Theoretical Profile Shapes for Optically Thin X- Ray Emission Lines from Spherical Stellar Winds.

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1 East Tennessee State University From the SelectedWorks of Richard Ignace March 1, 2001 Theoretical Profile Shapes for Optically Thin X- Ray Emission Lines from Spherical Stellar Winds. R. Ignace, East Tennessee State University Available at:

2 The Astrophysical Journal, 549:L119 L12, 2001 March The American Astronomical Society. All rights reserved. Printed in U.S.A. THEORETICAL PROFILE SHAPES FOR OPTICALLY THIN X-RAY EMISSION LINES FROM SPHERICAL STELLAR WINDS R. Ignace 20 Van Allen Hall, Department of Physics and Astronomy, University of Iowa, Iowa City, IA 52242; ri@astro.physics.uiowa.edu Received 2000 November 6; accepted 2001 January 19; published 2001 February 2 ABSTRACT One of the major outstanding problems in hot star wind theory is an understanding of the observed X-ray emissions from the early-type B, O, and Wolf-Rayet (WR) stars. The latest X-ray satellites Chandra and XMM- Newton are providing key new observations of resolved emission profiles to advance that understanding. This study presents a derivation of the expected emission-line profiles, assuming optically thin line emission and spherical symmetry, with a proper treatment of the attenuation of X-rays by the dense cool wind component. Examples of line profile variability for a narrow outflowing shell are presented. Then the case of embedded hot gas existing throughout the wind flow is discussed. It is shown that for the special case of constant expansion, emission profile shapes can be derived analytically, and in the limit of strong wind attenuation, the profile achieves a self-similar form. The results of this Letter provide a framework in which to model X-ray line profiles and analytic results to serve as a benchmark for more sophisticated numerical evaluations. Subject headings: line: profiles stars: early-type stars: winds, outflows X-rays: stars 1. INTRODUCTION The observations and subsequent theory of X-ray emission from hot stars has had an interesting history. Many early-type Wolf-Rayet (WR), O, and B stars have been observed to be sources of X-ray emission by Einstein and ROSAT (Harnden et al. 1979; Seward et al. 1979; Seward & Chlebowski 1982; Pollock 1987; Pollock, Haberl, & Corcoron 1995; Kudritzki et al. 1996; Berghöfer et al. 1997). Attempts were made to explain these X-rays as arising from stellar coronae (e.g., Cassinelli & Olson 1979; Waldron 1984), but the model which has seemed most consistent with the data is one based on stellar wind shocks (e.g., Lucy & White 1980; Lucy 1982; Owocki, Castor, & Rybicki 1988). There remain several outstanding problems associated with the theory. The main one is that the latest shock models have some difficulty explaining the amount of observed X-ray luminosity (Feldmeier, Puls, & Pauldrach 1997b). Although these simulations are very sophisticated in many ways, they are also restricted to spherical symmetry, and so far the results suggest that a single high-temperature, high-density shell will dominate the observed emission. Such a feature should lead to significant variability which has not been observed. (Note that these authors suggest that perhaps the emission arises in independent sectors, thereby suppressing the variability.) The newest X-ray satellites Chandra and XMM-Newton possess much larger collecting area and superior spectral resolution as compared to past missions. Keys to resolving present theoretical difficulties will come from observations that better constrain the levels of X-ray variability amplitude and timescale versus energy and resolving strong X-ray emission lines. Already a rich X-ray line spectrum is evident in observations of several hot star winds made by Chandra and XMM-Newton: the O star v 1 Ori C (Schulz et al. 2001), the O star z Ori (Waldron & Cassinelli 2001), the O star z Pup (Cassinelli et al. 2001; Kahn et al. 2001), and the WR star HD 9162 (pwr 25; Maeda & Tsuboi 1999). P Cygni profiles have also been observed in the X-ray binary Cir X-1, presumably arising from a disk wind centered on the compact object (Brandt & Schulz 2000). With the advent of these new data, it seemed pertinent L119 to undertake a study of the expected line profile shapes and variability from X-ray emitting gas in expanding spherical wind flows, which is the focus of this Letter. To do so, the work of MacFarlane et al. (1991) on single discrete shells is extended to account for the emission from a flow that is embedded with zones of X-ray emitting gas. 2. X-RAY EMISSION-LINE PROFILE SHAPES In deriving profile shapes, it shall be assumed that (a) the line emission is optically thin, (b) the wind is spherically symmetric, and (c) the hot gas is characterized by a temperature TX and volume filling factor fx, both of which may in principle vary with radius in the wind. In relation to point a, some strong permitted lines in the X-ray band could show optical depth effects. Although limited space precludes a derivation here, I have made a rough estimate of the line optical depth based on Sobolev theory and reasonable assumptions for typical stellar parameters and the density profile of hot gas with radius. Here I am concerned only with line emission that is not strongly attenuated. I find that X-ray lines will likely be thin for B stars, owing to the rather small wind mass-loss rates, and for WR stars, because the line emission emerges only from large radius (except for hard X-ray energies) owing to high mass loss and enhanced metal abundances. With intermediate mass-loss rates and solar abundances, lines from O stars may be the most likely to show optical depth effects, but even in this case optical depths might not exceed unity by much. A more careful future analysis of this issue is certainly warranted. Given the assumptions a c, and adopting standard spherical coordinates (r, v, f) centered on the star, the line-of-sight Doppler shift observed at any point in the expanding envelope will be given by vz p vrcos v, where v is measured from the line joining the observer and the star, which is also taken to be the z-axis. Two cases are considered for the line emission profile: in the first, simplified expressions are derived for constant expansion with both TX and fx also treated as constant, whereas in the second case, a framework for including more general variations of these parameters is presented.

3 L120 X-RAY EMISSION LINES FROM SPHERICAL STELLAR WINDS Vol Derivation for Spherical Constant Expansion For constant expansion with v r(r) p v0, the zones of constant observed Doppler shift (i.e., isovelocity zones) will be conical surfaces of fixed opening v given by vz p v0 cos v for a finite shell of radius r. Consequently, the luminosity of line emission per unit velocity interval from a thin shell of radial width Dr will be given by dl dv 2pj (r) t n (r, z) 2 (r, v z) p e r Dr, (1) v l n v z 0 where j n is the line emissivity. The attenuation by the wind is given by the formula (MacFarlane et al. 1991) n z v t n(r, v z) p krdz p t 0(r), (2) sin v where z p r cos v. The parameter t 0 represents the optical depth up to the shell of radius r along the line of sight to the star. The angular dependence accounts for the variation of the optical depth around the star to the back side. Implicit in the 2 expression is that the density r r, and also that the wind attenuation of X-rays is primarily due to photoabsorption with kr n r (i.e., constant opacity per gram). The optical depth parameter is thus given by n r R r t (r) p krdr p t. () 0 Thus, combining equations (2) and () and substituting into equation (1) yields the emission profile of a single finite spherical shell. Indeed, allowing the shell to move outward according to the relation r(t) p R0 v0t, the time evolution of the X-ray emis- sion profile can be calculated. This is relevant since the best spherically symmetric models of Feldmeier et al. (1997b) indicate that a single spherical shell could dominate the wind X- 2 ray production. Specifying the emissivity as jn p j 0(r/r 0) 4 r, with j0 p j 0(T X, f X), illustrative results of the calculation are shown in Figure 1 for t p 1,, and 10 at radii of r p 1R,2R,4R, and 16R. Showing results at a sequence of radii is equivalent to the time evolution of the line profile as a single shell coasts with the wind. Stellar occultation is included, which in the case of r p 1R eclipses all of the receding hemisphere of the shell. Along the ordinate, y refers to the normalized line luminosity, such that y p 1 occurs for r p R and t p 0. For the abscissa, w p v /v is the normalized z z 0 t observed velocity shift. For low, the blueshifted line emission drops monotonically with radius. At small radius, the occultation severely alters the profile shape. Owing to the greater absorption column to the receding part of the shell, the red wing is more attenuated than the blue wing, as found by MacFarlane et al. (1991). At large radius, the profile becomes flat-topped as expected. With increasing t, the variability takes on a different nature. The line emission from small r is almost entirely attenuated, so that as the shell moves outward, the line emission initially increases owing to the reduced attenuation and then decreases owing to the plummeting emissivity. So the type of X-ray line variability is sensitive to the total optical depth of the cool wind component. Now suppose that the hot gas is embedded in the cooler ambient gas in a time-steady way throughout the wind flow. Fig. 1. Emission profile shapes y plotted in log values against the normalized observed Doppler shift wz for the indicated value of t as a thin spherical shell moves through the wind. The different profiles are snapshots at times when the shell resides at r p 1R (solid line), 2 R (dotted line), 4 R (short-dashed line), and 16 R (long-dashed line). At large radius where the attenuation is small, the profiles become flat-topped in shape. The emergent X-ray profile as a function of observed velocity shift results from integrating over radius, leading to v z )( ) 4 2 l 0 t R v/r sin v z ( 4 2 z 0 R min(v) dl 2pj R R r dr (v ) p e, (4) dv v r R R where R min is the radius of the star for blueshifted emission, or R /sinvto account for stellar occultation of redshifted emission. If instead the hot gas exists in a series of discrete shells, the integral can be changed to reflect that scenario. To evaluate the integral, a change of variable is introduced with u p R /r, giving u min(v) l 2pj0R t uv/ sin v z z 0 dl (v ) p e du (5) dv v 0 t v/ sin v 2pj0R sin v (1 e ) for w z! 0 p ( )# t v (6) v t v { (1 e ) for w z Note that in the limit of t k 1, the exponential factor tends toward zero. For t K 1, equation (6) reduces to dl l/dvz p 2pj0R/v 0 for the blue wing and sin v # 2pj0R/v 0 for the red wing, the leading factor accounting for stellar occultation. Defining g(w z) p dl l/dvz, plots for different values of t are shown in Figure 2. Interestingly, at large optical depths t k 1, the profile shape reduces to a fixed contour form given by ) z 2 2pj R sin v jr 1 wz 0 0 g(w ) p p 2p z (. (7) v t v v t cos (w ) This profile shape is shown in Figure, with g 0 p 2p j R /t. 0

4 No. 1, 2001 IGNACE L121 Fig. 2. Example emission profile shapes g plotted against the normalized observed Doppler shift w z. The profiles are for winds in constant spherical expansion with density being the only radially varying quantity. The parameter t represents the total line-of-sight optical depth to the star owing to photoabsorption in the cool wind component Generalized Expressions for Optically Thin Line Emission The major simplifications of the previous section are (a) treating the wind as being in constant expansion and (b) allowing only the wind density to vary with radius. In regard to point a, for Wolf-Rayet stars and also some O stars, the X-ray emission does emerge from large radius over a broad range of X-ray energies (e.g., see Hillier et al. 199; Ignace, Oskinova, & Foullon 2000). This simply means that t k 1, so that X-ray line emission arising from deeper layers where the wind is accelerating is entirely absorbed by the overlying cool wind flow. However, this is not the case for B stars, where the mass loss is much smaller than for WR stars or O stars. As a result, X-ray emission can emerge from the wind with little attenuation even from the inner accelerating portion of the wind. In relation to point b, Owocki & Cohen (1999) have argued that the volume filling factor of hot gas may vary with radius. Feldmeier et al. (1997a) discuss X-ray production due to wind shocks and the expected range of temperatures that may exist within a shock zone. Likewise, the distribution of shocks throughout the wind flow will vary in strength, and then the whole ensemble varies in time, and thus so do the shock temperatures. Retaining the assumptions of spherical symmetry and optically thin line emission, the effects associated with case b are addressed first with case a following. Variation of the hot gas temperature and the volume filling factor depends on the dominant cooling processes and will likely be time dependent in regard to T X. Consequently, the ion fraction for the species under consideration may also vary with radius and time. And then, in addition, the flow is probably inhomogeneous, although perhaps spherically symmetric in a statistical sense with time and direction. Any intrinsic asphericity in the flow geometry would constitute additional complexities. As a result, there are simply too many potentially Fig.. In the limit of large optical depths t k 1, the emission profiles of Fig. 2 reduce to an asymptotic profile shape as plotted here. The parameter g 0 is a convenient normalization. varying parameters that depend on the detailed physics to be included in a general yet rigorous way. So a power-law representation is adopted as a parametric approach for including 4 q 4q these effects by writing jn p j 0(R /r) (R /r) p ju 0, for q 0. Inserting this new emissivity relation into the integral of equation (6) yields u max(v) l 2pj0R q t uv/sinv z z 0 dl (v ) p ue du. (8) dv v 0 Note that with a suitable change of variable, the integral in this equation can be transformed to yield an incomplete gamma function. Defining a 0 p (t v)/(sin v), the integral part becomes (1q) a 0 G(q 1, au 0 max). It is important to note that variations in TX, fx, ion fraction, and so on, will to good approximation not much affect the wind attenuation parametrization described in 2.1, as long as the constant spherical expansion regime is still being considered. The other simplification to be lifted is that of constant expansion. In this case severe new challenges arise. As an example, a standard wind velocity law is assumed with ) b br v r p v ( 1, (9) r with b! 1 so that v r is nonzero at r p R and b a constant that controls the wind acceleration (although in fact the velocity law for the hot gas might actually be nonmonotonic). With equation (9), the wind density becomes 2 b r p r0u (1 bu). (10) The implication is that the optical depth for the wind attenuation of X-rays will no longer have the simple separable form of a function of radius and of angle. Instead, the expression to be

5 L122 X-RAY EMISSION LINES FROM SPHERICAL STELLAR WINDS Vol. 549 solved is dz 2 b z (r/r )(1bR /r) t p kr. (11) n n 0 Adopting cylindrical coordinates (p, f, z), then r 2 p p 2 z 2 and tan v p p/z. Employing a change of variable from z to v, the integral becomes v dv b 0 (1 sin vbr /p) R tn p t, (12) p with impact parameter p p r sin v a constant for a given line of sight through the envelope. This integral has analytic solutions in special cases, as for example when b p 1, but not in general cases. As a result, emission-line profile shapes where the wind velocity must be considered can only be evaluated numerically. Indeed, even for an analytic case like b p 1, the optical depth will be a complicated function of v. Thus v p v(r) since the b Doppler formula is given by wz p (1 br /r) cos v, and hence the isovelocity surfaces intersect spherical shells of constant density in a manner that varies with radius. Owocki & Cohen (2001) have now conducted a study of the profile shapes that result when a realistic wind velocity is included, but again, these effects are relevant if the wind attenuation (which is a strong function of energy) is weak enough that a substantial flux of X-rays can emerge from the region where the wind is accelerating.. DISCUSSION Useful expressions for emission profile shapes relevant to X-ray lines that form in stellar winds have been derived assuming optically thin line emission, spherical symmetry, and a stationary flow. Initially the wind was treated as being in constant expansion. Based on the results of MacFarlane et al. (1991), it was shown that line profiles should be highly variable, in both shape and total emission, for the case of a single discrete shell that moves through the wind flow. Although the data are sparse, most single early-type stars are not variable X-ray sources of any significance (one exception being z Pup; Berghöfer et al. 1996), hence a large number of shells is implied. Although work is in progress to model the expected variability of the pseudocontinuum X-ray emission, it would also be useful to expand the present theoretical framework to incorporate large numbers of outflowing hot gas shells and model statistical variations expected in resolved line profiles. Treating the X-ray gas as being well mixed throughout the normal wind and also time independent, an analytic expression for the X-ray profile shape formed in a constant expansion flow was derived. If the wind photoabsorption optical depth is quite large (relevant to WR stars especially, but also to O stars at low X-ray energies), implying that the bulk of line emission emerges only at large radius, then the profile shapes obtain a self-similar form, with the total emission scaling as jr/t 0. These formulae are especially useful as new high-resolution data become available from the latest X-ray satellites. Some of the important and strongest X-ray emission lines come from He-like ions. These ions produce a multiplet of somewhat closely spaced emission lines involving a forbidden component, an intercombination component, and a resonance component. The relatively small separations of these lines (typically tens of ev) combined with the high speeds of stellar winds (typically km s 1 ) indicate that these three components will usually be blended. If the theory of 2.1 is applicable, it may be relatively straightforward to decompose the blended profile into its separate line components. As a final point, it is worth commenting on the effects of clumping, which appear to be of increasing concern for hot star winds, especially for the WR winds (e.g., Hillier 1991; Hamann & Koesterke 1998; Lépine & Moffat 1999; Li et al. 2000; Dessart et al. 2000; Howk et al. 2000; Harries 2000; Rodrigues & Magalhães 2000). As long as the inhomogeneous structure of the wind is not dominated by relatively few massive discrete blobs or compact bullets, so that the mean free path of a photon is significantly smaller than the clump size, then the wind attenuation being proportional to density will not be affected very much. Indeed, the line shape may allow one to find t 0 ; since this depends on density and not density squared, one will have t 0 M but insensitive to the wind clumping. On the other hand, the line emissivity could be affected, depending on the physics of the clumping and how the hot gas is being generated. Specific models are needed to assess these effects. 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