The ISM component has a transmission factor of 0. The wind-local model of emission and absorption is discussed in more detail in Section 3 and Appendix A.
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We show this region of the zeroth-order spectrum in Figure 2. Figure 2. The X-ray emission line strengths and profiles from stellar winds are very sensitive to the wind structure and dynamics. MacFarlane et al.
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Ignace studied the line profile for a shell undergoing constant spherical expansion, including the limit of high continuum opacity. They also modeled the effect of a minimum radius of line formation, showing how a central void in both velocity and emission measure can strongly affect the line profile by flattening the peak, since the volume of higher density, low-velocity plasma has been reduced.
With the higher resolution of HETG, we find that a Gaussian is a very poor approximation to the profile. Instead, we find that a spherical constant velocity expansion model, similar to that described by Ignace , gives a very good fit. As a baseline for modeling the lines, we assumed a provisional plasma model based on the RGS analysis of Oskinova et al.
This is primarily a phenomenological model because it does not entail distributed emission and absorption throughout the wind, but is a slab model, with an underlying multithermal plasma, overlying wind absorption with ionized wind edges, and an interstellar foreground absorption component. This serves as a basis for identifying and characterizing spectral features and incorporates blends from the basic plasma model. Absorption components, which were determined from XMM-Newton analysis, were left fixed, since they are better constrained by the longer wavelength data.
A model summary is given in Table 3 and more details are given in Appendix A. They are fin-shaped, with a sharp blue edge, upward-convex, sloping down to the red.
PPLATO | FLAP | PHYS A particle model for light
Figure 1 bottom panel shows detail for a narrow spectral region and demonstrates the near-vertical blue edge and maximum blueward of the line center. The profiles are very much like those shown in Ignace or of the profiles from optically thick in the continuum winds with a large central cavity e. The analytic solutions of Ignace were developed for just this case. We have adopted the analytic form of Ignace for the line profile, f w z , valid in the limit of large optical depth:. In this formula the emissivity per volume is assumed to vary as the square of density.
This q parameter serves to modify the shape of the line profile from a pure density-squared result. Physically, one can think of this accompaniment to the line emissivity as representing perhaps a number of factors, including a volume filling factor to accommodate clumping, or a radius-dependent X-ray temperature distribution to accommodate variations in shock strength. In this sense different q values are to be expected from fits to different lines in contrast to seeking a single value of q that applies to all lines.
As an illustration, a family of model line profiles is shown in Figure 3 for a range of q values.
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Note that the photoabsorption optical depth is assumed to be large, meaning that X-rays from near the stellar photosphere are strongly absorbed and that the X-ray line profile is formed generally in the vicinity of optical depth unity in photoabsorption, although this depends in detail on the value of q. Figure 3. Example intrinsic line profiles for constant spherical expansion for several values of q as defined by Equation 1. In using the adopted form of Equation 1 to model line profiles, we have made several assumptions.
First, that the X-ray line emission is taken to be well-described by collisional ionization equilibrium in which every collisional excitation is from the ground state and results in a radiative transition with negligible optical depth in the lines. This serves as the basis of the density-squared emissivity. Second, the continuum opacity of the WR 6 wind is very large. In soft X-rays, the large radial optical depth prevents us from seeing down to the acceleration zone, so constant expansion is a reasonable assumption for the visible plasma.
In this limit, if we assume that all X-ray emission lines have the same profile, then it means they all sample the same terminal velocity with the same temperature or same temperature distribution. To relax the latter assumption, that all lines have a common thermal origin with a similar hot gas filling factor, we allow the exponent, q , to be non-zero. In this way, we can fit individual profiles to explore trends in expansion velocity or shape. For example, the continuum opacity is lower at shorter wavelengths; if it is significantly smaller such that we can see deeper, where conditions may be different, we might expect the shortest wavelength lines to have a different shape from the longer wavelength lines even though all the lines form in the asymptotic terminal speed flow.
We have implemented the model line profile as a parametric fit function, but also as a global intrinsic line profile in the APEC model evaluation that is, our APEC model, in addition to the usual parameters of temperatures, normalizations, abundances, and Doppler shift, has wind-profile parameters. We adopted a line of sight Doppler velocity of The differences between the observations are negligible considering the resolution and the line width, though very important to set a priori because the terminal velocity and Doppler shift are degenerate parameters.
We take the Doppler velocity as a given and do not fit the line center. One must be careful to distinguish the line center from a line centroid.
PHYS 10.1: A particle model for light
A common diagnostic of stellar winds is often referred to as a "blueshifted profile. Here we specifically refer to the theoretical line profiles' centers when we indicate the line position. In Figure 4 , we show observed and model profiles for relatively clean portions of the spectrum to demonstrate the "fin"-shape and relative position of the line center. The models were APEC plasmas fit within the narrow regions as described above.
Figure 4. The line centers of the strongest lines in the region are marked; other model lines in the regions shown are an order of magnitude fainter than the brightest. The lower panels show the fit residuals.
The density dependence was computed with APEC, then parameterized for use as a line emissivity modifier. This is less physically constrained than the plasma-model approach since, for instance, there is no a priori relation between the forbidden and intercombination line.
Uncertainties in this approach tend to be large because the lower limit on the intercombination line can be very small, and the forbidden-to-intercombination ratio arbitrarily large; hence we favor the APEC-based results. The fitted values are given in Table 4. Wavelengths are given for the resonance lines. R 0 is for the four-temperature model and so may differ slightly from the value at the temperature of maximum emissivity for each ion.
The lines are well resolved. Figure 5. The top panel shows the photon flux spectrum black , the model red , and the residuals below in the small sub-panel. The bottom plot shows the same counts black and model gray , but also shows the components for the resonance red , intercombination green , and forbidden blue lines also labeled as r , i , and f.
Line centers not centroids are marked in the upper panel. In Figure 6 we show a broader region of the spectrum along with the same spectral region as observed with RGS bottom panel pair. This clearly demonstrates the character of the profiles' vertical blue edge and the great advantage of the HETG's higher resolution for determining the profile shape. With RGS, we could determine that the lines are broad, but a near-Gaussian profile was sufficient to fit them.
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Figure 6. In each case, the black histogram is the observed spectrum with two bins per resolution element and the red histogram is the model. Below each are the residuals. In this region it is clear that the HETG has resolved the sharp blue edges on the profiles, which were not evident in the lower-resolution RGS data.
We show the determinations for each feature measured in Figure 7 , and the values are listed in Table 5. Figure 7. Line width and shape parameters against wavelength. On the left, there is no apparent trend of width with wavelength. On the right, there is a weak trend in the shape with wavelength. To examine the time history of the X-ray emission, we binned count-rate light curves from both the dispersed and zeroth-order events.
For dispersed events, we used the program, aglc 10 which handles the dispersed photon coordinates and CCD exposure frames over multiple CCD chips. Count rates from the dispersed spectrum were slightly higher than from zeroth order. Figure 8 shows the combined zeroth and first order rates for each of the three observations. Table 2 lists the mean count rates per observation and the overall means for dispersed and zeroth orders. Hardness ratios were also computed, but since these showed no trend, we have not included them in the figure.
Figure 8. Light curves; Top: HETG dispersed first order plus zeroth-order light curve for each observation as labeled. The inset axis gives the phase according to the ephemeris of Georgiev et al. The horizontal gray line is the mean count rate for the observation. See Table 2 for observation dates. Empirical uncertainties were estimated from the variance in flat portions of VGuide light curves for several stars over a range of magnitudes.