We present a dynamical model of the high mass X-ray binary LMC X-1 based on high-resolution optical spectroscopy and extensive optical and near-infrared photometry .
From our new optical data we find an orbital period of P = 3.90917 \pm 0.00005 days .
We present a refined analysis of the All Sky Monitor data from RXTE and find an X-ray period of P = 3.9094 \pm 0.0008 days , which is consistent with the optical period .
A simple model of Thomson scattering in the stellar wind can account for the modulation seen in the X-ray light curves .
The V - K color of the star ( 1.17 \pm 0.05 ) implies A _ { V } = 2.28 \pm 0.06 , which is much larger than previously assumed .
For the secondary star , we measure a radius of R _ { 2 } = 17.0 \pm 0.8 R _ { \odot } and a projected rotational velocity of V _ { rot } \sin i = 129.9 \pm 2.2 km s ^ { -1 } .
Using these measured properties to constrain the dynamical model , we find an inclination of i = 36.38 \pm 1.92 ^ { \circ } , a secondary star mass of M _ { 2 } = 31.79 \pm 3.48 M _ { \odot } , and a black hole mass of 10.91 \pm 1.41 M _ { \odot } .
The present location of the secondary star in a temperature-luminosity diagram is consistent with that of a star with an initial mass of 35 M _ { \odot } that is 5 Myr past the zero-age main sequence .
The star nearly fills its Roche lobe ( \approx 90 \% or more ) , and owing to the rapid change in radius with time in its present evolutionary state , it will encounter its Roche lobe and begin rapid and possibly unstable mass transfer on a timescale of a few hundred thousand years .