as templates. Spectra were resampled onto a logarithmic wavelength scale without using interpolation, but with masking of the bad pixels. The effect of template mismatch was reduced by simultaneously fitting the template with a ∼17-order Legendre polynomial. Our results depend only slightly on the choice of the order of the polynomial (Appendix A). Together with the measured velocity dispersion, the fit also gives us the line-of-sight velocity, and thus z_spec.

We also look at dependence of the velocity dispersion on the template choice. In particular, for the younger galaxies in our sample that show a clear signature of A-type stars, we find a dependence of the measured velocity dispersion as a function of template age. A more stable fit is obtained when restricting the wavelength range to 4020 Å < λ < 7000 Å, which excludes the Balmer break region (see also Appendix A).

The errors on the velocity dispersion were determined in the following way. We subtracted the best-fit model from the spectrum. Residuals are shuffled in wavelength space and added to the best-fit template. We then determined the velocity dispersion of 500 of these simulated spectra. Our quoted error is the standard deviation of the resulting distribution of the measured velocity dispersions. When we include the Balmer break region in the fit, the formal random error decreases, but the derived dispersion becomes more dependent on the chosen stellar template. In total, we have three high-quality measurements, and two with medium quality. We note that if we exclude the two galaxies with medium-quality measurements from our sample, our main conclusions would not change.

The velocity dispersion found here for NMBS-C7447 agrees well with the results from van de Sande et al. (2011). For UDS-19627, we find a slightly lower value as compared to Toft et al. (2012). However, they use a different method for constructing the template for the velocity dispersion fit. When we fit the spectrum of UDS-19627 in the same way as was described in Toft et al. (2012), we find a similar answer to theirs.

All dispersions are corrected for the instrumental resolution (σ = 23 km s⁻¹) and the spectral resolution of the templates (σ = 85 km s⁻¹). Furthermore, we apply an aperture correction to our measurements as if they were observed within a circular aperture of radius r_e. In addition to the traditional correction for the radial dependence of velocity dispersion (e.g., Cappellari et al. 2006), we account for the effects of the non-circular aperture, seeing, and optimal extraction of the 1D spectrum. The aperture corrections are small with a median of 4.8% (see Appendix B). The final dispersions and corresponding uncertainties are given in Table 4.

Table 4. Compilation of Masses and Structural Parameters for High-redshift Galaxies

Referencea	ID	zspec	re	nSersic	b/a	σe	σe/σap	β(n)	log Mdyn	log M*, corr	Filter
0	7447	1.800	1.75 ± 0.21	5.27 ± 0.23	0.71 ± 0.02	287$^{+ 55}_{- 52}$	1.048	5.16	11.24$^{+0.13}_{-0.12}$	11.22	HF160W
0	18265	1.583	0.97 ± 0.12	2.97 ± 0.06	0.26 ± 0.02	400$^{+ 78}_{- 66}$	1.065	6.61	11.38$^{+0.13}_{-0.11}$	11.32	HF160W
0	7865	2.091	2.65 ± 0.33	4.82 ± 0.15	0.83 ± 0.02	446$^{+ 54}_{- 59}$	1.031	5.42	11.82$^{+0.09}_{-0.09}$	11.64	HF160W
0	19627	2.036	1.32 ± 0.17	3.61 ± 0.73	0.48 ± 0.06	304$^{+ 43}_{- 39}$	1.059	6.18	11.24$^{+0.10}_{-0.09}$	11.20	K
0	29410	1.456	1.83 ± 0.23	2.59 ± 0.03	0.54 ± 0.02	371$^{+114}_{- 90}$	1.045	6.88	11.61$^{+0.19}_{-0.15}$	11.24	HF160W
...	...	...	...	...	...	...	...	...	...	⋅⋅⋅ ⋅⋅⋅

Notes. This table can also be downloaded from http://www.strw.leidenuniv.nl/~sande/data/. Spectroscopic redshifts z_spec are obtained from the velocity dispersion fit as described in Section 3.2. Structural parameters r_e, n_Sersic, and b/a are derived using GALFIT on available imaging, as described in Section 3.3. σ_e is the velocity dispersion within a circular aperture of size r_e from Section 3.2, and σ_e/σ_ap is the aperture correction we apply to the observed velocity dispersion as described in Appendix B. From Equation (2) we calculated β(n), and dynamical masses are derived using Equation (1). Stellar masses as given here are corrected to account for the difference between the catalog magnitude and our measured magnitude. The filter in which the structural parameters are measured is given in the last column. ^aReferences: (0) this work; (1) Bezanson et al. 2013; (2) van Dokkum et al. 2009; (3) Onodera et al. 2012; (4) Cappellari et al. 2009; (5) Newman et al. 2010; (6) van der Wel et al. 2008 and Blakeslee et al. 2006; (7) Toft et al. 2012.

Only a portion of this table is shown here to demonstrate its form and content. A machine-readable version of the full table is available.

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Radial profiles are measured for all galaxies on all available imaging as described in Section 2.2. Galaxies are fitted by 2D Sérsic radial surface brightness profiles (Sérsic 1968), using GALFIT (ver. 3.0.2; Peng et al. 2010). Relatively large cutouts of 25'' × 25'' were provided to GALFIT to ensure an accurate measurement of the background, which was a free parameter in the fit. All neighboring sources were masked using a segmentation map obtained with SExtractor (Bertin & Arnouts 1996). In the case of UDS-19627, the close neighbor was fitted simultaneously. Bright unsaturated field stars were used for the PSF convolution. All parameters, including the sky, were left free for GALFIT to determine.

Even though galaxies at low redshift are well-fitted by single Sérsic profiles (e.g., Kormendy et al. 2009), this does not necessarily have to be true for galaxies at z ∼ 2. Therefore, we correct for missing flux using the method described in Szomoru et al. (2010). We find very small deviation in residual-corrected effective radii, with a median absolute deviation of 3.4%. Color images and measured profiles are shown in Figure 2.

We repeated the measurements using a variety of PSF stars (N ∼ 25). We find an absolute median deviation in the half-light radius of ∼3% for HST-WFC3, ∼3.5% for HST-ACS, and ∼10% for the ground-based UDS-UKIDSS data, due to variations in the PSF. The largest source of uncertainty in the measured profiles is, however, caused by the error in the sky background estimate. Even though these galaxies are among the brightest at this redshift, using the wrong sky value can result in large errors for both r_e and n. We determine the error in the sky background estimate by measuring the variations of the residual flux in the profile between 5 and 15 arcsec. For sizes derived from HST-ACS, the absolute median deviation in the effective radius due to the uncertainty in background is ∼13%, and for HST-WFC3 ∼12%. Due to the deeper ground-based UDS-UKIDSS data, the uncertainty for UDS-19627 due to the sky is ∼8%. All of our results are summarized in Table 4.

We note that we find a smaller size and larger n for UDS-19627 as compared to Toft et al. (2012), which cannot be explained within the quoted errors. We have compared our results with the size measurements from Williams et al. (2009) and R. J. Williams (2012, private communication), who also use UDS-UKIDSS data for measuring structural parameters. They too find a smaller size in the K band of r_e = 1.63 kpc, with a similar axis ratio of q = 0.53, while keeping the Sérsic index fixed to n = 4. Furthermore, we compare the size of UDS-29410 obtained from the ground-based UDS-UKIDSS data to the size from HST-WFC3 to test how reliable the ground-based data are for measuring structural parameters. From the ground-based UDS H band we find r_e = 1.97 ± 0.11 kpc, and n = 2.47 ± 0.22 for UDS-29410 which is consistent with the measurements using the HST-WFC3 data within our 1σ errors. From these two independent results, we are confident that our size measurement for UDS-19627 is correct.

In what follows, we will use the mean effective radius and Sérsic n from the band which is closest to rest-frame optical r'. The effective radii reported here are circularized, $r_{e}=\sqrt{ab}$.

Combining the size and velocity dispersion measurements, we are now able to estimate dynamical masses using the following expression:

$$\begin{equation} M_{\rm dyn}=\frac{\beta (n) \sigma _{e}^2 r_e }{G}. \end{equation} \tag{ 1 }$$

Here, β(n) is an analytic expression as a function of the Sérsic index, as described by Cappellari et al. (2006):

$$\begin{equation} \beta (n) = 8.87 - 0.831n + 0.0241n^2. \end{equation} \tag{ 2 }$$

This is computed from theoretical predictions for β from spherical isotropic models described by the Sérsic profile, for different values of n, and integrated to one r_e (cf. Bertin et al. 2002). The use of a Sérsic-dependent virial constant β(n) gives a better correspondence between M_dyn and M_* for galaxies in the SDSS (Taylor et al. 2010a). This does require, however, that the total stellar masses are also derived using the luminosity of the derived Sérsic profile. Thus we correct our total stellar mass, as derived from the total magnitude as given in the catalogs (measured with SExtractor), to the total magnitude from the Sérsic fit. We note that the values for β that we find are all close to 5, a value often used in the literature (e.g., Cappellari et al. 2006). Our dynamical masses and corrected stellar masses are given in Table 4.

At low redshift, we select galaxies from the SDSS DR7. Stellar masses are based on MPA⁶ fits to the photometry following the method of Kauffmann et al. (2003) and Salim et al. (2007). Star formation rates (SFRs) are based on Brinchmann et al. (2004). Structural parameters are from the NYU Value-Added Galaxy Catalog (NYU-VAGC; Blanton et al. 2005). For all galaxies, velocity dispersions were aperture corrected as described in Section 3.2, and stellar masses are calculated with a Chabrier (2003) IMF. We furthermore correct the stellar masses using the total magnitude from the best Sérsic fit. All dynamical masses were derived using Equation (1). For making an accurate comparison between low- and high-redshift galaxies, we only select non-star-forming galaxies, i.e., sSFR <0.3/t_H (see Williams et al. 2009), where t_H is the age of the universe at the given redshift.

Our high-redshift sample consists of a collection of both optical and NIR spectroscopic studies of individual galaxies. van der Wel et al. (2008) present a sample of quiescent galaxies at z ∼ 1, which itself is a compilation of three studies in the following fields: Chandra Deep Field South (CDF-S; van der Wel et al. 2004, 2005), the Hubble Deep Field North (HDF-N; Treu et al. 2005a, 2005b), and cluster galaxies in MS 1054−0321 at z = 0.831 (Wuyts et al. 2004). We derive stellar masses for this sample by running the stellar population code FAST on available catalogs, i.e., FIREWORKS (Wuyts et al. 2008) for the CDS-S, R. Skelton et al. (in preparation) for the HDF-N, and FIRES (Förster Schreiber et al. 2006) for MS 1054−0321. For CDF-S and HDF-N, the stellar masses are corrected using the total magnitude from the best n = 4 fit to be consistent with the structural parameters from van der Wel et al. (2008). For MS 1054−0321, we use structural parameters and stellar mass corrections based on the results by Blakeslee et al. (2006), who fit Sérsic profiles with n as a free parameter. We note that Martinez-Manso et al. (2011) also study a sample of four z ∼ 1 galaxies, but find dynamical masses that are significantly lower than their stellar masses, in contrast to the result by van der Wel et al. (2008).

Other high-redshift results included here are from Newman et al. (2010) and Bezanson et al. (2013), who use the upgraded red arm of LRIS on Keck to obtain UV rest-frame spectra of galaxies at z ∼ 1.3 and z ∼ 1.5, respectively. Velocity dispersions for two galaxies at z = 1.41 are presented by Cappellari et al. (2009) and have been observed with VLT-FORS2 (see also Cenarro & Trujillo 2009). Using NIR spectrographs, Onodera et al. (2012, Subaru-MOIRCS) and van Dokkum et al. (2009, GNIRS) obtained velocity dispersions for two galaxies at z = 1.82 and z = 2.186. Similar to the current study, Toft et al. (2012) study UDS-19627 using VLT X-Shooter. Dynamical masses were derived using Equation (1). Note that for the studies of Cappellari et al. (2009), Onodera et al. (2012), van Dokkum et al. (2009), and Toft et al. (2012) no stellar mass corrections were applied due to the absence of the necessary information. All structural and kinematic properties of our high-redshift sample are listed in Table 4.

As noted in Section 2.1, this sample is biased toward young quiescent galaxies. Therefore, we will first investigate whether our sample and that of Bezanson et al. (2013) are biased in size as compared to other high-redshift galaxies. We gathered structural properties of galaxies from two studies that use CANDELS data in the UDS and GOODS-South fields (Patel et al. 2013; Szomoru et al. 2012). We compare to a subsample of these galaxies that are determined to be quiescent from their rest-frame U − V and V − J colors (see, e.g., Figure 1(a)). When comparing the effective radii versus the stellar mass in Figure 8(a), we find that our galaxies (red circles) and those of Bezanson et al. (2013, cyan circles) are in general more compact than the high-redshift CANDELS galaxies (small green circles).

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For low-redshift galaxies, we parameterize the mass–size relation by

$$\begin{equation} r_e=r_c \left(\frac{M_{\rm *}}{10^{11}\ M_{\odot }}\right)^b \end{equation} \tag{ 5 }$$

(Shen et al. 2003; van der Wel et al. 2008). Using a linear least-squares fit in log–log space, we find best-fitting values of r_c = 4.32 kpc and b = 0.62. This is slightly different from the fit by Shen et al. (2003) who find r_c = 4.16 kpc and b = 0.56. The difference may be explained by different selection criteria and their use of an older release version of SDSS. Figure 8(b) shows the evolution in effective radius by comparing galaxies with similar mass at different redshifts. Using both the SDSS and the CANDELS data, we examine the amount of evolution in size by fitting the following relation:

$$\begin{equation} r_e \propto (1+z)^{\alpha }. \end{equation} \tag{ 6 }$$

We find α = −1.02 ± 0.05 (linear fit in log–log space). Our spectroscopic targets and those of Bezanson et al. (2013) are mostly below this best-fit relation, being smaller by a factor of ∼1.28 as compared to median in redshift bins (big green squares). This is especially clear from Figure 8(c), where we correct for the evolution in size.

This bias might be explained by the method with which our targets are selected. As our selection is based on the magnitude within a fixed aperture of 15, instead of the total magnitude, we create a bias toward compact galaxies. For galaxies with similar total magnitudes, the smaller galaxies will be brighter within a photometric aperture, and thus make it into our sample. In what follows, we correct for this bias by increasing our sizes and those of Bezanson et al. (2013) by a factor of 1.28, and decreasing the velocity dispersion by a factor of $\sqrt{1.28}$.

In Figure 9(a), we plot effective radius versus dynamical mass. Symbols are the same as in Figure 7. For the low-redshift galaxies, we parameterize the mass–size relation according to the following equation:

$$\begin{equation} r_e=r_c \left(\frac{M_{\rm dyn}}{10^{11}M_{\odot }}\right)^b, \end{equation} \tag{ 7 }$$

and find r_c = 3.23 kpc and b = 0.56 (dashed line). This is in good agreement with b = 0.56 and r_c = 3.26 kpc as found by van der Wel et al. (2008). At fixed dynamical mass, we see that all our galaxies have smaller effective radii as compared to low redshift. This finding is further illustrated in Figure 9(b), where we compare the effective radii at fixed dynamical mass to the mass–size relation at z ∼ 0. The solid line is the best fit as described by Equation (6), with α = −0.97 ± 0.1. This result is in agreement with what has been found in previous kinematical studies (van der Wel et al. 2008; Newman et al. 2010). The scatter between different studies is considerable, with the work by van Dokkum et al. (2009) having the largest size difference while that by Onodera et al. (2012) has the smallest. Our sample falls in between these two extremes, i.e., we find smaller sizes than Onodera et al. (2012), but larger effective radii than van Dokkum et al. (2009).

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Instead of comparing galaxies' sizes at fixed dynamical mass, we will now take into account that galaxies do grow in mass (e.g., Patel et al. 2013). In van Dokkum et al. (2010) they find that, for a sample selected at a constant number density, the stellar mass evolves as

$$\begin{equation} \log M_{n}/M_{\odot } = 11.45-0.15z. \end{equation} \tag{ 8 }$$

The number density on which this result is based, n = 2 × 10⁻⁴ Mpc⁻³, corresponds to an average mass of log M_*/M_☉ ∼ 11.2 at z ∼ 2, similar to our sample. Assuming that the mass evolves as Δlog M/M_☉ ∼ 0.15z, we will compare effective radii for galaxies at different redshifts. For example, a galaxy with log M_dyn/M_☉ = 11 at z ∼ 2 will be compared with a z ∼ 0 galaxy with log M_dyn/M_☉ = 11.3.

However, the evolution in stellar mass is determined for a complete sample of both star-forming and quiescent galaxies, while in this paper we only look at quiescent galaxies. Therefore, we assess whether the evolution in size at constant cumulative number density is different for the quiescent population as compared to the full population. This was already done for galaxies in CANDELS by Patel et al. (2013) at n_cum = 1.4 × 10⁻⁴ Mpc⁻³, which corresponds to a median mass of log M_*/M_☉ ∼ 10.9 at z ∼ 1.8. The sample studied here, however, has a median mass of log M_*/M_☉ ∼ 11.2 at z ∼ 1.8, which corresponds to n_cum = 2.5 × 10⁻⁵ Mpc⁻³. Thus, we repeat the analysis by Patel et al. (2013), but now using our CANDELS sample (Section 6.1) at this lower constant cumulative number density. Our results are similar to Patel et al. (2013), i.e., the quiescent population has smaller effective radii as compared to full population, but the difference in size between the quiescent and star-forming populations is slightly smaller at z < 1.8 as compared to Patel et al. (2013). The difference is due to the fact that at our lower constant cumulative number density, we find a higher fraction of quiescent galaxies. Therefore, the effective radii of the full population will be closer to the effective radii of the quiescent population, as compared to Patel et al. (2013). To correct for this difference in size, we increase the effective radii of all the quiescent galaxies in our combined sample by the size difference of the quiescent population as compared to the full. We correct each galaxy individually by finding the correction factor at this particular redshift. Below z < 1.8, the median correction factor is ∼1.05, while at z ∼ 2.1 the correction factor is ∼1.6.

Figure 9(c) shows the evolution in size at fixed number density as a function of redshift. Not surprisingly, the evolution in effective radii is more extreme, as we are now comparing z ∼ 2 galaxies to more massive, and therefore bigger, galaxies at z ∼ 0. Using Equation (6), we find that α = −1.16 ± 0.1 provides the best fit. In conclusion, assuming that galaxies evolve in both mass and size, we find that the effective radii have to grow by a factor ∼4 from z ∼ 2 to the present day.

In Figure 10(a), we compare the stellar velocity dispersion within one r_e versus the dynamical mass for both low- and high-redshift samples. The dashed line is the parameterization of the σ_e–M_dyn relation for low-z galaxies using the following equation:

$$\begin{equation} \sigma _e=\sigma _c \left(\frac{M_{\rm dyn}}{10^{11}M_{\odot }}\right)^b. \end{equation} \tag{ 9 }$$

We find that σ_c = 148.9 km s⁻¹ and b = 0.24. Our high-redshift sample is clearly offset from low-redshift galaxies in the SDSS, i.e., at fixed mass they have higher velocity dispersions. Comparison of the velocity dispersion at fixed dynamical mass, as seen in Figure 10(b), shows a clear evolution in σ_e, such that velocity dispersion decreases over time. From this figure, the increase in accuracy for the velocity dispersion measurements of this study, as compared to other studies at similar redshift, is also clearly noticeable. Again, we use the following simple relation to quantify the amount of evolution:

$$\begin{equation} \sigma _e \propto (1+z)^{\alpha }, \end{equation} \tag{ 10 }$$

and find that α = 0.49 ± 0.08. From z ∼ 2 to z ∼ 0 the stellar velocity dispersions decrease by a factor ∼1.7. Again, we note that we apply a correction to the velocity dispersions in our sample, in order to correct for the bias toward more compact galaxies (Section 6.1).

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If we now compare low- and high-redshift galaxies using an evolving mass function as described above, we find that the velocity dispersion decreases less with cosmic time than when compared at fixed dynamical mass. Here, we also take into account that quiescent galaxies at constant cumulative number density are smaller as compared to the full sample, and therefore they have higher velocity dispersions. Similar to the description above, we therefore corrected the velocity dispersions of all galaxies in our combined sample. At z < 1.8, velocity dispersions on average decrease by a factor of ${\sim} \sqrt{1.05}$, while at z ∼ 2.1 they decrease by a factor of ${\sim} \sqrt{1.6}$. If we use Equation (10), we find that α = 0.31 ± 0.08. In other words, the velocity dispersion within one r_e decreases by a factor ∼1.4 from z ∼ 2 to the present day.

Next, we will focus on the central and effective mass densities using a similar approach as described in Saracco et al. (2012). In short, using the intrinsic Sérsic profile we can calculate the fraction of the luminosity that is within 1 kpc as compared to the total luminosity. For a Sérsic profile this ratio is given by (Ciotti 1991)

$$\begin{equation} \frac{L_{1\,{\rm kpc}}}{L_{{\rm tot}}} =\frac{\gamma (2n,x)}{\Gamma (2n)}. \end{equation} \tag{ 11 }$$

Here, Γ(2n) is the complete gamma function, γ(2n, x) is the incomplete gamma function, and x = b_n(r_1 kpc/r_e)^1/n, with b_n = 1.9992n − 0.3271. Using this ratio we can now calculate the dynamical mass within 1 kpc and within r_e from the total mass:

$$\begin{equation} {M_{1\,{\rm kpc}}} =\frac{L_{1\,{\rm kpc}}}{L_{{\rm tot}}} M_{{\rm dyn}}. \end{equation} \tag{ 12 }$$

Here we assume that the dynamical mass profile follows the light profile, and furthermore that the mass-to-light ratio of the galaxy is radially constant. The detection of small color gradients in our galaxies indicates, however, that this is not the case, but the effect on the derived densities is small (Saracco et al. 2012; see also Szomoru et al. 2012). Finally, the densities are calculated as follows:

$$\begin{equation} {\rho _{1\,{\rm kpc}}} = \frac{M_{1\,{\rm kpc}} }{(4/3) \pi r_{1\,{\rm kpc}}^3}, \end{equation} \tag{ 13 }$$

and

$$\begin{equation} {\rho _{e}} = \frac{0.5\, M_{{\rm dyn}}}{(4/3) \pi r_e^3 }. \end{equation} \tag{ 14 }$$

As for r_e and σ_e, we now compare the density as a function of dynamical mass (see Figure 11). The top row shows the results for the mass density within one effective radius, while the bottom row compares the central density within 1 kpc. The first thing to notice is the large scatter for low-redshift galaxies when looking at ρ_e versus M_dyn, while ρ_1 kpc versus M_dyn shows a tight relation. The density–mass relation can be parameterized by

$$\begin{equation} \rho =\rho _c \left(\frac{M_{\rm dyn}}{10^{11}M_{\odot }}\right)^b. \end{equation} \tag{ 15 }$$

For the density within r_e we find ρ_{c, e} = 4.7 × 10⁸ M_☉ kpc⁻³ and b_e = −0.68 ± 0.15, and for the central density within 1 kpc ρ_{c, 1 kpc} = 6.6 × 10⁹ M_☉ kpc⁻³ and b_1 kpc = 0.56.

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When we compare the galaxies in our high-redshift sample to galaxies in the SDSS, we find that they have higher densities within r_e. Comparison at equal dynamical mass shows that the effective densities are higher by a factor of ∼50 (Figure 11(b)) for our sample. The same comparison, but now for the central density within 1 kpc, reveals only mild evolution, approximately a factor of ∼3 from z ∼ 2 to the present. When fitting

$$\begin{equation} \rho \propto (1+z)^{\alpha }, \end{equation} \tag{ 16 }$$

we find that α_e = 2.86 ± 0.15, while α_1 kpc = 0.74 ± 0.15.

Instead of comparing galaxies at fixed mass, we again take into consideration that galaxies evolve in mass when comparing low- and high-redshift galaxies. Again, we correct for the fact that quiescent galaxies at constant cumulative number density are smaller as compared to the full sample. This time, we find that ρ_e evolves even faster as compared to the equal mass comparison, α_e = 2.93 ± 0.15. The density within 1 kpc, however, requires a decrease less than a factor of ∼2, with α_1 kpc = 0.42 ± 0.15, from z ∼ 2 to the present.

Our sample spans a redshift range of 1.4 < z < 2.1, which means that different parts of the rest-frame spectra will be affected by skylines and atmospheric absorption for each galaxy. This can be seen from Figure 4: we often lose strong absorption features in our spectrum, which affects the region of the rest-frame spectrum that we can fit. Here we investigate how stable the measured velocity dispersion is as a function of the wavelength range.

For the velocity dispersion fitting in this paper, the lower wavelength limit is set by stellar libraries and models as no systematic high-resolution observations exist below 3550 Å. The higher wavelength limit is set by lower S/N in our spectra in the observed K band. Our approach for testing the wavelength dependence of the fit is as follows. First, we use the full-range spectrum to determine a best-fit polynomial (1 order per 10,000 km s⁻¹), which is used to correct for the difference between the observed and the template continuum. Next, we repeat the measurement with a zeroth-order polynomial while changing the start or end wavelength. The polynomial is not a free parameter in this fit, as this would make it impossible to separate between the effect of the polynomial and the wavelength range. Note also that we use a single template for all fits as determined from the full visual+NIR spectrum together with broadband and medium-band data (see Section 3.1).

Figure 12 shows the results for the different sources. The left panel shows the result where we change the starting wavelength, i.e., the wavelength range is λ_start < λ < 7000 Å, whereas the panels on the right show the effect of changing the end wavelength, 3600 Å < λ < λ_end. The first thing to notice is that the measured dispersions are remarkably stable, even when most of the absorption lines have been excluded from the fit. The two galaxies with the youngest ages and strong Balmer absorption lines (NMBS-C7447 and UDS-19627) show a change in the velocity dispersion in the region of the Balmer break. NMBS-C7447 shows an increase for λ_start > 3800 Å, but decreases after Ca H&K have been removed from the fit. UDS-19627 shows an increase of 50 km s⁻¹ when the Balmer break is excluded. When we reduce the red part of the spectrum (Figure 12, right panels), we also find a stable fit, except for NMBS-C18265. After excluding Na D from the fit, the velocity dispersion increases by ∼100 km s⁻¹. We think this is because NMBS-C18265 has more evolved stellar populations than say, for example, NMBS-C7447. With the Ca H&K lines masked out due to atmospheric absorption, Na D is one of the strongest lines in the spectrum, and its exclusion could explain the sudden change in the measured dispersion.

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To summarize, for most galaxies we find only a mild dependence on the wavelength range that is used in the fit. We do find that with decreasing wavelength range, the random error increases. Finally, even in the absence of strong absorption features like Ca H&K, we find similar velocity dispersions as compared to the full-range fit. Due to template mismatch, we only fit λ > 4020 Å in our final results, as is explained in the next section.

Next, we study how different templates may influence the measured velocity dispersion. We use a sample of BC03 τ-models, as presented in Section 3.1. In particular, we are interested in the effect of template age and metallicity. In Figure 13, we show the reduced χ² from the SPS modeling versus the velocity dispersions measured using this template. The different points represent different ages of the templates, with the minimum χ² corresponding to the best-fit age as listed in Table 3. The different colors indicate the different metallicities. We show the χ² from the SPS modeling instead of the χ² from the dispersion fit, as the former is derived from the full visual and NIR spectrum plus all the broadband and medium-band photometric data. This large wavelength range yields better constraints for the stellar population, thus larger relative ranges in χ² as compared to the χ² from the dispersion fit. Also, as we add high-order additive polynomials to the templates from fitting the velocity dispersion (in this case one order per 6000 km s⁻¹), the effect of different template ages is mostly washed out, and we get a small relative χ².

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In Figure 13, we see that for most galaxies the velocity dispersion for templates allowed within 2σ gives consistent results. Different metallicities do give different velocity dispersions at their minimum χ², but this is a mere reflection of the age–metallicity degeneracy. Different metallicities have different best-fitting ages, which in turn give different velocity dispersion. At a 1σ level, we only have a handful of best-fitting templates for which we obtain similar velocity dispersions. UDS-19627 is the exception, which shows a large dependence of the measured velocity dispersion as a function of template age within the 1σ allowed range. At the 1σ level, we find a range of ∼30 km s⁻¹ due to template uncertainty, while the random error is one of the lowest, only ∼30 km s⁻¹. Even though templates with χ² of 1.20 or higher are statistically considered a bad fit, the large range in the velocity dispersion is worrying. If we could not have constrained the best-fitting template from the full-range spectrum and broadband data, the measured velocity dispersion would be highly uncertain.

However, if we exclude the Balmer break from the velocity dispersion fit, the dependence on template age almost completely disappears (Figure 14). UDS-19627 shows the most dramatic change, where we suddenly see a tight range of best-fitting velocity dispersions. We do note that the velocity dispersion has increased, as was already shown in Figure 12. UDS-29410 appears to have a slightly higher template uncertainty if we only fit for λ > 4020 Å, but this is driven by the lower S/N of the spectrum as we now use a shorter wavelength range.

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To conclude, we find a systematic uncertainty due to templates with different ages. This is caused by the Balmer break, present in the relatively young galaxies in our sample. By only fitting for λ > 4020 Å, the uncertainty due to template mismatch almost completely disappears. In that case, templates with different metallicities do give different velocity dispersions, but this is most likely caused by the underlying age–metallicity degeneracy.

In order to correct for stellar continuum emission differences between the observed galaxy spectrum and the template, we use an additive Legendre polynomial. If we did not apply such a correction, the fitting routine could try to correct for this discrepancy by changing the velocity dispersion. Values that are typically used in the literature vary from 5000 to 15,000 km s⁻¹ per order. We do not use multiplicative Legendre polynomials because the S/Ns of the spectra are too low, and it would add another degree of uncertainty to the fit. Here, we test the influence of the additive polynomial on our measured velocity dispersions. Again, we use the best-fit τ-model as a template while varying the additive polynomials from 0 to 50. We fit both the full-wavelength range and the wavelength range with λ > 4020 Å.

Figure 15 shows the results, with the blue line representing the full-wavelength fit, while the red line shows the results for λ > 4020 Å. The vertical dashed line indicates the polynomial with one order per 10,000 km s⁻¹. Overall we find that by increasing the additive polynomial, the measured velocity dispersion increases. In general, we find that between polynomials of 10 and 30, the smallest increase in the velocity dispersion occurs, and this appears to be the most stable region. For this reason, we use one order per 10,000 km s⁻¹ for our science results.

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While it is clear that a decrease in the S/N of the spectra will increase the random errors on the velocity dispersion, it could also cause systematic offsets (e.g., see Franx et al. 1989). A concern would be that low S/N would lead to an overestimate of the velocity dispersions. Therefore, in this section we test if there are systematic effects when changing the S/N for each source. The S/N we refer to in this section is derived in the region in which we also fit the velocity dispersion, i.e., 4020 < λ_rest frame[Å] < 7000.

For each source, we calculated the effect of S/N in a similar fashion as we derived the uncertainty on the velocity dispersion in Section 3.2. First, we subtracted the best-fit model from the spectrum. This residual now corresponds to the noise for this particular S/N of this source. We multiply this residual by a certain factor to obtain a new (higher or lower) S/N as compared to our reference S/N. These residuals are then randomly rearranged in wavelength space and added to the best-fit template that has been convolved to a velocity dispersion of 300 km s⁻¹. We then determine the velocity dispersion of simulated spectra with this S/N, and repeat this shuffling 100 times. Lastly, we repeat this procedure for a large range in S/N from 0.1 to 100 Å⁻¹. We note that uncertainties due to template mismatch are not included in this analysis.

In Figure 16 we show the results, with the different values of the S/N on the horizontal axis. We see that increasing the S/N will also decrease the random error in the velocity dispersions. On average, we find that in order to reach an uncertainty of less than 10% on the velocity dispersion measurement, you need an S/N of 8 Å⁻¹ or higher. NMBS-COS 7865 shows the lowest random error at fixed S/N, which is the older stellar population of the galaxy. No systematic offset in the velocity dispersion is found for an S/N of ≳ 1 Å⁻¹. Below this value, however, we find that the velocity dispersion is systematically underestimated as the S/N decreases, though the offset is still smaller than the random errors. We note that Franx et al. (1989) already presented simple arguments for predicting the systematic error. They show that the systematic offset of the velocity dispersion can be derived from error in the measured velocity, and varies as Δσ∝0.5dv²/σ. We verified from these simulations that this is indeed the case. From this analytic approximation, we furthermore conclude that the systematic uncertainties of our measured velocity dispersion are small.

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Here, we test how our velocity dispersions would change if we make different choices for the SPS model, and test the difference between a single τ-model and a template constructed from different combinations of single stellar population (SSP) models.

The left and middle panels of Figure 17 show what would happen if we would choose the Flexible-SPS models by Conroy et al. (2009, CO09) or the models by Maraston & Strömbäck (2011, MA11). These models are based on a different stellar library with slightly higher resolution as compared to BC03, i.e., MILES (Sánchez-Blázquez et al. 2006) versus STELIB (Le Borgne et al. 2003). If a systematic uncertainty in the measured velocity dispersion is present, for example, due to the resolution or details that go into the SPS models, it would show up in this comparison. We determined a best-fit τ-model using the CO09 and MA11 models in exactly the same way as was done for the BC03 models (see Section 3.1). When comparing the velocity dispersions derived by using the BC03 and CO09 τ-models, we do not find any significant difference (left panel of Figure 17). In the middle panel, we compare MA11 and BC03. Here too, we find a good correspondence. For UDS19627, the MA11 models give a lower velocity dispersion, though still within the 1σ error. These results confirm that our measurements are stable against the choice of SPS model and template with different resolution.

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Finally, in Figure 17 (right panel), we compare the velocity dispersion from the best-fit τ-model versus an optimal template constructed by pPXF. This optimal template is built from a full spectral library from BC03 SSP models, with full range in age and metallicity. We note that the optimal template construction only uses the wavelength regime provided in the dispersion fit (3600 Å < λ < 7000 Å) and does not take into account the effects of dust. The velocity dispersions from the optimal templates are slightly higher as compared to the single τ-model, although it is well below the random error. Interestingly, the galaxy with the largest dust contribution, NMBS-C18265, also shows the largest difference between the two different fitting techniques. In this paper, we choose to use the best-fitting τ-model as the dispersion template, as this is the best representation of the stellar population. As the stellar mass is also based on this τ-model, we use the same stellar population when comparing the stellar to the dynamical mass.