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We present a general procedure for deriving a line profile model for massive star X-ray spectra that captures the dynamics of the wind more directly. The basis of the model is the analytic solution to the problem of variable jets in Herbig-Haro objects given by \citet{Canto2000}. In deriving our model, we generalize this jet solution to include flows with a prescribed nonzero acceleration for the context of radiatively driven winds. We provide example line profiles generated from our model for the case of sinusoidal velocity and mass ejection variations. The example profiles show the expected shape of massive star X-ray emission lines, as well as interesting but complicated trends with the model parameters. This establishes the possibility that observed X-rays could be a result of temporal variations seeded at the wind base, rather than purely generated intrinsically within the wind volume, and can be described via a quantitative language that connects with the physical attributes of those variations, consistently with the downstream momentum-conserving nature of radiatively cooled shocked radial flows., Comment: 18 pages, 6 figures, 1 table

We fit the high resolution \textit{Chandra} X-ray spectra of the O supergiant $\zeta$ Puppis using the variable boundary condition (VBC) line model to test the stability of its mass-loss rate between two epochs of observation: 2000 March and 2018 July -- 2019 August. At issue is whether the observed variations are induced by global changes in the cool (unshocked) wind itself or are isolated to the local pockets of hot gas (i.e., changes in the frequency and location of the shocks). Evidence in the literature favored the possibility of a 40 per cent increase in the mass flux of the entire stellar wind, based on X-ray reabsorption from a line-deshadowing-instability-inspired parameterization, whereas our fit parameters are consistent with a constant mass flux with a change in the velocity variations that determine the locations where shocks form. Our results suggest the shocks in the more recent data are formed at somewhat larger radii, mimicking the enhanced blueshifts and increased line fluxes interpreted in the previous analysis as being due to increases in both the X-ray generation and reabsorption from an overall stronger wind., Comment: 18 pages, 13 figures, 4 tables

We fit a new line shape model to \textit{Chandra} X-ray spectra of the O supergiant $\zeta$ Puppis to test the robustness of mass-loss rates derived from X-ray wind line profiles against different assumed heating models. Our goal is to track the hot gas by replacing the common assumption that it is proportional to the cool gas emission measure. Instead of assuming a turn-on radius for the hot gas (as appropriate for the line-deshadowing instability internal to the wind), we parametrize the hot gas in terms of a mean-free path for accelerated low-density gas to encounter slower high-density material. This alternative model is equally successful as previous approaches at fitting X-ray spectral lines in the 5 -- 17 \AA\ wavelength range. We find that the characteristic radii where the hottest gas appears is inversely proportional to line formation temperature, suggesting that stronger shocks appear generally closer to the surface. This picture is more consistent with pockets of low-density, rapid acceleration at the lower boundary than with an internally generated wind instability. We also infer an overall wind mass-loss rate from the profile shapes with a technique used previously in the literature. In doing so, we find evidence that the mass-loss rate derived from X-ray wind line profiles is not robust with respect to changes in the specific heating picture used., Comment: 11 Pages, 5 Figures, Published in MNRAS

We have obtained a very deep exposure (813 ks) of $\zeta\,$Puppis (O4 supergiant) with the Chandra/HETG Spectrometer. Here we report on analysis of the 1-9 \r{A} region, especially well suited for Chandra, which has a significant contribution from continuum emission between well separated emission lines from high-ionization species. These data allow us to study the hottest plasma present through the continuum shape and emission line strengths. Assuming a powerlaw emission measure distribution which has a high-temperature cut-off, we find that the emission is consistent with a thermal spectrum having a maximum temperature of 12 MK. This implies an effective wind shock velocity of $900\,\mathrm{km\,s^{-1}}$, well below the wind terminal speed of $2250\,\mathrm{km\,s^{-1}}$. For X-ray emission which forms close to the star, the speed and X-ray flux are larger than can be easily reconciled with strictly self-excited line-deshadowing-instability models, suggesting a need for a fraction of the wind to be accelerated extremely rapidly right from the base. This is not so much a dynamical instability as a nonlinear response to changing boundary conditions., Comment: 18 pages, 4 figures; accepted for publication in the Astrophysical Journal

We attempt to determine the driver for clumping in hot-star winds by extending the measure of the spectral variability level of Galactic Wolf-Rayet stars to by far the hottest known among them, the WN2 star WR 2 and the WO2 stars WR 102 and WR 142. These three stars have T* = 140 kK and 200 kK, the last two being well above the bulk of WR stars with T* ~ 40-120 kK. This full temperature range for WR stars is much broader than that of their O-star progenitors (~30-50 kK), so is better suited to look for any temperature dependence of wind clumping. We have obtained multiple observations with high signal-to-noise, moderate-resolution spectroscopy in search of smallscale variability in the strong emission lines from the dense winds of these three extreme stars, and find a very low-level of variability in both stars. Temperature and terminal velocity are correlated, so faster winds show a lower variability, though this trend goes against any predictions made involving Line Deshadowing Instability (LDI) only, implying that instabilities intrinsic to LDI are not the main source of wind clumping. Instead, it could be taken as support for the suggestion that clumps are caused by a sub-surface convection zone (SSCZ) at T ~ 170 kK, since such an SSCZ would have little opportunity to operate under the hydrostatic surface of these hottest WR stars. It is still possible, however, that an SSCZ-related driver could interact with nonlinear line instability effects to enhance or possibly even produce clumps., Comment: 17 pages, 6 figures, 5 tables, ApJ, in press

We have obtained a very deep exposure (813 ks) of $\zeta\,$Puppis (O4 supergiant) with the Chandra/HETG Spectrometer. Here we report on analysis of the 1-9 \r{A} region, especially well suited for Chandra, which has a significant contribution from continuum emission between well separated emission lines from high-ionization species. These data allow us to study the hottest plasma present through the continuum shape and emission line strengths. Assuming a powerlaw emission measure distribution which has a high-temperature cut-off, we find that the emission is consistent with a thermal spectrum having a maximum temperature of 12 MK. This implies an effective wind shock velocity of $900\,\mathrm{km\,s^{-1}}$, well below the wind terminal speed of $2250\,\mathrm{km\,s^{-1}}$. For X-ray emission which forms close to the star, the speed and X-ray flux are larger than can be easily reconciled with strictly self-excited line-deshadowing-instability models, suggesting a need for a fraction of the wind to be accelerated extremely rapidly right from the base. This is not so much a dynamical instability as a nonlinear response to changing boundary conditions., Comment: 18 pages, 4 figures; accepted for publication in the Astrophysical Journal

Observations of free-free continuum radio emission of four young main-sequence solar-type stars (EK Dra, Pi1 UMa, Chi1 Ori; and Kappa1 Cet) are studied to detect stellar winds or at least to place upper limits on their thermal radio emission, which is dominated by the ionized wind. These stars are excellent proxies for representing the young Sun. Upper limits on mass loss rates are calculated using their observational radio emission. Our aim is to re-examine the faint young Sun paradox by assuming that the young Sun was more massive in its past, and hence to find a possible solution for this famous problem. The observations of our sample are performed with the Karl G. Jansky VLA with excellent sensitivity, using the C-band and the Ku-band. ALMA observations are performed at 100 GHz. For the estimation of the mass loss limits, spherically symmetric winds and stationary, anisotropic, ionized winds are assumed. We compare our results to 1) mass loss rate estimates of theoretical rotational evolution models, and 2) to results of the indirect technique of determining mass loss rates: Lyman-alpha absorption. We are able to derive the most stringent direct upper limits on mass loss so far from radio observations. Two objects, EK Dra and Chi1 Ori, are detected at 6 and 14 GHz down to an excellent noise level. These stars are very active and additional radio emission identified as non-thermal emission was detected, but limits for the mass loss rates of these objects are still derived. The stars Pi1 UMa and Kappa1 Cet were not detected in either C-band or in Ku-band. For these objects we give upper limits to their radio free-free emission and calculate upper limits to their mass loss rates. Finally, we reproduce the evolution of the Sun and derive an estimate for the solar mass of the Sun at a younger age.

Observations of free-free continuum radio emission of four young main-sequence solar-type stars (EK Dra, Pi1 UMa, Chi1 Ori; and Kappa1 Cet) are studied to detect stellar winds or at least to place upper limits on their thermal radio emission, which is dominated by the ionized wind. These stars are excellent proxies for representing the young Sun. Upper limits on mass loss rates are calculated using their observational radio emission. Our aim is to re-examine the faint young Sun paradox by assuming that the young Sun was more massive in its past, and hence to find a possible solution for this famous problem. The observations of our sample are performed with the Karl G. Jansky VLA with excellent sensitivity, using the C-band and the Ku-band. ALMA observations are performed at 100 GHz. For the estimation of the mass loss limits, spherically symmetric winds and stationary, anisotropic, ionized winds are assumed. We compare our results to 1) mass loss rate estimates of theoretical rotational evolution models, and 2) to results of the indirect technique of determining mass loss rates: Lyman-alpha absorption. We are able to derive the most stringent direct upper limits on mass loss so far from radio observations. Two objects, EK Dra and Chi1 Ori, are detected at 6 and 14 GHz down to an excellent noise level. These stars are very active and additional radio emission identified as non-thermal emission was detected, but limits for the mass loss rates of these objects are still derived. The stars Pi1 UMa and Kappa1 Cet were not detected in either C-band or in Ku-band. For these objects we give upper limits to their radio free-free emission and calculate upper limits to their mass loss rates. Finally, we reproduce the evolution of the Sun and derive an estimate for the solar mass of the Sun at a younger age.

With a deep Chandra/HETGS exposure of WR 6, we have resolved emission lines whose profiles show that the X-rays originate from a uniformly expanding spherical wind of high X-ray-continuum optical depth. The presence of strong helium-like forbidden lines places the source of X-ray emission at tens to hundreds of stellar radii from the photosphere. Variability was present in X-rays and simultaneous optical photometry, but neither were correlated with the known period of the system or with each other. An enhanced abundance of sodium revealed nuclear processed material, a quantity related to the evolutionary state of the star. The characterization of the extent and nature of the hot plasma in WR 6 will help to pave the way to a more fundamental theoretical understanding of the winds and evolution of massive stars., Comment: Accepted by the Astrophysical Journal

With a deep Chandra/HETGS exposure of WR 6, we have resolved emission lines whose profiles show that the X-rays originate from a uniformly expanding spherical wind of high X-ray-continuum optical depth. The presence of strong helium-like forbidden lines places the source of X-ray emission at tens to hundreds of stellar radii from the photosphere. Variability was present in X-rays and simultaneous optical photometry, but neither were correlated with the known period of the system or with each other. An enhanced abundance of sodium revealed nuclear processed material, a quantity related to the evolutionary state of the star. The characterization of the extent and nature of the hot plasma in WR 6 will help to pave the way to a more fundamental theoretical understanding of the winds and evolution of massive stars., Comment: Accepted by the Astrophysical Journal

We present a new method for using measured X-ray emission line fluxes from O stars to determine the shock-heating rate due to instabilities in their radiation-driven winds. The high densities of these winds means that their embedded shocks quickly cool by local radiative emission, while cooling by expansion should be negligible. Ignoring for simplicity any non-radiative mixing or conductive cooling, the method presented here exploits the idea that the cooling post-shock plasma systematically passes through the temperature characteristic of distinct emission lines in the X-ray spectrum. In this way, the observed flux distribution among these X-ray lines can be used to construct the cumulative probability distribution of shock strengths that a typical wind parcel encounters as it advects through the wind. We apply this new method (Gayley 2014) to Chandra grating spectra from five O stars with X-ray emission indicative of embedded wind shocks in effectively single massive stars. Correcting for wind absorption of the X-ray line emission is a crucial component of our analysis, and we use wind optical depth values derived from X-ray line-profile fitting (Cohen et al. 2014) in order to make that correction. The shock-heating rate results we derive for all the stars are quite similar: the average wind mass element passes through roughly one shock that heats it to at least $10^6$ K as it advects through the wind, and the cumulative distribution of shock strengths is a strongly decreasing function of temperature, consistent with a negative power-law of index $n \approx 3$, implying a marginal distribution of shock strengths that scales as $T^{-4}$, and with hints of an even steeper decline or cut-off above $10^7$ K., Comment: 11 pages, 6 figures (5 color); MNRAS in press

We present a new method for using measured X-ray emission line fluxes from O stars to determine the shock-heating rate due to instabilities in their radiation-driven winds. The high densities of these winds means that their embedded shocks quickly cool by local radiative emission, while cooling by expansion should be negligible. Ignoring for simplicity any non-radiative mixing or conductive cooling, the method presented here exploits the idea that the cooling post-shock plasma systematically passes through the temperature characteristic of distinct emission lines in the X-ray spectrum. In this way, the observed flux distribution among these X-ray lines can be used to construct the cumulative probability distribution of shock strengths that a typical wind parcel encounters as it advects through the wind. We apply this new method (Gayley 2014) to Chandra grating spectra from five O stars with X-ray emission indicative of embedded wind shocks in effectively single massive stars. Correcting for wind absorption of the X-ray line emission is a crucial component of our analysis, and we use wind optical depth values derived from X-ray line-profile fitting (Cohen et al. 2014) in order to make that correction. The shock-heating rate results we derive for all the stars are quite similar: the average wind mass element passes through roughly one shock that heats it to at least $10^6$ K as it advects through the wind, and the cumulative distribution of shock strengths is a strongly decreasing function of temperature, consistent with a negative power-law of index $n \approx 3$, implying a marginal distribution of shock strengths that scales as $T^{-4}$, and with hints of an even steeper decline or cut-off above $10^7$ K., Comment: 11 pages, 6 figures (5 color); MNRAS in press

X-ray satellites since Einstein have empirically established that the X-ray luminosity from single O-stars scales linearly with bolometric luminosity, Lx ~ 10^{-7} Lbol. But straightforward forms of the most favored model, in which X-rays arise from instability-generated shocks embedded in the stellar wind, predict a steeper scaling, either with mass loss rate Lx ~ Mdot ~ Lbol^{1.7} if the shocks are radiative, or with Lx ~ Mdot^{2} ~ Lbol^{3.4} if they are adiabatic. This paper presents a generalized formalism that bridges these radiative vs. adiabatic limits in terms of the ratio of the shock cooling length to the local radius. Noting that the thin-shell instability of radiative shocks should lead to extensive mixing of hot and cool material, we propose that the associated softening and weakening of the X-ray emission can be parametrized as scaling with the cooling length ratio raised to a power m$, the "mixing exponent". For physically reasonable values m ~= 0.4, this leads to an X-ray luminosity Lx ~ Mdot^{0.6} ~ Lbol that matches the empirical scaling. To fit observed X-ray line profiles, we find such radiative-shock-mixing models require the number of shocks to drop sharply above the initial shock onset radius. This in turn implies that the X-ray luminosity should saturate and even decrease for optically thick winds with very high mass-loss rates. In the opposite limit of adiabatic shocks in low-density winds (e.g., from B-stars), the X-ray luminosity should drop steeply with Mdot^2. Future numerical simulation studies will be needed to test the general thin-shell mixing ansatz for X-ray emission., Comment: 12 pages, 4 figures, accepted for publication in MNRAS

X-ray satellites since Einstein have empirically established that the X-ray luminosity from single O-stars scales linearly with bolometric luminosity, Lx ~ 10^{-7} Lbol. But straightforward forms of the most favored model, in which X-rays arise from instability-generated shocks embedded in the stellar wind, predict a steeper scaling, either with mass loss rate Lx ~ Mdot ~ Lbol^{1.7} if the shocks are radiative, or with Lx ~ Mdot^{2} ~ Lbol^{3.4} if they are adiabatic. This paper presents a generalized formalism that bridges these radiative vs. adiabatic limits in terms of the ratio of the shock cooling length to the local radius. Noting that the thin-shell instability of radiative shocks should lead to extensive mixing of hot and cool material, we propose that the associated softening and weakening of the X-ray emission can be parametrized as scaling with the cooling length ratio raised to a power m$, the "mixing exponent". For physically reasonable values m ~= 0.4, this leads to an X-ray luminosity Lx ~ Mdot^{0.6} ~ Lbol that matches the empirical scaling. To fit observed X-ray line profiles, we find such radiative-shock-mixing models require the number of shocks to drop sharply above the initial shock onset radius. This in turn implies that the X-ray luminosity should saturate and even decrease for optically thick winds with very high mass-loss rates. In the opposite limit of adiabatic shocks in low-density winds (e.g., from B-stars), the X-ray luminosity should drop steeply with Mdot^2. Future numerical simulation studies will be needed to test the general thin-shell mixing ansatz for X-ray emission., Comment: 12 pages, 4 figures, accepted for publication in MNRAS

Given that dynamically significant magnetic fields in at least some massive stars have now been measured, our contribution addresses the question, to what extent can fields be directly detected in circumstellar gas? The question speaks directly to the very interesting topic of line-driving physics coupled with magnetized plasmas, and how this coupling produces structure in the wind flow. We focus our attention on weak-field diagnostics. These come in two main types: the Hanle effect, which pertains to coherence effects for linear polarization from line scattering, and the weak longitudinal Zeeman effect, which pertains to circular polarization in lines., Comment: to appear in the proceedings of Clumping in Hot Star Winds

Given that dynamically significant magnetic fields in at least some massive stars have now been measured, our contribution addresses the question, to what extent can fields be directly detected in circumstellar gas? The question speaks directly to the very interesting topic of line-driving physics coupled with magnetized plasmas, and how this coupling produces structure in the wind flow. We focus our attention on weak-field diagnostics. These come in two main types: the Hanle effect, which pertains to coherence effects for linear polarization from line scattering, and the weak longitudinal Zeeman effect, which pertains to circular polarization in lines., Comment: to appear in the proceedings of Clumping in Hot Star Winds

Given that dynamically significant magnetic fields in at least some massive stars have now been measured, our contribution addresses the question, to what extent can fields be directly detected in circumstellar gas? The question speaks directly to the very interesting topic of line-driving physics coupled with magnetized plasmas, and how this coupling produces structure in the wind flow. We focus our attention on weak-field diagnostics. These come in two main types: the Hanle effect, which pertains to coherence effects for linear polarization from line scattering, and the weak longitudinal Zeeman effect, which pertains to circular polarization in lines., Comment: to appear in the proceedings of Clumping in Hot Star Winds

The hypothesis that CAK-type line driving is responsible for the large observed Wolf-Rayet (W-R) mass-loss rates has been called into question in recent theoretical studies. The purpose of this paper is to reconsider the plausibility of line driving of W-R winds within the standard approach using the Sobolev approximation while advancing the conceptual understanding of this topic. Due to the multiple scattering required in this context, of particular importance is the role of photon frequency redistribution into spectral gaps, which in the extreme limit yields the statistical Sobolev-Rosseland (SSR) mean approximation. Interesting limits to constrain are the extremes of no frequency redistribution, wherein the small radii and corresponding high W-R surface temperature induces up to twice the mass-loss rate relative to cooler stars, and the SSR limit, whereby the reduced efficiency of the driving drops the mass flux by as much as an order of magnitude whenever there exist significant gaps in the spectral line distribution. To see how this efficiency drop might be sufficiently avoided to permit high W-R mass loss, we explore the suggestion that ionization stratification may serve to fill the gaps globally over the wind. We find that global ionization changes can only fill the gaps sufficiently to cause about a 25% increase in the mass-loss rate over the local SSR limit. Higher temperatures and more ionization states (especially of iron) may be needed to achieve optically thick W-R winds, unless strong clumping corrections eliminate the need for such winds., Comment: 39 pages, 8 figures, 7 tables, accepted for publication in ApJ (fixed typos and grammar)

The hypothesis that CAK-type line driving is responsible for the large observed Wolf-Rayet (W-R) mass-loss rates has been called into question in recent theoretical studies. The purpose of this paper is to reconsider the plausibility of line driving of W-R winds within the standard approach using the Sobolev approximation while advancing the conceptual understanding of this topic. Due to the multiple scattering required in this context, of particular importance is the role of photon frequency redistribution into spectral gaps, which in the extreme limit yields the statistical Sobolev-Rosseland (SSR) mean approximation. Interesting limits to constrain are the extremes of no frequency redistribution, wherein the small radii and corresponding high W-R surface temperature induces up to twice the mass-loss rate relative to cooler stars, and the SSR limit, whereby the reduced efficiency of the driving drops the mass flux by as much as an order of magnitude whenever there exist significant gaps in the spectral line distribution. To see how this efficiency drop might be sufficiently avoided to permit high W-R mass loss, we explore the suggestion that ionization stratification may serve to fill the gaps globally over the wind. We find that global ionization changes can only fill the gaps sufficiently to cause about a 25% increase in the mass-loss rate over the local SSR limit. Higher temperatures and more ionization states (especially of iron) may be needed to achieve optically thick W-R winds, unless strong clumping corrections eliminate the need for such winds., Comment: 39 pages, 8 figures, 7 tables, accepted for publication in ApJ (fixed typos and grammar)

We examine radiatively driven mass loss from stars near and above the Eddington limit (Ledd). We begin by reviewing the instabilities that are expected to form extensive structure near Ledd. We investigate how this "porosity" can reduce the effective coupling between the matter and radiation. Introducing a new "porosity-length'' formalism, we derive a simple scaling for the reduced effective opacity, and use this to derive an associated scaling for the porosity-moderated, continuum-driven mass loss rate from stars that formally exceed Ledd. For a simple super-Eddington model with a single porosity length that is assumed to be on the order of the gravitational scale height, the overall mass loss is similar to that derived in previous porosity work. This is much higher than is typical of line-driven winds, but is still only a few percent of the photon tiring limit--for which the luminosity becomes insufficient to carry the flow out of the gravitational potential. To obtain still stronger mass loss that approaches observationally inferred values near this limit, we introduce a power-law-porosity model in which the associated structure has a broad range of scales. We show that the mass loss rate can be enhanced by a factor that increases with the Eddington parameter Gamma, such that for moderately large Gamma (> 3-4), mass loss rates could approach the photon tiring limit. Together with the ability to drive quite fast outflow speeds, the derived mass loss could explain the large inferred mass loss and flow speeds of giant outbursts in eta Carinae and other LBV stars., Comment: 17 pages, 6 figures, to appear in ApJ

We examine radiatively driven mass loss from stars near and above the Eddington limit (Ledd). We begin by reviewing the instabilities that are expected to form extensive structure near Ledd. We investigate how this "porosity" can reduce the effective coupling between the matter and radiation. Introducing a new "porosity-length'' formalism, we derive a simple scaling for the reduced effective opacity, and use this to derive an associated scaling for the porosity-moderated, continuum-driven mass loss rate from stars that formally exceed Ledd. For a simple super-Eddington model with a single porosity length that is assumed to be on the order of the gravitational scale height, the overall mass loss is similar to that derived in previous porosity work. This is much higher than is typical of line-driven winds, but is still only a few percent of the photon tiring limit--for which the luminosity becomes insufficient to carry the flow out of the gravitational potential. To obtain still stronger mass loss that approaches observationally inferred values near this limit, we introduce a power-law-porosity model in which the associated structure has a broad range of scales. We show that the mass loss rate can be enhanced by a factor that increases with the Eddington parameter Gamma, such that for moderately large Gamma (> 3-4), mass loss rates could approach the photon tiring limit. Together with the ability to drive quite fast outflow speeds, the derived mass loss could explain the large inferred mass loss and flow speeds of giant outbursts in eta Carinae and other LBV stars., Comment: 17 pages, 6 figures, to appear in ApJ