# Inference of hot star density stream properties from data on rotationally recurrent DACs

### J.C. Brown^{1}, R. K. Barrett^{1}, L.M. Oskinova^{3},
S.P. Owocki^{1,4}, W.-R. Hamann^{3}, J. A. de Jong^{2,5}
L. Kaper^{2}, H. F. Henrichs^{2}

^{1}Department of Physics and Astronomy, University of Glasgow, Scotland, UK

^{2}Astronomical Institute, University of Amsterdam, Netherlands

^{3}Universität Potsdam, Institut für Physik, Bereich Astrophysik, Germany

^{4}Bartol Research Institute, University of Delaware, Newark, USA

^{5}Leiden Observatory, University of Leiden, Netherlands

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
v_{r}, 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
v_{r}(r) and \Omega(r) of the absorbing stream matter but
also
by the density profile across the stream, determined by the azimuthal
(\phi_{o}) distribution function F_{o}(\phi_{o}) of mass loss rate
around the stellar equator. When F_{o}(\phi_{o}) is fairly wide in
\phi_{o}, 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 v_{r}(r), \Omega(r) and F_{0}(\phi_{o}) 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
v_{r}(r), \Omega(r) and F_{o}(\phi_{o}) from data on
\tau(w,\phi)
of sufficiently high precision and resolution since v_{r}(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.

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