Inference of hot star density stream properties from data on rotationally recurrent DACs
J.C. Brown1, R. K. Barrett1, L.M. Oskinova3, S.P. Owocki1,4, W.-R. Hamann3, J. A. de Jong2,5 L. Kaper2, H. F. Henrichs2
The information content of data on rotationally periodic recurrent discrete absorption components (DACs) in hot star wind emission lines is discussed. The data comprises optical depths \tau(w,\phi) as a function of dimensionless Doppler velocity w=(\Delta\lambda/\lambda_0)(c/v_\infty) and of time expressed in terms of stellar rotation angle \phi. This is used to study the spatial distributions of density, radial and rotational velocities, and ionisation structures of the corotating wind streams to which recurrent DACs are conventionally attributed. The simplifying assumptions made to reduce the degrees of freedom in such structure distribution functions to match those in the DAC data are discussed and the problem then posed in terms of a bivariate relationship between \tau(w,\phi) and the radial velocity vr, transverse rotation rate \Omega(r) and density \rho(r,\phi) structures of the streams. The discussion applies to cases where: the streams are equatorial; the system is seen edge on; the ionisation structure is approximated as uniform; the radial and transverse velocities are taken to be functions only of radial distance but the stream density is allowed to vary with azimuth. The last kinematic assumption essentially ignores the dynamical feedback of density on velocity and the relationship of this to fully dynamical models is discussed. The case of narrow streams is first considered, noting the result of Hamann et al. (2001) that the apparent acceleration of a narrow stream DAC is higher than the acceleration of the matter itself, so that the apparent slow acceleration of DACs cannot be attributed to the slowness of stellar rotation. Thus DACs either involve matter which accelerates slower than the general wind flow, or they are formed by structures which are not advected with the matter flow but propagate upstream (such as Abbott waves). It is then shown how, in the kinematic model approximation, the radial speed of the absorbing matter can be found by inversion of the apparent acceleration of the narrow DAC, for a given rotation law. The case of broad streams is more complex but also more informative. The observed \tau(w,\phi) is governed not only by vr(r) and \Omega(r) of the absorbing stream matter but also by the density profile across the stream, determined by the azimuthal (\phio) distribution function Fo(\phio) of mass loss rate around the stellar equator. When Fo(\phio) is fairly wide in \phio, the acceleration of the DAC peak \tau(w,\phi) in w is generally slow compared with that of a narrow stream DAC and the information on vr(r), \Omega(r) and F0(\phio) is convoluted in the data \tau(w,\phi). We show that it is possible, in this kinematic model, to recover by inversion, complete information on all three distribution functions vr(r), \Omega(r) and Fo(\phio) from data on \tau(w,\phi) of sufficiently high precision and resolution since vr(r) and \Omega(r) occur in combination rather than independently in the equations. This is demonstrated for simulated data, including noise effects, and is discussed in relation to real data and to fully hydrodynamic models.