SUBARU SPECTROSCOPY OF THE GLOBULAR CLUSTERS IN THE VIRGO GIANT ELLIPTICAL GALAXY M86^*

We selected the spectroscopic targets from a photometric sample of the GC candidates in M86 identified in Washington CT₁ images (158 × 158) taken at the KPNO 4 m telescope (Park 2012). The GCs in M86 appear as point sources in the ground-based KPNO images, and we first selected point sources around M86 with colors of 0.9 < (C − T₁) < 2.1 as GC candidates. This (C − T₁) color selection criterion is effective in selecting GC candidates in early-type galaxies: the success rate of the photometric search for GCs is about 90% in the case of M87 (Côté et al. 2001), M49 (Côté et al. 2003), and NGC 4636 (Park et al. 2010). We then selected from the bright sources with magnitudes 19 < T₁ < 21.5 as the spectroscopic targets. A total of 67 targets were chosen, which included the M86 nucleus and two known faint galaxies (NGC 4406B, VCC 0833). We also observed two red, bright point sources with (C − T₁) ∼ 3 to fill the mask gaps.

We obtained spectra of the 67 targets from observations in the Multi-Object Spectroscopy (MOS) mode of the Faint Object Camera and Spectrograph (FOCAS; Kashikawa et al. 2002) at the Subaru 8.2 m telescope on 2002 April 21. We observed two circular masks with diameters of 6'. Figure 1 presents a gray-scale map of the T₁ image of M86 taken at the KPNO 4 m telescope, which shows the positions of the spectroscopic targets as well as the observed masks. We subtracted the M86 stellar light from the original image using the IRAF/ELLIPSE³ task to show the point sources clearly.

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The observational log is given in Table 2. We used a medium-dispersion blue grism (300B) with a dispersion of 1.34 Å pixel⁻¹ and an order-cut filter L600 covering 3700–6000 Å. Seeing during the observation was 06. To make the masks, we obtained R-band pre-images with exposure times of 180 s in the FOCAS camera mode under a seeing condition of ∼10 on 2002 March 9. Using these images, we constructed masks with Mask Design Pipeline, a software utility for MOS (Saito et al. 2003). The slit width along the dispersion axis was 08, and the resulting spectral resolution was R ∼ 500.

We used three 1200 s exposures in mask-C and one 1800 s exposure in mask-1. We also obtained the comparison spectra with Th–Ar lamps before and/or after each exposure. We used the FOCAS long-slit spectroscopy mode to calibrate the flux, radial velocity, and metallicity. We observed a standard star, BD+33d2642, for the flux calibration and five Milky Way (MW) GCs (M5, M13, M92, M107, and NGC 6624) for the velocity and metallicity calibrations. We observed the MW GCs with stepping scan mode by moving the slit along the dispersion direction to sample an area larger than that covered by the slit. We observed these calibration targets during the same run and also used their spectra in the study of the NGC 4636 GCs (Park et al. 2010). Details of the long-slit mode observation are given in Section 2.2 and Table 2 of Park et al. (2010).

We first applied basic processing techniques (overscan correction, bias subtraction, and cosmic-ray rejection) to the CCD images using IRAF tasks; the CCD images were obtained with a pair of 4K × 2K CCDs. We used the FOCASRED/bigimage task in IDL (Saito et al. 2003) to produce a large single image from a pair of CCD images and correct distortions in the optics. We then clipped the two-dimensional spectrum of each target out of the single image and applied a flat-field correction. The spectrum from each two-dimensional image was traced, extracted, and sky-subtracted using the IRAF/APALL task. We could not extract the spectra of seven faint targets because of a low signal-to-noise ratio (S/N). Wavelength calibration was performed using Th–Ar lamp spectra with ∼40 useful emission lines in the 3800–6000 Å range. The typical rms error for this calibration is ∼0.8 Å. The flux of the target spectra was calibrated using the flux standard star.

Sample flux-calibrated spectra of an M86 GC, a foreground star, and the M86 nucleus are shown in Figure 2. We classified the target (ID 81) in panel (b) as a K2III star because of the broad (∼300 Å) absorption feature around the Mgb index. This feature is typically seen in the K giant star templates (Santos et al. 2002) but not in GC templates. Several absorption lines typically seen in old stellar systems including the G band, Hβ, and Mgb are clearly visible in the M86 GC spectrum in panel (a). Absorption features in the spectrum of the M86 nucleus are much broader than those in the GC and star because of its large velocity dispersion.

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We measured the radial velocities for the targets using the Fourier cross-correlation task, IRAF/FXCOR (Tonry & Davis 1979). A wavelength range of 4200–5400 Å was used for the cross-correlation because of the low S/N at ∼4000 Å and the strong night sky emission line [O i] at 5577 Å. During the cross-correlation, we fit the continuum of the spectra using a spline function with a 2σ clipping for the low level and a 4σ clipping for the high level. Radial velocities were measured for each target using the five MW GC templates; velocities for M86 GCs derived from the M5, M13, M92, and NGC 6624 templates are consistent within 1σ, but those from the M107 template differ by 2σ. Therefore, an error-weighted average of the first four measurements was taken to give the final radial velocity for each target. The error in the measured radial velocity is estimated as 〈_v〉 = (Σ⁻²_i)^−1/2, where _i is the error in each measurement.

The final number of targets with measured radial velocities is 31. The radial velocities for 36 objects among the original 67 could not be determined because of the poor quality of the spectra. We determined the radial velocity of the M86 nucleus to be $\upsilon _{p}$ = −234 ± 41 km s⁻¹, which is consistent with the previous measurement $\upsilon _{p}$ = −244 ± 5 km s⁻¹ (Smith et al. 2000). We also measured the radial velocities for two faint galaxies: $\upsilon _{p}$ = 777 ± 38 km s⁻¹ for VCC 0833 and $\upsilon _{p}$ = 949 ± 23 km s⁻¹ for NGC 4406B. These values are also consistent with previous measurements (Sloan Digital Sky Survey and Strauss et al. 1992). Errors in our velocity measurements range from 20 to 90 km s⁻¹ with a mean error of 49 ± 16 km s⁻¹.

To identify genuine M86 GCs among the 31 objects with measured velocities, we used (C − T₁) colors and radial velocities. The (C − T₁) colors are plotted as a function of the radial velocity in Figure 3; note that the M86 nucleus and two faint galaxies are not included in the plot. The velocity distribution in panel (b) shows that all objects have velocities of −900 to +300 km s⁻¹ except for one object with a very large velocity of $\upsilon _{p}$ ∼2400 km s⁻¹. We consider the objects with −900 kms⁻¹ < $\upsilon _{p}$ <300 km s⁻¹and 0.9 ⩽ (C − T₁) <2.1 to be genuine GCs bound to M86 (the velocity of the M86 nucleus is −234 km s⁻¹). The (C − T₁) color range is equivalent to a metallicity range of −2.24 dex ≲ [Fe/H] ≲ 0.33 dex (Lee et al. 2008c).

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Among the 28 objects (excluding the M86 nucleus and two faint galaxies), two do not satisfy the GC selection criteria. One target (ID 226) is classified as a star because its color is too red, (C − T₁) >2.1, even though it satisfies the velocity criterion. The other target (ID 446) satisfies the color criterion but does not satisfy the velocity criterion: $\upsilon _{p} = 2434 \pm 52$ km s⁻¹. This object seems to be an intracluster GC in the Virgo Cluster.

Because the systemic velocity of M86 is similar to the radial velocities of the MW stars, there could be some MW stars that satisfy our GC selection criteria. To remove these stars, we performed a careful visual inspection of the target spectra. We found one object (ID 81) with stellar spectral features (see Section 2.3 and Figure 2(b)) and rejected it from the M86 GC catalog. Thus, 25 genuine M86 GCs out of 28 targets, excluding the M86 nucleus and two faint galaxies, were identified. We note that there may be one or two more stars among the 25 GCs considering the selection efficiency for the GC candidates (see Section 2.1).

The reliability of our M86 GCs was further checked against the ACS Virgo Cluster Survey (ACSVCS) source catalog (Jordán et al. 2009). This catalog provides the classification probability for GC candidates in 100 Virgo early-type galaxies. We found that eight GCs (ID = 448, 284, 65, 270, 316, 107, 430, and 324) among the 25 M86 GCs are included in the ACSVCS catalog, and all of the eight have a GC probability larger than 90%, which confirms our classification. We show these eight GCs as open circles in Figure 3.

Table 3 lists the photometric and spectroscopic data including the metallicities of 16 GCs using the BH (Brodie & Huchra) method and the ages and metallicities of eight GCs using the grid method (see Section 5). The first column represents the identification number. The second and third columns give the right ascension and declination (J2000), respectively. The galactocentric radius and position angle (P.A.) are given in Columns 4 and 5, respectively. The magnitude and color information in Columns 6 and 7 are from Park (2012). Column 8 gives the radial velocity and its error measured in this study. Columns 9 and 10 give the age and metallicity derived from the grid method. Column 11 gives the metallicity derived from the BH method. The final column indicates the corresponding mask from Table 2. The M86 GCs are listed first, followed by foreground stars, a probable intracluster GC, and the faint galaxies and M86 nucleus.

The M86 GCs in our sample are found in a radial range of 42''–446'' (i.e., 3.4–36.1 kpc, see Figure 1). The mean radial velocity for the 25 GCs determined with the biweight location of Beers et al. (1990) is $\overline{v_p}=-354^{+81}_{-79}$ km s⁻¹, which is smaller than the radial velocity for the M86 nucleus (v_gal = −234 ± 41 km s⁻¹). The radial velocity distribution for the M86 GCs is shown in the top panel of Figure 4; the distribution appears to be Gaussian. The I statistics (Teague et al. 1990) gave an I value of 1.022, which is smaller than the critical value for rejecting the Gaussian hypothesis at the 90% confidence level, I_0.90 = 1.176. This suggests that the velocity distribution of the M86 GCs follows a Gaussian distribution. A Gaussian fit yields a peak at v_p = −343 km s⁻¹ with a width of σ_p = 279 km s⁻¹.

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The bottom panel of Figure 4 shows the radial velocities as a function of the projected galactocentric radius R. Mean velocities of the GCs in the inner (42'' ⩽ R < 240'') and outer (240'' ⩽ R < 446'') regions are similar to that of the GCs in the entire region. The velocity difference between the GC samples and the M86 nucleus is more visible in this panel.

The velocity dispersion of the M86 GCs determined with the biweight scale of Beers et al. (1990) is σ_p = 292⁺³²_{− 32} km s⁻¹. The velocity dispersion of the GCs in the inner region, σ_p = 292⁺³⁹_{− 39} km s⁻¹, is similar to that in the outer region, σ_p = 314⁺⁹⁰_{− 90} km s⁻¹. It is noted that the velocity dispersion determined in this study could be contaminated by the inclusion of possible MW stars because the systemic velocity of M86, v_gal = −234 km s⁻¹, is in the velocity range of the MW stars.

The plot of the radial velocities as a function of the (C − T₁) colors, T₁ magnitudes, and P.A.s Θ (measured from the north to east) in Figure 5 shows that the mean values of the radial velocities do not change with either the magnitude or the color. However, the mean values of the radial velocities seem to depend on the P.A., showing a minimum value at Θ ≈ 60° and ≈300°. This suggests a rotation of the M86 GC system (discussed in detail in the next section).

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In Figure 6, we show the spatial distribution of the M86 GCs with measured velocities. Although the spatial coverage is neither uniform nor large, the spatial segregation of the high-velocity and low-velocity GCs relative to the velocity of the M86 nucleus can be seen, which indicates a rotation of the GC system (see also Figure 5(c)).

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The amplitude and axis of rotation for the M86 GC system was measured with the following assumptions: (1) the GC system is spherically symmetric and (2) the rotation axis of the GC system lies in the plane of the sky. If the GCs follow any overall rotation, the radial velocities will depend sinusoidally on the azimuthal angles. Thus, we can then determine the amplitude and axis of rotation by fitting the radial velocities (v_p) to the function (Côté et al. 2001, 2003; Hwang et al. 2008; Lee et al. 2010a),

$$\begin{equation} v_p (\Theta) = v_{\rm sys} + (\Omega R) \sin (\Theta - \Theta _0), \end{equation} \tag{ 1 }$$

where ΩR is the rotation amplitude and v_sys is the systemic velocity of the GC system.

Figure 7 plots the radial velocities of the GCs as a function of the P.A. and an overlay of the best-fit rotation curve. We use an error-weighted, nonlinear fit of Equation (1) with v_sys fixed to the value of the M86 nucleus velocity. This gives a rotation amplitude of ΩR =228⁺⁷¹_{− 80} km s⁻¹, which suggests a rotation of the M86 GC system. However, as this result is based on a small sample size, it will need to be examined again with a larger number of GCs in future studies.

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The orientation of the rotation axis (Θ₀) is estimated to be 91°⁺¹⁹_{− 21}, and this appears to be closer to the photometric major axis of M86 (Θ_phot = 120°) than to the minor axis. The rotation of the M86 GC system around the major axis is consistent with the result based on stellar kinematics, although the spatial coverage of the stellar kinematics was much smaller than this study (Krajnović et al. 2011).

We summarize the kinematic results of the M86 GC system in Table 4: the range of the galactocentric radius of the GCs in arcseconds, the mean value of the radial distance in arcseconds, the number of GCs, the mean projected velocity and velocity dispersion about the mean velocity (σ_p), the P.A. of the rotation axis and rotation amplitude estimated using Equation (1), the velocity dispersion about the best-fit rotation curve (σ_{p, r}), and the absolute value of the ratio of the rotation amplitude to the velocity dispersion about the best-fit rotation curve. The uncertainties of the values represent 68% (1σ) confidence intervals that were determined from the numerical bootstrap procedure following the method of Côté et al. (2001).

Table 4. Kinematics of the M86 Globular Cluster System

R	〈R〉	N	$\overline{v_p}$	σp	Θ0	ΩR	σp, r	ΩR/σp, r
(arcsec)	(arcsec)		(km s−1)	(km s−1)	(deg)	(km s−1)	(km s−1)
42–446	203	25	−354+81− 79	292+32− 32	91+19− 21	228+71− 80	282+36− 33	0.81+0.32− 0.30

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Here, we investigate the existence of an extended dark matter halo in M86 by comparing the observed velocity dispersion profile (VDP) of the GCs with the VDP expected from the stellar mass profile (Côté et al. 2001, 2003; Hwang et al. 2008; Lee et al. 2010a). The stellar mass profile is derived from the surface brightness profile of M86, which is then used to compute the VDP.

Assuming that the M86 GC system is spherically symmetric in the absence of rotation, we apply the spherical Jeans equation (e.g., Binney & Tremaine 1987) to a dynamical analysis of the GC system. The spherical Jeans equation is

$$\begin{equation} {d\over {dr}}\, n_{\rm cl}(r) \sigma _r^2(r) + {{2\,\beta _{\rm cl}(r)}\over {r}}\, n_{\rm cl}(r) \sigma _r^2(r) = - n_{\rm cl}(r)\,{{G M_{\rm tot}(r)}\over {r^2}}\, \end{equation} \tag{ 2 }$$

where r is the three-dimensional radial distance from the galactic center, n_cl(r) is the three-dimensional density profile of the GC system, σ_r(r) is the radial component of velocity dispersion, β_cl(r) ≡ 1 − σ²_θ(r)/σ²_r(r) is the velocity anisotropy, G is the gravitational constant, and M_tot(r) is the total gravitating mass contained within a sphere of radius r. The tangential component of velocity dispersion, σ_θ(r), is equal to the azimuthal component of the velocity dispersion, σ_ϕ(r), in the absence of rotation. The total mass M_tot(r) interior out to any radius is the sum of the dark matter mass M_dm(r) and stellar mass M_s(r).

To solve Equation (2), n_cl(r) and M_s(r) are obtained from the observation. If we fix β_cl(r) as a constant and assume M_dm(r), we can predict the profile of σ_r(r). A comparison of this with the observed VDP will provide information about the existence of a dark matter halo. Because the observed VDP for the GCs is projected, we also need to compute the projected VDP, σ_p(R):

$$\begin{equation} \sigma _p^2(R) = {2\over {N_{\rm cl}(R)}}\, \int _R^{\infty } n_{\rm cl} \sigma _r^2(r) \left(1 - \beta _{\rm cl}\,{{R^2}\over {r^2}} \right)\, {{r\,dr}\over {\sqrt{r^2 - R^2}}}, \end{equation} \tag{ 3 }$$

where R is the projected galactocentric distance and the surface density profile, N_cl(R), is the projection of the three-dimensional density profile n_cl(r).

4.3.1. Density Profile of the M86 GC System

The three-dimensional density profiles are derived from the surface number density of the M86 GCs using the Navarro–Frenk–White (NFW) profile (Navarro et al. 1997) and the Dehnen profile (Dehnen 1993). We adopt the surface density profile given in Park (2012), which was derived using data from the HST/ACSVCS (Jordán et al. 2009) for GCs at R < 2'' and from the KPNO CT₁ data for GCs at R > 2''. The surface number density profile N_cl(R) is shown in Figure 8. We fit the surface number density profile to the projections of the NFW profile, n_cl(r) = n₀(r/b)⁻¹(1 + r/b)⁻² and the Dehnen profile, n_cl(r) = n₀(r/a)^−Γ(1 + r/a)^{Γ − 4}. The profile N_cl(R) is derived from an integration of the three-dimensional density profile n_cl(r) as follows:

$$\begin{equation} N_{\rm cl}(R) = 2\int _{R}^{\infty } n_{\rm cl}(r) {{r\,dr}\over {\sqrt{r^2-R^2}}}. \end{equation} \tag{ 4 }$$

The results of the fit are summarized as follows:

$$\begin{eqnarray} \hbox{\fontsize{9}{11}\selectfont $ n_{\rm cl}^{\rm NFW}(r)$} & = & \hbox{\fontsize{9}{11}\selectfont $ 0.052\,{\rm kpc}^{-3}(r/14.58\,{\rm kpc})^{-1}(1+r/14.58\,{\rm kpc})^{-2}$} \nonumber \\ \hbox{\fontsize{9}{11}\selectfont $ n_{\rm cl}^{\rm Dehnen}(r)$} & = & \hbox{\fontsize{9}{11}\selectfont $ 0.028\,{\rm kpc}^{-3}(r/27.94\,{\rm kpc})^{-0.93}(1+r/27.94\,{\rm kpc})^{-3.07}.$} \nonumber \\ \end{eqnarray} \tag{ 5 }$$

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4.3.2. Stellar Mass Profile

Figure 9 gives the radial profile of the M86 surface brightness, derived from the KPNO R-band images (Park 2012), and stellar mass. This surface brightness profile agrees well with the results in Peletier et al. (1990), which are based on R-band photometry, and with those in Caon et al. (1990), which are converted from B-band photometry.

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A fit of the surface brightness profile of Park (2012) to the projection of the three-dimensional luminosity density profile (Côté et al. 2003; Hwang et al. 2008; Lee et al. 2010a) represented by

$$\begin{equation} j(r) = {{(3-\gamma)(7-2\gamma)}\over {4}}\, {L_{\rm tot}\over {\pi a^3}}\, \left({r\over {a}}\right)^{-\gamma }\, \left[1+\Bigl ({r\over {a}}\Bigr)^{1/2} \right]^{2(\gamma -4)}\ \end{equation} \tag{ 6 }$$

gives γ = 1.75, L_tot = 1.60 × 10¹¹L_{R, ☉}, and a = 29.94 kpc. The projected best-fit curve is overlaid in Figure 9(a).

Figure 9(b) shows a three-dimensional stellar mass density profile, ρ_s(r) = ϒ₀j(r), derived with an R-band mass-to-light ratio of ϒ₀ = 6.5 M_☉ L⁻¹_{R, ☉} (determined in the next section). From this profile, we obtain an M86 stellar mass profile represented by

$$\begin{eqnarray} M_{\rm s}(r) & = &\int _{0}^{r}{4{\pi }x^2\,\rho _s(r)}dx = \Upsilon _0\int _{0}^{r}{4{\pi }x^2\,j(x)}dx \nonumber \\ & = & {\Upsilon _{0}}L_{\rm tot}\left[\! {(r/a)^{1/2} \over 1+(r/a)^{1/2}} \!\right]^{2(3-\gamma)}\left[\! {(7-2\gamma)+(r/a)^{1/2} \over 1+(r/a)^{1/2}} \!\right]. \,\,\,\quad \end{eqnarray} \tag{ 7 }$$

4.3.3. Extended Dark Matter Halo in M86

If we adopt the stellar mass profile as the total mass profile (i.e., M_tot(r) = M_s(r)) and ρ_s(r)∝j(r) instead of n_cl(r), we can derive the radial component of VDPs for the stellar system from the Jeans equation. We also assume R-band mass-to-light ratios (ϒ₀) and velocity anisotropies of the stellar halo (β_s(r)). The projected VDPs are then computed from the radial component of VDPs through Equation (3).

Figure 10 plots the M86 GC VDP measured in this study and the observed M86 stellar VDP given in Bender et al. (1994). The stellar velocity dispersion is almost constant around 220 km s⁻¹ in the inner region and is smoothly connected to the GC velocity dispersion at R ≈ 3 kpc. We also plot the predicted VDPs derived with ϒ₀ = 5.0 M_☉ L⁻¹_{R, ☉}, β_s(r) = 0.4 (radially biased) and ϒ₀ = 6.5 M_☉ L⁻¹_{R, ☉}, β_s(r) = 0.0 (isotropic), which fit the observed stellar kinematic data at R < 2 kpc well. For comparison, the predicted VDPs for the M86 GCs derived with the same stellar mass profile, the GC number density profiles (n^NFW_cl(r) and n^Dehnen_cl(r)) determined in the previous section, and several velocity anisotropies (β_cl(r) = +0.99 (radially biased), 0.0 (isotropic), and −99 (tangentially biased)) are also shown. The VDPs based on the NFW and Dehnen density profiles are not significantly different from each other. Note that none of the models agree with the observed VDPs of the GCs at R > 3 kpc, which suggests that the mass-to-light ratio is not constant over the galactocentric radius but should increase with the radius. This clearly suggests the existence of an extended dark matter halo in the outer region of M86.

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We determined the metallicities and ages of the M86 GCs using two methods: (1) the BH method, which determines the metallicity through an empirical relation between the absorption line index and metallicity developed by Brodie & Huchra (1990), and (2) the grid method, which derives metallicity and age from a comparison of the Lick line index of the spectrum with that of a single stellar population (SSP) model. We explain each method here.

5.1.1. BH Method

5.1.2. Lick Index Grid Method

Lick absorption line indices are useful for determining the metallicity and age of old stellar systems by comparing the indices derived from the spectra with the line index grids predicted from SSP models (Tripicco & Bell 1995; Trager et al. 2000, 2008; Thomas et al. 2004; Puzia et al. 2005; Beasley et al. 2008; Woodley et al. 2010). For this grid method, we adopt the SSP models given by Thomas et al. (2003, 2004, 2005) and follow the technique described in Puzia et al. (2005) and Park et al. (2012).

We calibrate our absorption line indices to the Lick index system as follows. The spectra are smoothed with the Lick resolution (Worthey & Ottaviani 1997) after shifting each spectrum into the rest frame. Lick line indices are then derived from the spectra of the M86 GCs, following the definitions given in Worthey (1994) and Worthey & Ottaviani (1997). Here, the line index errors are derived from the photon noise in the spectra before the flux calibration. The resulting line indices are then calibrated to the Lick system with the zero-point offset, Index(Lick) = Index(Subaru) + offset, determined from the spectra of five MW GCs that are common to this study, Trager et al. (1998), and Kuntschner et al. (2002). The offsets we derived are 0.184 ± 0.140 for Hβ, 0.165 ± 0.187 for Mgb, 0.443 ± 0.306 for Fe5270, and −0.103 ± 0.162 for Fe5335 (see Table 1 in Park et al. (2012) for the offsets of other indices). We determined the Lick line indices for eight GCs in M86, which are listed with errors in Table 5.

Table 5. Lick Line Indices and Errors

ID	Hβ	Mg2	Mgb	Fe5270	Fe5335
	(Å)	(mag)	(Å)	(Å)	(Å)
65	2.423 ± 0.340	0.120 ± 0.006	2.765 ± 0.340	0.996 ± 0.359	1.074 ± 0.459
270	2.527 ± 0.641	0.075 ± 0.012	3.316 ± 0.623	2.496 ± 0.659	−2.172 ± 0.909
107	2.142 ± 0.458	0.179 ± 0.008	3.635 ± 0.444	2.037 ± 0.457	−0.147 ± 0.602
143	0.427 ± 0.474	0.102 ± 0.008	3.255 ± 0.448	2.762 ± 0.478	2.387 ± 0.574
58	2.173 ± 0.301	0.109 ± 0.005	0.814 ± 0.307	2.114 ± 0.312	2.031 ± 0.399
79	2.164 ± 0.368	0.072 ± 0.007	0.916 ± 0.363	2.585 ± 0.380	5.069 ± 0.475
96	2.487 ± 0.312	−0.005 ± 0.006	1.128 ± 0.314	3.116 ± 0.319	1.920 ± 0.420
149	1.705 ± 0.432	0.043 ± 0.007	1.164 ± 0.404	1.955 ± 0.430	0.134 ± 0.556

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The composite index [MgFe]', defined by $[{\rm MgFe}]^\prime = \sqrt{\rm {Mgb} \times \ (0.72 \times \rm {Fe5270} + 0.28 \times \rm {Fe5335})}$, is a good metallicity tracer because of its low sensitivity to [α/Fe]. The index Hβ is an age indicator and the least sensitive to [α/Fe] among the Balmer lines (Thomas et al. 2003). Thus, we determine the metallicity and age of each GC in the Hβ versus [MgFe]' grids provided by Thomas et al. (2003). Figure 11 shows the observational indices of the M86 GCs in comparison with the SSP model grids for Hβ versus [MgFe]', which indicate SSP models with [α/Fe] = 0.2, [Z/H] = −2.25, −1.35, −0.33, 0.0, 0.35, and 0.67 dex, and ages of 0.4, 0.6, 0.8, 1, 2, 3, 5, 8, 10, and 15 Gyr. All the GCs seem to be older than ∼5 Gyr and metallicities smaller than the solar abundance. For the GCs inside the envelope of the model grid, we take the [Z/H] value and age at the nearest model grid interpolated with bins of 0.01 dex for [Z/H] and 0.1 Gyr for age. For the GCs outside the envelope, we take the values of the nearest envelope of the model grid in the direction of the error vector as done in Puzia et al. (2005). The two outliers with a large difference from the model envelope along the Hβ axis might be due to a limit of the low-resolution-integrated spectroscopy or of the model grids because these objects are also often shown in studies of MW GCs and M31 GCs derived from high S/N spectra (Puzia et al. 2002, 2005; Schiavon et al. 2012) as well as other gE GCs (Cenarro et al. 2007; Woodley et al. 2010). To estimate the errors in the age and metallicity, we calculate the ages and metallicities of four data points composed of Hβ ± error and [MgFe]' ± error in the grid. The difference between the average of these four values and the estimate calculated directly from the index is taken as the final error.

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The metallicities of eight GCs derived from the grid method are compared with those from the BH method in Figure 12(a). The two measurements are broadly consistent within the uncertainty. Here, the total metallicity ([Z/H]) derived from the grid method was converted into [Fe/H](grid) using the relation [Fe/H] = [Z/H] − 0.94 [α/Fe] (Thomas et al. 2003). For this conversion, we adopted [α/Fe] = 0.2, which is the mean [α/Fe] for the GCs in gEs (Park et al. 2012). We also compared the observational (C − T₁)₀ colors of the M86 GCs (Park 2012) with the model colors derived from the [Z/H] and age using the SSP model in Marigo et al. (2008). Among the eight GCs, five GCs agree well within their errors as shown in Figure 12(b).

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The metallicities and ages of the M86 GCs are listed in Table 3, and Figure 13 shows (1) [Fe/H] versus R, (2) age versus R, (3) [Fe/H] distribution, and (4) age distribution of the M86 GCs. It is not easy to derive any systematic trend in the data because of small number statistics. However, several features are noted. First, the metallicities of the 16 GCs based on the BH method show a wide range of −2.0 < [Fe/H] <−0.2 with a mean value of −1.13 ± 0.47, which is similar to that based on the grid method (−0.91 ± 0.44). Second, metal-rich GCs ([Fe/H] >−0.9) are found only in the inner region (R ⩽ 4'). Third, the age of the M86 GCs also shows a wide range from 4 to 15 Gyr with a mean of 9.7 ± 4.0 Gyr. This is similar to the GCs in other gEs (e.g., Woodley et al. 2010; Park et al. 2012).

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Figure 14 shows the metallicity as a function of age for the M86 GCs. This figure hints at an age–metallicity relation, meaning that the younger GCs are more metal-rich. This relationship in gEs has also been seen for the GCs in M60 (Pierce et al. 2006) and in NGC 5128 (Woodley et al. 2010). However, Woodley et al. (2010) did not strongly confirm its existence because of large biases in their selected GC sample (e.g., extremely bright clusters). A more complete analysis with a more comprehensive data set of gE GCs is necessary to draw a strong conclusion about the age–metallicity relation.

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