STRUCTURE AND DYNAMICS OF THE GLOBULAR CLUSTER PALOMAR 13^*

We select spectroscopic targets in Pal 13 based on CFHT CFH12K and Keck/Low Resolution Imaging Spectrometer (LRIS) imaging as described in C02. We supplement these data at large radius with wider field photometry from P. Stetson's online database.⁷ Candidate Pal 13 member stars were identified in V- and I-bands, and V- and R-band when I-band was not available. This isochrone was used for target selection only; a slightly better fitting isochrone is used in the subsequent analysis (see Figure 2). Targets were selected based on proximity to a Girardi et al. (2002) theoretical isochrone for an age of 12 Gyr and a metallicity of [Fe/H] =−1.7 (Zinn & Diaz 1982; Friel et al. 1982; Côté et al. 2002). This isochrone was used for target selection only; a slightly better fitting isochrone is used in the subsequent analysis. We determine a star's distance to the isochrone via the quantity $d = \sqrt{[(V-I)_* - (V-I)_{\rm iso}]^2 + (V_* -V_{\rm iso})^2}$. Stars on the red giant branch (V ⩽ 20.5) within d = 0.1 mag of the isochrone were given the highest priority for spectroscopy; stars within 0.2 mag of the isochrone were given lower priority. On the subgiant branch and main sequence (V > 20.5) and on the horizontal branch the selection region was widened to within 0.2 and 0.3 mag of the isochrone for the higher and lower priority categories, respectively. Candidate BSSs were chosen in a box blueward of and brighter than the main-sequence turnoff/subgiant branch region. Following the photometric selection, spectroscopic targeting priorities were set to favor known radial-velocity members from C02 and Blecha et al. (2004), and likely proper motion members from Siegel et al. (2001).

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The spectroscopic data were taken with the Keck II 10 m telescope and the DEIMOS spectrograph (Faber et al. 2003). Eight multislit masks were observed in Pal 13 in 2008 September, with a ninth mask obtained in 2009 October. Exposure times, observation dates, and other observing details are given in Table 1. The masks were observed with the 1200 line mm⁻¹ grating covering a wavelength region 6400–9100 Å. The spectral dispersion of this setup was 0.33 Å pixel⁻¹, equivalent to R = 6000 for our 07 wide slitlets, or an FWHM of 1.37 Å. The spatial scale was 012 pixel⁻¹. The minimum slit length was 5'' which allows adequate sky subtraction; the minimum spatial separation between slit ends was 04 (3 pixels).

Spectra were reduced using a modified version of the spec2d software pipeline (version 1.1.4) developed by the DEEP2 team at the University of California-Berkeley for that survey. A detailed description of the reductions can be found in Simon & Geha (2007). The final one-dimensional spectra were rebinned into logarithmic wavelength bins with 15 km s⁻¹ pixel⁻¹. Radial velocities were measured by cross-correlating the observed science spectra with a series of high signal-to-noise stellar templates. We employ the telluric absorption lines in the stellar spectra to calculate and apply a correction for the radial-velocity errors that arise from a miscentering of the star in the spectrograph slit. Each science spectrum is cross-correlated with a hot stellar template where the telluric absorption lines are the dominant spectral features. Since these lines are illuminated by the stellar flux, this defines the star's photocenter within the slit in velocity space. We applied both a telluric and heliocentric correction to all velocities presented in this paper.

We determined the random component of our velocity errors using a Monte Carlo bootstrap method. Noise was added to each pixel in the one-dimensional science spectrum, we then recalculated the velocity and telluric correction for 1000 noise realizations. Error bars are defined as the square root of the variance in the recovered mean velocity in the Monte Carlo simulations. The systematic contribution to the velocity error was determined by Simon & Geha (2007) to be 2.2 km s⁻¹ based on repeated independent measurements of individual stars and we refer the reader to this paper for further details. The systematic error contribution is expected to be constant as the spectrograph setup and velocity cross-correlation routines are identical. We added the random and systematic errors in quadrature to arrive at the final velocity error for each science measurement. Radial velocities were successfully measured for 482 of the 566 extracted spectra across the nine observed DEIMOS masks. The fitted velocities were visually inspected to ensure reliability. We identified 90 sources as background galaxies or QSOs and obtained repeat measurements for 62 stellar sources. Our final stellar sample consists of 392 measurements of 306 unique stars.

2.2.1. Deep CFHT MegaCam Imaging of Palomar 13

2.2.2. Improved Structural Parameters for Palomar 13

Using our CFHT/MegaCam catalog, we recalculated structural parameters for Pal 13 using a Maximum Likelihood method as described in Muñoz et al. (2010). In this method, a fixed analytic profile is assumed and its free parameters are fit using all the available stars in the color-magnitude diagram (CMD) selection window. This avoids the need to bin or smooth data, providing more reliable estimates for systems with low numbers of stars (Martin et al. 2008). We simultaneously estimated the scale length, the central coordinates, the ellipticity, the position angle, and the background density of Pal 13. We assumed an empirical King profile (King 1962), for which we calculated a King core (r_c) and tidal (r_t) radius. We additionally fit a Navarro–Frenk–White (NFW; Navarro et al. 1997) and a Plummer profile (Plummer 1911), a good representation for dwarf galaxies, but found that these were poor fits to the light distribution shape of Pal 13. The best-fitting King parameters are r_c = 042  ±  006 (3.0  ±  0.4 pc) and r_t = 139  ±  15 (98  ±  11 pc). The two-dimensional half-light radius of this King profile is r_1/2 = 127  ±  016 (9.0  ±  1.1 pc). The overall ellipticity of Pal 13 is consistent with being zero and therefore its position angle is poorly constrained.

In the left panel of Figure 3, we show the projected density profile for Pal 13, overplotting our best-fit King profile. Error bars on the data points include uncertainty in determining the background level which dominates outside r ∼ 7'. We emphasize that the King profile shown is not a direct fit to the data points, but constructed with the best-fit parameters obtained through the maximum likelihood estimator. We compared our best-fitting King profile to that determined in C02. If we parameterize the outer slope of the surface brightness profile as Σ∝r^−η, then typical globular clusters have slopes of η ∼ 4 (McLaughlin & van der Marel 2005). We measure a shallow slope for Pal 13 of η = 2.8 ± 0.3. This is consistent with our inferred King tidal radius which is two times larger than the calculated tidal radius of Pal 13 based on its stellar mass and distance from the Milky Way (see Section 3). Our inferred slope is steeper than the C02 estimate of η = 1.8 ± 0.2. These two profiles are compared in Figure 3. A key difference between the two King profiles is the CMD region in which they were determined. While C02 determined the profile using stars inside a box in CMD space, we have used a window around the best-fitting isochrone. At faint magnitudes, there is significant contamination from unresolved galaxies, and our CMD window should reduce this contamination.

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In the right panel of Figure 3, we show the spatial density map of Pal 13. The outer isophotes in this figure are 3σ contours above the background, corresponding to r ∼30 mag arcsec⁻². The red dashed line/circle in both panels of Figure 3 are drawn at the King tidal radius r_t = 139 (98 pc). The photometric isophotes are regular out to the half-light radius of the cluster and appear mildly irregular outside this radius. This is in contrast to other faint globular clusters such as Palomar 1 and Palomar 5 which show evidence for tidal tails at much brighter surface brightness levels (Niederste-Ostholt et al. 2010; Odenkirchen et al. 2003), although similar to that observed for Palomar 14 (Sollima et al. 2011). We interpret the slightly irregular isophotes and shallow density profile as mild evidence for tidal disruption and discuss further in Section 5.

Finally, we estimated the total luminosity of Pal 13. The published values for Pal 13 range between 1.2 × 10³ and 3.5 × 10³ L_☉ (Siegel et al. 2001; Côté et al. 2002), making traditional methods of adding stars' individual fluxes too sensitive to the inclusion (or exclusion) of potential members (outliers). To alleviate these shot noise issues, the method used here relies solely on the total number of stars belonging to the satellite and not on their individual magnitudes (Muñoz et al. 2010). To estimate an absolute magnitude, we modeled the satellite's population with the best-fitting theoretical luminosity function, in this case a 12 Gyr population with [Fe/H]=−1.6 (Girardi et al. 2002). We then integrated the theoretical luminosity function to obtain the total flux down to a given magnitude limit. The last step was to scale this flux using the total number of Pal 13 stars in our catalog down to the same magnitude limit. To estimate the uncertainty in the absolute magnitude we carried out a bootstrap analysis generating 10,000 realizations of Pal 13 from the photometric data (for more details see Muñoz et al. 2010). We assumed a Chabrier initial mass function for the estimation (Bruzual & Charlot 2003). We obtained M_V = −2.8 ± 0.4, or equivalently L_V = 1.1^+0.5_−0.3 × 10³ L_☉. The central surface brightness is μ_{0, V} = 23.9^+0.6_−0.7 mag arcs⁻². While our results are 1σ lower than published values, it is a more robust estimate of the true luminosity of Pal 13 as it better accounts for shot noise in this quantity. We list our measured properties of Pal 13 and those of C02 in Table 2 for comparison.

Table 2. Palomar 13 Properties

Property	Côté et al. (2002)	This Work
Core radius (King model), rc	0.23 ± 0.03 arcmin	0.42 ± 0.06 arcmin
Tidal radius (King model), rt	26 ± 6 arcmin	13.9 ± 1.50 arcmin
Half-light radius (King model), r1/2	N/A	1.27 ± 0.16 arcmin
Luminosity, LV	(2.8 ± 0.4) × 103 L☉	(1.1+0.5−0.3) × 103 L☉
Absolute magnitude, MV	−3.8 mag	−2.8 ± 0.4 mag
Metallicity, $\rm [Fe / \rm H]$	−1.9 ± 0.2 dex	−1.6 ± 0.1 dex
Right ascension (center, King model), R.A.	3466867	34668519 ± 000063
Declination (center, King model), decl.	127717	12771539 ± 000068
Distance modulus, (m − M)0	16.93 ± 0.10 mag	N/A
Distance, d	24.3+1.2−1.1 kpc	N/A

Notes. We list properties of Pal 13 compiled by C02 and as independently measured by this work for comparison.

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We obtained numerous repeated velocity measurements of stars in Pal 13 in order to investigate the frequency of unresolved binary stars and their influence on the velocity dispersion. We measured two or more velocities for 62 out of 306 total (36 out of 89 member) stars. We further fold in historical data from C02, taken 10 years prior to our DEIMOS observations, providing multiple measurements for 72 out of 306 total (42 out of 89 member) stars. We list both DEIMOS and HIRES velocity measurements in Table 3. Binary fractions inferred from our data are necessarily lower limits on the true binary fraction and are most sensitive toward short (few day) periods, but also include one year to ten year periods depending on the available data. We also note that the repeated stars are heavily biased toward bright magnitudes.

Using the error-normalized velocity difference plotted in Figure 6, we define a star to be variable if individual independent measurements are more than 3σ discrepant in this quantity. We acknowledge that the number of "identified" binaries depends sensitively on this choice. We find that 5 out of 89 in our full member sample are velocity variables (6%). These five stars are also contained in our cleaned sample. Given the large velocity differences (a few to tens of km s⁻¹), we assume that these systems are unresolved binary stars, as single variable stars are unlikely to be in this region of color–magnitude space. As expected, the fraction of binary stars identified in our incomplete sample is far less than 30%  ±  4% measured by Clark et al. (2004), but does provide spectroscopic confirmation that unresolved binary stars exist in Pal 13.

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For all stars with multiple measurements, we use the error-weighted mean velocity in the calculations below. Because we are measuring the five binary stars at a random points in their orbits, it is not guaranteed that this will be the true systemic velocity of the star. We will therefore calculate, in Section 4.2, the velocity dispersion of Pal 13 with and without the five identified binary systems.

The velocity dispersion of Pal 13 measured by C02 of 2.2 ± 0.4 km s⁻¹ is significantly larger than that predicted based on the stellar mass alone of ∼0.3–0.4 km s⁻¹. If this velocity dispersion accurately traces the motion of stars in a gravitational potential, it implies over 95% of mass in Pal 13 is not luminous, i.e., that Pal 13 has a significant dark matter component. We measured Keck/DEIMOS velocities for all 21 stars in the Keck/HIRES C02 sample and, in most cases, have multiple measurements separated by up to one year.

Throughout this paper, we determine dispersions using a Monte Carlo Markov Chain (MCMC) method (Lane et al. 2010; Gregory 2005) assuming uniform priors on the velocity and velocity dispersion. This method produces the same results as the likelihood maximization technique described in Walker et al. (2006) when the error distributions are Gaussian. In the regime where the velocity dispersion approaches the observational errors on this quantity, the MCMC method allows us to calculate confidence intervals directly from posterior distributions without the assumption of Gaussianity. We confirmed that our MCMC algorithm reproduces the same dispersion and errors reported by C02 for their HIRES data set.

Using a single epoch of velocities and the 21 star sample defined in C02, we measured a Keck/DEIMOS velocity dispersion of 2.5 ± 0.7 km s⁻¹, in good agreement with the Keck/HIRES measurement. We next combined our 47 independent measurements of these 21 stars (six stars have only a single measured velocity, while three stars have four or more epochs) and redetermine the velocity dispersion. Combining our repeat measurements, we find a lower velocity dispersion of 1.6 ± 0.7 km s⁻¹, suggesting that unresolved binary stars are indeed inflating the single-epoch dispersion, although not to a level consistent with the stellar mass alone.

We next examined the C02 stars for evidence of variability. There are two stars in this sample which we suspect, ORS-38 and ORS-91, identified as orange symbols in Figure 4. ORS-38 is near the red giant branch (RGB) and varies in velocity by 17 km s⁻¹ between the DEIMOS and HIRES measurements. This is one of the five variables identified in Section 3.1. While it does not inflate the HIRES velocity dispersion, ORS-38 is the primary star inflating our calculations. ORS-91 is on the horizontal branch and lies in the instability band identified in Ivezić et al. (2005). Although the DEIMOS and HIRES velocities agree, and including/removing this stars does not affect our results, we chose to exclude this and other horizontal branch stars from our sample. Removing ORS-38 and ORS-91 does not affect the HIRES-determined dispersion, but the DEIMOS-based velocity dispersion falls to 0.7^+0.8_−0.6 km s⁻¹. This dispersion is consistent with a normal stellar mass-to-light ratio, although our measurement uncertainties are too large to rule out a higher than expected mass-to-light ratio. We plot the error-normalized differences in velocity between the HIRES and DEIMOS samples in Figure 6, noting the position of these two rejected stars.

We determine the velocity dispersion of Pal 13 first using the cleaned sample of 66 stars defined above. This sample has minimal Milky Way foreground contamination and contains five known velocity variable stars. Using only a single epoch of DEIMOS observations, we calculate a dispersion of 2.3 ± 0.5 km s⁻¹. Using a weighted average of all our repeat measurements, the velocity dispersion decreases to 1.6 ± 0.5 km s⁻¹. By removing the five known variable stars, the velocity dispersion drops close to our measurable limits: σ = 0.6^+0.7_−0.5 km s⁻¹. Finally, adding in quadrature measurements from C02 where available, the velocity dispersion of Pal 13 is σ = 0.4^+0.5_−0.4 km s⁻¹. These results demonstrate that unresolved binaries or variables, even multiply measured and averaged, can still contribute significantly to the dispersion given our sample size.

We take as the final velocity dispersion, the value determined without binaries combining both DEIMOS and HIRES measurements (61 stars, σ = 0.4^+0.5_−0.4 km s⁻¹). The average cluster distance of this sample is 15, slightly larger than the Pal 13 half-light radius of r_1/2 = 13, as seen in the right panel of Figure 5. We do not have a sufficient number of stars to produce a binned velocity dispersion as a function of radius, however, splitting the sample at the radius enclosing half the measured stars, we find that both the inner and outer bin have similar velocity dispersions, but the error on each value doubles.

We calculate the mass and mass-to-light ratio of Pal 13 using the velocity dispersion determined via our 61 member clean sample without binary stars, σ = 0.4^+0.5_−0.4 km s⁻¹. We determine the dynamical mass enclosed within the half-light radius applying the formula from Wolf et al. (2010): $M_{1/2}(\rm r_{\rm 1/2}) \simeq 4G^{-1}\langle \sigma _{\rm los}^2 \rangle \rm r_{\rm 1/2}$, where M_1/2 is the mass contained within the three-dimensional projected half-light radius, σ_los is the line-of-sight velocity dispersion. This mass estimator is based on the Jeans equation and is less sensitive to uncertainties in the velocity anisotropy than other simple mass estimates. We calculated an enclosed mass of M_1/2 < 1.3^+2.7_−1.3 × 10³ M_☉. Using the luminosity determined in Section 2.2.2, L_V = 1.1^+0.5_−0.3 × 10³ L_☉, we calculated a mass-to-light ratio of ϒ_V = 2.4^+5.0_−2.4 M_☉/L_☉ within the half-light radius.

We alternatively calculate the mass-to-light ratio of Pal 13 assuming velocity isotropy and that the cluster is well fit by a King profile. Following C02, the mass-to-light ratio is given by ϒ_V = σ²₀/(0.003r_cμ_{0, V}) (Richstone & Tremaine 1986), where r_c is the King core radius in parsecs, μ_{0, V} is the central surface brightness (L_☉ pc⁻²), and σ₀ is the central velocity dispersion. Using the best-fitting King parameters from Section 2.2.2, we determine ϒ_V = 1.9.

Within the limits of our velocity precision, the mass-to-light ratio of Pal 13 is consistent with its stellar mass. Our estimates are consistent with the average dynamical mass-to-light ratio of old Milky Way globular clusters of ϒ_V = 1.45 (McLaughlin 2000). Since Pal 13 is slightly more metal-poor than the average Milky Way globular cluster in this sample, we computed the theoretical mass-to-light ratio, based on a Chabrier initial mass function, an age of 12 Gyr and [Fe/H] =−1.6 (Bruzual & Charlot 2003) to be ϒ_V = 2.0. We therefore conclude that the dynamics of Pal 13 are consistent with its stellar mass alone.

We measured spectroscopic metallicities for individual stars in Pal 13 using the spectral synthesis method introduced by Kirby et al. (2008a) and refined by Kirby et al. (2010). Metallicities based on this method are reliable for stars with log g < 3.6 (at the distance of Pal 13 roughly r < 20). In Figure 7, we plot metallicities of the 20 stars passing this criterion in our full sample. As discussed in Section 3, four stars have clearly deviant metallicities from the main sample and we have excluded these stars from Pal 13 membership. The weighted average metallicity for the 16 member stars is 〈[Fe/H]〉 = −1.6 ± 0.1 dex. This is comparable to previous spectroscopic estimates of the metallicity based on medium resolution spectroscopy, [Fe/H] = −1.67 ± 0.15 (Zinn & Diaz 1982) and [Fe/H] = −1.9 ± 0.1, (Friel et al. 1982), as well as [Fe/H] = −1.98 ± 0.31 dex based on a single Pal 13 star observed at high resolution (Côté et al. 2002).

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Using the same sample of 16 stars above, we estimated an internal metallicity dispersion of 0.1 ± 0.1 dex. While the 1σ errors are consistent with no internal metallicity dispersion, as expected for a single stellar population globular cluster, the measurement does allow a modest metallicity dispersion. This dispersion is smaller than observed in some luminous globular clusters (e.g., NGC 2419; Cohen et al. 2010) and is significantly smaller than observed for stellar systems containing dark matter, e.g., dwarf galaxies show clear metallicity dispersions on the order of 0.5 dex (Kirby et al. 2010). Dwarf galaxies also exhibit a tight correlation between luminosity and metallicity (Kirby et al. 2008b). Given Pal 13's low luminosity (M_V = −2.8), this relationship predicts a metallicity of [Fe/H] ∼ − 3, significantly more metal-poor than observed. Both the average metallicity of Pal 13 and the low internal metallicity dispersion are further evidence that Pal 13 is a globular cluster.

We show above that the dynamics of Pal 13 are influenced by the presence of unresolved binary stars. This is unlike other Milky Way globular clusters whose single-epoch velocity dispersions are consistent with their stellar mass (Baumgardt et al. 2009; Lane et al. 2010; Hankey & Cole 2011). We next investigate the BSSs population of Pal 13 as a means to determine whether the binary fraction in Pal 13 is indeed higher than typical globular clusters. BSSs are positioned brighter and blueward of the primary main-sequence turnoff. In globular clusters, BSSs mimic a younger stellar population, but are interpreted as systems which formed via stellar collisions (Hills & Day 1976) or mass transfer in close binary systems (McCrea 1964). We have not included BSSs in our kinematic analysis, however, our spectroscopic observations of these stars provide some clue to their formation, and their frequency provides further evidence for an enhanced binary fraction in Pal 13.

In our spectroscopic sample, 12 stars pass the BSS criterion in color–magnitude (Figure 4). Two of these stars do not pass our velocity criterion for membership, and their proper motions measured by Siegel et al. (2001) confirm they are not Pal 13 members. The remaining 10 stars have proper motions consistent with Pal 13 membership. We obtained repeated velocity measurements for only one member star; the repeat measurements are consistent with a constant velocity. However, the velocity distribution of the 10 stars identified as BSS members of Pal 13 is broader than the overall velocity distribution of Pal 13. This is reflected in the velocity dispersion of the BSSs, 4.1 ± 1.4 km s⁻¹,  substantially larger than the main Pal 13 sample. This hints that at least some of these BSSs may be close binary systems.

A common way to express the BSS abundance is by measuring the ratio, of BSSs to RGB stars, F^bss_rgb. Using our CFHT photometric data, we estimated the frequency (F^bss_rgb) of BSSs following the method of Leigh et al. (2007) by counting the number of BSSs and RGB stars within well-defined regions of the CMD. These regions are similar to Leigh & Sills, with the main difference being our BSS box is located closer to the main-sequence turnoff. The reason for this is that Pal 13 is a low density cluster and overcrowding near the center is not an issue and, therefore, the scatter in the main sequence is lower.

We measured F^bss_rgb in both a region within the half-light radius (13) and three times this value obtaining F^bss_rgb = 1.0 ± 0.4 and 0.8  ±  0.3, respectively. Our results are directly comparable to the values of F^bss_rgb determined by Leigh et al. (2007) for 57 Milky Way globular clusters (their sample did not include Pal 13). These authors found an average value of F^bss_rgb = 0.14 with a standard deviation of 0.19. In comparison, Pal 13 presents a remarkably high frequency of BSSs. Given the binary fraction measured by Clark et al. (2004) of 30% ± 4% based on stars redward of the main sequence, the high frequency of BSSs can be interpreted as due to mass transfer between primordial binaries. The high BSS frequency is consistent with Pal 13 having a higher binary fraction than typical Milky Way globulars.