Low-mass X-ray binaries ( LMXBs ) form efficiently in globular clusters ( GCs ) .
By combining Chandra X-ray Observatory and Hubble Space Telescope -Advanced Camera for Surveys observations of early-type galaxies , we probe the LMXB-GC connection using the most accurate identification of LMXBs and GCs to date .
We explore the optical properties of 270 GCs with LMXBs and 6,488 GCs without detectable X-ray emission from a sample of eleven massive early-type galaxies in the Virgo cluster .
Globular clusters that are more massive , are redder , and have smaller radii are more likely to contain LMXBs .
Using known structural scaling relations for GCs , the latter implies that denser GCs are more likely to hold LMXBs .
Unlike Galactic GCs , a large number of GCs with LMXBs have half-mass relaxation times > 2.5 { Gyr } ; GCs do not need to survive for more than five relaxation timescales to produce LMXBs .
By fitting the dependence of the expected number of LMXBs per GC , \lambda _ { t } , on the GC mass { M } , color ( g - z ) , and half-mass radius r _ { h, { cor } } , we find that \lambda _ { t } \propto { M } ^ { 1.24 \pm 0.08 } 10 ^ { 0.9 ^ { +0.2 } _ { -0.1 } ( g - z ) } r _ { h, { cor% } } ^ { -2.2 \pm 0.2 } .
This rules out that the number of LMXBs per GC is linearly proportional to GC mass ( 99.89 % confidence limit ) .
We derive an expression to estimate the number of multiple LMXB sources in GCs and predict that most GCs with high X-ray luminosities contain a single LMXB .
The detailed dependence of \lambda _ { t } on GC properties appears mainly due to a dependence on a combination of mass and radius , and a dependence on color , that are essentially equivalent to a dependence on the encounter rate \Gamma _ { h } and the metallicity { Z } , \lambda _ { t } \propto \Gamma _ { h } ^ { 0.82 \pm 0.05 } { Z } ^ { 0.39 \pm 0.07 } .
Our analysis provides strong evidence that dynamical formation and metallicity play the primary roles in determining the presence of an LMXB in extragalactic GCs .
The shallower than linear dependence of GC sources requires an explanation by theories of dynamical binary formation ; however , we note that our use of \Gamma _ { h } as a proxy for the encounter rate , particularly if core-collapsed extragalactic GCs preferentially contain LMXBs , needs further testing in nearer galaxies .
A metallicity-dependent variation in the number of neutron stars and black holes per unit GC mass , effects from irradiation induced winds , or suppression of magnetic braking in metal-poor stars may all be consistent with our derived abundance dependence ; all three scenarios require further development .