Wind Inhomogeneities in WR Stars. II. Investigation of Emission-Line

WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. INVESTIGATION OF EMISSION-LINE PROFILE VARIATIONS SE BASTIEN LE PINE1 AND ANTHONY F. J. MOFFAT2 De partementde Physique, Universite deMontre al, and Observatoire du Mont-Me gantic, C.P. 6128, Succ. Centre-Ville, Montre al, QC, CANADA H3C 3J7; lepine=astro.umontreal.ca, mo†at=astro.umontreal.ca Received 1997 September 9; accepted 1998 August 4 ABSTRACT We present high-resolution spectroscopic monitoring of the line-proÐle variations (LPVs) in the He II j5411 emission line of four Wolf-Rayet (WR) stars of the WN sequence (HD 96548, HD 191765, HD 192163, and HD 193077) and in the C III j5696 emission line of Ðve WR stars of the WC sequence (HD 164270, HD 165763, HD 192103, HD 192641, and HD 193793). The LPVs are shown to present systematic patterns: they all consist of a number of relatively narrow emission subpeaks that tend to move from the line centers toward the line edges. We introduce a phenomenological model that depicts WR winds as being made up of a large number of randomly distributed, radially propagating, discrete wind emission elements (DWEEs). This working model is used to simulate LPV patterns in emission lines from a clumped wind. General properties of the LPV patterns are analyzed with the help of novel numerical tools (based on multiscale, wavelet analysis), and simulations are compared to the data. We investigate the e†ects on the LPVs of local velocity gradients, optical depths, various numbers of discrete wind elements, and a statistical distribution in the line Ñux from individual elements. We also investigate how the LPV patterns are a†ected by the velocity structure of the wind and by the extension of the line-emission region (LER). Eight of the stars in our sample are shown to possess strong similarities in their LPV patterns, which can all be explained in terms of our simple model of local wind inhomogeneities. We Ðnd, however, that a very large number(Z104) of DWEEs must be used to account for the ~1 LPV. Large velocity dispersions must occur within DWEEs, which give rise to thepm D 100 km s line-of-sight velocity dispersions. We Ðnd evidence for anisotropy in the velocity dispersion within DWEEs withpvr D 4pvh, wherepvr andpvh are the velocity dispersions in the radial and azimuthal directions, respectively. We Ðnd marginal evidence for optical depth e†ects within inhomogeneous features, with the escape probability being slightly smaller in the radial direction. The kinematics of the variable features reveals lower than expected radial accelerations, with20 \bR (R ) \ 80, where b andR are \ [ ~1 b * _ * parameters of the commonly used velocity lawv(r) v=(1 R r ) , withv= the terminal wind velocity. The mean duration of subpeak events, interpreted as the crossing* time of DWEEs through the LER, is found to be consistent with a relatively thin LER. As a consequence, the large emission-line broadening cannot be accounted for by the systematic radial velocity gradient from the accelerating wind. Rather, emission-line broadening must be dominated by the large ““ turbulent ÏÏ velocity dispersion pvr suggested by the LPV patterns. The remaining WR star in our sample (HD 191765) is shown to present signiÐcant di†erences from the others in its LPV pattern. In particular, the associated mean velocity ~1 ~1 dispersion is found to be especially large(pm D 350 km s , compared topm D 100 km s in other stars). Accordingly, the LPV patterns in HD 191765 cannot be satisfactorily accounted for with our model, requiring a di†erent origin. Subject headings: instabilities È line: proÐles È stars: mass loss È stars: Wolf-Rayet È turbulence

1. INTRODUCTION that emission lines from di†erent atomic transitions are formed at di†erent depths in the wind. The spectra of Wolf-Rayet (WR) stars are dominated by Current models of WR atmospheres rely on a set of sim- broad emission lines of Helium, with lines of nitrogen in plifying assumptions within the framework of the so-called stars of the WN sequence, or carbon and oxygen in stars of ““ standard model ÏÏ (see, e.g., Hillier 1995; Hamann 1995, the WC/WO sequence. These lines are sometimes accompa- and references therein), which describes the formation of nied by blueshifted P Cygni absorption features. The gener- emission lines in a dense wind photoionized by a hot core. ally accepted interpretation is that the lines are formed in In the standard model, it is assumed that the wind is spher- extended regions of a fast (D103 km s~1), dense (D10~5 ~1 ically symmetric, homogeneous, and stationary. Radiative M_ yr ) stellar wind (see, e.g., Willis 1991). The spectral and statistical equilibrium are adopted, and a monotonic analysis of WR stars is made difficult by the fact that the wind velocity law is Ðxed a priori. These assumptions have usual assumptions of local thermodynamic equilibrium and been used to perform spectral analyses of WR spectra and plane-parallel atmospheres do not apply. Moreover, WR to predict e†ective temperatures and luminosities. However, winds are believed to be stratiÐed in ionization (see, e.g., several systematic deÐciencies have been observed (see, e.g., Schulte-Ladbeck, Eeenens, & Davis 1995), which means Howarth & Schmutz 1992; Hamann, Wessolowski, & Koesterke 1994; Hillier 1996; Schmutz 1997), suggesting 1 Present address: Space Telescope Science Institute, 3700 San Martin that at least some assumptions in the standard model are Drive, Baltimore, MD 21218; lepine=stsci.edu. invalid. 2 Killam Research Fellow of the Canada Council for the Arts. The picture of a smooth, homogeneous wind in WR stars 909 910 LEŠ PINE & MOFFAT Vol. 514 is being challenged by several lines of observational evi- e.g., in HD 191765 (Vreux et al. 1992; McCandliss et al. dence (see Mo†at 1996). Since it is believed that inhomoge- 1994) and in EZ CMa (Robert et al. 1992; St-Louis et al. neous winds might lead to a downward revision of the 1995; Morel, St-Louis, & Marchenko 1997). It is suspected mass-loss rate (Mo†at & Robert 1994), this would have that these periodic variations are linked either to the pres- important implications for evolutionary models of massive ence of a compact companion or to the rotation of global stars (Maeder 1991). Recently, some attempts have been wind structures. However, whereas such a periodic behavior made to relax certain assumptions of the standard model is observed in only a few stars, stochastic variations seem to (Hillier 1996, and references therein), and spectral analysis be present in every WR star observed so far. using clumped wind models has been attempted (Schmutz A phenomenological model has been suggested to explain 1997; Hillier & Miller 1998). the LPV patterns in terms of local overdensities (termed The difficulty in establishing realistic, inhomogeneous ““ blobs ÏÏ or ““ clumps ÏÏ) in WR winds, as being related, e.g., wind models is that clues about the degree of clumping have to compressible, supersonic turbulence (Robert 1992; mainly come from indirect observational methods. For Mo†at et al. 1994). Each clump presumably follows the example, continuum-emission excess in the infrared and general wind expansion, giving rise to one emission-line radio, which is interpreted as excess free-free emission due subpeak as it moves in, through, and out of the LER. A to wind overdensities (Lamers & Waters 1984), was similar inhomogeneous wind model had previously been observed in 18 early-type stars (Runacres & Blomme 1996). investigated (see Antokhin et al. 1992) in relation to devi- Though models of inhomogeneous winds were shown to ations in the shape of an emission-line proÐle for a clumped reproduce the IR and radio excess well (Blomme & wind. A review of the simultaneous photometric, spectro- Runacres 1997), only limited constraints on the detailed scopic, and polarimetric variations expected from such a density structure could be obtained, since observations clumped wind model has been presented elsewhere (Brown mostly depend on the global e†ects of clumping. Stochastic et al. 1995). variations in polarization and photometry of single WR In the Ðrst paper of this series(Le pine, Mo†at, & Henrik- stars were also interpreted in terms of residuals from a sen 1996, hereafter Paper I) we performed a preliminary clumped wind (Robert et al. 1989). Other examples include analysis of LPVs in emission lines of WR stars using the measures in the intensity of the electron-scattering wings in clumped-wind hypothesis as a working model. We investi- some WR emission lines (Hillier 1991), the relative intensity gated the e†ects on the LPV patterns of a hierarchy of of the IR lines of He I and He II (Nugis & Niedzielski 1995), inhomogeneous elements, which might result from the pres- and j-dependence in the secondary eclipse in V444 Cyg ence of self-structured chaos (e.g., compressible turbulence) (Cherepashchuk, Khaliullin, & Eaton 1984). in the wind (Robert 1994; Henriksen 1994). This involved a Although we can directly observe details in the clumpy study of the LPVs in a static situation (i.e., snapshots of the structure of wind-blown bubbles around Wolf-Rayet stars emission line). (see Marston 1997), we cannot yet directly resolve regions In this second paper, we investigate the LPV patterns in a close to the star, where emission lines are formed and the more general way, in terms of a complete, phenomenologi- wind is presumably driven by the intense radiation Ðeld. In cal model of radially propagating, discrete wind elements. the near future, it might be possible to resolve this wind We Ðrst present a database that comprises sets of high- region in c2 Vel, the closest WR star at D250 pc (van der resolution spectra of optical emission lines from nine WR Hucht et al. 1997; Schaerer, Schmutz, & Grenon 1997), stars (° 2). Our phenomenological formalism, which allows using optical interferometric imaging techniques (see Vakili for a simpliÐed but general description of the inhomoge- et al. 1997). Still, most WR stars will remain out of reach, at neous structure from a WR wind, is then presented (° 3). We least initially. use this working model to generate synthetic spectral time Another technique that can provide limited resolution of series, which are analyzed in order to determine which the inhomogeneous structure of the wind, and that is inde- information can be obtained from the LPV patterns (° 4). pendent of the distance, is high-resolution spectroscopy of We are helped in this by the introduction of two numerical broad emission lines. Because of the large Doppler shifts tools that provide a systematic analysis of the LPV pattern. induced by the high wind velocity, regions in the wind We perform a comparative analysis between the data and having large di†erences in their line-of-sight velocities can the simulations, and obtain various constraints on the inho- be resolved on the emission-line spectrum. Early obser- mogeneous wind structure of the WR stars in our sample. vations showed rapidly Ñuctuating features in the broad The results are discussed in ° 5. A summary is presented in C III j5696 emission-line proÐle of the WC7 component in the Ðnal section (° 6). the binary star HD 152270 (Schumann & Seggewiss 1975). SPECTROSCOPIC OBSERVATIONS These were interpreted as spectroscopically resolved, 2. moving clouds or knots in the expanding wind. Later, high- A set of high signal-to-noise ratio (S/N [ 200), high- resolution spectroscopic monitoring of the He II j5412 resolution (j/*j D 30,000) spectra of emission lines from emission line in the WN6 star HD 191765 revealed many nine WR stars was obtained from two observing runs at the narrow, moving emission features superposed on the Canada-France-Hawaii Telescope (CFHT) in 1987 and emission-line proÐle (Mo†at et al. 1988). These appeared to 1988 and one at the European Southern Observatory (ESO) be the trace signature of clumps being accelerating along in 1989. Reduction and preliminary analysis was carried out with the wind and emitting for a few hours as they passed by C. Robert as part of her Ph.D. thesis (Robert 1992). Basic through the line-emission region (LER). Extensive monitor- characteristics of the selected targets are presented in Table ing of nine WR stars of various subtypes showed this phe- 1. The CFHT spectra cover the optical region around the nomenon to be present in each star observed (Robert 1992, C III j5696 line for the WC stars WR 135, WR 137, and WR 1994), and to be apparently stochastic. Similar, but periodic, 140, and around the He II j5411 line for the WN stars WR LPVs were also observed in some apparently single stars, 134, WR 136 and WR 138. The ESO spectra of stars WR 40, No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 911

TABLE 1 SAMPLE OF WOLF-RAYET STARS OBSERVED

Orbital b Period v= Number of a ~1 HD WR Subtype (yr) (km s ) Spectra S/Ncont 96548 ...... 40 WN8 ... 975 16 350 164270 ...... 103 WC9 ... 1190 30 210 165763 ...... 111 WC5 ... 2415 28 150 191765 ...... 134 WN6 ... 1905 36 190 192103 ...... 135 WC8 ... 1405 26 180 192163 ...... 136 WN6 ... 1605 26 200 192641 ...... 137 WC7]OB 12.6c 1885 25 240 193077 ...... 138 WN5]OB 4.2d 1345 18 230 193793 ...... 140 WC7]O4. . .5 7.94e 2900 23 380

a WR number from the van der Hucht et al. 1981 catalog. b Terminal wind velocities from Prinja, Barlow, & Howarth 1990. c From Williams et al. 1996. d From Annuk 1990. e From Williams et al. 1990.

WR 103, and WR 111, cover a broader range with several star, the wind has approximately constant radial velocity v emission lines (5200È5900A ), but here we chose to study the and accelerationv5 , which yieldsm5 P m, similar to what is same lines in each case as observed at CFHT, since these observed. lines are the strongest isolated emission lines in the spectra On the gray-scale plots in Figures 1, 2, and 3, the eye of WC and WN stars. The temporal resolution and cover- catches what appear to be a few moving subpeaks. It is age (one spectrum every hour or so over an interval of D8 tempting to assume each of these apparent features to arise hr on each of three or four consecutive nights) make these in a single wind event (cloud or clump). It then appears spectral time series well suited for a comprehensive study of possible to enumerate how many clump events instantane- the LPVs on short timescales. ously occur in the LER. Unfortunately, this is an overly We present a compact display of the observed LPVs: simplistic interpretation of the data. In Paper I, we have time-wavelength gray-scale plots of the residuals after sub- shown that the few apparently individual subpeaks tracting the mean line proÐle are shown in Figures 1, 2, and uncovered either by eye or by any other kind of structure 3. These clearly reveal the presence of moving spectral fea- identiÐer (such as multiÈGaussian proÐle Ðtting), may tures (subpeaks). The most apparent subpeaks have full actually result from the sum of a large number of indepen- widths at half-maximum (FWHMs) ranging from D2È10 A , dent wind events. Only a few features actually appear with a typical amplitude of about 2%È8% of the line inten- because of superposition and blending e†ects. Therefore, sity. Residuals show subpeaks with positive (negative) the characteristics of apparent subpeaks extracted from the amplitudes, which most likely indicate excess (lack) of emis- data can be biased, because they depend on the way in sion. Absorption is unlikely, since subpeaks are exclusively which subpeaks superpose in line-of-sight velocity space. observed in the spectral range spanned by the line-emission This is why we have introduced numerical tools, such as the proÐle (Robert 1992). Subpeaks systematically move from wavelet transform, which analyze the LPV pattern as a line center toward line edges (Mo†at et al. 1988; Robert whole instead of a few apparently independent components. 1992). This behavior rules out both nonradial pulsations, or Wavelet analysis is analogous to a windowed Fourier any rotation/orbital motion hypothesis, since one-half of transform in that it yields simultaneous information on the subpeaks on average would then be expected to move both the scale and the location of features in a signal. We toward the line center. Rather, these data are consistent have already discussed the advantages of the wavelet trans- with wind features accelerated outward along radial trajec- form over the Fourier transform (see Paper I). We provide tories. in Appendix A a brief review of our wavelet analysis The central wavelength of a subpeak corresponds to the method. One powerful tool is the wavelet spectrum, which mean line-of-sight velocity of one local wind feature (via the is like a multiscale analyzer. We present in Figures 1, 2, and Doppler e†ect). The subpeak width presumably reÑects a 3 the mean wavelet spectrumSR3 (m, pm)T of the residual dispersion in velocity within the feature. Such internal spectra from each star. The wavelet spectra show striking velocity dispersions may arise from thermal or turbulent patterns that can be interpreted as consisting of two com- motions, but they may also arise from a systematic velocity ponents: (1) a more or less uniform response over location gradient throughout the feature (e.g., from a shock m, which gradually decreases from small to medium scale pm, discontinuity). Subpeaks near the edges on the emission-line on which is superposed (2) a triangular pattern with proÐle have the largest apparent line motion (i.e., line-of- maximum response nearp D 102 km s~1 (except for WR m 2.5 ~1 sight acceleration), whereas subpeaks near the center 134, where the maximum is nearer topm D 10 km s ) appear to be almost stationary. This can be naturally and centered on m \ 0. The Ðrst component is the wavelet explained as a consequence of the radial motion of wind response to instrumental noise, which is uniform over the features. With line-of-sight velocity m \ vk, where k \ cos(h) spectrum and yields the highest response at the smallest is the projection factor relative to the line of sight (h \ 0), scales (pixel-to-pixel variation). The second component is one sees that a radial motion having k \ const implies a the wavelet response to intrinsic LPVs, as evidenced from spectral motionm5 \ v5 k. In a shelllike region centered on the the fact that the response correlates strongly with the 912 LEŠ PINE & MOFFAT Vol. 514

FIG. 1.ÈPlots of the LPVs observed in HD 191765 (\ WR 134), HD 192163 (\ WR 136), and HD 193077 (\ WR 138). Bottom panels: Minimum, mean, and maximum measured values in the intensity in the He II j5411 emission line for the whole observing run. Center panels: Time-resolved plots of the residuals after subtracting o† the mean proÐle. Color displays: Wavelet spectrum from each time series. The wavelet spectrum identiÐes the typical location (m) and scale(pm) of the variable features in the LPVs. Note the outstanding behavior of WR 134, where the variable subpeaks are broader and a 2.3 day period occurs. Narrow subpeaks in the other stars appear to be stochastic. emission-line proÐle and shows up on scales that are much 80 \p \ 150 km s~1. Stars WR 134 and WR 40 show m ^ ~1 larger than pixel-to-pixel variations. broader features, withpm 350 km s . We suspect that The mean wavelet spectra provide clear evidence that the the broad subpeaks in WR 40 might result from variations scale (width) of emission subpeaks depends on their loca- in the whole line proÐle that can be confused, on residual tion on the line; LPVs are made up of narrow subpeaks spectra, with broad subpeaks having the width of the emis- near the line center and broader ones near the edges. This sion line. In any case, the wavelet spectrum of WR 40 in behavior shows up especially well for the broadest emission Figure 4 suggests the occurrence of narrow features on a lines (see WR 140). We note that the location where the scale similar to that found in other stars. On the other hand, narrowest intrinsic subpeaks are found coincides with the LPVs in WR 134 seem to be dominated by broad sub- m \ 0. The pattern is also symmetrical around m \ 0, indi- peaks; there is no clear evidence for a distinct population of cating that the width of a subpeak does not depend on the narrow subpeaks. Thus, WR 134 appears to be a special sign of its line-of-sight velocity. case. Indeed, whereas observations suggest that LPVs in the The general, multiscale properties of LPVs can be studied other WR stars are stochastic, recurrence in the LPV pat- with the mean wavelet power spectrum. We have plotted in terns have been detected for WR 134 (McCandliss et al. Figure 4 the mean wavelet power spectrumSR3TLPV of the 1994), conÐrmed and interpreted as being related to the residuals averaged over the spectral domain m (see Appen- rotation of a structured wind (Morel et al. 1999). dix A). The maximum inSR3 (pm)TLPV yields an approximate We note that the small-scale behavior of SR3 (pm)TLPV measure for the mean line-of-sight velocity dispersionpm of depends on the observing run. In the Ðrst CFHT run (WR LPVs subpeaks. Most stars show a maximum in the range 134, WR 136, and WR 138), the wavelet power in the range No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 913

FIG. 2.ÈAs in Fig. 1, but of the LPVs observed in the C III j5696 emission line in HD 192103 (\ WR 135), HD 192641 (\ WR 137), and HD 193793 (\ WR 140). One can see that subpeaks tend to propagate from lower to higher o m o . Note how the number of apparent subpeaks increases with the emission line width, an artifact that is explained in ° 4.1.

~1 0.5 \ log [pm (km s )] \ 1.2 is more or less uniform, may occupy several neighboring pixels. The wavelet spec- whereas in the second run (WR 135, WR 137, and WR 140), trum may thus e†ectively separate the generally small-scale the wavelet power increases proportionally withpm. It instrumental noise from the intrinsic large-scale features, appears that the scaling properties of the CFHT instrumen- yielding a straightforward detection of the latter. tal noise are epoch dependent. In the ESO run (WR 40, WR 103, and WR 111), there is a sharp drop in power below 3. PHENOMENOLOGICAL, CLUMPED WIND MODEL ~1 log [pm(km s )] D 0.8; this arises because a smoothing procedure was applied to this particular data set in the 3.1. Discrete W ind Emission Elements reduction procedure, to remove a contaminating high- The intrinsic, stochastic subpeaks can be described as frequency signal (Robert 1992). statistical Ñuctuations arising in a distribution of discrete Our wavelet analysis technique has some advantages emission features. The amplitude of the Ñuctuations should over other methods for the identiÐcation of intrinsic varia- depend on the number of discrete elements as well as on bility, such as the temporal variance spectrum (TVS) tech- their individual Ñux. Assume that the line emission can be nique (Fullerton 1990; Robert 1992). The TVS compares represented as arising in a Ðnite numberNe of ““ discrete the variability at each pixel with that expected from instru- wind emission elements ÏÏ (DWEEs). Line emission from a mental noise, accounting for the quality of each individual single DWEE arises as the associated wind feature passes spectrum. However, while TVS works on pixel elements through the LER, where the ionization balance is such as to individually, wavelet analysis separates the variable features stimulate the emission from the given atomic transition. according to their scale (in wavelength space) and thus can The LER is expected to span some well-deÐned range from make use of the fact that some intrinsic subpeak features the underlying WR star. Line variability is expected to 914 LEŠ PINE & MOFFAT Vol. 514

FIG. 3.ÈAs in Figs. 1 and 2, but of the LPVs observed in the He II j5411 emission line in HD 96548 (\ WR 40) and in the C III j5696 emission line in HD \ \ 164270 ( WR 103) and HD 165763 ( WR 111). The wavelet spectra reveal that, in all nine WR stars, the scalepm of variable subpeaks is smaller near the line center m \ 0 and larger near the edges.

occur because the discrete wind elements will successively c the speed of light. We assume thatsi(m) is of the form move in and out of this LER. Our model of LPVs is there- [ [ 2 fore very simple: every DWEE has an assigned location in \ Fi (m mi) si(m) exp 2 . (1) the wind; as this location changes (according to the wind p Jn pmi velocity law) the DWEE enters the LER and yields a dis- mi crete subpeak in the emission-line proÐle; this subpeak van- Here,Fi is the integrated emission-line Ñux from the ishes as the DWEE leaves the LER. This is sufficient to DWEE;mi andpmi are the spectral location and the width of reproduce the observed LPV patterns with reasonable Ðdel- the subpeak feature, respectively. ity. The parametermi corresponds to the mean line-of-sight Let[r , k , / ] be the coordinates of the ith DWEE, in a velocity of the emitting material in the DWEE, i.e., m \ i i i \ i spherical coordinate system centered on the star, with ki k v . Since the DWEE is moving, it is implicitly assumed i i \ cos hi. We assume that the wind is radially expanding and thatvi vi(t). The value forvi at any given time can be that the expansion follows a monotonically increasing obtained from the wind velocity law v \ v(r) (see ° 3.2 velocity law v(r). It simpliÐes things to calculate the problem below). Since we assume that DWEEs move only radially, in the directly observable bulk wind velocity space and use k is a constant. The velocity dispersionp reÑects the \ \ i mi the coordinates[vi, ki, /i], withvi v(ri). Letsi si(m) be motion of atoms associated with the DWEE. This includes the spectral emission (subpeak) arising from the ith DWEE, thermal and turbulent motions, as well as possible system- where we use the line-of-sight Doppler velocity m 4 j~1(j atic velocity gradients within the element. Values for p [ 0 mi j0)c, withj0 the rest wavelength of the line radiation and might vary with distance from the star, and the velocity No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 915

smallNe will result in high levels of variability. The patterns of LPVs, however, will be critically dependent on the wind velocity law and on the geometry of the LER, as well as on the statistical distribution of Ñuxes from individual DWEEs.

3.2. T he V elocity L aw The velocity law describes the bulk motion of the accelerated wind material as a function of distance from the star. The most popular velocity law typically used in models of hot-star winds is the so-called b-law (Castor & Lamers 1979), R b v(r) \ v A1 [ *B , (3) = r

which is given here in its simplest version, withv= the terminal velocity of the wind andR some spatial dimension of the order of the stellar radius. This* law is a parameterized generalization of a law that arises from the theory of radiation-driven winds (Castor, Abbott, & Klein 1975; the so-called CAK model). Di†erent values of b, typically in the range [0.5, 4.0], have been used in the literature for di†erent models and stars. In the approximation of point source stars, the CAK theory predicts a velocity law with b \ 0.5. When Ðnite disk e†ects are considered, the CAK theory yields b \ 0.8 (Friend & Abbott 1986). In modeling the winds of OB and WR stars, the velocity Ðeld is usually assumed to follow the b-law, but the value of b is left as a free parameter. A value of b ^ 1 has been successfully used for models of stellar winds from OB stars (Puls et al. 1996), while observations of LPV subpeaks in WR winds suggest much larger values of b (Robert 1994). This is supported by recent spectral analysis of WR spectra with a clumped wind model (Schmutz 1997), which yields Ðts of the velocity law with b ^ 4È8. The b-law yields an ““ acceleration law ÏÏ in the form dv dv v2 R 2 R 2b~1 a(r) 4 \ v \ = A *B bA1 [ *B . (4) dt dr R r r * The acceleration a can also be expressed as a function of the FIG. 4.ÈMean wavelet power spectrumSR3 (p )T for all the spectral local, bulk wind velocity v: m LPV ^ ~1 time series in our data set. All stars show a maximum atpm 100 km s , except for WR 134, which has a maximum nearp ^ 350 km s~1. The v2 v ~1@2b v 1@2b 2 m a(v) \ b CA B [ A B D . (5) width of the stochastic subpeaks is apparently independent of the star R v v observed, but the periodic subpeaks in WR 134 are signiÐcantly broader. * = = Note how the behavior ofSR3 (pm)TLPV at smallpm depends on the observing It turns out that the function ba(v) converges very fast for run, indicating variable noise quality and statistics. b ] O such that, for b[1, we may use the approximation ^ \ ~1 2 ba(v) lim ba(v) R [v ln (v/v=)] . (6) dispersion might be anisotropic. As we will show (° 3.3), the b?= * LER does not usually span a large domain in wind bulk Figure 5 shows the a(r) and ba(v) functions for di†erent velocity, and possible variations inpmi with depth can be values of the parameter b. One sees that the approximation neglected. However, we do need to consider possible aniso- in equation (6) yields values of ba(v) to better than D5% for tropic e†ects (° 3.4). allb Z 2. This result has important implications, because it The observed global emission-line proÐle S(m) will result means that unless there exists a reliable estimate of R * from the combination of a large numberNe of DWEEs: (which is usually not the case for WR stars), it is not possible Ne to determine b from the magnitude of the acceleration \ S(m) ; si(m) . (2) alone, since we havea(v) P b~1R~1. A measure of the wind i/1 acceleration as a function of the* velocity, combined with a We expect the resulting emission-line proÐle to be variable, knowledge ofv=, yields only a constraint onbR . On the depending on the mean number of DWEEs. An inhomoge- other hand, any large divergence from the very speciÐc* a(v) neous wind with a very largeNe should yield a line proÐle function rules out the b-law as an adequate representation very close to that of a smooth wind, whereas a wind with a of the motion of inhomogeneous wind features. 916 LEŠ PINE & MOFFAT Vol. 514 ^ one getsfe(v) 'e[r(v)], where r(v) is obtained by inversion of the velocity law v(r). However, here we are interested in obtaining the most information from observations alone. It is possible to derive an approximate expression for the total line ÑuxFe(v) as a function of the radial velocity v from the shape of the emission-line proÐle (see Brown et al. 1997). The technique is based on the assumptions that the wind is roughly homogeneous, spherically symmetric, and optically thin. An emission-line proÐle can be imagined to consist of a sum of proÐles arising from inÐnitesimally thin, concentric wind shells. It is easily shown that the emission dF of such a shell formed by wind material having a velocity between v and v ] dv yields a spectral proÐle dS(m, v) in the form F (v) q e [ \ tF0 dv , v \m\v dS(m, v) r a(v) (7) t0 , otherwise , s where a(v) is the ““ acceleration law ÏÏ (see eq. [5]) andF0 is some normalization constant. The global line proÐle S(m)is the sum of all shells: v/v= S(m) \ P dS(m, v) . (8) v/0 It is possible to solve the inverse problem (see Brown et al. 1997) to obtainFe(v) from the shape of the line proÐle S(m): P ~1AdSB Fe(v) a(v) v . (9) dm @ m @/v This is equivalent to performing a ““ spherical deprojection ÏÏ of the line emission, starting with the emission as a function of line-of-sight velocity m to obtain emission as a function of radial velocity v. We note that the observedF (v) is expected \ e to deviate from the theoreticalfe(v) 'e[r(v)], because Fe(v) is a†ected by the existence of ““ turbulent ÏÏ motions, which tend to increase the apparent range of line emission in velocity space, whereasfe(v) should reÑect the extension of the LER in distance space only. Both sides of the emission FIG. 5.ÈIllustration of the b-law, the commonly adopted velocity law used in hot star wind models. Upper panel: Dimensionless acceleration line proÐle (m\0 and m[0) can in principle be used to plotted as a function of radius. In velocity space, however, the b-law has estimateFe(v). However, the redshifted side of the emission the same general shape for all relevant values of b (lower panel). This is line (m[0) may be a†ected by stellar occultation (see crucial in the interpretation of our data, since spectroscopy measures only Ignace et al. 1998), and the blueshifted side may also be events in velocity space. a†ected by P CygniÈtype absorption. We therefore use either the blueshifted or redshifted side of the line, depend- 3.3. L ine Emission from DW EEs ing on whether these biases are assumed to be signiÐcant or The line ÑuxF from each DWEE depends on its distance not. i In Figure 6 we plotF (v), as obtained from equation (9), from the star or, equivalently, on its velocityvi. The DWEE e will be emitting line radiation while being accelerated for the emission lines of the WR stars in our sample, using a v through the LER. LettingF be the emission-line Ñux from ( ) from equation (6) as the velocity law. We then Ðtted the i F v a DWEE, we may writeF \ f f (v ), wheref (v ) is the e( ) using a Gaussian-like function, which appears to be i i e i e i quite appropriate: dependence of the line Ñux with the locationvi of the DWEE in the wind andf is a constant that allows for the [ [ 2 i P (v ve) relative emission from that DWEE (no two are alike). Fe(v) exp 2 , (10) *ve The most rigorous way to obtainfe(vi), or more generally fe(v), would be to use reliable theoretical WR models. In with the free parametersve, the velocity regime where standard models of WR winds, recombination-line emission maximum emissivity occurs, and*ve, the extension of the in WR winds arises in concentric ““ shells, ÏÏ whose depth and LER in velocity space. Although this parameterization is, at thickness depend on the particular line transition (see, e.g., best, a crude approximation of the actual emissivity func- Hillier 1988, 1989). The classical approach to obtaining an tion, it serves our purpose reasonably well here, since we are estimate offe(v) for one speciÐc line transition would be to mainly interested in the approximate size of the region perform standard spectral analysis and derive'e(r), the where most of the emission occurs, and not in the detailed emission as a function of distance from the star, from which line emission, e.g., in regions closer to the star where it is No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 917

Since the velocity dispersion*ve reÑects the radial extension of the LER in bulk wind velocity space as well the turbulent motions within the LER region, we use the relation ^ ] *ve pve pvr , (11)

wherepvr is a measure of the radial ““ turbulent ÏÏ motions within the LER, andpve is the spatial extension of the LER translated in bulk wind velocity space. We may estimate pvr from the widths of LPV subpeaks, assuming these to be a reliable measure of local velocity dispersions in the wind. The e†ective emission functionfe(v) for individual DWEEs becomes [ [ 2 P (v ve) fe(v) exp 2 . (12) pve As we will show in ° 4.4, we can obtain an independent estimate ofpve from an analysis of LPV patterns. This will allow us to verify the consistency ofpve as estimated from the shape of the emission-line proÐle. DWEEs are assumed not to have the same relative emis- IG F . 6.ÈDeprojected emission functionFe(v) for the WR stars in our sion. For practical purposes, and inspired by previous sample, obtained with the method discussed in ° 3.3. This yields the relative studies (see Paper I), we will use a statistical power-law emission from discrete wind elements as a function of their radial velocity. distribution in the form n( f )df D f ~a df and let the individ- Dotted lines show the best-Ðt Gaussian proÐles, whose parameters are listed in Table 2. The mean emission-line proÐles (see Figs. 1, 2, and 3) were ual Ñuxes f be in a rangef0 \ f \gf0, with g[1. ForFe the mean total Ñux in the line emission andN the mean smoothed with a low-pass Ðlter before computation ofFe(v), in order to e reduce instrumental noise e†ects. Normalization to the maximum value number of DWEEs in the LER (i.e., clumps in the ve [F ] facilitates the comparison between stars. [ ] e max pve \ v \ ve pve velocity regime, where most of the Ñux is found), we get the statistical distribution function, ~a relatively small. Best Ðts are plotted in Figure 6, and results \ A f B n( f ) Kf , (13) are listed in Table 2. For the C III j5696 line (WC stars), the f0 values obtained should be reliable estimates of the size and extension of the LER (within the validity of the b-law), since whereKf andf0 are functions of the parameters Fe, Ne, a, the line is optically thin. On the other hand, the He II j5411 and g. These are determined with the relations line in the WN stars is likely not to be optically thin, which gf0 \ P means that the estimated size and location of the LER will Ne n( f ) df (14) be biased by the fact that optically thick lines tend to be f0 naturally rounded (Castor 1970). In this case,*ve will over- and estimate the actual extension in velocity space of the LER. gf0 As it turns out, the*v obtained from the WN stars are \ P e Fe fn( f ) df . (15) signiÐcantly larger (except in the case of the WN8 star WR f0 40) than those measured in the WC stars. The Ñux distribution is therefore deÐned by four independent parameters (Fe, Ne, a, and g). Of these, the total line TABLE 2 ÑuxFe is determined by the emission-line equivalent width. IND ROPERTIES FROM THE HAPE OF THE MISSION INE ROFILES W P S E -L P The distribution therefore uses three free parameters (Ne, a, and g). EW ve *ve ~1 ~1 ~1 WR Star Transition (km s ) (km s ) (km s ) 3.4. Anisotropy WR40...... He II j5411 [460 315 ^ 10 175 ^ 10 The wavelet analysis has revealed that the widths of emis- WR103...... C III j5696 [15500 545 ^ 10 280 ^ 10 sion subpeaks depend on their location on the line proÐle. WR111...... C III j5696 [3580 1610 ^ 10 255 ^ 10 II [ ^ ^ This suggests that DWEEs have anisotropic velocity disper- WR134...... He j5411 3860 1090 10 430 20 sions. We model this anisotropy by expressing the velocity WR135...... C III j5696 [13040 890 ^ 10 240 ^ 10 WR136...... He II j5411 [3480 905 ^ 10 360 ^ 15 dispersion of DWEEs as a vector rv : III [ ^ ^ WR137...... C j5696 4410 1220 10 260 10 r (r) \ p rü ] p hü ] p /ü , (16) WR138...... He II j5411 [740 620 ^ 10 350 ^ 15 v vr vh vÕ WR140...... C III j5696 [4090 2310 ^ 10 265 ^ 10 wherepvr, pvh, andpvÕ are the velocity dispersions in the NOTES.ÈThe line equivalent width (EW) is translated into km s~1 radial, latitudinal, and longitudinal directions r, h, and /, units, as the line is plotted in projected velocity space m. Parametersve and respectively. The scalar, line-of-sight velocity dispersion pmi *ve are derived from the shape of the line proÐle and correspond to the of one DWEE will be mean wind velocity of line-emitting material and the emission-line broadening (dispersion in velocity space), respectively. \ J 2 2] [ 2 2 pmi ki pvr (1 ki )pvh , (17) 918 LEŠ PINE & MOFFAT Vol. 514 \ where we assumepvh pvÕ. Consider a spherical shell con- local velocity dispersions (such as atomic thermal motions Ðning a region of approximately constant wind expansion or turbulence), which blend the emission from regions velocityvc. Within this region, we get a relation between pm having di†erent spatial locations, but similar line-of-sight and m, namely, velocities. Let an emission line arise in a spherically expanding LER with mean radial velocityve, and mean local line- m2 m2 p (m) \ S (p )2]A1 [ B(p )2 . (18) of-sight velocity dispersionpm (see ° 3). We deÐne the m 2 vr 2 vh ~1 vc vc spectroscopic wind resolving powerRw 4 ve pm . Higher values ofRw mean that a larger number of wind regions can Mo†at & Robert (1992) already surmised this in their pre- be resolved with spectroscopy. The best results should liminary analysis of emission-line subpeaks of the WR star therefore be obtained for winds having large bulk velocities WR 140, supporting evidence for anisotropic local velocity and relatively small turbulent motions. dispersion. We present in Figure 7 simulations of LPV patterns Similarly, we may allow for the emission from individual obtained from the phenomenological model presented in ° DWEEs to be anisotropic as well. This can happen because, 3. Values of the parameters used in each simulation are e.g., of optical depth e†ects. Following the same reasoning listed in Table 3. The Ðrst series (SIM 1, SIM 2, SIM 3) as forp , we will now write the ÑuxF from the ith DWEE m i shows how the LPV pattern is a†ected by a change inve. All as three simulations used exactly the same distribution of \ J 2 2] [ 2 2 DWEEs; only the values ofv andR were adjusted. One Fi fi fe(vi) ki f r (1 ki ) f h , (19) = notices that the number of apparent* subpeaks is pro- wheref andf deÐne escape probabilities in the directions r portional to the width of the line (i.e., tov ). Note how the r h \ \ e and h. In the optically thin case, we usef f 1, whereas apparent subpeaks in SIM 1 break up into many resolved r hD the optically thick case can be modeled with fr fh. components in SIM 3. Similar e†ects on the LPV patterns 4. COMPARATIVE ANALYSIS can be obtained by varyingpm . One example is shown by the other three simulations (SIM 4, SIM 5, SIM 6) where, 4.1. Spectroscopic Resolution of W ind Elements again, the same spatial distribution of DWEEs was used, The large velocities of spherically expanding WR winds, but with di†erent values for their pm. combined with the Doppler e†ect, generate very broad line In previous studies of LPV patterns in WR emission lines proÐles. Spectroscopy therefore allows the observer to (Robert 1994), it was found that the number of subpeaks resolve parts of the wind in emission-line radiation, i.e., to observed on the emission line was proportional to the ter- observe wind regions having di†erent line-of-sight bulk minal velocityv= of the wind. In the light of our current velocities separately. However, a practical limit is set by investigation, this correlation Ðnds an easy explanation.

TABLE 3 PARAMETERS OF THE SIMULATIONS

Simulation R v v p F p p = e ve e vr vh * ~1 ~1 ~1 ~1 Number (R_) (km s ) b (v=)(v=) (km s ) Ne ag(km s ) (km s ) fr/fh SIM1 ...... 2.5 750 4 0.65 0.15 2000 103 4104 200 75 1 SIM2 ...... 5 1500 4 0.65 0.15 2000 103 4104 200 75 1 SIM3 ...... 10 3000 4 0.65 0.15 2000 103 4104 200 75 1 SIM4 ...... 5 1700 4 0.65 0.10 2000 103 4104 100 50 1 SIM5 ...... 5 1700 4 0.65 0.10 2000 103 4104 200 100 1 SIM6 ...... 5 1700 4 0.65 0.10 2000 103 4104 400 200 1 SIM7 ...... 5 2000 4 0.65 0.10 2000 104 2103 200 100 1 SIM8 ...... 5 2000 4 0.65 0.10 2000 104 2102 200 100 1 SIM9 ...... 5 2000 4 0.65 0.10 2000 104 2101 200 100 1 SIM10...... 5 2000 4 0.65 0.10 2000 104 3103 200 100 1 SIM11...... 5 2000 4 0.65 0.10 2000 104 4103 200 100 1 SIM12...... 5 2000 4 0.65 0.10 2000 104 6103 200 100 1 SIM13...... 5 2000 4 0.65 0.10 2000 102 3103 200 100 1 SIM14...... 5 2000 4 0.65 0.10 2000 103 3103 200 100 1 SIM15...... 5 2000 4 0.65 0.10 2000 105 3103 200 100 1 SIM16...... 5 2000 4 0.65 0.10 2000 103 3102 100 100 3 SIM17...... 5 2000 4 0.65 0.10 2000 103 3102 130 65 3 SIM18...... 5 2000 4 0.65 0.10 2000 103 3102 160 40 3 SIM19...... 5 2000 4 0.65 0.10 2000 103 3102 100 100 1 SIM20...... 5 2000 4 0.65 0.10 2000 103 3102 130 65 1 SIM21...... 5 2000 4 0.65 0.10 2000 103 3102 160 40 1 SIM22...... 5 2000 4 0.65 0.10 2000 103 3102 100 100 0.3 SIM23...... 5 2000 4 0.65 0.10 2000 103 3102 130 65 0.3 SIM24...... 5 2000 4 0.65 0.10 2000 103 3102 160 40 0.3 OTES N .ÈR : wind velocity-law parameter (roughly stellar core radius);v= : terminal wind velocity; b: velocity-law power index; v : location of* the LER in wind-velocity space;p : extension of the LER in wind velocity space;F : mean emission-line Ñux e ve e (equivalent width in units of projected Doppler velocity);Ne : mean number of DWEEs in the LER; a: power index in the statistical distribution of DWEE Ñuxes; g: total range in the Ñux from individual DWEEs;p : velocity dispersion in the direction of vr propagation (radial);p : velocity dispersion perpendicular to propagation; andf /f : escape probability ratio. vh r h No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 919

FIG. 7.ÈSimulations of LPVs from our model of radially propagating DWEEs. The LPV pattern from one DWEE distribution is shown to depend (upper panels) on the wind terminal velocityv= and (lower panels) on the average width of emission subpeakspm. Note how the number of apparent subpeak features is proportional to the ratiov=/pv. Simulated spectral time series include 60 spectra spread over a 48 hr time base. The minimum, mean, and maximum line proÐles are shown. Gray-scale plots show the residuals obtained after subtracting o† the mean proÐle. The same DWEE distribution was used in SIM 1, SIM 2, and SIM 3, and another one in SIM 4, SIM 5, and SIM 6. Model parameters are listed in Table 3.

Becausepm is approximately the same in all WR stars (see IfNe is the mean number of DWEEs in the emission-line Fig. 4), the number of apparent subpeaks should be pro- region, theR spectroscopically resolved sectors consist of w ~1 portional to the line width, which in turn is proportional to the emission fromNe Rw wind elements. LetFj be the v=. We wish to emphasize that the number of apparent sub- mean emission-line Ñux from the jth sector. The mean Ñux peaks in L PV patterns is not related to the actual number of variationpFj induced by the statistical variation in the DW EEs in the L ER; it simply reÑects the spectroscopic number of DWEEs in this sector will be resolving power for the emission line. Bothv andp are e m p P F N~1@2 R1@2 . (20) intrinsic to the star. Therefore, one cannot hope to get Fj j e w higherRw by instrumental means; there is a fundamental We can verify this relation by comparing the wavelet limit in the spectroscopic investigation of inhomogeneous spectra for the simulations shown in Figure 7. In Figure 8, wind structure. Sincepm is apparently independent of the we present the normalized, mean wavelet power averaged stellar subtype, the best WR candidates for a study of the over the emission-line regionSR3 (pm)TLPV. These show that, wind structure are those with the largest v=. for similar values ofNe, the magnitude of the variability in 920 LEŠ PINE & MOFFAT Vol. 514

In the case in which all the DWEEs have approximately the same Ñux (e.g., if g ] 1), p(a, g) converges to 1, and we get \ ~1@2 the familiar relation pFe Fe Ne . A plot of p(a, g) as a function of a for di†erent g is presented in Figure 9. One sees that signiÐcant deviations from \ ~1@2 thepFe Fe Ne relation occur for small positive values of a and for large g. Deviations occur when the statistical distribution is such that the total ÑuxFe is dominated by low-Ñux elements, while the Ñuctuations are mainly caused by high-Ñux elements making a small but nonnegligible contribution toFe. Thus, p(a, g) converges to 1 for small values of g and for large values of a, because these cases represent situations in which the number of high-Ñux elements is negligible For an emission line proÐle S(m) the standard deviation in the amplitudepS will be proportional topFe but will also depend on the spectroscopic wind resolution, P ~1@2 1@2 pS p(a, g)Ne Rw . (24) So far, we have used the normalized wavelet power spec- trumSR3 (pm)TLPV to measure the amplitude in the LPVs. We used a large number of simulations to calibrate the relation between the maximumSR3 (pm )TLPV andpS, leading to ^ ~1@2 1@2 [SR3 (pm )TLPV]max 0.3p(a, g)Ne Rw . (25) We present a synthesis of these results with simulations of LPV patterns from inhomogeneous winds obtained with various values of a, g, andNe. Some of the resulting LPVs IG are shown in Figure 10. In Figure 11, we plot the wavelet F . 8.ÈWavelet power spectrumSR3 (pm)TLPV for the six simulations in Fig. 7. Dotted horizontal lines, spaced by 1/2 log 2 intervals, show that the power spectrumSR3 (pm)TLPV, obtained from each of the 1@2 1@2 simulations. The maxima in the wavelet spectra are com- LPV amplitude is proportional tov= and inversely proportional to pv . Note that the locations of the maxima inSR3 (pm)TLPV depend on the mean pared with the values predicted from equation (25), shown scalepm of LPV subpeaks. as dotted lines. The amplitudes of the maxima fall within D10% of the predicted values. However, except for the amplitude of the LPVs, the patterns look extremely similar. 1@2 ~1@2 \ 1@2 It is virtually impossible to guess which pattern comes from the emission line is proportional tove pm Rw , as predicted in equation (20). which simulation. This strongly suggests that speciÐc values

4.2. Statistical Distribution of DW EE Fluxes Let an emission line with total line ÑuxFe be made up of a mean numberNe of DWEEs. If n( f ) is the statistical distribution in Ñux of individual DWEEs, the standard deviationpFe in the total line Ñux will be given by gf0 2\P 2 pFe f n( f ) df . (21) f0 Under the power-law distribution given in equation (13), pFe has the general form \ ~1@2 pFe Fe Ne p(a, g) . (22) We can calculate the function p(a, g) from equations (13), (14), and (15): g1~a [ 1 1@2 g2~a [ 1 ~1 g3~a [ 1 1@2 p(a, g) \ A B A B A B , 1 [ a 2 [ a 3 [ a a D 1, 2, 3 ;

ln g g ] 1 1@2 IG p(1, g) \ p(3, g) \ A B ; F . 9.ÈVariability induced by a distribution of DWEEs with a power- 2 g [ 1 law distribution in their individual Ñux will depend on the power index a and Ñux range g. This is demonstrated here with a plot of p(a, g)asa A g [ 1 B function of a, for di†erent values of g, where p(a, g) is a correction factor \ used in determining the total Ñux variabilityp from a set onN individ- p(2, g) . (23) Fe e Jg ln g ual DWEEs (see ° 4.2). No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 921

IG F . 10.ÈSimulations of LPVs, showing the dependence of the LPV patterns on the mean numberNe, power-law Ñux index a, and Ñux range g of DWEEs. Model parameters used in these simulations are listed in Table 3. The general patterns of the LPVs are indiscernible: unique determinations of Ne, a, and g cannot be obtained from such a spectral time series. Only the amplitude in the LPV varies, yielding constraints on a combination of these parameter values. of a, g, andN cannot be estimated from the L PV patterns might be required to account for the LPVs. We note that N e ~1@2 e alone; only an estimate of p(a, g)Ne can be obtained. represents only DWEEs that lie in the LER, i.e., a relatively This is a serious limitation to the study of inhomogeneous small fraction of the whole wind. We conclude that the structure based on emission-line variability, although we structure of WR winds is probably extremely fragmented. may obtain constraints on possible ranges of values of a, g, 4.3. Anisotropic Emission and Ne. We have used the wavelet power spectrum SR3 (pm)TLPV We have run several simulations of line emission from obtained from the data for our nine WR stars (see Fig. 4), inhomogeneous winds, with various values of the radial to ~2 combined with estimates ofRw, to evaluateNe p(a, g) for angular velocity dispersion ratiopvr/pvh and of the radial to each star (see Table 4). They indicate that one would need at angular Ñux emission ratiofr/fh. Nine simulations were per- 3 4 \ \ least10 [ Ne [ 10 DWEEs in the LER at any time to formed with combinations ofpvr/pvh [1, 2, 4] and fr/fh account for the LPVs in these WR stars. If there is a power- [3.3, 1, 0.3]. White noise was added to simulate the presence law distribution in the Ñux of individual DWEEs such that of some instrumental noise. In Figure 12, we show the mean p(a, g) [ 1, several orders of magnitude more DWEEs proÐles obtained from a 48 hr spectral time series, along 922 LEŠ PINE & MOFFAT Vol. 514

the He II j5411 line could be a†ected by optical-thickness e†ects. The wavelet spectrum appears to be mainly sensitive to the ratiopvr/pvh, but it is also a†ected by changes in the ratio fr/fh. One has to be careful in discriminating both e†ects. The ratiofr/fh a†ects the amplitude of the wavelet response as a function of m, whereas the ratiopvr/pvh a†ects both the amplitude and the scale of the wavelet response. Statistical Ñuctuations also arise because of the limited sampling of the time series. The wavelet spectra from SIM 21 and SIM 24 are those that resemble most the wavelet spectra obtained from the data. Overall, we have found that the best Ðts are obtained with3 \pvr/pvh \ 5, and with fr/fh [ 1. The fact that we obtainfr/fh [ 1 from Ðtting the wavelet spectra of the LPV seems to be in contradiction with the impression that the C III line is optically thin. However, this could be explained if individual DWEEs have di†erent values offr/fh. For example, high-Ñux DWEEs may arise from wind elements with higher column density and might be more subject to opacity e†ects than low-Ñux DWEEs. As mentioned in ° 4.2, one can have a line whose global emission is dominated by the low-Ñux DWEEs, while the LPV arises because of the high-Ñux DWEEs. This would explain the observed e†ect on the wavelet spectra. The addition of white noise in the simulations does not critically a†ect the wavelet response to the LPV. Since the noise varies on a pixel-to-pixel level, it consists only of very narrow features that show up in the Wavelet spectrum as a response near the smallest scale (in Fig. 12; this is the ~1 uniform response nearlog [pm(km s ) D 1]. The noise is not a serious problem in our analysis because the spectral resolution is such that intrinsic features are at least several pixels in size. On the other hand, since the quality of the Wavelet spectrum depends on the size of the sample, larger spectral time series would improve the quality of the wavelet spectra, which could be used to investigate more subtle e†ects such as occultation of DWEEs by the stellar disk. 4.4. Kinematics of the Subpeaks We have seen that variable subpeaks tend to persist on emission line proÐles for a certain period of time and that IG F . 11.ÈWavelet power spectrumSR3 (pm)TLPV for the simulations SIM subpeak features show a systematic motion from the line 7 to SIM 15, which were made using di†erent numbers and Ñux distribu- center to the line edges. This behavior is suggestive of radial tions of DWEEs (see Table 3). T op panel: g is varied whileNe and a are expansion of wind features. The general evolution in time of Ðxed. Center panel: a is varied, while g andNe Ðxed. Bottom panel:isNe varied, while a and g Ðxed. Horizontal lines show the amplitude of the the LPV pattern depends on the wind dynamics and line- maxima inSR3 (pm)TLPV predicted from eq. (A2). All measurements fall emission structure. The simplest interpretation for the char- within 10% of the predictions. acteristic timescale of a subpeak event relates it to the time it takes for the wind feature to cross the LER, whereas subpeak motion reÑects the wind acceleration in the LER. with the maximum and minimum spectrum proÐles; the In order to keep the analysis as objective as possible (i.e., two-dimensional wavelet spectrum of the residuals is also without having to rely on the identiÐcation of ““ individual ÏÏ plotted for each of the nine simulations. subpeak events), we introduce the ““ degradation function ÏÏ The shape of the line proÐle is found to be strongly as an analysis tool. The degradation function, denoted pD(a, a†ected by the ratiofr/fh, but it seems almost una†ected by a *t) (see Appendix B) is similar to a correlation function and reasonable degree of anisotropy inpv. The proÐle is espe- is used to estimate the mean radial accelerationae of cially sensitive tofr/fh [ 1, where it yields something remi- DWEEs in the LER and the velocity rangepve of this LER niscent of a proÐle generated by an emission ring. In the from the LPV pattern. The quality ofpD(a, *t) is pro- \ case in whichfr/fh 1, the proÐle is ““ Ñat topped, ÏÏ whereas portional to the size of the data sample, which must include it looks somewhat rounded whenfr/fh \ 1. This works in pairs of spectra with temporal separations shorter than the the sense expected from optically thick line emission in a subpeak lifetimes. This requirement is necessary to obtain a spherical, radial wind Ñow. Comparison with the observed reliable estimate for ae. III mean proÐles conÐrms that the shape of the C j5696 line The minimal degradation value ofpD(a, *t) is obtained \ \ is consistent with optically thin emission( fr/fh 1), whereas when parameter a is set toa ae. More reliable estimates No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 923

TABLE 4 WIND PROPERTIES FROM THE LPV PATTERNS

WR p a p p m e ve vr ~1 ~2 ~2 ~1 ~1 Number (km s ) Ne p(a, g) (m s ) (km s ) (km s ) WR40...... 90 ^ 20 1.6 ^ 0.6 ] 103 3.5 ^ 1.5 60 ^ 15 115 ^ 18 WR103...... 120 ^ 25 3.5 ^ 1.4 ] 103 4.5 ^ 1.5 70 ^ 15 210 ^ 22 WR111...... 90 ^ 20 1.5 ^ 0.6 ] 104 20.0 ^ 5.0 120 ^ 20 135 ^ 22 WR134...... 350 ^ 75 2.0 ^ 0.8 ] 103 25.0 ^ 3.0 285 ^ 40 145 ^ 49 WR135...... 120 ^ 25 9.5 ^ 3.8 ] 103 11.0 ^ 1.5 100 ^ 25 140 ^ 27 WR136...... 90 ^ 20 4.1 ^ 1.7 ] 104 5.5 ^ 1.0 80 ^ 20 280 ^ 25 WR137...... 75 ^ 15 2.4 ^ 1.0 ] 104 8.5 ^ 3.5 95 ^ 20 165 ^ 22 WR138...... 110 ^ 25 6.9 ^ 2.7 ] 103 9.0 ^ 1.5 135 ^ 30 215 ^ 34 WR140...... 90 ^ 20 3.6 ^ 1.4 ] 104 12.0 ^ 2.5 90 ^ 25 175 ^ 27 OTES N .ÈAll values are estimated from the LPV patterns:pm is the average width of subpeak features,Ne is the mean number of DWEEs in the LER (see ° 4.2),ae is the radial acceleration of DWEEs,p is the estimated size of the LER for one DWEE. The values listed forp are the ve vr radial velocity dispersions needed to account for the large*v (see Table 2) provided thatp are e ve reliable measures for the size of the LER.

can be obtained fromSpD(a)T*t:te, i.e., the mean degrada- values forpve (long-dashed lines), which are listed in Table 4. tion obtained from all pairs of spectra separated by *t \ te, From these, we evaluated the radial wind velocity disper- wherete is shorter than the estimated duration of subpeak sionpvr, which would account for the dispersion*ve in the events. We present in Figure 13 the mean degradation emission-line proÐle (see eq. [11]). SpD(a)T, estimated from pairs of spectra separated by For most of the stars, the estimatedpvr are consistent *t \ 3 hr, for each of the WR stars in our data set. The with what has been suggested from the wavelet analysis, locations of the minima yield estimates ofa , which are namely thatpvr D 4pvh. Wavelet analysis of simulations ~2e listed in Table 4; they range from D3.5ms in WR 40 to made with these values typically yieldpm Bpvr/2. The star ~2 D25ms in WR 134. To facilitate the comparison WR 134 again makes the exception, having pvr \pm between the stars, we have plotted in Figure 13 the normal- instead. One also sees thatpve is apparently much larger for ized SpD(a)T/SpD(ae)T. this star, although the emission line proÐle is not very di†er- We summarize the results in Figure 14, where we plot the ent from that in other WR stars. This is further evidence nondimensional stellar wind accelerationa R v~2 as a that the LPVs in WR 134 cannot be accounted for by radi- e~1_ = function of the normalized wind velocityve v= . This plot ally propagating wind features. The large, apparent pve indicates the relative wind acceleration as a function of would be consistent with a wind structure having a motion wind depth, where we use the relative radial wind velocity inhü as well as inrü . For the other WN stars, because of the ~1 ve v= as a depth measure. We compare the measurements possible optical depth e†ects, we suspected that*ve would from the stars to the values expected from idealized b-type overestimate the actual dispersion in the line emission (see ° wind velocity laws having di†erent values of the parameter 3.3). One sees that this might indeed be the case for some of bR (see ° 3.2). The estimates all correspond to b-laws with the WN stars: e.g., for WR 136, we Ðndpvr D 3pm , which can * ~1 20 \bR R_ \ 80. be reproduced with a ratiopvr/pvh D 9, much larger that The duration* of subpeak events can be estimated from a what is inferred from the wavelet spectrum. III plot ofpD(ae, *t) (e.g., Fig. 15). One sees in Figure 15 that The Ñat-topped C line in the WC stars stands out as pD(ae, *t) increases steadily with *t until it asymptotically the line that is best suited for a comparison with our simula- reaches some constant level. From that moment onward, tions. We note that the size of the LER contributes only a any correlation is lost, which indicates that the time interval minor part of the emission-line broadening*ve. It is the has become larger than the subpeak duration. All structures ““ turbulent ÏÏ motions, as revealed by the width of variable initially present in the signal have disappeared, which subpeaks, which dominate the line broadening. It appears explains the loss of correlation. that the size of the LER would be relatively small, at least Using the estimated values ofbR obtained from the for the C III transition. * accelerationsae, we performed simulations of LPVs using the model described in ° 3. For each star in our data set, we 5. DISCUSSION performed independent simulations using the same time 5.1. Physical Origin for the V ariable Emission Elements base and selecting values ofae to match those obtained from Figure 13. For each star, the degradation function Our analysis of the LPVs in WR emission lines strongly pD(ae, *t) from the data was compared to that obtained suggests that a very similar phenomenon is at work in all from the matching simulation; the results are shown in the stars, independent of the spectral sequence and subclass. Figure 15. We performed the simulations by Ðrst trying Except for the peculiar (and periodic) case of HD 191765, \ pve *ve (i.e., neglecting the turbulent broadeningpm ; see the LPVs can be reproduced using the same model of radi- eq. [11]), as the extension of the LER expressed in wind ally propagating, stochastic inhomogeneities. The fact that velocity space (short-dashed line). This attempt resulted in the physical aspect of the subpeaks, i.e., their velocity dis- poor agreement with the data, the longer lifetime of the persion, does not depend on the geometry of the LER, the subpeak features revealing that the size of the LER had magnitude and acceleration of the wind velocity, or the been overestimated. Better Ðts were obtained with smaller depth in the wind at which the line is formed, is suggestive 924 LEŠ PINE & MOFFAT Vol. 514

FIG. 12.ÈSimulated LPV showing the e†ects of anisotropic velocity dispersions and photon escape probabilities. The parameters of the simulation are listed in Table 3. The minimum, mean, and maximum proÐles from 48 hr long spectral time series are shown. The shape of the mean proÐle is sensitive to the emissivity ratiof /f , especially whenf /f [ 1. The mean wavelet spectrum of the spectral time series (top panels) is sensitive to the ratiop /p . A comparison r h r h vr vh with Figs. 1, 2, and 3 suggests that inhomogeneous features in WR winds havep /p ^ 4. We can rule out the possibility thatf /f [ 1 in WR stars, because of vr vh III r h the shape of WR emission-line proÐles. The LPV patterns seem to be consistent withfr/fh [ 1, though the Ñat-topped C proÐles seem to argue against this possibility (see ° 4.3). of some universal process within the inhomogeneous WR local systematic motions such as from shocks in large-scale wind structure. stream interactions (see Cranmer & Owocki 1996). One conclusion to be drawn is that variable subpeaks We believe that the observedpvr/pvh D 4 reÑects a true must reÑect strong, local velocity gradients. One cannot anisotropy in the velocity dispersion. The large ratio might explain easily the largepv in terms of the extension in space be reproducible from radiative transfer e†ects in optically of an overdense region, for which the velocity dispersion thick wind clumps, but such e†ects would likely show up in would simply reÑect the range in radial bulk wind velocity the shape of the emission-line proÐle as well, in the form of within the region. One reason is that one would then expect deviations from a Ñat-topped proÐle. Since we do not see III to observe di†erentpv in di†erent stars, depending on the any such large deviations, at least for the C j5696 line in wind velocities and accelerations. Another reason is that the the WC sequence, the only possibility would be that only very large number (?103) of wind elements that are some DWEEs (likely in the high-Ñux regime) would show required to model the LPVs suggests that individual these optical depth e†ects. However, in this case, we should DWEEs occupy relatively small volumes. The large velocity also detect narrow (although weaker) subpeaks near the dispersions are more easily explained in terms of large local edges of the line; this is apparently not the case. Thus, random motions such as from macroturbulence, or large although we believe that optical depth e†ects might be No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 925

FIG. 14.ÈComparison between the dimensionless acceleration ~2 ~1 ae R_ v= and the relative velocity rangeve v= in the wind, as estimated from the LPVs in the WR stars of our sample. Dashed lines show the ratios expected from b-laws for various values of the combinedbR parameter. The kinematics of the WR LPV subpeaks are consistent* with 20 \ ~1 bR R_ \ 80, much larger than the values typically used in theoretical WR* wind models. The large dispersion inbR for the stars in the sample might reÑect either di†erent values in their core* radii or simply indicate the inadequacy of one general b-law as a good representation of WR wind velocity Ðelds.

supersonic turbulence, or self-structured chaos (see Henrik- sen 1994), is that coherent wind features could survive for only a limited time, possibly yielding a subpeak timescale that is shorter than expected from the crossing time through IG F . 13.ÈRadial accelerationae of wind inhomogeneities, as obtained the LER. from the mean degradation functionSpD(a)T of LPV patterns, from pairs The supersonic/anisotropic behavior could also be of spectra separated by *t \ 3 hr. The minima yield estimates of the radial accounted for by the existence of radially propagating wind accelerationae in the LER (see Appendix B). We obtain estimates in the range4 \ a \ 25 m s~2 (see Table 4). shocks. It has been shown that the radiation line-driving e mechanism is unstable and that strong shocks can form and propagate in a radiatively driven wind (Owocki, Castor, & a†ecting some DWEEs, yieldingfr/fh [ 1, thepvr/pvh ratio Rybicki 1988). These shocks are usually thought to be initi- most likely depends on the real velocity dispersion within ated by hydrodynamical perturbations at the base of the DWEEs. wind, such as stellar pulsations or surface inhomogeneities Anisotropic turbulence is believed to exist in the chromo- in conjunction with stellar rotation (Owocki, Cranmer, & spheres of some stars (see, e.g., Carpenter & Robinson Fullerton 1995). These mechanisms may account for the 1997). However, the latter imply motions ofpv D 20È30 km periodic wind Ñuctuations such as in the discrete absorption s~1, substantially lower than what we measure (we note component (DAC) phenomenon in OB stars (Cranmer & that our observedpv correspond to highly supersonic Owocki 1996) but could be hard to reconcile with the sto- motions). It is not clear how coherent wind features with chastic behavior of the WR LPVs. However, it has been such large internal random motions can persist for long shown that the growth rates of instabilities in WR winds are enough to be observed as discrete wind features. Should likely to be extremely large, such that small, random Ñuc- hydrodynamical viscous dissipation occur, a hierarchy of tuations could grow into the stochastic shocks that we clumps will also be expected to form, governed by speciÐc observe (Gayley & Owocki 1995). scaling laws (Henriksen 1991). This means that smaller However, it has been claimed that shock propagation in clumps, i.e., shocks having smaller velocity dispersions, winds driven by line scattering should be ““ polarized ÏÏ in the should be present in the wind. This makes the analysis of radial direction (Rybicki, Owocki, & Castor 1990), i.e., the variable subpeak patterns still more complex, since our damped in other directions. Our results provide direct evi- assumption that all clumps have the same velocity disper- dence supporting this claim. We note, however, that even sion would be only approximate. This speciÐc problem of though wind perturbations are such thatpvh > pvr, we might scaling relations in the clumps was investigated in Paper I, still get relatively large apparent measures ofpvh because of in which we presented marginal observational support for the Ðnite angular extension of the perturbation. For the existence of such scaling laws and obtained constraints example, a radially propagating shock with angular size ^ on the possible scaling properties. One consequence of *) 0.2 sr and radial velocity dispersionpvr would yield 926 LEŠ PINE & MOFFAT Vol. 514

usually assumed in modeling the winds and atmospheres of WR stars (see Hillier 1996). The degradation function provides reliable estimates ofae, the radial acceleration of DWEEs. However,ae might not be equal to the radial acceleration of the wind material: if DWEEs arise, e.g., from shock compressions, the corresponding density enhance- ments (compressions patterns) might be moving relative to the mean motion of wind material. The question is whether this motion relative to the bulk wind velocity would bias the ae measures. If we believe this e†ect to be unimportant (or nonexistent), there is still a basic uncertainty in the determi- nation of the velocity law from emission subpeaks. Spec- troscopy provides only information in velocity space, and because the magnitude of the acceleration depends linearly on the productbR , it is not possible to evaluate b or R separately. * * Nevertheless, our estimates ofbR may provide inter- esting constraints on the b-law in WR* winds. So far, we obtainbR values that are not consistent with the values commonly* used in WR wind models. For instance, values of \ \ b 1 and a core radiusR 3 R_ have been used to model the wind from WR 111* (Hillier 1989), whereas we ÐndbR D 30 for this same star. Non-LTE spectral analysis * \ \ of the WN star WR 136 yieldedR 6.4 R_ with b 1 (Hamann et al. 1994), whereas we* ÐndbR ^ 70. These results were obtained from using the ““ standard* model ÏÏ hypotheses, including homogeneity. More recently, Schmutz (1997) has performed a spectral analysis of HD 50896, this time using a clumped wind model. He obtained a velocity law that was Ðtted (in the outer wind regions) with \ \ b 8,R 3.5 R_. This appears to be in better agreement with our* analysis, though a study of the LPVs in HD 50896 would be required to confront this result. Still, thebR that we measure in some stars is signiÐcantly larger than what* is obtained from most models. The source of this discrepancy might reside in the fact that the parameterR in the b-law is not related to the stellar core radius. Close* binary systems provide upper limits for the core radii of the components, since the size of the orbit must always be larger than the sum of the radii of the component stars. There exist a few WR stars in very short period WR]O binaries, such as CQ Cep with a IG F . 15.ÈComparison between the degradation function pD(ae, *t) period of 1.6 days. Upper limits to the core radii of six WN obtained from the data (squares) with that from simulated LPVs. For each stars (including CQ Cep) in close binary systems have been star, we present one simulation that uses*v as the full rangep of the e ve LER (dotted lines), i.e., that assumes that there is no turbulent broadening estimated to be in the range 2È10R_ (Mo†at & Marchenko 1996). It is not clear whether the results from binary stars in the proÐle and that*ve yields a correct estimate of the DWEEs velocity range (see ° 3.3). ThepD(ae, *t) for these simulations are found to disagree should apply to our sample of single WR stars, since single with the data. A better Ðt was obtained using much smaller depths for the WR stars may have evolved di†erently from WR stars in LER (dashed lines). These smaller values suggest a radial turbulent binary systems (Dalton et al. 1995; see however Mo†at broadeningp D 2p . This is consistent with the results obtained from the vr m wavelet analysis that suggested anisotropic velocity dispersion (see ° 4.3). 1995). Therefore, either we adopt large values for b or we abandon the idea thatR is related to the core radius. Using values ofR that are* larger than the core radius will result in a failure for* the b-law to describe the velocity Ðeld close to the stellar radius. an apparentp ^ p /4 when observed to propagate per- vh vr In any case, we can still attempt to Ðt the velocity struc- pendicular to the main propagation axis. Such large ture at larger v with a b-law. Depending on the actual veloc- angular sizes (about 1/50 of a spherical shell) could be ity structure near v \ 0, we will obtainR values that can N reconciled with a largee, since shocks could be relatively be either larger or smaller than the core radius.* The analysis shallow in the radial direction. It is not clear, however, why of HD 50896 by Schmutz (1997) appears to be consistent shocks would then tend to have similar angular sizes. with this view and suggests that there is a divergence from the b-law form at small distances from the stellar surface. 5.2. T he W ind Velocity Structure On the other hand, models of radiation-driven winds (see, The wind acceleration is a crucial quantity for testing the e.g., Gayley, Owocki, & Cranmer 1995; Springmann 1994) velocity law. This test is critical because the velocity law is suggest a more extended wind-acceleration region than that No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 927 from the b-law, which could actually result in a smaller than on the simulations and did not Ðnd signiÐcant deviations in expected acceleration near to the star for a given terminal the measure of pve. velocity. On the other hand, there are physical e†ects that would There remains the possibility that the apparent motion of yield an underestimate ofpve from the degradation function. DWEEs in velocity space does not reÑect the actual mean These would be responsible for the short timescale of vari- hydrodynamical wind Ñow. If the inhomogeneous structure able subpeaks while accounting for the large*ve in the line is in the form of radially propagating shocks, and if we rely proÐle without having to rely on the existence of large local on the smaller, theoretical estimates ofbR , this implies turbulent motions: (1) a large random motion of DWEEs that shocks have smaller radial acceleration* than the wind relative to one another, (2) variable locations of the LER for material. This might be explained with shocks propagating individual DWEEs, and (3) rapid Ñuctuations in the density toward the star, in the wind rest frame, like the ““ reverse structure. shocks ÏÏ found by Owocki et al. (1988) in one-dimensional The width of one LPV subpeak depends on the velocity simulations of radiatively driven winds. In this hydrody- dispersion within the region occupied by the corresponding namical model, it was found that shocks of di†erent sizes DWEEs. Large random motions could exist that would not seemed to propagate at di†erent rates, denser shocks having be reÑected in the width of LPV subpeaks, e.g., the motion a lower radial acceleration. Such a behavior might even- of one clump relative to others. These ““ macroturbulent ÏÏ tually be veriÐed, by testing whether the LPV subpeaks are motions would a†ect the shape of the emission line proÐle, consistent with intense DWEEs having low acceleration increasing the*ve, but would not be detectable from the (highbR ), and weaker DWEEs having higher acceler- LPV pattern. ations (low* bR ). We have assumed so far that each and every DWEE has * the same emissivity function, i.e., that the location of the 5.3. T he Size of the L ER LER was an independent quantity. However, the emissivity We have interpreted the timescale of LPV subpeaks as of one DWEE as a function of wind depth might depend on the crossing of one DWEE through the LER. Using our its density. Since DWEEs represent stochastic wind Ñuctua- estimates for the radial acceleration of DWEEs, we have tions, it is reasonable to assume that inhomogeneous com- estimated the size in velocity spacepve of the LERs for the ponents will not all be emitting from the same wind region. WR stars in our sample. We have found relatively small While individual subpeaks could emit only within a narrow values, which suggest thatpve is not the main factor region from the star, they could, as a whole, be emitting responsible for the emission-line broadening *ve. from a much larger region. The large*ve, estimated from the shape of the emission- Finally, the inhomogeneous wind structure might be line proÐles (see ° 3.3), can be accounted for by the large rapidly Ñuctuating, as in models of self-structured chaos, or radial velocity dispersionpvr within the DWEEs, provided turbulent energy dissipation. The relative density of thatpvr D 2pm , wherepm is the mean velocity dispersion DWEEs may also be changing because of pressure gra- measured from the LPV subpeaks. This is consistent with dients. It is therefore possible that ourpve would not the conclusions from the wavelet analysis, which suggest measure the crossing time of one DWEE through the LER, large anisotropies in the velocity dispersion, with pvr D 4pvh. but rather measure the lifetime of inhomogeneous wind fea- A smallpve, however, does more than just corroborate the tures. The actual size of the LER could therefore be larger, anisotropic velocity dispersion hypothesis; it also supports although we may still account for the smallpve measured the view that the whole line emission comes from DWEEs from the degradation function method. having largepvr. We have assumed in our model that the SUMMARY AND PERSPECTIVES whole line arose from DWEEs and that the velocity disper- 6. sion vectorrv (see ° 3.4) was the same for all DWEEs. We have presented a set of high-resolution spectra for However, the LPV patterns could not provide any support nine WR stars. The spectra show time series of the C III to this view, because one cannot know whether the LPV j5696 emission line for Ðve stars of the WC sequence and subpeaks reÑect the physical properties of all DWEEs, or the He II j5412 emission line for four stars of the WN only a few, especially large ones. sequence. The time series each cover 3È4 consecutive nights, For example, in the case where there is a power-law dis- with spectral sampling every hour or so. The time series tribution in the DWEE Ñuxes, most of the line emission Fe reveal the presence of line proÐle variations (LPVs) on the might arise from the low-Ñux DWEEs, while the variability order of 5% of the line intensity. The LPVs show character- pFe is dominated by the high-Ñux components. In Paper I, istic patterns, with narrow emission subpeaks moving from we have investigated the possibility that there could be a line center toward line edges on both halves of the line. The proportionality relation between the Ñux and the velocity universality of the phenomenon is supported by recent dispersion of DWEEs. If this were the case, then the large pm detection of similar LPVs in an O-star wind (Eversberg, might reÑect only the velocity dispersion of the high-Ñux Le pine, & Mo†at 1998). DWEEs. However, the small values ofpve indicate that We have investigated the hypothesis that these LPVs largepvr must prevail for the whole wind emission. could be due to radially propagating wind inhomogeneities. One could blame the smallpve on a systematic bias in the We developed a simple phenomenological model that uses a degradation function, such as e†ects of standing but vari- random distribution of discrete wind emission elements able features (e.g., atmospheric lines, noise). These might (DWEEs). These DWEEs are assumed to propagate radi- bias the measure of the acceleration to smaller values, in ally according to some monotonically increasing velocity turn leading to underestimates inpve. The degradation in law v(r) and to be emitting line radiation as they cross a the signal might also be ““ accelerated ÏÏ by the noise, which corresponding line-emission region (LER), which has the might induce underestimates in the characteristic time form of a spherical shell with some arbitrary thickness. We duration. However, we veriÐed the e†ect of synthetic noise used the so-called b-law in the form v(r) \ v (1 [ R r~1)b, = * 928 LEŠ PINE & MOFFAT Vol. 514 withv= the terminal wind velocity. Both the location of events). Data analysis revealed smaller accelerations than individual DWEEs and the location/thickness of the LER expected. We found that the data could be Ðt with a b-law are expressed in wind velocity space. The DWEEs are with20 \bR R~1 \ 80, an order of magnitude larger than * _ assigned a certain velocity dispersion, which represents the predicted by homogeneous models of WR winds. local deviations from the wind velocity law, such as turbu- We compared the data to simulations, to determine the lent motions or shocks. The model also allows for some size of the LER that best reproduced the duration of optical depth e†ects within individual DWEEs, although subpeak events. These Ðts resulted in relatively small sizes these are assumed to be optically thin to one another. for the LER, much smaller than the observed line-proÐle We used this model to generate time series of synthetic broadening*ve. This suggests that most of the line emission-line spectra, which were shown to reproduce well broadening arises from a large, ““ turbulent ÏÏ radial velocity the observed LPV patterns. Synthetic spectra were used to dispersion, which must bepvr D 2pm . Simulations show this investigate how model parameters a†ect the shape of the result to be consistent with large anisotropic velocity dis- ^ emission-line proÐle and the pattern of LPVs. We found persion withpvr 4pvh. This result thus yields independent that a Ðnely structured inhomogeneous wind, requiring a evidence for anisotropy, while suggesting thatpvr must be huge number of DWEEs, could yield LPV patterns in a†ecting the whole line emission and not just a few intense which only a few apparent subpeak events are observed. DWEEs. Other possible interpretations were also discussed The actual number of subpeak events detected at any time (° 5.3). ~1 was shown to depend onRw 4 ve pm , the ratio of the wind One of the stars (HD 191765) is shown to yield results velocityve (related to the emission-line width) to the mean that are inconsistent with our model of radially propagating line-of-sight velocity dispersionpm of variable subpeaks. wind features. The large-scale, LPV pattern from this star We used the continuous wavelet transform as a multi- was already known to be somehow peculiar, being recurrent scale analysis tool for the study of LPV patterns. This rather than stochastic, as in other WR stars. A better model numerical tool is shown to be very useful in many ways, in for this peculiar star might involve the rotation of a struc- that it can also be used to distinguish intrinsic, variable tured wind (Morel et al. 1997, 1999). features in the signal (large-scale response) from instrumen- Overall, this new method for the analysis of LPV patterns tal noise variability (small-scale response). We found intrin- has yielded several new constraints on the inhomogeneous sic features with mean velocity dispersionsp D 100 km and dynamical structure of WR winds. Since our numerical ~1 m ^ s , with only one exception (HD 191765, where pm 350 tools use statistical measures that combine the information km s~1). Wavelet analysis showed that LPV subpeaks are from several consecutive spectra, we expect that the preci- narrower near the line center than near the edges, i.e., that sion and quality of the constraints should increase with the the velocity dispersion is larger for DWEEs that are propa- number of spectra obtained. Continuous observations of gating along the line of sight. This suggests that the velocity bright WR stars, such as from multisite spectroscopy or dispersion of DWEEs is larger in the radial direction than from space, could provide a data sample allowing for more in the azimuthal direction, withpvr D 4pvh. Wavelet analysis precision on the acceleration and clump duration. Further- also provided marginal evidence for optical depth e†ects, more, an in-depth investigation of the wind velocity struc- with an escape probability within DWEEs possibly smaller ture would necessitate measuring the acceleration in in the radial direction. However, the Ñat-topped shape of di†erent regions of the wind. This could be provided by the C III j5696 line in stars of the WC sequence suggests detailed, simultaneous spectroscopy of many emission lines that only some of the DWEEs might be a†ected by optical from the same star, since the LER from di†erent species thickness e†ects. occur at di†erent distances from the star. Systematic temporal variations in the LPV patterns were investigated with the use of the so-called degradation func- The authors wish to thank the referee, W. Schmutz, tion, which Ðnds correlations between di†erent spectra in whose suggestions and comments have stimulated a sub- the time series. The degradation function is used to deter- stantial improvement of the manuscript. S. L. acknowledges mine the magnitude of the radial acceleration of DWEEs the support provided by Post-Graduate Scholarships from and the characteristic duration of the subpeak events. These NSERC of Canada and from FCARQue bec. A. F. J. M. is sources of information are used to investigate both the grateful to the same two organizations, as well as the Killam velocity law (from the acceleration) and the size in velocity program of the Canada Council for the Arts, for Ðnancial space of the LER (relating it to the duration of subpeak aid.

APPENDIX A

MULTISCALE WAVELET ANALYSIS Wavelet analysis is in many ways analogous to Fourier analysis: it provides information about the scale of certain features in a signal. However, its advantage over Fourier analysis with periodically repetitive trigonometric functions is that, because of their discrete nature, wavelets also provide information about the location of features in the signal (see Paper I, and references therein). Wavelets are deÐned as functions that follow two speciÐc criteria. First of all, a wavelet function t(m) must be such that it has zero mean, i.e., = P t(m) dm \ 0 . (A1) ~= No. 2, 1999 WIND INHOMOGENEITIES IN WOLF-RAYET STARS. II. 929

Furthermore, the wavelet must be localized in space, a condition that is satisÐed if = P [t(m)]2 dm \ C , (A2) ~= where C is some Ðnite value. One very simple, popular wavelet function is the so-called ““ Mexican-hat wavelet, ÏÏ which comes from the second derivative of a Gaussian function: t(m) \ (1 [ m2)e~m2@2 . (A3)

The continuous wavelet transform uses, as a basis, a wavelet family, denotedtm,pm(m@), which is obtained by continuous translation (parameter m) and dilation (parameterpm) of one wavelet function: 1 Am@ [ mB tm,pm(m@) 4 t . (A4) pm pm Let R(m, t) be some emission-line time series of residual spectra plotted in line-of-sight velocity space m. The continuous wavelet transformR3 (m, pm, t) is obtained by convolution of each spectrum in the series with the wavelet family, e.g., n P = Am@ [ mB dm@ R3 (m, pm, t) 4 R(m, t)t . (A5) 8 ~= pm pm The wavelet transform as deÐned in equation (A4) yields a simple form for the inverse transformation, the so-called ““ wavelet reconstruction theorem, ÏÏ which gives back the data from its wavelet transform. It is possible to show that if we use the Mexican hat as the wavelet in equation (A4), the reconstruction is obtained from = \ P dpm R(m, t) R3 (m, pm, t) . (A6) 0 pm This relation considerably simpliÐes the interpretation of the wavelet transform. One now sees that the original signal is recovered from its wavelet transform by a simple sum over the scale parameterpm. This suggests that the wavelet transform is simply a representation of the data in which signal features have been separated according to their characteristic scalepm. This is very similar to a scale decomposition using a passband Fourier Ðlter. We may synthesize the results obtained from the wavelet analysis of the spectral time series to get a statistical measure of the mean scaling properties. We call this the ““ mean wavelet power spectrum, ÏÏ denotedSR3 (m, pm)T. If the time series consists of n measures at timesti, the mean wavelet power spectrum is given by n 2 1 2 SR3 (m, pm)T 4 ; [R3 (m, pm, ti)] , (A7) i n This will be useful in determining the general location and scale of variable subpeak components in the spectra, and especially for uncovering any relation between the width (scale) and position of emission features on the line proÐle, for an ensemble of data. We may also be interested in obtaining a measure for the amplitude of signal features as a function of their location but independent of their scale, or as a function of their scale but independent of their location. We obtain these by integration over one independent parameter or the other: pm2 2 P 2 [SR3 (m)T*pm1,pm2+] 4 SR3 (m, pm)T dpm ; (A8) pm1 1 m2 [SR3 (p )T ]2 4 P SR3 (m, p )T2 dm . (A9) m *m1,m2+ [ m m2 m1 m1 We call the latter measure (eq. [A9]) the ““ wavelet power spectrum, ÏÏ by analogy with the Fourier power spectrum of a signal, since it yields an average measure over m in the scalepm of signal features. For an emission line arising in a spherical shell from a radially expanding wind, we will use a normalization: SR3 (p )T \ 2(v ] p )EW ~1SR3 (p )T , (A10) m LPV e ve m m *~ve~pve,ve`pve+ ] whereEWm is the line equivalent width (in units of the Doppler line-of-sight velocity), and2(ve pve) is the characteristic width of a line arising in a wind region with mean radial velocityve and velocity dispersionpve (see ° 3.3). This normalization allows for a comparison in the variability from lines with di†erentEWm ; SR3TLPV yields a measure of the absolute emission-line variability as a function of the scale of the variable subpeak features.

APPENDIX B THE DEGRADATION FUNCTION Consider a measure of the degradation of the LPV pattern in a spectral line with time, i.e., a measure of the changes in the pattern induced by the evolution of DWEEs. We will characterize this measure by deÐning a degradation function in the 930 LEŠ PINE & MOFFAT Vol. 514 following way. Letp (*t) be a measure for the degradation in a time series of residuals R(m,t ), with LPVs in the spectral [ D i region ve \m\ve : n~1 n ve 2\ P [ 2 ~2 [pD(*t)] ; ; [R(m, tj) R(m, ti)] [R(m, tj)] dm , (B1) i/1 j/i ~ve comparing all pairs of spectra with*t \ t [ t . One can select speciÐc intervals of *t and use the averagep (*t) obtained for [ j i D all the pairs ij that havetj ti falling in this interval. The functionpD is expected to increase with *t as the pattern slowly changes, and it reaches an asymptotic maximum value after some characteristic time for the whole pattern to have changed completely. Spectra that have a time separation larger than the characteristic time di†er in the same way as randomly generated signals. Mainly two e†ects will yield a pattern degradation with time: (1) the appearance and disappearance of emission subpeaks and (2) the motion of the subpeaks on the line. The two e†ects should be distinguished, since they have di†erent physical origins. The Ðrst e†ect is related to the time te necessary for a wind feature to cross the emission-line region. The second e†ect is related to the radial acceleration a of wind features. It is however possible to estimate the radial acceleration of the inhomogeneous wind components from the LPV pattern. Consider the functionpD(a, *t), deÐned as ve 2 P ] [ ~1 [ 2 ~2 [pD(a, *t)] 4 ; [R(m a(tj ti)ve m, ti) R(m, tj)] [R(m, tj)] dm , (B2) 0:tj~ti:*t ~ve which Ðnds the mean standard deviation between pairs of spectra separated by a time interval *t, when the former spectrum in the pair is ““ stretched ÏÏ in order to imitate the e†ects on the LPVs of features radially accelerating at a rate a in a wind region with mean radial velocityve. The minimum value ofpD(a, *t) should be obtained for a matching the actual mean radial acceleration of the wind features. One drawback of this technique is that the shapes of individual subpeak features are distorted by the stretching. This distortion may become especially important for broad features, or for pairs of spectra with large *t separations. This is why one has to be careful not to compare spectra separated by too large *t and to account for possible biases in LPVs where broad features are found. 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