A LOCAL BASELINE OF THE BLACK HOLE MASS SCALING RELATIONS FOR ACTIVE GALAXIES. II. MEASURING STELLAR VELOCITY DISPERSION IN ACTIVE GALAXIES

Objects were observed with the Low Resolution Imaging Spectrometer (LRIS) at Keck I using a 1'' wide long slit, the D560 dichroic (for data taken in 2009) or the D680 dichroic (for data taken in 2010), the 600/4000 grism in the blue, and the 831/8200 grating in the red with central wavelength 8950 Å. This setup allows us to infer M_BH from the Hβ line and cover three spectral regions commonly used to determine stellar velocity dispersions. In the blue we cover the region around the Ca H&K λλ3969,3934 (hereafter CaHK) and around the Mg Ib λλλ5167,5173,5184 (hereafter MgIb) lines; in the red we cover the Ca ii λλλ8498,8542,8662 (hereafter CaT). The instrumental resolution (this is not the FWHM, but the square root of the second moment, which is approximately FWHM/2.355 for a Gaussian) is ∼90 km s⁻¹ in the blue and ∼45 km s⁻¹ in the red. We cover 0.63 Å pixel⁻¹ in the blue and 0.58 Å pixel⁻¹ in the red (0.915 Å pixel⁻¹ for data taken before 2009 September). The long slit was aligned with the host galaxy major axis as determined from SDSS ("expPhi_r"), allowing us to measure the stellar velocity dispersion profile and rotation curves.

Observations were carried out on 2009 January 21 (clear, seeing 1''–15), 2009 January 22 (clear, seeing ∼11), 2009 April 15 (scattered clouds, seeing ∼1''), 2009 April 16 (clear, seeing ∼08), 2009 September 20 (low clouds, seeing ∼12), 2010 January 14 (clear, seeing ∼12), 2010 January 15 (low clouds, seeing ∼1''), 2010 March 14 (scattered clouds, seeing ∼08), and 2010 March 15 (clear, seeing ∼07; see also Table 1).

Note that the 2009 January and 2009 April data were obtained before the LRIS upgrade and were presented in Paper I (a total of 25 objects). The 2009 September, 2010 January, and 2010 March data (79 objects) benefited from the upgrade with higher throughput and lower fringing. While we observed a total sample of 104 objects, for 20 objects the spectra did not allow a robust measurement of the stellar kinematics, due to dominating AGN flux and redshift (12 objects) or problems with the instrument (8 objects), so our final kinematic sample consists of 84 objects, while the spectroscopic sample consists of 96 objects. Out of the 84 objects in the kinematic sample, 22 were already presented in Paper I. We revisited these measurements for this paper, and therefore they are again included in Tables 1 and 2 and Figures 1–8; the measurements presented here supersede those given in Paper I.

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The data were reduced using Python-based scripts that include the standard reduction steps such as bias subtraction, flat fielding, and cosmic-ray rejection. Arclamps were used for wavelength calibration in the blue and sky emission lines in the red. A0V Hipparcos were used to correct for telluric absorption and to perform relative flux calibration; to minimize overhead, these stars were observed immediately after a group of objects close in coordinates.

Throughout this paper, we use S/N (pixel⁻¹) as a measurement of spectrum quality, particularly the quality of absorption features. Therefore in our determination of S/N we use the same (or similar) wavelength ranges used for stellar velocity dispersion fitting (see Section 4)—5050–5450 Å in the blue and 8480–8690 Å in the red. The blue chip physically ends around 5600 Å, so many spectra do not actually extend to 5450 Å (in which case we use 5050 Å to the end of the spectrum). We determine the S/N from spectra before Fe ii subtraction (Section 3.3) so for objects with strong Fe ii emission the blue S/N does not necessarily reflect absorption line quality, but should be a good estimator for the Fe ii-subtracted spectra used for the fit.

We caution the reader that our higher-offset measurements of S/N and σ may be biased for late-type galaxies with bright star-forming regions, because we align the slit with the host galaxy major axis (which contains, when present, the galaxy's bright spiral arms). In these cases, the S/N is increased by the young stars to a level that does not accurately reflect the older stellar populations that are of interest to our study. These younger stars may also bias our measurements of σ, although our fitting procedure can incorporate young stellar templates when needed (Section 4).

From the reduced two-dimensional spectra, one-dimensional spectra were extracted either as "spatially resolved" spectra or "aperture" spectra in the following manner.

"Spatially resolved" spectra were created to measure the variation of σ as a function of radius. We extracted a central spectrum with a width of 054 (043) in the blue (red) for nights before 2009 September but a width of 054 in both the blue and red for 2009 September onward (i.e., after the red CCD chip upgrade). Off-center spectra were extracted by stepping out from the center in both directions, at every step increasing the extraction window by 1 pixel (above and below the trace) and choosing the step size such that there is no overlap with the extraction window of the previous step.

Spatially resolved spectra do not represent the way in which most studies extract one-dimensional spectra from which to measure σ. Instead, one-dimensional spectra are usually created from a broad extraction window centered on the galactic center. In the following, we refer to a spectrum created in such a way as an "aperture spectrum." If made from a two-dimensional spectrum, the width of the extraction window is chosen according to some criterion, e.g., to obtain a certain S/N. However, fiber spectra such as SDSS spectra come from a fixed circular aperture of 3'' diameter.

Our study is concerned with the effect of the width of the extraction window on σ when creating aperture spectra. Therefore, in addition to the spatially resolved spectra, we create a series of aperture spectra of increasing aperture width, ranging from ∼05 to ∼7'' in 027 increments (26 spectra per object). With this range of aperture widths we can not only study the effect of increasing aperture width on σ but also directly compare our measurements to measurements in the literature and to the results that would be derived from Sloan fiber spectra. For the latter, we determine σ_{ap, SDSS}, measured from aperture spectra within the central 15 radius as a proxy for what would have been measured with the 3'' diameter Sloan fiber. Note, however, that in fact, our σ_{ap, SDSS} corresponds to a rectangular region of dimension 3'' by 1''  given the width of the long slit.

With both types of extraction we can test whether the agreement between the three different spectral regions used to determine σ depends on offset from galactic center or aperture width (Section 6).

The majority of objects (58 out of the 96 in the spectroscopic sample) display broad nuclear Fe ii emission in their spectra (∼5150–5350 Å) that interferes with σ measurements from the MgIb region. For those objects, we simultaneously fit a set of IZw1 templates (varying width and strength) and a featureless AGN continuum to the central (spatially resolved) spectrum. The best fit was determined by minimizing χ² and was then subtracted from the five central offset positions in spatially resolved spectra, and from all aperture spectra. Fe ii-corrected spectra were not used in the CaHK measurements, since Fe ii features in this region are too broad and weak to have an effect on σ (Greene & Ho 2006b). Fe ii subtraction is illustrated in Figures 2 and 3 of Paper I, and details can be found in Woo et al. (2006).

Here, we present a brief overview of our fitting method. Tests of the method are provided in the Appendix.

As described in Section 5.2 of Suyu et al. (2010), we use a Python-based code that employs the algorithm of van der Marel (1994) but expanded to use a linear combination of template spectra. The code simultaneously fits a linear combination of broadened stellar templates and a polynomial continuum to the data, using a Markov Chain Monte Carlo (MCMC) routine to find the best-fit velocity dispersion (σ) and velocity (v). The stellar template library is composed of seven G and K giants of various temperatures as well as spectra of A0 and F2 giants from the Indo-US survey. Before fitting, template spectra are rebinned to the instrumental resolution determined from the science spectrum. Our reported measurement of σ (or v) is the median of the MCMC distribution, and measurement errors are the semi-difference of the values at the 16 and 84 percentile values (see the Appendix for a discussion of the relative merits of the median, mean, and most likely values as estimates). Systematic errors due to template variations are accounted for in our fitting routine since the template weights are fitted simultaneously with the velocity dispersion and marginalized over.

Our fitting routine employs Gaussian broadening kernels (range 30–500 km s⁻¹) to the templates. This is the standard quantity measured for the host galaxy of black holes and it is therefore the quantity that needs to be measured in order to compare our results with the previous literature (e.g., Ferrarese & Merritt 2000; Barth et al. 2002; Onken et al. 2004; Greene & Ho 2006a; Gültekin et al. 2009; Woo et al. 2010). We should note, however, that line profiles are not necessarily Gaussian and a more accurate determination of σ can be obtained by relaxing this assumption (see, e.g., van der Marel & Franx 1993). This is left for future work.

For each measurement we need to set a few parameters. These determine whether A and F stars are required, set the wavelength range to fit, and define masks and the order of the continuum fit. We now discuss how we determine each of these parameters from the spatially resolved spectra and how we apply them to the aperture spectra.

Every fit is initially performed with all seven G and K stellar templates, without the A0 and F2 templates. Some spectra show strong Hδ absorption in one or more of the noncentral spatially resolved spectra (presumably due to a region of star formation) and therefore could not be fit well by G and K templates alone. In these cases, the A and F stellar templates are also included in the fitting procedure for both the spatially resolved and aperture σ measurements.

As anticipated in Section 3.1, to test the dependency of stellar velocity dispersion on observed spectral regions we independently fit to three different regions corresponding to commonly used deep stellar absorption features: (1) around the Ca ii NIR triplet, 8480–8690 Å (CaT), (2) around the Mg Ib triplet, 5050–5330 Å (MgIb), and (3) around Ca H&K, 3910–4300 Å.

In region (1), if the broad AGN emission line O i λ8446 is present and superposed with the first of the CaT lines, we exclude the first CaT line and used 8520–8690 Å. In some cases, imperfect subtraction of telluric absorption lines or cosmic rays requires that the third line be excluded. Thus, σ_CaT may be determined from all three lines, the second and third lines, the second and first lines, or the second line alone. In region (2), though the longest rest-frame wavelength used is 5330 Å, the wavelength range available for a given object depends on its redshift, as the spectra end at the observed wavelength 5600 Å. At z ≳ 0.08, there is not enough information in the MgIb region to recover σ. In region (3), if the H AGN emission line filled the CaH stellar absorption line, we exclude the Ca H&K lines and instead use only weaker absorption features in the range 4150–4300 Å. For all but two objects, excluding Ca H&K lines was necessary for the central three spatially resolved spectra. The effect of these wavelength range variations on σ is discussed in Section 5.

Note that the G-band feature redward of 4300 Å that usually provides a good measure of galaxy kinematics is excluded from the CaHK region, to avoid the broad Hγ λ4341 emission feature.

Within the chosen region, we apply masks to ensure that we fit only the stellar contributions to each spectrum. In the blue regions, we check for narrow AGN emission features such as [Fe v] λ4227.49; [Fe vi] λλλ5145.77, 5176.43, 5335.23; [Fe vii] λλ5158.98, 5277.67; [N i] λλ5197.94,5200.41; [Ca v] λ5309.18 (wavelengths taken from Moore 1945; Bowen 1960), and various broad He i and Balmer lines. In the red, we may mask wavelength ranges that are affected by telluric absorption rather than shorten the region, if the affected range is narrow or does not extend to the end of the spectrum. In any case, a mask is only applied to known contaminating features when doing so changes the measurement more than the uncertainty on the original measurement.

As for the order of the polynomial continuum, we choose the lowest order that preserves the quality of the fit; most commonly, this is a linear continuum (i.e., a first-order polynomial). However, in the bluest regions, AGN emission features at either end of the wavelength range or sudden continuum breaks (e.g., near the Ca H&K feature) require the use of a third-order polynomial (or second-order, in a few cases).

Each of these parameters (template set, wavelength range, masks, polynomial order) is determined by visual inspection of each spatially resolved spectrum with an S/N ≳ 10 pixel⁻¹. In effect, this S/N restriction is a restriction on the maximum offset observed, as S/N decreases as a function of offset (galaxies are dimmer at larger radii). Every spatially resolved spectrum must be inspected because the contribution of the AGN changes with offset in a unique way for every target. Additionally, noise features such as bad pixels or cosmic-ray subtraction residuals affect only certain offsets.

It is infeasible to apply this same vigilance to the aperture spectra because we make nearly 100 aperture σ measurements per object. Furthermore, there is a physical reason to expect that such vigilance is unnecessary, since these parameters depend predominantly on AGN emission features, which are shared among all aperture spectra. We therefore expect that the central fit parameters determined from the spatially resolved spectra are sufficient for all aperture spectra, so we apply the parameters determined from an object's central spatially resolved spectrum to all of its aperture spectra.

However, our treatment of the aperture fits is not completely blind. We visually inspect the aperture spectra at the Sloan fiber radius for all measurements. Additionally, if A and F templates were required for any spatially resolved measurements, then we also apply these templates to the aperture spectra that would be similarly affected, i.e., aperture widths greater than or equal to the offset position of the spatially resolved spectra in which the Hδ absorption feature is observed. In some cases the aperture spectra include noise features not seen in the spatially resolved spectra, because the aperture spectra cover a larger spatial extent than the spatially resolved spectra due to S/N constraints on the spatially resolved spectra discussed above. In such cases we visually inspect the target's widest aperture spectrum to determine the mask for all aperture spectra.

The resulting σ for all objects are provided here in Figures 1–7 (spatially resolved) and Figure 8 (aperture). Examples of our spectral fits are provided in Figures 4–7 of Paper I and will not be repeated here for conciseness. No corrections (e.g., for wavelength range used, Section 5) have been applied to these measurements. As mentioned in Section 4.1, our code determines the v required to precisely align the templates with the rest-frame spectra (a minor correction to the Sloan redshift), which we use to determine systemic velocities, shown in the lower panels (relative to 〈v〉). We also plot the velocity dispersion and velocity profiles for all the objects analyzed in Paper I for completeness; the measurements presented here supersede those given in Paper I.

As noted in Paper I, in Figures 1–7, measurement errors tend to be larger for spectra with low S/N (i.e., higher offsets) as well as—for CaHK and MgIb—spectra with higher AGN contribution (i.e., central offsets). In many cases central CaHK measurements (blue points) are missing since σ_CaHK could not be faithfully determined due to AGN contamination; in some cases, the AGN predominance also requires the exclusion of central σ_MgIb and σ_CaT measurements. Note also that σ_MgIb may be missing from the objects with highest redshift because the 5600 Å instrumental limit did not leave strong enough features for a measurement of σ_MgIb, and objects observed on 2010 March 14 do not have σ_CaT due to problems at the instrument.

Without the photometric determination of the spheroid effective radius, we can only make a few qualitative and general comments about σ spatial profiles from Figures 1–8. From the spatially resolved profiles, we see that in all regions σ tends to either decrease with increasing offset or remain relatively constant. We see that higher concavity in these profiles correlates with a higher slope in the rotation curve, i.e., σ decreases more rapidly with offset from the center in galaxies with strong rotational signatures. In the aperture profiles, we see that profiles are relatively flat, having in general small variations compared to those seen in the spatially resolved dispersions. These conclusions are qualitative estimations of the results we expect to see when photometry is incorporated (Paper III), constraining the spheroid effective radius (r_{eff, sph}). We can then plot the dispersion profiles in units of r_{eff, sph} to make more meaningful comparisons between the galaxies, and to have a physical benchmark for quantitative analysis of trends in the σ profiles.

Finally, we provide the results of our fits to the aperture spectra at the Sloan fiber radius in Table 2, along with the S/N per pixel of the spectrum. Column 3 of this table lists measurements without any corrections applied, from the CaT region. If we could not fit to the CaT region we provide the measurement from the MgIb or CaHK region, again at the Sloan fiber radius. If we were not able to fit to any aperture spectra—due to AGN contamination—we provide the spatially resolved measurement closest to the Sloan fiber radius. Column 4 of this table provides our best measurement of σ, which takes the result of Column 3 and applies the corrections for both wavelength range and spectroscopic region found in Sections 5 and 6. Note that by "best" we refer to the best measurement we can deduce without photometric results. With r_{eff, sph} we will be able to calculate a luminosity-weighted σ that removes the effect of the disk, as in Paper I, which we consider to be a more accurate σ.

As discussed in Section 1, we consider 10% to be the upper limit on the desired accuracy of σ measurements, but 5% is our goal. Our σ_best have an average error of 5.9% ± 0.6% and median error of just 4.5%. Of the 84 σ, 9 have errors higher than 10% but come from the lower-quality spectra (mean S/N of 43 pixel⁻¹ compared to the mean of the total sample, 78 pixel⁻¹).

All the measurements shown in Figures 1–7 are provided in Table 3

Although the CaT region is generally considered the best region for measuring σ—for active galaxies in particular, due to the scarcity of AGN emission features in this wavelength range—it nevertheless can be affected by AGN contamination or, at higher redshifts, by telluric absorption. A typical situation occurs when one has to exclude the bluest and reddest lines of the triplet (hereafter, first and third, respectively); the first line may need to be excluded due to O i contamination and the third line may need to be excluded when it is redshifted into strong telluric features.

Unlike the CaHK and MgIb regions discussed below, the CaT region does not have nearby (strong) absorption lines from which we can measure σ. Measurements rely on the Ca ii triplet lines alone. Therefore we test the effect of line exclusion, making independent measurements of σ using various wavelength ranges in the CaT region. For brevity we denote the velocity dispersions measured using the first and second lines (8480–8580 Å), the second line alone (8520–8580 Å), and the second and third lines (8520–8690 Å) as σ_{1, 2}, σ₂, and σ_{2, 3}, respectively. We refer to these generically as σ_cut, and our diagnostic for agreement is log (σ_cut/σ).

It may be that σ_cut/σ is a function of AGN contamination (i.e., spatial offset) or S/N—or a combination of the two—so we study σ_cut/σ as a function of each. An illustrative example of our results is the top panel of Figure 9 (errors are statistical only, i.e., without our 0.01 dex systematic uncertainty).

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We find no trends with offset or S/N. As expected, scatter greatly increases at S/N <20 pixel⁻¹. As illustrated by the cyan bars in the right panel of Figure 9, high-offset spectra are dominated by low-S/N trends. As described in the Appendix, independent of σ, measurements are most reliable for S/N⩾20 pixel⁻¹ and we therefore impose this requirement on the measurements used to calculate biases. From the high S/N measurements (108 of 203), we find no bias in any of the three most common cases: 〈log (σ_{1, 2}/σ)〉 = 0.001 ± 0.003 ± 0.01, 〈log (σ₂/σ)〉 = 0.001 ± 0.003 ± 0.01, and 〈log (σ_{2, 3}/σ)〉 = 0.002 ± 0.003 ± 0.01.

Of particular importance is our finding that σ₂ recovers σ measured from the full CaT region accurately. As we see in Figure 9, the recovery is perfect with extremely low (≈1%) statistical dispersion. At "intermediate" S/N (10 pixel⁻¹ < S/N <40 pixel⁻¹) σ₂ is only ≈2.5% higher than σ, that is, equal within the 2.5% statistical dispersion.

This is a critical result for studies of the M_BH–σ relation because it broadens the candidate pool for target selection, especially at higher redshifts where the third line is blanketed by telluric features. As an illustrative example, when compiling their target sample Greene & Ho (2006a) required that at least two of the three Ca ii IR lines be available; but we show that this two-line constraint is not necessary. Had the two-line constraint been removed from the Greene & Ho (2006a) sample selection process, the study may have been able to use a greater number of high-S/N spectra, increasing both its sample size and data quality.

Our default wavelength range for the MgIb region is 5050–5330 Å, though redshift may truncate the red end. To investigate potential systematics due to excluding parts of the region, we study the effect of measuring σ from 5050 to 5250 Å and 5100 to 5330 Å. The first choice is motivated by redshift—to avoid atmospheric contamination, we set the dichroic to 5600 Å (observed) and therefore could not recover the full range for z ≳ 0.051. Because it excludes the Fe λ5270 absorption feature, we refer to this range as σ_noFe. The second choice is motivated by contamination from AGN emission—the red wing of [O iii]λ5007 or other AGN emission features such as He iλ5016,5048 may contaminate the blue end of the MgIb region. Because it represents the red end of the MgIb region, we refer to this range as σ_Mgred. Our results for σ_noFe are illustrated in the middle row of Figure 9. These comparisons were made only among objects for which the full region was available.

In both cases, we find no strong trend with S/N. However, interestingly, the central spectra give different results from the higher-offset spectra. We report only 〈log (σ_noFe/σ)〉 because—regardless of offset—it is equal to 〈log (σ_Mgred/σ)〉. Averaging over measurements with S/N >20 pixel⁻¹ (100 of 188 measurements), we find 〈log (σ_noFe/σ)〉 = −0.016 ± 0.004  ±  0.01.

Since there are clear spatial trends we analyze different offsets separately (still requiring S/N >20 pixel⁻¹). Central spectra (22 measurements) give 〈log (σ_noFe/σ)〉 = −0.04 ± 0.01  ±  0.01 and spectra within 07 (42 measurements) give 〈log (σ_noFe/σ)〉 = −0.0191 ± 0.006  ±  0.01 and non-central spectra (36 measurements) give 〈log (σ_noFe/σ)〉 = 0.001 ± 0.007  ±  0.01.

The small bias in σ_noFe is in the same direction as that reported by Barth et al. (2002) who noticed that including Mg (or, in our case, increasing its significance by eliminating Fe) can bias σ high at the 5% level (10% for central measurements). This bias is mass-dependent, so we note that our mass range (or, equivalently, stellar velocity dispersion range) is similar to the Barth et al. (2002) sample (see Section 6.3 and Figure 11). We thus recommend that measurements based on spectral ranges that are dominated by Mg i be corrected by multiplying by 1.05 ± 0.03 if they cannot be avoided.

Our default wavelength range for the CaHK region is 3910–4300 Å. If strong H fills the Ca H feature we use 4150–4300 Å, excluding both the Ca H and K features (σ measured from this range is named σ_noHK). The standard region includes the Hδ feature, which might be biased by the presence of thermally broadened absorption from A and F stars if those are not properly represented in the templates. Thus, we also test a third region that excludes the Hδ but includes Ca H&K (3900–4090 Å; σ measured from this range is named σ_noHδ). This is approximately the wavelength range used in Greene & Ho (2006a). As always, these tests are only run for spectra with available Ca H. Our results for σ_noHK are illustrated in the bottom row of Figure 9.

Similarly to the case for CaT, the narrower wavelength ranges accurately recover the same σ as the full range, though dispersion increases greatly for S/N < 20 pixel⁻¹. Using measurements with S/N >20 pixel⁻¹ (68 of 278 measurements), 〈log (σ_noHK/σ)〉 = −0.002 ± 0.005± 0.01 and 〈log (σ_noHδ/σ)〉 = −0.002  ±  0.005  ±  0.01, corresponding to no bias for either.

We find that reducing the size of the wavelength range does not affect σ_CaT or σ_CaHK although it decreases marginally σ_MgIb, seemingly as a function of AGN contribution. Our results in the MgIb region confirm and strengthen the findings by Barth et al. (2002), showing that excluding the Fe lines in the MgIb region lowers σ by 5% ± 3%.

For our study of the fidelity of the CaHK and MgIb regions compared to CaT (Section 6), MgIb measurements are corrected for wavelength range bias according to the findings described in this section, and errors on these corrections are incorporated into the measurement error.

These findings have good implications for future studies: we consistently find that good data quality (high S/N) is the driving requirement for σ accuracy. Studies must understand the accuracy with which their fitting program recovers σ as a function of S/N; but provided spectra are above this S/N threshold, the unavoidable exclusion of strong lines due to telluric or AGN contamination does not affect results.

We consider the effect of S/N before considering spatial effects, as the higher-offset spatially resolved spectra tend to fall in the lower S/N regime and therefore are affected by S/N trends, as seen in Section 5.

The desired S/N of a spectrum is a factor in determining exposure time and also the aperture width when extracting the spectra, so for higher-redshift studies that rely on the bluer regions, there is a strong motivation to know if σ_MgIb/σ_CaT or σ_CaHK/σ_CaT varies significantly with S/N. The concern is that there is an S/N threshold below which scatter is too high for a precise calibration of MgIb and CaHK to CaT.

We find no clear trends in 〈log (σ_CaHK/σ_CaT)〉 or 〈log (σ_MgIb/σ_CaT)〉 with S/N. As expected, scatter increases at S/N lower than 20 pixel⁻¹ (see the Appendix).

Having investigated possible trends with S/N and finding none, we now investigate whether σ_CaT/σ_CaHK or σ_CaT/σ_MgIb depend on the distance from the center or the aperture width, which are both affected by the AGN contribution in the center of the galaxy. A quantitative comparison of these measurements is shown in Figure 10. As in Section 5, for easy reference we draw dashed lines at 5% and 10% accuracy (5% accuracy is the target for this study). In this figure, if applicable, measurements have been corrected for the wavelength bias discussed in Section 5.

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For the spatially resolved analysis (panels (a) and (c)), measurements are put into 1'' bins (plotted at the center of the bin); bins containing less than 10 measurements are omitted because we expect our averaging procedure to underestimate the uncertainty in such cases. Note that for this analysis, the (arbitrary) sign of the spatial offset is disregarded, and 4-Δ clipping is used in each bin (see Section 5).

We find that 〈log (σ_CaHK/σ_CaT)〉 is constant within the errors, with values of −0.01 ± 0.01 ± 0.01 at 0'', −0.02 ± 0.01 ± 0.01 at 1'', and −0.03 ± 0.03 ± 0.01 at 2''. In contrast, 〈log (σ_MgIb/σ_CaT)〉 decreases marginally, dropping from 0.01 ± 0.02 ± 0.01 at 0'' to −0.019 ± 0.009 ± 0.01 at 1'' then −0.07 ± 0.03 ± 0.01 at 2''. In both cases, the 1'' bin contains the most measurements (≈85) while the 2'' contains the fewest (≈13) and the 0'' bin contains ≈35 measurements. These trends are qualitatively the same as were found in Paper I. In Section 5, we also found that MgIb results are somewhat more sensitive to offset than CaHK.

The aperture comparisons show that 〈log (σ_CaHK/σ_CaT)〉 and 〈log (σ_MgIb/σ_CaT)〉 are independent of aperture width by ∼25. The CaHK comparison (b) rises from its central value (same as spatially resolved central value) to a constant 0.00  ±  0.01  ±  0.01, and reaches this point by the Sloan fiber radius. The MgIb comparison (d) shows the same trend seen in the spatially resolved case: spectra with a higher fraction of galaxy light give a lower 〈log (σ_MgIb/σ_CaT)〉 than the central spectra. Here, the asymptotic value is −0.02 ± 0.01 ± 0.01.

With the spheroid effective radii for our objects we can revisit this spatial comparison in physical units, which may clarify the cause and strength of these trends.

In Table 2 and in Section 8, we use the results of this section to make σ_CaT-equivalent measurements from σ_CaHK and σ_MgIb.

Barth et al. (2002) investigated the fidelity of the MgIb region compared to the CaT region, although the MgIb wavelength range of those measurements extends from 5040–5430 Å, 100 Å greater in the red which may be significant as Greene & Ho (2006a) find 5250–5820 Å to be a superior measurement region.

Figure 11(a) mimics Figure 4 in the paper by Barth et al. (2002), comparing blue-side σ to σ_CaT as a function of σ_CaT(top panels). We use a linear plot here to show errors. Our data—σ_MgIb at the Sloan fiber radius, corrected for wavelength range bias—are plotted as filled circles and the Barth et al. (2002) data as open circles in panels (a) and (c). Barth et al. (2002) does not measure σ_CaHK so there is no data from that study in panels (b) and (d).

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Our sample and the Barth et al. (2002) sample have approximately the same distribution of measurements in both comparisons. The Barth et al. (2002) data return 〈log (σ_CaT/σ_MgIb)〉 = 0.003 ± 0.009, which agrees with our value (over all points, no S/N limits) of 0.00 ± 0.01. This agreement encourages an expansion upon the work presented in Barth et al. (2002) to the CaHK region, shown in Figure 11(b). As with MgIb, we see no clear trend with σ_CaT.

We also look for trends with blue-side σ (panels (c) and (d)), since the blue-side spectra have about half the kinematic resolution of the red and blue-side σ are generally considered to be less reliable. Plotting this way we see that most outlying measurements—measurements that are higher than the bulk of the measurements—occur at low σ_blue, in particular for σ_blue ≲ 125 km s⁻¹, i.e., when we approach the resolution of the blue spectra. This is consistent with the limits of the fitting program, discussed in the Appendix.

In this section, we explored the fidelity of the CaHK and MgIb regions compared to CaT. We made the comparison as a function of S/N, offset/aperture width, and σ.

S/N is a common diagnostic of spectrum quality. In the Appendix, we show that our fitting program is most reliable (independent of σ) at S/N above 20 pixel⁻¹. Using only data above this threshold, we find no trends with S/N; below, scatter is high and trends are dominated by fitting complications.

Spatial offset and aperture width correlate with fraction of galaxy light in the spectrum. We find no significant spatial trend in σ_CaHK/σ_CaT, and for aperture widths greater than ≈1'', σ_CaHK is equal to σ_CaT. On the other hand, we find that σ_MgIb/σ_CaT decreases by about 10% between the 0'' and 2'' offsets; by an aperture width of ≈2'' 〈log (σ_MgIb/σ_CaT)〉 reaches its asymptotic of −0.02 ± 0.01 ± 0.01, a 5% ± 3% bias high in σ_MgIb.

Finally, we have made the comparison as a function of σ, following the MgIb analysis of Barth et al. (2002) who showed σ_MgIb/σ_CaT as a function of σ_CaT (note that study did not measure σ_CaHK). We see that the ratios are more strongly a function of the blue-side σ than σ_CaT; the data from both studies show that the scatter and value of σ_MgIb/σ_CaT and σ_CaHK/σ_CaT increase at low σ. We believe this trend reflects the limitations of fitting programs at σ close to the instrumental resolution (which is lower in the blue), as demonstrated in the Appendix.

Our analysis shows that σ_CaHK recovers σ_CaT faithfully, for sufficiently large S/N and velocity dispersion (see Appendix). However, even when the fitting program is in its most accurate regime, σ_MgIb is not equal to σ_CaT. In the central spectra, σ_MgIb is slightly lower than σ_CaT and in the highest offsets it is significantly higher.

Motivated by the need to determine systematic uncertainties at the level of a few percent, we ran extensive Monte Carlo simulations to quantify how accurately our fitting program recovers the input value (σ_in) as a function of S/N and of σ. Consistent with previous work (e.g., Treu et al. 2001), we expect that for large values of S/N the code will return the input value, while as the S/N degrades significant biases might appear. Quantitatively, the amount of bias as a function of S/N and σ will depend on the exact instrumental configuration, chiefly the adopted wavelength range and resolution. To ensure a realistic test, the Monte Carlo realizations are obtained by adding random noise to the best-fit results of a subset of our galaxies, spanning a range in σ and each of the three wavelength regions.

The results of our tests are shown in Figure 15. For conciseness, we show only the test for the CaT region; the MgIb and CaHK regions give similar results. Gray bands of varying darkness show 1%, 5% (ideal target), and 10% accuracy. We use the median σ but include the results for the mean and most likely (maximum posterior) values for completeness; the median is generally most accurate, as can be seen in the figure. For the higher σ_in the code is well within our target accuracy, even at low S/N. As expected σ is least accurate for the lowest σ_in, as σ_in approaches the instrument resolution; however, we find that with S/N  >  40 pixel⁻¹ the code can recover σ to the desired accuracy even for the lowest σ_in. The difficulty of accurately recovering σ at low values of σ most likely causes the low-σ trends seen in panels (c) and (d) of Figure 11 (see Section 6.3).

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