Mo delling Variability in Hot-Star Winds
1
Stanley P.Owocki
Bartol Research Institute University of Delaware, Newark, DE 19716 USA
Abstract. I review 2-D hydro dynamical simulations of rotating hot-star winds with
azimuthal structure induced by mo dulation of the radiative driving force near the wind
base. As a rst step toward examining more realistic p erturbation mechanisms (e.g.,
nonradial pulsations, or magnetic elds), the driving mo dulation here is taken to arise
from bright and dark sp ots in the stellar photophere. These sp ots induce decreases or
increases in wind ow sp eed, and as the star rotates, spiral \Co-Rotating Interaction
Regions" (CIRs) form, much as in the solar wind, from from interaction b etween fast
and slow ow streams. A new feature unique to line-driven owisavelo city-gradient
kink that propagates inward from interaction fronts at a fast radiative-acoustic mo de
sp eed. The slowly evolving velo city plateaus that form b ehind suchkinksgiveriseto
absorption features with a slow apparent acceleration, muchlike the Discrete Absorp-
tion Comp onents (DACs) often observed in UV wind lines from hot-stars. In simulation
mo dels with base driving sinusoidal ly mo dulated b etween increases and decreases, there
arise alternating spiral streams of enhanced or decreased density, asso ciated resp ec-
tively with decreased or increase ow sp eeds. These sp eed variations have substantial
impact on the line pro le, and so these dynamical simulations are not as successful as
analogous kinematic mo dels of corotating density streams in repro ducing the \phase-
bowing" of p erio dic absorption mo dulations observed in the recent IUE `Mega' pro ject.
1 Intro duction
Several pap ers at this meeting (e.g. by Prinja, Kap er, Kaufer, Massa, and Wolf )
have summarized the extensive observational evidence for explicit variability,
cyclical or otherwise, in the winds from hot, luminous, early typ e (OB) stars. A
general challenge for theory is to understand the nature of the physical changes
and/or structures asso ciated with this variability.To b e visible as direct varia-
tions in line-pro les formed from globally integrated radiative ux, the asso ciated
ow structure must b e on a relatively large scale, on order the stellar radius. This
makes it unlikely that suchvariations could stem from pro cesses entirely intrin-
sic to the wind out ow, such as the inherent instability in the line-driving of the
wind to small-scale p erturbations (Rybicki 1987; Owocki, Castor, and Rybicki
1988; Owocki1994b). The dynamical evolution of such large-scale structure can
be simulated using the lo cal, computationally ecient, CAK/Sob olev expres-
sions for the line force (Castor, Abb ott, and Klein 1975; Sob olev 1960), making
it feasible to carry out multidimensional simulations of the wind structure re-
sulting from large-scale p erturbations from the underlying, rotating star.
In this review, I will fo cus on recent e orts to develop initial dynamical
mo dels for two distinct classes of suchvariability, namely the `classical' Discrete
2 Stanley P.Owocki
Absorption Components (DACs) and the more recently identi ed Periodic Ab-
sorption Modulations (PAMs) discovered in the IUE `Mega' pro ject (Massa et al.
1995). Unlike the quasi-episo dic, slowly evolving, net absorption enhancements
that characterize most DACs, the PAMs recur regularly at p erio ds a loworder
fraction of the rotation p erio d, include b oth reductions and enhancements of
absorption, and evolve relatively quickly over the line pro le. Indeed, in contrast
to the slow blueward evolution of DACs, the PAMs in one case (BO I star HD
64760) show a \phase-b owing" that re ects apparentredward as well as blue-
ward propagation (Owocki, Cranmer, & Fullerton 1995; ; Fullerton et al. 1997).
These distinct observational characteristics likely re ect di erences in the under-
lying p erturbation mechanisms, p erhaps, for example, with the more sto chastic
DACs b eing induced by magnetic activity, and the regular PAMs b eing initiated
by Non-Radial Pulsations (NRPs). But given the present uncertainty, the initial
simulations here simply induce wind variations rather arti cially, through direct
mo di cation of the radiative driving in the inner wind, much as might o ccur
from \sp ots" on the underlying star (Cranmer and Owocki 1996, hereafter CO).
I rst (x 2) describ e the e ect of isolated sp ots, b oth brighter and darker than
the ambiant photophere. As the star rotates, Co-rotating Interaction Regions
(CIRs) form along spiral patterns by collision b etween faster and slower wind
streams originating from di erent longitudes relative to the sp ot. These simula-
tions thus represent the rst dynamical test of the original prop osal by Mullan
(1984a,b; 1986) that the wind density enhancements in such CIRs could cause
the DACs. A central goal here is to determine whether key characteristics of ob-
served DACs, particularly their apparent slow acceleration, can b e repro duced
in synthetic line-pro les generated from dynamical mo dels with CIRs.
I also examine (x 3) the e ect of a sinusoidal modulation of the radiativedriv-
ing near the wind base, assuming a xed number (m = 4) no des around the star.
This is intended as a dynamical version of the simple kinematic picture prop osed
to explain the \phase b owing" of the PAMs in HD 64760 (Owocki, Cranmer, &
Fullerton 1995.) In this picture the PAMs arise from corotating streams of al-
ternating increased or decreased density, within the key simpli cation that the
velo city is xed to a sp eci c law, una ected by the density p erturbations. The
dynamical simulations here self-consistently include suchvelo cityvariations, and
so allow us to examine their e ect in the line-formation.
2 CIRs Induced by Isolated Sp ots
Let us rst review the CO simulations of wind structure induced by isolated
sp ots on a rotating hot-star. The aim is to mimicphysical pro cesses { e.g. mag-
netic eld, NRPs { that might increase or decrease the mass ux emerging from
some lo calized region of the star, and then study howsuchvariations are prop-
agated through the radiatively driven stellar wind. To reduce the complexity
and computational costs, the simulations are con ned to 2-D variations in ra-
dius and azimuth (r , ) within the equatorial plane. The line-driving force is computed using the standard CAK/Sob olev formalism (Castor, Abb ott, Klein
Mo dellin g Variability in Hot-Star Winds 3
Fig. 1. Grayscale p olar plots showing the spatial dep endence of the deviations from
a steady mo del for a. density, b. radial velo city,c.azimuthal velo city, and d. Sob olev
optical depth.
1975; Sob olev 1960), mo di ed by the nite-disk correction factor for a spherical
out ow(Friend and Abb ott 1986; Pauldrach, Puls, and Kudritzki 1986). This
ignores the azimuthal force that should arise from sideview p ersp ectives of the
sp ot, and simply mo di es the radial force by a xed enhancement/reduction
factor determined by the relativeproximity to the sp ot.
The sp ot parameterization allows sp eci cation of b oth a longitudinal width
and an amplitude A, with 1
sp ots. Of the mo dels listed in Table 1 of CO, I con ne attention here to the rst
o
two, for which =20 and A = 0:5, representing a standard bright and dark
sp ot. The stellar parameters represent those for the canonical O-typ e sup ergiant
Puppis, with a rotation sp eed V = 230 km/s, corresp onding to a rotation
rot
p erio d of P 4:2days. For convenience, let us assume two identical sp ots
p ositioned on opp osite hemispheres, allowing restriction of the computation to
o
an azimuth range of just 180 with p erio dic b oundary conditions.
For the bright sp ot case, gure 1 shows grayscale plots of the resulting density,
radial velo city, azimuthal velo city, and radial Sob olev optical depth, all relative
to values in the corresp onding steady, spherically symmetric, CAK wind mo del
without anyspot.Thespothereiscentered on the x-axis, at the origin the
spiral pattern of enhanced densitythatcharacterizes the resulting CIR. Note
that regions of enhanced density generally corresp ond to regions of lower radial
velo city. Because of the enhanced brightness over the sp ot, the wind driven from
there initially has a higher mass ux, and thus higher density. As the enhanced
brightness fades with increasing heightabove the sp ot, the higher densityofthis
4 Stanley P.Owocki
Fig. 2. left panels: Radial variation of velo city (upp er panel) and density (lower panel)
along selected, xed azimuthal angles in the bright sp ot mo del.
Fig. 3. right panels: Same as g. 2, but for the dark sp ot mo del.
material makes it harder to accelerate, re ecting the characteristic scaling of the
line-acceleration with the inverse of the density ,
1 @v
r
g (1)
lines
@r
where is the usual CAK exp onent. The lower velo city gradient @v =@ r asso-
r
ciated with the lower acceleration also contributes, through eq. (1), to a further
reduction in the acceleration. As the stellar rotation brings this higher-density,
lower-sp eed material into interaction with faster ambient wind originating away
from the sp ot, there results a sho ck compression of the gas into the dense spiral
stream that characterizes the CIR.
Though the general CIR formation here is analogous to that o ccuring in the
solar wind (Hundhausen 1972; Zirker 1977), there are imp ortant di erences. The
much higher density of hot-star winds makes radiative co oling very ecient, and
so the sho cks are e ectively isothermal. Unlike the nearly adiabatic sho cks in
the solar wind, whichhave a maximum compression ratio of four in the strong
sho ck limit, the density compression in such isothermal sho cks scales with the
square of the Machnumb er, and so can b e arbitarily large. Moreover, whereas
the nearly adiabatic sho cks of the solar wind propagate backward into the faster
wind at roughly a quarter of the sho ck sp eed, these isothermal sho cks remain in
close proximity to the compressed layer.
Mo dellin g Variability in Hot-Star Winds 5
Another signi cant di erence arises when we consider the nature of the pre-
shock ow. In ordinary gas-dynamics (or even MHD) sho cks, the presho ck ow
is assumed to b e completely una ected by the imp ending sho ck, simply b ecause
the sp eed of this material exceeds the sound sp eed (or in MHD, the fast-mo de
sp eed), preventing anyupwind information propagation that a disturbance lies
ahead. However, as rst shown by Abb ott (1980), the characteristic sp eeds c in
a line-driven ow are substantially mo di ed by the op eration of the line force,
given by
p
2 2
c = U=2 (U=2) + a ; (2)
where a is the ordinary (isothermal) sound sp eed, and U @g =@ (@v =@ r )
lines r
is a new characteristic sp eed, nowadays often called the \Abb ott sp eed". From
eqn. (1) we ndU g =(@v =@ r ), but from the radial equation of motion,
lines r
we exp ect v @v =@ r g , which suggests U v , i.e. this Abb ott wave sp eed is
r r r r
typically of order the radial ow sp eed. Since most the wind is sup ersonic, we
nd v U a, implying that inward wave propagation can now o ccur at a
r
sp eed c U v that is much faster than the propagation of an ordinary
r
sound wave (Owocki and Rybicki 1985, 1986; Rybicki, Owocki, & Castor 1990).
Figure 2 shows line plots of the radial variation of (a) velo city and (b) den-
sity along selected, xed azimuthal angles. The lo cation of the compressed CIR
is apparent from the velo cityminima and density maxima. But note that the
velo city has a decreasing outward gradientover a broad region ahead of this
CIR, extending backtoavelo city gradient discontinuity, or \kink", that marks
the connection to the unp erturb ed, outward-accelerating wind. This weak, kink
discontinuity propagates inward (relative to the wind out ow) at roughly the
characteristic sp eed c U , and since this is nearly as fast as the lo cal out-
ow sp eed, it has only a slow net outward propagation in the xed stellar frame.
Such fast inward-propagation of a velo city-gradientkinkisanovel and quite
unique feature of a line-driven ow, and it has p otentially imp ortant conse-
quences for interpreting observed line-pro le features, particularly DACs. To
identify which wind structures should yield the most prominent pro le varia-
tions, let us examine the radial Sob olev optical depth,
v (r; )
th
: (3) (r; )=
r
j@v =@ r j
r
The line absorption co ecient and the ion thermal sp eed v are assumed to
th
have the same, constantvalue in the p erturb ed and unp erturb ed wind. Thus
the variations in optical depth ratio plotted in gure 1d re ect changes in the
ratio of densityover velo city-gradient. Surprisingly,we see from comparison
with gure 1a that the regions of strongest optical depth enhancements do not
o ccur within the dense CIR compression, but rather within the shallowvelo city-
gradient region immediately after the kink.
The emergent ux line-pro les from this wind structure can b e readily syn-
thesized using the standard SEI (Sob olev source function with Exact Integration)
metho d of Lamers et al. (1987), generalized for 3-D integration as describ ed in
CO. For this purp ose, the 3-D latitudinal structure is derived byinterp olation
6 Stanley P.Owocki
Fig. 4. Line pro le variation for marginally thick line as a function velo city and time.
In the middle b ox, the solid line shows the unp erturb ed wind pro le, and the dashed
lines show the extrema for eachvelo city. The b ottom b oxshows the pro le standard
deviation.
from the equator to a p olar mo del characterized by a 1-D, unp erturb ed wind so-
lution. For a mo derate strength line that is marginally optically thickinthewind,
gure 4 shows a grayscale plot of the pro le variation as a function of frequency
(in velo city units) and time. The latter is derived from simple mapping of the
azimuthal co ordinate using the star's rotation frequency ,i.e.t =( )= ,
o
where is de ned such that the observer is directly over the sp ot at the initial
o
time t =0.Following observational convention, the gray scale here is scaled to
the range from mininum to maximum ux at each frequency, with the maxi-
Mo dellin g Variability in Hot-Star Winds 7
Fig. 5. Apparent acceleraton vs. velo city for bright-sp ot mo del DAC, compared to
velo citylaws with various exp onents .
mum ux (i.e., minimum absorption) corresp onding to the lightest shade. This
convention is suitable for describing the observed DACs that represent mainly
enhanced absorption relative to the background wind. But the synthetic pro le
variations here also showintervals of reduced absorption, re ecting the relative
reduction of the density in the region just b eyond the CIR.
As noted ab ove, the absorption enhancements here arise primarily from re-
gions of shallowvelo city-gradient near the reverse mo de kinks. Because of the
slow net outward propagation of these kinks, the asso ciated synthetic DACs mi-
grate quite slow ly blueward across the pro le over roughly 3:9day, in general
agreement with the slow apparent acceleration of observed DACs. Often this
slow accleration is characterized in terms of large values of the exp onent in
the canonical velo citylawgiven by v (r )=v (1 R =r ) . Figure 5 plots the ap-
1
parent acceleration vs. velo city for the synthetic DAC compared with results for
velo citylaws with various . Whereas an unp erturb ed wind is exp ected to have
avelo citylaw with 1, g. 4 shows that the DAC in this mo del corresp onds
more closely to higher exp onents 2 4, much as inferred for observed DACs.
Let us next brie y consider results for the case of a \dark" sp ot, wherein
the radiative driving is reduced at base of the wind out ow. In this case, the
mass ux emerging from the sp ot region is now reduced, and as the radiative
driving recovers at greater heights ab ove the sp ot, the reduced density causes an
enhanced acceleration, which then results in a very fast stream. Figure 3 shows
the radial variation of velo city and density at selected xed angles. Note the
velo cities ranging up to 5000 km/s, more than double the maximum sp eed in
8 Stanley P.Owocki
Optical Density Velocity Depth
-0.5 0.0 0.5 -40 -20 0 20 40 -1.0 -0.5 0.0 0.5 1.0
ρ ρ V-Vo (km/s) τ/τ -1
log [ / o] o
Fig. 6. Spatial variation of density,velo city and Sob olev optical depth, normalized
relative to the unp erturb ed mo del, for the case with m =4 sinusoidal mo dulation in
azimuth.
the unp erturb ed wind, and muchhigherthanisever observed in line-pro les of
such an O-sup ergiant. This implies that such reduced mass ux regions cannot b e
very prominent in hot-star winds. In particular, it argues against the imp ortance
of hot-star analogs of solar coronal holes (Zirker 1977), with very high-sp eed
wind streams arising from regions of rapidly diverging op en magnetic eld (cf.
MacGregor 1988).
3 Co-rotating Streams from Sinusoidal Mo dulations
An imp ortant result of the IUE 'Mega' campaign was the identi cation of the
clearly cyclical PAMs in wind line pro les of several stars. In the case of the
B0 I star HD 64760, these PAMs show a \phase-b owing" that suggests appar-
ent redward as well as blueward propagation. Owocki, Cranmer, and Fullerton
(1996) have shown that suchbowing can b e repro duced from a simple kinematic
mo del with co-rotating spiral streams of alternating regions of increased and re-
duced density. This kinematic mo del ignores any dynamical e ect these density
variations mighthave in inducing corresp onding velo citychanges. As an ini-
tial attempt to extend this kinematic picture into a dynamically self-consistent
mo del, let us next brie y consider the resp onse of a line-driven wind to a p er-
turbation that varies sinusoidal ly in azimuth b etween enhanced and reduced
radiative driving.
Figure 6 shows grescale plots of the changes in density,velo city, and Sob olev
o
optical depth for such a dynamical mo del with an azimuthal p erio d of 90 and
Mo dellin g Variability in Hot-Star Winds 9
Fig. 7. As in g. 4, but for the sinusoidal ly mo dulated mo del.
p erturbations amplitdue of A =0:5. Note that there are substantial e ects on the
velo cityaswell as density, and that the optical depth variations no longer just
follow the density,aswas assumed in the heuristic corotating stream mo del for
the observed phase-b owing. Figure 7 shows the quite complex pro le variability
for the line pro le of a mo deratel strong in this sinusoidal mo dulation mo del. The
complexity re ects the intricate interdep endence of owvariations in density,
velo city, and velo city gradient. The comparitively simple phase-b owing pattern
of the kinematic mo del is not apparent here. Indeed, it is not clear what sort of
dynamically consistent structure could give the observed phase-b owing.
10 Stanley P.Owocki
4 Concluding Remarks
The simulations here demonstrate that the physical dep endencies of line-driving
play an imp ortant role in the dynamical evolution of wind structure, and lead
to phenomena, e.g., the reverse-mo de kink, that have no analog in ordinary gas
dynamics. The inward propagation of this kink yield the slow net outward prop-
agation of a velo city plateau ow pattern that causes the enhanced absorption,
providing one p ossible explanation for the relatively slow apparent acceleration in
observed DACs. However, suchmodelstypically also predict line-pro le features
that not commonly seen, most notably reduced absorption comp onents. Thus,
an alternativemechanism for pro ducing slowDACs is a substantial increase,
i.e. factor of two or more, in the mass b eing ejected into the wind, without a
comp ensating increase in the driving. The acceleration of such material can b e
esp ecially slow, and so the extra absorption from its higher density could also
form slowly evolving DACs (Owocki, Fullerton, and Puls 1994).
As for the PAMs observed in the IUE mega pro ject, we nd that dynamical
mo dels with p erio dic mo dulation typically show substantial velo cityvariations
that complicate the line-pro le signature. In particular, the simple, phenomena-
logical, kinematic picture develop ed to explain phase b owing cannot b e easily
adapted to a fully self-consistent dynamical mo del.
The work describ ed here represents only the rst steps in trying to develop
dynamically self-consistent mo dels of the ow structure assoicated with the ob-
served variability of hot-star winds. In the future, mo dels should address more
directly the twomostobvious p ossible p erturbation mechanisms, namely non-
radial pulsations and surface magnetic elds, with particular emphasis on iden-
tifying observatonal characteristic that could help in distinguishing whichmech-
anism is most resp onsible for eachofvarious kinds of observed wind variability.
Within such studies, it will remain helpful to contrast the inferred owstruc-
tures with those derived for the sun and other co ol stars, and in particular to
identify whichcharacteristics are p eculiar to the dynamics of radiative driving
by line-scattering. The cyclical variability of hot-stars thus provides an unique
p otential for studying this imp ortant asp ect of radiation hydro dynamics.
Acknowledgements: This work was supp orted in part by NASA grant
NAGW-2624. Supp orting computations were carried out using an allo cation of
sup ercomputer time from the San Diego Sup ercomputer Center. Muchofthe
research reviewed here was carried out in collab oration with S. Cranmer, as part
of his Ph. D. thesis. I also acknowledge numerous helpful discussions with A.
Fullerton, K. Gayley, H. Henrichs, L. Kap er, and J. Puls.
Mo dellin g Variability in Hot-Star Winds 11
References
Abb ott, D. C. 1980, Ap. J., 242, 1183
Castor, J. I., Abb ott, D. C., & Klein, R. I. 1975, Ap. J., 195, 157 (CAK)
Cranmer, S. R., & Owocki, S. P. 1996, Ap. J., 469, 469
Friend, D. B., & Abb ott, D. C. 1986, Ap. J., 311, 701
Fullerton, A. W., & Owocki, S. P. 1992, in
Fullerton, A. W., Massa, D. L., Prinja, R. K., Owocki, S. P., and Cranmer, S. R. 1997,
Astron. & Ap., 327, 699
Hundhausen, A. J. 1972, Coronal Expansion and Solar Wind (Berlin: Springer-Verlag)
Lamers, H. J. G. L. M., Cerruti-Sola, M., & Perinotto, M. 1987, Ap. J., 314, 726
MacGregor, K. B. 1988, Ap. J., 327, 794
Massa, D., et al. 1995, Ap. J.l, 452, L53
Mullan, D. J. 1984a, Ap. J., 283, 303
Mullan, D. J. 1984b, Ap. J., 284, 769
Mullan, D. J. 1986, Astron. & Ap., 165, 157
Owocki,S.P. 1994, in IAU Symp. 162, Pulsation, Rotation, and Mass Loss in Early-
Typ e Stars, ed. L. A. Balona, H. F. Henrichs, & J. M. Le Contel (Dordrecht: Kluwer),
475
Owocki,S.P. 1994, Ap. Sp. Sci., 221, 1
Owocki,S.P., Castor, J. I., & Rybicki, G. B. 1988, Ap. J., 335, 914
Owocki,S.P., Cranmer, S. R., & Fullerton, A. W. 1995, Ap. J.l, 453, L37
Owocki,S.P., Fullerton, A. W., & Puls, J. 1994, Ap. Sp. Sci., 221, 437
Owocki,S.P., & Rybicki, G. B. 1985, Ap. J., 199, 365
Owocki,S.P., & Rybicki, G. B. 1986, Ap. J., 209, 127
Pauldrach, A. W. A., Puls, J., & Kudritzki, R.-P. 1986, Astron. & Ap., 164, 86
Pizzo, V. J. 1982, J. Geophys. Res., 87, 4374
Rybicki, G. B. 1987, in Instabilities in Luminous Early Typ e Stars, ed. H. J. G. L. M.
Lamers & C. W. H. de Lo ore (Dordrecht: Reidel), 175
Rybicki, G. B., Owocki, S. P., & Castor, J. I. 1990, Ap. J., 349, 274
Sob olev, V. V. 1960, Moving Envelop es of Stars (Cambridge: Harvard University Press)
Zirker, J. B., ed. 1977, Coronal Holes and High Sp eed Wind Streams (Boulder: Colorado
Asso ciated University Press)
Henrichs: We, as observers, would like to thank you very muchforthesomany
years of fundamental calculations you did, that help ed enormously to interpret
the data on stellar wind variability.You showed even an example that is far
ahead of any observations I have seen so far. For obvious reasons you put your
p erturbing sp ot on the stellar equator, but in practice such a sp ot is most likely
at higher latitude. Would that not change the predicted b ehavior drastically?
Could you comment?
Owocki: Perturbations away from the equator obviously require a 3-D simula-
tion, which with mo dern computers is readily doable, though not yet attempted.
In the context of CIRs in the solar wind, V. Pizzo in the 1970's already devel-
op ed 3-D mo dels, showing some interesting new e ects not p ossible in 2-D, e.g.
ow around the CIR at other latitudes. In rapidly rotating hot stars, an addi-
tional e ect may b e de ection toward or away from the equator. This remains
a problem to b e investigated.