- Inference of hot star density stream properties from data on rotationally recurrent DACs
- 1 Introduction
- 2 Kinematic formulation of the DAC interpretation problem
- 3 Properties of narrow streams
- 4 Wide streams and the general DAC inversion problem
- 5 Numerical inversion results for artificial data sets
- 6 Discussion and application to real data
- Appendix: The continuity equation
- References
- 8 Online Material

A&A 413, 959-979 (2004)

DOI: 10.1051/0004-6361:20031557

**J. C. Brown ^{1,2} - R. K. Barrett^{1} - L. M. Oskinova^{1,3} -
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, Glasgow, G12 8QQ, Scotland, UK

2 -
Astronomical Institute "Anton Pannekoek'', University of Amsterdam,
Kruislaan 403, 1098 SJ Amsterdam, The Netherlands

3 -
Professur Astrophysik, Universitat Potsdam, Am Neuen Palais 10,
14469 Potsdam, Germany

4 -
Bartol Research Institute, University of Delaware, Newark, DE 19716, USA

5 -
Leiden Observatory, University of Leiden, Niels Bohrweg 2, 2333 CA Leiden,
The Netherlands

Received 16 June 2003 / Accepted 15 September 2003

**Abstract**

The information content of data on rotationally periodic recurrent
discrete absorption components (DACs) in hot star wind emission lines
is discussed. The data comprise optical depths
as a
function of dimensionless Doppler velocity
and of time expressed in
terms of stellar rotation angle .
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
and the radial velocity
,
transverse rotation rate
and density
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
is governed not only by
and
of the absorbing stream matter but also
by the density profile across the stream, determined by the azimuthal
()
distribution function
of mass loss rate
around the stellar equator. When
is fairly wide in
,
the acceleration of the DAC peak
in *w* is
generally slow compared with that of a narrow stream DAC and the
information on
,
and
is
convoluted in the data
.

We show that it is possible, in this kinematic model, to recover by inversion, complete information on all three distribution functions , and from data on of sufficiently high precision and resolution since and 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.

**Key words: **stars: early-type - stars: winds, outflows - stars: mass-loss - line: profiles

1 Introduction

The phenomenon of Discrete Absorption Components (DACs) moving (often
recurrently) in the broad emission line profiles of hot star winds has
been discussed extensively in the literature - see e.g., Prinja &
Howarth (1988), Owocki et al. (1995), Henrichs et al. (1994),
Fullerton et al. (1997) and recent overviews of data and theoretical
interpretation by Kaper (2000) and by Cranmer & Owocki (1996)
respectively. In the present paper we summarise some major aspects of
DAC interpretation *in relation to the information content of the
data*, and examine how analytic aspects of a simplified kinematic
model enable formulation of DAC modelling from a diagnostic inverse
problem viewpoint (Craig & Brown 1986). This may provide a useful tool
in the quantitative non-parametric interpretation of recurrent DAC
data sets from specific stars.

The present situation can be summarised as follows.

- 1.
- DACs are attributed to structures, accelerating outward in stellar winds, which have enhanced optical depth over a rather narrow range of Doppler wavelengths compared to the overall absorption line width.
- 2.
- This is attributed to enhanced density and/or reduced velocity gradient along the line of sight; that is, the increase in the number of absorbers per unit velocity can be attributed to an actual increase in spatial density, or to an increase in the spatial volume over which the absorbers have that velocity.
- 3.
- In the case of recurrent DACs the periodicity is commonly attributed to stellar rotation and the distribution pattern of absorbing material is taken to be time-independent in the stellar rotation frame, though the matter itself moves through this pattern (that is, the flow is stationary in the corotating frame, but not static). This is the type of DAC phenomenon we will consider here.
- 4.
- There are suggestions (e.g., Hamann et al. 2001) that some periodicity should be attributed to causes other than rotation (e.g., non-radial pulsation), and even debate over whether "DAC'' is the correct terminology for some Component features, even though they are Discrete and in Absorption!
- 5.
- The existence of such corotating patterns of enhanced density/reduced velocity gradient is often attributed to corotating interaction regions (Mullan 1984). These CIRs arise where outflows with different radial speeds from azimuthully distinct regions collide.
- 6.
- Cranmer & Owocki (1996) have modelled the creation of CIRs physically by studying the hydrodynamic response of a radiatively driven wind to empirical imposition of bright spots azimuthally localised on the stellar surface. Their simulations predict DAC profiles and time dependence generally similar to data and have provided the best insight yet into the interpretation of DACs, such as the relative importance in the absorbing matter patterns of deviations in density and in velocity gradients from the mean wind. It is central to these dynamical models that the absorbing pattern is created by variation in the outflow speed with azimuth as well as radius and that the inertia of the enhanced density reacts back on the velocity field.
- 7.
- On the other hand, progress has been made in the interpretation of specific DAC data sets by use of a purely kinematic approach (Owocki et al. 1995; Fullerton et al. 1997). In this the absorption is attributed to a rotating density pattern following radial (and rotational) velocity laws which are the same at all azimuths (though not in general the same velocity laws as the mean wind).
- 8.
- Neither the kinematic nor the dynamical approach is entirely satisfactory. The former ignores the dynamical feedback of density on velocity. The latter, on the other hand, necessarily involves non-monotonic velocity variation along the line of sight, creating ambiguity in identifying Doppler velocities with distances (cf. Brown et al. 1997 discussion of emission line profiles). Secondly, matching the DAC data set from a particular object requires the hydro code to be run for sets of radiative driver (e.g., hotspot) properties occupying a large range of parameter space.

We first (Sect. 2) discuss the information content of recurrent DAC data and the need to make assumptions concerning the wind structure in order to reduce the degrees of freedom in the model to match those in the data, whether by forward fitting or inverse inference. With these assumptions we then derive the basic equations relating the DAC properties in the kinematic approximation to the density and velocity distributions with radius for a general CDR which may be broad in azimuth (cf. discussions in Hamann et al. 2001 of the narrow stream case).

In Sect. 3 we summarise the analytic properties of the narrow-stream CDR case, first in the forward modelling approach (cf. Hamann et al. 2001), then as an inverse problem of inferring the CDR matter velocity law non-parametrically from DAC acceleration data. We set out the narrow-stream inversion procedure given a wind rotation law and without the need to use mass continuity, and discuss the fact that our model assumption of an axisymmetric wind velocity law is not strictly necessary in this case. The relationship of narrow-stream to wide-stream inversions is considered. In Sect. 4 we tackle the problem of a general wide CDR showing how its density and velocity distibutions are reflected in the DAC profile and its recurrent time variation. Note that Hamann et al. (2001) only addressed the time dependence of the narrow DAC wavelength and not the DAC profile, and that in most treatments (e.g., Prinja & Howarth 1988; Owocki et al. 1995) addressing the DAC profile, only a parametric fit (e.g., Gaussian) is used, rather than the full information present in the profile. We then address the inverse problem of inferring non-parametrically the CDR/stream velocity and density structure from full data on the time-varying wide DAC profile, and illustrate in Sect. 5, using synthetic data, the success of the method within the restrictions of the kinematic approach. Finally in Sect. 6 we discuss how future work may integrate this inverse CDR diagnostic formalism with full dynamical modelling to enhance our ability to model the true CIR structure of specific stars from their recurrent DAC data sets.

2 Kinematic formulation of the DAC interpretation problem

The blue wing of the P Cygni profile of a hot-star wind spectral line contains an absorption component (from moving material in the wind absorbing the stellar continuum) and a scattered component (from the wind volume). Since we are interested in DACs we want to remove the scattered light leaving only the absorption component. This requires a careful treatment (Massa et al. 1995, 2003). In addition, it will be seen in this section that for the kinematical model that we develop the optical depth profile depends linearly on the surface density (i.e., mass loss rate) variation, for a given wind velocity law. It follows from this that we could examine the absorption component of the whole wind, or we could consider only the optical depth excess related to the DAC overdensity itself. The latter may be simpler in practice to obtain, by subtraction of a "least absorption'' wind absorption line profile, say (e.g., Kaper et al. 1999). In the results presented here we generally assume that we have the optical depth excess corresponding to the DAC, but this is not necessary for the formalism we develop.

In any event, we suppose here that high resolution spectral line data
can be processed so as to extract the absorption component (or
the absorption component of a single recurrent DAC feature) from the
overall line profile. Then the recurrent DAC data can be
expressed in terms of the DAC optical depth
as a function of
Doppler shift
and time *t* related to the observer
azimuth ,
measured relative to a convenient reference point in
the frame of the star, rotating at angular speed ,
by
.

The data function
of two variables for a
single line is clearly incapable of diagnosing the full 3-D structure
of even a steady state general wind which involves at least the mass
density ,
velocity
and temperature *T* as functions of
3-D vector position ,
and the inclination *i*. Clearly,
the structure inference inversion problem to determine these four
functions of three variables (along with the inclination) from the
single function,
,
of two variables is
massively under-determined. Correspondingly, in any forward modelling
of the
data from a theoretical structure,
there may be a multiplicity of ,
,
*T* distributions
which fit the data. Progress in either approach can only be made then
by introducing a number of simplifying assumptions about the
geometry etc., and by utilising the physics of the situation.

In the work presented here we make one important physical approximation regarding the nature of the wind velocity law, and a number of assumptions (mostly geometrical) of lesser importance, which do not critically affect to the conclusions we reach but greatly simplify the presentation.

Note that the principal consequence of this assumption is that all streamlines in the corotating frame are obtained from a single streamline (see Eq. (3) by shifting in azimuth: the pattern of streamlines is also axisymmetric.

Although dynamical simulations do not satisfy monotonic and independent of (Cranmer & Owocki 1996), the deviations from these conditions are not very large, at least for weak DACs, and our assumption seems a reasonable first approximation. Moreover, as we will show, it allows us to make considerable progress with the structure inference problem; without this model assumption the inference of the wind structure is far from trivial (although see Sects. 3.4 and 4.2, where we discuss inferring the structure of general, nonaxisymmetric winds).

The assumption (cf. Point 7 in Sect. 1) that the DAC can be described approximately by a "corotating dense region'' or "CDR'' with definite radial flow speed independent of and originating at some inner surface is akin to the DAC data analysis modelling by Owocki et al. (1995) and Fullerton et al. (1997) in which, as they put it "the hydrodynamical feedback between density and velocity is ignored''.

What the observer sees is then the time-dependent profile of the
stream absorption over the range of Doppler speeds presented along the
observer line of sight at each moment under the combined effects of
radial matter flow and rotation of the matter pattern. It is vital to
note that (a): we do not assume the spatial stream flow speed
of DAC producing dense matter to be the same as that (
)
of matter in the mean wind producing the overall stellar
profile; (b) the apparent pattern/phase speed
of the absorption
features seen as a function of time ()
is not the same as that
of either
nor of
since rotation
sweeps matter at larger *r* into the line of sight. Indeed our
approach is aimed at inferring
for the stream material
and comparing it with the velocity law
of the
general wind. It can encompass any monotonic form of velocity law
,
including for example that used for narrow streams by
Hamann et al. (2001) with the usual -law form plus a
superposed inward pattern speed, intended as a qualitative
representation of an Abbott wave (Abbott 1980 - cf. Cranmer & Owocki
1996; Feldmeier & Shlosman 2002). By not restricting
to some parametric form, we should be able to recover, from DAC
profile data, information on such features as plateaux in
,
i.e., regions where
is small, which can be
(see Eq. (1)) at least as important in determining
DAC profiles as local density enhancements (Cranmer & Owocki 1996).
By formulating the data diagnostic problem in a non-parametric way in
our general treatment (e.g. not enforcing a -law) we show that
it is possible to infer kinematically the form of
and
hence the presence both of density enhancements and of velocity
plateaux from DAC profile data, and to compare these with dynamical
model predictions.

In addition to our model assumption we make the following geometrical and physical idealisations, mostly to clarify the relationship between the intrinsic physics of the wind and its observational characteristics. These simplifications are similar to those used by previous authors in similar regimes of the problem (e.g., Fullerton et al. 1997; Kaper et al. 1999 and references therein).

Firstly, we neglect variations with radius of the ionisation or
excitation state of the absorbing ion. Such variations would
correspond to an effective source or sink term in the continuity
equation (reflecting the fact that the number of absorbers in any
fluid element does not remain constant as it moves through the wind),
and the function
in Eq. (6) would consequently be
modified. In principle, such variations can be accounted for by
analysing lines from a range of ions and levels. This lets us, for
example, drop *T*(*r*) as an unknown. The absorbing ion density variation
with *r* and
is then effectively controlled solely by steady
state continuity. We recognise that, in reality, application of
continuity to a single ion without allowing for varying ionisation
could give very misleading results since variations in ionisation are
often observed (Massa et al. 1995; Fullerton et al.1997; Prinja et al. 2002).
Determining the ionisation balance throughout the wind - and correcting
our continuity equation in light of this - is a separate inference
problem that we do not consider here.

Secondly, we make the "point-star'' approximation:

- 1.
- The system is assumed to be seen at and the wind stream structure is approximated as constant (or averaged) across the stellar disk. This essentially reduces the structure problem from 3-D to 2-D, eliminating the spherical coordinate .
- 2.
- We consider only absorption features formed at large enough
distances
*r*compared to the (continuum) stellar radius*R*so that (a) absorption layers are essentially plane parallel (perpendicular to the line of sight*z*) and (b) the line of sight speed of absorbing matter is essentially the radial speed of stream matter away from the star. Though this can hardly apply to the H and other Balmer lines, according to Kaper et al. (1999), DACs in the UV are typically not detected until at least , and even higher for Main Sequence stars, so for these our approximations should be reasonable. - 3.
- We assume the time variation of the DAC to arise from the rotation of
the perturbation pattern through the line of sight. However we assume
that the rotation speed
,
where
is the radial physical speed of the stream matter, so
that the
*z*-component of does not significantly affect the observed Doppler shifts.

With the above assumptions we now get, in the Sobolev approximation,
that for a transition of oscillator strength *f*_{0} and rest wavelength ,
the optical depth at shift
is

where

In fact, however, we show in Sect. 4.3 that, rather
surprisingly, it is not actually necessary to make an assumption on .
Due to the separable/self-similar form of the dependence
of
on
,
and
it proves possible to
recover all three functions
,
*n* and
from
.
This means that
combined with the continuity equation contain more information than
just
and that we are able to use it to infer not only
but information on
and .

[In the numerical treatment of the DAC data modelling problem by de Jong (2000), the continuity equation is not considered but a form
is adopted and it is further assumed that
is known (in fact it is taken to be the same as the
mean wind speed). For specified
they could invert
Eq. (1) to get the radial density profile at each ,
viz

and so build up a picture of the stream structure. However, by specifying and , their approach ignores the steady state continuity equation which the stream material , and must satisfy and in fact doubly over-determines the problem since information on both and is present in . In other words, with specified, it may not be possible to find a satisfactory solution of Eq. (2) for from DAC data, or to satisfy the continuity equation, unless the adopted and happen to be in fact the true ones.

The most convenient way to express continuity is to link to the stream density
at some inner boundary
surface *r*=*R* where the flow speed
.
The original
azimuth
of the stream when at *r*=*R* is related to its
azimuth
when in the line of sight at distance *r* by
where
is determined from
so that

Here is the azimuth angle through which a parcel of stream matter has moved between leaving the surface

The continuity equation (see Appendix) then gives

where we have assumed the flow is 3-D. (If it were strictly 2-D then would be replaced by ). Thus, once we have determined and , we can derive everywhere using the continuity Eq. (4).

Before discussing the forward and inverse properties of the problem,
we introduce a set of dimensionless variables and parameters:

Then with and the important function defined by

the dimensionless optical depth equation becomes, using Eqs. (1), (4), and (5),

for dimensionless base density function

and the stream azimuth shift Eq. (3) becomes

The significance of the function derives from the fact that, when the wind density is axisymmetric ( ) we have

so that is the (time-independent) optical depth profile of the stellar line, in the absence of scattering from the volume of the wind (that is, it is the absorption component of the P Cygni profile).

Using Eq. (7) above we can now consider the DAC diagnostic
problem as determining as much as possible about the velocity laws
,
*W*(*x*) and the mass loss angular distribution function
from optical depth data
using steady state
continuity to reduce the number of degrees of freedom and so make the
problem determinate. We will address this from both the forward
predictive viewpoint (
)
and the inverse deductive one (
).

At this point we note a very important property of expression (7) which is the basis for our later inverse solution of
the problem but which also describes the limitation of the purely
kinematic model we are using. The time ()
evolution of the
optical depth line profile function
is a direct reflection
of the azimuthal distribution
of the surface mass loss
density subject only to a scaling factor
and a phase shift
wholly determined by the velocity laws
, *W*(*x*). With the
scaling factor removed the time profile
of *f* should look the same at all *w* apart from a phase shift. This
is a restrictive property of the kinematic model and is not satisfied
by -periodic functions
in general. It arises
because
is time independent and since the kinematic
model approximates
only. If
,
as in
dynamical CIR models where the density stream variation with feeds back on the
,
then in general
will
not have the
-scaled,
-phase-shifted invariance
property we are using here. So, as already noted, we are using a
kinematic approximation to the real situation. The extent to which
this approximates to dynamical models and to real DAC data is the
subject of a future paper but is briefly discussed in
Sect. 6.

We will mainly discuss the problem in terms of general functional
forms rather than assumed parametric ones though we will discuss the
forward problem in some particular parametric cases of
and for *W*(*x*) given by the parametric form

where would imply full corotation of a straight stream with the star, is the case usually considered of constant angular momentum (e.g., de Jong 2000; Hamann et al. 2001), and corresponds to rapid damping of the angular momentum with distance, stream matter moving in purely radial lines in the observer frame.

3 Properties of narrow streams

3.1 Predicted DAC properties for general and

By a narrow stream we mean one in which the dense outflow at the inner boundary is not very extended in azimuth so that the spread in consequent DAC the overall width of the absorption line.

This results in a narrow range of stream *x* at each
and so in
a narrow range ()
of *w*(*x*) in the line of sight at any given
time
(see Fig. 1). In the limiting case we can
describe this by
where
is the delta function and we arbitrarily adopt the
mass loss point as .
It is obvious physically and from
Eq. (1) that at any observer azimuth (i.e., time) the DAC will appear as a sharp feature in *f* at a single
Doppler shift *w* (Fig. 1). Although the stream density
pattern is time independent in the stellar frame, it is carried by
rotation across the line of sight as shown in Fig. 1
and so the Doppler shift changes with
and at a rate determined
by the stream geometry as well as by the physical flow speed
of the stream matter. We describe this in more detail in
Sect. 3.4.1. What the observer sees is an
acceleration due to the changing view angle of the density pattern and
we will use the terms "pattern speed
'' and "pattern
acceleration
'' for this (cf. Fullerton et al. 1997; Hamann
et al. 2001). To find the value *w* of this Doppler shift speed as a
function of "time''
we equate the observer direction
to
the angle
at which the dense stream matter passing
through the line of sight at that time has spatial speed *w*.

Since we have adopted
as the stream base point this yields,
using Eq. (3) and recalling that
is measured in
the corotating stellar frame,

Thus defines the pattern (or phase) speed with which the matter

Expression (12) is valid for any form of *W*,
and, as first noted by Hamann et al. (2001) for specific *W*,
,
reveals a surprising and important property of such kinematic DAC
models which appears to have received little mention hitherto though
the essential equations are contained in e.g., Fullerton
et al. (1997).
is a measure of time so
in Eq. (12) measures the time it takes the DAC produced by
the rotating pattern to accelerate (apparently) from
to
*w* - i.e., for the observer to rotate from
to the
azimuth where matter in the line of sight has speed *w*. This can be
compared with the "time''
it takes for a single actual
element of stream matter moving with the same radial flow speed law
,
to accelerate from *w*_{0} to *w*, namely

In the terminology of Prinja & Howarth (1988) this would be termed the behaviour of a "puff''.

Comparison of Eqs. (12) and (13) immediately shows
that for any rotation law satisfying
(which
is the case for all plausible )

(this follows because , so the

This generalises the result of Hamann et al. (2001) and is surprising
in two ways. First the DAC seen from a corotating structure in which
the matter follows a radial flow speed law
takes
*less* time to reach (i.e., accelerates *faster* up to) any
observed Doppler speed *w* than would an absorption feature produced
by a transient puff of material following the same flow law
.
Second, the enhancement of the apparent acceleration of the
DAC from the corotating stream pattern over that for the puff is
independent of the absolute value
of the rotation rate
(though it does depend on the relative variation *W*(*x*) of with *x*). This can also be expressed in terms of *dimensionless*
accelerations

Evaluating for and given by Eqs. (12) and (13) we obtain the actual,

(which is trivial), and the

so that

(for

These results are rather counter-intuitive. It is tempting to think
that DACs are observed to accelerate more slowly than the mean wind
because the long rotation period of the star (compared with wind flow
time
)
carries the absorbing stream across the line of
sight only slowly. In fact Eq. (18) shows that
precisely the opposite is true. As time passes any rotation brings
into the line of sight stream matter which left the star progressively
earlier. This *increases* the rate at which higher Doppler speeds
are seen above the rate due to material motion alone (which is the
rate exhibited by a puff of the same material speed) - that is,
is the phase acceleration of a pattern (cf. Hamann
et al. 2001). This is very important because it means that, at least
for narrow streams (but see also Sect. 4.1), for the
slow observed acceleration
of DACs (compared to the mean
wind acceleration
)
to be attributed to a corotating
density pattern the actual flow acceleration
of the matter
creating that pattern must be *lower* than
since
the apparent
is in fact higher than the physical
acceleration
of the stream matter. That is, for the
observed DAC (pattern) acceleration to be slow compared with the mean
wind, the stream matter acceleration must be *very slow* compared
with the wind. For example, Hamann et al. (2001), addressing the
forward problem, added a constant inward speed plateau to the general
outflow to represent empirically the presence of an Abbott wave (Abbott
1980). In the dynamical modelling results of Cranmer & Owocki
(1996), denser material is accelerated more slowly because of its
greater inertia per unit volume. This lends motivation to our aim of
providing a means of inferring the true flow speed of dense stream
matter direct from recurrent DAC data. The result may also provide a
partial explanation for why de Jong (2000) found difficulty in fitting
data with a parametric stream density model
since they
assumed a flow speed
equal to that of the mean
wind. Such a flow speed model should, from the above results, predict
apparent DAC (pattern) accelerations *higher* than those of the
mean wind and so could never properly fit the observed slow
accelerations. (Recall also that de Jong 2000 did not ensure that
their ,
*v*(*r*), *W*(*r*) satisfied the continuity equation.)

The second surprise, that
is independent of the absolute
rotation rate *S*, can be understood by the fact that although higher
sweeps the dense matter pattern across the line of sight
faster, the pattern itself is more curved for higher .
The
effects of higher rate and of greater stream curvature cancel out. It
is also instructive to note the two limiting cases of
.
For
(
)
we get
because all stream elements
observed are moving directly toward the observer, and for
,
(
)
because the density stream is straight and
radial and all points (*w*) along it are swept into the line of sight
at the same moment.

The finding that the ratio of the observed apparent stream pattern
acceleration
compared with the true matter acceleration
should be independent of
does not contradict the data
(Kaper et al. 1999) which suggest a correlation between observed
acceleration and .
This is because (see
Fig. 5, Sect. 4.1) the translation from
data on
to
involves the value of *S*. In
addition, only a wide-stream analysis is adequate fully to describe the
situation, since the acceleration of the peak of a DAC from a
*wide* stream depends on the density function
which
may be affected by the rotation rate - see
Sect. 4.2.2.

3.2 Explicit expressions for -law parametric form of

Though we are mainly seeking to address the DAC problem
non-parametrically, explicit expressions for some of the results in
Sect. 3.1 for particular forms of *w*(*x*) are useful
for illustrating properties of the kinematic DAC model such as the
dependence of streak and stream line shape on rotation and
acceleration parameters (e.g., , ).

Here, for reference, we restate some results of Hamann et al. (2001) for the
-law (with *w*_{0}=0)

For this -law and for form Eq. (11) of the rotation

( ). In general Eq. (20) has to be evaluated numerically, which is rather inconvenient, especially if (cf. below) one wants to invert to get , but the integration is analytic for some specific cases (so long as we approximate

while the time at which a puff moving radially with speed would reach speed

For we get

and

(a logarithmically divergent term has been neglected in Eq. (24): this simply indicates that the velocity law is singular at the stellar surface; fluid elements take an infinitely long time to accelerate to a finite velocity).

Note that for arbitrary ,
with
(no
stream angular momentum) the second term in the integrand in
Eq. (20) vanishes and stream material moves purely
radially, rotation serving only to "time-tag'' the part of the stream
in the observer line of sight. The observed stream Doppler speed is
then just the actual matter speed and
for all *w*.

Secondly, for arbitrary ,
with
(rigid stream
corotation), stream matter corotates rigidly with the star and all
points along it are seen simultaneously, corresponding to infinite
apparent acceleration or
for all *w*.

3.3 A new -law parametrisation of

Though we do not use it explicitly in the present paper we suggest
here a new parametric form of velocity law which should prove useful
in future studies of hot-star winds, particularly from an inferential
point of view, as it makes it easier to obtain analytic results. This
"alpha-law'' parametric form for
,
in contrast
to the -law, allows exact analytic integration to give
and
for any value of a
continuously variable acceleration parameter
and for any
finite *w*_{0}, namely

Note that for this form it is essential to retain non-zero

Figure 2:
Comparison of approximate best fit -law
parametric forms of w(x) with -law w(x)(Eqs. (19) and (25)) for w_{0}=0.1. Solid lines
represent -law and dotted lines represent -law.
Parameters
and
are indicated within each panel. |

3.3.1 Explicit expressions for -law

Using Eq. (25) we obtain the following expressions

(27) |

(28) |

Then for the times to reach speed

and

= | |||

(30) |

where is given by Eq. (26).

3.4 Inversion to find stream flow speed from observed DAC pattern speed for a narrow stream

We have seen in Sect. 3.1 that the actual stream
matter flow acceleration
in a corotating density
pattern must be slower than the apparent (pattern) acceleration
(and much slower than typical wind acceleration
)
in order to match typical narrow DAC observations -
cf. results in Hamann et al. (2001) for -laws. More generally it
is of interest to see whether it is possible to infer the actual flow
speed
from sufficiently good data on the apparent DAC
acceleration. We do so here assuming *W*(*x*) is known. What we observe
is a pattern speed
as a function of time
.
What we really want is the true matter flow speed law

3.4.1 Forward problem

The forward problem is to determine how the observed line-profile
variations are determined by the physical properties of the stream,
i.e., to find the observed
given the wind
law
.
This is illustrated in Fig. 3.

Figure 3:
Obtaining the narrow-stream DAC dynamic
spectrum
for given v(r) and :
i) v and fix the shape of the physical CDR spiral
in the corotating frame
(see text); ii) converting radius to radial velocity gives the "velocity
spiral''
;
iii) unwrapping
gives
,
showing how the position of the absorption feature
varies as a function of rotational phase (the three lines correspond to
successive windings of the spiral - see Fig. 1).
Note that for the narrow-stream inversion we only use the position of
the absorption feature in the spectrum,
,
not the
actual value of the optical depth.
Figures are for a
law with constant angular momentum (). |

The wind velocity and rotation laws and

( ), which can be integrated to give (see Eq. (3)). This is monotonic, by assumption (), and so can be inverted (in general numerically) to obtain the spiral law . Once we have the spiral law for the streamline of the CDR we can use the (monotonic) velocity law to describe the spiral in terms of the variation of radial velocity with , as shown in Fig. 3:

(32) |

as was done for -laws in Hamann et al. (2001). This is the actual doppler velocity that is observed when matter at is in front of the star, so unwrapping the velocity spiral directly gives the observed dynamic spectrum, as shown Fig. 3.

3.4.2 Narrow-stream inversion given

The inverse problem is to find
given observations of the
time-dependent Doppler shift in the form of the monotonic "velocity
spiral'' function

Here we present the solution to this inverse problem when we assume that we know the rotation law . In this case, the observed line profile variations contain enough information to determine without the need to use the continuity equation. This fact has important consequences for the significance of the inversion, as we discuss below.

In order to recover
we need to determine the spatial
spiral law
(see Fig. 3), because then we
will know, at any ,
the distance of the absorbing material from
the star *and* its radial velocity
,
which immediately gives us the wind velocity law. In other words we
need to translate
from the observed time variable
to
the real spatial variable
defining the distance at which
the absorbing matter lies when it has speed *w*.

This translation is most easily achieved as follows. Imagine we did not
have the CIR (or CDR) model of the narrow DAC feature, but instead
thought that the DAC results from a spherical shell of material emitted
from the star at some instant (a "puff''). Then at each time the
we observe would be the radial velocity of this shell as it
accelerates through the wind. Assuming that the shell is ejected from
the surface, *x*=1, we can integrate up
to obtain the
actual spatial position
of the shell at each time:

(where, as usual, we have chosen the time unit to be related to the rotation period of the star, so that ; the 1/

This is then the wind law that

However, we really believe that the observed DAC results from a CIR or
CDR and, whereas the puff is a time-dependent, axisymmetric
disturbance in the wind density, the CDR spiral is a stationary (in
the corotating frame), nonaxisymmetric density perturbation. As a
result, the observed DAC feature does not directly trace the actual
motion of matter through the wind, but rather reflects the shape of the
spiral pattern. As the star rotates (i.e., as
increases) we see
fluid elements at parts of the spiral further out in the wind
(
increases), where the wind velocity
is larger:
effectively, the velocity of the actual *material* doesn't increase
as fast as that of the spiral itself (see the discussion of DAC
acceleration in Sect. 3.1, after
Eq. (18)). We must somehow account for the fact that we
do not see the same (or, at least, equivalent) fluid elements at
different times.

In order to understand the relationship between the material velocity
and the "spiral'' velocity we must stop identifying time with ,
because the difference between the puff and CDR interpretations
derives precisely from the difference between the way material moves
in time and the spiral moves in .
If at time *t* the absorbing
material in the spiral is at
and
(and so has velocity
), after a short time *dt*this *fluid element* will have moved through a distance

exactly as for the puff model. However, we will now be seeing absorption from another fluid element in a different part of the spiral, at , with position . So the actual change in radius of the spiral is

Now, we want to determine , which is the physical spiral radius, but we cannot as it stands because from Eq. (37) depends on , which we don't know. We do, however, know in Eq. (36), because we obtained directly from the observed in Eq. (34), and we can relate to through

Since this only depends on the given rotation law

Inserting this into Eq. (35) gives the solution we seek:

To illustrate this, suppose we observe a Doppler shift variation of the form which would arise if we were seeing absorption by a puff of matter moving toward us with a velocity law. From Eq. (22) this would produce

We have computed from Eq. (41) for

or

In Fig. 4 we show the resulting actual matter speed required for a rotating stream to give the same observed as that from a law puff motion. In line with our earlier discussion the resulting has a slower acceleration than the law, looking more like a law. These results show just how important it is to include the effect of pattern rotation when interpreting apparent DAC accelerations.

It will pay to think a little more deeply about the narrow-stream inversion
procedure we have just set out. At no stage did we make use of the
*value* of the optical depth along the streamline, only the
doppler velocity at which the absorption occurs. It is for this reason
that the continuity equation is not required (and, indeed, cannot be
used as a constraint). Furthermore, we required knowledge only of
the fluid flow along the single CDR streamline of the narrow DAC, not
of any neighbouring streamlines. Most importantly, we did not directly
use the assumption that the wind velocity is axisymmetric: the velocity
law that we derive *does not depend on our model assumption* of
Sect. 2. The only sense in which we use the
assumption of axisymmetry is to allow us to apply our inferred
velocity-radius relation Eq. (40) to the entire wind
(i.e., to every azimuth). Without the assumption of an axisymmetric
wind velocity we can only say that *along this particular
streamline* velocity varies with radius according to
Eq. (40), but on other streamlines the velocity-radius
relation may be different: for streamlines originating from the
surface at azimuth
we have a velocity-radius
relation
.
Were we to observe several discrete
narrow DACs simultaneously we could use the inversion procedure of
this section to infer
for each of them
(i.e., for each ), *potentially recovering a
non-axisymmetric wind velocity law*. We discuss the implications of
this in relation to wide-stream DAC inversions in Sect. 4.2.

3.4.3 Narrow stream inversion using the continuity equation

Can we extract more information from narrow-stream DAC observations by
making use of the optical depth of the absorption feature and how it
varies with phase, possibly allowing
to be inferred rather
than assumed? It turns out that we can, but at a cost. Whereas, as we
have just discussed, the narrow-stream inversion procedure of
Sect. 3.4.2 requires no knowledge of streamlines in
the vicinity of the CDR stream, to interpret optical depth information
requires the continuity equation and therefore knowledge of the
variation of the wind velocity around the CDR stream, since the continuity
equation relates the divergence of neighbouring streamlines to the
change in density along them, and thus, through
Eq. (1) to the change in optical depth of the
corresponding DAC. It follows, therefore, that we *must* employ
our model assumption on the axisymmetry of the velocity law (or some
alternative) to take advantage of narrow-stream optical depth variations.

If we consider a narrow DAC generated by a -function surface
density function
,
then from
Eq. (7) the dimensionless optical depth function becomes

and we can think of the DAC either as a -function in velocity at any time , or as a -function in phase at any velocity.

The inversion procedure of Sect. 3.4.2 was based on the observed DAC velocity as a function of phase , which is precisely the inverse function of in Eq. (44). Can we use optical depth measurements to determine instead the function in Eq. (44), and then use this to further constrain the parameters of the wind?

As we mentioned in Sect. 2, represents the line profile of the absorption component of the wind in the absence of DACs, and we show in Sect. 4.3 that it can be used to determine the wind law without knowledge of

To determine
from observations, given the dynamical
spectrum from Eq. (44), we must integrate over the
spectrum to obtain the amplitude *A* of the -function DAC feature.
As we discuss in Sect. 4.2.2, it is advantageous to
think of the variation of optical depth with phase at fixed *w*,
rather than in terms of the spectrum at fixed phase. This is seen
clearly here if we integrate over
in
Eq. (44) to obtain
.
Integrating over
gives

which is what we want, whereas integrating over

= | |||

= | |||

= | (46) |

So from the DAC amplitude of Eq. (45) we can infer the velocity law of the wind as in Sect. 4.3, and can compare this with the law inferred in Sect. 3.4.2 to examine the consistency of our choice of rotation law, and ultimately to infer . We will not pursue this further here, since it is just a limiting case of the wide-stream DAC inversion that we present in Sect. 4.3.

4 Wide streams and the general DAC inversion problem

4.1 Acceleration of DAC peak from a wide stream

In Sect. 3.1 we discussed the apparent Doppler
acceleration of the narrow DAC feature arising from a stream which is
narrow in
(and therefore in *w*). In reality streams do have
substantial widths, as evidenced by the finite Doppler width of DACs
and the fact (Kaper et al. 1999) that they must have sufficient spatial
extent to cover a large enough fraction of the stellar disk for DAC
absorption to be important. Thus narrow stream analysis must be
treated with caution, as indeed must analyses (e.g., Kaper et al. 1999)
that make restrictive parametric assumptions (e.g., Gaussian) on the
shape of the profile of either the DAC *f*(*w*) or of the stream density
.
For wide streams there is no unique
but
rather a profile
which depends (Eq. (7))
not only on
but also on *W*(*x*) and
.
For these one has to discuss the acceleration of a feature (or of the
mean over some *w* interval) - for example of the value
of the Doppler speed at which
maximises. In general the
apparent (pattern) acceleration *a*_{*} of
may depend on the
mass loss flux profile function
as well as on
and *W*. Here we examine the acceleration of
for general
,
and *W*(*x*)to see how much
affects our earlier narrow stream result
that the apparent DAC pattern acceleration from a narrow stream exceeds
that from an absorbing puff moving radially with the same matter speed.

We will denote by
the dimensionless Doppler "acceleration''
in the "time'' coordinate
of the DAC peak. For a very narrow stream
where there is a unique Doppler speed this will just be the pattern
acceleration
we derived earlier, viz.

For a broad stream

Defining

is given by

Differentiating Eq. (50) for

(51) |

where denotes . Solving for

where is given by Eq. (48). Finally we note from Eq. (47) that where is the apparent (pattern) "acceleration'' for a narrow stream at Doppler speed

where

The first property we note about this expression is that if is constant then and for all

For the case constant (anything other than a velocity law), whether or , i.e., whether the peak of a DAC from a wide stream accelerates faster or slower than that from a narrow stream, depends on whether or in Eq. (53).

We see that, in general,
depends on
as well as on
,
*W*(*x*), so that the acceleration *a*_{*} of a spectral
peak in the DAC optical depth *f* from a broad stream is *not*
the same as that (
)
from a narrow one. This is because the
shape of *F*_{0} causes the spectral shape of *f*, including the
behavior of its peak, to change with time. Now in Eq. (54)
is always >0 while the sign of
describes the concavity of .
Noting that

we also see that, since anywhere in the neighbourhood of a peak in , so the sign of , and hence of , is opposite to the sign of .

To proceed further we need to adopt definite forms for
and
for
.
Taking a -law *w*(*x*) as an example we have

while is given by Eqs. (12) and (17). Thus we find

the sign of which is simply fixed by the sign of and is always <0 for all . By Eq. (54) this means that if is concave down at the relevant to the peak at

To see how large this effect is we consider for convenience the particular
form of *F*_{0}

This resembles a Maxwellian function, is continuous across , and has an asymmetric peak at a value, and with a sharpness, which depend on constant dimensionless parameters

Inserting this and from Eq. (57) in Eq. (53), we obtain an explicit expression for as a function of

Thus using the peak optical depth point *w*_{*} as if it were the unique
for a narrow stream is a good approximation and so can be
used to deduce
from
as described in
Sect. 3.4. For streams with *F*_{0} of considerable
width in ,
corresponding to those DACs which have rather broad in *w* at small *w* (cf. figures of data in Massa
et al. 1995 and of simulations in Cranmer & Owocki 1996) the
results can be very different and quite complex since the evolution of
is strongly influenced by the stream density profile
function
.
(Note that the DAC from a stream of any width
in
always becomes narrow in *w* as
since all
the material eventually reaches terminal speed.) Results are shown for
various
values in Fig. 5 for *A*=0.1, *B*=10 which
correspond to the fairly extreme case of a stream with a half width
of about 0.25 in .

Figure 5:
Ratio of apparent accelerations of a
wide stream DAC peak (a_{*}(w)) to a narrow stream DAC (
)
for various -laws for the input mass loss function with constants
A, B specified in the text |

We see that in such cases the wide stream peak acceleration *a*_{*} can
be much less than the narrow stream result
especially for
smaller values of *w*, and particularly for
close to but
greater than 0.5 which is thus a singular case. This means that, for
DACs which are wide in *w* at any stage in their development,
estimating
from, or even just fitting a value of
to, data by applying narrow stream results to the acceleration
of the DAC peak can be very misleading. The essential point here is
that recurrent DAC data
contain much more information than
on just
but also on
and *W*(*x*). To
utilise this information content fully we have to treat the inverse
problem, using both *w* and distributions of .
We
show how this can be done in the next section.

4.2 Inversion of wide-stream for the wind characteristics

In Sect. 3.4.2 we addressed the problem of
recovering the *w*(*x*) law from the observed
pattern
in the dynamic spectrum for a narrow DAC (representing matter flowing
along a single streamline, from a single point on the stellar surface),
and we showed that if we assume a rotation law (constant angular momentum,
,
say) we can recover the spiral pattern
and velocity law along that streamline without any
consideration of neighbouring streamlines (i.e., without taking advantage
of mass continuity or knowing
). Since a wide-stream DAC can be
thought of as a collection of narrow streams from many (all) points on
the stellar surface, surely we can apply the narrow-stream inversion
procedure for each streamline, using the
from
each
on the surface to obtain the spatial spiral law
for that streamline and the velocity-radius relation
along each spiral, thus recovering the
(in general non-axisymmetric) wind law
*without using our model assumption of an axisymmetric wind
velocity* (Sect. 2). In fact, we could do better
even than this, because, if we obtain the velocity-radius relation
along every streamline from the surface we have the wind velocity
*everywhere in the wind*; we then know how it varies in the
vicinity of every streamline and can calculate the derivatives
necessary to apply the continuity equation and thus make use of
the optical depth variations along streamlines as we discussed in
Sect. 3.4.3. These variations would only be
consistent with the *observed*
if the rotation
law
that we used to find the streamlines was correct,
allowing us, in principle, to infer
as well as the
velocity law, thus giving all required wind parameters for a general
non-axisymmetric wind.

Why don't we apply this procedure to the wide-stream DAC inversion problem? The answer is obvious: dynamic spectra do not come with the "velocity spirals'' drawn on. It may be possible in general to draw many different spiral patterns on top of the dynamic spectrum of a wide DAC that give consistent inversions for , and it is certainly not obvious how, given just the dynamic spectrum, such a set of streamline spirals could be unambiguously chosen. As a result, the inference of a general azimuthally varying velocity law from recurrent DAC data is not a simple matter. We sidestep this issue here by introducing our model assumption of Sect. 2 (namely axial symmetry) to reduce the wide-stream inverse problem effectively to the narrow stream procedure (in a certain sense), but with the inclusion of mass continuity (Sect. 3.4.3). In fact, with our model assumption allowing us to make use of the continuity equation, wide-stream inference closely parallels the narrow-stream problem with continuity of Sects. 3.4.2 and 3.4.3. As we will show, with this simple model assumption we are able to recover all characteristics of the wind velocity and CDR density.

4.2.1 Forward problem

The forward problem for wide-stream DACs involves the calculation
of the dynamical spectrum
from
,
and *W*(*x*):

- 1.
- From
and
*W*(*x*) find the phase-shift function , which is just the inverse function of the "velocity spiral'' found for the narrow-stream forward problem in Sect. 3.4.1; - 2.
- Calculate from according to Eq. (6);
- 3.
- For each
*w*calculate in Eq. (7) by phase-shifting through and multiplying by .

4.2.2 Inversion of to find for given

In Sect. 3.4 we showed how the actual stream
matter speed
could be derived from the apparent DAC speed for
a narrow stream. For a wide stream one might think of a similar
method, using the apparent speed of DAC peak optical depth (i.e., the
motion in
of the *w*=*w*_{*} at which
).
However a better approach here is actually to consider rather the
variation with *w* of the time
at which the optical
depth at *w* maximises, i.e., the
at which
.
In fact, as a moment's thought shows, if we can identify any
feature in the surface density profile
,
such at its
peak at
,
say, and follow it as it flows out through
the wind then we are precisely determining the velocity spiral for
the single streamline emanating from the point
.
We can then apply the narrow-stream inversion procedure of
Sect. 3.4.2 to this spiral to infer the wind velocity
law (strictly, to infer
,
but this
is universal, i.e., independent of ,
by our model assumption).
Owing to the axisymmetry of the streamlines (Sect. 2)
the way to trace the movement of the peak is to examine the variation
of the optical depth with phase at each *w*, since, from Eq. (7),
at fixed *w* the variation of *f* with
is just proportional to *F*_{0} phase-shifted by
.

The position of the peak can be found from

(cf. Eq. (7)). We know the peak occurs at , so the solution of (60) is simply

The result is thus independent of the form of

An alternative way to see explicitly how
can be derived from Eq. (61) is to
note that a monotonic function *Z*(*w*) can be constructed from the
data on
and related to *W*,
by Eq. (72), viz.

(62) |

For a known , the right side is a known monotonic function of

(63) |

with solution

(64) |

4.3 General inversion of to find

We now show that it is not necessary to know *W* and that it is
actually possible to recover all three functions *w*(*x*), *W*(*x*)and
from data on ,
via the basic relationship

This somewhat surprising result holds essentially because is periodic and because and

We pointed out in Sect. 3.4.3 that use of quantitative optical depth information boils down ultimately to the determination of the "line profile function'' , which contains information about mass continuity contraints (Sect. 2). Here we show that may easily be found from the observed dynamic spectrum, and can be used to infer without considering the rotation law at all.

Denoting by
the mean value of *f* at any fixed *w* over
any -range of
we have, from Eq. (7),

with

(67) |

so that within a factor we have

We have now found directly from the optical depth data by averaging

From Eq. (68) it follows that

Putting

so we can determine the inverse solution for from namely

from which, by inversion (since is monotonic) the matter velocity law can be recovered. Note that Eq. (71) only involves integrals of averages of the data funcion

Next we show that we can use the combination of the streamline-based
and
-based determinations of
(Eqs. (40) and (71)) to find the rotation
law .
From Eq. (9) (see also
Sect. 3.4.2) we can
see that
depends on *W*(*x*) and
.
Since
we have just determined
from
independently
of *W*(*x*) in Eq. (71) we can use this to determine *W*(*x*).
Differentiating
with respect to *w* gives

We know the left-hand side and from the data, and so we can rearrange Eq. (72) for

This solution for

Finally, having found
and
from the
data we can, at each *w*, rescale the optical depth by
and remove the phase shift
that results from the
spiral shape to obtain the mass loss flux distribution function at the
inner boundary, *F*_{0}, from :

On the face of it,

5 Numerical inversion results for artificial data sets

In Sect. 4.3 we have shown that, in principle,
it is possible in the context of our kinematic wide-stream model
to recover
,
and *W*(*x*) from DAC
optical depth data .
We now investigate the extent to
which it is actually possible to use this procedure to infer
numerically the properties of a stellar wind from dynamical
spectra, both in the case of perfect data and in the realistic
case where there are errors on the observed dynamical spectrum.

To this end, we have tested the inversion procedure of Sect. 4.3 for a variety of artificial datasets, including -laws with , velocity laws with a plateau (see below), rotation laws with different , and various surface density profiles (wide and narrow Gaussians and sinusoidal modulations). In addition we have examined the effects on the inferred quantities of adding noise to the dynamical spectrum and also of "smearing'' in velocity of the spectrum, simulating the influence of thermal and turbulent broadening intrinsic to the source.

Here we present the results of several representative inversions, choosing those that are most relevant to hot-star winds. Inversions for other wind parameters have comparable stability and accuracy.

1 | |||

Wide plateau | 0.03 | 0.07 | 0.12 |

Narrow plateau | 0.004 | 0.007 | 0.010 |

We concentrate on examining the dependence of DAC inversions on the
characteristics of the underlying velocity law, and on the quality of
the DAC data; the following rotation law and surface density
profiles were used for all of the inversions shown in
Figs. 6-18^{}:

- We take a constant angular momentum rotation law
*W*(*x*)=1/*x*^{2}(i.e., in Eq. (11)). - We adopt a Gaussian surface density profile

with (dimensionless) amplitude*D*=0.2 and width , centred on . (Strictly speaking, we consider given by folding Eq. (75) with period , adding the contributions in each period together to ensure continuity across .)

where is simply the chosen width of the plateau region, and the amplitude

(see the velocity-law panels of Figs. 6-8b, c). In all of the inversions shown we centre the plateau on

Figure 8:
[,
noise free] As in Fig. 6,
but for an underlying
velocity law. |

Figure 11:
[,
10% errors] As in Fig. 6,
but for an underlying
velocity law and 10% errors. |

To summarise, three types of velocity laws are shown in the inversions of Figs. 6-14:

- 1.
- pure -law (Eq. (19));
- 2.
- -law with a wide plateau (
*x*_{0}=5, ); - 3.
- -law with a narrow plateau (
*x*_{0}=5, ).

Figure 14:
[,
10% errors, 5% smearing] As in
Fig. 6, but for an underlying
velocity law,
10% errors and "velocity blurring'' at 5% of the terminal velocity. |

For the rotation law, surface density profile, and velocity
laws set out above we calculated the resulting dynamical spectrum
at *N*_{v}=300 uniformly-spaced velocity values between *v*=0 and
and
times (i.e., rotational
phases) throughout one rotation period by determining
and
from
Eqs. (6) and (9) respectively and then using
Eq. (7) to obtain .
(Although it is not
necessary to use 300 velocity points to obtain acceptable resolution
in general, the strong plateaux that we consider here give rise
to extremely sharp features in the line profile, and failure to resolve
these leads to significant truncation error in the calculation of
the dynamical spectrum, and a consequent bias in the inversion.)
These dynamical spectra were then inverted using the procedure described
in Sect. 4.3 to infer
,
*W*(*x*),
(as well as
and
).

Figure 17:
[]
As in
Fig. 15, but for an underlying
velocity law. |

Figure 18:
Inferred wind rotation law, W(x) for the dynamical spectrum
shown in the left panel of Fig. 6a, that is, for a pure
velocity law, with 10% errors added to the dynamical
spectrum. |

Figures 6, 7 and 8 show inversions for , and 1, respectively when no noise was added to the dynamical spectrum: the left panels show the dynamical spectra (data to be inverted), the right panels the inferred , , and (along with their "true'' input values, which are virtually indistinguishable). Subfigures a), b) and c) are the inversions for each of the three velocity law types (-law, wide plateau and narrow plateau, respectively.

Figures 9, 10 and 11 show inversions of exactly the same models as Figs. 6-8, but with 10% noise added to the dynamical spectrum.

Finally, we illustrate in Fig. 14 an example of the effect of smearing the dynamical spectrum in velocity by convolving the error-free spectrum with a Gaussian blur with width 5% of the terminal velocity to artificially simulate thermal and turbulent broadening of the absorption from material at each radius. The inversions in Fig. 14 are for an underlying velocity law (results for each of the three velocity law types are shown, as usual). 10% errors are added to the smeared dynamical spectra. It is clear from the inversions that it is perfectly possible to infer the velocity law and surface density profile using the inversion procedure.

We have not yet discussed the rotation inference, however.
We show in Figs. 15-17 the rotation
law inferred for the *noise-free* inversions of
Figs. 6-8. There are errors (mainly
resulting from truncation error in the calculation of the model
dynamical spectrum), but the quality of the inversion is good.
Unfortunately, to calculate *W*(*x*) requires differentiation of quantities
derived from the data. This amplifies the errors significantly, and
precludes accurate inference of *W*(*x*) in the presence of errors
as shown in Fig. 18. Looking at this more positively,
though, it does indicate that the details of the wind's rotation velocity
do not strongly influence the observed line profiles (physically this is
because the angular velocity of fluid elements tends to zero quite quickly
in general, so they follow the same path once they are beyond a few
stellar radii), and uncertainty in
does not prevent accurate
inference of the other wind parameters. This means that it will not
generally be neccessary to account accurately for the rotation law of
the wind to make useful inferences about its structure.

6 Discussion and application to real data

The analysis above sheds much light on inferential wind stream diagnostics within the context of the kinematic model, and so is a large step forward on the parametric forward fitting approach mainly used before in analysing data (Prinja & Howarth 1988; Fullerton et al. 1997; Owocki et al. 1995) using the kinematic description. It is also a useful approach in bringing out the crucial importance of finite stream width and DAC feature profile in interpreting acceleration of features and in obtaining the fullest information from data on stream structure from DAC data which goes well beyond the velocity of feature peaks discussed in Hamann et al. (2001). However, to take the model method forward it will be important to consider how well they match up to real data and to dynamical models (Cranmer & Owocki 1996). These are the subjects of ongoing work and future papers (Krticka et al. 2003) but we consider them briefly here.

Recurrent hot star DAC features can be most clearly seen in
UV resonance lines, such as SiIV and NV, though also present in
(de Jong 2000; de Jong et al. 2001), reflecting the presence
of structure quite close to the stellar surface. There are a number of
such datasets available from the IUE SWP instrument and we have made
first attempts to apply our technique to the SiIV spectra of HD64760
(Massa et al. 1995). Results proved confusing mainly because the data
contain distinct components (cf. Hamann et al. 2001). In particular
there are strong slow DAC features lasting several rotation periods
and crossing the weaker modulations at high *w* corresponding to the
CIR/CDRs which we have been discussing. Direct application of our
solution to the full data set fails because the strong slow feature
in no way conforms to the rotationally periodic character typical of
the data features we set out to model. To progress,
further work will be needed to try to isolate the two components and
then apply our method to the rotationally periodic features only.
As emphasised by Fullerton et al. (1997) exploration of data sets
for the presence of several effects can be facilitated by phase
binning all the data over a single period when effects not included
in the model may show up as systematic residuals.

Another approach we are developing (Krticka et al. 2003) is to test
the method against simulated data generated by dynamical models
(Cranmer & Owocki 1996) in which the dense wind streams are truly
rotationally recurrent features. We do not expect to get wholly
accurate recovery of
since, as discussed
earlier, the dynamical models, with the density feeding back on the
veocity law though the radiative driver, produce structures in which
and not just
.
Such features will not fully satisfy the assumption
of kinematic treatments
used here and by others. The issue is to test how well the
kinematically based inversion method we have developed enables
*approximate* recovery of the density and velocity structure
of the CIRs in the dynamical models. If reasonable first
approximations to the forms
of the dynamical
models are recovered then we will be able to use the method on real
data to provide first approximation inputs to dynamical simulations
of the wind perturbation giving rise to the radiatively driven CIR.
This will then enable dynamical modelling of real data sets to
proceed faster than trial and error input of base perturbations
to the dynamical code.

The other approximation which may limit the applicability of our method to real data is the "point-star'' approximation. However, for a law speeds in the usefully observable range correspond to which should be reasonably approximated by the plane assumption , though again this can be tested by comparison of results with those of simulations incorporating curvature and finite star effects, to be undertaken in future work.

We wish to acknowledge the financial support of UK PPARC Research and Visitor Grants as well as University of Amsterdam Visitor funds. This paper has benefitted from numerous discussions with A.W. Fullerton, T. Hartquist, R. Prinja and J.P. Cassinelli. We are grateful for the comments of the referee, D. Massa.

Appendix: The continuity equation

The continuity equation was used in Sect. 2 to give Eq. (4). This can be proved in two ways - cf. Bjorkman (1992) and Bjorkman & Cassinelli (1993).

- 1.
- Consider equatorial streamlines which are equally spaced in
at the stellar surface. The streamline shape is given by

(78)

which is independent of if and are. Thus the streamlines are equally spaced in at any*r*. So the material flowing radially through an arc at the equator later flows radially through arc length at*r*(but displaced in by amount due to the streamline curvature - Eq. (3)). If the surface density is perpendicular to the equator then steady flow requires that

(79)

so that

(80)

But in 3-D flow near the equator , so

(81)

This simply means that the radial mass flow must be constant across all corresponding azimuthal elements. - 2.
- More rigorously: the 3-D continuity equation at
with
is

Consider the variation of the quantity

(83)

On a streamline this is, using (82),

(84)

and, again using the continuity Eq. (82), this is

= = (85)

which is zero if and are independent of . Thus is constant along streamlines. Equation (4) is just equivalent to

*X*(*r*)=*X*(*R*) on any streamline.

- Abbott, D. C. 1980, ApJ, 242, 1183 NASA ADS
- Bjorkman, J. E. 1992, The effects of rotation on the winds from hot stars, Ph.D. Thesis, U. Wisconsin
- Bjorkman, J. E., & Cassinelli, J. P. 1993, ApJ, 409, 429 NASA ADS
- Brown, J. C., Richardson, L. L., Ignace, R., & Cassinelli, J. P. 1997, A&A, 325, 677 NASA ADS
- Craig, I. J. D., & Brown, J. C. 1986, Inverse Problems in Astronomy (Adam Hilger Ltd)
- Cranmer, S. R., & Owocki, S. P. 1996, ApJ, 462, 469 NASA ADS
- de Jong, J. A. 2000, Ph.D. Thesis, University of Amsterdam
- de Jong, J. A., Henrichs, H. F., Kaper, L., et al. 2001, A&A, 368, 601 NASA ADS
- Feldmeier, A., & Shlosman, I. 2002, ApJ, 564, 385 NASA ADS
- Fullerton, A. W., Massa, D. L., Prinja, R. K., Owocki, S. P., & Cranmer, S. R. 1997, A&A, 327, 699 NASA ADS
- Hamann, W.-R., Brown, J. C., Feldmeier, A., & Oskinova, L. M. 2001, A&A, 378, 946 NASA ADS
- Henrichs, H. F., Kaper, L., & Nichols, J. S. 1994, A&A, 285, 565 NASA ADS
- Kaper, L., Henrichs, H. F., Nichols, J. S., & Telting, J. H. 1999, A&A, 344, 231 NASA ADS
- Kaper, L. 2000, in Thermal and Ionization Aspects of Flows from Hot Stars, ed. H. Lamers, & A. Sapar, ASP Conf. Ser., 204, 3
- Krticka, J., Barrett, R. K., Owocki, S., & Brown, J. C. 2003, A&A, submitted
- Massa, D., Prinja, R. K., & Fullerton, A. W. 1995, ApJ, 452, 842 NASA ADS
- Massa, D., Fullerton, A. W., Sonneborn, G., & Hutchings, J. B. 2003, ApJ, 586, 996 NASA ADS
- Mullan, D. J. 1984, ApJ, 283, 303 NASA ADS
- Owocki, S. P., Cranmer, S. R., & Fullerton, A. W. 1995, ApJ, 453, L37 NASA ADS
- Prinja, R. K., & Howarth, I. D. 1988, MNRAS, 233, 123 NASA ADS
- Prinja, R. K., Massa, D., & Fullerton, A. W. 2002, A&A, 388, 587 NASA ADS

8 Online Material

Figure 7:
[
,
noise free] As in Fig. 6,
but for an underlying
velocity law. |

Figure 9:
[
,
10% errors] As in Fig. 6,
but with 10% errors added to the dynamical spectrum. |

Figure 10:
[
,
10% errors] As in Fig. 6,
but for an underlying
velocity law and 10% errors. |

Figure 12:
[
,
10% errors, 5% smearing] As in
Fig. 6, but with 10% errors added to the dynamical
spectrum and "velocity blurring'' at 5% of the terminal velocity. |

Figure 13:
[
,
10% errors, 5% smearing] As in
Fig. 6, but for an underlying
velocity law,
10% errors and "velocity blurring'' at 5% of the terminal velocity. |

Figure 15:
[
]
Inferred wind rotation law, W(x)(i.e., )
for the error-free dynamical spectra shown in the
left panels of Fig. 6, that is, for pure -law,
wide-plateau and narrow-plateau velocity laws, with an underlying
velocity law. The true rotation law (constant angular momentum,
W(x)=1/x^{2}) is shown as a dashed line). |

Figure 16:
[
]
As in
Fig. 15, but for an underlying
velocity law. |

Copyright ESO 2004