arXiv subjects

R. K. Prinja

Publications and source records attributed to R. K. Prinja.

At least 19 recordsLinked to original sources

arXiv2026

First measurement of wind line formation regions in an early O-type star

Massive stars with their strong ionizing radiation and strong stellar winds are the key feedback agents of the universe. Stellar winds of massive stars are often measured by fitting resonance lines in the UV using non-LTE stellar atmosphere models. So far, the line formation regions of these lines have not been measured empirically, preventing a comparison to the model's structures. We aim to conduct the first measurement of the resonance line formation regions in an early-type eclipsing binary in the SMC, namely AzV 75. We employ TESS and ASAS-SN photometry in combination with radial velocity measurements from multi-epoch HST UV spectra to derive the ephemeris. We examine the intensity changes in the C IV and N V resonance lines in the UV and combine them with a light-curve analysis to estimate the region in the wind where these lines are formed. AzV 75 has an orbital period P=165.66d, eccentricity e=0.42, mass ratio q=0.72, and inclination i=85.77{\deg}. With this orbital configuration, no secondary eclipse is expected. We report that the optically thick UV resonance lines exhibit flattening and shortening of the absorption trough, and weakening of their emission features, as they approach the phase of the expected secondary eclipse, while the continuum UV flux appears to remain unaffected. We illustrate that this can be explained by the primary's optically thick wind eclipsing the secondary star. The C IV and N V resonance line formation regions in the primary star extend up to 316 Rsol. The measured extend of the formation regions of resonance lines in a stellar wind are important benchmarks for 1D as well as 3D non-LTE stellar atmosphere models. A first comparison to 1D-stellar atmosphere models indicates that a classical beta-law with an exponent of beta=0.5 instead of beta=0.8 might be favoured for the primary star's velocity field.

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arXiv2024

Determining stellar properties of massive stars in NGC346 in the SMC with a Bayesian statistic technique

NGC 346 is a young cluster with numerous hot OB stars. It is part of the Small Magellanic Cloud (SMC), and has an average metallicity that is one-seventh of the Milky Way's. A detailed study of its stellar content provides a unique opportunity to understand the stellar and wind properties of massive stars in low-metallicity environments, and enables us to improve our understanding of star formation and stellar evolution. The fundamental stellar parameters defining a star's spectral appearance are its effective surface temperature, surface gravity, and projected rotational velocity. Unfortunately, these parameters cannot be obtained independently from only H and He spectral features as they are partially degenerate. With this work we aim to overcome this degeneracy by applying a newly developed Bayesian statistic technique that can fit these three parameters simultaneously. Multi-epoch optical spectra are used in combination with a Bayesian statistic technique to fit stellar properties based on a publicly available grid of synthetic spectra of stellar atmospheres. The use of all of the multi-epoch observations simultaneously allows the identification of binaries. The stellar parameters for 34 OB stars within the core of NGC 346 are derived and presented here. By the use of both $\mathrm{He}\textsc{i}$ and $\mathrm{He}\textsc{ii}$ lines, the partial degeneracy between the stellar parameters of effective surface temperature, surface gravity, and projected rotational velocity is overcome. A lower limit to the binary fraction of the sample of stars is found to be at least 46%. Based on comparisons with analysis conducted on an overlapping sample of stars within NGC 346, the Bayesian statistic technique approach is shown to be a viable method to measure stellar parameters for hot massive stars in low-metallicity environments even when only low-resolution spectra are available.}

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arXiv2024

X-Shooting ULLYSES: Massive Stars at low metallicity IX: Empirical constraints on mass-loss rates and clumping parameters for OB supergiants in the Large Magellanic Cloud

Context. Current implementations of mass loss for hot, massive stars in stellar evolution models include a sharp increase in mass loss when blue supergiants become cooler than Teff 20-22kK. This drastic mass-loss jump has been motivated by the potential presence of a so-called bistability ionisation effect, which may occur for line-driven winds in this temperature region due to recombination of important line-driving ions. Aims. We perform quantitative spectroscopy using UV (ULLYSES program) and optical (XShootU collaboration) data for 17 OB-supergiant stars in the LMC (covering the range Teff 14-32kK), deriving absolute constraints on global stellar, wind, and clumping parameters. We examine whether there are any empirical signs of a mass-loss jump in the investigated region, and we study the clumped nature of the wind. Methods. We use a combination of the model atmosphere code fastwind and the genetic algorithm code Kiwi-GA to fit synthetic spectra of a multitude of diagnostic spectral lines in the optical and UV. Results. We find no signs of any upward mass loss jump anywhere in the examined region. Standard theoretical comparison models, which include a strong bistability jump thus severely over predict the empirical mass-loss rates on the cool side of the predicted jump. Additionally, we find that on average about 40% of the total wind mass seems to reside in the diluted medium in between dense clumps. Conclusions. Our derived mass-loss rates suggest that for applications like stellar evolution one should not include a drastic bistability jump in mass loss for stars in the temperature and luminosity region investigated here. The derived high values of interclump density further suggest that the common assumption of an effectively void interclump medium (applied in the vast majority of spectroscopic studies of hot star winds) is not generally valid in this parameter regime.

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arXiv2023

X-Shooting ULLYSES: massive stars at low metallicity. I. Project Description

Observations of individual massive stars, super-luminous supernovae, gamma-ray bursts, and gravitational-wave events involving spectacular black-hole mergers, indicate that the low-metallicity Universe is fundamentally different from our own Galaxy. Many transient phenomena will remain enigmatic until we achieve a firm understanding of the physics and evolution of massive stars at low metallicity (Z). The Hubble Space Telescope has devoted 500 orbits to observe 250 massive stars at low Z in the ultraviolet (UV) with the COS and STIS spectrographs under the ULLYSES program. The complementary ``X-Shooting ULLYSES'' (XShootU) project provides enhanced legacy value with high-quality optical and near-infrared spectra obtained with the wide-wavelength coverage X-shooter spectrograph at ESO's Very Large Telescope. We present an overview of the XShootU project, showing that combining ULLYSES UV and XShootU optical spectra is critical for the uniform determination of stellar parameters such as effective temperature, surface gravity, luminosity, and abundances, as well as wind properties such as mass-loss rates in function of Z. As uncertainties in stellar and wind parameters percolate into many adjacent areas of Astrophysics, the data and modelling of the XShootU project is expected to be a game-changer for our physical understanding of massive stars at low Z. To be able to confidently interpret James Webb Space Telescope (JWST) spectra of the first stellar generations, the individual spectra of low Z stars need to be understood, which is exactly where XShootU can deliver.

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arXiv2022

Stellar wind properties of the nearly complete sample of O stars in the low metallicity young star cluster NGC346 in the SMC galaxy

Massive stars are among the main cosmic engines driving the evolution of star-forming galaxies. Their powerful ionising radiation and stellar winds inject a large amount of energy in the interstellar medium. Furthermore, mass-loss ($\dot{M}$) through radiatively driven winds plays a key role in the evolution of massive stars. Even so, the wind mass-loss prescriptions used in stellar evolution models, population synthesis, and stellar feedback models often disagree with mass-loss rates empirically measured from the UV spectra of low metallicity massive stars. The most massive young star cluster in the low metallicity Small Magellanic Cloud galaxy is NGC346. This cluster contains more than half of all O stars discovered in this galaxy so far. A similar age, metallicity ($Z$), and extinction, the O stars in the NGC346 cluster are uniquely suited for a comparative study of stellar winds in O stars of different subtypes. We aim to use a sample of O stars within NGC346 to study stellar winds at low metallicity. We mapped the central 1" of NGC346 with the long-slit UV observations performed by the Space Telescope Imaging Spectrograph (STIS) on board of the {\em Hubble Space Telescope} and complemented these new datasets with archival observations. Multi-epoch observations allowed for the detection of wind variability. The UV dataset was supplemented by optical spectroscopy and photometry. The resulting spectra were analysed using a non-local thermal equilibrium model atmosphere code (PoWR) to determine wind parameters and ionising fluxes. The effective mapping technique allowed us to obtain a mosaic of almost the full extent of the cluster and resolve stars in its core. Among hundreds of extracted stellar spectra, 21 belong to O stars. Nine of them are classified as O stars for the first time. We analyse, in detail, the UV spectra of 19 O stars... (continued)

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arXiv2020

COBRaS: The e-MERLIN 21 cm Legacy survey of Cygnus OB2

The role of massive stars is central to an understanding of galactic ecology. It is important to establish the details of how massive stars provide radiative, chemical, and mechanical feedback in galaxies. Central to these issues is an understanding of the evolution of massive stars, and the critical role of mass loss via strongly structured winds and stellar binarity. Ultimately, massive stellar clusters shape the structure and energetics of galaxies. We aim to conduct high-resolution, deep field mapping at 21cm of the core of the massive Cygnus OB2 association and to characterise the properties of the massive stars and colliding winds at this waveband. We used seven stations of the e-MERLIN radio facility, with its upgraded bandwidth and enhanced sensitivity to conduct a 21cm census of Cygnus OB2. Based on 42 hours of observations, seven overlapping pointings were employed over multiple epochs during 2014 resulting in 1 sigma sensitivities down to ~21microJy and a resolution of ~180mas. A total of 61 sources are detected at 21cm over a ~0.48deg x 0.48deg region centred on the heart of the Cyg OB2 association. Of these 61 sources, 33 are detected for the first time. We detect a number of previously identified sources including four massive stellar binary systems, two YSOs, and several known X-ray and radio sources. We also detect the LBV candidate (possible binary system) and blue hypergiant (BHG) star of Cyg OB2 #12. The 21cm observations secured in the COBRaS Legacy project provide data to constrain conditions in the outer wind regions of massive stars; determine the non-thermal properties of massive interacting binaries; examine evidence for transient sources, including those associated with young stellar objects; and provide unidentified sources that merit follow-up observations. The 21cm data are of lasting value and will serve in combination with other key surveys of Cyg OB2.

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arXiv2019

A Radio Census of the Massive Stellar Cluster Westerlund 1

Massive stars and their stellar winds are important for a number of feedback processes. The mass lost in the stellar wind can help determine the end-point of the star as a NS or a BH. However, the impact of mass-loss on the post-Main Sequence evolutionary stage of massive stars is not well understood. Westerlund 1 is an ideal astrophysical laboratory in which to study massive stars and their winds in great detail over a large range of different evolutionary phases. Aims: We aim to study the radio emission from Westerlund 1, in order to measure radio fluxes from the population of massive stars, and determine mass-loss rates and spectral indices where possible. Methods: Observations were carried out in 2015 and 2016 with the Australia telescope compact array (ATCA) at 5.5 and 9 GHz using multiple configurations, with maximum baselines ranging from 750m to 6km. Results: 30 stars were detected in the radio from the fully concatenated dataset, 10 of which were WRs (predominantly late type WN stars), 5 YHGs, 4 RSGs, 1 LBV star, the sgB[e] star W9, and several O and B supergiants. New source detections in the radio were found for 5 WR stars, and 5 OB supergiants. These detections have led to evidence for 3 new OB supergiant binary candidates, inferred from derived spectral index limits. Conclusions: Spectral indices and index limits were determined for massive stars in Westerlund 1. For cluster members found to have partially optically thick emission, mass-loss rates were calculated. Under the approximation of a thermally emitting stellar wind and a steady mass-loss rate, clumping ratios were then estimated for 8 WRs. Diffuse radio emission was detected throughout the cluster. Detections of knots of radio emission with no known stellar counterparts indicate the highly clumped structure of this intra-cluster medium, likely shaped by a dense cluster wind.

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arXiv2018

Diffuse X-ray emission in the Cygnus OB2 association

We present a large-scale study of diffuse X-ray emission in the nearby massive stellar association Cygnus OB2 as part of the Chandra Cygnus OB2 Legacy Program. We used 40 Chandra X-ray ACIS-I observations covering $\sim$1.0 deg$^2$. After removing 7924 point-like sources detected in our survey, background-corrected X-ray emission, the adaptive smoothing reveals large-scale diffuse X-ray emission. Diffuse emission was detected in the sub-bands Soft [0.5 : 1.2] and Medium [1.2 : 2.5], and marginally in the Hard [2.5 : 7.0] keV band. From X-ray spectral analysis of stacked spectra we compute a total [0.5 : 7.0 keV] diffuse X-ray luminosity of L$_{\rm x}^{\rm diff}\approx$4.2$\times$10$^{\rm 34}$ erg s$^{-1}$, characterized with plasma temperature components at kT$\approx$ 0.11, 0.40 and 1.18 keV, respectively. The HI absorption column density corresponding to these temperatures has a distribution consistent with N$_{\rm H}$ = 0.43, 0.80 and 1.39 $\times$10$^{22}$ cm$^{-2}$. The extended medium band energy emission likely arises from O-type stellar winds thermalized by wind-wind collisions in the most populated regions of the association, while the soft band emission probably arises from less energetic termination shocks against the surrounding Interstellar-Medium. Super-soft and Soft diffuse emission appears more widely dispersed and intense than the medium band emission. The diffuse X-ray emission is generally spatially coincident with low-extinction regions that we attribute to the ubiquitous influence of powerful stellar winds from massive stars and their interaction with the local Interstellar-Medium. Diffuse X-ray emission is volume-filling, rather than edge-brightened, oppositely to other star-forming regions. We reveal the first observational evidence of X-ray haloes around some evolved massive stars.

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arXiv2018

An ALMA 3mm continuum census of Westerlund 1

Massive stars play an important role in both cluster and galactic evolution and the rate at which they lose mass is a key driver of both their own evolution and their interaction with the environment up to and including their SNe explosions. Young massive clusters provide an ideal opportunity to study a co-eval population of massive stars. We performed 3mm continuum observations with the Atacama Large Millimetre/submillimetre Array of the Galactic cluster Westerlund 1, to study the constituent massive stars and determine mass-loss rates for the diverse post-main sequence population. We detected emission from 50 stars in Westerlund 1, comprising all 21 Wolf-Rayets within the field of view, eight cool and 21 OB super-/hypergiants. Emission nebulae were associated with a number of the cool hypergiants while, unexpectedly, a number of hot stars also appear spatially resolved. We measured the mass-loss rates for a unique population of massive post-main sequence stars at every stage of evolution, confirming a significant increase as stars transition from OB supergiant to WR states. The range of spectral types exhibited provides a critical test of radiatively driven wind theory and the reality of the bi-stability jump. The extreme mass-loss rate inferred for the interacting binary Wd1-9 in comparison to other cluster members confirmed the key role binarity plays in massive stellar evolution. The presence of compact nebulae around a number of OB and WR stars is unexpected; by analogy to the cool super-/hypergiants we attribute this to confinement and sculpting of the stellar wind via interaction with the intra-cluster medium/wind. Given the morphology of core collapse SNe depend on the nature of the pre-explosion circumstellar environment, if this hypothesis is correct then the properties of the explosion depend not just on the progenitor, but also the environment in which it is located.

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arXiv2018

Asymmetric Ejecta of Cool Supergiants and Hypergiants in the Massive Cluster Westerlund 1

We report new 5.5 GHz radio observations of the massive star cluster Westerlund 1, taken by the Australia Telescope Compact Array, detecting nine of the ten yellow hypergiants (YHGs) and red supergiants (RSGs) within the cluster. Eight of nine sources are spatially resolved. The nebulae associated with the YHGs Wd1-4a, -12a and -265 demonstrate a cometary morphology - the first time this phenomenon has been observed for such stars. This structure is also echoed in the ejecta of the RSGs Wd1-20 and -26; in each case the cometary tails are directed away from the cluster core. The nebular emission around the RSG Wd1-237 is less collimated than these systems but once again appears more prominent in the hemisphere facing the cluster. Considered as a whole, the nebular morphologies provide compelling evidence for sculpting via a physical agent associated with Westerlund 1, such as a cluster wind.

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arXiv2017

Variable millimetre radiation from the colliding-wind binary Cyg OB2 #8A

In the colliding-wind region of massive binaries, non-thermal radio emission occurs. This non-thermal radio emission (due to synchrotron radiation) has so far been observed at centimetre wavelengths. At millimetre wavelengths, the stellar winds and the colliding-wind region emit more thermal free-free radiation, and it is expected that any non-thermal contribution will be difficult or impossible to detect. We aim to determine if the material in the colliding-wind region contributes substantially to the observed millimetre fluxes of a colliding-wind binary. We also try to distinguish the synchrotron emission from the free-free emission. We monitored the massive binary Cyg OB2 #8A at 3 mm with the NOrthern Extended Millimeter Array (NOEMA) interferometer of the Institut de Radioastronomie Millimetrique (IRAM). The data were collected in 14 separate observing runs (in 2014 and 2016), and provide good coverage of the orbital period. The observed millimetre fluxes range between 1.1 and 2.3 mJy, and show phase-locked variability, clearly indicating that a large part of the emission is due to the colliding-wind region. A simple synchrotron model gives fluxes with the correct order of magnitude, but with a maximum that is phase-shifted with respect to the observations. Qualitatively this phase shift can be explained by our neglect of orbital motion on the shape of the colliding-wind region. A model using only free-free emission results in only a slightly worse explanation of the observations. Additionally, on the map of our observations we also detect the O6.5 III star Cyg OB2 #8B, for which we determine a 3 mm flux of 0.21 +- 0.033 mJy. The question of whether synchrotron radiation or free-free emission dominates the millimetre fluxes of Cyg OB2 #8A remains open. More detailed modelling of this system, based on solving the hydrodynamical equations, is required to give a definite answer.

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arXiv2017

Mass loss rates from mid-IR excesses in LMC and SMC O stars

We use a combination of BVJHK and Spitzer [3.6], [5.8] and [8.0] photometry to determine IR excesses for a sample of 58 LMC and 46 SMC O stars. This sample is ideal for determining IR excesses because the very small line of sight reddening minimizes uncertainties due to extinction corrections. We use the core-halo model developed by Lamers & Waters (1984a) to translate the excesses into mass loss rates and demonstrate that the results of this simple model agree with the more sophisticated CMFGEN models to within a factor of 2. Taken at face value, the derived mass loss rates are larger than those predicted by Vink et al. (2001), and the magnitude of the disagreement increases with decreasing luminosity. However, the IR excesses need not imply large mass loss rates. Instead, we argue that they probably indicate that the outer atmospheres of O stars contain complex structures and that their winds are launched with much smaller velocity gradients than normally assumed. If this is the case, it could affect the theoretical and observational interpretations of the "weak wind" problem, where classical mass loss indicators suggest that the mass loss rates of lower luminosity O stars are far less than expected.

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arXiv2016

ALMA observations of the supergiant B[e] star Wd1-9

Mass-loss in massive stars plays a critical role in their evolution, although the precise mechanism(s) responsible - radiatively driven winds, impulsive ejection and/or binary interaction -remain uncertain. In this paper we present ALMA line and continuum observations of the supergiant B[e] star Wd1-9, a massive post-Main Sequence object located within the starburst cluster Westerlund 1. We find it to be one of the brightest stellar point sources in the sky at millimetre wavelengths, with (serendipitously identified) emission in the H41alpha radio recombination line. We attribute these properties to a low velocity (~100 km/s) ionised wind, with an extreme mass-loss rate 6.4x10^-5(d/5kpc)^1.5 Msol/yr. External to this is an extended aspherical ejection nebula indicative of a prior phase of significant mass-loss. Taken together, the millimetre properties of Wd1-9 show a remarkable similarity to those of the highly luminous stellar source MWC349A.We conclude that these objects are interacting binaries evolving away from the main sequence and undergoing rapid case-A mass transfer. As such they - and by extension the wider class of supergiant B[e] stars - may provide a unique window into the physics of a process that shapes the life-cycle of ~70% of massive stars found in binary systems.

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arXiv2015

On the Origin of Wind Line Variability in O Stars

We analyze 10 UV time series for 5 stars which fulfill specific sampling and spectral criteria to constrain the origin of large-scale wind structure in O stars. We argue that excited state lines must arise close to the stellar surface and are an excellent diagnostic complement to resonance lines which, due to radiative transfer effects, rarely show variability at low velocity. Consequently, we splice dynamic spectra of the excited state line, N IV1718, at low velocity to those of 1393 component of the Si IV 1400 doublet at high velocity in order to examine the temporal evolution of wind line features. These spliced time series reveal that nearly all of the features observed in the time series originate at or very near the stellar surface. Further, we positively identify the observational signature of equatorial co-rotating interaction regions in two of the five stars and possibly two others. In addition, we see no evidence for features originating further out in the wind. We use our results to consolidate the fact that the features seen in dynamic spectra must be huge, in order to remain in the line of sight for days, persisting to very large velocity and that the photospheric footprint of the features must also be quite large, ~ 15 - 20% of the stellar diameter.

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arXiv2014

CIR Modulation of the X-ray Flux from the O7.5 III(n)((f)) Star xi Persei?

We analyze a 162 ks HETG Chandra observation of the O7.5 III(n)((f)) star xi Per, together with contemporaneous H alpha observations. The X-ray spectrum of this star is similar to other single O stars, and not pathological in any way. Its UV wind lines are known to display cyclical time variability, with a period of 2.086 days, which is thought to be associated with co-rotating interaction regions (CIRs). We examine the Chandra and H alpha data for variability on this time scale. We find that the X-rays vary by about 15% over the course of the observations and that this variability is out of phase with variable absorption on the blue wing of the H alpha profiles (assumed to be a surrogate for the UV absorption associated with CIRs). While not conclusive, both sets of data are consistent with models where the CIRs are either a source of X-rays or modulate them.

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arXiv2014

X-ray emission from massive stars in Cyg OB2

We report on the analysis of the Chandra-ACIS data of O, B and WR stars in the young association Cyg OB2. X-ray spectra of 49 O-stars, 54 B-stars and 3 WR-stars are analyzed and for the brighter sources, the epoch dependence of the X-ray fluxes is investigated. The O-stars in Cyg\,OB2 follow a well-defined scaling relation between their X-ray and bolometric luminosities: log(Lx/Lbol) = -7.2 +/- 0.2. This relation is in excellent agreement with the one previously derived for the Carina OB1 association. Except for the brightest O-star binaries, there is no general X-ray overluminosity due to colliding winds in O-star binaries. Roughly half of the known B-stars in the surveyed field are detected, but they fail to display a clear relationship between Lx and Lbol. Out of the three WR stars in Cyg OB2, probably only WR144 is itself responsible for the observed level of X-ray emission, at a very low log(Lx/Lbol) = -8.8 +/- 0.2. The X-ray emission of the other two WR-stars (WR145 and 146) is most probably due to their O-type companion along with a moderate contribution from a wind-wind interaction zone.

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arXiv2013

Kepler photometry and optical spectroscopy of the ZZ Lep central star of the Planetary Nebula NGC 6826: rotational and wind variability

We present three years of long-cadence and over one year of short-cadence photometry of the central star of the Planetary Nebula NGC 6826 obtained with the Kepler spacecraft, and temporally coinciding optical spectroscopy. The light curves are dominated by incoherent variability on time scales of several hours, but contain a lower-amplitude periodicity of 1.23799 d. The temporal amplitude and shape changes of this signal are best explicable with a rotational modulation, and are not consistent with a binary interpretation. We argue that we do not observe stellar pulsations within the limitations of our data, and show that a binary central star with an orbital period less than seven days could only have escaped our detection in the case of low orbital inclination. Combining the photometric and spectroscopic evidence, we reason that the hourly variations are due to a variable stellar wind, and are global in nature. The physical cause of the wind variability of NGC 6826 and other ZZ Leporis stars is likely related to the mechanism responsible for wind variations in massive hot stars.

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arXiv2012

The 2.35 year itch of Cyg OB2 #9. II. Radio monitoring

Cyg OB2 #9 is one of a small set of non-thermal radio emitting massive O-star binaries. The non-thermal radiation is due to synchrotron emission in the colliding-wind region. Cyg OB2 #9 was only recently discovered to be a binary system and a multi-wavelength campaign was organized to study its 2011 periastron passage. We report here on the results of the radio observations obtained in this monitoring campaign. We used the Expanded Very Large Array (EVLA) radio interferometer to obtain 6 and 20 cm continuum fluxes. The observed radio light curve shows a steep drop in flux sometime before periastron. The fluxes drop to a level that is comparable to the expected free-free emission from the stellar winds, suggesting that the non-thermal emitting region is completely hidden at that time. After periastron passage, the fluxes slowly increase. We introduce a simple model to solve the radiative transfer in the stellar winds and the colliding-wind region, and thus determine the expected behaviour of the radio light curve. From the asymmetry of the light curve, we show that the primary has the stronger wind. This is somewhat unexpected if we use the astrophysical parameters based on theoretical calibrations. But it becomes entirely feasible if we take into account that a given spectral type - luminosity class combination covers a range of astrophysical parameters. The colliding-wind region also contributes to the free-free emission, which can help to explain the high values of the spectral index seen after periastron passage. Combining our data with older Very Large Array (VLA) data allows us to derive a period P = 860.0 +- 3.7 days for this system. With this period, we update the orbital parameters that were derived in the first paper of this series.

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