A {\it Chandra} Survey of Milky Way Globular Clusters I: Emissivity and Abundance of Weak X-ray Sources
Zhongqun Cheng Zhiyuan Li Xiaojie Xu Xiangdong Li



^1 Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China 
^2 School of Astronomy and Space Science, Nanjing University, Nanjing 210023, China 
^3 Key Laboratory of Modern Astronomy and Astrophysics (Nanjing University), Ministry of Education, Nanjing 210023, China
lizy@nju.edu.cn; lixd@nju.edu.cn



Based on archival Chandra data, we have carried out an X-ray survey of 69, or nearly half the known population of, Milky Way globular clusters (GCs), focusing on weak X-ray sources, mainly cataclysmic variables (CVs) and coronally active binaries (ABs).  
Using the cumulative X-ray luminosity per unit stellar mass (i.e., X-ray emissivity) as a proxy of the source abundance, we demonstrate a paucity (lower by 41%±27% on average) of weak X-ray sources in most GCs relative to the field, which is represented by the Solar neighborhood and Local Group dwarf elliptical galaxies.  
We also revisit the mutual correlations among the cumulative X-ray luminosity (L_X), cluster mass (M) and stellar encounter rate (Γ), finding L_X∝ M^0.74± 0.13, L_X∝Γ^0.67±0.07 and Γ∝ M^1.28 ± 0.17. 
The three quantities can further be expressed as L_X∝ M^0.64±0.12 Γ^0.19 ±0.07, which indicates that the dynamical formation of CVs and ABs through stellar encounters in GCs is less dominant than previously suggested, and that the primordial formation channel has a substantial contribution. 
Taking these aspects together, we suggest that a large fraction of primordial, soft binaries have been disrupted in binary-single or binary-binary stellar interactions before they can otherwise evolve into X-ray-emitting close binaries,
whereas the same interactions also have led to the formation of new close binaries.
No significant correlations between L_X/L_K and cluster properties, including dynamical age, metallicity and structural parameters, are found.






§ INTRODUCTION
 
Globular clusters (GCs) are aged and self-gravitationally bound systems that evolve with a high stellar density.
Since the launch of the first X-ray satellite, Uhuru, it has been recognized that the abundance (i.e., number per unit stellar mass) of luminous X-ray binaries (with luminosity L_ X≳ 10^36 erg s^-1) in GCs is ∼100 times higher than in the Galactic field <cit.>. 
This over-abundance was attributed to the efficient formation of low-mass X-ray binaries (LMXBs) by stellar dynamical interactions in the dense core of GCs, where an isolated neutron star can be captured by a main sequence star through tidal force <cit.>, by a giant through collision <cit.>, or by a primordial binary through exchange with one of the constituent stars <cit.>. 
The fundamental parameter quantifying these dynamical interactions is the so-called stellar encounter rate, Γ, which is related to the stellar density (ρ) and velocity dispersion (σ), as Γ∝∫ρ^2/σ, an integral over the cluster volume. 

With its superb angular resolution and sensitivity, the Chandra X-ray Observatory has resolved a large number of low-luminosity (L_X≲ 10^34 erg s^-1) source in GCs. 
When deep HST optical/ultraviolet images are available, the majority of these weak X-ray sources are found to be cataclysmic variables (CVs) and coronally active binaries (ABs), the rest being quiescent LMXBs (qLMXBs) and millisecond pulsars (MSPs) <cit.>.  
These stellar systems either are experiencing the drastical binary evolution stage (i.e., qLMXBs, CVs, ABs) or are the immediate remnants of close binaries (i.e., MSPs), hence their formation could also have been affected by dynamical processes in GCs. 
Indeed, previous work revealed a correlation between the number of detected X-ray sources, in particular CVs, and the stellar encounter rate <cit.>, which was interpreted as a dominant fraction of these sources originating from dynamical processes <cit.>.

However, it remains an open question of which dynamical process(es) is primarily responsible for the above correlation. 
In GCs, binaries tend to sink to the dense cluster core due to equipartition of kinetic energy, where dynamical processes including two-body and three-body encounters would take place with competing effects: binaries can be created in two-body interactions, but also can be destroyed or modified in three-body interactions <cit.>. 
In particular, by binary-single interactions soft binaries (with bound energy |E_b| less than the average stellar kinetic energy E_k) tend to be softer or even disrupted, whereas hard binaries (with |E_b| > E_k) become harder[Similar processes will take place in four-body (binary-binary) interactions, provided that the GC binary fraction is sufficiently high <cit.>.]<cit.>.
While it is generally accepted that the total number of primordial binaries in GCs would decrease with time under dynamical interactions, it is far less clear whether the same processes would have a net effect of producing or destructing close binary systems such as CVs and ABs. 
The abundance of weak X-ray sources in GCs relative to the field offers a crucial diagnostics to this problem.

To date, few studies exist to quantify the relative abundance of weak X-ray sources in GCs. 
Based on ROSAT observations, <cit.> showed that most GCs have lower cumulative X-ray emissivities (i.e., L_X per unit stellar mass) than that of the old open cluster M67. 
In a Chandra study of ω Centauri (NGC 5139), <cit.> found that the abundance of CVs in this cluster is at least 2-3 times lower than that of the field. 
On the other hand, using the K-band specific rate of classical novae detected in external galaxies as a proxy, <cit.> estimated that the CV abundance of an old stellar population
is compatible with the number of CVs detected in 47 Tuc.
More recently, Ge et al. (2015) compared the cumulative X-ray emissivities of four Galactic GCs (including 47 Tuc and ω Cen) with the stellar X-ray emissivity averaged over several Local Group dwarf elliptical galaxies as well as the Solar neighborhood (Sazonov et al. 2006), but no firm conclusion could be drawn due to their limited GC sample.

In the present work, we study the largest sample of Galactic GCs observed by Chandra so far, to determine the abundance of X-ray sources and to examine its relation to various physical properties of the host cluster.
Unlike previous work (e.g., Pooley & Hut 2006) that focused on the individually resolved sources, our approach, similar to Verbunt (2001) and Ge et al. (2015), is to use the cumulative X-ray emissivity[The contribution of luminous LMXBs, if present, are excluded. See Section 2.1 for details.] as a proxy of the source abundance.
The source-counting method is inevitably subject to the strongly varied detection sensitivity among different GCs, resulting in a limited sample size. This method is also subject to contamination of foreground/background interlopers, which were often not properly accounted for. 
In contrast, our approach of measuring the total X-ray emissivity is generally insensitive to the exposure time or distance of a given GC, and thus in principle can be applied for a large GC sample in a highly uniform fashion. 
Moreover, the derived GC X-ray emissivities can be directly compared to the cumulative stellar X-ray emissivity of other galactic environments, e.g., the Galactic field and dwarf elliptical galaxies, which are crucial to evaluating the relative source abundance in GCs.

The limitation of our approach lies in that we do not distinguish the various X-ray populations (except that luminous LMXBs are precluded). However, it has been demonstrated that CVs and ABs together dominate the X-ray emission from GCs <cit.>.  
For example, <cit.> have detected ∼300 X-ray sources in NGC 104 using deep Chandra observations. They estimated that roughly 70 are background sources, 5 are qLMXB candidates, 25 are MSP candidates, and the remaining majority (∼200) are CVs and ABs. 
As we will show below, the average GC X-ray emissivity clearly indicates a paucity of X-ray sources. Any minor contribution by qLMXBs and MSPs to the measured emissivity only strengths our conclusion. 
Moreover, in practice there is no clear cut between CVs and ABs in their X-ray appearance, perhaps except that CVs are on average harder and more luminous.         
CVs and ABs are also closely related to each other from the veiwpoint of binary evolution.

The remainder of this paper is organized as follows. Section 2 describes data reduction and analysis that lead to a uniform measurement of the X-ray luminosity and stellar mass of individual GCs.  
Section 3 explores correlations between the X-ray luminosity or X-ray emissivity and various physical properties of the GCs. Discussion and summary of our results are given in Section 4 and Section 5, respectively.    
Throughout this work we quote 1 σ errors unless otherwise stated.



§ DATA PREPARATION AND ANALYSIS
    



 §.§ X-ray data and sample selection


To date, 157 Galactic GCs have been discovered and tabulated in the catalogue of <cit.>. We searched the Chandra archive for all GCs with ACIS-I or ACIS-S observations taken by May 2014.
Some GCs are known to harbor luminous LMXBs that easily dominate the total X-ray emission from the host cluster. We consulted the LMXB catalogue of <cit.> and visually inspected the Chandra images for the presence of luminous LMXBs. 
We found that in most such cases even just the PSF-scattered halo of the LMXB would severely affect our analysis, and hence we decided to remove all GCs hosting luminous LMXBs from our sample.
Our final sample thus consists of 69 GCs (Table <ref>), nearly half of the known Galactic GC population. Among them, 21 are dynamically old and have been designatated “core-collapsed" GCs in the catalogue of <cit.>. We refer to the rest (48) as dynamically normal GCs.
Notably, the size of our sample is ∼6 and ∼3 times that investigated by <cit.> and <cit.>, respectively[The preliminary analysis of source abundance in <cit.> considered 63 GCs, but a large fraction of this sample was necessarily binned by the encounter rate to ensure a significant number of detected sources in each bin.]. 

We used CIAO v4.5 and the corresponding calibration files to reprocess the data, following the standard procedure[http://cxc.harvard.edu/ciao]. Because a substantial fraction of the total X-ray flux of a given GC may come from its unresolved emission, care was taken to filter background flares using lc_clean. 
A log of the Chandra observations analyzed in this work is given in Table <ref>. 



 §.§ X-ray flux measurement
 

We took the following steps to measure the net flux for each GC. First, we defined the cluster and background regions in a uniform fashion. Due to the mass segregation effect, a large fraction of the X-ray sources are likely to concentrate within 2-3 r_c, where r_c is the core radius (Harris, 2010 edition). 
We adopted the half-light circle, of radius r_h, as the cluster region for all sample GCs.
This is a reasonable choice for the dynamically normal GCs, in which r_h is typically a few times of r_c. For core-collapsed GCs, the ratio of r_h/r_c are much larger (Table <ref>), thus contamination from foreground/background sources can be substantial. Nevertheless, any contribution from the interlopers would be statistically subtracted (see below). 
Our choice of the cluster region also facilitates direct comparison with previous work <cit.>.



Since the Chandra field-of-view (FoV) is not always large enough to cover the half-light circle (Figure <ref>), we defined for each observation a correction parameter k_1=S_h/S_ FoV, where S_h=π r_h^2 and S_ FoV≤ S_h is the actual area allowed by the FoV. 
In this regard, the total flux is calculated as F_ X=k_1× F_ X, FoV, where F_ X, FoV is the flux within S_ FoV. 

For the background region, we adopted an annulus with inner-to-outer radii of 2r_h-3r_h when the FoV was large enough, but for those GCs with a large r_h, the annulus was set as r_h-2r_h. 
We required that the background region fall within the same CCDs (AICS-I or ACIS-S3) as of the cluster region, but for NGC 6121, the background was chosen from ACIS-S2. We also avoided CCD edges and gaps where detector response may have a substantial uncertainty. We defined for each observation a second parameter k_2=S_ FoV/B_ FoV, where B_ FoV is the actual background area allowed by the FoV (Figure <ref>). In most cases, k_2 was less than 0.5 (Table <ref>).

Next, we calculated for each observation the cluster net counts as N_S-N_B'=N_S-k_2N_B, where N_S and N_B are the 0.5–8 keV counts within S_ FoV and B_ FoV, respectively.
The signal-to-noise ratio, S/N=(N_S-N_B')/√(N_S+N_B'), was derived for each observation and listed in Table <ref>.
Among the 69 GCs, 51 have S/N ≥ 3 and are considered as solid detections.

For most of our sample GCs, the total X-ray luminosity is not expected to be dominated by any single source. However, a source caught in a rare outburst may affect the total X-ray luminosity substantially. 
Therefore, for each observation we ran the CIAO wavdetect script to detect sources. 
A source located in S_ FoV was referred to as an outbursting source if its 0.5-8 keV unabsorbed luminosity, derived with a power-law spectral model, is greater than 3×10^33 erg s^-1. 
Such bright sources, found in only 3 GCs (Table <ref>), in which they each contribute more than 70% of the total net count rate, were subsequently removed for flux calculation. 

Spectra were then extracted from the cluster and background regions for each GC. The background-subtracted cluster spectra were grouped to have at least 20 counts and a minimum signal-to-noise ratio of 3 per bin. 
We performed spectral analysis with Xspec v12.8.0. The models adopted to fit the spectra were either an absorbed power-law (phabs*powerlaw), an absorbed bremsstrahlung (pabs*brem), or a combination of the two (phabs*(powerlaw+brem)). 
We calculated the absorption column density (N_ H) of each GC from their color excess E(B-V) and fixed this parameter in the fit. 
If a GC had more than one observations, a joint fit was performed to minimize the statistical uncertainty, allowing the normalization to vary but having the other parameters linked among the observations.
The 0.5–8 keV unabsorbed flux was derived from the best-fit model. 
For the 18 GCs with S/N < 3, we derived for them an upper limit in the net count rate (at 95% confidence) using the CIAO tool aprates[By assuming an non-informative prior distributions for the background and source intensity, aprates uses Bayesian statistics to compute the background-marginalized, posterior probability distribution for source intensity, which can be used to determine the intensity value and confidence bounds or intensity upper limit.], and converted it into an unabsorbed flux assuming an intrinsic power-law model with the photon-index fixed at 2.0. 
We then calculated the 0.5–8 keV luminosity or upper limit by adopting the cluster distance from <cit.>. Our spectral analysis results are summarized in Table <ref>.

Direct subtraction of the local background is expected to be a sufficient treatment for most GCs. For 13 GCs with a relatively large angular extent (r_h ≳ 2'; marked by “*" in Table <ref>), however, 
the effect of vignetting may bias low the background level, which is necessarily estimated at large off-axis angles.   
For such GCs, we corrected for vignetting following the `double-subtraction' procedure (e.g., Li et al. 2011): a first subtraction of the non-vignetted instrumental background followed by a second subtraction of the vignetted cosmic background. 
Briefly, we first generated the instrumental background spectra for both the cluster and background regions, using the Chandra “stowed" background files. 
We then characterized the local cosmic background spectrum (i.e., instrumental background-subtracted) with a phenomenological model, being either an absorbed power-law or a power-law plus APEC thermal plasma with absorption. 
The hence derived cosmic background was added as a fixed component to the total spectral model, after scaling with the sky area. 

Lastly, we examined the effect of source variability on the derived total X-ray luminosity, focusing on three GCs (Terzan 5, NGC 6626 and NGC 6397) with multiple observations. The spectra extracted from individual observations were fitted independently with an absorbed power-law model. 
The results, listed in Table <ref>, indicate that in all three cases the derived X-ray luminosity varies by less than 30 % over a timespan of years. This mild variability should have little effect on our statistical analysis and conclusions below.



 §.§ 2MASS Data
  

To obtain an accurate measurement of the X-ray emissivity, a well determined cluster stellar mass is required. 
We adopted the K-band image of the Two Micron All Sky Survey (2MASS; ), which is expected to be a better proxy of old stellar populations in GCs than the optical bands such as those quoted in <cit.>. 
We downloaded the archival 2MASS images of the individual GCs and derived their K-band luminosity, L_K, as follows. The cluster and background regions for photometry were the same as those used in the X-ray analysis (Section <ref>). 
If a single 2MASS image did not cover the defined regions, we calibrated several adjacent images and merged them into a mosaic image with astrometry correction. 
We then obtained the K-band apparent magnitude of the cluster region after subtracting the background and correcting for extinction with E(B-V). 
Finally, we calculated L_K according to the apparent magnitude and cluster distance, which is expressed in units of solar K-band luminosity L_K,⊙=5.67×10^31 erg s^-1 (for solar K-band absolute magnitude M_K,⊙=+3.31). 
The results are listed in Table <ref>.

 

To establish the use of L_K as a proxy of the cluster mass M, we first calculated M using the V-band magnitude listed in <cit.>, following the empirical relation by <cit.>, M/ M_⊙=10^-0.4(M_V-5.764 ± 0.063), which is based on multi-band photometry of 297 GCs in M 31 and stellar population synthesis models. 
Figure <ref> shows that L_K is tightly correlated with M. Most GCs follow a fitted power-law function L_K/L_K,⊙=10^-0.38± 0.23×(M/ M_⊙)^1.00 ± 0.04, which suggests a quasi-linear relation between L_K and M. 

In the above fit, we have excluded four significant outliers (Terzan1, Terzan 5, Terzan 9 and Glimpse 01). We note that all these four GCs suffer from strong foreground extinction, thus their V-band fluxes may have been underestimated. 
For example, the V-band magnitude-based masses of Terzan 5 and Glimpse 01 are 1.88 × 10^5 M_⊙ and 4.67 × 10^4 M_⊙, respectively, whereas their K-band luminosities, 9.67 × 10^5 L_K,⊙ and 2.41 × 10^5 L_K,⊙, predict 
∼12 times larger masses according to the above L_K-M relation. 
Hereafter, we adopt a cluster mass of 2×10^6 M_⊙ for Terzan 5 and 3 × 10^5 M_⊙ for Glimpse 01, which were obtained by <cit.> and <cit.>, respectively, and are consistent with our L_K-M relation. 
Because there is no reliable source of cluster mass for Terzan 1 and Terzan 9, we do not include these two core-collapsed clusters in the following statistical analysis. 



§ STATISTICAL RELATIONS
 

As discussed in Section <ref>, in the absence of luminous LMXBs, the bulk of X-ray emission from GCs arises from a collection of CVs, ABs, qLMXBs and MSPs, hence the cumulative X-ray luminosity (L_X) should be roughly scaled with the total number of such sources, and more fundamentally, scaled with the cluster mass (M). 
If dynamical interactions are prone to create X-ray sources, a correlation between L_X and the stellar encounter rate (Γ) is also expected. 
Other physical properties, such as metallicity, stellar density, age and structural parameters, of the GCs may also affect the formation and evolution of the X-ray populations. 
In this section, we examine the dependence of the cumulative X-ray emissivity, and hence the source abundance, on the various cluster properties. 
We define the cumulative X-ray emissivity as ϵ_X = 2L_X/M, where the factor of 2 accounts for the fact that L_X as given in Table <ref> has been measured within the half-light circle.
The quasi-linear L_K-M relation derived in Section <ref> also allows us to adopt the quantity L_X/L_K as a proxy of the X-ray emissivity, which has the virtue of being distance-independent.
The GC parameters are taken from Harris (2010 edition); errors, when available and relevant, are also quoted. 



 §.§ Correlations with cluster mass


In Figure <ref>, we plot L_X versus L_K for all GCs. 
A correlation between L_X and L_K is evident, for which we find the Spearman's rank correlation coefficient r=0.694 ± 0.078, with the p-value of 7.4 × 10^-11 for random correlation. 
We fit a power-law function to the correlation and obtain L_X∝ L_K^0.74 ± 0.13, which is marked by the red solid line in Figure <ref>. 
Here and below we exclude the GCs with S/N < 3 (highlighted with open symbols in the figures) from the fit to the correlations.
There is no significant difference between the dynamically normal and core-collapsed GCs. A power-law fitting function for the dynamically normal GCs gives L_X∝ L_K^0.79 ± 0.12 (purple line in Figure <ref>).
Notably, the fitted indice imply a sub-linear correlation.
      
        
      
In Figure <ref>, we plot the GC X-ray emissivity ϵ_X versus M. Here the latter has been derived from M/ M_⊙=10^0.38× (L_K/L_K,⊙) rather than the V-band magnitude-based masses (Section 2.2). 
It can be seen that ϵ_X has a substantial scatter ranging from 10^27 to a few times 10^28 erg s^-1 M^-1_⊙. 
Nevertheless, a marginally significant negative correlation between ϵ_X and M for the full sample is suggested by the Spearman's rank correlation coefficient r=-0.323 ± 0.077, with p=0.007 for random correlation. 
The best fitting power-law function is ϵ_X∝ M^-0.30 ± 0.11 (red solid line in Figure <ref>). 
However, if NGC 6397 and NGC 5139 (labelled in Figure <ref>) are not included in the fit, the ∼ 3 σ anti-correlation becomes ∼ 2 σ, with ϵ_X∝ M^-0.19 ± 0.10.
We note that this anti-correlation is consistent with the sub-linear correlations found in Figure <ref>.

 

We also measure the average X-ray emissivity of all 67 GCs to be (6.9±2.6)×10^27 erg s^-1M^-1_⊙, as marked in Figure <ref> by the dashed horizantal lines. If only GCs with S/N≥ 3 were considered, the average emissivity is (7.4±2.8)×10^27 erg s^-1M^-1_⊙.
If the three outburst sources were taken back into account, the X-ray emissivity of their host GCs would increase substantially (Table <ref>), but they have little effect in the average GC emissivity, which becomes (7.6±2.7)×10^27 erg s^-1M^-1_⊙. 
The average X-ray emissivities of the core-collapsed and dynamically normal GCs are marginally consistent with each other (9.4±3.4 versus 6.2±2.3×10^27 erg s^-1M^-1_⊙). 


For comparison, also shown as a blue strip in Figure <ref> is the cumulative X-ray emissivity of CVs and ABs detected in the Solar neighborhood, with a value of ϵ_X = (11.7 ± 3.1) × 10^27 erg s^-1M^-1_⊙, which was based on the X-ray luminosity function of CVs and ABs with L_X ranging from 10^27 to 10^34 erg s^-1<cit.>[To derive the 0.5–8 keV emissivity, we have converted Sazonov et al.'s 2–10 keV emissivity, (3.1 ± 0.8) × 10^27 erg s^-1M^-1_⊙), into the 2–8 keV band, by assuming a power-law spectrum with a photon-index of 2.1, suitable for the Galactic Ridge X-ray emission, and added the 0.5–2 keV emissivity, (9 ± 3) × 10^27 erg s^-1M^-1_⊙), as given in <cit.>.].
Also compared in Figure <ref> (magenta strip) is the average stellar X-ray emissivity of four gas-poor dwarf elliptical galaxies (M 32, NGC 147, NGC 185 and NGC 205), with a value of ϵ_X= (11.9 ± 2.7) × 10^27 erg s^-1M^-1_⊙ derived from <cit.>. 
In their work, discrete sources with X-ray luminosities greater than ∼ 10^34 erg s^-1 have been removed to ensure that the unresolved X-ray emission from these galaxies is dominated by CVs and ABs. 
We note that the Solar neighborhood and the dwarf elliptical galaxies exhibit highly similar stellar X-ray emissivities, indicating a quasi-universal emissivity in normal galactic environments, i.e., where stellar dynamical effects are not important <cit.>. 
On the other hand, as clearly shown in Figure <ref>, most GCs have a cumulative X-ray emissivity lower than the field level represented by the Solar neighborhood and the dwarf ellipticals. This strongly suggests a dearth rather than over-abundance of weak X-ray sources in GCs relative to the field [The power-law luminosity function dN∝ L_X^-γd  ln L_X of GC X-ray sources has a typical slope of γ <1 <cit.>, while the slope of X-ray sources in the Solar neighborhood is γ≈ 1.22 in the luminosity range 10^30-10^34 erg s^-1 <cit.>. Therefore, this arguement is insensitive to the number of X-ray sources at the faint-end.]. 

For completeness, we place in Figure <ref> two old open clusters, M 67 and NGC 6981. Their cumulative X-ray emissivities, derived from <cit.>, are significantly higher than that of most GCs and the field. 
Such a trend was previously noted by Verbunt (2001) based on ROSAT observations, and by Ge et al. (2015) based on only four GCs observed by Chandra.



 §.§ Correlations with stellar encounter rate


Traditionally, the stellar encounter rate, Γ∝∫ρ^2/σ, has been evaluated as Γ∝ρ_c^2 r_c^3/σ_c or Γ∝ρ_c^1.5 r_c^2 <cit.>, with the assumption that the spatial distribution of stars in GCs can be characterized by the King model <cit.>.
Here r_c is the core radius and ρ_c the average stellar density within r_c. The central velocity dispersion follows σ_c∝ρ_c^0.5 r_c for a virial system.
 
   
 
To quantify the dynamical interactions in GCs, we adopt the updated stellar encounter rates by <cit.>. These authors have reconstructed the stellar density profiles of GCs based on their surface brightness profiles, thus the dynamically normal and core-collapsed GCs can be treated equally. 
More importantly, with Monte-Carlo simulations, errors in Γ can also be properly estimated from the uncertainty in the observables such as distance, reddening and surface brightness. 
Specifically, we adopt Γ_1∝ 4πσ_c^-1∫ρ^2(r)r^2dr from Table 4 of <cit.>, which is integrated over the entire cluster and normalized to a value of 1000 for NGC 104, and is compatible to our measurement of the X-ray luminosity. 

In Figure <ref>a, we plot L_X as a function of Γ for each GC. It is evident that a strong correlation between L_X and Γ exists for the dynamically normal GCs. 
The Spearman's rank correlation coefficient is r=0.769 ± 0.093, with the p-value of p=1.7 × 10^-10 for random correlation. This correlation is still significant when taking the core-collapsed GCs into account, with r=0.627 ± 0.078 and p=1.4 × 10^-8 for the total GCs. 
The best fitting functions for the dynamically normal and total GCs can be written as L_X∝Γ^0.62 ± 0.06 and L_X∝Γ^0.67 ± 0.07, respectively (purple and red lines in Figure <ref>a). 
These relations are consistent with the finding of <cit.> that, above a limiting luminosity of 4×10^30 erg s^-1, the number of detected X-ray sources in 12 GCs is proportional to the encounter rate, with N_X∝Γ^0.74 ± 0.36. 
A similar result has been obtained by <cit.>, with N_X∝Γ^0.55 ± 0.09 again based on only 12 GCs. 

The core-collapsed GCs show no significant correlation among themselves, but notably many of them are located below the best-fitting correlation for the dynamically normal GCs. 
This is contrary to the work of <cit.>, <cit.> and <cit.>, in which most core-collapsed GCs appeared abundant in X-ray sources and were located above the best-fitting correlation in their N_X-Γ diagram. 
We suggest that this discrepancy is most likely due to their different adoption of Γ, which, based on the tranditional method, might have been underestimated for core-collapsed GCs.
This is hinted in Figure <ref>b, in which Γ from <cit.> is plotted against M. Evidently, most core-collapsed GCs have a larger Γ than the dynamically normal GCs at a given cluster mass. 
With this updated Γ, <cit.> first noted the lower abundance of X-ray sources in core-collapsed GCs than in their dynamically normal counterparts, which is consistent with our results in Figure <ref>a. 

The two significant outliers in the L_X-Γ diagram, Terzan 1 and Terzan 9, deserve some remarks. Both show an X-ray luminosity significantly higher than expected from their Γ. 
<cit.> detected 14 X-ray sources in the central 1.4 arcmin of Terzan 1, about 20 times larger than expected from the N_X-Γ relation of <cit.>. 
For Terzan 9, with the 15.2 ks Chandra observation, we detected 4 X-ray sources within the half-light radius, also more than expected from the N_X-Γ relation[The faintest X-ray source in Terzan 9 has a luminosity of ∼ 4×10^31 erg s^-1. 
For its encounter rate of 1.71^+1.67_-0.959 <cit.>, only ≲1 source with a luminosity greater than 4×10^30 erg s^-1 is predicted by the N_X-Γ relation of <cit.>]. 
As noted in Section <ref>, at face value the K-band luminosities of these two clusters are not compatible with their V-band magnitudes, which are likely due to strong foreground extinction. Therefore Γ, evaluated based on V-band data in <cit.>, might have been underestimated for both clusters, 
and it is premature to claim an over-abundance of X-ray sources in these two clusters. 



Now it becomes clear that L_X depends on both M and Γ, and so does N_X, the number of detected X-ray sources in a given GC. 
To minimize the dependence on cluster mass, <cit.> and <cit.> have studied the specific number of detected sources (n_x≡ N_X/M_6) as a function of the specific encounter rate (γ≡Γ/M_6), where M_6 is the cluster mass in units of 10^6 M_⊙. 
<cit.> found that the two parameters are correlated, with a fitting function n_x = C+A γ^α, where C, A, and α are free parameters. In this regard, the origin of X-ray sources in GCs can be attributed to two channels: descendants of primordial binaries (C) and a dynamically formed population (A γ^α). 

Similarly, we plot L_X/L_K versus γ in Figure <ref>, showing all 69 GCs in Figure <ref>a, but only those 23 GCs considered by <cit.> in Figure <ref>b to facilitate a direct comparision.
Since Γ were estimated based on the V-band data by <cit.>, here we have calculated γ (γ≡Γ/M_6) using the V-band-based mass (Figure <ref>) for consistency.
Both Figure 2d of <cit.> and our Figure <ref>b show a moderate increase of X-ray source abundance (in terms of n_x or L_X/L_K) with increasing γ[Besides the aforementioned difference in the estimation of γ for core-collapse GCs between the two diagrams, Terzan 5 in Figure 2d of <cit.> shows a very high specific number of X-ray sources, which we suggest is overestimated because the mass of Terzan 5 calculated from its V-band magnitude is likely an underestimate.]. 
However, the much larger sample in Figure <ref>a does not support a significant correlation;
the Spearman's rank correlation coefficients for the dynamically normal, core-collapse and total GCs are r=0.298 ± 0.092, r=-0.149± 0.149 and r=0.300 ± 0.078, with random correlation p-value of p=0.040, p=0.542 and p=0.013, respectively. 

We end this subsection by emphasizing that a significant correlation exists between Γ and M (Figure <ref>b). The Spearman's rank coefficients for the dynamically normal, core-collapsed and total GCs are r = 0.754 ± 0.092, 0.660 ± 0.149 and 0.611 ± 0.078, respectively, with the p-value of 5.365 × 10^-11, 0.002 and 8.543 ×10^-8 for random correlation.
We fit the Γ-M correlation with a power law function, which gives Γ∝ M^1.43 ± 0.17 for dynamically normal GCs, Γ∝ M^1.01 ± 0.26 for core-collapsed GCs, and Γ∝ M^1.28 ± 0.17 for total GCs. 
The fitting functions are also plotted in Figure <ref>b.



 §.§ Correlations with other physical parameters


Observationally, LMXBs are more likely to be found in old, metal-rich GCs, rather than in young, metal-poor ones <cit.>. 
This indicates that the formation and evolution of LMXBs in GCs are affected by stellar metallicity. 
It has been suggested that a smaller convection zone in metal-poor stars relative to metal-rich stars at a given mass may help turn off the magnetic braking during binary evolution, leading to a lower formation efficiency of LMXBs in metal-poor GCs <cit.>. 
Alternatively, a stronger irradiation-induced wind from metal-poor stars, due to less efficient line cooling and energy dissipation, will speed up the evolution of LMXBs in the host cluster and reduce their number <cit.>. 



In Figure <ref>a and <ref>e, we test the dependence of L_X and L_X/L_K on metallicity. No clear correlation exists for these quantities.
Presumably the same physical processes such as magnetic braking and irradiation also happen in CVs and ABs, but the influence of metallicity in these processes appears less important than in the case of LMXBs. 

In Figure <ref>b, we study the dependence of L_X on the cluster central luminosity density ρ_c. It is evident that the dynamically normal GCs have a larger L_X with increasing ρ_c.
The core-collapsed GCs have higher central stellar densities, but they exhibit no similar correlation between L_X and ρ_c. 
On the other hand, dependence of L_X/L_K on ρ_c is not evident in Figure <ref>f for either the dynamically normal or core-collapsed GCs. This suggests that cluster mass is the more fundamental parameter underlying the L_X-ρ_c relation. 


In Figure <ref>c and <ref>d, a positive correlation exists between L_X and the cluster central concentration c or the central velocity dispersion σ. We note that similar relations between GC V-band absolute magnitude and c or σ have been found by <cit.>. 
These two correlations may again reflect the more fundamental dependency on cluster mass. Indeed, Figure <ref>g and <ref>h show no significant correlation between L_X/L_K and c or between L_X/L_K and σ.

In Figure <ref>a-d, we plot L_X/L_K as a function of GC core relaxation time t_c, median relaxation time t_h, relative age t_r and absolute age t_a. 
In general, GCs with smaller t_c or t_h are dynamically older. It appears that the GC X-ray emissivity increases with the dynamical age, as suggested by a marginally significant negative dependence of L_X/L_K on t_c and t_h (Figure <ref>a and <ref>b, respectively).
The marginal dependence of L_X/L_K on the relative age t_r or the absolute age t_a (Figure <ref>c and <ref>d) is also consistent with such a mild trend. 

In Figure <ref>e-g, we plot L_X/L_K as a function of the GC distance to the Galactic center R_gc, ellipticity of optical isophotes e, tidal radius r_t and main sequence binary fraction f_b. 
In principle, with increasing distances away from the Galactic center, GCs will suffer from weaker tidal force and grow their tidal radius, affecting its dynamical and geometrical structure and potentially the abundance of binaries. 
This is suggested by the mild anti-correlations in Figure <ref>e and <ref>g. 
However, we find no statistically significant correlation between L_X/L_K and ellipticity of optical isophotes e in Figure <ref>f. Although the main sequence binary fraction (f_b) may be closely related to the abundance of the weak X-ray sources, Figure <ref>h shows no significant correlation between L_X/L_K and f_b.



To conclude this section, we summarize in Table <ref> the Spearman's rank coefficients for all the tested correlations. 



§ DISCUSSION


It has been known for over 40 years that the abundance of luminous LMXBs in GCs exceeds that of the field by orders of magnitude, which is generally accepted as the consequence of efficient dynamical formation of neutron star binaries in the dense environment of GCs. 
Hut & Verbunt (1983) were among the first to predict that dynamical processes would lead to the formation of as many white dwarf binaries as neutron star binaries in GCs. 
Pooley and colleagues, upon finding a correlation between the number of weak X-ray sources and the stellar encounter rate for a moderate sample of GCs, argued that the X-ray populations, in particular CVs, are over-abundant in GCs. 
Our analysis in Section 3, however, points to an opposite trend: most GCs exhibit a lower cumulative X-ray emissivity than found in the field, which is represented by the Solar neighborhood and the Local Group dwarf elliptical galaxies. 
Because in all these environments the cumulative X-ray emissivity is a reasonable proxy of the source abundance, the immediate conclusion is that the weak X-ray populations, primarily CVs and ABs, are under-abundant in GCs as compared to the field.
In the following, we address the implications of this result.



 §.§ Dependence on stellar encounter rate

The under-abundance of weak X-ray sources in GCs demands for a revisit of the N_X-Γ <cit.> and L_X-Γ (Figure <ref>a) correlations as evidence for a predominantly dynamical origin of the X-ray populations. 
As shown in Figure <ref>b, the stellar encounter rate Γ is strongly correlated with the cluster mass M, thus it is not unreasonable to raise the question of whether cluster mass is the more fundamental parameter underlying the N_X-Γ and L_X-Γ relations.   
Indeed, in Section 3.1 we have demonstrated a correlation between L_X and L_K (hence M), which is statistically as significant as the L_X-Γ relation, according to the Spearman's rank cofficients. 
Moreover, our larger GC sample disfavors a strong positive correlation between the source abundance (traced by L_X/L_K) and the specific encounter rate, γ (Figure <ref>a), as previously suggested by <cit.>. 

To further compare the influence of the cluster mass and stellar encounters on the X-ray populations, we test a correlation between L_X, M and Γ using the following form: log L_X = a_K  log L_K + a_Γ  log Γ + c. 
Giving equal weights to L_K and Γ in the regression, we find a_K=0.64 ±0.12, a_Γ=0.19±0.07 and c=29.4±0.52 (Figure <ref>). 
It appears that the cluster mass, rather than the stellar encounter rate, is the predominant factor in determining the amount of weak X-ray sources. 
This implies that a substantial fraction of the X-ray populations, in particular CVs and ABs, are descendants of primordial binaries, although dynamical processes must have played at least a partial role in the formation of the X-ray populations.
In any case, taking the N_X-Γ and L_X-Γ relations as solid evidence for a dynamical origin of the X-ray populations may be an over-simplification. Below, we argue that not all dynamical interactions in GCs will lead to the formation of X-ray sources. 





 §.§ Efficiency of the primordial channel of close binary formation


In general, when considering the formation of close binary systems in a dense stellar environment such as GCs, three competing and coupled effects are relevant.
The first is normal stellar evolution that eventually turns wide, primordial binaries into close binaries, which is referred to as the primordial channel. 
The second is the dynamical formation of close binaries due to two-body or three-body interactions.
The third and a negative effect is the dynamical disruption of primordial binaries mainly due to three-body (binary-single) or four-body (binary-binary) interactions <cit.>.
For LMXBs, because of the rarity of neutron stars and the relatively short lifetime of their progenitor stars, the first and third effects are generally negligible.
However, for CVs and ABs, because their progenitors are mainly low-mass stars, they should evolve on a timescale comparable to or even greater than the GC relaxation time, making all the above three effects relevant.
Therefore, we argue that the under-abundance of CVs and ABs in GCs reflects a low efficiency of the primordial channel, i.e., a substantial fraction of the primordial binaries are dynamically disrupted before they can otherwise evolve into close binaries. 
Such a scenario is supported by both theoretical and observational studies.

Theoretically, the fate of binaries in GCs is governed by both normal stellar evolution and stellar dynamical interactions.
In the former case, a number of stellar processes, such as Roche lobe msss tranfer, steller wind and supernova kick, can substantially modify the binary oribits and even lead to orbit disruption or binary merger.
Dynamical interactions can alter binaries in a more abrupt manner. 
For example, most “soft" binaries in GCs will be destroyed if they suffer from a strong encounter with other stars. 
Even for the “hard" binaries, their interactions with other stars can exchange one of the primordial members with the intruding star, in the meantime causing the orbit to expand or shrink, modifying the orbit eccentricity, enhancing the systemic velocity via gravitational recoil, or leading to physical collision. 
Numerical simulations have confirmed that binaries can be dynamically disrupted in GCs <cit.>. 
The probability for a primordial binary to survive from dynamical destruction depends on the binary hardness and the collision timescale in GCs. 
Comparing with simulations that only consider normal stellar evolution, simulations taking into account binary destruction result in a much lower binary fraction in GCs <cit.>. 

Observationally, the main sequence binary fraction is found to be on average much lower in GCs than in the Solar neighborhood <cit.> or open clusters <cit.>. 
Recent numerical simulations suggest that these present-day binary fractions are consistent with a universal, near-unity initial binary fraction in GCs <cit.>. 
If this were the case, dynamical disruption of primordial binaries must have been efficient. This is also suggested by the observed trend that the main sequence binary fraction of GCs decreases with increasing cluster age <cit.>.
Furthemore, as shown in Figure <ref>, a negative correlation is evident between the main sequence binary fraction and the specific stellar encounter rate, strongly suggesting dynamical disruption of primordial binaries and consequently the dearth of X-ray-emitting close binaries.

 



 §.§ Dynamical formation versus dynamical disruption

 
Nevertheless, dynamical formation of CVs and ABs must be taking place in GCs. 
In reality, it is most likely that dynamical formation and destruction of binaries occur simultaneously and persistently in GCs. 
Fortunately, the two competing effects may be unified in one process, namely, the binary-single interaction. 
In general, such interactions obey the Hills-Heggie law, which states that hard binaries (with |E_b| > E_k) evolve into smaller orbits, while soft binaries (with |E_b| < E_k) tend to be softer and even disrupted <cit.>.  

More quantitatively, we may express the abundance of X-ray binaries as, 

    L_X/L_K ≈ N_X L̅_X / (N_*L̅_K)
    = (N_b/N_*)(N_X/N_b)L̅_X/L̅_K,

where L̅_X (L̅_K)
is the characteristic X-ray (K-band) luminosity of a binary (star), N_b/N_*≈ f_b is the GC binary fraction, and N_X/N_b is the fraction of binaries being an X-ray-emitting close binary.  
Recall that the average GC X-ray emissivity is ∼40% lower than that of the cumulative X-ray emissivity of the field populations (6.9×10^27 erg s^-1M^-1_⊙ vs. 11.8×10^27 erg s^-1M^-1_⊙; Section 3.1).
On the other hand, the main sequence binary fraction in the Solar neighborhood is ∼40% <cit.>, while most measurements of GC binary fraction range between 1-20% (Figure 7h). 

If we assume that GCs have an initial binary fraction at least comparable to that of the Solar neighborhood – in fact, a near-unity initial binary fraction has been suggested for GCs (Leigh et al. 2015) – and that the Solar neighborhood binaries had evolved little due to the lack of dynamical interactions,
from the above we may infer that the reduction in binary fraction (by a factor of 2-40 in N_b/N_*) due to dynamical disruption of “soft” primordial binaries has been partially compensated by the dynamical formation of X-ray-emitting “hard” binaries (i.e., an enhanced N_X/N_b as compared to the field). 
The statistical outcome of this competition appears to be a mild anti-correlation between the abundance of the X-ray sources and the cluster mass, ϵ_X∝ M^-0.31 ± 0.09 (Section 3.1), 
which may be qualitatively understood in the sense that dynamical disruption of primordial binaries is more efficient in more massive GCs, because of their higher encounter rate (Γ∝ M^1.28 in Figure <ref>b) and higher stellar velocity dispersion (E_k∝σ^2). 
On the other hand, the competition may allow the abundance of weak X-ray sources in some dynamically old GCs to catch up with, or even exceeds, the field level. For example, NGC 6397 has the highest X-ray emissivity among all GCs (Figures <ref> and 5), and the binary fraction in this cluster is also very low (f_b=2.4±0.6%, ), which suggests that the compensation of weak X-ray sources by dynamical formation channel have surpassed the dynamical disruption channel, and many “hard" main sequence binaries have been dynamically transformed into weak X-ray sources in this cluster.
We leave a more quantitative study of the effect of binary-single interaction on the X-ray source abundance to a future work.  
 


§ SUMMARY


In this work, we have surveyed the cumulative X-ray emission of the largest-so-far sample (69) of Milky Way globular clusters observed by Chandra. Our main findings are as follows.

1. The X-ray emissivity of most GCs is lower than that of the Solar neighborhood and dwarf elliptical galaxies, indicating that a paucity of weak X-ray sources, mainly CVs and ABs, in GCs with respect to the field populations. 
This under-abundance of X-ray-emitting close binaries suggests that formation of such systems through the primordial channel is suppressed in GCs, which is likely due to dynamical disruption of primordial binaries before they can evolve into CVs and ABs. 

2. The GC X-ray luminosity, stellar encounter rate and cluster mass are highly correlated among each other, with L_X∝Γ^0.67 ± 0.07, L_X∝ M^0.74 ± 0.13 and Γ∝ M^1.28 ± 0.17. 
Furthermore, the GC X-ray luminosity can be expressed as a function of cluster mass and stellar encounter rate, with Log L_X=(29.40±0.47) + (0.64±0.12)  Log L_K + (0.19±0.07)  Log Γ, 
which suggests that dynamical formation of CVs and ABs in GCs is less dominant than previously thought, while the primordial channel may still have a substantial contribution. 

3. Binary-single interactions, which would lead to both the disruption of soft primordial binaries and the formation of close binaries, seem to provide a natural explanation of the above results. The net outcome is the present-day abundance of weak X-ray sources in GCs, which appears no higher than that of the field.

4. Correlations between the GC X-ray emissivity and global cluster properties such as metallicity, dynamical age and structural parameters have been examined, but no statistically significant trends can be revealed.


We thank the anonymous referee for valuable comments that help improve our manuscript. This work is supported by the National Natural Science Foundation of China under grants 11133001, 11333004 and 11303015, and the Strategic Priority Research Program of CAS under grant XDB09000000. Z.L. acknowledges support from the Recruitment Program of Global Youth Experts.


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lrrrrrrrrrrrrrr

15
0.75
0pc

BASIC PROPERTIES OF GLOBULAR CLUSTERS

Name     M     Γ     -δ     +δ     γ     -δ     +δ     D     E(B-V)     r_h     r_h/r_c     Fe/H     ρ_c     t

(1)     (2)     (3)     (4)     (5)     (6)     (7)     (8)     (9)     (10)     (11)     (12)     (13)     (14)     (15)
Normal GCs:
NGC 104       118.5     1000      134      154      844      113      130       4.5      0.04     3.17      8.8      -0.72     4.88     13.06 

NGC 288       10.13     0.766     0.205    0.284    7.56     2.02     2.80      8.9      0.03     2.23      1.7      -1.32     1.78     10.62 

NGC 2808      115.2     923       82.7     67.2     801      71.8     58.3      9.6      0.22     0.8       3.2      -1.14     4.66     10.8  

NGC 3201      19.3      7.17      2.27     3.56     37.1     11.8     18.4      4.9      0.24     3.1       2.4      -1.59     2.71     10.24 

NGC 5024      61.6      35.4      9.6      12.4     57.5     15.6     20.1      17.9     0.02     1.31      3.7      -2.1      3.07     12.67 

NGC 5139      256.8     90.4      20.4     26.3     35.2     7.94     10.2      5.2      0.12     5         2.1      -1.53     3.15     11.52 

NGC 5272      72.04     194       18       33.1     269      25       45.9      10.2     0.01     2.31      6.2      -1.5      3.57     11.39 
 
NGC 5286      63.33     458       60.7     58.4     723      95.8     92.2      11.7     0.24     0.73      2.6      -1.69     4.1      12.54 

NGC 5824      70.08     984       155      171      1400     221      244       32.1     0.13     0.45      7.5      -1.91     4.61     12.8  

NGC 5904      67.55     164       30.4     38.6     243      45       57.1      7.5      0.03     1.77      4.0      -1.29     3.88     10.62 

NGC 5927      26.89     68.2      10.3     12.7     254      38.3     47.2      7.7      0.45     1.1       2.6      -0.49     4.09     12.67 

NGC 6093      39.59     532       68.8     59.1     1340     174      149       10       0.18     0.61      4.1      -1.75     4.79     12.54 
 
NGC 6121      15.19     26.9      9.56     11.6     177      62.9     76.4      2.2      0.35     4.33      3.7      -1.16     3.64     12.54 

NGC 6139      44.63     307       82.1     95.4     688      184      214       10.1     0.75     0.85      5.7      -1.65     4.67     –    

NGC 6144      11.11     3.14      0.85     1.07     28.3     7.65     9.63      8.9      0.36     1.63      1.7      -1.76     2.31     13.82 

NGC 6205      53.16     68.9      14.6     18.1     130      27.5     34        7.1      0.02     1.69      2.7      -1.53     3.55     11.65 

NGC 6218      16.97     13.0      4.03     5.44     76.6     23.8     32.1      4.8      0.19     1.77      2.2      -1.37     3.23     12.67 

NGC 6287      17.77     36.3      7.74     7.70     204      43.6     43.3      9.4      0.6      0.74      2.6      -2.1      3.78     13.57 
  
NGC 6304      16.81     123       22       53.8     732      131      320       5.9      0.54     1.42      6.8      -0.45     4.49     13.57 

NGC 6333      30.59     131       41.8     59.1     428      137      193       7.9      0.38     0.96      2.1      -1.77     3.78     –    

NGC 6341      38.87     270       29       30.1     695      74.6     77.4      8.3      0.02     1.02      3.9      -2.31     4.3      13.18 

NGC 6352      7.83      6.74      1.3      1.71     86.1     16.6     21.8      5.6      0.22     2.05      2.5      -0.64     3.17     12.67 

NGC 6362      12.18     4.56      1.03     1.51     37.4     8.46     12.4      7.6      0.09     2.05      1.8      -0.99     2.29     13.57 

NGC 6366      3.996     5.14      1.76     2.75     129      44       68.8      3.5      0.71     2.92      1.3      -0.59     2.39     13.31 

NGC 6388      117.4     899       213      238      766      181      203       9.9      0.37     0.52      4.3      -0.55     5.37     12.03 

NGC 6401      29.21     44        10.7     11       151      36.6     37.7      10.6     0.72     1.91      7.6      -1.02     3.79     –    

NGC 6402      88.23     124       30.2     31.8     141      34.2     36        9.3      0.6      1.3       1.6      -1.28     3.36     –    

NGC 6440      63.91     1400      477      628      2193     746      983       8.5      1.07     0.48      3.4      -0.36     5.24     –    

NGC 6517      40.33     338       97.5     152      838      242      377       10.6     1.08     0.5       8.3      -1.23     5.29     –    

NGC 6528      8.58      278       49.5     114      3240     577      1330      7.9      0.54     0.38      2.9      -0.11     4.77     –    

NGC 6535      1.61      0.388     0.192    0.389    24.2     12       24.2      6.8      0.34     0.85      2.4      -1.79     2.34     10.5  

NGC 6539      41.84     42.1      15.3     28.6     101      36.6     68.4      7.8      1.02     1.7       4.5      -0.63     4.15     –    

NGC 6553      25.92     69        18.8     26.8     266      72.5     103       6        0.63     1.03      1.9      -0.18     3.84     –    

NGC 6569      41.46     53.6      20.8     30.2     129      50.2     72.8      10.9     0.53     0.80      2.3      -0.76     3.63     –    

NGC 6626      37.12     648       91.1     83.8     1750     245      226       5.5      0.4      1.97      8.2      -1.32     4.86     –    

NGC 6637      22.99     89.9      18.1     36       391      78.7     157       8.8      0.18     0.84      2.5      -0.64     3.84     13.06 

NGC 6638      14.24     137       27.1     38.6     962      190      271       9.4      0.41     0.51      2.3      -0.95     4.09     –    

NGC 6656      50.77     77.5      25.9     31.9     153      51       62.8      3.2      0.34     3.36      2.5      -1.7      3.63     12.67 

NGC 6715      198.4     2520      274      226      1273     138      114       26.5     0.15     0.82      9.1      -1.49     4.69     10.75 

NGC 6717      3.71      39.8      13.7     21.8     1070     369      587       7.1      0.22     0.68      8.5      -1.26     4.58     13.18 

NGC 6760      27.64     56.9      19.4     26.6     206      70.2     96.2      7.4      0.77     1.27      3.7      -0.4      3.89     –    
 
NGC 6809      21.56     3.23      1        1.38     15       4.64     6.4       5.4      0.08     2.83      1.6      -1.94     2.22     12.29 

NGC 6838      3.55      1.47      0.138    0.146    4.15     3.89     4.12      4        0.25     1.67      2.7      -0.78     2.83     13.7  

NGC 7089      82.72     518       71.4     77.6     626      86.3     93.8      11.5     0.06     1.06      3.3      -1.65     4        11.78 

Glim 01       30.00     400       200      200      8560     4280     12800     4.2      4.85     0.65      5.6      –        5.0      –    

Pal 10        4.18      59        35.5     42.8     1410     848      1020      5.9      1.66     0.99      1.2      -0.1      3.51     –    

Ter 3         1.71      1.18      0.37     0.65     68.9     21.4     38.1      8.2      0.73     1.25       1.1     -0.74     1.82     –    
 
Ter 5         200.0     6800      3020     1040     36200    16100    5540      5.9      2.38     0.72      3.4      -0.23     5.14     –    

Core Collapse GCs:
NGC 362       47.6      735       117      137      1540     246      288      8.6      0.05     0.82      4.6      -1.26     4.74     10.37 

NGC 1904      28.16     116       44.7     67.6     412      159      240      12.9     0.01     0.65      4.1      -1.6      4.08     11.14 

NGC 5946      15.05     134       44.6     33.6     890      296      223      10.6     0.54     0.89      11.1     -1.29     4.68     11.39 

NGC 6256      14.64     169       60.4     119      1150     413      813      10.3     1.09     0.86      43       -1.02     5.59     –    

NGC 6266      94.97     1670      569      709      1760     599      747      6.8      0.47     0.92      4.2      -1.18     5.16     11.78 

NGC 6293      26.16     847       239      337      3240     914      1440     9.5      0.36     0.89     17.8      -1.99     5.31     –    

NGC 6325      12.29     118       45.6     44.7     960      371      364      7.8      0.91     0.63      21       -1.25     5.52     –    

NGC 6342      7.48      44.8      12.5     14.4     599      167      193      8.5      0.46     0.73      14.6     -0.55     4.97     12.03 

NGC 6355      34.17     99.2      25.7     41.1     290      75.2     120      9.2      0.77     0.88      17.6     -1.37     5.04     –    

NGC 6397      9.15      84.1      18.3     18.3     919      200      200      2.3      0.18     2.9       58       -2.02     5.76     12.67 

NGC 6453      15.62     371       88.7     128      2380     568      820      11.6     0.64     0.44      8.8      -1.5      4.95     –    

NGC 6522      23.21     363       98.5     113      1560     424      478      7.7      0.48     1         20       -1.34     5.48     –    

NGC 6541      51.71     386       63.1     95.2     746      122      184      7.5      0.14     1.06      5.9      -1.81     4.65     12.93 

NGC 6544      12.07     111       36.5     67.8     920      302      562      3        0.76     1.21      24.2     -1.4      6.06     10.37 

NGC 6558      7.61      105       19.3     26.2     1380     253      344      7.4      0.44     2.15      71.7     -1.32     5.35     –    

NGC 6642      9.32      97.8      24.5     31.3     1050     263      336      8.1      0.4      0.73      7.3      -1.26     4.64     –    

NGC 6681      14.24     1040      192      267      7300     1350     1870     9        0.07     0.71      23.7     -1.62     5.82     12.8  

NGC 6752      24.98     401       126      182      1610     504      729      4        0.04     1.91      11.24    -1.54     5.04     11.78 
 
NGC 7099      19.3      324       81.2     124      1680     421      642      8.1      0.03     1.03      17.2     -2.27     5.01     12.93 

Ter 1         1.17      0.292     0.17     0.274    24.9     14.5     23.3     6.7      1.99     1.4       95.5     -1.03     3.85     –    
 
Ter 9         0.616     1.71      0.959    1.67     278      256      271      7.1      1.76     0.78      26       -1.05     4.42     –    



Column 1-2: Name, total globular cluster mass (in units of 10^4M_⊙) derived from V-band magnitude. Column 3-5: Stellar encounter rate from <cit.>, except for Glimpse 01, which is adopted from <cit.>. ±δ is the 1σ upper and lower limits of Γ. Column 6-8: specific encounter rate, which is calculated with γ=Γ/M_6, where M_6 is the GC mass in units of 10^6M_⊙. Column 9-10: GC distance in units of kpc, foreground color excess. Column 11-12: half-light radius in arcmins, and ratio of half-light radius to core radius. Column 13-15: metallicity, central luminosity density in units of log_10(L_⊙ pc^-3) and absolute age of GCs in units of Gyr <cit.>. 




llllll


6

OBSERVATION LOG

GC Name     Chandra Obs.ID     Effect. exposure     Background     Cluster region     Source to Background 

          t_exp(ks)     Region     factor k_1     region factor k_2

(1)     (2)     (3)     (4)     (5)     (6)
Normal GCs:
NGC 104      953, 955              30.7, 30.7           2, 2        1.21, 1.16           0.249, 0.252        

NGC 288      3777                  45.6                 1           1.04                 0.471               

NGC 2808     7453, 8560            45.4, 10.8           2, 2        1.71, 1.71           0.148, 0.147        

NGC 3201     11031                 82.9                 1           1                    0.894               

NGC 5024     6560                  24.2                 2           1                    0.316               

NGC 5139     13726, 13727          171, 47.6            1, 1        1.11, 1.10           0.440, 0.415        

NGC 5272     4542, 4543, 4544      9.9, 9.8, 9.4        1, 1, 1     1.05, 1.05, 1.05     0.942, 0.890, 0.914 

NGC 5286     8964, 9852            10, 3.3              2, 2        1, 1                 0.232, 0.235        

NGC 5824     9026                  10.7                 2           1                    0.200               

NGC 5904     2676                  41.8                 2           1                    0.412               

NGC 5927     13673, 8953           46.5, 0.77           2, 2        1, 1                 0.276, 0.290        

NGC 6093     1007                  36.1                 2           1                    0.200               

NGC 6121     946, 7446, 7447       25.2, 47.9, 44.8     1, 1, 1     1.21, 1.22, 1.21     0.811, 0.841, 0.825 

NGC 6139     8965                  17.7                 2           1                    0.260               

NGC 6144     7458                  53.4                 2           1                    0.330               

NGC 6205     5436, 7290            26.5, 26.6           2, 2        1, 1                 0.464               

NGC 6218     4530                  22.8                 2           1                    0.447               

NGC 6287     13734                 39.7                 2           1                    0.268               
  
NGC 6304     11073, 8952           96.2, 4.9            2, 2        1.52, 1              0.143, 0.321        

NGC 6333     8954                  7.8                  2           1                    0.288               

NGC 6341     5241, 3778            19.7, 29.3           2, 2        1, 1                 0.246, 0.241        

NGC 6352     13674                 19.5                 2           1                    0.595               

NGC 6362     11024, 12038          28.8, 8.7            1, 1        1, 1                 0.411, 0.404        

NGC 6366     2678                  20.1                 1           1                    0.784               

NGC 6388     5505                  44.6                 2           1                    0.200               

NGC 6401     8948                  11.1                 2           1.04                 0.578               

NGC 6402     8947                  11.8                 2           1                    0.299               

NGC 6440     947, 3799, 11802      22.6, 23.4, 4.91     2, 2, 2     1, 1, 1              0.200, 0.200, 0.200 

NGC 6517     9597                  23.3                 2           1                    0.200               

NGC 6528     8961, 12400           12.2, 52.7           2, 2        1, 1                 0.200, 0.200        

NGC 6535     11025                 52.4                 2           1                    0.200               

NGC 6539     8949                  14.4                 2           1                    0.440               

NGC 6553     13671, 8957           30.9, 4.9            2, 2        1, 1                 0.255, 0.285        

NGC 6569     8974                  11.1                 2           1                    0.245               

NGC 6626     2683, 2684, 2685      8.9, 12.4, 12.5      2, 2, 2     1, 1, 1              0.632, 0.602, 0.606 

             9132, 9133            141, 52.5            2, 2        1, 1                 0.568, 0.561        

NGC 6637     8946                  7.4                  2           1                    0.253,              

NGC 6638     8950                  8.3                  2           1                    0.200               

NGC 6656     5437, 14609           15.8, 84.6           1, 1        1.02, 1.26           1.219, 0.799        
           
NGC 6715     4448                  29.1                 2           1.59                 0.150               

NGC 6717     13733                 16.9                 2           1                    0.240               

NGC 6760     13672                 50.7                 2           1                    0.277               

NGC 6809     4531                  33.7                 1           1.04                 0.622               

NGC 6838     5434                  50.8                 2           1                    0.453               

NGC 7089     8960                  11.2                 2           1                    0.326               

Glim 01      6587                  44.6                 2           1                    0.200               

Pal 10       8945                  10.8                 2           1                    0.299               

Ter 3        11026                 64.3                 2           1                    0.231               

Ter 5        13706,10059,13705     45.5, 35.9, 13.2     2, 2, 2     1, 1, 1              0.225, 0.244, 0.243 

             14339,14625,15615     34.1, 48.9, 82.9     2, 2, 2     1, 1, 1              0.235, 0.262, 0.266 

             3798,13225,13252      24.9, 29.7, 39.2     2, 2 ,2     1, 1, 1              0.200, 0.234, 0.241 

             14475,14476,14477     29.3, 28.0, 28.3     2, 2, 2     1, 1, 1              0.426, 0.428, 0.428 

             14478                 28.3                 2           1                    0.449               

Core Collapse GCs:
NGC 362      4529, 5229            51.1, 8              2, 2        1, 1                 0.218, 0.218        

NGC 1904     9027                  10                   2           1                    0.200               

NGC 5946     9956                  24.3                 1           1                    0.349               

NGC 6256     8951                  9.4                  2           1                    0.268               

NGC 6266     2677                  60.7                 2           1                    0.200               

NGC 6293     8962                  9.6                  2           1                    0.271               

NGC 6325     8959                  16.7                 2           1                    0.237               

NGC 6342     9957                  15.8                 2           1                    0.236               

NGC 6355     9958                  22.2                 2           1                    0.274               

NGC 6397     79, 2668, 2669        48.3, 28.1, 26.7     1, 1, 1     1.19, 1.04, 1.03     0.213, 0.708, 0.708 
       
             7460, 7461            146, 88.9            1, 1        1.11, 1.10           0.596, 0.671        

NGC 6453     9959                  20.4                 2           1                    0.200               

NGC 6522     8963                  8.33                 2           1                    0.272               

NGC 6541     3779                  43.2                 2           1                    0.274               

NGC 6544     5435                  8.96                 2           1                    0.266               

NGC 6558     9961                  11                   1           1.08                 0.516               

NGC 6642     8955                  6.4                  2           1                    0.236               

NGC 6681     9955, 8958            67.8, 6.9            2, 2        1, 1                 0.233, 0.237        

NGC 6752     948, 6612             27.5, 37             2, 2        1, 1                 0.501, 0.585        

NGC 7099     2679                  43                   2           1                    0.217               

Ter 1        5464                  18.5                 1           1.24                 0.317               
                                       
Ter 9        9960                  15.2                 2           1                    0.239               

Column 1-3: Name, Chandra observation ID and effective exposure time. Column 4: background regions, 1 and 2 denote the use of annulus with radii r_h-2r_h and 2r_h-3r_h, respectively. Column 5-6: value of S_h/S_fov and S_fov/B_fov.




llllllll

8
0.65
0pc

GCs with X-ray sources in outburst

Name     ID     N_s-N_b'      S1      L_X,1     L_X     L_X,tot     2L_X,tot/M 

(1)     (2)     (3)     (4)     (5)     (6)     (7)      (8) 
NGC 5272     4542      419      297      3.02^+0.23_-0.26     0.70^+0.38_-0.26     3.63^+0.41_-0.30     0.92^+0.38_-0.33 

             4543      750      628      5.84^+0.35_-0.62     1.05^+0.29_-0.29     8.18^+0.93_-0.80     2.07^+0.86_-0.89 

NGC 6717     13733     865      643      5.49^+0.23_-0.56     0.97^+0.14_-0.15     5.82^+0.35_-0.41     19.8^+9.08_-9.28 

NGC 6453     9959      286      255      3.17^+0.32_-0.28     0.46^+0.13_-0.10     3.41^+0.29_-0.37     4.37^+0.37_-0.47 



Column 1-2: Cluster name, Chandra observation ID. Column 3-4: Cluster net counts and counts of the outburst source. Column 5-8: Luminosities (0.5-8 keV) of the outburst source, GC cumulative luminosity when the outburst source has been rejected and included separatively, with a units of 10^33 erg s^-1. Column 9: The total X-ray emissivity of GCs in units of 10^28 erg s^-1M_⊙^-1.



lllllllllll


11
0pc

DERIVED X-RAY PROPERTIES

Name     Net counts     S/N     Model     nH     PI     KT     χ^2_ν     L_X,0.5-8     L_K     L_X/L_K 

(1)     (2)     (3)     (4)     (5)     (6)     (7)     (8)     (9)     (10)     (11)
Normal GCs:
NGC 104^*     17246     102.5    PL+Brem    0.023     0.42^+0.04_-0.05    0.65^+0.05_-0.08    376.3/352     58.9^+7.65_-8.56     74.6      7.90^+1.03_-1.15
 
NGC 2808        700       19.2      PL        0.128     1.92^+0.16_-0.16                              21.3/38       28.9^+2.53_-1.86     46.91     6.16^+0.54_-0.40

NGC 288^*     230       3.2       PL        0.017     2.15^+1.72_-1.51                              42.8/22       1.28^+1.19_-1.16     3.42      3.74^+3.48_-3.40
 
NGC 3201        93        0.48      PL        0.139     2(frozen)                                                       < 1.49                   7.21      < 2.07 

NGC 5024        184       4.2       PL        0.012     2(frozen)                                         3.8/3         12.1^+2.56_-2.59     23.1      5.24^+1.11_-1.12

NGC 5139^*    12012     32.4     PL+Brem    0.069     1.52^+0.16_-0.14    0.21^+0.07_-0.08    171.3/167     14.1^+1.56_-1.09     112.2     1.26^+0.14_-0.10

NGC 5272^*    365       5.1       PL        0.006     2.34^+0.81_-0.64                              58.9/46       9.14^+3.09_-2.63     32.98     2.77^+0.94_-0.80
 
NGC 5286        134       6.1       PL        0.139     2.1^+0.52_-0.49                               2.8/8         13.5^+3.55_-2.6      26.1      5.17^+1.36_-0.99

NGC 5824        14        1.2       PL        0.075     2.0(frozen)                                                     < 27.8                   35.98     < 7.73 

NGC 5904        513       6.9       PL        0.017     1.8^+0.42_-0.41                               9.5/10        5.06^+1.37_-0.83     22.75     2.22^+0.60_-0.37

NGC 5927        330       6.0       PL        0.261     2.0^+0.58_-0.47                               9.4/10        4.86^+1.23_-0.83     17.25     2.82^+0.71_-0.48

NGC 6093        960       23.3      PL        0.104     1.78^+0.12_-0.11                              44.4/52       23.7^+1.39_-1.81     16.05     14.8^+0.87_-1.13
 
NGC 6121^*    3791      12.2     PL+Brem    0.203     0.60^+0.43_-0.73    0.73^+0.72_-0.30    385.3/279     5.26^+0.03_-0.70     6.14      8.56^+0.05_-1.14

NGC 6139        288       9.1       PL        0.435     2.30^+0.37_-0.34                              9.3/13        24.9^+3.88_-3.48     18.87     13.2^+2.06_-1.84

NGC 6144        428       4.9       PL        0.209     3.2                                               11.7/6        4.95^+1.35_-0.81     4.67      10.6^+2.89_-1.73

NGC 6205        1373      15.0      PL        0.012     2.14^+0.17_-0.17                              50.4/51       9.15^+0.62_-0.68     19.4      4.72^+0.32_-0.35

NGC 6218        380       6.3       PL        0.11      1.55^+0.51_-0.49                              5.3/7         4.01^+0.97_-0.81     5.1       7.87^+1.90_-1.59

NGC 6287        232       6.9       PL        0.348     1.15^+0.41_-0.41                              5.4/9         9.75^+1.61_-1.93     6.68      14.6^+2.41_-2.89
  
NGC 6304        1561      18.4      PL        0.313     3.2^+0.22_-0.21                               57.7/56       17.2^+0.9_-0.6       11.55     14.9^+0.78_-0.52

NGC 6333        23        1.1       PL        0.22      2.0(frozen)                                                     < 5.14                   10.58     < 4.86 

NGC 6341        643       12.4      PL        0.012     2.16^+0.22_-0.22                              48.4/30       5.42^+0.51_-0.46     13.23     4.10^+0.38_-0.34

NGC 6352^*    337       5.4       PL        0.128     3.17^+0.65_-0.69                              35.1/25       4.14^+0.84_-0.87     6.07      6.82^+1.38_-1.43

NGC 6362        106       1.1       PL        0.052     2.0(frozen)                                                     < 3.88                   6.14      < 6.32 

NGC 6366^*    445       4.9       PL        0.412     1.93^+0.64_-0.63                              68.6/57       3.17^+1.04_-0.98     1.06      30.0^+9.84_-9.28

NGC 6388        3682      54.9      PL        0.215     2.05^+0.05_-0.05                              176.2/136     88.8^+1.67_-2.80     58.38     15.2^+0.29_-0.48

NGC 6401        74        1.5       PL        0.418     2.0(frozen)                                                     < 23.2                   17.58     < 13.2 

NGC 6402        125       3.7       PL        0.348     0.99^+0.9_-0.93                               1.44/3        17.7^+6.38_-6.11     33.16     5.34^+1.92_-1.84

NGC 6440        2672      46.9      PL        0.621     2.78^+0.08_-0.08                              133/122       72.5^+1.73_-2.11     36.51     19.9^+0.47_-0.58

NGC 6517        17        0.9       PL        0.626     2.0(frozen)                                                     < 4.53                   9.82      < 4.61 

NGC 6528        262       8.7       PL        0.313     1.98^+0.33_-0.31                              7.1/14        3.73^+0.59_-0.69     6.23      5.99^+0.95_-1.11

NGC 6535        30        0.7       PL        0.197     2.0(frozen)                                                     < 1.15                   0.607     < 19.0 

NGC 6539        333       6.8       PL        0.592     3.37^+0.69_-0.65                              10.3/11       31.7^+6.41_-4.41     18.34     17.3^+3.49_-2.40

NGC 6553        627       12.4      PL        0.365     2.88^+0.32_-0.31                              25.9/26       9.96^+1.03_-0.63     32.1      3.10^+0.32_-0.20

NGC 6569        18        0.8       PL        0.307     2.0(frozen)                                                     < 7.58                   18.17     < 4.17
 
NGC 6626        37119     123.4     PL        0.232     1.98^+0.03_-0.03                              1135.5/658    58.3^+0.58_-0.53     19.75     29.5^+0.29_-0.27

NGC 6637        99        4.8       PL        0.104     1.0^+0.53_-0.49                               4.7/5         16.6^+5.57_-3.74     15.98     10.4^+3.49_-2.34

NGC 6638        90        5.7       PL        0.238     1.35^+0.52_-0.48                              1.24/6        14.7^+3.39_-2.44     7.51      19.6^+4.51_-3.25

NGC 6656^*    3629      17.6      PL        0.197     1.66^+0.69_-0.34                              223.5/155     4.70^+2.18_-0.96     20.81     2.26^+1.05_-0.46

NGC 6715        271       11.6      PL        0.087     2.38^+0.24_-0.23                              18.04/15      124^+12.5_-7.08      117.3     10.6^+1.07_-0.61

NGC 6717        221       8.9       PL        0.128     0.99^+0.33_-0.32                              1.49/10       9.67^+1.48_-1.51     2.45      39.4^+6.04_-6.16

NGC 6760        85        1.5       PL        0.447     2.0(frozen)                                                     < 3.03                   20.63     < 1.47 

NGC 6809        10        0.09      PL        0.046     2.0(frozen)                                                     < 1.26                   7.02      < 1.80 

NGC 6838        1063      12.9      PL        0.145     1.97^+0.29_-0.28                              15/18         2.67^+0.51_-0.36     2.33      11.5^+2.19_-1.55

NGC 7089        101       3.8       PL        0.035     1.54^+0.54_-0.48                              3.1/5         14.2^+6.37_-4.01     33.16     4.28^+1.92_-1.21

Glim 01         1107      25.3      PL        2.813     1.58^+0.21_-0.21                              65.9/48       15.6^+0.72_-0.97     26.98     5.78^+0.27_-0.36

Pal 10          15        0.7       PL        0.963     2.0(frozen)                                                     < 3.54                   2.98      < 11.9 

Ter 3           -106      -1.5      PL        0.423     2.0(frozen)                                                     < 1.07                   0.37      < 29.0 

Ter 5           39938     170.5     PL        1.380     1.93^+0.02_-0.02                             2320.6/1687    117^+0.84_-1.13      96.65     12.1^+0.09_-0.12



Core Collapse GCs:
NGC 362         1368      26.3      Brem      0.029                               1.15^+0.16_-0.13    62.7/61       9.20^+0.34_-0.46     20.63     4.46^+0.17_-0.22

NGC 1904        58        3.7       PL        0.006     2.04^+0.93_-0.72                              1.7/4         7.42^+2.48_-2.01     7.83      9.48^+3.17_-2.57

NGC 5946        16        0.5       PL        0.313     2.0(frozen)                                                     < 4.73                   15.94     < 2.97 
 
NGC 6256        135       6.1       PL        0.632     1.81^+0.51_-0.46                              8.2/7         29.4^+4.3_-8.09      11.64     25.3^+3.69_-6.95

NGC 6266        5894      63.9     PL+Brem    0.273     1.15^+0.29_-0.36    0.74^+0.15_-0.13    238.6/176     63.9^+1.02_-4.99     39.86     16.0^+0.27_-1.25

NGC 6293        49        2.1       PL        0.209     2.0(frozen)                                       0/1           < 10.1                   10.67     < 9.47

NGC 6325        82        4.1       PL        0.528     2.0(frozen)                                       3.06/2        3.17^+1.18_-0.90     3.79      8.36^+3.11_-2.37

NGC 6342        84        3.5       PL        0.267     0.97^+0.66_-0.61                              1.5/4         8.47^+2.97_-1.87     3.9       21.7^+7.62_-4.80

NGC 6355        108       3.2       PL        0.447     1.70^+1.10_-0.84                              0.34/3        7.28^+2.77_-2.56     9.47      7.69^+2.92_-2.70

NGC 6397^*    58210     133.5    PL+Brem    0.104     1.19^+0.02_-0.02    0.29^+0.08_-0.07    1315/1274     13.3^+0.48_-0.47     1.98      67.3^+2.41_-2.38
       
NGC 6453        284       12.2      PL        0.371     2.0(frozen)                                       3.2/3         5.13^+1.41_-1.14     7.2       7.13^+1.96_-1.58

NGC 6522        64        2.5       PL        0.278     2.0(frozen)                                                     < 10.1                   11.48     < 8.8

NGC 6541        2870      39.5      PL        0.081                               1.43^+0.13_-0.11    139.3/103     22.6^+0.07_-0.08     13.28     17.0^+0.50_-0.59

NGC 6544        132       4.2       PL        0.441     2.0(frozen)                                       3.67/3        1.71^+0.44_-0.38     3.28      5.21^+1.35_-1.15

NGC 6558        87        1.6       PL        0.255     2.0(frozen)                                                     < 12.62                   3.35      < 37.6 

NGC 6642        19        1.2       PL        0.232     2.0(frozen)                                                     < 5.17                   4.06      < 12.7 

NGC 6681        72        1.4       PL        0.041     2.0(frozen)                                                     < 1.38                   4.62      < 2.99

NGC 6752        4640      35.4      PL        0.023     1.47^+0.06_-0.06                              221.7/153     11.1^+0.38_-0.42     7.25      15.3^+0.52_-0.58
  
NGC 7099        1045      18.0      PL        0.017     2.32^+0.12_-0.12                              68.2/50       8.71^+0.60_-0.56     5.22      16.7^+1.15_-1.07

Ter 1           633       12.2      PL        1.154     1.41^+0.33_-0.32                              18.5/19       24.5^+2.12_-3.06     27.34     8.96^+0.77_-1.12

Ter 9           275       8.9       PL        1.021     1.17^+0.32_-0.34                              8.9/13        23.9^+2.37_-3.06     6.26      38.2^+3.79_-4.89

Column 1-3: Name, observation net counts (0.5-8 keV) and signal-to-noise ratio. Column 4-5: X-ray spectral fitting model and the fixed absorption column in units of 10^22 cm^-2, which is calculated with N_H=0.58 × E(B-V) × 10^22 cm^-2. GCs marks with “*" have a large angular extent (r_h ≳ 2'), we adopted the “Double Subtraction" procedure to correct the vignetting effect during spectral fitting. Column 6-8: Photon-index, plasma temperature in units of keV, and χ^2/degree of freedom. Column 9-11: 0.5-8 keV X-ray luminosity of GCs in units of 10^32 erg s^-1, K-band luminosity in units of 10^4L_K,⊙ and X-ray to K-band luminosity ratio, in units of 10^27 erg s^-1L_K,⊙^-1.




lllllllll

9
0pc

X-RAY VARIABILITY OF THREE GCs

Obs.ID     Exp.     Date     Model     nH     Photon          χ^2_ν      L_X 

     (ks)               (10^22cm^-2)     Index          (d.o.f.)     (10^32erg/s)
Terzan 5:
3798       24.9     2003-07-13     PL          1.322     1.82^+0.08_-0.08                               116.7/109     137.3^+3.85_-4.43  

10059      35.9     2009-07-15     PL          1.322     1.81^+0.10_-0.09                               129.3/135     100.4^+2.50_-3.13  

13225      29.7     2011-02-17     PL          1.322     1.99^+0.09_-0.09                               101.9/116     119.2^+3.62_-3.71  

13252      39.2     2011-04-29     PL          1.322     1.98^+0.09_-0.08                               151.2/122     101.0^+3.28_-3.59  

13705      13.2     2011-09-05     PL          1.322     1.99^+0.14_-0.13                               66.8/57       116.0^+4.67_-5.84  

13706      45.5     2012-05-13     PL          1.322     1.82^+0.06_-0.06                               202.9/187     145.6^+2.82_-3.17  
                    
14339      31.4     2011-09-08     PL          1.322     1.98^+0.09_-0.09                               120.1/121     111.7^+2.61_-3.89  

14475      29.3     2012-09-17     PL          1.322     2.15^+0.10_-0.10                               111.2/105     116.4^+3.42_-3.54  

14476      28.0     2012-10-28     PL          1.322     2.03^+0.10_-0.10                               112.7/99      110.2^+4.01_-3.39  

14477      28.3     2013-02-05     PL          1.322     2.02^+0.10_-0.10                               117.9/100     112.5^+3.08_-5.12  

14478      28.3     2013-07-16     PL          1.322     1.88^+0.09_-0.09                               103.7/104     115.4^+3.83_-3.76  

14625      48.9     2013-02-22     PL          1.322     1.91^+0.07_-0.07                               229/175       125.1^+2.40_-2.49  

15615      82.9     2013-02-23     PL          1.322     1.88^+0.05_-0.05                               336.4/236     113.1^+1.72_-1.59  

Joint      -        –             PL          0.232     1.93^+0.02_-0.02                              2320.6/1687    116.9^+0.84_-1.13  

1-9
NGC 6626:
2683      8.9      2002-09-09     PL          0.232     2.03^+0.15_-0.15                               52.8/52       45.3^+2.11_-0.28  
                    
2684      12.4     2002-07-04     PL          0.232     1.93^+0.12_-0.12                               59.3/65       45.0^+2.59_-1.61  

2685      12.5     2002-08-04     PL          0.232     2.14^+0.14_-0.13                               50.3/62       38.4^+2.28_-2.04  

9132      141      2008-08-07     PL          0.232     1.96^+0.03_-0.03                               339.5/296     63.8^+0.90_-0.82  

9133      52.5     2008-08-10     PL          0.232     2.00^+0.06_-0.06                               261/185       58.8^+1.17_-1.24  

Joint     –       –             PL          0.232     1.98^+0.03_-0.03                              1135.5/658     58.3^+0.58_-0.53  

1-9 
NGC 6397:
79        48.3     2000-07-31     PL          0.104     1.42^+0.04_-0.04                               226.8/234     16.0^+0.28_-0.33  

2668      28.1     2002-05-13     PL          0.104     1.19^+0.06_-0.06                               202.4/202     13.8^+0.27_-0.37  

2669      26.7     2002-05-15     PL          0.104     1.21^+0.07_-0.07                               227.2/184     11.4^+0.37_-0.27  

7460      146      2007-07-16     PL          0.104     1.20^+0.03_-0.03                               302.3/335     13.0^+0.15_-0.16  

7461      88.9     2007-06-22     PL          0.104     1.14^+0.03_-0.03                               349.9/319     15.9^+0.31_-0.18  



Column 1-3: Chandra observation ID, effective exposure time in units of keV and observation date. Column 4-6: X-ray spectral fitting model, the fixed absorption column in units of 10^22 cm^-2 and Photon-index. Column 8-9: χ^2/degree of freedom and GC 0.5-8 keV X-ray luminosity in units of 10^32 erg s^-1.





lrrrrrrrrr

10
0pc

TESTED CORRELATIONS

1c          2cTotal GCs         2cNormal GCs          2cCore-Collapsed GCs

3-4 6-7 9-10 

Relations          r     p(>|r|)          r     p(>|r|)          r     p(>|r|) 
L_X-L_K                0.694        7.2×10^-11         0.754        5.9×10^-10            0.366          0.123 

L_X-Γ               0.627        1.4×10^-8         0.769        1.7×10^-10            0.194          0.426  

Γ-M                   0.611        4.1×10^-8         0.754        5.4×10^-11            0.660          0.002  

γ-M                   0.111        0.374                       0.337        0.019                          0.125          0.611  

L_X/L_K-L_K          -0.321       0.008                       -0.272       0.061                          -0.228         0.348  

L_X/L_K-γ         0.300        0.013                       0.298        0.040                          -0.149         0.542  

L_X-Fe/H                 0.043        0.734                       0.061        0.685                          0.015          0.952  

L_X/L_K-Fe/H           0.132        0.291                       0.234        0.114                          0.014          0.955  

L_X-σ               0.781        2.8×10^-9         0.867        2.0×10^-8            0.370          0.174  

L_X/L_K-σ         -0.055       0.737                       -0.042       0.844                          -0.089         0.751  

L_X-ρ                 0.422        3.8×10^-4         0.721        7.2×10^-9            0.008          0.974  

L_X/L_K-ρ           0.291        0.017                       0.234        0.109                          0.053          0.830  

L_X-c                    0.317        0.009                       0.584        1.3×10^-5            -0.019         0.938  

L_X/L_K-c              0.247        0.044                       0.138        0.358                          0.123          0.615  

L_X-t_c                -0.124       0.320                       -0.331       0.023                          0.184          0.450  

L_X/L_K-t_c          -0.423       4.0×10^-4          -0.457       0.001                          -0.191         0.432  

L_X-t_h                0.270        0.027                       0.238        0.103                          0.542          0.016  

L_X/L_K-t_h          -0.390       1.1×10^-3          -0.456       0.001                          0.191          0.435  

L_X-t                    0.062        0.696                       0.019        0.921                          0.275          0.414  

L_X/L_K-t              0.376        0.014                       0.457        0.010                          0.545          0.083  

L_X-t_r                -0.021       0.906                       0.014        0.942                          -0.152         0.774  

L_X/L_K-t_r          0.417        0.013                       0.474        0.009                          0.152          0.774  

L_X-r_gc               -0.093       0.454                       -0.117       0.426                          -0.066         0.789  

L_X/L_K-r_gc         -0.402       7.5×10^-4          -0.454       0.001                          -0.134         0.587  

L_X-e                    -0.118       0.374                       -0.058       0.713                          -0.233         0.386  

L_X/L_K-e              -0.161       0.222                       -0.210       0.176                          0.009          0.974  

L_X-r_t                  0.103        0.405                       0.166        0.260                          0.159          0.516  

L_X/L_K-r_t            -0.321       0.008                       -0.398       0.005                          -0.307         0.201  

L_X-f_b                -0.721       6.9×10^-6         -0.742       3.3×10^-5             -0.580         0.228  

L_X/L_K-f_b          -0.097       0.609                       0.114        0.597                          -0.493         0.321  



Spearman's rank correlation test for each parameter. The p(>|r|) values show the probability that a correlation arises randomly.







