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\centerline{\hfill\large\tt Hydrodynamics of astrophysical winds}
\bigskip

\centerline{\hfill\large\tt driven by scattering in spectral lines}
\bigskip
\bigskip
\bigskip
\centerline{\hfill\tt Achim Feldmeier}
\bigskip
\bigskip
\bigskip
\centerline{\hfill\tt Habilitation thesis in Astrophysics}
\centerline{\hfill\tt Potsdam University, June 2001}

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\vskip18truecm
\centerline{\hfill Meinen Eltern}
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\centerline{\hfill Instead of a quote$\qquad\qquad$}
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\small

\noindent{\bf Abstract.} Line driven winds are accelerated by the
momentum transfer from photons to a plasma, by absorption and
scattering in numerous spectral lines. Line driving is most efficient
for ultraviolet radiation, and at plasma temperatures from 10$^4$~K to
$10^5$~K. Astronomical objects which show line driven winds include
stars of spectral type O, B, and A, Wolf-Rayet stars, and accretion
disks over a wide range of scales, from disks in young stellar objects
and cataclysmic variables to quasar disks. It is not yet possible to
solve the full wind problem numerically, and treat the combined
hydrodynamics, radiative transfer, and statistical equilibrium of
these flows. The emphasis in the present writing is on wind
hydrodynamics, with severe simplifications in the other two areas. I
consider three topics in some detail, for reasons of personal
involvement. 1.~{\it Wind instability}, as caused by Doppler
de-shadowing of gas parcels. The instability causes the wind gas to be
compressed into dense shells enclosed by strong shocks. Fast clouds
occur in the space between shells, and collide with the latter. This
leads to X-ray flashes which may explain the observed X-ray emission
from hot stars. 2.~{\it Wind runaway}, as caused by a new type of
radiative waves. The runaway may explain why observed line driven
winds adopt fast, critical solutions instead of shallow (or breeze)
solutions. Under certain conditions the wind settles on overloaded
solutions, which show a broad deceleration region and kinks in their
velocity law. 3.~{\it Magnetized winds}, as launched from accretion
disks around stars or in active galactic nuclei. Line driving is
assisted by centrifugal forces along co-rotating poloidal magnetic
field lines, and by Lorentz forces due to toroidal field gradients. A
vortex sheet starting at the inner disk rim can lead to highly
enhanced mass loss rates.

\bigskip\medskip

\noindent{\bf Zusammenfassung.} Liniengetriebene Winde werden durch
Impuls\"ubertrag von Photonen auf ein Plasma bei Absorption oder
Streuung in zahlreichen Spektrallinien beschleu\-nigt. Dieser Prozess
ist besonders effizient f\"ur ultraviolette Strahlung und
Plasmatempe\-ra\-turen zwi\-schen 10$^4$~K und 10$^5$~K. Zu den
astronomischen Objekten mit liniengetriebenen Winden geh\"oren Sterne
der Spektraltypen O, B und A, Wolf-Rayet-Sterne sowie
Akkretions\-scheiben verschiedenster Gr\"o{\ss}enordnung, von Scheiben
um junge Sterne und in kataklys\-mischen Ver\"anderlichen bis zu
Quasarscheiben. Es ist bislang nicht m\"oglich, das vollst\"andi\-ge
Windproblem numerisch zu l\"osen, also die Hydrodynamik, den
Strahlungs\-transport und das statis\-ti\-sche Gleichgewicht dieser
Str\"omungen gleichzeitig zu behandeln. Die Betonung liegt in dieser
Arbeit auf der Windhydrodynamik, mit starken Ver\-ein\-fachungen in
den beiden anderen Gebieten. Wegen pers\"onlicher Beteiligung
betrach\-te ich drei Themen im Detail. 1.~{\it Windinstabilit\"at}
durch Doppler{\it de-shadowing} des Gases. Die Instabilit\"at bewirkt,
dass Windgas in dichte Schalen komprimiert wird, die von starken
Sto{\ss}fronten begrenzt sind. Schnelle Wolken entstehen im Raum
zwischen den Schalen und sto{\ss}en mit diesen zusammen. Dies erzeugt
R\"ontgen{\it flashes}, die die beobachtete R\"ontgenstrahlung
hei{\ss}er Sterne erkl\"aren k\"onnen. 2.~{\it Wind runway} durch
radiative Wellen. Der {\it runaway} zeigt, warum beobachtete
liniengetriebene Winde schnelle, kritische L\"osungen anstelle von
Brisenl\"osungen (oder {\it shallow solutions}) annehmen. Unter
bestimmten Bedingungen stabilisiert der Wind sich auf
masse\"uberladenen L\"osungen, mit einem breiten, abbremsenden Bereich
und Knicken im Geschwindigkeitsfeld. 3.~{\it Magnetische Winde} von
Akkretionsscheiben um Sterne oder in aktiven Galaxienzentren. Die
Linienbeschleunigung wird hier durch die Zentrifugalkraft entlang
korotierender poloidaler Magnetfelder und die Lorentzkraft aufgrund
von Gradienten im toroidalen Feld unterst\"utzt.  Ein Wirbelblatt, das
am inneren Scheibenrand beginnt, kann zu stark erh\"ohten
Massenverlustraten f\"uhren.

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{\bf Contents}
\bigskip
\bigskip

 \S 1 Introduction \hfill \pageref{par1}

\medskip
{\bf Chapter 1. The line force}
\medskip

 \S 2 Pure absorption line force. Sobolev approximation \hfill\pageref{par2}

 \S 3 Scattering in SSF and EISF approximation \hfill\pageref{par3}

\bigskip
{\bf Chapter 2. Unstable winds}
\medskip

 \S 4 The de-shadowing instability \hfill \pageref{par4}

 \S 5 Evolved wind structure \hfill \pageref{par5}

 \S 6 Wind structure and line profiles \hfill \pageref{par6}

 \S 7 Including energy transfer \hfill \pageref{par7}

 \S 8 X-rays and clouds \hfill \pageref{par8}

\bigskip
{\bf Chapter 3. Runaway winds}
\medskip

 \S 9 Solution topology \hfill \pageref{par9}

 \S 10 Abbott waves \hfill \pageref{par10}

 \S 11 Abbott wave runaway \hfill \pageref{par11}

 \S 12 Overloaded winds \hfill \pageref{par12}

\bigskip
{\bf Chapter 4. Disk winds}
\medskip

 \S 13 Analytical model \hfill \pageref{par13}

 \S 14 Numerical model \hfill \pageref{par14}

 \S 15 Magnetized line driven winds \hfill \pageref{par15}

\bigskip
{\bf Outlook and Acknowledgments} \hfill \pageref{ackno}

\medskip
{\bf Appendix} \hfill \pageref{pprs}

%---------------------------------------------------------------------

 \clearpage
 \bigskip
 {\sc \S 1 Introduction\footnote{Only some key references are cited in
the introduction.}}
 \bigskip
 \label{par1}

\noindent{\bf Why line driven winds?} The study of winds from
astronomical objects started with Parker's work on the solar wind in
1958, one of the landmark theories in astronomy. At present, four
important types of hydrodynamic winds are known in astronomy: the
thermal wind from Sun; dust driven winds from red supergiants; line
driven winds from blue stars and accretion disks; and
magnetocentrifugal winds from accretion disks, either around stars or
in quasars.

Thermal winds are accelerated by gas pressure in a hot corona.
Parker's breakthrough idea was that the solar wind can be described as
a transsonic hydrodynamic flow, instead of a discrete particle flux.
Dusty winds are driven by continuum radiation pressure acting on dust
grains in a relatively cool environment. Line driven winds are also
driven by radiation pressure, yet, in numerous ultraviolet spectral
lines. Finally, magnetocentrifugal winds are launched from accretion
disks, via centrifugal forces acting along poloidal field lines, or
Lorentz forces caused by the toroidal field.

By contrast to geological winds which are essentially horizontal flows
caused by pressure gradients and the Coriolis force, the above four
astronomical winds directly oppose gravity, and carry away mass and
momentum from the central object. For line driven winds, both the mass
loss rate, $\dot M$, and the momentum rate, $\dot M v_\infty$ (with
terminal wind speed $v_\infty$), are large. These winds are therefore
important in two respects:

{\it Stellar evolution.} Hot, massive stars lose a large fraction of
their initial mass through winds, and the winds control stellar
evolution. Unfortunately, the phases of strongest mass loss, the LBV
(luminous blue variable) and Wolf-Rayet phase, are not well
understood, and empirical formulae have to be used in evolutionary
calculations. The LBV phase may be characterized by the star reaching
its Eddington limit (radiation pressure on electrons larger than
gravity; Langer et al.~1999).

{\it Star formation.} Hot stars often reside in environments rich of
gas and dust. The stellar wind enriches the interstellar medium with
metals and triggers (bursts of) star formation (Leitherer et
al.~1999). Figure~1 shows an aspect of this process which was recently
discovered. Shown is a Hubble Space Telescope image, where radiation
from an O7 main sequence star may prevent planet formation. The
stellar wind is seen blowing off matter from the protoplanetary
accretion disk.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=7cm
 \epsffile{nasa.ps}
 \caption{\footnotesize {\bf NASA press release, April 26, 2001.}
``{\it Hubble Watches Planetary Nurseries Being Torched by Radiation
from Hot Star.} Planet formation is a hazardous process. This
snapshot, taken by HST, shows a dust disk around an embryonic star in
the Orion Nebula being `blowtorched' by a blistering flood of
ultraviolet radiation from the region's brightest star. Within these
disks are the seeds of planets. Evidence suggests that dust grains in
the disk are already forming larger particles, which range in size
from snowflakes to gravel. But these particles may not have time to
grow into full-fledged planets because of the relentless `hurricane'
of radiation from the nebula's hottest star, Theta~1~Orionis~C. In the
picture, the disk is oval near the center. Radiation from the hot star
is heating up the disk, causing matter to dissipate. A strong `stellar
wind' is propelling the material away from the disk.''}
 \end{center}
 \end{figure}

We add more reasons why the hydrodynamics of line driven winds is an
important and interesting new research area.

{\it Spectroscopy.} Fundamental parameters of hot stars like radius
and mass can be derived from quantitative spectroscopy of spectral
lines forming in their winds (Pauldrach, Hoffmann, \& Lennon~2001;
Kudritzki \& Puls 2000; Hamann \& Koesterke 1998a; Hillier \& Miller
1998). Most importantly, the star's luminosity follows from the
wind-momentum luminosity relation (Kudritzki, Lennon, \& Puls
1995). In the near future, hot, massive stars may compete with
Cepheids as primary distance indicators. Wind hydrodynamics affects
spectral line formation and therefore quantitative spectroscopy in a
fundamental way. Observed spectral line features which are indicators
for time-dependent flow features are (we mention only the terms,
without going into explanations): black troughs, discrete absorption
components, enhanced electron scattering wings, bowed variation
contours, variable blue edges, and discrete emission components.

{\it New force.} Line driving is a new hydrodynamic force, and defines
a new class of `radiating fluids'. In astrophysics, radiation
hydrodynamics is of similar importance as magnetohydrodynamics. The
unique quality about line driving is `Doppler tuning'. Each spectral
line can absorb photons in a narrow frequency band only, its width
being determined by the thermal speed of ions. After a minute
acceleration of the gas by the line force, the Doppler-shifted
spectral line can absorb at bluer frequencies, from the `fresh'
stellar continuum. The ratio of terminal to thermal speed, which is
roughly the maximum Mach number, Ma, of the flow, is therefore of
order the width of the UV frequency band divided by the line width:
${\rm Ma} > 100$! By contrast, flows driven by continuum absorption of
radiation, the solar wind, and MHD flows (if the Mach number with
respect to \Alfven~waves is considered) have ${\rm Ma} \lea 10$ in
their accelerating regions.

{\it New waves.} Actually, ${\rm Ma} \lea 10$ also holds for line
driven winds, with respect to a new type of line driven or {\it
Abbott} waves (Abbott 1980). These waves are discussed controversially
in the literature, and a strict proof of their existence is still
missing.  They may play a central role in the ubiquitous `discrete
absorption components' observed in unsaturated P~Cygni line profiles
(Cranmer \& Owocki 1996); and they may drive line driven winds towards
a unique, critical solution.

{\it Range of objects.} The importance of line driving is also clear
from a list of astronomical objects which share this flow type. Line
driven winds occur in O and B stars (Lucy \& Solomon 1970), Wolf-Rayet
stars (Lucy \& Abbott 1993; Gayley, Owocki, \& Cranmer 1995; Hamann \&
Koesterke 2000), central stars of planetary nebula (Koesterke \&
Hamann 1997), and in A~supergiants (Kudritzki et al.~1999). The latter
are the {\it optically} brightest stars, and are central in
extragalactic distance determination. A relatively new idea is that
line driven winds are launched from {\it accretion disks}. Relevant
cases span a huge range, reaching from active galactic nuclei (quasars
and Seyfert galaxies; Weymann, Turnshek, \& Christiansen 1985) to
cataclysmic variables (white dwarfs with a late-type, main-sequence
companion; Heap et al.~1978) and young stellar objects (bright
B~protostars; Drew, Proga, \& Stone~1998).

{\it Interaction of line and magnetic driving.} Line driven winds from
magnetized accretion disks may show an intricate interplay of three
driving forces: the radiative, Lorentz, and centrifugal force. The
assistance of line driving may help to overcome problems encountered
with pure magnetocentrifugal driving. This should be relevant for
accretion disks in quasars and young stellar objects, where magnetic
fields and radiation fields are strong.

After this general motivation, and before we go over to more detailed
discussions in the main text, we give an overview of the results and
ideas treated in the following.

\medskip\noindent{\bf Wind instability and X-ray emission.} Line
driven winds are subject to a new radiation-hydrodynamics instability
(Lucy \& Solomon 1970). If a fluid parcel experiences a small,
positive velocity perturbation, it gets Doppler-shifted out of the
absorption shadow of gas lying closer to the radiation source (the
photosphere). It sees more light and experiences stronger driving,
hence, is further accelerated: an amplification cycle results, termed
{\it de-shadowing} instability. Since the flow is highly supersonic,
perturbations are expected to quickly grow into shocks.

In an important paper, Owocki, Castor, and Rybicki (1988; OCR from
now) calculated for the first time the unstable flow structure
numerically, along a 1-D, radial ray assuming spherical symmetry. The
initially smooth flow is transformed into a sequence of dense shells.
Since the shells are accelerated outwards, they are Rayleigh-Taylor
unstable and should fragment. In linear approximation, the
de-shadowing instability has no lateral component (Rybicki, Owocki, \&
Castor 1990), and one may expect that the R-T debris maintains a
relatively large, lateral scale. However, this is presently mere
speculation, since 2-D simulations are lacking due to computational
limitations. The fragmented shells are separated radially by broad
regions of rarefied, steeply accelerating gas. The thin, fast gas is
eventually decelerated in strong reverse shocks on the inner, starward
(or disk) facing rim of the shells. The shells propagate outwards at a
speed similar to that of smooth, stationary flow. Figure~2 shows the
evolved wind structure. The origin of X-ray emission in cloud-shell
collisions is also indicated in this sketch.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=12cm
 \epsffile{fig2.ps}
 \caption{Expected structure of an unstable, line driven wind. Dense
shells fragment via Rayleigh-Taylor instability. Fast clouds collide
with the shell fragments, creating X-ray flashes.}
 \end{center}
 \end{figure}

The detection of X-rays from early-type stars was one of the major
discoveries of the {\sc Einstein} satellite (Seward et al.~1979;
Harnden et al.~1979). For O~stars, the ratio of X-ray and total
luminosity is $L_x/L_{\rm bol} \sim 10^{-7\pm 1}$. X-ray temperatures
are 10$^6$ to 10$^{7.5}$ Kelvin, two or three orders of magnitude
above photospheric temperatures. Early-type stars have no envelope
convection zones and are thus quite different from solar-type stars
with convection zones, magnetic fields, and hot coronae. It is
generally believed that the X-ray emission from hot stars originates
in their winds, possibly in shocks which result from de-shadowing
instability.

While this idea is consistent with many observational facts,
especially the absence of K-shell absorption edges, theoretical
modeling encounters severe problems. In a phenomenological model of
strong forward shocks in the wind (Lucy 1982b), $L_x$ was found to be
a factor of 100 smaller than derived from {\sc Einstein} data
(Cassinelli \& Swank 1983). The same problem occured for the
hydrodynamic wind models of OCR, which show an X-ray flux deficiency
by factors of 10 to 100 (Hillier et al.~1993).

Instead of a continuous X-ray emission from quasi-steady wind shocks,
we propose X-rays flashes of short duration. These flashes occur when
fast wind clouds collide with a dense, cold shell. The clouds are
created via turbulent ablation from a `mildly' dense gas reservoir
lying ahead of the overdense, highly compressed shells. At the wind
base, roughly one half of the upstreaming gas is quickly compressed
into shells. The remaining gas is available for cloud formation at
larger heights in the wind. The clouds are accelerated through empty
intershell space, until they collide with the next outer shell. Time
averaged X-ray spectra synthesised from these models match {\sc Rosat}
observations well. Variations between individual snapshots are large,
however, in contrast to the observed constancy of X-ray fluxes.

Since the clouds are of turbulent origin, we speculate that their
lateral scale is much smaller than that of shell fragments. In future
2-D modeling, X-ray flux constancy may be achieved via angle averaging
over independent, radial wind rays, each with its own cloud-shell
collisions taking place.

\medskip\noindent{\bf Wind runaway caused by Abbott waves.} After the
discussion in Chapter~2 of {\it localized} flow features due to
de-shadowing instability, we turn in Chapter~3 to the {\it global}
solution topology of steady, line driven winds. The question we pose
is: why do line driven winds from stars and accretion disks adopt a
unique, critical solution? In a first attempt to answer this question,
we suppress de-shadowing instability by adopting the simplifying {\it
Sobolev approximation} in the line force. Future work has to unify
both aspects, global and local. (A relevant question could then be:
can local, unstable flow structure provide the seed perturbations
required for global solution transformation?)

In their fundamental work on line driven winds, Castor, Abbott, and
Klein (1975; CAK from now) showed that the stationary Euler equation
has an infinite number of possible solutions. They come in two
classes, `shallow' and `steep'. Steep solutions are supersonic, and
cannot connect to the subsonic wind base. Shallow solutions fail to
reach large radii, as they cannot perform the required spherical
expansion work. CAK concluded that the wind adopts the unique,
critical solution which starts shallow and ends steep, switching
smoothly between the classes at some critical point. This defines the
critical or CAK solution.

Evidence mounts that the argument given by CAK is too restrictive. The
question arises whether the CAK solution is still unique if
discontinuities are allowed for in {\it derivatives} of flow
quantities (i.e., kinks). Is the CAK solution a dynamical attractor in
the sense of mechanical system theory? And how does the transition
between shallow and the critical solution occur?

Radiative or Abbott waves are the key to answer these questions. We
show that shallow solutions are sub-abbottic, and Abbott waves
propagate inwards towards the photosphere from any point in the wind.
Numerical wind simulations published so far assume pure outflow
boundary conditions, which apply if all characteristics leave the
mesh. The standard argument is that the outer boundary is highly
supersonic, at ${\rm Ma}>100$. Instead, we see now that ${\rm Ma}<1$
for Abbott waves along shallow solutions. We show that, by chosing
outflow boundary conditions, the numerical scheme is forced to relax
to a super-abbottic solution. If the wind is sub-abbottic (yet,
supersonic), outflow boundary conditions cause numerical runaway.

Abbott waves define the characteristics of line driven winds, and as
such have to be included in the Courant time step, to prevent
numerical runaway. This has not be done so far. Accounting for Abbott
waves in the Courant time step and applying non-reflecting Riemann
conditions at the outer boundary, we find that the numerical scheme
converges to shallow solutions from a wide class of initial
conditions.

Will line driven winds in nature adopt a shallow solution? We suggest
that this is not the case, and identify a new, physical runaway which
drives shallow solutions towards the critical one. This runaway
depends on new and strange dispersion properties of Abbott waves:
negative velocity gradients propagate downstream and outwards, as
opposed to upstream propagating, positive velocity gradients. This
asymmetry causes systematic evolution of the wind towards larger
speeds. The runaway stops when a critical point forms in the flow, and
prevents waves to penetrate downwards to the wind base. This is the
case for the CAK solution.

A new solution type occurs with generalized critical points. If the
wind is perturbed in the sub-abbottic region below the CAK critical
point, runaway does not stop on the CAK solution. Instead, the wind
becomes {\it overloaded}. A new critical point forms, showing up as a
{\it kink} in the velocity law at which the wind starts to decelerate.
This is a direct consequence of overloading, since super-CAK mass loss
rates cannot be {\it accelerated} through the CAK critical point.
Increasing the overloading further, negative flow speeds result, and a
steady solution is no longer possible. Shocks and shells form in the
wind, and propagate outwards. There is some observational evidence
suggesting that the wind of the LBV star P~Cygni has indeed a broad,
decelerating region.

\medskip\noindent{\bf Winds from accretion disks.} The last, forth
chapter also deals with a simple Sobolev line force, however, in a
more complicated flow geometry. Observations indicate that line driven
winds are launched by the ultraviolet radiation field in certain types
of accretion disks. For example, P~Cygni line profiles from broad
absorption line quasars show terminal speeds of 10\% the speed of
light (Turnshek 1984). The flow is rather massive, with mass fluxes of
order one solar mass per year in luminous quasars. Both facts are
suggestive of a line driven wind. However, the ionizing radiation from
the central source poses a serious problem. Wind gas which is not
highly compressed or shielded from this radiation becomes fully
ionized, and line driving stalls. Shielding could be provided by a
cold disk atmosphere. The wind is launched vertically from the disk,
reaches large speeds already within the atmosphere, and escapes on
ballistic orbits after being exposed to the central radiation
(Shlosman et al.~1985). Alternatively, dense regions of hot, ionized
gas (of unspecified origin) may occur between the central source and
the wind (Murray et al.~1995). This gas may block ionizing X-rays, but
could be transparent to UV radiation. The flow is again launched
vertically by local disk radiation, but quickly bends over and makes a
shallow angle with the disk when irradiated by the central
source. Numerical simulations show that the shielding region may
consist of highly-ionized, failed wind (Proga, Stone, \& Kallman
2000). Strong observational support that winds from broad absorption
line quasars are driven by resonance line scattering comes from ``the
ghost of Ly$\alpha$'' (Arav 1996).

While very fascinating, line driven quasar winds are still a matter of
debate. On the other hand, observational evidence for line driven disk
winds in cataclysmic variables (CVs) is unambiguous (Heap et al.~1978;
Krautter et al.~1981). These systems consist of a white dwarf and a
late-type, main sequence companion. The latter fills its Roche lobe
and feeds an accretion disk onto the white dwarf. Observed P~Cygni
line profiles clearly indicate a biconical outflow from the disk, not
a spherical outflow from the white dwarf itself (Vitello \& Shlosman
1993). In Chapter~4, we discuss numerical simulations and a
semi-analytical model for CV winds. We encounter a two-dimensional
eigenvalue problem, for mass loss rates and wind tilt angles with the
disk. The derived mass loss rates are much smaller than was hitherto
expected.  We discuss why the latter expectations were overestimates.
Still, a large discrepancy remains, and it is not yet clear why CV
disk winds are so efficient in nature. Possibly, magnetic fields
assist line driving. This leads over to the last topic of this
writing, magnetized winds.

Magnetocentrifugal winds could occur in young stellar objects and in
quasars. In the classical model of Blandford \& Payne (1982; see
Fig.~3 for a realistic scenario), outflow occurs along poloidal
magnetic field lines. If the magnetic pressure in the disk corona is
much larger than gas pressure, the field lines co-rotate with the
disk. The field lines act as lever arms and, at inclination angles
$<60$ degrees with the disk, flung the gas outwards. Above the
Alfv\'en point, ram pressure dominates magnetic pressure in the
wind. The gas parcels start to conserve angular momentum, and lack
behind the disk rotation. The poloidal field gets wound-up into a
toroidal field. Hoop stresses of the latter confine the centrifugal
outflow to bipolar jets, and jets are indeed often found associated
with young stellar objects (Eisloeffel et al.~2000). There is a large
body of literature on the theory of magnetocentrifugal disk
winds. Some key references include: Pudritz \& Norman (1983; 1986),
K\"onigl (1989), and Heyvaerts \& Norman (1989). An excellent review
is K\"onigl \& Pudritz (2000).

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=12cm
 \epsffile{fig3.ps}
 \caption{Realistic scenario for a magnetocentrifugal wind from a
quasar disk. Gas clouds are centrifugally driven outwards along
sufficiently inclined, co-rotating magnetic field lines. From
Emmering, Blandford, \& Shlosman (1992).}
 \end{center}
 \end{figure}

Recently, Drew, Proga, \& Stone~(1998) suggested that line driving
could assist magnetocentrifugal driving in {\it bright} young stellar
objects (young B~stars). The cold accretion disk is assumed to be
illuminated by the hot central star, and either scatters the incident
UV radiation or is itself heated to temperatures resulting in strong
UV emission. The disk radiation field launches a flow. The terminal
speed of line driven winds scales with the escape speed at their base,
and is much smaller from outer disk regions than from the central
star. This scenario could therefore explain the small observed outflow
speeds from certain objects.

At the end of Chapter~4, we present explorative 2-D simulations of
magnetized line driven winds. The resulting flow dynamics above
accretion disks with small Eddington factor is rather intricate. For
realistic magnetic field strengths, mass loss rates may be
dramatically increased. We find that dominant magnetic driving is {\it
not} via the centrifugal force along poloidal field lines, but via the
Lorentz force caused by toroidal field gradients. This scenario,
complementary to the Blandford \& Payne model, was proposed by
Contopoulos (1995). Still, we find that the poloidal field is
mandatory for increased mass loss. A vortex sheet forms in the
poloidal wind velocity and magnetic field, and carries the toroidal
field to such heights that it can assist in driving the enhanced mass
loss through the critical point (the `bottleneck' of the flow).

Much work remains to be done on magnetized line driven winds, in order
to understand the relevant physics and rule out numerical artefacts.
This is the reason why no paper or preprint on this subject is
appended to the present writing.

%-------------------------------

\clearpage

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfysize=19cm
 \epsffile{wegmarke1.ps}
 \label{wegma1}
 \end{center}
 \end{figure}

\clearpage

 \begin{figure}[h]
 \begin{flushleft}
 \leavevmode
 \epsfysize=19cm
 \epsffile{wegmarke2.ps}
 \label{wegma2}
 \end{flushleft}
 \end{figure}

\clearpage
\bigskip
\noindent{\bf CHAPTER 1. THE LINE FORCE}
\bigskip
\label{chap1}

\noindent{\sc \S 2 Pure absorption line force. Sobolev approximation}
\bigskip
\label{par2}

{\bf For one line.} Line driven winds stand and fall with the
formulation of the radiative force. The force due to momentum transfer
by absorption and re-emission of photons in spectral lines is termed
`line force' from now. The basic problem in calculating the line force
is the inclusion of scattering in the wind. The absorbed radiation
determines together with particle collisions the ionization degree and
occupation numbers of the plasma, which in turn determine the emitted
radiation field. For a stationary and smooth flow, the feedback
between radiative transfer and statistical equilibrium of the gas can
be solved using ALI techniques (Cannon 1973; Scharmer 1981; Hamann
1985). However, coupling the radiative transfer and thermal
equilibrium to time-dependent hydrodynamics over $\approx 10^5$ time
steps (instead of `1' for stationary winds) and for $\approx 5000$
spatial grid points in a highly non-monotonic velocity law (instead of
50 for smooth flow) is not yet possible. Hydrodynamic simulations have
to severely approximate the radiative transfer and thermal equilibrium
of the wind. The line force from all driving lines (between 10 and
10$^5$) is calculated from the force on a single line via a power law
line distribution function with two parameters (Castor, Abbott, \&
Klein 1975; CAK in the following). For dense O~supergiant winds, these
parameters are constrained within relatively narrow margins (Gayley
1995; Puls, Springmann, \& Lennon 2000). The radiative transfer in the
remaining, `generic' spectral line is further simplified. In
Chapter~2, we consider pure line absorption and a simplified approach
to scattering.  In Chapters~3 and 4, Sobolev approximation is used.

The force which acts on a gas absorbing radiation of intensity $I$ is
derived in the textbooks by Chandrasekhar (1950) and Mihalas (1978).
To avoid angle integrals at the present step, we consider a radial,
spherically symmetric flow with coordinate $r$, which is accelerated
by absorption (no scattering) of radiation from a point source located
at $r=0$. The line force per mass, $g_l$, is
 %
 \begin{equation}
 \label{eq150201_01}
 g_l(r) = g_t(r) \int_{-\infty}^\infty dx \,\phi(x-v(r)/\vth)
e^{-\tau(x,r)},
 \end{equation}
 %
 where $\tau$ is the line optical depth,
 %
 \begin{equation}
 \label{eq150201_02}
 \tau(x,r)=\int_0^r dr' \kappa(r') \phi(x-v(r')/\vth))
 \end{equation}
 %
 and 
 %
 \begin{equation}
 g_t(r)= \kappa\Delta\nu_D F_{\nu_0}(r)/c
 \end{equation}
 %
 is the radiative acceleration if the line were optically thin,
$\tau\ll 1$. A Doppler profile function is assumed, $\phi(x)=
\pi^{-1/2} \exp(-x^2)$, with normalized frequency variable
$x=(\nu-\nu_0)/\Delta \nu_D$. Here, $\nu_0$ is the line frequency and
$\Delta\nu_D= \nu_0 \vth/c$ the Doppler width of the line, with $c$
the speed of light. For simplicity, the thermal speed $\vth$ is
assumed to be constant. The radial wind velocity is given by $v$, and
$v/\vth$ in the profile function accounts for Doppler shifts. In an
accelerating wind, ions can absorb `blue' photons, $\nu > \nu_0$, in
the line transition $\nu_0$, since they appear redshifted. Finally,
$F_\nu$ is the radiative flux per frequency interval $d\nu$ at
frequency $\nu$, and $\kappa$ is the mass absorption coefficient of
the line, in units cm$^2$ per gram. For line transitions to the ground
state (resonance lines) and to metastable levels, which both dominate
the line force, $\kappa$ is to a good approximation constant, as shall
be assumed in all the following.

In hydrodynamic wind simulations of the de-shadowing instability,
eqs.~(\ref{eq150201_01}) and (\ref{eq150201_02}) have to be calculated
as they stand, by explicit quadrature over $dr$ and $dx$ (and over
angle; Owocki, Castor, \& Rybicki 1988; Owocki 1991; paper~[1]). The
$r$ and $x$ integrals are time consuming, since high spatial and
frequency resolution is required: the thermal band has to be resolved
in both $r$ and $x$. The thermal speed of metal ions is a few km/s in
hot star winds, whereas the terminal wind speed is thousand km/s and
larger. Hence, thousands of frequency and spatial grid points are
required. The angle integral, on the other hand, is cheap for stellar
wind simulations with a high degree of symmetry. A one- or two-ray
quadrature may be sufficient. This is no longer true for winds above
accretion disks, due to the radial temperature stratification of the
disk and the complicated flow geometry (Proga, Stone, \& Drew 1998;
[6]).

\medskip\noindent{\bf Sobolev approximation.} In Sobolev
approximation, the profile function $\phi$ is replaced by a Dirac
$\delta$ function: the wind is assumed to accelerate so steeply that a
spectral line is Doppler-shifted into resonance with a photon over a
narrow spatial range only. This region is called a resonance or
Sobolev zone. The quantities $\rho$, $\kappa$ and $dv/dr$ are assumed
to be constant over the Sobolev zone. By definition, $v$ increases by
a few thermal speeds over the zone.

Radiative transfer takes place then on a `microscopic' scale within
the zone, hydrodynamics (i.e., changes in $\rho$ and $dv/dr$) on a
`macroscopic' scale (Rybicki \& Hummer 1978). The fact that the wind
speed $v$ is neither a truly microscopic nor macroscopic variable
causes some difficulties in our understanding of radiative waves and
critical points in line driven flows [10]. A characteristic analysis
in Chapter~3 shows that in Sobolev approximation, $dv/dr$ and not $v$
is a fundamental hydrodynamic quantity besides $\rho$.

To calculate the line force in Sobolev approximation from a point
source of radiation, substitute $r \rightarrow \tilde x$ in the
optical depth (\ref{eq150201_02}), where $\tilde x= x-v(r)/\vth$.
This is allowed for monotonic $v(r)$, and gives
 %
 \begin{equation}
 \label{soboptdep}
 \tau(\tilde x) = \tau_0\, \Phi(\tilde x),
 \end{equation}
 %
 where $\Phi(\tilde x)=\int_{\tilde x}^\infty dy\,\phi(y)$, and
$\tau_0=\rho\kappa\vth/(dv/dr)$ is the total optical depth of the
Sobolev resonance zone. The line force at $r$ is caused by lines which
absorb at $r$, hence $\tilde x(r)=0$ and $\tau_0=\tau_0(r)$.
Furthermore, $\tilde x \equiv \infty$ is assumed at the wind
base. Substituting $x\rightarrow \Phi$ in the line force
integral~(\ref{eq150201_01}) gives
 %
 \begin{equation}
 \label{soboabs}
 g_l(r)=g_t(r) b(r),\qquad\quad{\rm with}\quad\qquad
 b(r)={1-e^{-\tau_0} \over \tau_0}
 \end{equation}
 %
 the so-called `photon escape probability.' For $\tau\ll 1$ and
$\tau\gg 1$, the line force scales as $\kappa$ and $\rho^{-1} dv/dr$,
respectively.

The expression (\ref{soboabs}) holds also for the Sobolev force when
line scattering is included, and the diffuse radiation field is
fore-aft symmetric, i.e., the latter does then not contribute to the
line force. We will apply a Sobolev force in Chapters~3 and 4. In
Chapter~2, we consider the more complicated SSF and EISF forces, which
are extensions of the general force (\ref{eq150201_01}) to the case of
line scattering, using ingredients from Sobolev approximation to
calculate the radiative source function.

\medskip\noindent{\bf Force from all lines.} To calculate the line
force from thousands of spectral lines, the CAK line distribution
function is used throughout. By assumption, there is no wind-velocity
induced line overlap, and each photon is scattered in one line at
most. The CAK line distribution function is given by
 %
 \begin{equation}
 \label{eq240401_1607}
 N(\nu, \kappa) = {1\over \nu}\;{1\over \kappa_0}\,
 \left({\kappa_0 \over \kappa}\right)^{2-\alpha},
 \end{equation}
 %
 introducing two parameters, $\kappa_0$ and $\alpha$, where $0<\alpha
<1$. Integrating (\ref{eq150201_01}) over $\nu$ and $\kappa$ using
(\ref{eq240401_1607}), the total line force becomes (we keep the
symbol $g_l$ from the single line force),
 %
 \begin{equation}
 \label{eq200101_1}
 g_l(r) = {\Gamma(\alpha)\kappa_0^{1-\alpha} \vth \over c^2}\; F(r)\,
\int_{-\infty}^\infty dx\; {\phi(x-v(r)/\vth)\over \eta^\alpha(x,r)},
 \end{equation}
 %
 where $\Gamma$ is the Gamma function, $F$ is the frequency integrated
flux, and we introduced $\eta\equiv\tau/\kappa$ (constant $\kappa$ for
each line), with $\tau$ calculated from (\ref{eq150201_02}). Inserting
instead for a moment the Sobolev optical depth (\ref{soboptdep}), one
finds for the total line force in Sobolev approximation,
 %
 \begin{equation}
 g_l(r) = {\Gamma(\alpha)\kappa_0 \vth \over (1-\alpha) c^2}\; F(r)\;
\tau_0(r)^{-\alpha},
 \end{equation}
 %
 where we redefined $\tau_0=\kappa_0\rho\vth/(dv/dr)$ (in order to
avoid adding another subscript 0 to $\tau_0$). Precise values for
$\kappa_0$ (or, equivalently, the CAK parameter $k$) and $\alpha$ must
be obtained from a detailed NLTE treatment of the wind. For the
present purposes, some universal estimates are sufficient. Puls et
al.~(2000) derive $\alpha=2/3$ from Kramers' opacity law for
hydrogen-like ions; this value should apply for dense winds. In thin
winds, $\alpha\lea 1/2$ (Pauldrach et al.~1994). The absorption
coefficient $\kappa_0$ corresponds roughly to the strongest line in
the flow, $\kappa_0={\rm O}(10^8~{\rm cm}^2 \,{\rm g}^{-1})$ in dense
winds. Alternatively, $\kappa_0$ can be expressed in terms of an
effective oscillator number $Q$ (Gayley 1995),
 %
 \begin{equation}
 {\kappa_0 \vth \over \sigma_e c} = 
 Q \Gamma(\alpha)^{-{1\over 1-\alpha}},
 \end{equation}
 %
 where $\sigma_e$ is the absorption coefficient for Thomson scattering
on electrons. In a fully ionized hydrogen plasma, $\sigma_e=
0.4$~cm$^2$/g. For O~supergiant winds, $Q\approx 2000$ (Gayley 1995),
from which $\kappa_0$ can be calculated.

\newpage%play
\noindent{\sc \S 3 Scattering in SSF and EISF approximation}
\bigskip
\label{par3}

First time-dependent hydrodynamic simulations of unstable O~star winds
by Owocki et al.~(1988) assumed a pure absorption line force like that
in (\ref{eq150201_01},\ref{eq150201_02}) since the de-shadowing
instability vanishes in Sobolev approximation. This led to certain
unexpected results, most notably, that the time-averaged wind does not
adopt the CAK solution, but a steeper solution. (The stationary CAK
solution is treated in Chapter~3. Until then, it suffices to know that
this solution is unique, has a maximum mass loss rate, and a critical
point which is not the sonic point.) The defects of the pure
absorption model are still not fully understood (see Owocki \& Puls
1999 for new insights), but they vanish when line scattering is
included. (The diffuse force scales $\sim \vth/v$, and vanishes only
in Sobolev approximation.) Hence, we turn to a simplified treatment of
scattering now.

In the `smooth source function' or SSF method (Owocki 1991), a purely
local radiative source function from Sobolev aproximation is
assumed. The principal idea goes back to Hamann (1981a), and was
adopted in the SEI method of Lamers, Cerruti-Sola, \& Perinotto
(1987). The remaining `formal solution' of radiative transfer is no
more complex than for pure line absorption. Especially, the optical
depth for photons which are backscattered to the photosphere is by
symmetry related to the optical depth for photospheric photons. No
extra integrals are required in the SSF method, and the computational
costs are practically the same as for pure absorption. The SSF method
accounts for the important {\it line drag} effect (Lucy 1984), which
stabilizes the flow via the {\it mean} diffuse radiation field.

The next step of sophistication beyond pure absorption and SSF is the
`escape-integral source function' or EISF method (Owocki \& Puls
1996), which accounts for the {\it perturbed diffuse} radiation field.
Already linear stability theory becomes very complex when
perturbations in the diffuse radiation field are included (Owocki \&
Rybicki 1985). Yet, the basic idea of the EISF method is clear and
straightforward: in spherical symmetry, the direct radiative force due
to photon absorption in a single line is, including now angle
integrals explicitly,
 %
 \begin{equation}
 g_a(r) \sim \langle \mu I_\ast(\mu) b(\mu,r) \rangle.
 \end{equation}
 %
 Angle brackets indicate angle averages, $\mu$ is the cosine of the
angle of a photon ray with the radial direction, $I_\ast(\mu)$ is the
angle dependent, photospheric radiation field, and $b(\mu,r)$ is the
escape probability for direction cosine $\mu$
(cf.~eq.~\ref{eq150201_01}),
 %
 \begin{equation}
 b(\mu,r) = \int dx \,\phi(x-\mu v(r)/\vth)\, e^{-\tau(x,\mu,r)}.
 \end{equation}
 %
 The diffuse or scattering force, on the other hand, is
 %
 \begin{equation}
 g_s(r) \sim S(r) \langle \mu b(\mu,r) \rangle,
 \end{equation}
 %
 with isotropic source function $S$. For pure scattering lines in
Sobolev approximation, one derives from (\ref{soboabs}) and the
transfer equation (see Owocki \& Rybicki 1985 for details) that
 %
 \begin{equation}
 S(r)={\langle I_\ast(\mu) b(\mu,r) \rangle \over \langle b(\mu,r)
\rangle}.
 \end{equation}
 %
 In the SSF method, different expressions are used for the escape
probabilities in $g_a$, $g_s$ and in $S$: in the $g$'s, $b$ is
calculated from the actual, time-dependent flow structure; whereas in
$S$, purely local escape probabilities for a smooth flow are used. By
contrast, in the EISF method $b$ from the actual, structured flow is
used both in $g$ and $S$. No quantities are introduced in EISF which
did not already occur in SSF. Still, the calculational cost is much
larger in EISF since {\it two} spatial integrals are required. In the
first integral, the source function is calculated over the whole mesh,
the second serves for a formal solution. Furthermore, a quadrature
over frequency $x$ is required. Owocki \& Puls (1996) remark that
these 3 integrals can be reduced to 2 again for a single spectral
line; but not for a line ensemble.

The EISF approach allows for the first time to study phase reversal
between velocity and density fluctuations as a consequence of
perturbations of the diffuse radiation field. This subtle yet
important effect is discussed further on page~\pageref{gemma1}. The
simulations discussed in detail in the next chapter were calculated
using the SSF method.

%-----------------------------------------------------------------

\clearpage
\bigskip
\noindent{\bf CHAPTER 2. UNSTABLE WINDS}
\bigskip
\label{chap2}

It is now believed that line driven winds from single stars show
structure in all three spatial directions. In the polar ($r\theta$)
plane, wind-compressed disks may (Bjorkman \& Cassinelli 1993; Owocki,
Cranmer, \& Blondin 1994) or may not (Owocki, Cranmer, \& Gayley 1996)
form around rapidly rotating stars. In the equatorial ($r\phi$) plane,
co-rotating interaction regions may cause `discrete absorption
components' (Cranmer \& Owocki 1996) observed in non-saturated P~Cygni
line profiles. Most `simply', however, already in 1-D radial flow
shocks and dense shells develop due to a new hydrodynamic instability.
This instability is the subject of the present chapter.

To give an overview of the field, we start with a bibliography of
relevant papers. The instability mechanism is discussed, and results
from linear stability analysis are summarized. Because of the
complexities of line scattering, linear theory is not complete to the
present day. This is not just a mathematical curiosity, but the origin
of a fundamental debate. Namely, since the Green's function for the
case of pure line scattering is not yet known, the nature of signal
propagation in these unstable winds is -- mysterious. Signal
propagation plays a key role in understanding how the flow adapts to
boundary conditions. The main topic of this chapter is the evolved,
nonlinear wind structure found from numerical simulations, and its
relation to observed X-ray emission from O~stars.

\bigskip 
\noindent{\sc \S 4 The de-shadowing instability} 
\bigskip
\label{par4}

\noindent{\bf History.} A new, radiation hydrodynamics instability of
line driven flows was first suggested by Lucy \& Solomon (1970). The
mechanism is similar to one proposed by Milne (1926) for the solar
chromosphere. Approximate linear stability analysis was performed by
MacGregor, Hartmann, \& Raymond~(1979), Carlberg (1980), and Abbott
(1980). In the former two papers, unstable growth rates were derived,
whereas Abbott found a new, marginally stable, radiative-acoustic wave
mode. This contradiction was resolved by Owocki \& Rybicki (1984), who
showed that the opposing results apply on different length scales.
Lucy (1984) found that the diffuse radiation field from line
scattering causes a drag effect which could prevent the instability.
Owocki \& Rybicki (1985) derived that complete cancellation occurs
only close to the photosphere. A very puzzling result was derived by
Owocki \& Rybicki (1986), who showed that Abbott waves do occur in
absorption flows (they shouldn't), but as a pure mathematical
artefact. What are the consequences for Abbott waves if scattering is
included? Rybicki, Owocki, \& Castor (1990) proved that wind
instability occurs in flow or radiative flux direction only. In
lateral direction, line drag stabilizes the flow. A linear stability
analysis for Wolf-Rayet stars was performed by Owocki \& Rybicki
(1991) and Gayley \& Owocki (1995) in diffusion approximation. They
found that unstable growth rates are reduced by the multi-scattering
factor. Feldmeier (1998, paper~[5]) derived that the instability
occurs already in Sobolev approximation, if velocity curvature terms
are included. This issue of fore-aft asymmetries when crossing the
Sobolev zone is addressed in papers by Lucy (1975; Sobolev
vs.~Newtonian derivative); Owocki \& Zank (1991; radiative viscosity)
Gayley \& Owocki (1994; radiative heating), and Owocki \& Puls (1999;
source function depression). First numerical simulations of the
evolved wind structure were given in breakthrough papers by Owocki,
Castor and Rybicki (1988; OCR) for the case of pure line absorption,
and by Owocki (1991, 1992) for line scattering in SSF approximation.
Poe, Owocki, \& Castor (1990) suggested that the nodal topology of the
sonic point in line driven flows causes either solution degeneracy or
convergence to a steep, non-CAK solution. The classical paper on
steady wind solutions and the critical point topology is Castor,
Abbott, \& Klein (1975; CAK). The issue of solution topology is again
related to the inclusion of line scattering. The {\it perturbed}
diffuse radiation field was treated in linear analysis by Owocki \&
Rybicki (1985), and implemented in numerical simulations by Owocki \&
Puls (1996) via the EISF method. EISF simulations which clarified
deeper aspects of the Sobolev approximation were performed by Owocki
\& Puls (1999). Phenomenological wind shock models of X-ray emission
were suggested by Lucy \& White (1980), Lucy (1982b), Krolik \&
Raymond (1985), and MacFarlane \& Cassinelli (1989). The energy
equation and radiative cooling in post-shock zones was included in
hydrodynamic simulations of unstable winds by Feldmeier (1995, [1]),
and Feldmeier, Puls, \& Pauldrach (1997, [4]) suggested X-ray emission
from turbulent cloud collisions. P~Cygni lines from structured,
unstable winds were calculated by Puls, Owocki, \& Fullerton (1993)
and Puls et al.~(1994).

\medskip\noindent{\bf 1 The microscopic mechanism.}\footnote{The
numbers at the start of paragraphs correspond to the numbers in the
figure on pages~\pageref{wegma1} and \pageref{wegma2}.} The basic
mechanism of the instability is as follows. Consider the velocity law
$v(r)$ of a line driven wind. Because of its width of a few $\vth$ as
caused by thermal motions, $v(r)$ may be called a {\it thermal
band}. The thickness of the thermal band is characterized by the
Sobolev length, $L=\vth/dv/dr$. This is the {\it natural} length scale
for line driven winds. Instability occurs for perturbations with
wavelength $\lambda<L$ (we shall, however, find below that $\lambda>L$
is also unstable). An arbitrary, positive velocity fluctuation
$+\delta v$ Doppler-shifts a gas parcel out of the absorption shadow
of gas lying closer to the star (or the accretion disk). The enhanced
radiative flux on the parcel accelerates it to larger speeds and
de-shadows it further. Since the single-line force scales as $g_l \sim
e^{-\tau}$, the amplification cycle can be written as $\delta v
\rightarrow -\delta \tau \rightarrow \delta g_l \rightarrow \delta v$.

\label{10030101}

Once the parcel got `kicked out' of the thermal band, further
de-shadowing is impossible. Carlberg (1980) concluded that the
instability should {\it not} cause observable perturbations of the
velocity law, but microscopic fluctuations of order $\vth$ only. This
is correct for short scale perturbations $\lambda <L$. We shall see
below that long scale perturbations, $\lambda\gg L$, result in $\Delta
v\gg \vth$. It is therefore slightly misleading to say (as is
occasionally done) that the instability generally causes short scale
structure.

Milne (1926) described a runaway process which he held responsible for
the ejection of high-speed atoms from the static solar chromosphere.
The process is so similar to the de-shadowing instability that we give
a quotation from Milne's paper.

 \begin{quote}
 {\footnotesize ``An atom which, due to some cause or other, begins to
move outward from the sun with an appreciable velocity will begin to
absorb in the violet wing of the absorption line corresponding to the
same atom at rest, owing to the Doppler effect. It will therefore be
exposed to more intense radiation, and the atom will be accelerated
outwards. It will therefore move still further out into the wing,
where it will be exposed to still more intense radiation, and so on,
until it eventually climbs out of its absorption line.''}
 \end{quote}

Note that Milne describes a plasma instability for single ions,
whereas the de-shadowing instability is a hydrodynamic instability for
fluid parcels. The other difference is between a static atmosphere and
a wind.

\medskip\noindent{\bf 2 Linear instability for absorption lines.}
Owocki \& Rybicki (1984) gave the first, full derivation of
instability growth rates. For long scale perturbations $\lambda\gg L$,
the {\it imaginary} growth rates turn into a {\it real} dispersion
relation for Abbott waves (or, in the older literature,
`radiative-acoustic waves'). The derivation given in equations~(1) to
(30) of Owocki \& Rybicki (1984) is very compact. Hence we refer to
this paper, and quote only some results. Starting point of the
derivation is the line force from a point radiation source,
eqs.~(\ref{eq150201_01}, \ref{eq150201_02}). Perturbations $\delta v$
enter in {\it both} profile functions $\phi(x)$. After substitution
from spatial to frequency variables; applying Sobolev approximation
for the {\it mean} flow; and introducing harmonic perturbations
$\delta v(r)\sim \exp(ikr)$ obeying WKB approximation, one arrives at
the perturbed line force (a subscript 0 refers to the mean flow),
 %
 \begin{equation}
 {\delta g_l\over \delta v} = iK\omega_0\tau_0 \int_{-\infty}^\infty
dx \,\phi(x) e^{-\tau_0 \Phi(x)} \int_x^\infty dx' \phi(x')
e^{-iK(x'-x)}.
 \end{equation}
 %
 Here, $\omega_0=g_t/\vth$, $\tau_0=\kappa_0\rho_0\vth/(dv_0/dr)$,
$K=kL$ (the wavenumber in units of the Sobolev length, $L$), and
$\Phi(x)$ as above. The perturbed line force depends on the profile
function $\phi$. Using an ingenious integration trick, Owocki \&
Rybicki (1984) solved the double integral analytically for $\tau_0\gg
1$, where it becomes independent of $\phi$. The result is
 %
 \begin{equation}
 \label{eq160202_01}
 \qquad \qquad
 {\delta g_l\over \delta v} = \omega_b {ik \over \chi_b +ik}
\qquad\qquad (\tau_0\gg 1),
 \end{equation}
 %
 where (introducing an opacity $\chi=\rho\kappa$),
 %
 \begin{equation}
 \omega_b=\omega_0 \,\phi(x_b),\qquad\qquad
 \chi_b=  \chi_0   \,\phi(x_b),
 \end{equation}
 %
 and the {\it blue edge} frequency $x_b$ is defined by
 %
 \begin{equation}
 \Phi(x_b)\equiv 1/\tau_0.
 \end{equation}
 %
 For short scale perturbations, $k\rightarrow\infty$ and $\delta
g_l/\delta v= \omega_b$. This implies instability, since the phase
shift between velocity and force perturbations is 0. In the limit
$k\rightarrow 0$, on the other hand, $\delta g_l/\delta v=ik
\omega_0/\chi_0$. With 90 degrees phase lag between velocity and force
perturbations, this corresponds to marginally stable waves: Abbott
waves. For long scale perturbations with {\it finite} $\lambda$, one
has (slightly) unstable waves. Again from (\ref{eq160202_01}), the
growth rate drops as $(L/\lambda)^2$ for $\lambda\gg L$.

\medskip\noindent{\bf 3 Macroscopic linear instability: 2nd order
Sobolev.}  Instead of going through the above, `exact absorption'
analysis, Abbott (1980) applied Sobolev approximation to the mean flow
{\it and} to velocity perturbations. He finds marginally stable waves
which are not affected by the instability. This is odd since we saw
above that long scale waves, $\lambda\gg L$, {\it are} unstable, if
only at reduced growth rates. But for $\lambda\gg L$, Sobolev
approximation {\it should} apply. Indeed, instability occurs in {\it
second order} Sobolev approximation, including curvature terms $d^2
v/dr^2$ [5]. The second order Sobolev optical depth $\tau$ is found to
be,
 %
 \begin{equation}
 \label{eq240101_1}
 \tau(\tilde x,r) = \tau_0(r) \left[ \Phi(\tilde x) + {\vth\over 2v'}
\left( {v''\over v'}(r) - {\rho'\over\rho}(r) \right)
\bigl(\phi(\tilde x)-2\tilde x\Phi(\tilde x)\bigr)\right],
 \end{equation}
 %
 with $\tau_0$ as before, and primes indicating spatial derivatives.
Using (\ref{eq240101_1}) to calculate $\delta\tau$ in the perturbed
line force one finds {\it unstable} Abbott waves, with growth rate
$\sim (L/\lambda)^2$. Despite the different approximations made in the
two approaches, the latter growth rates agree to within 20\% with
those of Owocki \& Rybicki (1984).

Equation (\ref{eq240101_1}) offers an intuitive understanding of the
wind instability at {\it large} perturbation wavelengths, which
complements the picture of parcels being kicked out of the thermal
band described earlier (see Fig.~1 in [5]). When the velocity law
experiences a small {\it upward bend}, $v''<0$, the optical depth is
reduced (we assume that $v'$ averaged over the resonance zone is left
unchanged). This implies an increase in line force, $g_l\sim
e^{-\tau}$. The `elevated' region is accelerated to larger speeds.
This means it gets further elevated, and $-v''$ grows further. An
amplification cycle $-\delta v'' \rightarrow -\delta\tau \rightarrow
\delta g_l \rightarrow \delta v \rightarrow -\delta v''$ results. The
corresponding argument holds for a depression $+\delta v''$.

\medskip\noindent{\bf 4 Nonlinear steepening: a first look at evolved
wind structure.} We can already at this stage predict some features of
the evolved wind structure, before going into details of hydrodynamic
simulations. A depression in $v(r)$ as considered in the last
paragraph will eventually cause the {\it mean} velocity gradient
$v_0'$ to become smaller (assumed above to be left unchanged). Since
$\tau_0\sim 1/v_0'$ increases, the blue wing frequency $x_b$ becomes
more negative, and the growth rate $\omega_b\sim \phi(x_b)$ drops
steeply. The depression region will not evolve (`depress')
further. Depression regions will always remain close to the stationary
flow. On the other hand, elevations of the thermal band will continue
to grow until the flow becomes optically thin and no further
de-shadowing is possible. In the nonlinear regime, fast gas overtakes
slow gas ahead of it, and the velocity law evolves a triangular
sawtooth shape. The jumps in the `shark fins' decelerate the gas,
hence, are reverse shocks. The robust morphology of the wind velocity
law, which shows up in most numerical simulations, is therefore: a
sawtooth pattern of steep, accelerating regions and pronounced reverse
shocks.

\medskip\noindent{\bf 5 How large are the shock jumps?} Whereas short
scale perturbations $\lambda \le L$ saturate at the microscopic level
of a few $\vth$ (Carlberg 1980), large scale perturations $\lambda \gg
L$ give velocity jumps of order $v_\infty$, the wind terminal
speed. Overall, a short scale, noisy structure ($\vth$) is
superimposed on a large scale, coherent tilt of the wind velocity law
[5].

\medskip\noindent{\bf 6 Scattering: the line drag effect.} We consider
an effect which proves the fundamental importance of line scattering,
and that the pure absorption line case may be a degenerate limit
(Owocki \& Rybicki 1986). Lucy (1984) found that line scattering, via
a so-called line drag effect, may prevent wind instability! The origin
of the effect is simple. A wind parcel may again experience a
perturbation $+\delta v$. The parcel leaves the absorption shadow of
gas lying closer to the star, leading to instability. However, the
parcel experiences also a stronger, backscattered radiation field from
larger heights, because it is Doppler-shifted {\it into resonance}
with {\it faster} gas further out. Hence, the inward push
grows. Assuming a plane-parallel atmospheric slab and that the star
fills a hemisphere, and furthermore that scattering is fore-aft
symmetric, one shows that the direct and diffuse force perturbations
in an optically thick line cancel exactly.  We leave consideration of
the {\it outward} pointing diffuse force from gas lying at smaller
height than the perturbation site to the reader.

For spherically symmetric flow, the instability growth rate is back to
50\% of its pure absorption line value at one stellar radius above the
photosphere, and reaches 80\% of the absorption value at large radii
(Owocki \& Rybicki 1985).  Still, the line drag effect is of great
importance both numerically, preventing fast growth at the inner
boundary, and in nature, with regard to the much-speculated
`photospheric connection', i.e., whether wind structure grows from
photospheric perturbations or is wind intrinsic (Henrichs 1986;
Henrichs et al.~1990).

\bigskip
\noindent{\sc \S 5 Evolved wind structure}
\bigskip
\label{par5}

After decades of fascinating and frustrating mathematical research on
nonlinear growth of fluid instabilities (Landau equation, bifurcation,
chaos \& catastrophes), computers have shown a pragmatic way into
evolved, unstable flow via direct, time-dependent simulations. In this
and the next sections, we discuss the structure of fully developed,
unstable line driven winds. To model the evolution of the instability,
a standard Eulerian grid code is used. For the pure hydrodynamics
part, we coded a program following the detailed, technical
descriptions given in Hawley, Smarr, \& Wilson (1984), Norman \&
Winkler (1986), Reile \& Gehren (1991), Stone \& Norman
(1992a,b). Some of the techniques are also summarized in [1]. The
applied techniques comprise: `consistent' advection (Norman, Wilson,
\& Barton~1980) using van Leer (1977) or `piecewise parabolic'
interpolants (Colella \& Woodward 1984) on control volumes of
staggered grids; non-reflecting Riemann boundary conditions (Hedstrom
1979; Thompson 1987, 1990) ; tensor artificial viscosity (Schulz
1965); pressure predictor (Norman \& Winkler 1986).

As for the radiative line force, I followed Owocki et al.~(1988) which
includes technical subleties like line-list cutoff; a
Schuster-Schwarzschild photospheric layer to prevent unstable growth
close at the inner boundary; and the strict one-sidedness of the
direct force. The SSF method for treating the diffuse radiation field
is described in Owocki (1991, 1992), an unpublished draft from 1990,
and in Owocki \& Puls (1996). Except for one workshop paper (Owocki
1999), all simulations published so far are in radial direction only,
assuming spherically symmetric flow. CPU time requirements for 2-D
wind instability simulations are huge. The present work is no
exception, and deals with spherical symmetric simulations only.

\medskip\noindent{\bf 7 The basic entity of wind structure.} Figure~4
(taken from paper [1]) shows the evolved wind structure. The
pronounced features are: (1) broad rarefaction regions of accelerating
gas; this gas is (2) braked in strong reverse shocks, and fed into (3)
narrow, dense shells. (4) The shells propagate into gas which remains
close to stationary initial conditions. If forward shocks occur at all
at the outer shell edges, they are weak.

As a seed perturbation for wind instability, a coherent photospheric
sound wave of period 5,000 sec and pressure amplitude 1 percent is
introduced at the inner boundary of the model. The wave period
determines the spacing of wind shells. Namely, the perturbation
wavelength in the photosphere, $\lambda=aT=0.009\rast$ (with sound
speed $a$, period $T$, and stellar radius $\rast$), is stretched in
the accelerating flow by a factor $v_\infty/a=90$. Indeed, the shell
distance far out in the wind is $0.8\rast$, cf~the figure.

\medskip\noindent{\bf 8 Subshells and overtones.} Figure~4 of
paper~[1] shows that 50~(!)~overtones of the photospheric sound wave
can be clearly distinguished in the wind. There is no indication of
stochasticity at all in the wind, hence the structure is strictly
deterministic. This is rather unexpected from glancing at the velocity
law between 2 and $5\rast$, in which region the wind appears to be
rather chaotic (not meant in the new, technical sense of the word). Up
to these heights, collisions occur between multiple shells per
photospheric excitation period. As noted in [1], these sub-shells are
related to non-linear steepening and harmonic overtones. The dynamical
details are still not clear. Figure~\ref{figjko} shows the mass loss
rate in the wind as function of radius and time. After initial
transients have died out, the wind settles to a limit cycle. Per
photospheric perturbation period, three subshells are created and
mutually collide with each other up to around $3\rast$.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=15cm
 \epsffile{figjko.ps}
 \caption{\label{figjko} Wind mass loss rate as function of radius and
time. After initial transients have died out, the wind settles to a
limit cycle. Per photospheric excitation period, three subshells are
created and mutually collide.}
 \end{center}
 \end{figure}

\medskip\noindent{\bf 9 Two-stage instability.} Even for these
periodic models the de-shadowing instability is sufficiently delicate
that new details become visible when model parameters are
varied. Figure~5 below shows a run which differs from the above one
mainly by a shorter perturbation period of 1,000 seconds (besides
this, it was performed 7 years later). The difference is marked. In
the new model, sub-shell collisions terminate already around $2.5
\rast$. Afterwards, the shocks decay quickly, out to $6\rast$. Between
7 and $9\rast$, {\it new shocks} occur, which was not the case in the
old model.

 \label{1205011647}

How can new shocks occur after the instability went into saturation,
and left a fully developed flow? What happens here is a {\it second}
stage of the instability. We noted above that negative velocity
perturbations saturate quickly, and the wind remains close to
stationary initial conditions in regions ahead of pronounced
shells. Looking closely at the region from 4 to $6\rast$ in Fig.~5,
one sees a growing velocity perturbation: in the evolved wind
structure, eventually the pseudo-stationary regions become unstable,
too, via secondary perturbations. We expect that the latter are
related to overtones. The new perturbations steepen into new
shocks. The shocks accelerate, overtake and merge with the `old' shock
front from the first growth phase of the instability. This creates a
single, strong shock.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=12cm
 \epsffile{fig4.ps}
 \caption{Snapshot of an O~supergiant wind velocity law and density
stratification. A periodic sound wave of 5,000 sec period is applied
as photospheric seed perturbation for the de-shadowing instability.
Dots show individual mesh points. The numbers in the upper panel refer
to the sub-shells per excitation period. Taken from [1].}
 \end{center}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=12cm
 \epsffile{fig5.ps}
 \caption{Snapshot of the wind velocity law, assuming a photospheric
sound wave with period 1,000~sec. Two stages of unstable growth are
seen. In the second stage, perturbations grow within the
quasi-stationary gas ahead of dense shells, and lead to secondary
shocks.}
 \end{center}
 \end{figure}

\medskip\noindent{\bf 10 Lamb ringing.} Introducing coherent sound
waves as instability seeds allows to study the ultimate fate of linear
perturbations from harmonic stability analysis. But how relevant are
coherent perturbations physically? Below we shall argue that, to
understand the observed X-ray emission from hot star winds, one has to
refrain from periodic, coherent perturbations and consider random ones
instead. Still, relatively coherent perturbations may be expected from
stellar pulsations. They cannot be modeled in 1-D simulations, where
only pressure or $p$ modes (sound waves) can be excited. Internal
gravity waves or $g$ modes cannot propagate vertically through an
atmosphere, since lifting a planar atmospheric layer does not give a
buoyancy force.

Interestingly, coherent atmospheric perturbations do actually not
require an external piston like stellar pulsations, but could be
excited intrinsically in the atmosphere: an atmospheric resonance
frequency exists, Lamb's {\it acoustic cutoff}.

As the reader may already get weary of deflections from the straight
path brought about by `unexpected' hydrodynamic effects shooting in
from left and right, we add a historical note, trying to emphasize the
importance of the acoustic cutoff. Lamb built on earlier work by
Rayleigh (1890), who first derived the dispersion relation for sound
waves in an isothermal barometric density stratification. Rayleigh
failed to see the physical relevance of the frequency $\omega_a= a/2H$
defined by atmospheric parameters ($a$ the sound speed, $H$ the scale
height). Only after the detection of a single-frequency, atmospheric
response after the big Krakatao volcano eruption, Lamb realized the
importance of $\omega_a$ as an atmospheric resonance. His argument
(Lamb 1908), which is far from trivial, can be found on p.~544 of his
`Hydrodynamics' (1932). Especially, he proves that in an isothermal
atmosphere, a white-noise sound spectrum evolves into a one-spike
spectrum, the spike located at the acoustic cutoff period. The
acoustic cutoff is therefore indeed a resonance.

The acoustic cutoff was held responsible for the 5 minute oscillations
of Sun (Schmidt \& Zirker 1963; Meyer \& Schmidt 1967). But the
appropriate acoustic cutoff period is 3 to 4 min, not 5 min. Ulrich
(1970) developed the alternative and correct theory that the 5 min
oscillations correspond to acoustic waves trapped in a resonance
cavity reaching from the deep solar interior to the top of the
convection zone. This established the field of helioseismology.
Deubner (1973) observed a photospheric subsignal of 3 min period,
where the wave trains are correlated with the appearance of bright
granules. These granules are thought to struck the photosphere from
below, and excite Lamb ringing. For a review on solar 5 minute
oscillations, see Stein \& Leibacher (1974).

Simulations of line driven winds from O~stars show a self-excitation
of the atmosphere at the acoustic cutoff frequency. Its `perturbing'
influence on the outer wind structure was occasionally noted in the
literature (e.g., Blondin et al.~1990), but not traced back to its
physical cause. We close this excursion by expressing our belief that
`Lamb ringing' could be responsible for rather coherent trains of
pronounced shells in hot star winds.

\medskip\noindent{\bf 11 Reverse shocks.} Reverse shocks occur because
the instability steepens the velocity law of the wind, and the fast
gas is eventually decelerated. As a reminder, we add here the
distinction between reverse and forward shocks: forward shocks
overtake slow gas and accelerate it. In any reference frame, the shock
is faster than both the pre- and postshock gas. Forward shocks occur
in explosions. Reverse shocks, on the other hand, decelerate fast
gas. The shock is slower than the gas on both sides. A reverse shock
occurs if a supersonic stream hits a wall or an obstacle.

\newpage%play\medskip
\noindent{\bf 12 Rarefaction regions.} The rarefaction regions
and reverse shocks in numerical wind simulations are quasi-stationary.
Seen from a comoving frame, they change only little. Rarefaction
regions can therefore, to a good approximation, be identified with
(patches of) `steep' wind solutions (Owocki, priv.~comm.; Feldmeier et
al.~1997c). This type of solution to the stationary Euler wind
equation will be discussed in the next chapter.

If a stationary rarefaction region is to feed gas into a shell through
a reverse shock, a gas source must exist. Rarefaction regions lie
directly above depression regions of the velocity law (negative Abbott
half waves). Here, the flow does not evolve, but maintains the initial
conditions. The depression region extends inwards, to the outer edge
of the next shell.

\medskip\noindent{\bf 13 Contact discontinuity.} Assuming a Sobolev
line force and zero sound speed, a contact discontinuity separates the
rarefaction and depression regions. The velocity law is continuous
there but has a kink. Using terminology only introduced on
page~\pageref{kinkint}, the wind jumps from the critical, initial
conditions to a {\it steep} solution. Because of subcritical mass flux
in the latter, a density discontinuity occurs at the contact
discontinuity. Gas cannot penetrate through a contact discontinuity,
hence we are still left without gas source to feed the next outer
shell.

\label{hilbert1} Any discontinuity in a derivative of the fluid
variables $v$ and $\rho$ propagates at characteristic speed (Courant
\& Hilbert 1968). Indeed, the contact discontinuity moves at sound
speed. Yet, since $a=0$, it moves along with the fluid. Allowing for
finite sound speed instead, the `gate opens' and the rarefaction
region eats slowly into the depression. Constant mass flux is
maintained through the rarefaction region. The situation is sketched
in Fig.~6, which also shows earlier and later stages in the evolution
of the instability. Figure~7 shows results from a numerical
simulation.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=15cm
 \epsffile{fig6.ps}
 \caption{Evolution of winds structure from the de-shadowing
instability (schematic).}
 \end{center}
 \end{figure}

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=10cm
 \epsffile{fig7.ps}
 \caption{\label{pagefig7} How the rarefaction zone eats through the
mass reservoir. The plot shows subsequent snapshots of wind density
(logarithmic) as function of radius (linear). No scales are given on
the axes, since the same structures occur for small or large
perturbation periods, at small or large separation between pronounced
shells. The dotted lines show the underlying, stationary wind
model. The gas reservoir ahead of dense shells is unaffected by
instability, and has practically {\it stationary} densities.}
 \end{center}
 \end{figure}

\medskip\noindent{\bf 14 Forward shocks and EISF noise.} We come to
what may at first seem a rather subtle, technical issue. The central
importance of line scattering to line driven wind hydrodynamics will,
as we hope, become clearer throughout the subsequent
discussions. Owocki \& Rybicki (1985) derived from linear stability
analysis that the perturbed diffuse radiation field turns
anti-correlated density and velocity fluctuations into correlated
ones. Anti-correlated fluctuations steepen into reverse shocks,
correlated fluctuations steepen into forward shocks. In SSF, only the
mean diffuse radiation field is treated, and photospheric
perturbations evolve into reverse shocks. If, on the other hand, the
perturbed diffuse radiation field is included, the phase lag between
velocity and density perturbations may be inverted, and strong {\it
forward} shocks may occur instead of reverse shocks. This argument was
first made by Puls (1994), and stimulated development of the EISF
method.

\label{gemma1}

Owocki \& Puls (1999) proved from EISF simulations, which include the
perturbed scattered radiation field, that forward shocks are not
important in the unstable wind. EISF and SSF wind structure are
essentially identical, and are both dominated by reverse shocks. The
reason is that phase reversal occurs only for short scale fluctuations
below the Sobolev length (Owocki \& Rybicki 1985). Short scale
perturbations saturate at velocity amplitudes of order $\vth$ (see
page~\pageref{10030101}). Correlated perturbations and forward
`shocks' (if at all) appear therefore as short scale, small amplitude
noise superimposed on the long scale, large amplitude tilt of the
thermal band leading to reverse shocks.

\medskip\noindent{\bf 15 Are shells enclosed by forward shocks?}
Furthermore, forward shocks are {\it not} required to enclose the
shells on their outer edges, to prevent them from expanding away. The
argument is simple (Feldmeier et al.~1997c): shells are geometrically
thin and consist of subsonic post-shock gas. Hence, their internal
sound crossing time is small compared to the flow time, and
hydrostatic equilibrium can be assumed. According to the equivalence
principle, the outward directed acceleration of the shell by the line
force is indistinguishable from an inward gravity binding a static
atmosphere. Hence, the shells are held together without the necessity
of outer forward shocks.

\medskip\noindent{\bf 16 Shell densities.} This same argument can be
used to estimate shell overdensities with respect to smooth,
stationary gas densities.  The inner-shell gas expands via thermal
pressure. This thermal pressure and shell acceleration define a
(`gravitational') scale height, $H$, which measures the shell
thickness. We approximate the line acceleration by
$v_\infty^2/n\rast$, where $v_\infty$ is the wind terminal speed and
$n$ is `a few'; hence, $H=n\rast a^2/ v_\infty^2$. Assuming that all
gas from within a rarefaction region of average thickness $\lambda/2$
($\lambda$ being the perturbation wavelength) is fed into the shell;
and that $\lambda\approx\rast$ for the longest perturbations which can
still grow into saturation, the overdensity, $o$, of shell gas with
respect to stationary wind gas becomes,
 %
 \begin{equation} 
 o \approx {\lambda \over 2H}={1 \over 2n} \,\left({v_\infty
\over a}\right)^2.
 \end{equation}
 %
 For $v_\infty/a \approx 100$, this is of order $\ge 10^3$, in good
agreement with numerical simulations. At large radii, radiative
acceleration of the shell ceases, and $H$ grows. The shells should
have expanded away, and the wind be homogeneous again by 30 to
$100\rast$. This, too, agrees with numerical simulations, see Fig.~6
in [4].

\medskip\noindent{\bf 17 Inner shell velocity law.} At first
surprisingly, the velocity law has a {\it negative} gradient inside
shells (Owocki 1992; [3]). This is also true for {\it solar} wind
shells (Simon \& Axford 1966). The reason is that the velocity law
inside the shell reflects the gas history. Gas lying close to the
outer shell edge was shocked earlier than gas lying near the inner
edge, close to the reverse shock. The shell velocity law is close to a
stationary CAK velocity law, which means that the shell is constantly
accelerated on its trajectory. With the gas velocity at the reverse
shock increasing in course of time, a negative velocity gradient
results inside the shell.

\newpage%play 
\noindent{\sc \S 6 Wind structure and line profiles} 
\bigskip
\label{par6}

\noindent{\bf Motivation.} Quantitative spectroscopy of UV and optical
lines (especially of H$\alpha$, Puls et al.~1996) allows to determine
mass loss rates, terminal speeds, and metal abundances of the
wind. All this information (plus the stellar radius) flows into the
wind-momentum luminosity relation, from which the stellar luminosity
is inferred (Kudritzki et al.~1999). The premise is that hot stars
will in the near future become a primary distance indicator of the
same quality as Cepheids. It is therefore important to understand the
influence of wind structure on line formation.

\medskip\noindent{\bf P Cygni lines from 1-D, structured wind models.}
Puls et al.~(1993, 1994) calculated P~Cygni line profiles for
resonance lines from winds structured by the de-shadowing
instability. Radiative transfer in the highly non-monotonic velocity
law is solved for by iterating the source function, to account for
multiple resonance locations (Rybicki \& Hummer 1978). The resulting
line profiles agree well with profiles from stationary wind models.
This is at first surprising, given the `large amount' of structure in
the wind, cf.~Figures 4 and 5, and Figure~8 below. The reason for the
agreement is that most of the {\it mass} in the unstable, structured
wind still follows a smooth, CAK-type velocity law. The reasoning is
actually more subtle, and we refer to Puls et al.~(1993) for an
in-depth discussion. Some of the differences between line profiles
from stationary and unstable, time-dependent wind models resemble
observed variability features like black troughs, narrow absorption
components (NACs), and blue edge variability.

\medskip\noindent{\bf Observed line variability, and ideas of its
origin.}  Besides X-ray emission, which is discussed further below,
the major observational evidence for pronounced flow structure in
winds from hot stars comes from variability in optical and UV spectral
line profiles.  We give a brief, phenomenological overview of
variability features.  The books edited by Moffat et al.~(1994) and
Wolf et al.~(1999) are good entry points in the large body of
literature.

Of central importance are DACs (discrete absorption components), found
in unsaturated P~Cygni line profiles of OB stars, and so-called
discrete wind emission elements, observed in flat-topped emission line
profiles from W-R stars (Moffat 1994; L\'epine \& Moffat 1999) and
O~supergiants (Eversberg, L\'epine, \& Moffat 1998). The DACs come in
company of `bowed variation contours' or simply `bananas' (Massa et
al.~1995). DACs are pure absorption phenomena, while bananas are
modulative. Bananas are explained by dense spiral-arm structures in
the wind (Owocki et al.~1995; Fullerton et al.~1997), similar to CIRs
(co-rotating interaction regions) in the solar wind. Their period is an
even divisor (2 or 4) of the rotation period. The origin of the DACs
was first supposed in the de-shadowing instability, but this was later
excluded (Owocki 1994). Then CIRs were suspected as their origin, but
they can only explain the much faster bananas. Hence, DACs remain
enigmatic. A promising idea is that Abbott waves cause velocity
plateaus in the wind which lead to enhanced absorption and DACs
(Cranmer \& Owocki 1996). Interestingly, velocity plateaus and not
density enhancements were the first idea to explain DACs (Hamann
1980). Discrete wind emission elements, on the other hand, are thought
to be caused by compressible blob turbulence. Whereas turbulence in
incompressible fluids leads to an eddy cascade, supersonic or
compressible turbulence leads to a shock cascade. The eddy cascade is
direct, with big eddies feeding energy into small eddies. The shock
cascade, on the other hand, is inverse: strong, fast shocks overtake
weak, slow shocks, and merge with them into one strong and fast shock.
Hence, energy is transferred from small to large scales. A connection
may exist between this compressible turbulence and the wind clouds
thought to be responsible for X-ray emission from hot stars. This is
discussed in the following.

\medskip\noindent{\bf Wind clumping.} Terminal speeds and mass loss
rates are the central parameters of line driven winds. They determine
the metal entrichment of the interstellar medium, and the star
formation rate in starbursts.  Furthermore, wind mass loss determines
the evolution of hot, massive stars. And finally, knowledge of wind
parameters allows to derive stellar luminosities and distances, a
matter of prime importance in astronomy. Hence, reliable measurements
are needed for $v_\infty$ and $\dot M$. However, the mass loss rates
of Wolf-Rayet stars, and probably also of O~supergiants, are
fundamentally affected by instability-generated flow structure. This
was first realized by Hillier (1984), who introduced wind {\it
clumpiness} to explain observed electron scattering wings of emission
line profiles in Wolf-Rayet stars. The line core depends linearly on
gas density, whereas the wings develop with density squared. Hence,
line wings are pronounced in clumped winds. Quantitative profile fits
indicate that the wind gas only fills 10 to 30 percent of the
available volume around the star (Hamann \& Koesterke 1998b). This
leads to a reduction of mass loss rates for W-R stars by factors of 2
to 4, clearly demonstrating the importance of hydrodynamic wind
structure for quantitative spectroscopy of hot stars with winds. This
reduction in the formerly tremendous mass loss rates of W-R stars also
opened the way to new, quantitative modeling of their winds being
radiatively driven (Lucy \& Abbott 1993; Springmann 1994; Gayley et
al.~1995; Gr\"afener, Hamann, \& Koesterke 2000).

\bigskip
\noindent{\sc \S 7 Including energy transfer}
\bigskip
\label{par7}

\noindent{\bf 18 Radiative shocks.} Wind shocks are {\it radiative}
shocks, consisting of a narrow viscous layer in which the gas is
heated, and a subsequent cooling zone in which the gas cools again by
radiative losses. So far we assumed implicitly that radiative cooling
is very efficient in the wind, and that shock cooling zones are
narrow. Radiative shocks can then be viewed `from far' as isothermal
shocks, since both heating {\it and} cooling occurs on microscopic,
unresolved length scales. In this approximation, a solution of the
energy equation in the wind is not required. This is the reasoning
which led OCR to undertake {\it isothermal} wind calculations.

However, the numerical simulations discussed above make the assumption
of isothermality questionable at certain heights above the
photosphere, and isothermality seems not justified a posteriori. We
found that the wind gas is highly rarefied at the end of a rarefaction
region, before it undergoes the reverse shock transition. Efficient
radiative cooling {\it cannot} be assumed. To find a self-consistent
wind structure including the effects of radiative cooling (possibly on
long scales), Cooper \& Owocki (1992) included radiative cooling in
numerical wind simulations for the first time. This led to the
strange, unexpected results of unresolved cooling zones, and all
shocks were still isothermal. As the numerical mesh was chosen
sufficiently fine to resolve cooling zones, the conclusion was that
the latter got somehow {\it collapsed}.

\medskip\noindent{\bf Advective diffusion.} We offer in this and the
next paragraph two alternative explanations for cooling zone collapse.
The explanations are slightly technical, and the reader primary
interested in the main physical argument may want to skip two
paragraphs ahead. -- Owocki (1993, private communication; see also
Cooper 1994) explained shock collapse by {\it advective
diffusion}. This is a manifestation of Field's (1965) local thermal
instability. Consider a propagating temperature jump, i.e., a contact
discontinuity. Diffusive errors of the advection scheme spread the
jump over a few grid points. The gas at intermediate temperatures
cools better than hot gas: in pressure equilibrium, cold gas is denser
than hot gas, giving more collisions and stronger cooling. The
broadened jump is sharpened again by the different cooling rates, and
thereby gets slightly shifted into the hot gas. The jump introduces
new diffusive errors, and the cycle repeats. Advective diffusion
should occur within the viscous shock layer, which is spread out over
$\approx 3$ grid points by artificial viscosity. The shock front
`eats' then through its own cooling zone.  This argument assumes that
errors pile up, as they indeed do at a contact discontinuity where
always the same gas is located. But this is {\it not} the case in a
shock transition, where gas passes through.  A detailed calculation
shows that only a slight modification of the cooling zone results from
advective diffusion, but no collapse [1].

\medskip\noindent{\bf 19 Oscillatory thermal instability.} Besides
Field's local thermal instability, a second, {\it global} thermal
instability occurs in radiative shocks. This instability was found in
numerical simulations of accretion columns onto white dwarfs (Langer
et al.~1981, 1982) in magnetic (AM~Her) cataclysmic variables. The
linear stability analysis is due to Chevalier \& Imamura (1982), and
the instability mechanism is explained in detail in Langer et
al.~(1982), Gaetz, Edgar, \& Chevalier~(1988), and Wu, Chanmugam \&
Shaviv~(1992).  The instability is of oscillatory type, and causes
periodic contraction and expansion of the cooling zone. The
contractions are strong, and a fine grid is required to resolve
them. On a coarse grid, the contracted zone drops at some point below
grid resolution, and an {\it isothermal} shock remains [1]. Obviously,
there is no thermal instability for an isothermal shock: the shock
will not re-expand, and the collapse is permanent. We add as a side
remark that oscillating radiative cooling zones show rich
dynamics. For example, tiny condensations within the cooling zone can
grow into secondary shocks and propagate through the cooling zone
(Innes et al.~1987).

\medskip\noindent{\bf Altering the cooling function.} We assume in the
simulations below that radiative cooling is parameterized in power law
form, $\Lambda= A \rho^2 T^\delta$ (units ${\rm erg}\, {\rm s^{-1}}\,
{\rm cm^{-3}}$). The parameter $\delta$ is derived from fits to
calculated cooling functions (Cox \& Tucker 1969). Global thermal
instability occurs for $\delta\lea 1/2$. We find that the above
cooling zone collapse is prevented when an artificial, stable exponent
$\delta> \delta_c$ is used at low temperatures, as is demonstrated in
Fig.~6 of [1]. Typically, assuming $\delta =2$ at $T<5\times 10^5$~K
prevents cooling zone collapse in wind simulations. X-ray spectra of
O~stars, in which we are primarily interested, indicate temperatures
between $10^6$ and $10^{7.5}$~K.  Hence, X-ray emission should be
largely unaffected by the modified cooling function.

\medskip\noindent{\bf 20 Shock destruction.} How does radiative
cooling influence the wind structure? What happens at intermediate and
large radii in the wind, when radiative cooling ceases to be efficient
in rarefied gas undergoing a reverse shock transition? Using the above
method, we find in [1] that isothermality is a good approximation for
O~supergiant winds out to $\approx 5\rast$. At these radii, reverse
shocks are {\it suddenly destroyed}, in marked contrast to their
gradual decay in isothermal calculations, which continues out to
$\approx 20\rast$. A direct comparison is made in Figures~1 and 10 of
[1].

\medskip\noindent{\bf 21 Shocks becoming adiabatic.} The reason for
shock destruction is the depletion of intershell gas. Radiative
cooling becomes inefficient, and the cooling zones broaden as they
become adiabatic. This drives the shock front through the rarefaction
region, towards the next inner shell. Since the shock propagates into
the preshock gas, the postshock temperature raises, which makes
radiative cooling even more inefficient, pushing the shock {\it
further} into the gas. Actually, this is the very mechanism of the
global thermal instability of Langer et al.~(1981).  Eventually, the
shock merges with the next inner shell (this may create forward
shocks, cf.~page~\pageref{forwsho}). Tenuous, hot gas at temperatures
$>10^7$~K fills the whole space between shells.

Why does shock destruction occur so suddenly around $5\rast$? At this
height, the gas reservoir ahead of a shell, which remains first at
stationary conditions, is used up, i.e., was fully fed into the next
outer shell. Further sub-shells -- we term them `clouds' in the
following, for reasons which will become apparent -- cannot occur in
between shells. These clouds propagated outwards and pushed ahead of
them the adiabatic shock front which tried to expand in the opposite,
inward direction. Once the clouds cease, the shock front can expand
freely through empty intershell space, leaving hot, thin gas.

\medskip\noindent{\bf 22 Outer corona.} Volume filling factors of hot
gas at 10$^6$ to 10$^7$~K can almost reach unity between $5\rast$, the
location of shock destruction, and $20\rast$, where hot gas has
significantly cooled by adiabatic expansion [1, 4]. In winds from OB
supergiants, X-ray emission from this low density gas is negligible.
It is still possible that an X-ray emitting, outer corona occurs in
thin winds from B stars near the main sequence. Here, the first strong
wind shock can heat large fractions of the gas. UV line profiles show
indeed that the cold wind vanishes before it reaches terminal speed
(Lucy \& White 1980; Hamann 1981b).

\bigskip 
\noindent{\sc \S 8 X-rays and clouds}
\bigskip
\label{par8}

\noindent{\bf 23 Clouds.} The sub-shells or clouds are not quite the
accidental, secondary feature as which they were treated so far
(cf.~secondary shock formation on page~\pageref{1205011647}.) We will
argue in the following that clouds are a primary agent in explaining
X-rays from hot, massive stars [4]. We distinguish from now on
strictly between {\it shells} (or shell fragments) and {\it
clouds}. Shells (shell fragments) are instability generated, highly
overdense as compared to stationary wind densities, and have probably
large lateral scale even after Rayleigh-Taylor fragmentation (cf.~the
sketch in Figure~2). Clouds, on the other hand, are
turbulence-induced, have roughly stationary wind densities, and, if
turbulence is approximate isotropic, a tiny lateral scale.

Hillier et al.~(1993) estimated that OCR wind models should fail by
one or two orders of magnitude to produce the observed X-ray emission
from hot stars. The reason is the low density of shock heated gas
immediately behind the reverse shock. The gas density lies orders of
magnitude below stationary wind densities. Note that X-ray emission,
as a collisional process, scales with the density squared. We found in
[1] that large-amplitude, periodic perturbations in the photosphere
lead to chaotic wind structure, whereas strictly deterministic flow
results from low-amplitude perturbations (cf.~Figures~10 and 12 in
[1]). We speculated, therefore, that {\it random} or turbulent
boundary perturbations, still of small amplitude, could lead to an
`active' wind showing enhanced X-ray production. This is indeed the
case, and clouds are the means of enhanced X-ray emission. For random
boundary perturbations, wind clouds are prevalent, far above the level
of a few `overtone' sub-shells in models with coherent base
perturbations (Figure~4). Clouds are dense and collide with shells,
leading to strong X-ray emission.

\medskip\noindent{\bf 24 Langevin boundary conditions.} To mimic
photospheric turbulence, a velocity perturbation, $u$, is applied on
the inner simulation boundary [4]. The perturbation fulfills the
Langevin equation for a continuous Markov chain, $du/dt + u/t_c=
\Gamma(t)$. This equation is integrated in time-forward manner. The
correlation time, $t_c$, is a free parameter. We choose a value not
too far from the acoustic cutoff period. $\Gamma$ is the stochastic
force, with a white-noise correlation function. The force amplitude is
the second free parameter, typically chosen at 30\% the speed of
sound. This is well below the limit of measured turbulent velocity
dispersion in hot star atmospheres (Conti \& Ebbets 1977). The power
law index of this turbulence model is $-2$, not too different from the
Kolmogorov index $-5/3$ for eddy turbulence. Some classic papers on
the Langevin equation are collected in Wax (1954).

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=10cm
 \epsffile{fig8.ps}
 \caption{Clouds, marked with filled circles, in a snapshot of
the density stratification in an O~supergiant wind.}
 \end{center}
 \end{figure}

\newpage%play\medskip
\noindent{\bf 25 Cloud ablation.} We expect that the average
time interval between the passage of two dense shells is not too
different from the acoustic cutoff period. Clouds, on the other hand,
are excited at much shorter periods. They form in the dense gas ahead
of a shell, i.e., in the decelerating part of a harmonic perturbation
where unstable growth quickly saturates. The birth density of clouds
is the stationary wind density, and they roughly maintain this density
until collision with the next outer shell. In Fig.~8, clouds are
marked explicitly in a snapshot taken from a numerical
simulation. Figure~9 shows the cloud dynamics, their propagation
through empty rarefaction regions between shells, and collision with
the latter.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=6cm
 \epsffile{fig9.ps}
 \caption{Clouds, marked with +, $\times$, $\vartriangle$, $\square$,
and $\ast$, are ablated from the gas reservoir ahead of a pronounced
shell, propagate through the rarefaction region, and collide with the
next outer shell. Compare with Fig.~\ref{pagefig7} on
page~\pageref{pagefig7}.}
 \end{center}
 \end{figure}

\medskip\noindent{\bf A model case: $\zeta$~Ori.} The O9 supergiant
$\zeta$~Ori is a standard object for X-ray observations from
O~stars. Most recently, it was the first O~star for which an X-ray
line spectrum was taken with the {\sc Chandra} satellite. With
excellent {\sc Rosat} observations available at the time (1996), we
made $\zeta$~Ori the test case for the turbulent-cloud
scenario. Figure~13 in paper~[4] shows a snapshot of the wind
structure of $\zeta$~Ori. The X-ray spectrum from this snapshot is
shown in Fig.~14 of the same paper. The spectrum is calculated using a
formal integral approach on the 1-D, time-dependent wind. K-shell
opacities are assumed for cold gas, Raymond-Smith emissivities for hot
gas. The model flux is only a factor of 2 or 3 below the observed one,
but not by factors of 10 to 100 as estimated by Hillier et al.~(1993)
from OCR models.

\medskip\noindent{\bf 26 Cloud-shell collisions.} Strangely then,
Figure~15 in [4] shows that all the X-ray emission arises from a
single shock at $4.5\rast$. The gas density in this shock lies between
$\pm 1$~dex of the stationary wind density, far higher than in the
rarefaction region. What happens at this location? Figure~17 in [4]
shows that a collision of a fast cloud with a dense shell takes place
at this time, at this radius. The figure shows also that {\it any}
appreciable X-ray emission from the wind is due to cloud-shell
collisions.

\medskip\noindent{\bf 27 X-ray flux constancy.} From Fig.~17 in [4] it
is also evident that no more than one or two cloud collisions take
place at any time. This leads to strong variability of the X-ray flux,
$\pm 1$ dex around mean in spherically symmetric models. By contrast,
observed X-ray flux variations for $\zeta$~Ori are a mere few
percent. We expect that spherical symmetry is a very poor
approximation for turbulent structures. A realistic wind model should
instead consist of independent radial cones, each with its own
cloud-shell collisions taking place. A few thousand such cones would
guarantee the observed flux constancy.

\medskip\noindent{\bf 28 X-ray variability.} There are only few
detections of X-ray variability so far. Bergh\"ofer \& Schmitt (1994)
report on an episodic raise in the {\sc Rosat} X-ray count rate of
$\zeta$~Ori.  This gained wide attention, for example in the science
section of the New York Times. The possible origin for the raise in
count rates is the breakthrough of a dense shell through an X-ray
photosphere. The event was found in one energy band only. Oskinova et
al.~(2001) propose observations at higher signal-to-noise, to detect
different variability levels in soft and hard bands. This would allow
to test certain aspects of the shock model.

Bergh\"ofer et al.~(1996) find {\it periodic} X-ray variability in
$\zeta$~Pup, which maybe caused by absorption of X-rays in a
co-rotating interaction region (a spiral arm) in the wind. Oskinova
(2001, priv.~comm.) finds other examples of periodic X-ray variability
in ASCA observations of O~stars. We recently proposed a long (180
ksec) observation of the O~star $\xi$~Per with the {\sc Chandra}
satellite, to find first signatures of X-ray {\it line} variability.
The reason for this expectancy is that this star has amongst the
shortest recurrence time scale for DACs (Kaper et al.~1999; de Jong et
al.~2001).

\medskip\noindent{\bf O~stars observed with ROSAT.} Dedicated {\sc
Rosat} observations are available for 42 stars in the spectral range
from O3 to O9. These data were analyzed assuming a 2-component wind,
consisting of cold, X-ray absorbing gas and a random ensemble of X-ray
emitting shocks. Density and temperature stratifications of radiative
cooling zones behind shocks are included in this approach (Chevalier
\& Imamura 1982; [3]). These postshock stratifications allow to reduce
the number of free fit parameters to the observations from 4 (Hillier
et al.~1993) or 3 (Cohen et al.~1997) to 2 (see [3]). Application of a
simplified version of the method is described in Kudritzki et
al.~(1996). We find there that the scatter in the $L_x/L_{\rm bol}$
relation for O~stars is smaller than from {\sc Einstein} data. We
currently try to decide whether this scatter is caused by a dependence
of $L_x$ on a second wind parameter, besides $L_{\rm bol}$.

\medskip\noindent{\bf 29 Adiabatic shells.} Paper~[3] also contains an
analytic treatment of outer, adiabatic shocks, whose cooling length is
{\it not} small compared to the wind scale. We follow there a classic
paper by Simon \& Axford (1966) on solar wind shells.

\medskip\noindent{\bf 30 Forward shocks, again.} We discuss in the
appendix of [3] that shock destruction as described above can be
mimicked by a simple numerical test. Instead of calculating a fully
structured wind model using an unstable line force, we consider
multiple adiabatic shells in a flow expanding spherically symmetric,
and at constant speed. \label{forwsho} The merging of reverse shocks
with the next inner shell leads there to a train of {\it forward
shocks}. Hence, it is presently not quite clear whether insufficient
radiative cooling causes shock destruction or shock transformation
(reverse to forward). This question is of some relevance since, for
the O~supergiant $\zeta$~Pup, soft X-rays originate from very large
radii, far above the location of the final cloud-shell collisions
(Hillier et al.~1993; Schulz et al.~2000).

\medskip\noindent{\bf Future work on X-ray emission.} What determines
the azimuthal scale of shell fragments and clouds? This is presently
the most pressing question. The demand on computer time is huge for
2-D hydrodynamic modeling of the wind instability. One has to employ
computational tricks, like using a specially designed mesh for a 3-ray
radiative transfer (Owocki 1999) or 2nd order Sobolev approximation
[5]. Both methods have their caveats: the specially designed mesh
stretches too strongly as function of radius, and instability
generated structure may `fall between the mesh nodes'. For 2nd order
Sobolev approximation, inclusion of nonlocal couplings (shadowing of
one shell by another) may be impossible (Wegner 1999).

Another major question is: why are X-rays from O~stars universal? Why
does the intricate wind hydrodynamics and radiative transfer lead to a
simple $L_x/L_{\rm bol}=10^{-7}$? And why does universality break down
for thin winds from B~stars (Cassinelli et al.~1994) and for
Wolf-Rayet stars (Baum et al.~1992; Wessolowski 1996; Ignace,
Oskinova, \& Foullon~2000)?

\medskip\noindent{\bf 31 Symmetric X-ray lines.} With the launch of
the X-ray satellites {\sc Chandra} and {\sc Xmm}, observations of
X-ray lines from O~stars has become possible. First observations of
O~stars led to the unexpected result that X-ray emission lines in
$\zeta$~Ori und $\zeta$~Pup (the usual suspects) are symmetric and
almost not blue-shifted (Schulz et al.~2000; Kahn et al.\ 2001;
Waldron \& Cassinelli 2001). The line width is roughly half the
terminal wind speed. These results are puzzling if X-rays originate
indeed in a dense wind, where one expects different optical depths for
photons from the front and back hemisphere of the star, resulting in
asymmetry or effective blue-shift of the line profile. However, Owocki
\& Cohen (2001) have recently shown that even in a homogeneous wind,
observed line symmetry can be partially attributed to low instrumental
resolution. Even better, recent {\sc Chandra} observations of
$\zeta$~Pup ``show blue-shifted and skewed line profiles, providing
the clearest evidence that the X-ray sources are embedded in the
stellar wind'' (Cassinelli et al.~2001).

This should be the final `out' for coronal models of X-ray emission
from O~supergiants (Hearn 1972, 1973; Cassinelli \& Hartmann 1977).
Coronal model were practically excluded due to missing K~shell
absorption (Cassinelli \& Swank 1983; Cohen et al.~1997) and the
missing green coronal iron line (Baade \& Lucy 1987). Recently,
however, coronal models gained again some attention (Waldron \&
Cassinelli 2001).

We plan to perform line-synthesis calculations in a {\it clumped}
wind, where almost all gas is confined to narrow shell fragments. If
the lateral scale of the fragments is not too small, photons from the
stellar back hemisphere may escape to the observer by passing between
neighboring shells (cf.~Figure~2). Furthermore, on the back hemisphere
one looks onto the X-ray emitting, inner rim of the shell; whereas on
the front hemisphere, the shock is hidden behind a dense, absorbing
shell. Both effects (lateral escape between neighboring shells; inner
shell rim emission only) level off differences between the red and
blue line wing.

\medskip\noindent{\bf 32 Clouds and EISF structure.} We add another
technical remark. The turbulent wind clouds described above and in [4]
have short length scales of a few mesh points only. They were
calculated using the SSF method, which does not account for the
perturbed diffuse radiation field. On short scales, the latter may
turn anti-correlated fluctuations like clouds into correlated
ones. Future EISF simulations are therefore vital to the cloud
model. To achieve proper resolution of the thermal band, an adaptive
mesh technique will be used.

\medskip\noindent{\bf X-rays in stationary wind models.} The focus of
the present work is on wind hydrodynamics, with a highly simplified
treatment of NLTE (using a power-law line list; and treating {\it one}
resonance line only) and radiative transfer (using Sobolev
approximation, SSF, or EISF). The complementary viewpoint: full NLTE
and radiative transfer, and assuming parameterized hydrodynamics is of
central importance for quantitative spectroscopy of hot stars. Precise
UV fluxes from OB~stars are required to calibrate the wind
momentum-luminosity relation (Kudritzki et al.~1999); to determine
ionizing fluxes in H\,{\sc ii}-regions (Sellmaier et al.~1996); and
for population synthesis in starbursts (Leitherer et al.~1999) and in
blue galaxies at redshifts $z>3$ (Steidel et al.~1996). In
synthesising UV fluxes, the full statistical equilibrium problem of
the wind is treated, however, assuming stationary flow (Pauldrach et
al.~2001, and literature therein). To explain the observed
`superionization', inclusion of X-rays is still required (Pauldrach et
al.~1994, MacFarlane et al.~1994). The shocks are treated in
parameterized form, on a underlying, monotonic velocity law.
Furthermore, inclusion of an artifically high microturbulence is
necessary to reproduce observed UV line profiles (Hamann 1980). In
future calculations on structured wind models, microturbulence and
shock strength will no longer be independent, free parameters, but
will both appear as consequence of wind hydrodynamics. Lucy (1982a,
1983) and Puls et al.~(1994) showed that black troughs in saturated
P~Cyg profiles, so far explained by microturbulence, indeed result
from an instability-generated velocity law.

\newpage%play\medskip
\noindent{\bf High mass X-ray binaries.} Neutron stars or
black holes which orbit an O or B supergiant act as point-like probes
in the wind. Bondi-Hoyle accretion of the supersonic wind turns the
compact object into a strong X-ray source which ionizes a large volume
of the wind gas. Line driving stalls in this overionized region, and a
dense wake forms via the Coriolis force. The beautiful and intricate
wind dynamics is discussed by Blondin et al.~(1990) and Blondin,
Stevens, \& Kallman (1991). The presence of the dense wake can be
infered from asymmetries of X-ray lightcurves at ingress and
egress. For the system Vela X-1, the azimuthal extent of the wake is
consistent with the observed asymmetry [2]. Kaper,
Hammerschlag-Hensberge, \& Zuiderwijk (1994) suggest that observed,
short-term variability of X-ray lightcurves is due to fluctuations in
the wind velocity as caused by the de-shadowing instability. Hence,
high-mass X-ray binaries are another test case to diagnose line driven
flow structure.

%---------------------------------------------------------------------

\clearpage
\bigskip
\noindent{\bf CHAPTER 3: RUNAWAY WINDS}
\label{chap3}

\bigskip
\noindent{\sc \S 9 Solution topology}
\bigskip
\label{par9}

We turn now to a simpler level in our description of line driven wind
hydrodynamics, and consider the simplest line force possible, in first
order Sobolev approximation. The wind is then {\it no} longer subject
to de-shadowing instability. We consider why line driven winds from
stars and accretion disks adopt a unique, critical solution out of an
infinite number of possible solutions, the latter falling into two
classes: shallow and steep.

This unique solution is defined as follows (Castor, Abbott, \& Klein
1975): it starts in the photosphere as the fastest possible, shallow
solution, and crosses at some critical point smoothly and
differentiably to the slowest, steep solution. From all accelerating
wind solutions, this is the one with maximum mass loss rate. Shallow
solutions are discarded as global wind solutions because they break
down at large radii, becoming imaginary there. Steep solutions, on the
other hand, are everywhere supersonic. They cannot connect to the wind
base which is assumed to be subsonic, and are also discarded as global
wind solutions.

There are indications that this reasoning is too restrictive (see
paper~[6]): (1) Shallow solutions break down around $300\rast$.  Why
then do time-dependent hydrodynamic simulations which extend out to
$10\rast$ always adopt the critical solution, instead of remaining on
shallow initial conditions? (2) Spherical expansion work, as a
thermodynamic effect, scales with the gas temperature and sound
speed. It is not {\it inherent} to line driving. (3) For line driven
winds from accretion disks, neither shallow {\it nor} the critical
solution reach infinity, because the disk flux drops off steeper than
disk gravity [6]. Unavoidably, the wind has to decelerate at large
heights. This is unproblematic, since the wind speed is above the
local escape speed, both for the critical and for sufficiently fast,
shallow solutions. (4) Once the wind is allowed to decelerate in
certain regions, an initially shallow solution could jump to the
decelerating branch when the speed is larger than the escape speed,
and reach infinity. Wind deceleration is also discussed by Koninx
(1992) and Friend \& Abbott (1986), the latter authors modeling
velocity laws of rotating O~star winds.

\medskip\noindent{\bf A simple wind model.} We consider now a simple
model for line driven winds which allows to discuss the above
questions in detail. The model is used in numerical simulations [8, 9,
10] to study wind runaway caused by Abbott waves. In Sobolev
approximation, the line force scales as
 %
 \begin{equation}
 {\bf g}_l \sim \int d\omega\, {\bf n} \,I_n \bigl(\rho^{-1}\, \pa_n
v_n\bigr)^\alpha.
 \end{equation}
 %
 The integral is over solid angle, {\bf n} is the unit direction
vector, and $I$ is the frequency-integrated intensity of the radiation
field. A CAK line distribution function is assumed, and we adopt
$\alpha=1/2$ for simplicity in the following. Furthermore, the
velocity gradient is taken out of the integral, assuming that the
gradient of the flow speed in flow direction gives the main
contribution. This is equivalent to the CAK point star
approximation. Assuming 1-D, planar geometry,
 %
 \begin{equation}
 g_l(z) \sim F \sqrt{\rho^{-1} \pa v/\pa z},
 \end{equation}
 %
 with frequency-integrated, radiative flux $F$ in $z$ direction. We
assume that $F$ is constant, but $g(z)$ may be an arbitrary
function. Constancy of both $F$ and $g$ leads to solution degeneracy
for zero sound speed (Poe et al.~1990). If the disk is extended and
isothermal (the latter is a bad approximation), $F$ is indeed constant
at sufficiently small $z$. For $g$ we choose
 %
 \begin{equation}
 g(z)={GM\over q^2} {z/q \over 1+(z/q)^2},
 \end{equation}
 %
 which roughly resembles effective gravity (gravity minus centrifugal
force) close to a thin accretion disk. $M$ is the mass of the central
object, $q$ is the footpoint radius in the disk. Note that
$g(0)=0$. Furthermore, the sound speed is set to zero, which turns the
Euler equation into an algebraic equation in $vv'$. Note that the
Sobolev line force is independent of $\vth$, and therefore of $a$. The
continuity and Euler equation become,
 %
 \begin{gather}
 \label{continuity}
 C\equiv \dot\rho + (\rho v)'=0,\\
 %
 \label{euler} 
 E\equiv \dot v +vv' +g - \tilde K \sqrt{v'/\rho} =0.
 \end{gather}
 %
 Dots indicate temporal, primes spatial differentiation. The flux was
absorbed into $\tilde K$. The Euler equation was divided through by
$GM/q^2$, which means that speed is measured in units of the local
Kepler speed, $\sqrt{GM/q}$, height in units of $q$, and time in units
of a Kepler flow time through a distance $q$. We keep the symbols $v$,
$z$, and $t$ also for the normalized quantities. For a stationary
wind, the continuity equation becomes $\rho v={\rm const}$. We
introduce $m=\rho v/\rho_c v_c$ and $w'=vv'$. Subscripts $c$ refer to
the critical or CAK solution, to be introduced shortly. The Euler
equation turns into,
 %
 \begin{equation}
 \label{eulerstat}
 w' + g - K\sqrt{w'/m} =0,
 \end{equation}
 %
 For stationary winds, $m$ and $w'$ replace $\rho$ and $v$ as
fundamental hydrodynamic variables.

\medskip\noindent{\bf Shallow and steep solutions.} At each $z$,
(\ref{eulerstat}) is a quadratic equation in $\sqrt{w'}$, with
solutions
 %
 \begin{equation}
 \label{eulerstatsol}
 \sqrt{w'} = {K \pm \sqrt{K^2-4gm} \over 2\sqrt{m}}.
 \end{equation}
 %
 Solutions with `$-$' are termed {\it shallow}, solutions with `+' are
termed {\it steep}. \label{kinkint} For sufficiently small $m$,
shallow and steep solutions are globally defined. At the critical
point of the critical solution, termed $z_c$ from now, the square root
vanishes, and two {\it globally defined} shallow and steep solutions
merge. By definition, $m=1$ for the critical solution. For $m>1$,
shallow and steep solutions become imaginary in a neigborhood of
$z_c$. In this region, gravity $g$ overcomes line driving $\sim
K$. These failing winds are termed {\it overloaded} [8]. We may also
allow for $v'<0$, by introducing $\sqrt{|v'|}$ in the line force. This
expresses that the Sobolev force is blind to the sign of $v'$. A {\it
single} decelerating branch results from this generalization.
Figure~\ref{figwkv} shows the `solution topology' of the quadratic
equation (\ref{eulerstat}) in the $zw'$ plane (see also Fig.~3 in CAK;
Fig.~3 in Cassinelli 1979; Fig.~4 in Abbott 1980; Fig.~1 in Bjorkman
1995; and Fig.~6 in [6]).

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=11cm
 \epsffile{figwkv.ps}
 \caption{\label{figwkv} Solution topology of the equation of motion
(\ref{eulerstat}) for stationary winds. The right panel shows
acceleration solutions, the left panel decelerating ones. For the
latter, $\sqrt{|w'|/m}$ was assumed in the line force. Note the saddle
point at $x=1, w'=1/2$, which is the CAK critical point for Abbott
waves.}
 \end{center}
 \end{figure}

From (\ref{eulerstatsol}), the critical point is a saddle point of $E$
in the $zw'$ plane (Bjorkman 1995). Note the important difference that
the sonic points for the solar wind and the Lavalle nozzle are saddle
points in the $zv$ plane instead. The critical point being a saddle,
the shallow solution with $m=1$ has a discontinuity in $v''$ at $z_c$.
This discontinuity is avoided by switching to the steep solution.
Discontinuities in {\it derivatives} of the fundamental hydrodynamic
variables lie on {\it characteristics} (Courant \& Hilbert
1968)\label{hilbert2}. Characteristics are space-time curves along
which Riemann invariants or wave amplitudes are constant, or change
according to an ordinary, not a partial differential equation. Indeed,
the flow becomes super-abbottic at the CAK critical point (Abbott
1980).

The constant $K$ can be expressed in terms of flow quantities at the
critical point. Setting the square root in (\ref{eulerstatsol}) to
zero gives, for $m=1$,
 %
 \begin{equation}
 K = 2\sqrt{g_c},
 \end{equation}
 %
 where $g_c=g(z_c)$. Since the critical point is a saddle of $E$ in
the $zw'$ plane, $\pa E/\pa z_c= \pa E/\pa w'_c=0$ holds. The latter
(`singularity') condition leads to $w'_c=g_c$. The former
(`regularity') condition leads to $dg/dz_c=0$, hence the critical
point coincides with the gravity maximum. This reveals the role of the
critical point as bottleneck of the flow. For height-dependent $F$,
$g/F^2$ determines the nozzle function instead. If $F$ {\it and} $g$
are constant, the critical point degenerates, and each location $z$
becomes critical (for zero sound speed). Inserting $k$ and $w'_c$, the
general, stationary wind acceleration becomes,
 %
 \begin{equation}
 \label{cakwindsol}
 w'={g_c \over m} (1\pm\sqrt{1-mg/g_c})^2.
 \end{equation}
 %
 The velocity law $v(z)$ is found by (analytic or numerical)
quadrature of $w'=vv'$.

\bigskip
\noindent{\sc \S 10 Abbott waves}
\bigskip
\label{par10}

Since Abbott waves are of central importance in the following, we
derive them in three different ways: by dispersion analysis [9];
characteristic analysis [10]; and Green's function analysis [10]. The
first and last are necessarily linear, but the characteristic analysis
holds for arbitrary wave amplitudes. This should not mask the
fundamental limitation of our approach: a derivation of Abbott waves
including line scattering in any approximation (SSF, EISF) going
beyond Sobolev approximation was not given so far, due to mathematical
complexities.
 
\medskip\noindent{\bf Dispersion relation.} For all wind solutions,
$K=2\sqrt{g_c}$ holds. Inserting this into the time-dependent Euler
equation,
 %
 \begin{equation}
 \label{eulertimedep}
 \dot v + vv' = -g + \sqrt{2\rho\cri v\cri \; v'/\rho}.
 \end{equation}
 % 
 We consider small harmonic perturbations on a stationary wind
solution $\rho_0$, $v_0$ (not necessarily the critical solution),
$\rho=\rho_0+\rho_1 \exp[i(kz-\omega t)]$, $v=v_0+v_1 \exp[i(kz-\omega
t)]$. Linearizing the continuity and Euler equation gives,
 %
 \begin{gather}
 \label{continper}
 \left[-i {\omega \over v_0'}+i\chi +1 \right] {\rho_1 \over \rho_0} +
(i\chi -1) \; {v_1 \over v_0} =0,\\
 \label{eulerper}
 {1\over q_0} {\rho_1 \over \rho_0} + \left[-i {\omega \over v_0'} +1
+i\chi \left(1-{1\over q_0}\right) \right] {v_1 \over v_0} =0,
 \end{gather}
 %
 where $\chi\equiv k \,v_0/v_0'$ and $q_0=\sqrt{2 m_0 w_0'}$ were
introduced. Setting the determinant of the system to 0 gives the
dispersion relation $\omega(k)$. In the WKB approximation, $\chi\gg
1$, the phase speed and growth rate of the downstream mode (subscript
$+$) become, in the observers frame,
 %
 \begin{equation}
 v_\phi = {\rm Re}(\omega_+)/k = v_0,\qquad
 {\rm Im}(\omega_+)=-\epsilon.
 \end{equation}
 %
 The small damping term $-\epsilon$ is of no further consequence. In
the comoving frame, the wave speed is 0 in this zero sound speed
limit: the downstream mode consists of sound waves. For the upstream
or ($-$) mode,
 %
 \begin{equation}
 v_\phi = {\rm Re}(\omega_-)/k = v_0 \bigl( 1-1/q_0 \bigr),\qquad
 {\rm Im}(\omega_-)=0.
 \end{equation}
 %
 This new wave type is caused by radiation pressure, and is termed an
Abbott wave (after Abbott 1980). For shallow solutions, $q_0<1$, and
Abbott waves propagate towards smaller $z$: shallow solutions are the
analog to solar wind breezes. For steep solutions, $q_0>1$, and the
waves propagate towards larger $z$. At the critical point, $m=1$,
$w_c'=g_c$, and $q_0=1$, hence the critical point is a stagnation
point for Abbott waves. At smaller (larger) radii, the waves propagate
towards smaller (larger) $z$. For shallow solutions with $m_0\ll 1$
and $w'_0\ll w'_c$, Abbott waves propagate inward at arbitrary large
speed. Wave propagation along shallow solutions was not treated so far
in the literature, as a consequence of Abbott's (as it appears now:
erroneous) postulate that ``a line driven wind has no analog to a
solar breeze.'' (Abbott 1980)

\medskip\noindent{\bf Abandoning steep solutions.} Steep solutions are
super-abbottic everywhere, and cannot communicate with the wind base.
Hence, their mass loss rate cannot converge to an eigenvalue.
Numerical simulations show indeed that steep solutions are stable to
almost any perturbations [9]. The perturbations are advected to the
outer boundary and leave the mesh. Steep solutions are in the
following of not much interest. Still, they may have physical
significance, and we encountered them already before: the
quasi-stationary rarefaction regions in time-dependent simulations of
the de-shadowing instability are steep solution `patches'.

\medskip\noindent{\bf Characteristic analysis.} The above dispersion
analysis holds for linear waves. We turn now to a characteristic
analysis for arbitrary amplitudes. The equations of motion for
arbitrary $g$ are written as,
 %
 \begin{gather}
 \label{confun}
 C(\rho,v)= \dot\rho + v\rho' + \rho v' =0,\\
 %
 \label{eulfun}
 E(\rho,v)= \dot v + vv' + g(z) - 2\Gamma\sqrt{v'/\rho} =0,
 \end{gather}
 %
 with $\Gamma\equiv\sqrt{g_c \rho_c v_c}$. $C$ and $E$ are brought
into {\it advection} form without further approximation. A first-order
system of partial differential equations is called {\it quasi-linear}
if it is linear in all {\it derivatives} of the unknowns ($\rho$ and
$v$ here). Usually, characteristics are defined for quasi-linear
systems. Equations~(\ref{confun}, \ref{eulfun}) are {\it not}
quasi-linear, due to the presence of $\sqrt{v'}$. Courant \& Hilbert
(1968) show that characteristic directions, $a$, for general,
nonlinear systems are given by
 %
 \begin{equation}
 \label{mattrix}
 \begin{vmatrix}
 -a C_{\dot \rho} + C_{\rho'} & -a C_{\dot v} + C_{v'} \\
 %
 -a E_{\dot \rho} + E_{\rho'} & -a E_{\dot v} + E_{v'} 
 \end{vmatrix}
 =0.
 \end{equation}
 %
 The symbol $a$, so far reserved for the sound speed, refers now to
any characteristic speed. The distinction should be clear from the
context. For quasi-linear systems, the matrix in (\ref{mattrix})
becomes independent of differentials $\dot\rho$, $v'$, etc. For the
present case, however,
 %
 \begin{equation}
 \begin{vmatrix}
 -a+v & \rho \\
 0    & -a+v-\Gamma/\sqrt{\rho v'}
 \end{vmatrix} =0,
 \end{equation}
 %
 or, in the observers frame,
 %
 \begin{equation}
 a_+ = v,\qquad\qquad
 a_- = A = v-{\Gamma\over \rho v'},
 \end{equation}
 %
 with Abbott speed $A$. For small perturbations,
 %
 \begin{equation}
 \label{karfreitag}
  A = v\, (1-\sqrt{g_c/mw'})= A_0,
 \end{equation}
 %
 with $A_0$ from the dispersion analysis above. To bring the
hydrodynamic equations into characteristic form [10], the system must
be made quasi-linear, by applying the following trick (Courant \&
Hilbert 1968): the Euler equation is differentiated with respect to
$z$, and a new variable $f=v'$ is introduced,
 %
 \begin{equation}
 \label{0804010914}
 \dot f + vf' + f^2 - {\Gamma \over\sqrt{f\rho}}
 \left( f'-f{\rho' \over \rho}\right) =-g'.
 \end{equation}
 %
 This equation is linear in $\dot f$ and $f'$. The new system consists
then of the continuity equation, the Euler equation (\ref{0804010914})
for $f$, and the defining relation $f=v'$. Re-bracketing
(\ref{0804010914}) and multiplying by $\rho$,
 %
 \begin{equation}
 \rho\dot f + \rho Af' + \rho f^2 + {\Gamma \over\sqrt{f\rho}} f\rho'
=-\rho g'.
 \end{equation}
 %
 Replacing $\rho f$ using the continuity equation,
 %
 \begin{align}
 -\rho g'&= \rho\dot f + \rho Af' - f\dot\rho - \left(v- {\Gamma
\over\sqrt{f\rho}}\right) f\rho'\notag\\
 %
  &= \rho\dot f + \rho Af' - f\dot\rho - A f\rho'\\
 %
  &= \rho^2 \left(\pa_t + A \pa_z \right) {f\over \rho}.\notag
 \end{align}
 %
 The characteristic (or advection) form of the Euler equation is
therefore,
 %
 \begin{equation}
 \left(\pa_t + A \pa_z \right) {v'\over \rho}= -{g'\over \rho}.
 \end{equation}
 %
 We assume that WKB approximation applies, i.e., that the temporal and
spatial derivatives on the left hand side are individually much larger
than the right hand side, hence, the latter can be neglected. In a
frame moving at speed $A$, the function $v'/\rho$ is then constant,
and can be interpreted as a wave amplitude propagating with speed
$A$. The Sobolev optical depth is proportional to the inverse of
$v'/\rho$, which indicates that Abbott waves are indeed a radiative
mode. Introducing $f=v'$ in the continuity equation, the latter is
already in characteristic form,
 %
 \begin{equation}
  \left(\pa_t + v \pa_z \right) \rho =- f\rho.
 \end{equation}
 %
 Since it contains no derivatives of $\rho$ or $f$, $f\rho$ is an
inhomogeneous term. The Riemann invariant, $\rho$, is no longer
constant along the $v$ characteristic: WKB approximation, which would
mean $\rho'/\rho \gg v'/v$, does {\it not} apply. Since $\rho$ scales
with gas pressure, this mode can be identified with sound waves.

\medskip\noindent{\bf Green's function for Sobolev line force.}
Finally, we derive the Green's function for Abbott waves in Sobolev
approximation.  The Green's function gives the response of the wind to
a localized, delta function perturbation in space and time, and is
complementary to the harmonic dispersion analysis of Abbott (1980) and
Owocki \& Rybicki (1984). Since localized perturbations consist of
many harmonics, a Green's function describes wave interference. This
is clearly seen for water waves, whose Green's function is known from
Fresnel diffraction in optics (Lamb 1932, p.~386). For simplicity, we
consider a single, optically thick line only, with Sobolev force,
 %
 \begin{equation}
 g_l(z)=A\, {\pa v(z)\over\pa z}.
 \end{equation}
 %
 Density $\rho$ was absorbed into the constant $A$. We assume WKB
approximation to apply (slowly varying background flow), and consider
only velocity perturbations: the Abbott wave amplitude $v'/\rho$ from
characteristic analysis is not annihilated by this restriction. The
linearized Euler equation for small perturbations is
 %
 \begin{equation}
 \label{eq1}
 {\pa\over\pa t} \delta v(z,t) = \delta g_l(z,t)= A\, \delta v'(z,t).
 \end{equation}
 %
 The Green's function problem is posed by specifying as initial
conditions,
 %
 \begin{equation}
 \label{eq2}
 \delta v(z,0)=\delta(z-z_0).
 \end{equation}
 %
 The solution is obtained by Fourier transformation with respect to
$z$. We use the conventions, for an arbitrary function $F$ (bars
indicate Fourier transforms),
 %
 \begin{equation}
 \bar F(k,t)=\int_{-\infty}^\infty dz \, e^{-ikz} \, F(z,t),\qquad
 %
 F(z,t)={1\over 2\pi}\int_{-\infty}^\infty dk\,e^{ikz}\,\bar F(k,t).
 \end{equation}
 %
 Fourier transforming (\ref{eq1}) gives
 %
 \begin{equation}
 \label{eq3}
 {\pa\over\pa t} \overline{\delta v}(k,t) = ikA \overline{\delta
v}(k,t).
 \end{equation}
 %
 The right hand side was obtained by integration by parts, assuming
$\delta v(-\infty,t)= \delta v(\infty,t)=0$. This will be shown a
posteriori. The solution of (\ref{eq3}) is
 %
 \begin{equation}
 \overline{\delta v}(k,t)=be^{ikAt},
 \end{equation}
 %
 with constant $b$. Fourier transforming the initial conditions
(\ref{eq2}),
 %
 \begin{equation}
 \overline{\delta v}(k,0)=e^{-ikz_0}=b,
 \end{equation}
 %
 hence,
 %
 \begin{equation}
 \overline{\delta v}(k,t)=e^{ik(At-z_0)}.
 \end{equation}
 %
 Finally, Fourier transforming back to $z$ space,
 %
 \begin{equation}
 \delta v(z,t)= {1\over 2\pi} \int_{-\infty}^\infty dk \, e^{ikz}
e^{ik(At-z_0)}= \delta(z-z_0+At).
 \end{equation}
 %
 The delta function perturbation propagates without dispersion towards
smaller $z$, at a speed $-A$. Also, $\delta v=0$ at $z=\pm\infty$, as
assumed above. $A$ is the Abbott speed, as is seen by inserting a
harmonic perturbation, $\delta v=\overline{\delta v} \, e^{i(\omega
t-kz)}$, in (\ref{eq1}), giving for the phase and group speed,
 %
 \begin{equation}
 {\omega\over k} = {d\omega\over dk} = -A.
 \end{equation}
 %
 The result that a delta function peak propagates without dispersion
at speed $-A$ could have been foreseen from the phase speed of linear
Abbott waves: $v_\phi$ is independent of $\lambda$, hence no
dispersion occurs and no wave interference between different
harmonics. The Green's function, $G$, is defined by,
 %
 \begin{equation}
 \label{defgreen}
 F(z,t)=\int dz' \,G(z-z',t) \,F(z',0),
 \end{equation}
 %
 with some arbitrary function $F$, implying
 %
 \begin{equation}
 G(z,t)=\delta(z+At).
 \end{equation}

We have shown that the phase, group, characteristic and Green's
function speed for Abbott waves all agree.

The situation seems clear, but isn't.

\medskip\noindent{\bf No Abbott waves for pure absorption.} In a
landmark paper, Owocki \& Rybicki (1986) derived the Green's function
for pure absorption line flows subject to the force
(\ref{eq150201_01},\ref{eq150201_01}). Taking the Sobolev limit in
this Green's function, upstream propagating Abbott waves are found.

But this is impossible since, for pure absorption, photons propagate
only downstream and an upstream radiative mode cannot exist. Indeed,
the bridging law (Owocki \& Rybicki 1984) which is used in deriving
this Green's function is the sum of a delta function term, expressing
de-shadowing instability, and a Heaviside function, expressing {\it
downstream} shadowing of wind ions caused by velocity perturbations.
To derive Abbott's upstream mode from this downstream-only bridging
law is a contradiction.

The resolution was given by Owocki \& Rybicki (1986), who demonstrated
that two limits do not commute in the Green's function analysis: the
Sobolev limit $\lambda\rightarrow\infty$ for long perturbation
wavelengths; and the continuum limit $\sum_k\rightarrow\int dk$ in
Fourier transforms of localized functions. Taking the Sobolev limit
first, their eqs.~(C1, C2) give the same Green's function for inward
propagating Abbott waves as we derived above, $G(z,t)=\delta(z+At)$.
If, instead, the continuum limit is taken first (their eqs.~B10, C3,
C4) and only then the Sobolev limit,
 %
 \begin{equation}
 \label{paradox}
 G(z,t)=\sum\limits_{n=0}^\infty {(At)^n\over n!} \left({d\over
dz}\right)^n \delta(z).
 \end{equation}
 %
 Formally, this is the Taylor series expansion of a delta function,
$G(z,t)=\delta(z+At)$. The paradox is burried and resolved at this
point. The latter equation expresses $\delta(z+At)$, which is peaked
at $z=-At$, in terms of derivatives of $\delta(z)$, which {\it vanish
exactly} at $z=-At$ for $t>0$! The series (\ref{paradox}) must be
taken as it stands. Applying (\ref{paradox}) to an arbitrary initial
perturbation $F(z',0)$ using (\ref{defgreen}),
 %
 \begin{equation}
 F(z,t)=\sum\limits_{n=0}^\infty {(At)^n\over n!} \, {d^n F(z,0)\over
dz^n}.
 \end{equation}
 %
 Hence, $F(z,t)$ is constructed from $F(z,0)$, $F'(z,0)$, $F''(z,0)$,
$F'''(z,0)$, etc.~{\it alone}, from pure initial conditions at the
location $z$ under consideration. No paradox occurs, there are no
Abbott waves, only local information is used. The Abbott speed $A$ is
a mere constant, without deeper physical significance.

\medskip\noindent{\bf Including scattering.} To the present author,
this indicates that a bigger frame exists which contains the special
case of pure absorption, and gives physical meaning to $A$. This is
achieved when scattering is included. Indeed, numerical Green's
function experiments for SSF and EISF forces show an upstream
propagating Abbott front (Owocki \& Puls 1999). However, due to
mathematical difficulties, the Green's function for flows driven by
line scattering was not derived analytically so far. And some authors
doubt the physical relevance of Abbott waves and the CAK critical
point at all, suggesting that they are artefacts of the Sobolev
approximation (Lucy 1998). Using an ingenious argument, Lucy (1975)
avoided the critical point singularity (0/0) in a numerical quadrature
of the Euler equation. He used different numerical representatives for
$dv/dr$ in the advection term and in the Sobolev line force. The
argument is that $dv/dr$ in the advection term is a true differential,
whereas $dv/dr$ in the line force means the velocity difference over
the finite Sobolev zone. We adopt here the standpoint: in numerical
simulations which apply the Sobolev, SSF, or EISF approximation,
Abbott waves must be accounted for in the Courant time step and
(outer) boundary conditions. Yet, the physical relevance of Abbott
waves remains to be strictly proven by a Green's function analysis
including scattering.

\medskip\noindent{\bf Courant time step and outer boundary
conditions.} In simulations published so far, Abbott waves are not
included in the Courant time step, but only sound waves are. We saw
above that Abbott waves define the upstream characteristics of line
driven winds. Therefore, they determine the numerical time step. If
not properly accounted for, Abbott waves cause numerical runaway.

Furthermore, outflow boundary conditions are assumed in the literature
on the outermost mesh point. All flow quantities are extrapolated from
the interior mesh to the boundary. This is wrong for sub-abbottic
shallow solutions, for which Abbott waves {\it enter} through the
outer boundary. Applying outflow conditions may {\it drive} the
solution to the critical one, which is super-abbottic at the outer
boundary, hence consistent with outflow extrapolation [8].

To maintain shallow solutions, on the other hand, an outer boundary
condition must be applied instead of extrapolation. When Abbott waves
are included in the Courant time step and non-reflecting boundary
conditions are used, we find that shallow solutions are numerically
stable [9]. Non-reflecting boundary conditions annihilate any incoming
waves (the boundary condition is: `no waves'). Even initial conditions
which depart strongly from a shallow wind, e.g., a linear velocity
law, converge to a shallow solution.

We now turn the argument around: once {\it numerical} stability of
shallow solutions is achieved, this allows to explicitly introduce
flow perturbations on the interior mesh (away from boundaries) in a
{\it controlled} way, and to study their evolution, especially
stability. We find that a new runaway mechanism exists, which is
caused by Abbott waves [8, 10]. In older simulations which applied
outflow boundary conditions and did not account for Abbott waves in
the Courant time step, this physical runaway was totally outgrown by
numerical runaway, and therefore not detected.

\medskip\noindent{\bf Negative velocity gradients.} The new, physical
runaway occurs in regions where the wind decelerates, $v'<0$. The
Sobolev force is `blind' to the sign of $v'$. All what counts is the
relative Doppler shift between neighboring gas parcels, which
determines the width of the Sobolev resonance zone. The Sobolev line
force is then generalized to $g_l = 2\Gamma \sqrt{|v'| /\rho}$, where
$\alpha= 1/2$ is still assumed. However, flow deceleration implies a
non-monotonic velocity law, and multiple resonances occur. The
incident light is no longer given by the photospheric flux alone, and
the constants $K$ in (\ref{eulerstat}) or $2\Gamma$ above become
velocity dependent. Rybicki \& Hummer (1978) generalized Sobolev
theory to account for non-monotonic velocity laws. While introducing
interesting, non-local aspects to Abbott waves (action at a
distance!), the method of Rybicki \& Hummer is not analytically
feasible, and we do not consider it further here.  Instead, besides
the purely local force $g_l \sim \sqrt{|v'|}$ we also consider $g_l
\sim \sqrt{\max(v',0)}$: all incident light is assumed here to be
completly absorbed at the first resonance. The true line force should
lie between these extremes.

For $g_l\sim \sqrt{|v'|}$, the Euler equation has characteristic form
 %
 \begin{equation}
 \left(\pa_t + A \pa_z \right) {v'\over \rho}= -{g'\over \rho},
 \qquad{\rm where}\qquad
 A = v \mp {\Gamma\over \sqrt{\pm \rho v'}}
 \end{equation}
 %
 is the Abbott speed in the observers frame. The upper resp.~lower
sign applies for $v'>0$ resp.~$v'<0$. Therefore, if the flow switches
from acceleration to deceleration, Abbott waves turn from propagating
upstream ($-$ characteristic) to propagating downstream ($+$
characteristic).

For $g_l\sim\sqrt{\max(v',0)}$, the characteristic Euler equation for
$v'<0$ is,
 %
 \begin{equation}
 \left(\pa_t + v \pa_z \right) {v'\over \rho}= -{g'\over \rho} 
 \qquad\qquad (v'<0).
 \end{equation}
 %
 Since the line force vanishes, this is just the ordinary Euler
equation at zero sound speed. The upstream Abbott wave turns here into
an upstream sound wave. Note that a reversal into a downstream
(Abbott) wave does not occur. We conclude that for {\it both} types of
line force, {\it regions $v'<0$ cannot communicate with the base at}
$z=0$. Abbott (and sound) waves which originate in these regions
propagate only outward.

\bigskip
\noindent{\sc \S 11 Abbott wave runaway}
\bigskip
\label{par11}

Figure~10, which is taken from [8], shows runaway of a shallow wind
velocity law when a sawtooth-like perturbation of amplitude $\delta v$
and period $T$ is applied at a fixed position $z=2$ (at one mesh
point) at all times. The amplitude $\delta v$ is sufficiently large
that negative $v'$ results.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=10cm
 \epsffile{fig10.ps}
 \caption{Abbott wave runaway for a shallow wind velocity law (upper
panel), and stable Abbott wave propagation along the critical CAK
solution (lower panel). The critical point lies at $x=1$.}
 \end{center}
 \end{figure}

The sequence of events which lead to runaway should be as follows [8,
10]: during the half-period of negative velocity perturbations, the
slope $v'<0$ is to the left (at smaller $z$) of the slope
$v'>0$. Regions with $v'<0$ propagate outwards, regions with $v'>0$
inwards. Hence, the two slopes approach and annihilate each other,
leaving an essentially unperturbed velocity law. In some more detail:
the sawtooth perturbation $\delta v$ is established via many small,
roughly constant $dv \ll \delta v$ applied over many time steps $dt
\ll T$. During the negative half wave $-\delta v$, at each time step
$dt$, $-dv$ is applied, and largely annihilates itself during the
subsequent `advection' step of the numerical scheme. The total
velocity perturbation after $T/2$ is not the perturbation amplitude
$\delta v$, but roughly 0! Opposed to this, over the half-period $T/2$
when $\delta v>0$, the slope $v'<0$ is to the right of the slope
$v'>0$. The two slopes move apart. The gas in between gets rarefied
(is `stretched') at roughly constant speed. (This corresponds to a
centered rarefaction wave.) At the next time step, $dv$ is created
atop of this region. Over the half-period $T/2$, the full perturbation
amplitude $\delta v$ builds up.

This is shown in the figure: the negative half wave of the sawtooth
annihilates itself, leaving an unperturbed velocity law; the positive
half wave is excited at the intended amplitude, and spreads upstream
and downstream throughout the wind. Asymmetric evolution, or runaway,
of the wind velocity towards larger speeds results over a full period.
Note that the runaway is not caused by wave growth, as in a fluid
instability, but by missing negative velocity perturbations to
compensate for positive perturbations.

The same runaway is also found by introducing a train of sawtooth-like
perturbations in the initial conditions, and let them evolve freely,
without introducing further perturbations at later times. This is
shown in Figure~5
 %%true?
 of paper [10]. Neighboring slopes approach and
separate in the way described above, which leads to asymmetric
evolution of the wind velocity law towards larger speeds.

\bigskip
\noindent{\sc \S 12 Overloaded winds}
\bigskip
\label{par12}

\noindent{\bf Generalized critical points.} The runaway stops when
Abbott waves can no longer propagate inwards, after a critical point
formed in the flow. If the perturbation source is located {\it above}
the CAK critical point, the wind undergoes runaway until it reaches
the CAK solution. The perturbation site comes to lie on the
super-abbottic portion of the velocity law. Phases $v'>0$ and $v'<0$
of the perturbation cycle combine then to a smooth,
outward-propagating Abbott wave, as shown in the figure.

The perturbation source is not required to lie at fixed $z$. Even when
moving outwards with the local wind speed or faster, it can excite
Abbott waves which propagate inwards in the observers frame, and
therefore alter the whole inner flow.

For coherent perturbations below the CAK critical point, runaway does
not terminate at the critical solution [8]. The perturbation site is
sub-abbottic on the critical solution, and Abbott waves can further
penetrate to the wind base. The wind evolves towards a velocity law
which is steeper than the critical one. The wind becomes overloaded,
$m>1$. The runaway continues until a generalized (non-CAK) critical
point, $z_a$, forms as a barrier for Abbott waves.

What are these generalized critical points, $z_a$? To see this, we set
the (stationary) Abbott speed (\ref{karfreitag}) to zero, $A_a =
v_a(1-\sqrt{g_c/ m w'_a}) = 0$, giving $mw'_a=g_c$. This shows that
the square root in the solution for $w'$ in (\ref{cakwindsol})
vanishes, and a steep and shallow solution merge. Besides at the CAK
critical point, this happens only along overloaded winds, when a
shallow solution bends back towards smaller $z$ on the steep branch.
Below this location, the Euler equation has two real solutions, above
it has two imaginary solutions. Physically, this means that in the
proximity of the CAK critical point, which is the bottleneck of the
flow, the line force can no longer balance gravity, and the wind
starts to decelerate. The two {\it real} solution branches reappear at
some height above the critical point.

\medskip\noindent{\bf Stationary overloading: kinks.} Therefore, at
$z_a$ the wind jumps to the decelerating branch, $v'<0$. In Sobolev
approximation, the jump is sudden and causes a {\it kink} in the
velocity law. Since $v'<0$ on the decelerating branch, Abbott waves
propagate outwards beyond $z_a$, hence $z_a$ is a wave barrier [8].

We have arrived at the somewhat strange conclusion of a {\it
stationary} solution with {\it kinks}. As mentioned before (page
\pageref{hilbert1} and \pageref{hilbert2}), kinks and other high-order
(or weak) discontinuities of hydrodynamic variables lie on
characteristics, i.e., move at characteristic speed. This is true for
the present kink, which moves at characteristic Abbott speed $A_a=0$.

\medskip\noindent{\bf Time-dependent overloading: shocks and shells
again.}  Since $z_c$ lies at the gravity maximum, already small
super-CAK mass loss rates cause broad deceleration regions. In
practice, for mass loss rates only a few percent larger than the CAK
value, the deceleration regime is so broad that {\it negative} speeds
result.  Stationary solutions are then no longer possible, as upwards
streaming wind gas collides with falling gas. For a periodic
perturbation source in the wind, numerical simulations show the
occurence of a train of shocks and shells in the wind, which still
propagates outwards. This is shown in Fig.~8 of [10]. %%true?

\medskip\noindent{\bf Does overloading occur in nature?} The
discussion so far is idealized since it assumes {\it coherent}
perturbation sources in the wind. Perturbations with a finite life
time should still lead to `piecewise' runaway, each perturbation
lifting the wind to a slightly faster, shallow solution. The wind
waits on the new, stable shallow solution until the next perturbation
lifts it further.

The outer wind seems to be the natural seat for runaway perturbations,
since $v'<0$ is easily excited on the outer, flat velocity law. We
speculate that in nature, runaway perturbations are most prevalent
above the CAK critical point and drive the wind towards the critical,
not an overloaded solution [10]. In support of this we add that not
even the strong de-shadowing instability leads to significant wind
structure below the critical point, which could act as runaway
perturbation towards an overloaded solution.

Still, there remains a slight chance that overloading occurs in real
winds. Most notably, Lamers (1998, private communication) reports on a
broad region in the wind of the luminous blue variable P~Cygni, which
is indicative of flow deceleration; and therefore, possibly, of
overloading.

\medskip\noindent{\bf Instability vs.~runaway.} We close this chapter
with a brief summary of the `driving agents' of wind dynamics in the
last two chapters: de-shadowing instability and Abbott wave runaway.
The de-shadowing instability is a true fluid instability (an
amplification cycle) which acts on infinitesimal velocity
perturbations. Even perturbations of short duration will lead to
pronounced, non-linear flow structure which is advected outwards with
the wind. In Sobolev approximation, the instability occurs from second
order on (including velocity curvature terms). Abbott wave runaway, on
the other hand, requires finite amplitude perturbations, since $v'<0$
is required. There is no amplification cycle, but only a kind of
perturbation `filtering' as consequence of the asymmetry of the line
force with respect to the sign of $v'$. Negative velocity
perturbations annihilate themselves, and give way to systematic flow
acceleration towards the critical solution. The runaway requires
persistent perturbations. Yet, it occurs already in first order
Sobolev approximation.

We should also mention that the present, hydrodynamic runaway is not
related to the plasma runaway caused by frictional decoupling of line
driven metal ions and dragged-along protons (Springmann \& Pauldrach
1992). Still, a connection may exist between these two runaways:
Krticka \& Kub\'at (2000) report that frictional decoupling in thin
winds may actually be prevented by the wind jumping to a {\it slow}
solution with shallow velocity gradient. Issues of Abbott wave
propagation and multiple solution branches become again interesting in
these two-component fluids.

%---------------------------------------------------------------------

\clearpage
\bigskip
\noindent{\bf CHAPTER 4: DISK WINDS}
\bigskip
\label{chap4}

\noindent Line driven winds from accretion disks are a relatively new
research area, with quantitative modeling starting in the mid 80ies
(Shlosman et al.~1985; Weymann et al.~1985). Disk winds are
fascinating because of the richness of their environments, including
quasars, Seyfert galaxies, cataclysmic variables, and protostars. In
these environments, high-energy particle processes, magnetic fields,
coronae, jets, accretion-dominated advection, compact objects, hot
boundary layers, and dust formation play important roles.

Theoretical modeling of line driven disk winds is still in its
infancy. In the present chapter, we treat first a simple, analytic
model [6, 7] for line driven winds from cataclysmic variables.
Observational evidence for line driving is almost unequivocal in these
objects. In later sections, we discuss winds from magnetized accretion
disks, mainly aimed at young protostellar objects and cataclysmic
variables. Here, interplay of three driving forces (centrifugal,
Lorentz, and line force) leads to an intricate wind dynamics.

\bigskip
\noindent{\sc \S 13 Analytical model}
\bigskip
\label{par13}

\noindent{\bf 1-d description, flux and gravity.} We set up a
simplified, 1-D model for stationary, line driven winds from thin
accretion disks (no self-gravity) in cataclysmic variables (CVs).
These systems consist of a white dwarf and a late-type main-sequence
companion, the latter filling its Roche lobe. We consider first
non-magnetic systems (DQ~Her class), where the accretion disk should
(almost) reach the compact object. The clearest indication of line
driving comes from the fact that dwarf novae develop P Cygni line
profiles during outburst (Krautter et al.~1981; Klare et al.~1982;
C\'ordova \& Mason 1982). Hence, the disk radiation field and wind are
causally connected.

A number of approximations is made. First, we assume that the disk is
geometrically thin, a plane of zero width indeed. We assume that the
basic CAK formalism (Sobolev line force; statistical line distribution
function) applies for these objects. Our central assumption is that
each helical wind trajectory lies in a {\it straight} (yet, not
vertical) cone. This is sketched in Figure~\ref{fighcn}.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=10cm
 \epsffile{fighcn.ps}
 \caption{\label{fighcn} Flow geometry in a semi-analytic model of
line driven winds from accretion disks in cataclysmic variables. The
helical streamlines are assumed to lie in {\it straight} cones. Only
the velocity component $v_l$ along the cone is solved for. The tilt
angle, $\lambda$, of the cone with the disk is a function of radius,
and is derived as an eigenvalue.}
 \end{center}
 \end{figure}

A realistic approximation for the temperature run with radius $q$ in
the disk is $T\sim q^{-1/2}$ (Horne \& Stiening 1985; Rutten et
al.~1993), slightly shallower than the famos $T\sim q^{-3/4}$ of
Shakura \& Sunyayev (1973). We assume that the disk is optically
thick, and that the radiation field at each radius is that of a black
body. The radiative flux, $\vecF$, above a disk with $T\sim q^{-1/2}$
can be derived in closed form in cylindrical coordinates [6],
 %
 \begin{gather}
 \vecF(r,z)=(F_r,F_z)=\pi I(r,0)\,r^2\, {z\over r^2+z^2} \;\times
 \qquad\qquad\qquad\qquad\notag\\
 \hfill
 \left.\left(
 {3r^2-z^2-q^2\over 2r\sqrt{B}}-{r\over z^2+r^2}\ln C,\;
 {3z^2-r^2+q^2\over 2z\sqrt{B}}-{z\over z^2+r^2}\ln C
 \right)\right|_{q=r_{\rm wd}}^{r_{\rm d}},
 \end{gather}
 %
 where $r_{\rm wd}$ is the white dwarf radius, $r_{\rm d}$ the outer
disk radius, and
 %
 \begin{gather}
 B=(r^2+z^2-q^2)^2+4z^2 q^2,\notag\\
 C={(z^2-r^2)q^2+(z^2+r^2)^2+(z^2+r^2)\sqrt{B}\over q^2}.
 \end{gather}
 %
 The flux isocontours are shown in Figures~3 and 4 of [6]. We adopt
again the (`radial streaming') approximation that only velocity
gradients along the flow direction are important. A qualitatively new
feature of disk winds is their run of {\it effective} gravity with
height, i.e., gravity after subtraction of the centrifugal force. In
the disk plane, effective gravity is zero due to Kepler balance.
Effective gravity increases linearly with height, reaches a maximum,
and drops off with $r^{-2}$ far from the disk. The exact formula is
given in equation (9) of [6].

At first, it was thought that the existence of a gravity maximum makes
a standard CAK critical solution for disk winds impossible, and that
ionization gradients have to be included (Vitello \& Shlosman 1988;
[7]). The argument was that, because of zero effective gravity in the
disk, a too large mass is launched. Higher up in the wind, the line
force cannot carry the gas over the gravity hill, and the gas falls
back to the disk. From the discussion in the last chapter we see why
this argument does {\it not} apply. While the critical point (at
finite height) is indeed the bottleneck of the flow, Abbott waves can
adjust the disk (which is the wind base) to the correct, maximum
possible mass loss rate.

\medskip\noindent{\bf Wind tilt angle as an eigenvalue.} In the
terminology of the Lavalle nozzle, the effective `area function' along
a wind ray is $f^2/g$, if $\alpha=1/2$. Here, $f$ and $g$ are the
radiative flux and effective gravity, respectively, normalized to
their footpoint values in the disk. The tilt angle, $\lambda$, of the
straight wind cone (with length coordinate $l$) with the disk midplane
is found as a second eigenvalue, besides the mass loss rate. For each
disk ring, the 2-D eigenvalue problem is posed as,
 %
 \begin{equation}
 \label{eq120201_01}
 d\dot M= C \max_\lambda \min_l {f^2 \over g}.
 \end{equation}
 %
 Here, $d\dot M$ is the mass loss rate from the ring, and $C$ is a
constant. We search for the {\it maximum} mass loss rate with respect
to $\lambda$, which can be driven through the bottleneck. Hence, the
{\it minimum} must be taken with respect to $l$. Equation
(\ref{eq120201_01}) defines a new {\it saddle} point in the $l\lambda$
plane, besides the standard CAK saddle in the $lw'$ plane.

Isocontours of $f^2/g$ above an accretion disk are shown in Figures~6
and 9 of [6]. Opposed to the stellar wind case, the solution topology
is rather intricate here, with multiple saddles and extrema. This is a
familiar situation for complicated area functions, and is also
encountered when one allows for energy and momentum input at finite
height in the solar wind (Holzer 1977). 

We find that the accretion disk wind undergoes a rather sudden
transition, at about 4 white dwarf radii in the disk, from steep tilt
angles of only 10 degrees with the disk normal, to tilt angles of 30
to 40 degrees with the normal. The overall wind geometry is shown in
Fig.~10 of [6]. The wind is strongly bi-conical everywhere, which is
one of the main deductions from kinematic model fits to observed
P~Cygni line profiles (Shlosman \& Vitello 1993).

\medskip\noindent{\bf Mass loss rates.} The mass loss rate from the
disk is obtained by integrating over all rings. For $\alpha=1/2$, this
can be written as,
 %
 \begin{equation}
 \label{eq120201_02}
 \dot M= c_1(g) c_2(f) \; Q\Gamma \; {L\over c^2}.
 \end{equation}
 %
 Here, $L$ is the disk luminosity, and $L/c^2$ is the mass loss due to
a {\it single}, optically thick line (the photon mass flux!),
$Q\approx 2000$ is an effective oscillator strength (Gayley 1995), and
$\Gamma$ is the disk Eddington factor. The correction factors $c_1$
and $c_2$ account for the run of gravity and radiative flux.
$c_1=c_2=1$ gives the mass loss rate from a point star. For a thin
accretion disk, we find in [6] that $c_1= 3\sqrt{3}/2$. This is larger
than unity since centrifugal forces assist in launching the wind.
$c_2$ has to be calculated numerically, and is also $\gap 1$.

We encounter the problem that mass loss rates from the above formula
are one to two orders of magnitude smaller than values derived from
P~Cygni line fits ([7]; Vitello \& Shlosman 1993; Knigge, Woods, \&
Drew~1995). Here, the values for $\dot M$ obtained from fitting were
already revised downwards, for the following reason. In the original
line fit procedure, individual disk rings were assumed to have a
blackbody spectrum (as we did above). The strong Lyman continuum,
however, leads then to large ionization rates, which must be balanced
by large recombination rates, or large $\dot M$. If realistic spectra
(Long et al.~1991, 1994) with suppressed (or missing) Lyman continua
are used instead, $\dot M$ can be reduced by roughly one order of
magnitude, while maintaining the ionization parameter.

Still, there remains the discrepancy noted above, between calculated
and observationally deduced mass loss rates. One possible resolution
of this problem is that distances and luminosities of CVs are
systematically underestimated (Drew, Proga, \& Oudmaijer~1999).
Furthermore, the all-decisive parameter $Q$ was so far only calculated
for dense O~star winds. The different spectral shapes of accretion
disks, and ionization effects like the bistability jump (Pauldrach \&
Puls 1990) may cause larger $Q$ values in accretion disks than in
O~stars. The strongest candidate, however, for resolving the
discrepancy is the usual suspect: magnetic fields.

\bigskip
\noindent{\sc \S 14 Numerical model}
\bigskip
\label{par14}

The analytic model discussed so far is one dimensional, assuming
straight wind cones. First time-dependent, numerical 2-D simulations
of line driven winds from accretion disks were performed by Proga,
Stone, \& Drew (1998). Their wind rays are surprisingly straight in
the polar plane ($rz$ plane), and mass loss rates agree well with the
above values. A new result from numerical simulations is the occurence
of wind {\it streamers}. The wind above the disk is structured into
alternating dense and rarefied regions. The dense streams propagate
radially outwards, with a speed relative to the disk which is
proportional to the local Kepler speed. During this motion, the angle
between dense streamer and disk becomes ever shallower. This is shown
in Fig.~\ref{figdyd}. A movie can be found at internet URL {\tt
http://www.astro.physik.uni-potsdam.de/$\sim$afeld}). The occurence of
alternating dense and rarefied regions is most probably related to
mass overloading.

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=8cm
 \epsffile{figdyd.ps}
 \caption{\label{figdyd} Dense streamers in a line driven wind above a
cataclysmic variable disk (log density, dense regions in black). The
white cone around the disk axis is a nearly gas free region with very
fast parcels. }
 \end{center}
 \end{figure}

\bigskip
\noindent{\sc \S 15 Magnetized line driven winds}
\bigskip
\label{par15}

We turn to accretion disks which are threaded by external magnetic
fields. This could be the case in CVs (so-called polars or AM~Her
systems), young stellar objects (YSOs), and quasars. Our present
simulations aim at the first two classes of objects only. The results
are still preliminary, and much work remains to be done. The models
were calculated using the publically available {\sc Zeus 2-D} MHD code
(Stone \& Norman 1992a,b), augmented by an own, Sobolev line force
routine (Feldmeier, Drew, \& Stone, in preparation).

\medskip\noindent{\bf Boundary conditions.} The outcome of numerical
hydrodynamic simulations depends strongly on the applied boundary
conditions. We must, therefore, discuss boundary conditions in some
detail. The reader interested mainly in results on disk winds may skip
over the next few paragraphs, to the section entitled `magnetized wind
scenarios'.

For the present discussion, we chose again cylindrical coordinates (in
the code, however, spherical coordinates are used, to avoid
staircasing of the stellar surface). On the polar axis,
anti-reflecting boundary conditions are used. Empirically, one finds
that above the `dark star' (both in CVs and YSOs, stellar UV fluxes
are negligible), an essentially gas-free cone forms near the polar
axis. The highly rarefied gas in the cone is accelerated to large
speeds (a `steep' CAK solution is adopted), and the Alfv\'en speed,
$v_A=B/\sqrt{4\pi\rho}$, is very large in this cone. Both effects
limit the Courant time step. To avoid effective stopping of the code,
we let an artificial, vertical ram pressure jet originate on the star
(Krasnopolsky, Li, \& Blandford 1999). Along the stellar surface and
the outer mesh boundary circle, inflow resp.~outflow boundary
conditions are chosen. Note that outflow boundary conditions are wrong
close to the disk plane, below the \Alfven~surface, and exert
artificial forces on the wind (Ustyogova et al.~1999). Future work has
to account for this.

Remains the disk itself. The inner disk structure is not resolved in
our simulations (actually, zero sound speed is assumed), hence the
disk is a simple, planar boundary. Given are a total of 7 hydrodynamic
fields: $\rho, v_z, v_r, v_\phi, B_z, B_r, B_\phi$, minus one
constraint, ${\rm div}\,{\bf B}=0$. Hence, 6 wave modes result: 2
poloidal and 2 toroidal \Alfven~waves, and 2 magnetosonic waves (up-
and downstream mode in each case). Some of these waves are modified by
the radiative line force. For simplicity, however, we keep their above
names, with the one exception of the fast magnetosonic mode, which is
termed a magnetoabbottic mode from now on.

By assumption, the disk shall be supersonic, subabbottic, and
subalfv\'enic. Then 1 poloidal, 1 toroidal \Alfven~mode and the slow
magnetosonic mode enter the mesh from the disk, and 1 poloidal, 1
toroidal \Alfven~mode and the fast magnetoabbottic mode enter the disk
from the mesh. Hence, 3 extrapolations can be applied on the disk
boundary, and 3 boundary conditions must be specified. For the
poloidal components of the gas speed and magnetic field, we follow
largely Krasnopolsky et al.~(1999). Here, special care is taken to
avoid kinks when magnetic field lines enter the disk. These kinks
would cause artificial currents and forces. We do {\it not} follow
Krasnopolsky et al.~(1999) in case that gas falls back to the
disk. In contrast to their reasoning, this should only affect the slow
magnetosonic mode.

Remain the toroidal fields $v_\phi$ and $B_\phi$. With regard to the
toroidal \Alfven~mode, one extrapolation and one boundary condition
have to be applied. We {\it fix $v_\phi$ to be the Kepler speed}. This
avoids dramatic events of sub-Keplerian gas falling towards the
central object, which are encountered when $v_\phi$ is left free
(Uchida \& Shibata 1985; Stone \& Norman 1994). $B_\phi$ is
extrapolated. This causes a problem, since Lorentz force terms $\sim
\pa(r B_\phi)/\pa r$ occur in the Euler equation for $v_\phi$. For
$B_\phi\ne 0$, $v_\phi$ does generally not agree with the Kepler
speed. We proceed by postulating $B_\phi \sim 1/r$ in the disk (Ouyed
\& Pudritz 1997a,b). Whereas the latter authors leave the toroidal
disk field constant at all times, we calculate, at each time step,
$\bar B_\phi$ as average of $B_\phi$ on the first mesh row above the
disk. Within the disk (boundary), we set $B_\phi=\bar B_\phi R/r$,
where $R$ is a free parameter.

This introduces an infinite signal speed, since averaging $B_\phi$
over the whole disk surface, at each time step, implies instantaneous
couplings. Still, the above procedure works well empirically, and
$B_\phi$ evolves smoothly in both $r$ and $z$ direction. By contrast,
simulations assuming $B_\phi=0$ in the disk crash.

\medskip\noindent{\bf Magnetized wind scenarios.} As should be clear
from this lengthy discussion, $B_\phi$ is of central importance in our
simulations. We do not assume a magnetically dominated, co-rotating,
force-free corona a priori. If a strong, line driven flow is launched
from the disk, the angular-momentum conserving gas parcels may instead
drag the magnetic field along, building up a strong toroidal field.

There exist two complementary, magnetocentrifugal wind scenarios. In
the model of Blandford \& Payne (1982), rigid poloidal field lines in
a co-rotating corona act as lever arms transferring angular momentum
to the gas. There act {\it no} Lorentz forces in the launching region
of the wind. In the model of Contopoulos (1995), on the other hand,
vertical gradients of a toroidal field, $B_\phi$, cause non-vanishing
curl (pointing towards the central object), and a Lorentz force ${\rm
rot}\, {\bf B}_\phi \times {\bf B}_\phi$ which points vertically
upwards, see Figure~\ref{figiqb}. The generation of dominant, large
scale toroidal fields in disk-plus-corona simulations is treated in
Miller \& Stone (2000) and Elstner \& R\"udiger (2000).

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=12cm
 \epsffile{figiqb.ps}
 \caption{\label{figiqb} Two basic scenarios for magnetized disk
winds, launched either by poloidal (left) or toroidal (right) magnetic
fields. To apply the right-hand rule in the latter case, a constant
$B_{\phi,0}$ should be added to the toroidal field, which leaves the
curl unaltered.}
 \end{center}
 \end{figure}

\medskip\noindent{\bf Model 1: a pure Lorentz wind.} First, we
consider a pure Contopoulos scenario. The disk has an Eddington factor
of $10^{-2}$, which is appropriate for CVs and YSOs. The poloidal
field is set to zero. A {\it temporally constant}, toroidal field
which drops off as $r^{-1}$ is chosen in the disk. Note that a pure
toroidal field has no \Alfven~point. Physically, it is still
meaningful to assume that the flow is super\alfvenic, in that an
initially poloidal field got wound-up, above the \Alfven~point, into
the toroidal field. Therefore, both $B_\phi$ {\it and} $v_\phi$ are
now fixed in the disk. The toroidal field strength is taken from
model~2: the present model~1 serves mainly as a simple reference frame
for the more intricate model~2. From the calculation we find that just
above the disk, the vertical Lorentz force is a few times larger than
the line force. Hence, the wind is essentially magnetically launched,
i.e., a Contopoulos flow. Still, the mass loss rate is almost exactly
that of a {\it non-magnetic} model discussed in the last section. The
explanation is simple: since the flow starts super\alfvenic, the CAK
critical point is the only flow bottleneck, and determines the mass
loss rate -- whatever force launches the wind.

\medskip\noindent{\bf Model 2: magnetic eddies.} Next, we chose a
purely line driven models from the foregoing section as initial
conditions (regularly spaced, dense streamers alternating with
rarefied regions).  A poloidal magnetic field is switched on slowly,
assuming exponential growth. The field is initially axisymmetric,
$B_r=0$. However, the field lines get tilted in time and get wound-up,
creating $B_z$ and $B_\phi$ components. Typically, we find toroidal
fields which, at the rim where the accretion disk touches the stellar
surface, are 20 times stronger than the poloidal field. For a not-too
big disk, $B_\phi$ is still the dominant field component at the outer
disk rim, only dropping off as $r^{-1}$. With $B_\phi$ dominating, one
expects that the mass loss rate is still the one of model~1. Instead,
it is 20 times larger!

 \begin{figure}[h]
 \begin{center}
 \leavevmode
 \epsfxsize=8cm
 \epsffile{fig11.ps}
 \caption{Eddies in the poloidal magnetic field above an accretion
disk with small Eddington factor. The unit vector at the top right
corresponds to 1~Gauss.}
 \end{center}
 \end{figure}

The reason is apparently the following. In model~2, the $B_\phi$ bulge
or wedge above the disk reaches to significantly larger heights than
in model~1. The Lorentz force can assist then in overcoming the
critical point, and the mass loss rate increases.

Why is the $B_\phi$ bulge broader in model~2? This model shows a
pronounced vortex sheet in the toroidal magnetic field and gas speed,
as is shown in Fig.~11. The eddies are shed-off from the disk-star
rim, and carry the $B_\phi$ component to larger heights above the disk
than in model~1. Indeed, the $B_\phi$ bulge extends now to the top of
the spinning eddies.

\medskip\noindent{\bf The plasma gun, and other open questions.} What
is the origin of the vortex sheet? It is known from plasma physics
that toroidal fields are often unsteady and undergo periodic cycles of
field build-up and unloading. This is called a `plasma gun'. We
speculate that this is the origin of the present field eddies, too.
Future work has to clarify this issue.

Other issues which shall be addressed in future work are: 1.~Will a
Contopoulos wind still form when a Blandford \& Payne wind is launched
right from model start? Instead of an initially axisymmetric field,
tilt angles $>30$ degrees with the disk normal shall here be assumed
for the initial, poloidal field. 2.~In present simulations the
\Alfven~surface `dives' periodically into the disk before it reaches
the central star. This seems to be related to the shedding-off of
magnetic eddies. 3.~Riemann boundary conditions shall be applied along
$v_\phi, B_\phi$ characteristics, instead of the above, rather
artificial boundary conditions averaging $B_\phi$ over the disk.

As was already mentioned, the present models are first steps only.
They seem, however, to keep the basic premise that magnetic fields can
lead to increased mass loss rates in disk winds. These models shall
also help to clarify whether magnetic fields can provide confinement
of line driven quasar winds, in order to prevent over-ionization by
central source radiation (deKool \& Begelman 1995); and whether line
driving can overcome problems with launching magnetocentrifugal winds
from accretion disks. The latter problems were discussed in an
important paper by Ogilvie \& Livio (1998).

\clearpage
\centerline{\bf Outlook and Acknowledgments}
\label{ackno}
\bigskip
\bigskip

With the book ``Foundations of Radiation Hydrodynamics'' by Mihalas \&
Mihalas recently being added to the Dover series of classical science
texts, there seems no further need to motivate ``radiating fluids'' (a
term coined by Mihalas \& Mihalas). In astrophysical hydrodynamics,
radiation and magnetic fields are of similar importance.

Radiation hydrodynamics splits into two branches, according to whether
the flow is driven by photon absorption in the continuum or in
spectral lines. The former class includes flows with Thomson
scattering on free electrons and `dusty winds'. The latter class
includes winds from hot stars and accretion disks, and is subject of
the present writing. A simple, analytic approximation for the line
force $g_l$ from a point source of radiation is found by assuming a
power law line list and validity of the Sobolev approximation, giving
$g_l \sim F \bigl(\rho^{-1} dv/dr\bigr)^\alpha$, with radiative flux
$F$ and $0<\alpha<1$. Thus, $g_l$ depends {\it non-linearly} on the
velocity {\it gradient} $dv/dr$ and matter density $\rho$, and
represents a truly new, hydrodynamic force.

New waves (termed Abbott waves) and a new instability (termed
de-shadowing instability) result from this new force, and are
discussed in Chapters~3 and 2, respectively. Abbott waves could be
responsible for shaping the velocity law of line driven winds, by
causing a runaway towards the critical flow solution of Castor et
al.~(1975). The de-shadowing instability, on the other hand, is
responsible for the formation of strong shocks and dense shells in the
wind, and is probably the origin of observed X-ray emission from
O~stars; initial doubts after first {\sc Chandra} X-ray line
observations seem to be overcome. In Chapter~4, we gave a first glance
at a rather new area in line driven wind hydrodynamics: accretion disk
winds, and we discussed the interplay between radiative, Lorentz, and
centrifugal forces.

What could be interesting topics for future work? With regard to the
de-shadowing instability, all simulations so far are one-dimensional
(with one puzzling exception), and two-dimensional hydrodynamic models
are urgently missing. Line synthesis calculations using 2-D
hydrodynamic models will allow to test our understanding of X-ray line
formation and certain aspects of UV line variability. With regard to
Abbott waves, the controversy whether they are an artefact of Sobolev
theory is at present only partially overcome by post-Sobolev,
numerical simulations. A strict derivation of the Green's function in
presence of line scattering is missing. Finally, with regard to disk
winds, some central questions are the increase in mass loss rates by
combined magnetoradiative driving; magnetic flow confinement to
prevent over-ionization by central source radation; and the amount of
external disk viscosity caused by the wind.

With all these questions still being unanswered, line driven wind
hydrodynamics appears to be a promising field for future research.

{\it And now enough has been said} (Aquinas, Sum.~Theol., p.~II.1,
q.~114, a.~10). Almost. It is my pleasure to thank Janet Drew in
London, Wolf-Rainer Hamann in Potsdam, Rolf-Peter Kudritzki in Hawaii,
Colin Norman in Baltimore, Stan Owocki in Newark, Adi Pauldrach,
Joachim Puls, and Christian Reile in Munich, and Isaac Shlosman in
Lexington for many stimulating discussions, at blackboards and in
restaurants.

\clearpage
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\bigskip 

\centerline{Abbreviations of journals were adopted from the
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\bigskip

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\end{description}

\clearpage
\centerline{\bf Appended Papers}
\vskip2truecm
\label{pprs}

$\enspace[1]$ Feldmeier A.~1995, A\&A, 299, 523

\smallskip
$\enspace[2]$ Feldmeier A., Anzer U., B\"orner G., Nagase F.~1996, A\&A, 311, 793

\smallskip
$\enspace[3]$ Feldmeier A.,~Kudritzki~R.,~Palsa~R.,~Pauldrach~A.,~Puls~J.~1997,~A\&A,~320,~899

\smallskip
$\enspace[4]$ Feldmeier A., Puls J., Pauldrach A.~1997, A\&A, 322, 878

\smallskip
$\enspace[5]$ Feldmeier A.~1998, A\&A, 332, 245

\smallskip
$\enspace[6]$ Feldmeier A., Shlosman I.~1999, ApJ, 526, 344

\smallskip
$\enspace[7]$ Feldmeier A., Shlosman I., Vitello P.~1999, ApJ, 526, 357

\smallskip
$\enspace[8]$ Feldmeier A., Shlosman I.~2000, ApJ, 532, L125

\smallskip
$\enspace[9]$ Feldmeier A., Shlosman I.~2002, ApJ, 564, 385

\smallskip
       $[10]$ Feldmeier A., Shlosman I., Hamann W-R.~2002, ApJ, 566, 392


\end{document}
