A Simple Scaling Analysis of X-ray Emission and Absorption in Hot-Star Winds

Abstract

We present a simple analysis of X-ray emission and absorption for hot-star winds, designed to explore the natural scalings of the observed X-ray luminosity with wind and sstellar properties. We show that an exospheric approximation, in which all of the emission above the optical depth unity radius escapes the wind, reproduces very well the detailed expression for radiation transport through a spherically symmetric wind. Using this approximation we find that the X-ray luminosity $L_x$ scales naturally with the wind density parameter $\Mdot/\vinf$, obtaining $L_x \sim (\Mdot/\vinf)^2$ for optically thin winds, and $L_x \sim (\Mdot/\vinf)^{1+s}$ for optically thick winds with an X-ray filling factor that varies in radius as $f \sim r^s$. These scalings with wind density contrast with the commonly inferred empirical scalings of X-ray luminosity $L_x $ with bolometric luminosity $L_B$. The empirically derived linear scaling of $L_x \sim L_B$ for thick winds can however be reproduced, through a delicate cancellation of emission and absorption, if one assumes modest radial fall-off in the X-ray filling factor ($s \approx -0.25$ or $s \approx -0.4$, depending on details of the secondary scaling of wind density with luminosity). We also explore the nature of the X-ray spectral energy distribution in the context of this model, and find that the spectrum is divided into a soft, optically thick part and a hard, optically thin part. Finally, we conclude that the energy-dependent emissivity must have a high-energy cut-off, corresponding to the maximum shock energy, in order to reproduce the general trends seen in X-ray spectral energy distributions of hot stars.