Cross sections of the $^{186}$W, $^{187}$Re, $^{188}$Os($gamma,n$) reactions were measured using quasi-monochromatic photon beams from laser Compton scattering (LCS) with average energies from 7.3 to 10.9 MeV. The results are compared with the predictions of Hauser-Feshbach statistical calculations using four different sets of input parameters. In addition, the inverse neutron capture cross sections were evaluated by constraining the model parameters, especially the $E1$ strength function, on the basis of the experimental data. The present experiment helps to further constrain the correction factor $F_{sigma}$ for the neutron capture on the 9.75 keV state in $^{187}$Os. Implications of $F_{sigma}$ to the Re-Os cosmochronology are discussed with a focus on the uncertainty in the estimate of the age of the Galaxy.
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