We present a quantitative analysis of the X-ray spectrum of the eclipsing high mass X-ray binary Vela X-1 (4U 0900-40) using archival data from ASCA. The observation covers a time interval centered on eclipse of the X-ray pulsar by the companion. The spectrum exhibits two distinct sets of discrete features: (1) recombination lines and radiative recombination continua from mostly H- and He-like species produced by photoionization in an extended stellar wind; and (2) fluorescent K-shell lines associated with near-neutral species also present in the circumsource medium. Using a detailed spectral model that explicitly accounts for the recombination cascade kinetics for each of the constituent charge states, we are able to obtain a statistically acceptable (chi_r^2=0.88) fit to the observed spectrum and to derive emission measures associated with the individual K-shell ions of several elements. We find a best-fit mass loss rate of ~2.7 x 10^-7 M-solar/yr, which is approximately a factor of 10 lower than previous estimates of the mass loss rate for the Vela X-1 companion star, which have primarily been determined from P Cygni profiles, and X-ray absorption measurements. We argue that this discrepancy can be reconciled if the X-ray irradiated portion of the wind in Vela X-1 is structurally inhomogeneous, consisting of hundreds of cool, dense clumps embedded in a hotter, more ionized gas. Most of the mass is contained in the clumps, while most of the wind volume (>95%) is occupied by the highly ionized component. We show quantitatively, that this interpretation is also consistent with the presence of the X-ray fluorescent lines in the ASCA spectrum, which are produced in the cooler, clumped component.
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