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Magnetic fields and the variable wind of the early-type supergiant β Ori by Matthew Eric Shultz A thesis submitted to the Department of Physics, Engineering Physics and Astronomy in conformity with the requirements for the degree of Master of Science Queen’s University Kingston, Ontario, Canada April 2012 c Matthew Eric Shultz, 2012 Copyright Abstract Supergiant stars of spectral types B and A are characterized by variable and structured winds, as revealed by variability of optical and ultraviolet spectral lines. Nonradial pulsations and magnetically supported loops have been proposed as explanations for these phenomena. The latter hypothesis is tested using a time series of 65 high-resolution (λ/∆λ ∼ 65, 000) circular polarization (Stokes I and V ) spectra of the late B type supergiant Rigel (β Ori, B8 Iae), obtained with the instruments ESPaDOnS and Narval at the Canada-France-Hawaii Telescope and the Bernard Lyot Telescope, respectively. Examination of the unpolarized (Stokes I) spectra using standard spectral analysis tools confirms complex line profile variability during the 5 month period of observations; the high spectral resolution allows the identification of a weak, transient Hα feature similar in behaviour to a High Velocity Absorption event. Analysis of the Stokes V spectra using the cross-correlation technique Least Squares Deconvolution (LSD) yields no evidence of a magnetic field in either LSD Stokes V profiles or longitudinal field measurements, with longitudinal field 1σ error bars of ∼ 12 G for individual observations, and a mean field in the best observed period of 3 ± 2 G. Synthetic LSD profiles fit to the observations using a Monte Carlo approach yield an upper limit on the surface dipolar field strength of Bdip ≤ 50 G for most orientations of the rotational and magnetic axes, lowered to Bdip . 35 G if the mean LSD profile from the most densely time-sampled epoch (with an LSD SNR of ∼80,000) is used. A simple two-spot geometry representing the footpoints of a magnetic loop emerging from the photosphere yields upper limits on the spot magnetic fields of 60–600 G, depending on the filling factor of the spots. Given existing measurements of the mass loss rate and the wind terminal velocity, these results cannot rule out a magnetically confined wind as, for Bdip & 15 G, η∗ ≥ 1. However, the detailed pattern of line profile variability seems inconsistent with the periodic wind modulation characteristic of known magnetic early-type stars, suggesting that magnetic fields do not play a dominant role in Rigel’s variable winds. ii Acknowledgements I would first like to thank my supervisor, Gregg Wade, without whose guidance and most especially patience this work could not have proceeded. This work would not have been possible without numerous illuminating discussions with Jason Grunhut and Véronique Petit, who in addition with providing a great deal of useful software, were invaluable sources of knowledge regarding IDL programming; the assistance of James Silvester in the construction of line masks; and the advice and support provided by numerous members of the MiMeS Collaboration. This research used the facilities of the Canadian Astronomy Data Centre operated by the National Research Council of Canada with the support of the Canadian Space Agency, and those of the Vienna Atomic Line Database. I’d like to thank the dedicated work of staff at the Canada-France-Hawaii Telescope and the Bernard Lyot Telescope, who collected the observations analyzed here; a special thanks to Coralie Neiner, who was very patient with my efforts to access the Paris MiMeS server. Completion of this thesis was made possible with financial support from Queen’s University and the Natural Sciences and Engineering Research Council of Canada (NSERC). iii Statement of Originality The original work presented in this thesis comprises the spectral time series analysis described in Chapter 3 (radial velocity and equivalent width measurements, dynamic spectra, and temporal variance spectra), and the magnetic analysis described in Chapters 4 and 5 (extraction of LSD profiles and longitudinal field measurements, statistical analysis, and modeling of those profiles). Software written in the course of this thesis includes the radial velocity and equivalent width measurement programs (Chapter 3); the Temporal Variance Spectra program (Chapter 3); and the Stokes V LSD profile modeling software (Chapter 5). iv Table of Contents Abstract i Acknowledgements iii Table of Contents v List of Tables vii List of Figures viii Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 The lives and deaths of hot, massive stars . . . . . . . . . . . . . . . 1 1.2 The winds of massive stars . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Massive stars with magnetic fields . . . . . . . . . . . . . . . . . . . . 15 1.4 Magnetically confined winds . . . . . . . . . . . . . . . . . . . . . . . 22 1.5 BA supergiants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.6 Rigel in this context . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 Chapter 2: Observations . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.1 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v 53 2.2 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 Chapter 3: Spectroscopic Measurements and Analysis . . . . . . . . 66 3.1 Radial Velocities and Equivalent Widths . . . . . . . . . . . . . . . . 68 3.2 Dynamic Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3 Temporal Variance Spectra . . . . . . . . . . . . . . . . . . . . . . . . 80 3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 Chapter 4: Magnetic Field Diagnosis . . . . . . . . . . . . . . . . . . 86 4.1 Detection and diagnosis of magnetic fields using the Zeeman effect . . 86 4.2 Least Squares Deconvolution . . . . . . . . . . . . . . . . . . . . . . . 89 4.3 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Chapter 5: Modeling and Upper Limits . . . . . . . . . . . . . . . . . 107 5.1 Disk Integration and Synthetic LSD Profiles . . . . . . . . . . . . . . 108 5.2 Interpretation of measurements and upper limits . . . . . . . . . . . . 115 Chapter 6: Discussion and Conclusions . . . . . . . . . . . . . . . . . 127 References . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 vi List of Tables 1.1 Summary of Rigel’s Paramaters . . . . . . . . . . . . . . . . . . . . . 50 1.2 Summary of Rigel’s Wind Parameters . . . . . . . . . . . . . . . . . . 51 2.1 ESPaDoNS observations . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.2 Narval observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1 Radial velocity line list . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1 ESPaDoNS LSD Statistics and Longitudinal Field Measurements . . 98 4.2 Narval LSD Statistics and Longitudinal Field Measurements . . . . . 99 4.3 Longitudinal field measurements (coadded LSD profiles) 6.1 Upper limits for magnetic wind confinement . . . . . . . . . . . . . . 129 vii . . . . . . . 105 List of Figures 1.1 The Soul Nebula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Rigel (artist’s impression) . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 P Cygni profile formation . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 Wind line forming regions . . . . . . . . . . . . . . . . . . . . . . . . 13 1.5 Spherically symmetric wind model . . . . . . . . . . . . . . . . . . . . 14 1.6 HD 191612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.7 Fossil field stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 1.8 Magnetic wind confinement . . . . . . . . . . . . . . . . . . . . . . . 23 1.9 Equatorial magnetic wind confinement . . . . . . . . . . . . . . . . . 24 1.10 RFHD model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.11 Field aligned rotation of magnetically confined wind . . . . . . . . . . 26 1.12 η∗ vs W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.13 Hα dynamic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.14 Temporal variance spectra . . . . . . . . . . . . . . . . . . . . . . . . 35 1.15 HVA dynamic spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.16 Magnetic loop vs. CIR . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.17 Witch Head Nebula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.1 54 ESPaDOnS echelle orders . . . . . . . . . . . . . . . . . . . . . . . . viii 2.2 ESPaDOnS Stokes I and V spectrum of ξ ! CMa . . . . . . . . . . . . 58 2.3 MOST photometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.4 ESPaDOnS Stokes I spectrum . . . . . . . . . . . . . . . . . . . . . . 63 3.1 Hα . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.2 Radial velocity histograms . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Radial velocities and equivalent widths . . . . . . . . . . . . . . . . . 72 3.4 Radial velocities and equivalent widths . . . . . . . . . . . . . . . . . 73 3.5 Dynamic spectra (Balmer lines) . . . . . . . . . . . . . . . . . . . . . 75 3.6 Dynamic spectra (metal lines) . . . . . . . . . . . . . . . . . . . . . . 77 3.7 Hα and Si ii dynamic spectra . . . . . . . . . . . . . . . . . . . . . . 79 3.8 Temporal variance spectra . . . . . . . . . . . . . . . . . . . . . . . . 80 3.9 Temporal variance spectra (50-day bins) . . . . . . . . . . . . . . . . 82 3.10 Peak variability of Hα . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.1 Zeeman splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.2 LSD model spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.3 LSD profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.4 Stokes V LSD profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.5 Stokes V LSD profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.6 Stokes V LSD profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.7 Stokes V LSD profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.8 Longitudinal field measurements . . . . . . . . . . . . . . . . . . . . . 101 4.9 Kolmogorov-Smirnov test . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.10 Co-added LSD profiles . . . . . . . . . . . . . . . . . . . . . . . . . . 106 ix 5.1 Stellar disk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Synthetic Stokes V profiles from a disk-integrated magnetic field model 113 5.3 Probability distribution function for dipolar field strength upper limits 117 5.4 Synthetic profiles for dipolar field . . . . . . . . . . . . . . . . . . . . 118 5.5 Upper limits for dipolar magnetic field (nightly means) . . . . . . . . 123 5.6 Synthetic Stokes V profiles from for spotted field model . . . . . . . . 124 5.7 Upper Limits for dipolar and spot magnetic field (grand mean) . . . . 125 6.1 η∗ vs W . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 6.2 Lower limits for Fe convection zone spots . . . . . . . . . . . . . . . . 134 x Glossary AST: Automatic Spectroscopic Telescope BA SG: A supergiant star of spectral type B or A CFHT: Canada-France-Hawaii Telescope CIR: Corotating Interaction Region DAC: Discrete Absorption Component FeCZ: Fe opacity bump convection zone Hα: The hydrogen Balmer line n = 3 − 2 energy level transition HVA: High Velocity Absorption event HJD: Heliocentric Julian Date; the Julian date adjusted to the heliocentric reference frame. ISM: Interstellar Medium LPV: Line Profile Variability LSD: Least Squares Deconvolution LTE: Local Thermodynamic Equilibrium MOST: Microvariability and Oscillations in STars space telescope NIR: Near Infrared NLTE: Non-Local Thermodynamic Equilibrium NRP: Nonradial Pulsation xi xii NUV: Near Ultraviolet RSG: Red Supergiant, a supergiant star of spectral type K or M SED: Spectral Energy Distribution SG: Supergiant SNR: Signal to Noise Ratio TBL: Télescope Bernard Lyot, Pic du Midi Observatory TVS: Temporal Variance Spectrum ZAMS: Zero Age Main Sequence: the line on the H-R diagram at which hydrogen fusion begins. Chapter 1 Introduction 1.1 The lives and deaths of hot, massive stars Hot, massive stars have been described as the rock stars of the universe: they live fast and loud, die young in spectacular fashion, and their influence is out of all proportion to their numbers. The arms of spiral galaxies possess their characteristic blue color due to massive stars, despite the fact that they are outnumbered by orders of magnitude by their smaller, dimmer, redder kin. They are ‘cosmic engines’ (Bresolin et al., 2008): the sources of the majority of ionizing radiation in the interstellar medium (ISM); the furnaces in which the majority of the atomic elements are forged; and with their powerful stellar winds and even more powerful supernovae, a critical sculptor of the interstellar medium. After death, such stars continue to be of intense astrophysical interest, both for the black holes and neutron stars born in the incredible compressive force generated by the collapse of their stellar cores, and for the richly structured supernova remnants left behind in the wake of their supernovae. In the context of this thesis a massive star is one with an initial stellar mass M∗ & 1 CHAPTER 1. INTRODUCTION 2 8 M⊙ , where M⊙ is the solar mass. Such stars begin their main sequence (hydrogenburning) lives with a spectral type of O or B, with effective temperatures Teff ranging from 25, 000 − 50, 000 K, significantly hotter than the solar value of Teff ≃ 5780 K. Their high temperatures cause their spectral energy distributions (SEDs) to peak in intensity at blue or ultraviolet wavelengths, and at far a greater intensity than can be achieved by less massive stars: a main sequence OB star might have a bolometric luminosity of 103 solar luminosities L⊙ (for a B3 star) up to 105 L⊙ (e.g. ζ Puppis, O5Ia), rivaling entire globular clusters of 105 − 106 stars in integrated light. Thus, while OB stars make up a relatively low fraction of the stellar mass of spiral galaxies, and are insignificant by number within any stellar population, they dominate the light-to-mass ratio. The extraordinary temperatures and luminosities of OB stars are a consequence of the rapidity with which fusion takes place within their cores. This results in relatively short lifetimes despite a much greater initial allotment of fuel: while yellow dwarf stars such as the Sun burn hydrogen for billions of years on the main sequence, OB stars fully consume their H fuel and begin their evolution into red supergiants after mere millions of years. With the shortest ‘generations’ of any spectral type, measurements of their spatial distribution and chemical abundances track the most recent epochs of a galaxy’s star formation history. Such studies must be accompanied by an understanding of stellar evolution both on and after the main sequence. A massive star begins as a late O star, and burns over several Myr on the main sequence, cooling while increasing in luminosity; following core H exhaustion, the core contracts while the atmosphere expands and cools, increasing in radius from ∼ 6 R⊙ to ∼100 R⊙ and evolving over the subsequent CHAPTER 1. INTRODUCTION 3 ∼100,000 years or so from a late O supergiant to an early M type Ia supergiant with Teff ∼3500 K and a radius of ∼1100 R⊙ . Unlike the Sun and other cool stars, for which this traversal of the Hertzsprung-Russell diagram represents the penultimate stage before finishing their lives as white dwarves, OB stars with masses 9 M⊙ . M∗ . 40 M⊙ gain a new lease on life by continuing nuclear fusion well past the exhaustion of core H by burning nuclei of higher atomic numbers at ever higher temperatures in their cores, with the fusion of lighter elements continuing above in concentric shells. During the ∼100,000 years following ignition of the He core, the star travels back along the H-R diagram, increasing in effective temperature as its core re-expands and its atmosphere contracts. The He burning phase lasts for another ∼400,000 years after the star first returns to the approximate temperature and luminosity it possessed at the time of H exhaustion. When He is exhausted, little time remains: the C core will exhaust itself in a mere 30,000 years, and subsequent cores in progressively less time (Meynet et al., 2011). The process culminates with the formation of an Fe core, since Fe fusion is endothermic i.e. no net energy can be liberated through this process. With no radiation pressure to support the star against its own gravity, it detonates in a catastrophic, core collapse (type II) supernova. The energy unleashed in this final explosion (around 1050 ergs, equivalent to approximately 1 Gyr of solar radiance) provides the energy both to synthesize the higher-numbered elements of the periodic table, and to scatter the elements synthesized during the supernova and in the star’s lifetime throughout the local interstellar medium, expelling the matter with a velocity of up to 10% of the speed of light, c. Depending on the mass of the progenitor star, a neutron star (if M∗ . 20 M⊙ ) or a black hole may be born (see e.g. Heger et al., 2003). CHAPTER 1. INTRODUCTION 4 It has long been thought that the momentum unleashed by supernovae is a powerful influence on the ISM. Even as supernovae shockfronts enrich the ISM chemically, they also heat and displace it, carving an expanding bubble denuded of star-forming material while compressing the matter ahead (see Fig. 1.1). Where the nebulae are disrupted this of course quenches star formation, however the overlapping shockfronts trigger new bursts of star formation in compressed material. Thus, in addition to tracing star formation history, OB stars are themselves an important regulator of star formation. Precisely how much of their influence is due to supernovae, and how much due to radiation and winds, is a matter of debate: while early models assumed that the dramatic energy release of supernovae must be the most important contribution of OB stars to the star formation process, detailed modeling of turbulence in giant molecular clouds including the action of stellar winds by Harper-Clark & Murray (2011) has suggested that in fact it is the steady action of their supersonic winds which plays the most important role. Thus it may be not primarily in death, but from their births and throughout their lives that they act to sculpt and disrupt the star forming regions of galaxies. 1.1.1 Scope of the present study There is as yet no confirmed detection of a magnetic field in a BA supergiant. There are additionally no a priori reasons within the conventional theory of stellar evolution to expect that such stars will possess surface magnetic fields: their radiative envelopes should not allow magnetic dynamos to form, and the rise time of flux tubes from the convective cores (where significant fields are expected) is greater than the CHAPTER 1. INTRODUCTION 5 Figure 1.1: Spitzer Space Telescope image of IC 1848, the Soul Nebula in Cassiopeia. Blue (3.6 µm) and green (8 µm) show molecular clouds; red (24 µm) shows heated dust. The image spans ∼2000 ly (600 pc). The open cluster contains several late O-type stars (O9–O6) and a handful of B-type stars (B9–B0). Note the cavities surrounding the brightest stars (Koenig et al., 2008). CHAPTER 1. INTRODUCTION 6 Figure 1.2: This artist’s impression of Rigel shows its two binary companions framed within a large coronal loop, which is inferred from HVA activity. As discussed in the text, the β Ori BC binary system is quite far (2500 AU) from β Ori A, however as B9V stars they are quite luminous in their own right. The mottled appearance reflects the presence of low-amplitude, very high order nonradial pulsations. Sulehria (2005). CHAPTER 1. INTRODUCTION 7 estimated lifetimes of these stars. However, thin convective regions within the radiative envelope may form just beneath the stellar surface, thus allowing dynamos, from which weak magnetic fields might penetrate the photosphere and become detectable (Cantiello et al., 2009). A star of Rigel’s parameters is predicted to possess such an Fe convection zone (FeCZ). Furthermore, Cantiello & Braithwaite (2011) provide preliminary predictions that such an FeCZ in a star of 20 M⊙ and solar metallicity would produce a surface magnetic field of at least 10 G. At the same time, magnetic fields on the order of those predicted by the FeCZ model have been proposed by numerous authors (Kaufer et al. (1996b), Israelian et al. (1997), Markova et al. (2008)) as being necessary to support the corotating structures speculated to give rise to high velocity variability spectral lines which probe the base of the stellar wind, which offer circumstantial evidence for the hypothesis that weak or complex magnetic fields might in fact be present in the photospheres of BA supergiants. While early magnetometry seemed to offer evidence that Rigel possesses a magnetic field of ∼ 130 G (Severny, 1970), comfortably able to contain these possibilities, these results were not reproduced in further observations (Severny et al., 1974). Such fields are unlikely to be detectable in only a single spectrum, and the high apparent magnitude of β Orionis A means that spectropolarimetric measurements can be taken with exposure times of only a few seconds (rather than on the order of an hour for fainter objects, not at all uncommon in measurements of this kind). At the same time, the relatively low v sin i of the star leads to sharp spectral lines, thus increasing the information available to the Least Squares Deconvolution procedure by which the Zeeman signatures of magnetic fields are most reliably determined. With this in CHAPTER 1. INTRODUCTION 8 mind, in the context of the Magnetism in Massive Stars (MiMeS) large program, over a period of several months 66 high-resolution (R∼65000 at 550 nm) circular polarization (Stokes V ) spectra covering the entirety of the visual spectrum (370–1000 nm) were obtained of Rigel. The analysis of these spectroscopic and spectropolarimetric data, and the constraints they place on the magnetic field geometry of Rigel, are the subject of the present work. 1.2 The winds of massive stars The solar wind is thought to be a consequence of the expansion of 106 K plasma within the corona due to energy transferred from photospheric convection cells. The solar wind is optically quite thin, and had it not been for in situ measurement by spacecraft, might never have been detected directly (Owocki, 2001). Since it removes only a trickle of mass from the Sun, the solar wind decreases the mass of the Sun by an insignificant 10−4 M⊙ over the course of its main sequence life (Owocki, 2001); expressed in terms of annual rate of mass loss Ṁ, the wind mass loss rate is a mere Ṁ ≃ 10−14 M⊙ yr−1 . While of great interest in the study of space plasmas, such winds are almost invisible in Sun-like stars, and relatively unimportant in their main sequence evolution (although they become strong enough to remove a substantial fraction of the stellar mass during the red giant phase). The fundamental physics, as well as the gross properties, of the winds of earlytype stars are quite distinct. The luminosity of OB stars is so great that photon momentum alone, imparted to ions in the outer stellar atmosphere, accelerates the ions out of the gravity well and, ultimately, into the ISM (a concept first suggested by Milne, 1926). Since the essential driving mechanism is the coupling between photon CHAPTER 1. INTRODUCTION 9 momentum and atomic absorption lines, such mass flows are known as radiatively driven or ‘line-driven’ winds. The line-driven winds of OB stars are of far more significance to their evolution than is the solar wind to the Sun’s. They are much denser than the winds of Sunlike stars, with mass loss rates of 10−10 M⊙ yr−1 . Ṁ . 10−5 M⊙ yr−1 , and also much faster, with wind terminal velocities v∞ ≃ 1000 − 3000 km s−1 as compared to 400–700 km s−1 for cooler stars. These supersonic winds are able to remove a substantial fraction of the stellar mass, introducing significant modifications into both the duration of their lives on the main sequence, and the subsequent course of their evolution into blue and/or red supergiants. Particularly massive stars (M∗ & 30 M⊙ , Meynet et al., 2011) evolve into a Wolf-Rayet phase near the end of their lives, during which the winds become so strong that the photosphere essentially lies in the wind itself, manifesting in a spectrum rich in wind-broadened emission lines and abundant heavy elements (Owocki, 2010); these winds rapidly strip the H envelope of the star, exposing the nuclear-processed material within, and may be augmented by a brief (years to decades) Luminous Blue Variable stage during which mass loss rates can reach Ṁ ∼ 0.1 − 1 M⊙ yr−1 (Owocki, 2010). There are two key process enabling line-driven winds to reach such phenomenal strengths: the first is the bound-bound scattering of photons from electrons, and the second, the Doppler-shifting of the energy level transitions by the bulk motion of the matter flow (Owocki, 2001). Electrons bound to atoms primarily scatter photons at wavelengths corresponding to available energy level transitions, shuffling back and forth between unoccupied energy levels within the atom. Resonance at these allowed frequencies greatly amplifies the transfer of photon momentum. In a sufficiently CHAPTER 1. INTRODUCTION 10 abundant, motionless medium, the photon flux at frequency bands corresponding to these transitions would quickly become saturated; indeed, this is just what happens within the deeper layers of the star’s radiative envelope. However, in the moving portions of the atmosphere, the energy levels will be redshifted, and so can access the unattenuated stellar flux at lower frequencies. This enables the wind to make much more efficient use of the available photon flux, resulting in more efficient acceleration. In a rapidly accelerating flow, radiation propagates freely until Doppler shifting brings a line into a local resonance. Sobolev (1960) showed that in a supersonic flow the spatial extent of this resonance is much less than typical flow variations in density or velocity, thus enabling key parameters of the line scattering (such as the optical depth) to be described in terms of purely local conditions, without having to solve a non-local spatial integral. This allows a description of the optical depth in terms of the local density and velocity gradients, which, in the optically thick limit, yields a line acceleration that varies in proportion to the local velocity gradient. The simplest models of stellar winds treat the wind as spherically symmetric and time-invariant, yielding an isotropic ‘shell’ of circumstellar material around the star with constant Ṁ within which wind material obeys a velocity law that varies with radius but not time. Under this assumption the mass loss rate Ṁ can be defined (Kudritzki & Puls, 2000) simply as Ṁ = 4π r 2 ρ(r)v(r) (1.1) where r is the distance from the star, ρ(r) is the mass density of the wind and v(r) is the velocity of the wind, 11 CHAPTER 1. INTRODUCTION v(r) = v∞ β R∗ 1−b r (1.2) where v∞ is the asymptotic (terminal) velocity of the wind, R∗ is the stellar radius, b is a constant which fixes the velocity of the inner boundary of the wind to v(r∗ ) (usually the isothermal sound speed) and β is an exponent which, like v∞ , is obtained through spectral fitting (Kudritzki & Puls, 2000). This velocity law is derived from the theory of radiation-driven winds (Castor, Abbot & Klein, 1975; Pauldrach et al., 1986), and empirically justified through the quality of line profile fits. The simple wind model described in Eqns. 1.1 and 1.2 ignores deviations from spherical symmetry due to e.g. clumpiness or wind shocks, for which there is considerable evidence (e.g. Moffat & Robert, 1994; Kaper & Fullerton, 1998; Wolf et al., 1999). However, the simple model is still thought to give a reliable general description of the wind as the amplitudes of such deviations are generally small (even in cases of substantial spectral variability; Kudritzki et al., 1999). Stellar winds are inferred observationally from spectral absorption lines showing a ‘P Cygni’ profile (see Fig. 1.3), in which the red-shifted portion of the line is in emission relative to the photospheric profile. Circumstellar material is illuminated by the star, scattering a certain proportion of its light back into the line of sight and thus raising the overall intensity of the affected spectral lines. However, with a spherically symmetric, isotropic wind, there is a blueshifted absorption component arising from that portion of the wind projected in front of the stellar disk (where line scattering reduces the luminosity), together with an envelope surrounding the disk that is, on average, stationary with respect to the observer (since equal components are traveling towards and away from the line of sight). Since line scattering in the envelope raises CHAPTER 1. INTRODUCTION 12 Figure 1.3: Schematic illustrating formation of a P Cygni profile. Above: superposition of an emission component (symmetric about the rest wavelength) and an absorption component (blue-shifted with respect to the line centre). Below: regions of the wind projected in front of the stellar disk are both entirely blueshifted and in absorption, while the wind surrounding the disk is in emission, with equal proportions shifted to the red and blue halves of the line. the luminosity, it results in an emission component that is symmetrically distributed on the red and blue halves of the spectral line. The P Cygni profile results from the superposition of these components on top of the underlying photospheric profile. The wind terminal velocity, v∞ , is in general measured from the wavelength of the blue edge of the blueshifted absorption, while Ṁ requires more detailed modeling of the line profile. The behaviour of OB stellar winds is most easily studied in ultraviolet resonance lines, although in the optical spectral region the hydrogen Balmer-α line (Hα) is also a useful diagnostic. Different lines probe different regions in the stellar wind, as illustrated in Fig. 1.4. The sensitivity of recombination lines such as Hα, as well as subordinate UV lines, fall off as the square of density, thus providing a probe of CHAPTER 1. INTRODUCTION 13 Figure 1.4: Line forming regions in an OB stellar wind. Reproduced from de Jong et al. (2001). activity near the very base of the wind. The sensitivity of UV resonance lines, on the other hand, declines linearly with density, thus offering a window into activity at much greater circumstellar distances. The free-free or bremsstrahlung emission due to electron scattering also leads to a small excess in the radio and infrared (IR) regions of the spectrum. Excess radio emission is quite weak and has been measured for only a handful of stars (e.g. Drake & Linsky, 1989); the IR excess is somewhat stronger and has been studied in more detail by e.g. Barlow & Cohen (1977). Ṁ can be measured using fits to the infrared and radio continua, ultraviolet resonance lines or recombination lines such as Hα. Fits to these different diagnostics often yield results differing by orders of magnitude. Since the emission lines of OB stars are a result of scattering in the circumstellar environment, polarimetry can also be used to probe the structure of stellar winds. A spherically symmetric wind will result in no net polarization, while inhomogeneities in the wind will lead to a measureable linear polarization. Measuring the polarization angle in the Stokes Q, U plane can thus provide information on the distribution of scattering material within the wind (e.g. Hayes et al. (1986), Ignace et al. (2009)). While spherically symmetric, time-invariant winds are able to account for the CHAPTER 1. INTRODUCTION 14 Figure 1.5: Comparison of Hα profiles of 4 BA Ia stars (green) to synthetic profiles (black) combining photospheric absorption (representing the underlying spectrum of the stellar disk) with a time-invariant, spherically symmetric wind in emission around the disk (except for the portion projected in front). The stars are (right–left) HD 91619, HD 199478, HD 34085 (β Ori), and HD 96919. The generally poor fit of the models is interpreted as indicating time-dependent spatial structure within the wind. Reproduced from Markova et al. (2008b). gross features of spectral lines, in particular the presence of P Cygni profiles, such simplistic models fail to reproduce the detailed features of stellar winds. The obvious unsuitability of a simple, spherically symmetric, non-rotating wind model for late Btype supergiants is illustrated in Fig. 1.5, which compares Hα profiles of 4 such stars to synthetic profiles generated using the NLTE model atmosphere code fastwind (Markova et al., 2008). The variability of BA Ia stars is however particularly complex, and is discussed in greater detail in section 1.5. Many OB stars show evidence for time-variable winds with localized, inhomogeneous structures. Virtually all OB stars show Discrete Absorption Components (DACs) in their UV lines, regions of enhanced absorption which begin in the inner part of the blue absorption trough of P Cygni profiles and, over time, narrow while slowly accelerating to higher blue-shifted velocities (Howarth & Prinja, 1989). DACs are proposed to be the result of Corotating Interaction Regions (CIRs), azimuthally extended regions of enhanced density spiraling out from the photosphere. CIRs, resulting from the interaction of slow and fast streams of plasma, have been a CHAPTER 1. INTRODUCTION 15 known feature of the heliospheric environment since the early 1970s (see e.g. Hundhausen, 1973) and were soon inferred in circumstellar winds as an explanation for the Discrete Absorption Components (DACs) seen in the optical and UV P Cygni profiles of a large fraction of hot stars (first proposed by Mullan, 1984). In his discussion of hot star CIRs, Mullan (1984) showed that due to the difference in v∞ and the velocities of the DACs, CIRs should have a characteristic temperature of order 107 K, thus making them a good candidate to explain the X-ray emission of hot stars. Cranmer & Owocki (1996) performed two-dimensional hydrodynamic simulations of CIRs forming in the wind of a rotating O star, with the CIRs generated by dark and bright spots (as a stand-in for hypothetical photospheric disturbances due to non-radial pulsations or magnetic fields), showing that the bright spots were capable of generating low-density, high speed winds while dark spots lead to high-density, low-speed winds, with the CIRs naturally emerging from the collision between these two streams. The winds of OB stars also show evidence for small-scale turbulent structure in extended saturated absorption troughs. This is often accompanied by X-ray emission, thought to be a result of embedded wind shocks (Owocki, 2010). Clumping is thought to arise due to ‘line-shadowing instabilities’, in which line scattering drives wind material with a much greater acceleration than the mean outward acceleration. 1.3 Massive stars with magnetic fields Detectable magnetism in hot, massive stars appears to be relatively uncommon. Despite tremendous observational resources directed to large-scale surveys since the beginning of the 21st century, with error bars ranging from 15 to 135 G, only a few CHAPTER 1. INTRODUCTION 16 dozen magnetic OB stars have been firmly identified (Donati & Landstreet, 2009; Grunhut et al., 2011). Their rarity notwithstanding, it is apparent that those stars which do host detectable fields have certain common magnetic characteristics. The fields are usually topologically dipolar, with typical strengths of hundreds to thousands of gauss, and there often exists a significant obliquity between the magnetic and rotational axes: this is the ‘oblique rotator’ model. The rotation of the stars themselves seems to be systematically slower than non-magnetic coeval stars of comparable mass (likely due to coupling between the magnetic field and the stellar wind, which sheds angular momentum into the ISM and hastens rotational spin-down). The measured fields are remarkably stable, persisting over many rotational cycles with no detectable secular change of field strength or geometry – even for those stars, such as HD 37776 (Thompson & Landstreet, 1985) or τ Sco (Donati et al., 2006) whose relatively complex fields depart from the general rule of simple dipoles. The stars hosting these fields could not, however, be more diverse: they are both old and young, with strong and weak winds, and rotational periods varying from less than a day to decades. Some possess photospheric chemical peculiarities, others winds or circumstellar matter that interacts with the field, others experience pulsations, and some show combinations, or even all of these properties. While magnetic OB stars are rare, there are subclasses of OB stars that are often magnetic. Amongst B stars, chemically peculiar He-weak stars – particularly those showing enhanced Si or Sr and Ti lines – have been shown to regularly host magnetic fields (Borra, Landstreet & Thompson, 1983; Bohlender, Landstreet & Thompson, 1993). Magnetic fields are also common, and even stronger, in He strong stars (Borra & Landstreet, 1979). In essentially all cases the magnetic fields conform to the oblique CHAPTER 1. INTRODUCTION 17 rotator model. This seems to be a continuation of the trend seen for intermediate mass Ap/Bp stars (Donati & Landstreet, 2009). The Magnetism in Massive Stars (MiMeS) collaboration has conducted a highresolution spectropolarimetric survey of 146 stars with spectral types B3 and hotter, detecting magnetic fields in ∼8% of them: 10/98 B stars and 3/48 O stars (Grunhut et al., 2011) (the basic techniques employed in stellar magnetometry are described in Chapter 4 and the references contained therein). The O star detections have doubled the number of known magnetic O stars, which previously included only two stars: the Zero Age Main Sequence (ZAMS) O7V star θ1 Ori C (Donati et al., 2002) and the evolved Of?p star HD 191612 (Donati et al., 2006; Wade et al., 2011). Of the newly discovered magnetic O stars, 2 are also Of?p stars (peculiar (p) O stars notable for N iii and He ii in emission, denoted f, while the ? indicates C iii emission lines of comparable strength, Walborn et al., 2003): HD 108 (Martins et al., 2010) and HD 148937 (Hubrig et al., 2011; Wade et al., 2012). The last is HD 57682, a weak-wind O9V star (Grunhut et al., 2009, 2011). Fig. 1.6 shows the longitudinal magnetic field measurements, Hα equivalent widths and Hipparcos magnitudes of HD 191612, the most thoroughly studied of the five known examples of Of?p stars (Walborn et al., 2010). Howarth et al. (2007) found Hα equivalent widths and Hipparcos magnitudes to follow a 537.6 day period, later shown to phase with longitudinal field measurements by Wade et al. (2011). This suggests that the spectral variations likely arise due to confinement of the wind by the stellar magnetic field. This lockstep variation, maintained in this case on a timescale of 34 years or approximately 24 cycles (Howarth et al., 2007), is a characteristic common to all known magnetic O and B stars (Wade, 2011). CHAPTER 1. INTRODUCTION 18 Figure 1.6: Longitudinal magnetic field (top), Hα equivalent width (middle) and Hipparcos magnitude (bottom) of HD 191612. Reproduced from Wade et al. (2011). CHAPTER 1. INTRODUCTION 19 Despite the apparent stability of OB star magnetic fields, Landstreet et al. (2007, 2008) showed that they do, in fact, seem to evolve over time: using a large survey of Ap/Bp cluster members (Bagnulo et al., 2006) for which the ages were well determined, they found that the magnetic flux appears to decline by a factor of several throughout the main sequence lifetime, with the decline concentrated early in the star’s life. This, together with an apparent 300 G minimum threshold for the magnetic dipoles of Ap/Bp stars (Aurière et al., 2009) would seem to offer support for the ‘fossil field’ hypothesis, which explains hot star magnetism as primarily a result of magnetic flux in the protostellar nebular material being locked in and amplified with the stellar plasma. This is in contrast to the convective dynamos thought to power the magnetic fields of cool stars, which model fields that, like the Sun’s, are variable over relatively short time scales. While dynamo fields are continuously regenerated by convection within the photosphere, a fossil field would be expected to decrease monotonically over long time scales due to both the removal of magnetic flux through the wind, and the conservation of magnetic flux as the star increases in surface area. This suggests the possibility that the failure to detect significant magnetic fields in the photospheres of the majority of evolved blue giants or supergiants may be in part because they are inherently weak (another factor is the paucity of spectral lines available for analysis: whereas fields of 1 G or weaker are routinely detected in K or M giants, whose spectra possess tens of thousands of relatively sharp absorption lines, OB stars have only a few hundred lines, which tend to be relatively broad due to their more rapid rotation.) The fossil field hypothesis is also preferable on theoretical grounds: magnetic dynamos are not an expected feature of OB star photospheres, since their envelopes are CHAPTER 1. INTRODUCTION 20 Figure 1.7: The stable ‘twisted torus’ in the radiative envelope of a massive star. Yellow field lines are stronger than black. The left panel shows the view with the star’s magnetic axis perpendicular to the line of sight; in the right panel, the magnetic axis is parallel to the line of sight. Note that such structures contain the majority of their magnetic energy in the toroidal rather than poloidal component, rendering them difficult to detect directly. Reproduced from Braithwaite (2009). CHAPTER 1. INTRODUCTION 21 radiative rather than convective, and the magnetic flux tubes generated within their convective cores should have rise times much longer than the life of the star (Donati & Landstreet, 2009). Numerical simulations of the stability of fossil fields (Braithwaite & Nordlund, 2006) showed that while purely poloidal or toroidal fields are always unstable, fields with mixed poloidal and toroidal components can evolve into a stable twisted torus inside the star (see Fig. 1.7). Braithwaite (2009) investigated the ratios of toroidal to poloidal components and found that any field with more than an 80% poloidal component was unstable, but that the toroidal component could make up a very large fraction of the total magnetic energy, and that the allowable toroidal energy fraction increased with decreasing magnetic field strength. While it is primarily the poloidal component which is observed, the toroidal component might be inferred from other observations and could play an important role in stellar evolution. Cantiello et al. (2009) have pursued speculations regarding a ‘bump’ or localized increase in Fe opacity (an Fe Convection Zone or ‘FeCZ’) just beneath the surface of the photosphere, which may be able to sustain a weak dynamo from which flux tubes might be able to rise to the stellar surface in a much more tractable time frame than that required for transit from the convective core. The consequences include localized magnetic fields, possibly with the same ‘bright spots’ suggested by Cranmer & Owocki (1996) to be the engine driving CIRs, nonradial pulsational modes, and microturbulent broadening (Cantiello & Braithwaite, 2011). While this does not explain observed fields, the lower limits predicted for magnetic fields arising through the FeCZ mechanism range from a few to a few hundred G, increasing with luminosity class and spectral type (Cantiello & Braithwaite, 2011), and are challenging for current instrumentation to detect. 22 CHAPTER 1. INTRODUCTION 1.4 Magnetically confined winds The classical theory of line driven winds, discussed above, describes a balance between the radiative pressure pushing matter away from a star, and the gravitational field attempting to pull the wind back: once the gravitational potential has been overcome, the wind accelerates smoothly out until a terminal velocity is reached due to saturation of the available absorption lines. With the inclusion of magnetic fields in the model, however, the nature of the wind as a plasma must be taken into account. If the magnetic field is strong enough, the plasma wind will tend to follow magnetic field lines at least up to the Alfvén radius (i.e. the radius within which the magnetic energy density overpowers the wind ram pressure), rather than flowing out isotropically (ud-Doula & Owocki, 2002). In the case of a magnetic dipole, the wind is confined in closed loops around the magnetic equator. This is illustrated in Fig. 1.8. In the case of a magnetic dipole, the essential result is the concentration of wind material within a disk-like distribution locked into rigid co-rotation with the star. Magnetic fields do not have to be particularly strong to achieve this effect. ud-Doula & Owocki (2002) calculated a dimensionless ‘wind magnetic confinent parameter’, denoted η∗ , in order to express the interrelationship of R∗ , Ṁ , v∞ , and the surface magnetic field strength B: B 2 R∗2 η∗ = Ṁ v∞ (1.3) where if η∗ & 1 the wind is said to be magnetically confined, as shown in Fig. 1.8. It is intuitively obvious that a relatively weak magnetic field B can still result in a magnetically confined wind if either Ṁ or v∞ is sufficiently small, or if the star is CHAPTER 1. INTRODUCTION 23 Figure 1.8: Three models for 1D, non-rotating magnetic wind confinement with varying wind magnetic confinement parameters (top–bottom: η∗ = 0.1, 1, 10). The left-most column shows magnetic field lines, with the Alfvén radius (at which the radial flow velocity equals the Alfvén velocity) in bold. The remaining panels show, from left to right, contours of the log of density, the radial velocity, and the latitudinal velocity. η∗ = 1 is sufficient to form a magnetically confined disk at the magnetic equator. Note the increasing complexity and strength of the velocity fields, especially vθ , at higher η∗ . Reproduced from ud-Doula & Owocki (2002). CHAPTER 1. INTRODUCTION 24 Figure 1.9: Snapshots at arbitrary simulation times of the magnetic √ equatorial regions of three models with moderate to strong (left–right: η∗ = 0.1, η∗ = 10 and η∗ = 10) wind confinement, showing countours of logarithmic density (top) and magnetic field lines (bottom). Arrows indicate the directions of matter flow. Note the increasing complexity of the infall in more strongly confined winds, with the equatorial density enhancements formed at higher stellar radii being randomly deflected to the north or south. Reproduced from ud-Doula & Owocki (2002). sufficiently large. While η∗ = 1 is sufficient to confine the wind, as η∗ increases the wind is predicted to fall back onto the star in an increasingly complex fashion (see Fig. 1.9). As shown in Fig. 1.10, there can be a significant angle between the disk and the rotational equator, leading to distinct observational signatures as compared to Keplerian decretion disks: whereas Keplerian disks are viewed from the same angle at all rotational phases, magnetically confined disks show clear rotational modulation, leading to variability in photometry, emission and absorption lines, and linear CHAPTER 1. INTRODUCTION 25 Figure 1.10: A Rigid Field Hydrodynamics model of a stellar wind confined by a dipolar magnetic field, viewed at three different rotational phases. Note that, in contrast to a Keplerian disk, the plane of the disk is not perpendicular to the rotational axis; rather the plasma is confined where the magnetic and centripetal forces balance. An additional difference with a Keplerian disk is that the disk rotates as a solid body. Reproduced from Townsend, Owocki & ud-Doula (2007). polarization angle. While early work treated the non-rotating case for simplicity (ud-Doula & Owocki, 2002), the theory of magnetically confined winds has been extended to incorporate magnetic obliquity (Owocki & ud-Doula (2004)); to develop a Rigidly Rotating Magnetosphere model (Townsend, Owocki & Groote, 2005) in which the wind plasma is channeled into a corotating magnetosphere and centrifugally supported against gravity; to explore the possibilities of centrifugal breakout of magnetically confined winds (ud-Doula, Townsend & Owocki, 2006); to account for field-aligned rotation (ud-Doula, Owocki & Townsend, 2008); and to explore the influence on rotational spin-down due to angular momentum loss through coupling between the magnetic field and the wind (ud-Doula, Owocki, & Townsend, 2009). In order to explore the balance between centrifugal wind support and magnetic confinement, ud-Doula, Owocki & Townsend (2008) developed the rotational parameter, W : 26 CHAPTER 1. INTRODUCTION Figure 1.11: Logarithm of equatorial disk mass with radius (vertical axis, from 1–5 R∗ ) and time (horizontal axis, from 0–3000 ksec) for an array of models spanning a large dynamic range in the wind magnetic confinement parameter η∗ and the rotation parameter W . The dashed lines correspond to the Kepler radius and the dotted line to the Alfvén radius. Reproduced from ud-Doula, Owocki & Townsend (2008). W = vrot /vorb (1.4) where vrot is simply the equatorial rotational velocity of the star, and vorb is the circular velocity at the stellar surface at which the centrifugal and gravitational forces balance, inducing corotation, vorb = r GM∗ R∗ (1.5) where G is the gravitational constant, M∗ is the stellar mass, and R∗ is the stellar radius. Depending upon the relative magnitude of W and η∗ , and consequently the Kepler 1/4 and Alfvén radii RK ≃ W −2/3 R∗ and RA ≃ η∗ R∗ , a complex combination of infall, CHAPTER 1. INTRODUCTION 27 accumulation into corotating disks and centrifugal breakout events occurs. Material above RK will tend to be supported against infall, and if RK > RA it will simply be flung outwards; the plasma within RA , however, will tend to fall back towards the star. If RA > RK the wind of a star may be entirely confined within a rigidly corotating magnetospheric disk, with infall blocked by centrifugal support and escape denied by the magnetic field (although if W is high there will continue to be occasional breakout events). This often results in periodic Hα line profile variability that reoccurs like clockwork, in agreement with the observational phenomenology of known magnetic Be stars such as HR 5907 (Grunhut et al. , 2010a). Fig. 1.11, from ud-Doula, Owocki & Townsend (2008), plots the radial distribution of mass along a slice taken at the magnetic equator as a function of time and illustrates this combination of stable disk formation with periodic infall and breakout events for simulations across a matrix of W and η∗ strengths. Fig. 1.12 shows known magnetic massive stars on the logarithmic η∗ − W plane, with the line RK = RA separating dynamically from centrifugally supported magnetospheres. The Of?p star HD 191612 is an interesting case of a star with relatively high η∗ and low W . Due to the star’s slow rotation (itself likely a consequence of spindown due to coupling between the magnetic field and the ISM), there is no centrifugal support for the stellar wind plasma; clumping acts against the radiative support of a line-driven wind; thus gravity is able to ‘win’ against the radiation pressure and pull the wind material back to the stellar surface. Since the plasma cannot remain in the circumstellar environment for long, the continuous presence of the ‘disk’ implies that the plasma is being continually replenished, in essence dynamically rather than rotationally supported (Wade et al., 2011). This is in contrast to the magnetospheres CHAPTER 1. INTRODUCTION 28 Figure 1.12: The wind magnetic confinment parameter η∗ vs. the rotation parameter W . Known magnetic massive stars are labeled individually, with approximate spectral type given in the legend; stars with black points show Hα variability, while outlines indicate UV modulation; arrows indicate upper or lower limits for these stars. The diagonal dashed line indicates the boundary at which the Kepler radius is equal to the Alfvén radius, dividing the regions of centrifugally supported and dynamically supported magnetospheres. Original figure provided courtesy of Véronique Petit. CHAPTER 1. INTRODUCTION 29 of rapidly rotating magnetic OB stars such as σ Ori E (Townsend, Owocki & Groote, 2007) or HR 7355 (Oksala et al. (2010), Rivinius et al. (2010)). An example of a wind confined into a rotationally supported disk is illustrated in Fig. 1.10. 1.5 BA supergiants BA supergiants – evolved massive stars of spectral type late B or early A – range in mass from 9 – 25 M⊙ , in luminosity from ∼ 104−5 L⊙ , and have effective temperatures of around 10-12 kK (and so are sometimes known as ‘tepid supergiants’, Przybilla et al., 2010). They are a very rare class of star, with only about 100 known objects in the Milky Way (Verdugo et al., 2003). This rarity is thought to be a consequence of their very rapid evolution across the H-R diagram: BA SGs are a transitionary class, either in the process of evolving from the main sequence towards Red Supergiant (RSG) status, heading back again on a ‘blue loop’, or evolving again towards RSG status on the way to a core-collapse supernova (Meynet & Maeder, 2000). High-resolution spectroscopy with 8–10 m class telescopes and instruments such as the Keck I HIRES spectrograph have opened the possibility of using BA supergiants as a powerful tool in extragalactic astronomy. As the intrinsically brightest stars at visual wavelengths within spiral and irregular galaxies (their absolute magnitudes can range up to Mv ≃ −9.5, rivaling globular clusters and some dwarf spheroidal galaxies in integrated light (Przybilla et al., 2006)), they are valuable probes of the overall metallicity and the metallicity gradient within the Milky Way and other galaxies. Their high intrinsic brightness also gives them a long-recognized (Hubble, 1936), yet still un-realized potential as extragalactic distance indicators, as they are much more easily resolved as point sources in galaxies than dimmer stars (e.g. Kudritzki et al., CHAPTER 1. INTRODUCTION 30 2008). However, their use as distance indicators must be calibrated by an understanding of their nonlinear variability, especially that portion arising in their radiatively driven winds, known for complex non-axisymmetric structures and supersonic matter flows. While BA SGs show emission lines in their optical spectra (Kaufer et al., 1996a), the majority of their spectral lines are well-fit by photospheric models i.e. the line profiles are well-reproduced by models incorporating the effective temperature and surface gravity in the photosphere. The primary exception to this is Hα, which shows a complex, non-photospheric morphology. In a spectral atlas of a large sample of Galactic A-type SGs compiled by Verdugo et al. (1999) a strong correlation between the asymmetry of the Hα line and the luminosity class was found: as the luminosity class increases, Hα shifts from a symmetric profile (Ib) to an increasingly asymmetric and variable profile (Iab or Ia) characteristic of α Cygni variables. This asymmetry in the Hα line is thought to be related to significantly stronger stellar winds in stars with higher luminosities, and consequently higher mass loss. The asymmetric profiles also display much greater variability than the symmetric profiles. 1.5.1 α Cygni variability Perhaps the most fascinating property of BA supergiants is their intrinsic variability. Often called ‘α Cygni variables’, after the prototype of the class, α Cyg (Paddock, 1935), they are characterized by seemingly stochastic variations in radial velocity, low-amplitude photometric variability, and time-dependent spectral line profiles such as Hα which is often filled with emission and especially variable, exhibiting the full CHAPTER 1. INTRODUCTION 31 range of profile morphologhy: P Cygni, inverse P Cygni, pure absorption, and doublepeaked emission (see e.g. Kaufer et al., 1996a). As with most OB stars, the UV resonance lines of BA SGs show both P Cygni profiles and DACs (Lamers, Stalio & Kondo, 1978). Time series analyses of photometric and spectroscopic observations consistently find multiple periods with significant amplitudes, ranging from a few hours to ∼100 days (Lucy, 1976; Sterke, 1976; Maeder, 1980; Kaufer et al., 1996a, 1996b, 1997; Markova & Valchev, 2000; Percy, 2008). These periods are generally speculated to be the result of the superposition of multiple high-order, low-amplitude non-radial pulsational (NRP) modes. BA SGs exhibit macroturbulent spectral line broadening (due to large-scale turbulent motions within the stellar atmosphere; Gray, 1975) comparable to the broadening due to rotation, and it has been suggested that macroturbulence is in fact the result of surface motions arising from superimposed NRPs (see e.g. Aerts et al., 2009). The amplitude of variability seems to increase with luminosity (Maeder & Rufener, 1972), while the characteristic variability timescale or ‘semiperiod’ seems to increase with later spectral types (Wolf & Sterken, 1976; Burki, 1978). While the rotational periods Prot of these stars are generally unknown, upper and lower bounds for Prot can be estimated if the stellar radius R∗ , projected rotational velocity v sin i, and logarithmic surface gravity log(g) are known. The upper bound on the period comes from a straightforward calculation assuming the angle of inclination from the line of sight i = 90◦ , in which case the equatorial rotational velocity veq = v sin i, and Prot = 2πR∗ /v sin i. The minimum rotational period is obtained through calculating the breakup velocity (the velocity at which centripetal acceleration overpowers the CHAPTER 1. INTRODUCTION 32 star’s gravity) from log(g) and R∗ : since veq cannot be greater than the breakup velocity, this establishes a minimum value for i, and thus (with v sin i and R∗ ) a minimum Prot . Crucially, the predicted fundamental radial periods of BA SG stars (i.e. the longest period possible with radial pulsations) are much shorter than the minimum rotational periods, making it possible to clearly distinguish between variability arising due to radial pulsations and that from rotational modulation. Around 60% of BA SGs have semiperiods significantly longer than the theoretical fundamental radial pulsation modes (Lovy, 1984). This makes it unlikely that the semiperiodic variability is due to radial pulsations, and suggests that at least some of the variability might be a result of rotationally modulated surface features or non-spherical matter flows (e.g. Kaufer et al., 1996a). Kaufer et al. (1996a, 1996b, 1997) performed the most detailed study to date of α Cygni variability, conducting a long-term, densely time-sampled, intermediate resolution (λ/∆λ ≃ 20, 000) spectroscopic monitoring campaign of 6 α Cygni variables. All six of the BA SGs studied by Kaufer et al. displayed Hα emission wings extending out to ±1200 km s−1 , much broader than expected given the characteristic v∞ ∼ 200 −400 km s−1 . They suggested this might be attributed to electron scattering in deep atmospheric layers. The typical region of variability was within ±100 km s−1 of line centre, and the emission was found to be particularly variable close to the borders of this region. While half of the objects studied (HD 91619, β Orionis, HD 96919) displayed much more irregular Hα variability than the others (HD 92207, HD 100262, α Cygni), all showed variability localized symmetrically about the systemic velocity vsys , due to blue and red emission components superimposed on almost constant photospheric CHAPTER 1. INTRODUCTION 33 and/or wind profiles. In order to visualize the line profile variability, ‘dynamic spectra’ were constructed by means of subtracting a mean line profile from the individual profiles and representing the residual flux by mapping emission or absorption relative to the mean profile to an intensity gradient. This can then be plotted as a function of phase (if the period is known) or of time (if, as in these cases, it is not). This allows the relative variation between observations to be easily compared. The dynamic spectra of the Hα line of β Orionis in the 1993 and 1994 observing seasons are reproduced in Fig. 1.13. The behaviour of Hα in the two seasons is quite different, with no sign of periodicity, and, in 1994, a dramatic deformation in the blue-shifted half of the line dubbed a High Velocity Absorption (HVA) event (discussed in more detail below). To quantify the amplitude of line profile variability the authors made use of ‘temporal variance spectra’ or TVS (Fullerton, 1996), which measures the RMS deviation of each pixel across the line profile. The (T V S)1/2 curves for Hα (left) and the Si ii 634.7 line (right) of β Ori are shown in Fig. 1.14. In most stars, the peak region of Hα varies at a maximum of around 5–10% as compared the variability across the rest of the line, although in some stars peaks as high as 35% are seen (Kaufer et al., 1996a). In both lines variability tends to be localized around two peaks, shifted to the blue and red with respect to the systemic velocity. However the peaks of Hα show a systematic bias towards the blue which is absent in the Si ii 634.7 nm line. Variability in the blue lobe of Hα appears greater than in the red lobe; the photospheric TVS on the other hand shows slightly more variance in the red than the blue. The core of Hα is also much more variable than that of Si ii. The peaks of the Si ii 634.7 nm line’s TVS are located at around v = vsys ± v sin i, much lower velocities than CHAPTER 1. INTRODUCTION 34 Figure 1.13: Dynamic spectra of Rigel’s (β Ori) Hα line in 1993 (left) and 1994 (right). One-dimensional residual flux is shown in the top panels, together with color bars delineating the intensity scale in the dynamic spectra (bottom panels). Color figure provided courtesy of Andreas Kaufer (private correspondance). CHAPTER 1. INTRODUCTION 35 Figure 1.14: (Left) Temporal variance spectrum of Hα. Solid line corresponds to the 1993 observing season, dashed line to 1994. The dotted line indicates the systemic velocity, the horizontal lines the 95% significance probability. Small peaks at –260 km s−1 and +80 km s−1 are due to telluric lines. Reproduced from Kaufer et al.(1996a). (Right) The same for Si ii 634.7 nm. 1992 = dash-dotted; 1993 = short dashed; 1994 = solid; 1995 = dotted. Note the relative stability of the SI ii 634.7 nm line as compared to Hα. Reproduced from Kaufer et al.(1997). those seen in Hα TVS curves but showing that here too variability happens primarily in, but is not confined to the wings. This is interpreted as an indication of strong radial contributions to the velocity field (visible in the wings) along with non-radial contributions (resulting in line core variability). Time-scales of wind variability were explored by constructing equivalent width (Wλ ) curves, illustrating the integrated flux of a spectral line as a function of time. A period search conducted with the clean algorithm was used to extract frequencies from periodograms constructed using Lomb-Scargle statistics: in all cases a single dominant frequency was found , which – for those objects observed over a two-year period – were approximately reproduced from year to year. Periods typically ranged CHAPTER 1. INTRODUCTION 36 from around 10 days to three months, and were all well below the theoretically calculated fundamental radial pulsational periods; however, they were consistent with possible rotational periods. Timescales of photospheric variability were investigated using radial velocities by Kaufer et al.(1997). They found that radial velocities were not influenced by the depth of a given line’s formation within the photosphere. This, combined with very small variations in the equivalent width of these lines (.1% of the mean line strength), indicated that the observed photospheric variation was due to small-amplitude nonradial pulsations. cleaned period spectra constructed from radial velocity measurements of the metallic lines showed that – in stark contrast to the results for Hα – different periods, both longer and shorter than the fundamental radial pulsational period, were found for the same stars in different years. This suggests the presence of both nonradial modes and radial overtones. The work of Kaufer et al. (1996a, 1996b, 1997) was extended to HD 199478 (B8Iae) by Markova & Valchev (2000). They found a tendency of the Hα Wλ to anticorrelate with the equivalent width of C ii 658.2 nm, suggesting that at least some of the Hα variability might be assigned to photospheric changes. A Fourier analysis of Hα variability revealed periods (10–20 days) significantly longer than the wind time scale (3–4 days) but well within the upper and lower bounds of the estimated rotational period, suggesting that the wind variability in Hα is rotationally modulated and is maintained by photospheric processes. Markova et al. (2008b) utilized NLTE model atmospheres to analyze multiple lines in a more extended optical spectral time series of HD 199478, probing photospheric, near-star and outflow regions. Evidence CHAPTER 1. INTRODUCTION 37 was found for semi-modulation of the central velocities of photospheric lines over timescales of weeks to months, with one ‘stable’ period appearing in the variations of two lines. They found that with the exception of an HVA event, the wind profile could be divided into two components: a strong emission component, either centered in the rest frame or with a weak blue-biased asymmetry, and localized ‘bumps’ in emission variable in both depth and position. The former they attributed to a spherical, emission-only envelope; the latter to large-scale structured wind components with both outflows and infalls of matter, leading to both absorption and emission. 1.5.2 High Velocity Absorption events High Velocity Absorption events (HVAs), dramatic Hα line profile deformations appearing suddenly at a high velocity, have been observed to date in four stars: HD 91619 (B7Iae) and HD 96919 (B9Iae) (Kaufer et al., 1996a, 1996b), Rigel (B8Iae) (Kaufer et al., 1996b; Morrison et al., 2008) and HD 199478 (B8Iae) (Markova et al., 2008). All stars are similar in spectral type and luminosity class, although there has been some suggestion in unpublished work by Morrison et al. (2009) that nearby spectral types – A0-A2, Ia – also display these features, which would make HVAs a phenomenon common to all BA SG α Cyg variable stars. The dynamic spectra in Fig. 1.13 clearly shows the HVA in the 1994 observing season: it starts with a rapid increase in blue-shifted absorption, often preceded by a peak in blue-shifted emission (an inverse P Cygni profile). Initially the greatest absorption depth is located at a high velocity relative to the systemic velocity, and spread over a broad velocity range. Following this sudden and drastic Hα distortion, the absorption feature migrates redwards towards the systemic velocity over a CHAPTER 1. INTRODUCTION 38 Figure 1.15: Wλ of HVA events in the Hα lines of two stars: Rigel (left) and HD 96919 (right). In Rigel the feature persists for about 60 days, in HD 96919, for 150 days. Horizontal lines show the mean Wλ of the stars in epochs not including the HVA (solid) and the standard deviations (dotted lines). Reproduced from Kaufer et al. (1996b). timescale of weeks to months, becoming narrower and deeper as it goes; as this happens, a corresponding redshifted absorption feature appears, centred on the systemic velocity. Comparing an HVA observed in HD 199478 with those described by Kaufer et al. (1996b), Markova et al. (2008) found that weaker events tend to be more extended in velocity space, whereas stronger HVAs tended to achieve their maximum depth at a lower velocity. The maximum reshifted velocity is always lower than the maximum blueshifted velocity. Note also the remarkable similarity in the Hα equivalent width curves, reproduced in Fig. 1.15: rise times are consistently shorter than decay times, with duration dependent on the strength of absorption but the rise/decay ratio remaining constant. Photometric data also seem to show that, as the HVA initiated, HD 199478 was about 0.1 mag fainter than at the time of maximum line absorption (Percy et al., 2008). Kaufer et al. (1996b) suggested magnetically supported CIRs as the most likely CHAPTER 1. INTRODUCTION 39 explanation for this phenomenon. One key prediction of the CIR model is that HVAs should reoccur on timescales consistent with the rotational period of the star. This prediction was borne out by observations of HD 96919, with an HVA reappearing on precisely the day predicted for a corotating structure given the star’s suspected rotational period. However, in the case of Rigel, while the strongest observed HVA (in 1993) reoccurred at lower intensity in 1994, it failed to reoccur in the 1995 observations, indicating that if HVAs are in fact produced by CIRs the azimuthally extended structures must be relatively short-lived, surviving perhaps for only a few rotational cycles. Magnetic support of the CIR was invoked to enforce rigid corotation close to the star (see Fig. 1.16, top panels), in order to explain the sudden appearance of the feature at high velocity (Fig. 1.16, middle panels). Nothwithstanding their invocation by Kaufer et al. (1996b), classical CIRs cannot explain the observed properties of HVAs. Recall that classical CIRs have been proposed as an explanation for DACs. Whereas DACs tend to accelerate outwards from the star, HVAs appear at high blueshifted velocity and propagate inwards. Additionally, no DAC has ever been observed to exhibit a red-shifted absorption feature, whereas HVAs always develop these. This last is particularly significant as it is thought to be an indication of a matter infall accompanying the outflow. Israelian et al. (1997) argued that this infall could be explained by closed, cool coronal magnetic loops corotating with the star, with matter flowing up from one foot-point and down towards another (see Fig. 1.16, bottom panels). They showed that the free-fall time of matter from the top of the loop is significantly shorter than the time-scale of the appearance and disappearance of the redshifted component of the HVA (20–25 days), thus supporting the idea that the motion of material within CHAPTER 1. INTRODUCTION 40 the loop is retarded by the magnetic field, just as is observed with cool solar loops (Loughhead & Bray, 1984). At the same time, material in the upward leg is accelerated to highly supersonic velocities. Such a complex velocity field – with matter moving both towards and away from the observer within the loop, while the loop’s own motion is modulated by the rotation of the star – provides a potential explanation for the simultaneous blue- and red-shifted absorption, as well as the sudden appearance of the HVA over a wide velocity range. A magnetic loop implies the presence of a magnetic field. Based on magnetohydrodynamic calculations, Israelian et al. (1997) showed that a magnetic field of 1–10 G (if LTE models were used) or 25 G (with the NLTE model atmosphere code tlusty; Hubeny & Lanz, 1995) is required to support the magnetic loop. The notion of a magnetically confined equatorial structure composed of infalls and outflows of matter as the origin of HVA events also received support by Markova et al. (2008), who calculated that amongst the 4 stars in which HVAs have been observed, magnetic dipoles of a few to a hundred G would be sufficient to magnetically confine the wind (ud-Doula & Owocki, 2002). 1.6 Rigel in this context While often referred to simply as β Orionis, properly speaking the blue supergiant Rigel (B8Iae) is β Ori A. β Ori B, a spectroscopic binary consisting of two B9V stars separated by about 100 AU, orbits β Ori A at a distance of around 2500 AU, far enough that a full orbit about Rigel takes 25,000 years and thus no orbital motion has been observed; the binarity of β Ori A and B is established through their proximity and shared proper motions. With a magnitude V=10.4, β Ori BC would ordinarily CHAPTER 1. INTRODUCTION 41 Figure 1.16: (Top) a CIR in the equatorial plane, with magnetically enforced corotation up to 2 R∗ assumed in the right panel. (Middle) radial velocity vs rotational phase, with phase zero corresponding to the foot point crossing the central line of sight. Corotation results in a much sharper increase in the radial velocity as the foot point comes into view. Reproduced from Kaufer et al. (1996b). (Bottom) Schematic representation of a magnetic loop. Open arrow shows the direction of rotation; solid arrows show the direction of plasma flow within the loop. Reproduced from Israelian et al. (1997). CHAPTER 1. INTRODUCTION 42 Figure 1.17: Rigel illuminating the nearby Witch Head Nebula (IC 2118). Astrophotography by Andreo, 2009. be visible with a small telescope, however, the fierce illumination of the primary – with a visual magnitude V=0.12 – makes the companion a difficult target. Indeed, Rigel is so bright that the Witch Head Nebula (IC 2118), some 40 light years distant, is primarily visible due to the star’s scattered light (see Fig. 1.17). There is some suggestion that there might be a fourth star in the system, β Ori D, a K dwarf separated by 11,500 AU with an orbit of 250,000 years. CHAPTER 1. INTRODUCTION 1.6.1 43 Abundances and evolution The evolutionary status of Rigel is somewhat uncertain. While clearly an evolved massive star, whether Rigel is evolving towards the Asymptotic Giant Branch and has yet to commence core He burning, or has evolved back from the AGB on a blue loop and is currently in the core He burning phase, has been the subject of some debate. The high N/C abundance ratio determined by Przybilla et al. (2006) adds weight to the hypothesis that Rigel is directly evolved from the main sequence. Through comparing the spectroscopically determined mass (Mspec = 24±8 M⊙ ) with the evolutionary tracks of Meynet & Maeder (2003), the zero-age main sequence mass, evolutionary mass and evolutionary age are determined to be, respectively, MZAM S = 24±3 M⊙ , Mevol = 21±3 M⊙ , and τevol = 8±1 Myr (Przybilla et al., 2006). These masses are tabulated in Table 1.1. In their analysis, Rigel appears to have started on the main sequence with spectral type O8–O9 with an initial equatorial rotational velocity vrot,ini > 300 km s−1 . 1.6.2 Atmospheric parameters Using the total optical and UV flux and an angular diameter of 2.55 mas, Stalio et al. (1977) derived an effective temperature Teff = 12070±160 K. Takeda (1994) used Kurucz LTE model atmospheres to fit Hγ and Hδ lines and determined Teff ≃ 13000±500 K and log(g)≃ 2.0 ± 0.3, pointing out that the discrepancy between their effective temperature and that of Stalio et al. arose due to the latter’s choice of reference wavelength. Kaufer et al. (1996a) derived their own parameters from the equivalent width of Hγ, finding the significantly lower values of Teff ≃11200 K and log(g)≃ 1.67. Using the NLTE model atmosphere code tlusty (Hubeny & Lanz, 1995) to fit Hγ CHAPTER 1. INTRODUCTION 44 and Hδ lines from a spectrum with no obvious peculiarities, while simultaneously fitting several Si lines, and taking v sin i= 40 km s−1 and microturbulence ξ = 7 km s−1 , Israelian et al. (1997) found Teff ≃ 13000±500 K, supporting the analysis of Takeda et al., but found log(g)=1.6±0.1. More recent studies have indicated that the previous, lower values of Teff may be more accurate after all: Przybilla et al. (2006) found Teff = 12000 ± 200 K and log(g) = 1.75 ± 0.1; these are in excellent agreement with Teff = 12500 ± 500 K (found through Si ii/Si iii ionization balance) and log(g) = 1.7±0.1 (through fitting Hδ), found by Markova et al. (2008), who indicate that the close agreement with the parameters derived by Przybilla et al. and Israelian et al. suggest that line-blanketing/blocking effects (not considered by Israelian et al.) and wind effects (neglected by Przybilla et al.) are minimal for Rigel. 1.6.3 Velocity fields and rotational period While macroturbulence (ζ) was first suggested as a mechanism for line broadening almost 90 years ago (Evershed, 1922), due to the difficulty in disentangling the rotational and macroturbulent velocities most early studies have neglected the turbulent component and so have tended to significantly overestimate Rigel’s projected rotational velocity v sin i, with typical values in the range 50–60 km s−1 (e.g. Kaufer et al., 1996a). The first serious attempt at measuring v sin i together with ζ was by Gray (1975), who used the Mg ii 448.1 nm doublet to find v sin i = 42 km s−1 and 17 ≤ ζ ≤ 23 km s−1 . Takeda et al. (1995) used a multi-parameter fitting method on He i 667.8 nm to find v sin i = 40 km s−1 and ζ = 43 km s−1 ; oddly, when they checked their result against Gray’s using the same Mg ii doublet, they found an even CHAPTER 1. INTRODUCTION 45 smaller macroturbulence than Gray had found (ζ = 12 km s−1 ). Basing their analysis on a spectrum synthesis involving both line blends and single lines, limited to weak lines in order to avoid contamination due to the wind in strong lines, Przybilla et al. (2006) found v sin i= 36 ± 5 km s−1 , in good agreement with previous results, and ζ = 22 ± 5 km s−1 , in better agreement with the results of Gray (1975). Markova et al. (2008), again using the Mg ii doublet at 448.1 nm, but analysing it with a Fourier technique developed especially for OB stars by Simón-Diaz & Herrero (2007), found v sin i= 30 km s−1 and ζ = 35 km s−1 . Values for the microturbulent velocity ξ adopted by all authors quoted above have consistently been in the range 7–8 km s−1 ; microrurbulence is not thought to be generally significant for stars of Rigel’s spectral class (Markova et al., 2008). The angular diameter has been well-constrained through interferometry by Aufdenberg (2008), who found θD = 2.76 ± 0.01 mas, while the parallax determined through Hipparcos photometry (4.22±0.81 mas; Perryman, 1997) indicates a distance of 0.24 ± 0.05 kpc, yielding a physical radius of 71 R⊙ (ranging up to 148 R⊙ if previous distances are used, based on association with either the Ori OB1 complex (0.5 kpc; Humphreys, 1978) or the τ Ori R1 complex (0.34 kpc; Hoffleit & Jaschek, 1982). Taking R∗ = 71 R⊙ and log(g)= 1.75±0.10 (Przybilla et al., 2006) yields a breakup velocity of 167 km s−1 ; with v sin i= 36±5km s−1 (Przybilla et al, 2006), the minimum angle between the rotational axis and the line of sight is then ∼ 12◦ , yielding a minimum rotational period of 21 days. Taking the same v sin i, and the maximum physical radius based on the estimated distance to the Ori OB 1 complex, and further making the assumption that we are viewing the star equator on, we find a maximum CHAPTER 1. INTRODUCTION 46 period of 208 days. The assumption that Rigel is being viewed at a relatively high inclination angle is given some weight by interferometric observations of Hα and Brγ described by Chesneau et al. (2010), who a noted differential phase signal which is most easily explained if the star is viewed close to the rotational equator. 1.6.4 Wind parameters Using infrared photometry, Barlow & Cohen (1977) derived two mass loss rates, finding Rigel to be variable in the IR: the first, Ṁ = 8.6 × 10−7 M⊙ yr−1 using their own measurement of the 10 µm flux; the second, Ṁ =1.4×10−6 M⊙ yr−1 , using the 10 µm measurements of an earlier study by Gerhz et al. (1974). To do so they adopted a terminal wind velocity v∞ = 530 km s−1 derived from an examination of UV resonance lines by Snow & Morton (1976). A study of UV resonance lines by Bates et al. (1980) indicated a terminal wind velocity in the range 400 ≤ v∞ ≤ 600 km s−1 . Using multiple lines from the Atlas of Ultraviolet P Cygni Profiles compiled by Snow et al. (1994), Lamers et al. (1995) found v∞ = 350±50 km s−1 . Kaufer et al. (1996a) adopted a significantly lower value than previous authors, v∞ = 229km s−1 , indicating that as spectral types earlier than B9 show no sharp edge in the UV P Cygni blueshifted absorption, this value of v∞ is a lower limit. The first attempt to derive mass loss rates from radio observations was reported by Abbott et al. (1980), who were only able to establish an upper limit of the 6 cm flux and so derived Ṁ ≤ 9.1 × 10−7 M⊙ yr−1 . Drake & Linsky (1989) were more successful: they performed 6 cm VLA observations of 25 supergiants from B2–F8, of CHAPTER 1. INTRODUCTION 47 which only one – Rigel – was detected as a radio continuum source1 . The authors indicated that this was likely a result of β Ori’s bright apparent magnitude, since the upper limits established for the other stars of similar spectral type in the sample were of a similar magnitude as Rigel’s 6 cm luminosity L6 ∼ 7 × 1016 erg s−1 Hz−1 . They postulated that the radio emission could be interpreted as free-free radiation from the stellar wind, in which case the inferred mass loss rate would be Ṁ = 2.5×10−7 M⊙ yr−1 . Underhill et al. (1982) are reported by Israelian et al. (1997) to have estimated Ṁ = 1.3 × 10−7 M⊙ yr−1 , although as this work is not available online it is difficult to check their method. At any rate, mass loss rates for BA supergiants are notoriously difficult to measure, likely due to the variable nature of mass loss events in these stars, and Rigel is no exception. As such no single measurement can be taken as an absolute, and all that can be said with certainty is that the mass loss for Rigel lies somewhere in the range 10−7 −10−6 M⊙ yr−1 . Vink et al. (2000) calculated Ṁ using a grid of wind models across a wide range of stellar parameters: the predicted mass loss rates were consistent with those measured from radio observations, but not with IR or Hα derived values. It might therefore be concluded that Ṁ from Drake & Lisnky (1989) is the most trustworthy. Within the range of v∞ and Ṁ determined for Rigel, a straightforward calculation using Eqn. 1.3 shows that even a magnetic field on the order of 10 G is capable of yielding η∗ & 1, i.e. a relatively weak field can still achieve magnetic confinement of the wind, consistent with the suggestion by Israelian et al. (1997) that a relatively weak magnetic dipole might be able to produce HVA type events. 1 There was some suggestion in this paper that Rigel might also be a weak X-ray source, however, those results were ambiguous due to the limitations of the instrumentation. It was later indicated that this X-ray emission was in fact leakage from a nearby late-type star, likely the K dwarf mentioned above as a potential fourth star in Rigel’s extended system (Bergöfer et al., 1999). CHAPTER 1. INTRODUCTION 1.6.5 48 Variability As discussed in detail above, Rigel is classified as an α Cygni variable star. The variability of Hα in Rigel’s spectrum was first noted by Struve & Roach (1933), and the star was first found to display α Cygni variability by Sanford (1947). In addition to the spectroscopic monitoring described above, strong variability in Hα and Hβ was noted by Dachs et al. (1977, 1981), who included β Ori in a survey of Be star spectroscopic variability. As already discussed, the pattern of wind line variability in all stars of Rigel’s class is inconsistent with a steady-state, spherically symmetric wind but rather, strongly indicative of a localized density variations involving outflows and infalls of matter varying in a ‘semiregular’ (Abt, 1957) or ‘semiperiodic’ (Maeder & Rufener, 1972) fashion, which is to say, showing no evidence for periodic behavior per se but rather characteristic timescales, generally ranging from days to weeks, over which variability occurs (with the characteristic timescales themselves somewhat variable across epochs for a given star). Lamers et al. (1978) reported the detection of time-variable DACs in the P Cygni profiles of UV Mg ii and Fe ii resonance lines with characteristic velocities of – 195 km s−1 , while observations of these lines by Bates et al. (1980) provided further evidence for the time evolution of DACs. According to Kaufer et al. (1996a), Gilheany (1991) reported the detection of DACs out to –400 km s−1 in IUE spectra of Rigel. An extended time series of linear polarization measurements reported by Hayes (1986) provides further evidence for strong departures from spherical symmetry in the circumstellar environment. This study was uniquely valuable as, in addition to verifying the characteristic semiperiodic time-scales indicated by both previous and subsequent spectroscopic and photometric campaigns, it gave some indication of the CHAPTER 1. INTRODUCTION 49 direction of mass loss, which showed no sign of the colinearity expected from rotational modulation or from most global pulsational modes, either radial or nonradial, but was rather strongly anisotropic in both space and time, with no obvious preferred direction1 . HVA events in particular provide strong evidence for irregular matter flows, with the suggestion of rotational modulation, and evidence that the circumstellar structures giving rise to the HVAs persist for one or more rotational cycles. At the moment more HVAs have been observed in Rigel’s spectrum than in any other star: the first by Kaufer et al. (1996b), the second by Morrison et al. (2008), and a third, as yet unpublished, by Chesneau (personal communication). The most recent was acquired interferometrically and may reveal additional details about the spatial distribution of these mysterious events. 1 A similar, less detailed study of HD 92207 has been performed by Ignace et al. (2009), finding much the same results. CHAPTER 1. INTRODUCTION Parameter θD (mas) V (mag) (m − M)0 (mag) MV (mag) B.C. (mag) Mbol (mag) log(L/L⊙ ) Distance Radius (R⊙ ) Tef f (◦ K) logg (cgs) MZAM S (M⊙ ) Mevol (M⊙ ) Mspec (M⊙ ) τevol (Myr) Origin Measurement Aufdenberg (2008) 2.76±0.01 SIMBAD 0.12 Przybilla et al. (2006) 7.8±0.2 Przybilla et al. (2006) –7.84±0.20 Przybilla et al. (2006) –0.78 Przybilla et al. (2006) –8.62±0.20 Przybilla et al. (2006) 5.34±0.08 Ori OB 1 0.5 τ Ori R1 0.34±0.04 Hipparcos 0.24±0.05 Ori OB 1 148 τ Ori R1 101±12 Hipparcos 71±14 Stalio et al. (1977) 12070±160 Takeda (1994) 13000±500 Israelian et al. (1997) 13000±500 McEarleans et al. (1999) 13000±1000 Przybilla et al. (2006) 12000±200 Markova et al. (2008) 12500±500 Takeda (1994) 2.0±0.3 Israelian et al. (1997) 1.6±0.1 McEarleans et al. (1999) 1.75±0.20 Przybilla et al. (2006) 1.75±0.10 Markova et al. (2008) 1.75±0.20 Przybilla et al. (2006) 24±3 Przybilla et al. (2006) 21±3 Przybilla et al. (2006) 24±8 Przybilla et al. (2006) 8±1 Table 1.1: Summary of Rigel’s Parameters 50 CHAPTER 1. INTRODUCTION Parameter v rad (km/s) v sini (km/s) ζ (km/s) ξ (km/s) v ∞ (km/s) Ṁ (M⊙ yr−1 ) Origin Measurement SIMBAD 20.7±0.9 Gray (1976) 42 Takeda et al. (1995) 40 Przybilla et al. (2006) 36±5 Markova et al. (2008) 30 Gray (1976) 20±3 Takeda et al. (1995) 43 Przybilla et al. (2006) 22±5 Markova et al. (2008) 35 Przybilla et al. (2006) 7±1 Snow & Morton (1978) 530 Bates et al. (1980) 500±100 Lamers et al. (1995) 350±50 Kaufer et al. (1996a) 229 Barlow & Cohen (1977) 1.4×10−6 Barlow & Cohen (1977) 8.6×10−7 Underhill et al. (1982) 1.3×10−7 Drake & Linsky (1989) 2.5×10−7 CAK model 4.1×10−7 Table 1.2: Summary of Rigel’s wind parameters. 51 Chapter 2 Observations The time series of polarized spectra examined in this thesis were obtained using the ESPaDOnS spectropolarimeter at the Canada France Hawaii Telescope (CFHT), a 3.58 m telescope on the summit of Mauna Kea, Hawaii, at an altitude of 4204 m, and its clone Narval at the Téléscope Bernard Lyot (TBL), a 2 m Cassegrain telescope located on Pic du Midi in the French Pyrenees at an altitude of 2877 m. CFHT’s design allows the telescope to operate in Cassegrain focus mode or prime focus mode; while it no longer operates in Coudé mode, the Coudé room (located beneath the control room) houses the ESPaDOnS spectropolarimeter. As CFHT cannot operate in both Cassegrain and prime focus mode simultaneously, it switches between different modes and instruments throughout the year. To maximize the efficiency with which observing time can be allocated to multiple programs, CFHT operates in a Queued Service Observing (QSO) mode. This enables astronomers to submit targets through a web interface, specifying the right ascension, declination, and V magnitude of targets, limiting airmass, and the necessary SNR, along with optional parameters such as observation scheduling. Resident astronomers then collect 52 CHAPTER 2. OBSERVATIONS 53 the requirements of numerous programs utilizing the same instrument, and allocate telescope time so as to ensure as many targets are acquired as possible. 2.1 Instrumentation The Echelle SpectroPolarimetric Device for the Observations of Stars (ESPaDOnS) and its twin Narval1 are the workhorses of the Magnetism in Massive Stars (MiMeS) program. The two instruments are essentially identical, with the exception of longer exposure times required for Narval due to TBL’s smaller mirror as compared to CFHT; as such the following description is largely taken from the publically available ESPaDOnS documentation and is assumed equally valid for Narval. They are echelle spectropolarimeters (spectrographs combined with polarimeters), operating at high spectral resolution: λ/∆λ ∼65,000 at 500 nm when operating in spectropolarimetric mode, up to λ/∆λ ∼81,000 in spectroscopic mode. Their spectral range is 369–1048 nm, encompassing the entire visual spectrum along with the near ultraviolet and near infrared, projected over 40 overlapping spectral orders (Donati, 2008). The polarization unit is located at the Cassegrain focus of the telescope. Light entering through the pinhole aperture is channeled through a polarization analyzer consisting of a rotatable λ/2 (half-wave) Fresnel rhomb, a fixed λ/4 Fresnel (quarterwave) rhomb, and another rotatable λ/2 Fresnel rhomb. The Fresnel rhombs utilize total internal reflection, as opposed to the birefringence used by wave plates, to introduce polarization-dependent phase shifts (or retardations) of either 90◦ (quarter 1 French for ‘swordish’ and ‘narwal’, respectively. CHAPTER 2. OBSERVATIONS 54 Figure 2.1: The 40 orders of an ESPaDOnS flat field calibration frame projected on the CCD. Reproduced from Donati et al. (2008). wave) or 180◦ (half-wave) to incoming photons: the half-wave rhomb swaps orthogonal polarization states, while the quarter-wave rhomb converts circular into linear polarization (thus making it possible to measure). This can also be accomplished with wave plates, however the rhombs does so in an almost achromatic fashion, essential given the wide spectral range over which the spectrograph is designed to operate. The λ/2 Fresnel rhombs can rotate about the optical axis and, by varying the retardation given to incoming photons, can convert them to any desired pair of orthogonal polarizations. From the Fresnel rhombs, the beam enters a Wollaston prism: two right-triangle prisms with perpendicular optical axes, which separate the orthogonal polarizations into separate beams. The star image, now doubled, is then focused on the inputs of two 30 m long, 0.1 mm diameter optical fibres, which carry the output beams to the spectrograph, which sits on a mechanically stabilized bench in the thermally isolated Coudé room, at the CHAPTER 2. OBSERVATIONS 55 heart of the telescope. It then enters an image slicer, which separates the circular images from the fiber heads into three narrow slices. From here the beam travels to a parabolic collimator, which sends it to the diffraction grating: an R2 echelle grating with a 204×408 mm ruled area, 79 lines/mm, and a 63.4◦ blaze angle. The diffracted beam then enters a double prism cross-disperser, which separates the spectral orders before passing the beam through the f /2 camera which projects the 40 overlapping orders onto a 2000×4500 pixel CCD with 13.5 µm2 pixels. An ESPaDOnS flat field projected on the CCD is shown in Fig. 2.1. Although the process is nominally achromatic, the beam slicer visibly warps the order profile in a somewhat complex way. This, complicated further by the presence of two orthogonal polarization states, makes the extraction of data from the chip particularly complex. The unpolarized intensity (Stokes I ) component is formed by adding the two spectra, while the Stokes V component is obtained differentially. This is explained further below. Systematic errors arising due to small misalignments, differences in transmission, seeing, etc., are minimized by means of obtaining four successive spectra for each observation, with the polarization analyzer settings changing for each so as to switch the positions of the two spectra on the CCD through. This is accomplished by rotating Fresnel rhombs which introduce retardations of ±π/2 rad, switching the polarization of the light entering the Wollaston prism, which then splits the beams in opposite senses. The efficiency of the instruments is around 19% at 500 nm, declining to 2% at the edges of the spectral range. The combined efficiencies of the telescopes at the Cassegrain focus and the polarimeters are fairly steady at around 40% for much of the spectral range, but the efficiency of the spectrographs considerably lower. Taking into CHAPTER 2. OBSERVATIONS 56 consideration light lost at the instrument’s 1.6” aperture (10%) and normal atmospheric observing conditions, overall efficiencies of 15% are realistic. This contributes to the notoriously photon hungry nature of spectropolarimetry: high dispersion spectroscopy takes longer to reach the same SNR as lower resolution observations, since the photons are being spread over more pixels, and with polarization the situation is even worse since, if the magnetic field is particularly weak, the polarization fraction of the light will be very low, and the SNR required to detect correspondingly high. 2.2 Reduction All frames were processed using the CFHT’s Upena pipeline, feeding the automated reduction package Libre-ESpRIT (Echelle Spectral Reduction and Interactive Tool, Donati et al., 1997), which yields Stokes I and V spectra, along with a diagnostic null (or N ) spectrum created through combining the four sub-exposures in such a way as to cancel out all real polarisation, in order to control for spurious polarisation signals. The development of ESpRIT was necessitated by the particular challenges inherent in the reduction of ESPaDOnS data, as mentioned above. Extraction of the 40 unevenly spaced echelle orders spread over the chip (see Fig. 2.1) requires a series of 2D quadratic and cubic fits to define their directions and shapes along a coordinate system defined by the core of the order. Wavelength calibration is accomplished by means of a Th/Ar arc lamp frame, with the best line identification out of all orders used to calibrate the remainder; these pre-calibrated orders are then re-calibrated as a set, generating a dispersion relation for all orders in the spectrum. Intensity spectra are then extracted from the stellar spectrum with one flat-field 57 CHAPTER 2. OBSERVATIONS and one bias exposure. Before the pixels of the stellar spectrum are divided by those of the flat field, the flat is first divided by its mean across each order, pre-flattening it in order to preserve the relative variation of the pixels in the stellar frame. In order to avoid resampling aleady noisy data, rather than straightening the orders prior to invoking optimal extraction algorithms, a model is interpolated directly onto the data, adopting a method developed by Marsh (1989) that generates a polynomial fit to the fluxes in each order as a function of distance from the central axis; cosmic rays are automatically rejected during this process. This method ensures SNR optimization and spectrophotometric accuracy; produces output with no loss of spectral resolution; simplifies wavelength calibration; and enables automatic continuum normalization by fitting a high degree 1D polynomial to individual orders and a low degree 2D polynomial to the full set of orders. As a final step, the wavelength is automatically corrected to the heliocentric frame of reference. Polarization spectra are initially extracted individually in the same fashion, two orthogonal states from each of the four sub-exposures. The mean intensity Stokes I spectrum is created simply by adding the four subexposure frames, while the polarization rate P is given by P R−1 = I R+1 (2.1) i1,⊥ /i1,k i4,⊥ /i4,k I2,⊥ /I2,k I3,⊥ /I3,k (2.2) where R4 = where the ij,⊥ and ij,k are the orthogonal polarization spectra of the j th supexposure, CHAPTER 2. OBSERVATIONS 58 Figure 2.2: Normalized ESPaDOnS spectrum of ξ 1 CMa with Stokes I/Ic (below, black), V /Ic (above, red) and diagnostic null N/Ic (middle, blue), where Ic is the continuum intensity. The V and N spectra have been amplified by ×20. and the orthogonal polarization states are 1/2 and 4/3 (Donati et al., 1997). Deriving polarization using this double-ratio method provides a first-order protection against all spurious signals that might arise due to the impossibility of recording the subexposures at precisely the same time, under the same conditions, and on the same pixels. In order to be more certain that details in the polarization profile are indeed real, and not just noise, a diagnostic null N was created by switching subexposures 2 and 4 in equation 2.2. Fig. 2.2 shows the resulting Stokes I, V and diagnostic N in the spectrum of the magnetic pulsator ξ 1 CMa (B0.7IV), with the Zeeman splitting due to the magnetic field clearly in evidence in V , and N showing no Zeeman splitting but providing a good match to the gaussian noise seen in the continuum of the V spectrum. Note in particular that Stokes V does not scale in linear fashion with I, as each line has a different magnetic sensitivity. The Libre-ESpRIT routines are built directly into CFHT’s Upena pipeline, which manages the multiple programs ongoing in QSO mode, reducing all exposures and CHAPTER 2. OBSERVATIONS 59 outputting fits files with both normalized and un-normalized files spectra, together with all relevent metadata (observation time, target name, Stokes vector, etc) annotated to the fits header (fits is a standard file-type for astronomical images). 2.3 Observations Sixty-seven Stokes V observations of β Ori were acquired with ESPaDOnS and Narval in spectropolarimetric mode between 9/2009 and 02/2010. The densest time sampling (five or six observations per night) coincided with a 26-day high-precision photometric campaign with the Microvariability and Oscillations in STars (MOST) space telescope (Moravveji et al., 2012). MOST, a Canadian microsatellite, is a dedicated asteroseismology instrument, able to remain focused on a single star for up to two months within a pointing error of 1 arcsecond, enabling it to collect un-interrupted micromagnitude precision photometry (Walker et al., 2003). The MOST observations of Rigel are shown in Fig. 2.3, together with the window function of the ESPaDOnS and Narval observations. This is supplemented with a six-year spectroscopic monitoring campaign conducted with the Automatic Spectroscopic Telescope (AST), a 2.0 m Cassegrain telescope at the Fairborn Observatory, which collected over two thousand spectra with SNRs of 50–150, over half of them in the period coincident with spectropolarimetric monitoring. The window function of AST data coinciding with the MOST observations is also shown in Fig. 2.3. Observing a star of Rigel’s considerable brightness presents an interesting set of opportunities, as well as challenges. The necessary SNR in each sub-exposure can be obtained in seconds (2.0 s sub-exposures for ESPaDOnS, 5.0 s for Narval), making the observing time allocated to the project limited by read time rather than exposure CHAPTER 2. OBSERVATIONS 60 Figure 2.3: Top panel: photometric magnitude relative to an initial reference intensity acquired with the MOST space telescope. Note the gap around HJD 2455175: during this period contact was lost with the satellite. The magnitude appears to be monotonically decreasing during this time. Bottom panel: window functions of spectropolarimetric (red diamonds) and spectroscopic (blue triangles) observations. CHAPTER 2. OBSERVATIONS 61 time. This makes observations very cheap, and in particulr makes spectropolarimetric monitoring of a star not known to be magnetic feasible. It can also be potentially frustrating, since seeing can easily introduce large distortions when photons are being collected on such short time scales. As Rigel’s period is unknown there was no need to schedule observations. Instead, some were spaced apart by days or weeks, whilst on other nights (most notably those coincident with MOST photometry) multiple observations were collected. Linear spectropolarimetry especially was aquired on these nights, all with ESPaDOnS. The log of ESPaDOnS observations is given in Table 2.1, and the log of Narval observations in Table 2.2. The quality of the spectra are in general excellent: the peak signal-to-noise ratios (SNRs) per 1.8 km s−1 spectral CCD pixel in the reduced spectra range from about 450 to about 1300, with mean SNRs of 698 in the Narval spectra and 883 in the ESPaDOnS spectra (for a given observation, we report here the highest SNR for a given order). While the normalized spectra are available through the Upena pipeline, in this case the unnormalized spectra were normalized interactively order-by-order using custom IDL software2 . Each order was normalized first by fitting a third-order polynomial to the continuum over 250 pixel bins; the pre-normalized continuum was then renormalized with a fifth-order polynomial over 50 pixel bins. At both steps pixels departing from the continuum flux by greater than three standard deviations were removed automatically. The process was monitored to ensure that pixels in the wings of particularly broad lines (such as H Balmer or Paschen lines), or pixels in absorption or emission lines at the edge of the orders, were also removed, keeping the normalization as close as possible to the true continuum. The first ESPaDOnS observation from the time series is shown over various wavelength ranges in Fig. 2.4. Telluric lines 2 Originally programmed by Véronique Petit CHAPTER 2. OBSERVATIONS 62 are indicated in the top panel, which shows the full spectral range. Successive panels show successively smaller spectral ranges; in the final panel, resolved spectral lines, including numerous weak Fe ii lines, are visible across a 10 nm spectral range. CHAPTER 2. OBSERVATIONS 63 Figure 2.4: Normalized ESPaDOnS Stokes I spectrum of β Ori. The full spectrum is shown in the top panel; subsequent panels show successively smaller spectral ranges in the best resolved part of the spectrum. In the top panel telluric bands are identified by molecular species; in the bottom panel, spectral lines are identified by ionization species. Note the large number of well-resolved metal lines (short vertical lines correspond to Feii lines). CHAPTER 2. OBSERVATIONS Odometer Number 1145577 1145776 1146006 1146187 1146322 1146334 1146346 1146496 1146508 1146660 1146676 1146692 1146704 1146882 1146894 1146906 1146926 1147101 1147117 1147129 1147141 1147385 1147397 1147409 1147421 1158325 1158519 1158531 1158551 1162263 1162861 1163348 1164247 UT Date HJD Peak SNR MM/DD/YY HH:MM – 2455000 12/01/09 12:25 167.0174 714 12/02/09 11:43 167.9887 916 12/03/09 11:54 168.9962 967 12/04/09 08:23 169.8495 566 12/05/09 08:25 170.8510 793 12/05/09 10:05 170.9208 810 12/05/09 11:49 170.9930 714 12/06/09 09:53 171.9122 956 12/06/09 14:41 172.1125 819 12/07/09 07:40 172.8199 988 12/07/09 10:10 172.9242 939 12/07/09 12:39 173.0274 1070 12/07/09 14:21 173.0984 1014 12/08/09 06:40 173.7780 836 12/08/09 08:25 173.8510 672 12/08/09 10:08 173.9227 897 12/08/09 11:57 173.9984 936 12/09/09 06:52 174.7863 891 12/09/09 09:25 174.8926 1027 12/09/09 11:07 174.9639 978 12/09/09 12:49 175.0347 1055 12/10/09 06:36 175.7753 834 12/10/09 08:21 175.8485 936 12/10/09 10:04 175.9198 985 12/10/09 11:46 175.9908 972 01/05/10 12:06 202.0047 621 01/06/10 06:15 202.7610 858 01/06/10 07:02 202.7932 958 01/06/10 12:53 203.0372 963 01/23/10 06:24 219.7667 964 01/26/10 05:25 222.7260 964 01/28/10 04:38 224.6932 924 02/01/10 10:33 228.9399 694 Table 2.1: ESPaDOnS Observations 64 CHAPTER 2. OBSERVATIONS UT Date HJD Peak SNR MM/DD/YY HH:MM – 2455000 09/29/09 03:18 103.6395 724 10/03/09 02:12 107.5935 1001 10/04/09 02:39 108.6128 653 10/13/09 02:43 117.6162 996 10/17/09 05:18 121.7237 892 10/18/09 04:11 122.6778 555 10/19/09 04:58 123.7100 817 10/26/09 03:07 130.6339 786 10/26/09 03:11 130.6365 762 10/27/09 01:53 131.5821 710 10/27/09 01:56 131.5847 712 10/28/09 02:11 132.5947 735 10/30/09 05:31 133.6841 705 12/09/09 21:33 175.4035 884 12/10/09 20:56 176.3775 541 12/11/09 20:59 177.3795 797 12/11/09 21:02 177.3821 741 12/11/09 21:06 177.3847 828 12/11/09 21:10 177.3874 801 12/11/09 21:14 177.3900 824 12/15/09 02:45 180.6199 298 12/15/09 19:29 181.3170 509 12/15/09 19:35 181.3212 990 12/17/09 19:38 183.3238 824 12/20/09 18:38 186.2799 150 01/04/10 22:25 201.4391 904 01/06/10 20:50 203.3733 724 01/15/10 19:32 212.3183 165 01/22/10 20:25 219.3546 528 01/25/10 23:58 222.5027 1105 02/03/10 20:06 231.3406 135 02/14/10 18:47 242.2854 767 02/18/10 18:50 246.2872 464 Table 2.2: Narval Observations 65 Chapter 3 Spectroscopic Measurements and Analysis While the primary aim of the observing campaign was to investigate the magnetic properties of Rigel with the highest practical precision, the very high spectral resolution of the spectropolarimeters ESPaDOnS and Narval, combined with the light collecting power of the 3.6 m CFHT and the 2 m TBL, provides us with some of the highest quality spectra ever collected for β Ori. In this chapter we investigate the spectral variability of Rigel during this period through an examination of radial velocities, equivalent widths, dynamic spectra and temporal variance spectra. In particular we are interested in distinguishing between variability arising due to pulsations and that due to activity or structure in the wind, both to compare to previous observations, as context to the magnetic analysis which follows, and to ensure the results of the magnetic analysis are uncontaminated by unfit spectra or spectral lines showing particularly complex variability. The Balmer lines, once again most notably Hα, are quite variable: although the 66 CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 67 Figure 3.1: Top: Individual Hα observations in black; model atmosphere spectrum in red. Bottom: Difference between individual spectra and comparison spectrum. spectral deformations in the present time series are not as dramatic as behaviour seen in previous observations (e.g. those reported by Kaufer et al., 1996a, 1996b), the time evolution of the wind lines is still complex, showing P Cygni, inverse P Cygni, and absorption profiles over the period of observation. This is of no direct concern to the magnetic analysis, as Balmer lines are automatically excluded for reasons explained in Chapter 4. The top panel of Fig. 3.1 shows Hα line profiles for the entire 143 day observing period, with a synthetic profile shown for comparison. The synthetic profile was generated from an LTE model atmosphere, using atmospheric parameters derived from NLTE quantitative spectroscopy given by Przybilla et al. (2006) and in Table 1.1 of this thesis, and provides a good fit to the higher-numbered Balmer lines. The excess emission fills almost the entire line up to the continuum, with substantial variability within around ±100km s−1 of the mean radial velocity vsys = 16.8 ± 0.5km s−1 . The CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 68 bottom panel shows the residual flux when the theoretical profile is subtracted from the observational profiles, and suggests that the emission excess is stronger on the red-shifted half of the line, but more variable in the blue (this will be discussed further in conjunction with dynamic and temporal variance spectra). 3.1 Radial Velocities and Equivalent Widths In order to quantify the spectral variability, radial velocities and equivalent widths were measured. The former are a standard asteroseismological diagnostic of pulsational variability (Aerts, Christensen-Dalsgaard & Kurtz, 2010), while the latter shed some light on this subject but also carry potential information on rotationally modulated surface structures as well as the circumsteller environment. Radial velocities were measured by using IDL’s built-in gaussfit routine to find the core λc of a spectral line of rest wavelength λ0 using the standard Doppler equation: vrad λ0 c = 1− λc (3.1) where c is the speed of light in km/s. The number of measurements (29 lines across 78 spectra) requires that the process be automated. However, when applied blindly, in regions subject to telluric contamination and observations in which those lines are particularly strong, IDL’s gaussfit routine may fit the telluric line by mistake. This might introduce significant scatter in vrad . Due to the intrinsic variability of vrad , this could not be avoided by simply narrowing the integration range of the gaussian for all spectra, as a certain amount of continuum had to be included in the gaussian’s range so as to be sure of including CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 69 Figure 3.2: Left: Distribution of vrad for ESPaDOnS/Narval (red) and AST (blue) for all measurements in both data sets. Right: Distribution of vrad from individual line measurements across a single spectrum. Solid vertical line marks the mean; dotted lines indicate one standard deviation. Note the low-velocity outlier, discarded from the final measurement. the whole line. Thus the fitting process was monitored: when telluric lines did in fact shift the gaussian, the fitting range was narrowed to fit that particular spectrum so as to avoid the telluric line. In practice, no one spectral line is likely to provide an accurate measurement of a star’s radial velocity: lines are formed at different depths within the photosphere, and consequently may be subject to different velocity fields, especially if nonradial pulsations are at play. Blending of spectral lines may also lead to systematic errors in vrad , especially if the two lines are strongly blended, since the apparent centre of the absorption line will be shifted. This is avoided by using an ensemble of 29 spectral lines, whose rest wavelengths are given in Table 3.1. The line ensemble from most spectra yielded vrad measurements that appear to follow an essentially gaussian distribution (see Fig. 3.2), such that vrad was taken as the mean of vrad across the ensemble, with the error bar as the standard deviation (Aerts, Christensen-Dalsgaard & Kurtz, 2010). Individual line measurements deviating from the mean by more than two standard deviations were discarded, a cutoff chosen as a 3σ limit would include CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 70 all measurements, and a 1σ limit would include too few. This list of 29 lines was chosen to perform measurements as consistently as possible with those performed on data collected by the six-year AST campaign, provided courtesy of Guinan et al. (private communication). vrad was measured in the same fashion as described above, using the same 29 lines. Radial velocities from the present data are shown together with those measured from AST observations as a function of time in Fig. 3.3, while Fig. 3.4 shows the data over the most densely time sampled epoch. Comparing ESPaDOnS/Narval vrad measurements and those from AST data, the same general behaviour is apparent across both data sets: the radial velocity rises and falls with a ‘pseudo-period’ of perhaps 10–20 days, varying about a mean or systemic velocity (the component arising from Rigel’s motion through space) of vsys = 16.8 ± 0.3km s−1 within a range of approximately 10–25 km s−1 . Wavelength (nm) 501.572 501.845 503.244 504.099 504.769 505.604 512.755 516.904 526.422 531.655 542.865 543.281 545.388 547.360 550.973 Atomic Species He i Fe ii S ii Si ii He i Si ii Fe Fe ii Fe i Fe ii Si S ii S ii Si Si Wavelength (nm) 556.498 560.616 564.011 564.705 565.999 587.578 597.891 614.307 634.708 637.135 640.226 657.811 658.296 667.822 Atomic Species Si Si S ii S ii S ii He i Si i Ne Si ii Si ii Ne C ii C ii He i Table 3.1: Lines used in radial velocity measurements CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 71 Equivalent widths Wλ are calculated according to Wλ = Z (1 − Fλ )dλ (3.2) where Fλ is the flux at each wavelength interval, normalized by the mean flux immediately on either side of the line in order to account for small differences in normalization between spectra. As the distribution of the spectrum across pixels shifts somewhat with the heliocentric radial velocity of the telescope, the number of pixels subtending a given spectral range may vary by a small amount (or two pixels), leading to a small amount of scatter in Wλ . This was avoided by fixing the number of pixels along with the integration range. Error bars in Wλ are calculated from the error in flux, σ(Fλ ): 2 σ (Wλ ) = Z σ 2 (Fλ )dλ (3.3) Some of the spectra were strongly contaminated with telluric lines. In order to remove their potentially significant contribution, the Hα regions of the contaminated spectra were modified by excising the contaminated pixels and replacing them with a linear fit. This was performed on a spectrum-by-spectrum basis, in order to minimize modification to the spectra and because the location of the telluric lines was somewhat variable due to the heliocentric radial velocity correction. Wλ was also measured for other Balmer and metallic lines: the C ii lines at 657.81 nm and 658.30 nm, the stronger S ii doublet at 634.71 nm and 637.14 nm, together with the H i line at 667.80 nm. While the higher-numbered Balmer lines vary in a similar fashion to Hα, the amplitude is much less. The amplitude of variability in metallic lines is also much less than that of Hα: most of the nights are consistent with no change in Wλ for photospheric lines, regardless of how much vrad or Hα CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 72 Figure 3.3: (Top) vrad as a function of time, for ESPaDOnS/Narval (red circles) and AST (blue triangles) data; (middle) Wλ normalized to the mean measurement (0.06 nm) over the same interval for Si ii 634.7 nm; (bottom) Wλ normalized to the mean measurement (0.03 nm) as a function of time for Hα. Dotted lines indicate the period shown in Fig. 3.4. Wλ is changing (see Fig. 3.4). Wλ for the Hα line and the Si ii 634.7 nm line (both normalized to the mean measurement) are shown with vrad in Figs. 3.3 and 3.4. Hα is highly variable in Wλ , as expected from the complex line profile variability it is known to exhibit (see Fig. 3.1). It too shows a characteristic timescale of 10–20 days. Very little such variation is seen in the Si ii 634.7 nm line, where most of the apparent variation is well inside the error bars. A relatively static Wλ , despite large changes in vrad , is expected for lines that are primarily varying due to pulsational CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS Figure 3.4: As Fig. 3.3, for the most densely time sampled interval. 73 CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 74 modes rather than rotationally modulated photospheric or circumstellar structures, since the latter changes the total intensity of the line as the stellar disk is occulted or uncovered, while pulsations (to first order) simply redistribute the flux within the line (this is not strictly speaking true, as fluctuations in density and temperature can affect the luminosity, but it is not a bad approximation for small-amplitude, high-order NRPs such as those thought to characterize this star). 3.2 Dynamic Spectra Dynamic spectra map the difference between any given observation of a spectral line and a comparison spectrum, in this case a mean spectrum created from all observations, and are provided as an aid to visualization of LPV. Unless stated otherwise, when more than one observation was obtained on a single night a mean spectrum for that night is plotted in lieu of the individual observations. Emission relative to the comparison spectrum is shown in red and absorption in blue. One dimensional intensity profiles are shown beneath the dynamic spectra, with the comparison spectrum shown in red superimposed over the individual spectra in black. Below the intensity profiles the residual intensities are plotted, in order to explore small variations between spectra. Fig. 3.5 shows dynamic spectra for Hα through Hδ, while Fig. 3.6 shows dynamic spectra for two metallic lines, SiII 634.7 nm and HeI 667.8 nm (chosen for their strength relative to other lines in the spectrum, as well as their freedom from contamination by telluric lines). Of immediate note is that while the Hα clearly shows more emission than the higher-numbered Balmer lines, a visibly similar pattern of LPV can be seen in their dynamic spectra, in particular a blue-shifted absorption feature beginning around CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 75 Figure 3.5: Top: H Balmer lines of the individual spectra as compared with a mean spectrum. Top: Hα, Hβ; bottom: Hγ, Hδ. Colour corresponds to the absorption or emission relative to the mean. Where multiple spectra exist for one night, mean spectra created from the individual observations are shown. Middle: Individual Hα lines in black; mean spectrum in red. Bottom: Difference between individual spectra and mean spectrum. CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 76 HJD 2455165. However, their residual fluxes are fairly different from one another, with Hα and Hβ showing greater asymmetry between the blue and red halves of the line. This suggests that although the wind’s influence on Hγ and Hδ is not strong, at least some of the variability in these lines has its origin in circumstellar processes. Dynamic spectra of metallic lines (Fig. 3.6) show that in these lines, variability is confined more tightly within the line, to ±70km s−1 of the systemic velocity. Variability in the metal lines is substantially similar across different species, and while they are suggestive of some degree of correlation between metallic and Balmer line variability, a close examination reveals that the lines depart from their mean profiles in distinctive ways: for instance, around HJD 2455202, the Balmer lines show strong red-shifted pseudo-absorption, while the metal lines show blue-shifted pseudo-absorption. The O triplet at 777 nm shows particularly complex LPV. Dynamic spectra of Hα and Si ii are compared over the most densely time sampled epoch in Fig. 3.7. In Hα, an intriguing pseudo-absorption feature appears at the beginning of this period. While not nearly as dramatic as the HVAs reported by Kaufer et al. (1996b), Israelian et al. (1997) and Morrison et al. (2008), it shares many of the same properties: following an elevated blue-shifted emission, a blueshifted absorption feature appears at around –150 km s−1 , somewhat below the most recent upper limits for the wind terminal velocity suggested by Kaufer et al. (1996a) and Markova et al. (2008), v∞ ≤ 230km s−1 . Following this, the feature migrates redward, maintaining approximately the same intensity and width across the line until it reaches the systemic radial velocity vsys = 16 ± 0.3km s−1 , at which point it begins to dissipate. In Si ii 634.7 nm we see a pseudo-emission feature travelling from the blue to CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 77 Figure 3.6: As Fig. 3.5, for Si ii 634.7 nm (top left), He i 667.8 nm (top right), the O i triplet at 777 nm (bottom left) and the Fe ii 516.9 nm line (bottom right). CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 78 the red half of the line, while the blue-shifted half gradually enters into relative absorption. This is reminiscent of the behaviour expected for the activation of a low-mode nonradial pulsation, as examined by e.g. Kaufer et al. (1997) who showed that such a model approximately reproduced the photospheric LPV of HD 92207 (A0Ia). The presence of such pulsations are supported by both MOST photometry and AST radial velocities (Moravveji et al., 2012). Its coincidence with a weak HVA suggests that the two phenomena may be linked; indeed, the inner part of Hα (within ±70km s−1 of vsys ) shows a pseudo-emission component behaving in a similar fashion to that visible in Si ii 634.7 nm. Dynamic spectra revealed a curious feature at around HJD 2455175, in which a sudden departure from patterns of line profile variability both before and after the spectrum appears; this observation was omitted from the the dynamic spectra for the sake of clarity. The anomalous behaviour also appears in vrad and Si ii 634.7 nm Wλ (see Fig. 3.3). In order to take a closer look at the weak HVA, the radial velocity of its absorption feature was measured by finding the velocity of the point of lowest absorption within the residual flux. Fig. 3.7 shows a clearly monotonic evolution towards the systemic velocity, however its acceleration is uneven: at times the radial velocity of the feature seems to plateau, and then accelerate again towards the red, with a mean radial acceleration of −62 ± 43 cm s−2 and a peak radial acceleration of −136 cm s−2 . In principle, such measurements could be used to probe the kinematics of the circumstellar matter giving rise to the pseudo-absorption feature; however, the detailed modeling that would be involved exceeds the scope of this thesis. CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 79 Figure 3.7: Above: (Left) Hα (Right) Si ii 634.7 nm over the same period. Note that the weak HVA is preceded by an inverse P Cygni profile. Below : (Top) Radial velocity of the strongest absorption in the feature. (Bottom) Radial acceleration of the absorption minimum in the HVA-like feature in the Hα line. Open circles correspond to measurements from individual observations; filled squares are measured from nightly means. The mean acceleration is plotted with a dashed line, the standard deviation with dotted lines. CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 80 Figure 3.8: Temporal variance spectra for various Balmer (top) and metallic (bottom) linesa. The dotted line indicates vsys = 16.8 km s−1 . 3.3 Temporal Variance Spectra In order to visualize the line profile variability for the entire time series, Temporal Variance Spectra (TVS) were created for six spectral lines according to the method described by Fullerton et al. (1990). The TVS quantifies the amount of variability across the spectral line by computing for each pixel i: N 1 X T V Si = σ02 N − 1 j=1 Sij − S̄i p σjc Sij !2 (3.4) where Sij is the normalized intensity of the ith pixel of the j th spectrum in the time series, S̄i is the mean intensity of the ith pixel across all N spectra, and the contribution of each spectrum is weighted by σjc , the inverse of the SNR of the j th spectrum in a continuum band close to the spectral line, and σ0 , the inverse of the rms of the SNR of the time series. CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 81 Fig. 3.8 shows TVS for three H Balmer lines (Hα, Hβ and Hγ) and three metal lines (Si ii 634.7 nm, C ii 657.8 nm and He i 667.8 nm). The Narval observation on HJD 2455175, identified as anomalous in the vrad measurements and the dynamic spectra, was left out of the computation; this had no obvious effect on the results. In all cases the variability seems concentrated in two lobes located approximately symmetrically about vsys = 16.8 km s−1 . The peaks of variability are found at the maximum gradient of the photospheric profile, although variability in the core of the line is often substantial as well, a combination that indicates the simultaneous presence of nonradial and radial pulsational modes. Balmer lines of course show variability of a higher amplitude than seen in metal lines, not just at the peaks of the red and blue lobes but especially in the core. There are also substantial differences between Balmer lines, not entirely unexpected as an examination of the residuals in Figs. 3.5 shows that despite the similar pattern of LPV seen in the dynamic spectra, the amplitude of variability differs between lines. This is most likely due to the varying magnitude of contributions from the wind and nonradial pulsations. One puzzling aspect of the Hα TVS is that it seems to show greater variability in the red than the blue lobe, in contrast to the results reported by Kaufer et al. (1996a; see also Fig. 1.14), who found the blue-shifted component to be more variable than the red for almost all stars in almost all seasons. To investigate this further, Fig. 3.9 shows TVSs for Hα and Hβ in three 50 day bins: HJD 2455100–150, 150–200, and 200–250. In the first 50 day bin, the red-shifted lobe is more variable than the blue; in the second, and best-sampled, bin, the blue-shifted lobe is clearly more variable; while in the third bin variability seems to peak in the middle, with no clearly distinguishible lobes. CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS HJD 2455100 –2455150 HJD 2455150 –2455200 82 HJD 2455200 –2455250 Figure 3.9: Temporal variance spectra in successive seasons for Hα and Hβ. The dotted line indicates vsys = 16.8 km s−1 . To quantify the velocities of maximum variability, gaussians were fit to each lobe, with the integration ranges set interactively. Where the peaks converged to the point they were too blended to separate, only a single measurement was taken, and assigned to the red. Fig. 3.10 shows the velocities and amplitudes of these peaks in the Hα TVSs. In the red-shifted lobe, variability clusters around a mean of +56 km s−1 within a standard deviation of ±11 km s−1 , while in the blue lobe variability is more spread out, with a mean of –8.4 km s−1 and a standard deviation of 23 km s−1 . 3.4 Discussion This time series confirms previous observations of Rigel as a star variable in radial velocity and Hα equivalent width, with significant line profile changes on the scale CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 83 Figure 3.10: Amplitude of peak variability in the blue (diamonds) and red (circles) lobes of the TVS. Filled symbols denote ESPaDOnS/Narval measurements; open symbols correspond to the measurements reported by Kaufer et al. (1996a). vsys is indicated with the dotted line. of days to weeks. The blue-shifted halves of the Hα and Hβ lines seem to vary in a more complex fashion than either the red-shifted regions of those lines or the line profiles of lines with a pure photospheric profile, as seen in both the distribution of TVS peaks and the dynamic spectra; such blue-shifted variability is consistent with the influence of the stellar wind, especially in Hα which is formed at its base. At the same time, metallic lines exhibit only very small changes in Wλ for relatively large changes in vrad , suggesting a pulsational origin for the variability of these lines. Although the star appears to have been in a relatively quiescent phase in comparison with previous epochs, evidence is found for both the activation of pulsational modes and a persistent wind structure; in particular, a pseudo-absorption feature appearing at –100 km s−1 and migrating red-wards, reminiscent in behaviour (albeit much weaker than) the HVAs reported by e.g. Kaufer et al. (1996b). The feature’s CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 84 mean acceleration of −62 ± 43cm s−2 seems identical within error to Rigel’s surface gravity (log(g)= 1.75 ± 0.1 or 56 ± 12cm s−2 , Table 1.1, Przybilla et al. (2006)). However, even within the large error of these measurements, the peak acceleration of ∼ −136cm s−2 is larger than what one might expect from gravity alone. Since both radiation pressure and centripetal force would be expected to retard the downward acceleration on a cloud of infalling material, rather than speeding it up, this measurement is suggestive of other forces at work than just gravity, rotation, and radiation pressure: even if the maximum downward acceleration of the feature is discarded, the mean radial acceleration is still comparable to that of gravity, where one would expect it to be somewhat less. The progression of Wλ is also similar to the behaviour reproduced in the introduction for Rigel and HD 96919 in Fig. 1.15, albeit of lower amplitude and much shorter duration: where the HVAs lasted for ∼ 1–4 months and reached peak Wλ of 0.1–0.2 nm, this feature persists for ∼2 weeks and reaches a peak amplitude of ∼0.04 nm. As noted by Markova et al. (2008b), the rise time is roughly 1/2–2/3 the decay time: in this case, 3 days versus 5 days. Assuming a similar physical mechanism between HVAs and the weak HVA examined here, the shorter duration of the weak event suggests velocity fields more complex than simple rotational modulation. While spectral and photometric variability is a characteristic feature of magnetic early-type stars, the detailed phenomenology is quite different from the pattern seen here. Specifically, the variability of magnetic stars tends to follow a strictly periodic relationship, with Wλ and dynamic spectra synchronized to the same period as photometric brightness and, crucially, the longitudinal field variations. This synchronized variability in wind diagnostic lines arises due to the rotational modulation of wind CHAPTER 3. SPECTROSCOPIC MEASUREMENTS AND ANALYSIS 85 plasma confined into a magnetosphere by a stable (usually although not necessarily dipolar) magnetic field. While Rigel’s variability in vrad , Wλ and photometric brightness tends to share a common characteristic time scale, there is no apparent simple periodic relationship either within or between different types of measurements. Chapter 4 Magnetic Field Diagnosis 4.1 Detection and diagnosis of magnetic fields using the Zeeman effect In the presence of a magnetic field, atomic energy level transitions will be split and shifted from their rest energy (see Fig. 4.1). The split line components are called π and σ components, distinguished by their polarization: in the presence of a magnetic field aligned with the line of sight (a longitudinal field), π components vanish, while σ components attain opposite senses of circular polarization, left and right handed. The wavelength shift ∆λ (in nm) by which any given σ component will be displaced from its rest wavelength λ0 (in µm) in the presence of a magnetic field B (in G) is ∆λ = 4.67λ20 ḡB (4.1) where ḡ is the effective Landé factor of the transition, a unitless measure of the magnetic sensitivity of the line (Donati & Landstreet, 2009). ḡ is usually around 1.2, 86 CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 87 but ranging from 0 – i.e. a magnetic null line, for which no Zeeman splitting will be observed – up to about 3. The Zeeman effect can be used to detect magnetic fields using both polarized and unpolarized spectra. However, magnetic field strengths weaker than at least a few kG fail to broaden the line sufficiently for magnetic line splitting to be noticeable unpolarized stellar spectra, since the line broadening from rotation and macroturbulence (present even in slow rotators) is much stronger than the magnetic broadening. This effectively limits the utility of unpolarized high resolution optical spectroscopy for stellar magnetometry to the domain of slowly rotating stars with negligable turbulence and relatively strong magnetic fields. In stars with significant rotation or turbulence, Zeeman splitting can be detected by means of the difference of the right and left hand circularly polarized spectra, as illustrated in Fig. 4.1, where the actual splitting has been greatly exaggerated for illustrative purposes. The unpolarized intensity profile (black solid line) is broadened by both rotation and turbulence to ±50 km s−1 . The left and right hand circularly polarized spectra are shifted by a small amount from their rest wavelength, one blueshifted, the other red-shifted (dashed-dotted lines). Note that in the weak field regime (a few kG or less), the line splitting in Fig. 4.1 is greatly exaggerated and would be invisible on this scale: a 1 pm (0.001 nm) splitting at 500 nm corresponds to a Doppler velocity of 0.84 km s−1 , finer than can be measured by most spectrographs, including ESPaDOnS whose spectral resolution per pixel is 1.8 km s−1 . However, if the difference of the left and right handed circular polarization is taken, then (given the small wavelength shift involved) this is essentially the equivalent of taking the first derivative of the unbroadened profile (the top line in Fig. 4.1.) This is the ‘weak CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 88 Figure 4.1: Zeeman splitting of a spectral line Doppler-broadened by both rotation and turbulence. The right and left hand circularly polarized line profiles (red and blue dashed lines) are red- and blue-shifted from the rest wavelength (solid black line). Their difference is well-reproduced by the first derivative of the unsplit intensity profile (solid magenta line). field approximation’: V (v) ∝ gλ ∂I(v) ∂v (4.2) where v is the Doppler-shifted velocity, and I(v) and V (v) are the Stokes I and V profiles as functions of velocity (Donati et al., 1997). The weak field approximation is valid so long as the Doppler broadening is much greater than the magnetic broadening (Kochukhov et al., 2004). Polarization is described by the Stokes vector [I,Q,U,V ], where I is the total unpolarized intensity, Q and U represent the linear polarization, and V the circular polarization: CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 89 Q = hI0◦ − I90◦ i U = hI45◦ − I135◦ i V (4.3) = hIL − IR i where Iφ is the intensity the beam would have if filtered through a perfect polarimeter with a transmission axis set to φ relative to a reference direction, IL,R are the left and right hand circular polarizations, and h i denotes a temporal average. 4.2 Least Squares Deconvolution Early stellar magnetometry generally relied on information from a single spectral line (see Landstreet (1980) for a review of these methods), and allowed the detection of magnetic fields in some stars with error bars of at best 50 G (Donati & Landstreet, 2009). Modern methods utilizing multi-order echelle spectrographs are able to obtain the entire optical spectrum in a single exposure and so are able to collect polarization information from hundreds or thousands of spectral lines (depending on spectral type, with far fewer lines available for analysis in early-type stars in comparison to late-type stars.) The Least Squares Deconvolution (LSD) cross-correlation technique developed by Donati et al. (1997) allows information from all of the available medium-tostrong spectral lines to be combined into a single Zeeman signature with a noise level substantially below that achievable with single line techniques. LSD combines the information from spectral lines by means of the assumption that the spectrum can be reproduced by the convolution of a ‘mean’ line profile (the ‘LSD profile’) with an underlying spectrum of unbroadened atomic lines of given line CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 90 Figure 4.2: The observed (black) and LSD model (red) spectrum of ξ 1 CMa, showing Stokes V /Ic (above) and normalized I/Ic (below), where Ic is the intensity of the continuum. Horizontal blue lines mark the wavelengths and intensity weights of the line mask. CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 91 depth, Landé factor, and wavelength: the ‘line mask’, as described by Wade et al. (2000a) and illustrated in Fig. 4.2. In creating a single mean line profile, in essence assuming I(v) in equation 4.2 has the same shape for all lines, LSD necessarily ignores effects such as dependence on the depth of the lines’ formation within the photosphere or wavelength-dependent limb darkening (which simplifies integration over the surface of the stellar disk), and in addition assumes blended lines add linearly. The resulting LSD model of the Stokes I profile is thus a somewhat crude fit to any given line. However, so long as very strong lines with distinct shapes due to e.g. electron scattering or wind effects, such as H Balmer lines, H Paschen lines or He lines are excluded from the line mask, it provides a recognizable fit to the I profiles of those lines included in the analysis. In addition to finding the Zeeman signature which, when combined with the line mask, is the least squares solution to the reproduction of the Stokes V spectrum, a set of accurate error bars is achieved by propogating the photon-statistical uncertainties pixel-by-pixel through the process. LSD has the powerful advantage of greatly increasing the SNR of the Stokes V profile, thus enabling the detection of magnetic fields much weaker than can be probed using single-line methods. The objection might be raised however that, given the many assumptions LSD makes, the LSD profile may not be an accurate representation of the true line profile. This was investigated by Wade et al. (2000a), who compared the results obtained using LSD and those from single line methods for 129 hBz i measurements of 14 A and B type stars previously detected as magnetic. They found very good agreement between hBz i as measured from single lines and as obtained from LSD, however, the error bars resulting from LSD were much smaller: CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 92 Figure 4.3: (Left) An LSD profile for the magnetic pulsator ξ 1 CMa (B0.7IV). Above (red): Stokes V ; middle (blue): diagnostic N; Bottom (black): Stokes I. Zeeman splitting due to the magnetic field in ξ 1 CMa’s photosphere leads to a visible departure of Stokes V from gaussian noise. (Right) An LSD profile from Rigel. There is no apparent difference between V and N. whereas many of the stars were detected at only 3σ with typical 1σ error bars on the order of hundreds of G, in many cases LSD enabled measurements at the 50σ level, with error bars of 50 G or less. Comparing the distribution of reduced χ2 fits to longitudinal field measurements, they found essentially no difference between the results for the diagnostic N from the two methods, demonstrating that the LSD error bars are not underestimated. In a related study, Wade et al. (2000b) compared the Stokes V line profiles of 11 stars obtained from LSD to those for Fe ii lines at 492.4 nm and 501.8 nm, finding them to be in almost all respects quite similar. 4.2.1 Line Mask Development A custom LSD line mask was generated for Rigel using VALD (Vienna Atomic Line Database; Piskunov et al., 1995; Ryabchikova et al., 1997; Kupka et al., 1999, 2000) ‘extract stellar’ requests for a model star with Rigel’s surface gravity, effective temperature, abundances and microturbulence as determined through NLTE modeling CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 93 by Przybilla et al. (2006) (and given in Tables 1.1 and 1.2 of the present work). The original line mask was generated with a line threshold of 1% of the normalized continuum in order to include as many lines as possible (over 3000 spectral absorption lines) throughout the 370 nm – 1000 nm spectral range. Many of these lines were not, however, suitable for LSD, due for instance to blending with Balmer lines or contamination with telluric lines. Using custom IDL software1 the mask was cleaned by removal of these lines, as well as weak lines (those with a depth of less than 5% beneath the normalized continuum), which have been shown empirically to have a negligible contribution to the LSD profile. The oxygen triplet at 777 nm was also removed: while these lines are quite strong (∼60% of the normalized continuum) and uncontaminated by telluric or Balmer lines, the line profile variability in these lines is quite complex, as illustrated in Fig. 3.6. Following this filtering, using the same IDL software the depths of the remaining 90 spectral lines were then adjusted by hand (‘tweaked’) to match the observed line depths. This ad hoc procedure was performed without reference to the actual photospheric abundances or ionization balances, representing a purely empirical adjustment meant to maximize the agreement between the line mask and the observed spectrum and thus increase the SNR of the LSD profiles. To do this, the LSD profile was convolved with the line mask, and plotted over the stellar spectrum. The line weights were then adjusted interactively, and the LSD spectrum recomputed. With weights optimized to achieve a close fit to the line depths of the Stokes I spectrum, new LSD profiles, hopefully representing a better fit to the lines in the mask, are then computed. The new LSD profiles can then be used to tweak the mask again, and the process can be iterated as many times as necessary; an example of the resulting LSD 1 The code was written by Jason Grunhut CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 94 profiles is shown in the right panel of Fig. 4.3, with the magnetic star ξ 1 CMa shown for comparison, while the remainder of the LSD profiles can be found in Figs. 4.4–4.7. Figure 4.4: Individual ESPaDOnS LSD profiles are labelled with HJD - 2455000.000 and are presented in temporal order (left–right, top–bottom). Stokes I and V and diagnostic N are as in Fig. 4.3. V and N have been multiplied by a factor of 25. 4.3 Analysis Two methods were used to evaluate the presence of a photospheric magnetic field: a statistical test performed on the LSD Stokes V profile, and direct inference based on the significant detection of a longitudinal magnetic field. The former method, described by Donati, Semel, & Rees (1992) and Donati et al. (1997), employs the reduced χ2 of the signal in Stokes V within the bounds of the line profile. It reports the detection of a magnetic signature as ‘definite’ if the formal detection probability CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 95 Figure 4.5: ESPaDOnS LSD profiles, as Fig. 4.4. over several pixels within the line is greater than 99.999% (a level of at least five sigma). The detection probabilities outside the Stokes V line profile, and inside the diagnostic null (N ) line profile, should also both be negligible for a detection to be considered ‘definite’. A ‘marginal’ detection corresponds to a detection probability between 99.9% and 99.999%. Anything beneath this threshold is, of course, a nondetection. An example of a profile yielding a clear definite detection is shown in the right panel of Fig. 4.3, which shows the LSD profile of ξ 1 CMa (B0.7IV), a slowly pulsating magnetic B star: Stokes V shows clear Zeeman splitting, while N shows noise. Note CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 96 also the asymmetry in the I profile, reflecting the pulsational character of this star. To the right is an LSD profile for Rigel, created using the customized line mask described above. The Stokes I profile is much broader than ξ 1 CMa’s, as expected due to β Ori’s greater v sin i and macroturbulence. The Stokes V profile is also a clear non-detection. The second magnetic diagnostic applied to the LSD profiles is direct measurement of the longitudinal magnetic field hBz i in G from the first-order moment of the Stokes V profile within the line: R vV (v)dv R hBz i = −2.14 × 1011 λḡc [Ic − I(v)] dv (4.4) where v is the velocity in km s−1 within the profile measured relative to the centre of gravity (Mathys et al., 1989; Donati et al. 1997; Wade et al., 2000b), and λ and ḡ are the reference values of the wavelength (in nm) and Landé factor used in computing the LSD profiles. The longitudinal field hNz i of the null profile was also computed in order to provide a comparison to hBz i: in the presence of a detectable magnetic field, there should be a clear difference between hBz i and hNz i; otherwise the two distributions should be statistically equivalent. The uncertainties associated with hBz i and hNz i were determined by propagating the formal (photon statistical) uncertainties of each pixel through Eqn. 4.4. Due to spectral irregularities likely of an instrumental origin, two spectra – the observations on 30/10/2009 and 09/12/2009 – were left out of the magnetic analysis. The former observation was revealed as flawed shortly after its aquisition and was not validated. The latter, as discussed in Chapter 3, was discarded after spectral analysis revealed its LPV to be out of character with that of nearby lines. CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 97 Figure 4.6: Narval LSD profiles, as Fig. 4.4. 4.3.1 LSD profiles from individual spectra The LSD Stokes V profiles obtained from all 64 analyzed spectra are shown in Figs. 4.4 and 4.5. All profiles were extracted with common SNR-weighted mean Landé factor of ḡ = 1.29 and a mean wavelength of 528 nm, over a velocity grid of 2 km s−1 step size. The median LSD SNR is 6644 in the LSD profiles from ESPaDOnS spectra, 5166 in the Narval spectra, and 5896 overall. As can be seen in Figs. 4.4 and 4.5, none of the individual LSD Stokes V profiles can be distinguished by eye from the diagnostic nulls. The statistics associated with each profile are provided in Tables 4.1 (ESPaDOnS) and 4.2 (Narval). In only four cases is there a marginal detection associated with the LSD profile inside the lines; in no case is there a marginal detection outside the stellar lines. The marginal detections 98 CHAPTER 4. MAGNETIC FIELD DIAGNOSIS HJD –2455000 167.0174 167.9887 168.9962 169.8495 170.8510 170.9208 171.9122 172.1125 172.8199 172.9242 173.0274 173.0984 173.7780 173.8510 173.9227 173.9984 174.7863 174.8926 174.9639 175.0347 175.7753 175.8485 175.9198 175.9908 202.0047 202.7610 202.7932 203.0372 219.7667 222.7260 224.6932 228.9399 LSD SNR 4935 6412 6749 3853 5611 5476 6678 5585 6964 6717 7312 7132 5896 4669 6392 6542 6354 7314 6974 7351 6079 6713 7061 6822 6030 6029 6817 6688 6644 6770 6348 4865 DF FAP N N N N N N N N N N N N N N N N N N N N N M N N M M N N N N N N 0.3905 0.5943 0.1143 0.8773 0.7150 0.1152 0.2053 0.5807 0.6662 0.1108 0.6954 0.4809 0.3259 0.4302 0.7425 0.7061 0.7487 0.9767 0.0374 0.2553 0.0260 0.0074 0.5661 0.2743 0.0053 0.1834 0.1202 0.2152 0.0109 0.0244 0.2364 0.1902 hBz i (G) 15 -1 39 16 -2 16 9 31 16 2 25 -8 -9 3 -20 6 15 -7 -1 -9 -4 -17 7 -24 -1 -1 17 33 37 3 9 19 σB Bz /σB (G) 19 0.78 15 -0.10 14 2.79 25 0.64 17 -0.10 17 0.95 19 0.46 17 1.80 14 1.20 14 0.17 13 1.91 13 -0.61 16 -0.58 20 0.15 15 -1.33 15 0.40 15 0.97 13 -0.51 14 -0.09 13 -0.68 16 -0.25 14 -1.18 14 0.55 14 -1.71 15 -0.09 15 -0.09 14 1.25 14 2.42 14 2.69 13 0.22 15 0.62 18 1.05 hNz i σN Nz /σN (G) (G) 18 19 0.96 13 15 0.91 1 14 0.06 10 25 0.43 -10 17 -0.57 5 17 0.27 -9 19 -0.48 -10 17 -0.55 9 14 0.63 27 14 1.93 4 13 0.34 4 13 0.34 15 16 0.93 19 20 0.96 30 15 2.01 17 15 1.17 10 15 0.69 -20 13 -1.52 -5 14 -0.35 3 13 0.23 -24 16 -1.49 16 14 1.14 -2 14 -0.12 -1 14 -0.05 4 15 0.26 4 15 0.26 11 13 0.83 8 14 0.58 26 14 1.88 -18 13 -1.32 11 15 0.74 3 19 0.18 Table 4.1: ESPaDOnS LSD Statistics and longitudinal field measurements. DF is the detection flag (D = definite, M = marginal, N = no detection). FAP is the False Alarm Probability. V and N refer to measurements performed on the circular polarization and diagnostic null spectra, respectively. 99 CHAPTER 4. MAGNETIC FIELD DIAGNOSIS HJD LSD -2455000 SNR 103.6395 4972 107.5935 6707 108.6128 4396 117.6162 6819 121.7237 6198 122.6778 3807 123.7100 5407 130.6339 5131 130.6365 5197 131.5821 4753 131.5847 4773 132.5947 5054 133.6841 4877 176.3775 6174 177.3795 3720 177.3821 5358 177.3847 5166 177.3874 5528 177.3900 5586 180.6199 5556 181.3170 2036 181.3212 3334 183.3238 6755 186.2799 5842 201.4391 5996 203.3733 4692 212.3183 1064 219.3546 3611 222.5027 7584 231.3406 777 242.2854 5351 246.2872 3170 DF FAP N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N N 0.9998 0.9753 0.9907 0.8537 0.9455 0.2837 0.8703 0.9817 0.7991 0.9919 0.9034 0.7033 0.7833 0.9958 0.3341 0.8330 0.9458 0.9994 0.7336 0.9271 0.9828 0.2224 0.9852 0.1524 0.7250 0.2462 0.9836 0.9942 0.9083 0.8678 0.9043 0.8785 hBz i σB hBz i/σB (G) (G) -10 19 -0.52 -19 14 -1.42 2 22 0.08 29 14 2.10 10 15 0.68 1 25 0.06 -9 18 -0.51 -16 18 -0.86 -8 18 -0.44 7 20 0.35 -1 20 -0.03 8 19 0.41 -7 20 -0.34 -14 16 -0.89 11 26 0.41 -11 18 -0.61 -19 19 -1.02 1 18 0.08 -8 17 -0.47 11 18 0.61 56 48 1.17 41 28 1.44 -5 14 -0.39 -5 16 -0.35 -18 15 -1.21 -1 20 -0.06 35 89 0.39 -9 26 -0.35 7 12 0.63 104 119 0.88 -13 17 -0.74 16 30 0.54 hNz i σN hNz i/σN (G) (G) 12 19 0.66 0 14 0.01 24 22 1.09 22 14 1.57 -2 15 -0.12 5 25 0.22 -25 18 -1.41 -42 18 -2.36 -7 18 -0.39 -18 20 -0.90 -22 20 -1.14 29 19 1.48 -27 20 -1.33 0 16 0.03 -45 26 -1.72 -22 18 -1.23 -17 19 -0.92 -27 18 -1.54 10 17 0.55 15 18 0.85 46 48 0.96 26 28 0.92 8 14 0.54 8 16 0.48 17 15 1.15 4 20 0.20 -40 90 -0.44 -19 26 -0.75 3 12 0.29 101 119 0.84 7 18 0.39 12 30 0.41 Table 4.2: Narval LSD Statistics and longitudinal field measurements. DF is the detection flag (D = definite, M = marginal, N = no detection). FAP is the False Alarm Probability. V and N refer to measurements performed on the circular polarization and diagnostic null spectra, respectively. CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 100 Figure 4.7: Narval LSD profiles, as Fig. 4.4. are all for ESPaDOnS observations. In measuring the longitudinal magnetic field hBz i, the integration ranges employed in the evaluation of Eqn. 4.4 associated with each LSD profile were selected through visual inspection so as to include the entire span of the Stokes I profile. Rigel’s relatively small pulsations and lack of any detectable magnetic field meant that a common integration range could be adopted for all spectra, [−55, +80 ]km s−1 . The longitudinal magnetic field hBz i measurements from Stokes V profiles are shown as a function of HJD in Fig. 4.8 (top), while the measurements from the diagnostic nulls hNz i are shown in Fig. 4.8 (bottom); they are tabulated in Tables CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 101 Figure 4.8: Open blue triangles are measurements from individual V (above) and N (below) spectra; filled red circles correspond to measurements from nightly means. The right panels show in detail the most densely time sampled period. 4.1 and 4.2. Both sets of measurements are formally consistent with a null result, with a minimum error bar of 12 G and a median error bar of ∼20 G, implying 3σ upper limits of 36 G and 60 G, respectively. There are no hBz i measurements outside this latter range. The longitudinal field averaged across the time series is 2.5±1.6 G. In order to compare the Stokes V and N measurements in a more rigorous fashion, a two-sample Kolmogorov-Smirnov (K-S) test was performed (see Fig. 4.9). The K-S test analyzes the normalized cumulative distribution of hBz i/σ(hBz i) (the longitudinal field significance) in comparison with hNz i/σ(hNz i) (the null significance), testing for a statistically significant difference between the two distributions by comparing the maximum difference D between the two distributions to a reference probability Dα (Peacock, 1983). If D is greater than Dα , the null hypothesis (i.e. that the two CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 102 Figure 4.9: K-S tests comparing the cumulative distribution of hBz i/σ(hBz i) (red dashed line) to hNz i/σ(hNz i) (blue dash-dotted line). The solid green line denotes the maximum difference between hBz i/σ(hBz i) and hNz i/σ(hNz i). Top left: all measurements; bottom left: nightly mean measurements; top right: ESPaDOnS measurements; bottom right: Narval measurements. CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 103 distributions come from the same set) is rejected with confidence α. For distributions larger than those found in standard online reference tables, Dα can be approximated by Dα = c(α) r n1 + n2 n1 n2 (4.5) where n1 and n2 are the number of observations in each set and c(α) is a coefficient that increases with decreasing α. For the full set of 64 hBz i measurements, D0.01 = 0.29; if the ESPaDOnS and Narval measurements are treated as distinct sets, D0.01 = 0.41. For the full set of measurements we find, D = 0.09; for the ESPaDOnS measurements, D = 0.20; for the Narval measurements, D = 0.13. In all cases the K-S test shows no statistically significant difference between the Stokes V and diagnostic N measurements, allowing us to accept the null hypothesis at 99% confidence. We thus conclude from both the detection probabilities evaluated from individual LSD profiles, and from the longitudinal field measurements from those profiles, that there is no evidence for a photospheric magnetic field in the longitudinal fields of the Stokes V spectra examined here, with a 3σ upper limit of approximately 60 G. 4.3.2 LSD profiles from co-added spectra It is possible to raise the SNR significantly by co-adding the observations, either by creating a mean spectrum from some set of spectra and then extracting an LSD profile from that spectrum, or simply by creating a mean LSD profile by averaging the LSD profiles of individual spectra. In this section we co-add the spectra obtained on individual nights. We begin by interpolating each spectrum or LSD profile onto a CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 104 common velocity scale. The weighted mean flux hF i is then calculated by weighting the contribution of each spectrum or LSD profile by its error: hF i = 1 X fij hσi i,j σij2 (4.6) where fij is the flux of the ith observation in the j th velocity bin, σij is the corresponding error bar, and hσi is the weighted mean error bar, 1 hσi = sX 1 σij2 i,j (4.7) A maximum of 5 usable spectra were obtained on any given night. Mean profiles were created both by combining the LSD profiles of individual nights and by generating LSD profiles from mean spectra created directly from the original spectra; the results in either case are the same, yielding LSD SNRs ranging from 14,000 (2 spectra) to 30,000 (5 spectra) (see Table 4.3). While coadded spectra can be easily analyzed using the Libre-ESpRIT package, in order to analyze the detection probability of the directly coadded LSD spectra it was easier to implement the statistical test described above as an IDL program2 . Analysis of the detection probabilities revealed only non-detections within the LSD Stokes V profiles, while evaluation of Eqn. 4.4 obtained a minimum error bar of 7 G and a median error bar of 10 G. The hBz i and hNz i measurements from nightly co-added spectra are tabulated in Table 4.3, and shown as a function of time in Fig. 4.8. The average hBz i for the nightly coadded spectra is identical to that of the individual measurements; however, 2 Originally written by Véronique Petit. 105 CHAPTER 4. MAGNETIC FIELD DIAGNOSIS if we restrict our attention to the best-sampled period (HJD 2455170–181) we find average measurements of hB¯z i = 3.2 ± 2.2 G and hN¯z i = 0.0 ± 2.2 G. Evaluating hBz i and hNz i from the nightly co-added spectra with the K-S test (the results of which are shown in Fig. 4.9) yields D = 0.17, as compared to D0.01 = 0.65, confirming the lack of any statistically significant difference between Stokes V and N at 99% confidence. HJD NS –2455000 130.635 2 131.583 2 170.922 3 172.012 2 172.968 4 173.888 4 175.016 5 175.982 5 177.385 5 181.086 3 202.991 4 222.614 2 165–177 37 103–246 64 LSD DF FAP hBz i σB SNR (G) (G) 15155 N 0.9218 -12 13 13928 N 0.9788 4 14 19285 N 0.8445 7 10 18071 N 0.5420 22 11 28644 N 0.9140 9 7 24481 N 0.9895 -6 8 28883 N 0.7715 -2 7 29862 N 0.9785 -11 7 23533 N 0.9445 -7 9 14005 N 0.9803 22 14 24887 N 0.9998 15 8 21121 N 0.4847 6 9 73994 N 0.8904 3.0 1.9 95002 N 0.9887 2.7 1.5 hBz i/σB -0.93 0.29 0.66 1.97 1.27 -0.71 -0.36 -1.64 -0.79 1.52 1.94 0.66 1.58 1.80 hNz i σN hNz i/σN (G) (G) -25 13 -1.96 -19 14 -1.33 -5 10 -0.45 -6 11 -0.50 10 7 1.54 22 8 2.71 -4 7 -0.56 -1 7 -0.10 -17 9 -2.03 20 14 1.42 8 8 1.03 -5 9 -0.61 1.5 1.9 0.79 1.8 1.5 1.20 Table 4.3: Longitudinal field measurements from co-added spectra. The date corresponds to the average HJD of the individual spectra. The number of spectra used to create the profile is indicated. DF = Detection Flag (N for null), and FAP is the false alarm probability. 4.3.3 Grand mean LSD profile As a final step, a grand mean LSD profile was created for all 65 circularly polarized spectra in the data set (see Fig. 4.10). In order to compensate for radial velocity variation, as explored in Chapter 3, each line profile was brought to a common rest CHAPTER 4. MAGNETIC FIELD DIAGNOSIS 106 Figure 4.10: Black: an LSD profile from a single night; red: from 5 spectra taken on the same night; blue: from all spectra in the time series. wavelength before coaddition. This LSD profile has a SNR of ∼95,000, is a nondetection, and yields a longitudinal field of hBz i= 2.7 ± 1.5 G, in agreement with the average longitudinal field measurement of hB¯z i = 2.3 ± 1.6 G. The diagnostic null profile yields hNz i= 1.8 ± 1.5 G. It might be objected that the interval over which this LSD profile has been averaged is likely to be at least one and possibly several rotations, and so any slight Zeeman splitting arising due to magnetic fields could easily be smeared out over multiple cycles. However, the LSD profile computed from the nightly means restricted to the most densely observed period – a time span of 12 days – yields an only slightly smaller SNR of ∼74,000 and a comparable longitudinal magnetic field measurement of 3.0 ± 1.9 G (see Table 4.10). Unless the inclination of Rigel’s rotational axis from the line of sight is very small, this LSD profile is unlikely to sample a significant fraction of the rotational period and thus any smearing of a Zeeman signature over different phases of rotation should be negligible. Chapter 5 Modeling and Upper Limits The most sensitive diagnostic of a stellar magnetic field is the velocity-resolved Stokes V profile as supposed to the simple longitudinal field measurement hBz i. This is because there are null hBz imeasurements which still yield a detectable Stokes V signature within the line profile. Since a dipolar magnetic field (as well as higherorder fields such as quadrupoles or octopoles) varies longitudinally across the surface of the stellar disk, synthesizing the contributions of individual components of the Stokes V profile requires that disk integration be performed in two dimensions, as supposed to the 1-D integration sufficing for modeling the rotational broadening of the Stokes I profile (under the approximation that the star is not rotating differentially as the Sun does, thought to be true for early-type stars since their radiative envelopes presumably do not support convection). The remainder of this chapter concerns the mechanics of such a disk integration model, and its application to the LSD profiles described in the previous chapter in order to place upper limits on Rigel’s dipolar magnetic field, and thus constrain the possibilities for magnetic wind confinement in this star. 107 CHAPTER 5. MODELING AND UPPER LIMITS 5.1 108 Disk Integration and Synthetic LSD Profiles In order to model the various line-broadening mechanisms at work in stellar atmospheres, it is necessary to consider a star not as an unresolved point but as a resolved disk which can itself be subdivided into smaller sections. Each section of the disk will produce its own spectrum, modified due to its own own unique properties e.g. the line-of-sight component of the local rotational velocity. The disk sections are then combined, properly weighted for their total contribution to the intensity due to limb darkening and projection on the sky, in order to recover the theoretical line profile of the disk as an unresolved point source (see Fig. 5.1). One-dimensional disk integration is simple, but limited in its usefulness to modeling the Doppler broadening due to rotational motion in a star with no differential rotation. The disk is simply divided up into vertical strips. The Doppler velocity of a given strip is then vlos = xv sin i (5.1) where i is the inclination of the rotation axis from the line-of-sight and x = {−1, 1} is simply the horizontal location on the unit disk. Each strip is shifted by its Doppler velocity and then weighted according to its area dA. To model line broadening from complex velocity fields such as macroturbulence or pulsations, or from nonlinear phenomena such as magnetic fields, it is necessary to extend the disk integration to two dimensions, dividing the disk into a grid of points. While the disk itself is a two-dimensional artifact, it represents three-dimensional object which is most naturally represented with spherical coordinates (r, θ, φ)∗ . Since to a first approximation photospheric phenomena take place at the same distance CHAPTER 5. MODELING AND UPPER LIMITS 109 Figure 5.1: A pole-on view of a stellar disk. Points are scaled according to their fractional contribution to the total intensity of the disk, based upon their projection on the sky and stellar limb darkening. from the stellar core, and this radius is arbitrary, it simplifies the calculations to consider the star as having unit radius. dA is then dA = sin θdθdφ (5.2) The contribution of each point to the total intensity must also take into account foreshortening due to projection on the sky (see Fig. 5.1), which since we are working at unit radius can be achieved simply scaling by cos θ. We also weight the flux contribution of each point by limb darkening, which we represent here with a linear limb darkening law 1 − ǫ + ǫ cos θ, where ǫ is the limb darkening coefficient (Gray, 2005). Although more sophisticated approaches to limb darkening exist (e.g. Claret, 2000), the linear law is accurate to within a few percent and so is sufficient for our purposes here (Aufdenberg et al., 2008). The relative flux fi of the ith point is then fi = dA cos θ(1 − ǫ + ǫ cos θ) (5.3) It is next necessary to consider the inclination i of the star’s rotational axis with 110 CHAPTER 5. MODELING AND UPPER LIMITS respect to the line of sight, its rotational phase, and the obliquity of the magnetic field from the rotational axis. This is all much easier in Cartesian coordinates, so we convert (r, θ, φ)∗ into (x, y, z)∗ using the transformations x∗ = sin θ∗ cos φ∗ (5.4) y∗ = sin θ∗ sin φ∗ (5.5) z∗ = cos θ∗ (5.6) (5.7) Representing each (x, y, z)∗ of coordinates as a single-column matrix, we can now use simple rotation matrices to perform the necessary operations. If the z -axis is taken to correspond to the line of sight, then the rotation matrix Ri about the x -axis can be used to transform into the line-of-sight reference frame, thus 0 1 0 Rx = 0 cos i − sin i 0 sin i cos i (5.8) Denoting the rotational phase by ψ, the rotation matrix Rψ can be used to rotate about the new z -axis, transforming the coordinates to the appropriate phase: cos ψ − sin ψ 0 Rz = sin ψ cos ψ 0 0 0 1 (5.9) Since any given model will have a certain (i, ψ), we can save some time by combining the two matrices into one (really just a simplified Euler matrix, with the third 111 CHAPTER 5. MODELING AND UPPER LIMITS angle set to 0): Riψ cos ψ − cos i sin ψ sin i sin ψ = sin ψ cos i cos ψ − sin i cos ψ 0 sin i cos i (5.10) Points comprising the visible portion of the disk are simply those for which z ≥ 0. A further rotation into the reference frame of the magnetic dipole is given by a matrix similar to Ri : 0 0 1 Rβ = 0 cos β − sin β 0 sin β cos β (5.11) Using Rβ on (x, y, z)iψ we obtain the magnetic Cartesian coordinates, (x, y, z)B . Ultimately we wish to find the line of sight component of the magnetic field, Bz , at each point on the grid. If the magnetic field is a dipole, the Cartesian components of the field strength are given by Bdip 3 cos θB sin θB cos φB 2 Bdip 3 cos θB sin θB sin φB By = 2 Bdip Bz = (3 cos2 θB − 1) 2 Bx = (5.12) (5.13) (5.14) (5.15) Using coordinate transformations inverse to those of equation 5.10, we can obtain the angular components of the spherical magnetic coordinates from the Cartesian coordinates: CHAPTER 5. MODELING AND UPPER LIMITS cos θB = zB q sin θB = x2B + yB2 xB sin θB yB sin φB = sin θB cos φB = 112 (5.16) (5.17) (5.18) (5.19) (5.20) with which we solve equation 5.12, extracting Bz for those points which are visible. In the case of a magnetic spot the field can be modelled as purely radial: the field is normal to the surface of the star at every point in the spot. A centre (θ, φ) is defined, along with an angular radius Ω. Then, assuming a circular spot and using the law of cosines, any point on the grid whose angular distance from the centre than some angle α will be in the spot, where α = cos−1 (cos θ∗ cos θspot cos(φ∗ − φspot ) + sin θ∗ sin θspot ) (5.21) is simply the great circle distance from between any two points on a sphere. The line of sight component of the magnetic field for each point in the spot is then simply Bspot cos θ. To use this grid to find the resulting Stokes I and V profiles, a local line profile is required. In most cases, this local line profile is simply the thermally broadened absorption line profile of whatever atomic line the model is attempting to reproduce, and so we must know the atomic weight m, the temperature T and the rest wavelength λ0 in order to obtain a Gaussian with dispersion CHAPTER 5. MODELING AND UPPER LIMITS 113 Figure 5.2: An oblique rotator model with i = 60◦ , β = 60◦ , Bdip = 1000 G, and v sin i= 30 km s−1 , with ten rotational phases in phase steps of 0.1, clockwise from phase 0.0 in the top left panel. The points of the stellar disk have been scaled as in Fig. 5.1. Colour indicates longitudinal magnetic field strength. Note that, due to the particular combination of i and β, the south pole (phase 0.5) yields a stronger Zeeman signature than the north pole (phase 0). 114 CHAPTER 5. MODELING AND UPPER LIMITS λ0 ∆λD = c r 2kT m (5.22) where c is the speed of light and k is the Boltzmann constant. To account for turbulent velocity fields below either spectral or computational resolution, we can include a microturbulent broadening parameter, ξ: λ0 ∆λD = c r 2kT + ξ2 m (5.23) However, in the present exercise it is not a specific atomic line that we aim to reproduce but rather the output of a multiline analysis technique, whose properties do not preciseoly correspond to any given element or ion. Since 5.23 depends on the mass of the atom, and an LSD profile is created from atoms with widely divergent masses, it is not straightforward which mass to use. The problem can be avoided by considering that, in addition to the thermal broadening at the source, there is also an instrumental broadening at the observer’s end, related to the spectral resolution R, which will be as or more significant than the thermal broadening. Thus instead we have c + ξ2 σ= √ 2 ln 2R (5.24) with which we calculate the local Stokes I profile of the ith element: I= X i 2 /2σ 2 fi (1 − exp−(v−vi,z ) ) (5.25) where fi is the flux contribution of the ith element from Eqn. 5.3, vi,z is the Doppler velocity of this element from Eqn. 5.1, and v is the range in Doppler velocities, in km s−1 , spanned by the line. CHAPTER 5. MODELING AND UPPER LIMITS 115 Since we are utilizing the weak field approximation given by Eqn. 4.2, rather than solving the polarized equations of radiative transfer (requiring simultaneous solutions to Stokes Q, U, and V ) to calculate the local Stokes V profiles, we simply take the first derivative of the local Stokes I profiles, scaled by the Lorentz unit L, the Landé factor ḡ, and the local longitudinal magnetic field strength Bi,z from Eqn. 5.12. We then combine the local Stokes V profiles in the same fashion as the local Stokes I profiles: V =− where L is simply 2Lḡλ0 c X 2 2 fi Bi,z (v − vi,z )e−(v−vi,z ) /σ 2 σ i L= e × 10−8 4πme c2 (5.26) (5.27) where e and me are the electron charge and mass, respectively, c is the speed of light, and the factor 10−8 converts the Lorentz unit back to the km s−1 scale. This disk integrated model was implemented as an IDL program, and the source code is provided in the appendix. 5.2 Interpretation of measurements and upper limits While no circular polarization signature due to Zeeman splitting is visible in the LSD profiles, this does not necessarily mean that there is no magnetic field in the star: it remains possible that weak fields or fields with a complex topology might remain hiding in the noise. These fields cannot be detected, but the data is of high CHAPTER 5. MODELING AND UPPER LIMITS 116 enough quality to establish upper limits for various geometries. In principle a field of arbitrary complexity – dominated by high-order multipole components, strongly toroidal, or simply tangled – could be present, a potentially infinite parameter space that would be prohibitively difficult to search. In this work we consider only two relatively simple geometries: a dipolar field and a spotted field. These upper limits are set in two ways: by comparing the longitudinal field measurements to simple models (a zero field model, a static field model and a dipolar field model) and by constructing disk-integrated synthetic Stokes V profiles for various model parameters and comparing them to the observed Stokes V profiles. Since the latter method utilizes the detailed shape of the line profile it is considerably more sensitive than the former; the ability to utilize it represents one of the primary advantages of highresolution spectropolarimetry. 5.2.1 Constraints from longitudinal field measurements The simplest way to set upper limits is to utilize the hBz i measurements, which can be compared to basic models using χ2 significance tests. As a first step hBz i was compared to a zero field model and a mean field model; the reduced χ2 in either case was comparable, unsurprising as the mean longitudinal field measurement for the time series (hBz i = 2.5 ± 1.6 G) is formally equivalent to the zero field model. Variable fields were investigated by constructing a grid of longitudinal field curves for varying inclination i, obliquity β and dipolar field strength Bdip . For a given i, the rotational period can easily calculated as discussed in the Introduction. The individual hBz i measurements were then folded according to these periods, with 10 phase offsets of 0.1 tested at each inclination. The reduced χ2 for the entire data set CHAPTER 5. MODELING AND UPPER LIMITS 117 Figure 5.3: The probability that a configuration compatible with a given dipolar field strength at the three-σ level. The red and blue lines denote one- and two-σ upper limits; a three-σ upper limit, at ∼2840 G, is not shown. was calculated for each (i, β) at each phase offset, and the maximum Bdip compatible at the 3σ level recorded for each (i, β) combination. The result constrains Bdip . 50 G for most (i, β) although if β is small (. 15◦ ) or i is large (& 75◦ ) Bdip may range up to hundreds of gauss. Fig. 5.3 shows the resulting probability distribution function, strongly peaked around Bdip = 50 G. 5.2.2 Constraints from Stokes V profiles Nightly mean profiles A mean LSD profile created from the entire data-set, spanning perhaps three rotational periods, might easily smear out any Zeeman signature if there is rotational modulation of the profile. At the same time, even under the most extreme projected geometries of the rotation axis, the period of Rigel is not likely to be less than a few weeks (given v sin i and the inferred radius, see Tables 1.1, 1.2), and so it is unlikely that there would be much modulation of the Stokes V profile in any given night. CHAPTER 5. MODELING AND UPPER LIMITS 118 Figure 5.4: Four phases (clockwise from top left: 0.00, 0.25, 0.50, 0.75) of a model with i = 70◦ , β = 70◦ , Bdip = 50 G, and v sin i= 35 km s−1 . An ad hoc turbulent broadening parameter is added as discussed in the text. The points of the stellar disk have been scaled as in Fig. 5.1. Colour indicates the local strength and direction of Bz . To the right of each disk is the resulting synthetic (red) Stokes V (above) and Stokes I (below) profiles. Gaussian noise has been added to the synthetic V profile (blue), equivalent to an LSD SNR of ∼74,000 (matching the mean LSD profiles from the most densely observed period). At this SNR a 50 G field with this particular geometry would be easily detectable. The final magnetic analysis was performed with a mixture of individual LSD profiles (for those nights with only one observation) and nightly means (for those nights with multiple observations), providing a compromise between improved SNR and loss of signal due to co-adding of Stokes V profiles with different modulations. A Monte Carlo type approach was taken in exploring the parameter space. With the inclination i, the projected rotational velocity v sin i, and a radius R∗ (we adopt R∗ = 71 R⊙ , see Tables 1.1 and 1.2), a rotational period can be calculated. Zero CHAPTER 5. MODELING AND UPPER LIMITS 119 points for the period are then chosen pseudo-randomly, and the observations are phased according to these points. Model Stokes V spectra are created using the disk integration process described above, at a given dipolar field strength, with ḡ and λ0 set to the same values as those used for Rigel’s line mask, an instrumental broadening set to match that of ESPaDOnS, and limb darkening appropriate to an OB star (ǫ = 0.3). Turbulent broadening is added to the local profile in order to improve the fit between the model and observed LSD Stokes I profiles. The disk integrated profile is then scaled to the same depth as the Stokes I/Ic profile to which it is being fit, and resampled to the 2.0 km s−1 /pixel resolution of the LSD profiles. Following this, Gaussian noise identical to the N spectrum was added in by including the product of a one-dimensional array of pseudo-random numbers −1 ≤ n ≤ 1 and σV in the model V profile (see Fig. 5.4), where σV is the mean error bar, σV = 1 hSNRi (5.28) and the mean Signal to Noise Ratio is N 1 X 1 hSNRi = N i σV,i (5.29) where σV,i is the error bar of the ith velocity bin across the line profile and N is the number of velocity bins. In order to ensure that n remained as close as possible to true randomness, each iteration of the noise was used as a seed for the following iteration. nσV can be recorded as a synthetic null profile, which both allows the model spectrum to be analyzed using standard LSD profile analysis tools, and also allows the noise to be subtracted from the V profile, allowing a simple amplification factor to simulate the result of a stronger field at the same geometry without having to repeat the disk CHAPTER 5. MODELING AND UPPER LIMITS 120 integration. An illustration of this process is shown in Fig. 5.4. Each synthetic observation was then tested using the same statistical algorithm used on real LSD profiles, and flagged as either a definite, marginal, or non-detection. The average number of each detection flag at any given field strength was then calculated for 20 zero points randomly distributed throughout the duration of the time series. The number of such trials was determined experimentally: beyond this number of trials, the proportion of detection flags at a given field strength converged, making further synthetic observations pointless. As shown in Fig. 5.5, below the upper limit no model observations show detections; with increasing Bdip , the number of marginal detections increases, with the number of definite detections beginning to increase at slightly higher field strengths and soon surpassing the number of marginals, while non-detections of course continue to decrease, with the fraction of marginal detections thus reaching a maximum at a relatively low field strength. A few marginal detections were still observed in the LSD profiles generated using the final selected mask, so our upper limit must be consistent with the possibility for gaussian noise to lead to a marginal detection at any point on the grid i.e. we cannot set our upper limit to the highest field strength at which all observations result in non-detections. The criteria we adopt are that non-detections constitute more than 0.98 of the grid profiles, while marginal detections remain more numerous than definite detections. CHAPTER 5. MODELING AND UPPER LIMITS 5.2.3 121 Constraints from the grand mean LSD profile A slightly different method was used with the mean profile containing all spectra from the most densely time sampled period (HJD 2455165–177). Since the individual observations are binned into a single profile, there is little point in using a Monte Carlo type approach. Instead, representative models were created at 100 phases between 0 and 1. As before noise statistics were used from the observational LSD profile to generate pseudo-random noise for the synthetic profile, which was then subjected to the standard statistical test. The number of definite, marginal and non-detections across all phases was then tabulated. In binning 12 days of observations together we are implicitly assuming that the rotational period of Rigel is such that this span of time represents a relatively negligible fraction of the period. As this implies that the inclination angle i is relatively large, we restrict ourselves here to models with i = 90◦ , varying only β and Bdip . Since we have only a single observation to work with, we adopt more conservative criteria for an upper limit, taking the lowest field strength at which there is combined 100% chance of a definite or marginal detection. For the spotted models, this is relaxed to a ∼60% probability of a definite detection, reflecting the fraction of phases at which the spots are actually visible. 5.2.4 Dipolar field Upper limits derived from nightly mean LSD profiles are shown in Fig. 5.5. Nine representative models are shown, each with different combinations of (i, β), with the number of definite, marginal and non-detections plotted as a function of dipolar field strength and upper limits indicated with red lines. We find over most of the parameter CHAPTER 5. MODELING AND UPPER LIMITS 122 space that Bdip . 35 G. Only when the rotational axis is highly inclined from the line of sight and the dipole is of small obliquity to the rotational axis is the dipole sufficiently masked for a field of Bdip ∼ 100 G to remain undetected; only for β . 5◦ do we find Bdip . 300 G, a significant threshold as Auriére et al. (2007) found it to be an apparent lower limit of the dipolar field strength of magnetic early-type stars, with no magnetic dipoles weaker than 300 G found, and numerous stars undetected as magnetic constrained well below this limit. Magnetic spots Magnetically suspended loops corotating with the photosphere have been suggested as one possible explanation for HVAs (Israelian et al., 1997). Such a loop would have two footpoints anchoring the plasma flow in the photosphere; presumably, the magnetic polarity of the two footpoints would be opposite, however, there are few other constraints on how large or how far apart the spots might be, thus once again leaving a potentially large parameter space to search. In order to paramaterize the problem, each model contains two spots, each of which has a certain angular radius Ω and whose centres are separated by an angular distance δ. For simplicity, only cases for which the star rotates equator-on were considered, while Ω and δ were systematically varied in the same way i and β were for the dipole; δ, however, had negligible impact on the detection statistics. The Stokes V modulation of magnetic spots is quite distinct from that resulting from a dipolar field (see Fig. 5.6). In the absence of a turbulent broadening parameter, the magnetic signature is quite tightly confined to a narrow region in the line profile, travelling across it and increasing in amplitude as it moves from the limb to the centre CHAPTER 5. MODELING AND UPPER LIMITS 123 Figure 5.5: Upper limits for nine representative combinations of the inclination angle i, and the dipole’s obliquity β. The horizontal axis of each panel corresponds to the dipolar field of the models, while the vertical axis indicates the number of detections obtained, normalized to the total number of synthetic observations performed, as the normalized fraction of definite (dotted line), marginal (dash-dotted line) and nondetections (solid line). The red line indicates the upper limit, where essentially all of the observations are non-detections. CHAPTER 5. MODELING AND UPPER LIMITS 124 Figure 5.6: Six phases (clockwise from top left, in increments of 0.1, with φ = 0.0 corresponding to just before the first spot comes into view on the stellar limb) of a model with i = 90◦ , β = 90◦ , Bdip = 0.001 G, and v sin i= 35 km s−1 . An ad hoc turbulent broadening parameter is added as discussed in the text. As before the points on the disk are scaled by the net effect of limb darkening and projection on the sky on their total contribution to the integrated light, however in this representation shade too is scaled by these parameters as the dipole is not shown. Only the spots are magnetic, with radial fields of ±100 G and angular radii of 10◦ ; thus no field is detectable for half of the rotational period. Synthetic Stokes I and V profiles are shown as in Fig. 5.4. CHAPTER 5. MODELING AND UPPER LIMITS 125 Figure 5.7: Upper limits derived from the grand mean LSD profile for (top panels) dipolar models with three representative values of β, and (bottom panels) spotted models with three values of Ω. The line styles are as in Fig. 5.5. CHAPTER 5. MODELING AND UPPER LIMITS 126 of the disk, but not necessarily changing shape (unless more than one spot becomes visible). With the inclusion of turbulent broadening the magnetic signature is spread back over the full line profile, however, the centre of gravity of the Zeeman signature will still be offset from the line core of the intensity profile. Using the nightly means, for relatively large spots (Ω & 10◦ ) we achieve an upper limit of Bdip . 100 G, converging to the dipole upper limit of Bdip . 50 G for Ω & 20◦ . For smaller spots the detectable field strength diverges rapidly, with the smallest spot tested (Ω = 1◦ ) yielding upper limits around 1000 G. Since we are assuming, for simplicity’s sake, a model at high inclination, we can use the mean profile for HJD 2455165–177. We test upper limits with this co-added LSD profile in the same manner used for the dipolar field, i.e. testing a single observation against all possible phases. These upper limits are shown in Fig. 5.7. At this SNR our upper limits are of course significantly lower, ranging from Bdip . 600 G for the smallest spot tested (Ω = 1◦ ) to Bdip . 60 G for the largest spot tested with the mean LSD profile (Ω = 10◦ ). Larger spots of course converge to the upper limits for a dipolar field, in this case Bdip . 35 G (depending on the obliquity of the dipole from the rotational axis). Chapter 6 Discussion and Conclusions No evidence is seen is seen in 64 circular polarization spectra of Zeeman splitting due to magnetic fields, with upper limits using nightly mean LSD profiles on the dipolar field strength of Bdip . 50 G over most of the (i, β) parameter space. The interferometry reported by Chesneau et al. (2010) indicates that i may in fact be quite large, since a phase differential is observed which is generally seen only when the velocity field varies across the disk (no such phase differential is seen if the star is observed closer to pole-on). However, in the event that this is the case, we are justified in using the grand mean LSD profile, which establishes an upper limit of Bdip . 35 G for large obliquities and Bdip . 100 G even at small β. Israelian et al. (1997) estimate that a magnetic dipole of Bdip ≥ 25 G would be required to support a loop, a regime the dipole upper limits established in this study probe under the more optimistic assumptions regarding the orientation of Rigel’s rotational and magnetic axes. Markova et al. (2008), who base their calculations on the magnetically confined wind model of ud-Doula & Owocki (2002), show that a magnetic dipole in the range 5 G . Bdip . 100 G would be required. The upper 127 CHAPTER 6. DISCUSSION AND CONCLUSIONS 128 limits established here rule out most of this range but leave open the possibility of weaker magnetic dipoles. Using the upper limits set for dipolar fields, we can calculate upper limits for the wind magnetic confinement parameter, η∗ . Taking the stellar radius R∗ = 71 R⊙ (the value derived from Hipparcos photometry (Perryman, 1997) and interferometric angular diameter (Aufdenberg et al., 2008)), and the measured v sin i= 35 km s−1 (Pryzbilla et al., 2006), we can calculate the rotation parameter W for any given i with Eqn. 1.4. Using the same value for R∗ , and mass loss rates Ṁ and wind terminal velocities v∞ as summarized in Table 1.2 and discussed in detail in the introduction, and the dipolar magnetic field upper limits appropriate to any given i, we calculate η∗ with Eqn. 1.3. The results for representative combinations of i, v∞ , and Ṁ are tabulated in Table 6.1, where for i = 90◦ we have assumed the upper limits established using the 11 day mean profile. For each upper limit we show results for two (v∞ , Ṁ ) combinations: one maximal solution (the fastest, densest wind) and one minimal solution (the slowest, thinnest wind). With the exception of the model with i = 90◦ and maximal wind parameters, the upper limits for η∗ range from ∼ 2 − 60, indicating the possibility of a magnetically confined wind. η∗ is of course much larger for the minimal wind parameters, since a magnetic field is more easily able to confine a relatively weak wind. W falls in an intermediate range, from 0.14–0.55. If we adopt the largest possible stellar radius R∗ = 148 R⊙ (derived from association with the Ori OB 1 cluster), the effect is to increase both W and η∗ , as W scales with R∗ and η∗ scales with R∗2 ; for the i = 15◦ model, which has the highest equatorial velocity of those considered, we can obtain a relatively strong rotation parameter of W ≃ 0.80. CHAPTER 6. DISCUSSION AND CONCLUSIONS 129 Fig. 6.1 shows the positions of the various model upper limits on an η∗ − W diagram1 . The diagram shows the known magnetic OB stars on the η∗ − W plane, and indicates the regions of the two primary magnetospheric regimes: centrifugal magnetospheres, such as that of σ Ori E (Townsend, Owocki & Groote, 2007), and dynamic magnetospheres, such as that of HD 191612 (Wade et al., 2011). Strong wind Weak wind i Bdip W η∗ η∗ 15 50 0.55 2.90 42.59 45 60 0.20 4.18 61.33 90 25 0.14 0.72 10.65 Table 6.1: Upper limits for the wind magnetic confinement parameter η∗ for the dipolar magnetic field upper limits reported in this thesis. Two values of η∗ are given, one for a strong wind (v∞ = 600 km s−1 , Bates et al., 1980; Ṁ = 1.4 × 10−6 M⊙ yr−1 , Barlow & Cohen, 1977), and one for a weak wind (v∞ = 229 km s−1 , Kaufer et al., 1996a; Ṁ = 2.5 × 10−7 M⊙ yr−1 , Drake & Linsky, 1989). The intent is to illustrate the strong dependence of η∗ on the wind parameters v∞ and Ṁ. W of course depends strongly upon i. For high inclinations, W ∼ 0.2 (slightly higher or lower depending upon the stellar mass and radius), while for low inclinations W ∼ 1 can easily be achieved. Examining the W −η∗ diagram in Fig. 6.1, we see that Rigel’s circumstellar environment could easily contain either a centrifugally supported magnetospheric disk or a dynamically supported magnetosphere. Of the two, dynamic support is perhaps somewhat more likely: interferometry indicates a relatively high inclination, suggesting the grand mean upper limit to be permissible. In this case, even with the weakest wind parameters the upper limit is sufficient to rule out a centrifugally supported magnetic disk (as discussed in the introduction, the minimal mass loss rate may be slightly more reliable). If the maximal wind parameters are used, Rigel’s wind is of course not magnetically confined. 1 Provided by Véronique Petit. CHAPTER 6. DISCUSSION AND CONCLUSIONS 130 Figure 6.1: The wind magnetic confinment parameter η∗ vs. the rotation parameter W . The upper limits for the various models described in the text are shown in with red squares (strong wind) and blue squares (weak wind). Known magnetic massive stars are labeled individually, with approximate spectral type given in the legend; stars with black points show Hα variability, while outlines indicate UV modulation; arrows indicate upper or lower limits for these stars. The diagonal dashed line indicates the boundary at which the Kepler radius is equal to the Alfvén radius, dividing the regions of centrifugally supported and dynamically supported magnetospheres. Original figure provided courtesy of Véronique Petit. CHAPTER 6. DISCUSSION AND CONCLUSIONS 131 It is worthwhile to review the characteristic α Cygni type line profile variability, visible in Hα and other lines, that has characterized β Ori both in this study and historically. As explored in Chapter 3, this variability shows none of the highly periodic, synchronized behaviour characterizing the wind lines of stars with strong magnetospheres. On the contrary, there is little evidence for correlation between the variability of radial velocities or equivalent widths of metal lines, and the equivalent widths of H Balmer lines (Kaufer et al., 1996a, 1997; Markova et al., 2008; Moravveji et al., accepted). Radial velocities show evidence for multiple high-order, low-amplitude non-radial pulsations (Moravveji et al., accepted), with periodograms yielding substantially different solutions in different years (Kaufer et al., 1997; Moravveji et al., accepted), a pattern analysis of the present data confirms. Hα Wλ derived periods, on the other hand, seems to yield similar (but not identical) periods in any given season. Temporal variance spectra also reveal different patterns of variability in metallic vs H Balmer lines, with the Balmer lines showing much more variability in the core and at high velocities in the wings, with an especially variable blue wing, while metallic lines are characteristically peaked at vsys ±v sin i. There is also some suggestion of rotational modulation in Hα, i.e. the reoccurance of HVA events after an interval that might plausibly be ascribed to the rotational period given the upper and lower bounds that can be established for those stars in which the activity has been witnessed (Kaufer et al., 1996b). However, HVAs do not reoccur at the same amplitude from cycle to cycle, indicating that they might be relatively temporary structures in the lower stellar wind, surviving perhaps for only a few rotations. There is some evidence from photometric observations for brightening by approximately 0.1 mag during HVA events (Markova et al., 2008); in this CHAPTER 6. DISCUSSION AND CONCLUSIONS 132 case, too, the MOST photometry seems to indicate that the star was brightening, by approximately 0.03 mag, during the increase in Hα Wλ as the absorption feature migrated redwards. However, the photometric and Hα Wλ peaks do not coincide precisely. Finally, linear continuum polarimetry seems to show that Rigel’s circumstellar environment has no preferred scattering angle, as might be expected of a relatively stable structure such as a disk. This pattern of variability is clearly inconsistent with that of a centrifugally supported magnetic disk, whose variability – whether in Hα Wλ , the equivalent widths of other lines, photometry, UV wind lines, etc. – is synchronized across multiple cycles to very high precision. At first glance, it may seem similar to what might be expected of a dynamically supported magnetosphere, which shows continuous episodes of outbreak and infall. It is important to recall, however, that these events happen on a scale that is relatively small when compared to the scale of the stellar disk: we do not observe individual episodes of outbreak and infall, but rather many such events averaged over a large spatial extent which, taken together, exhibit synchronized patterns variability modulated primarily by the rotational period of the star, and not by the motion of the circumstellar plasma itself relative to the star. This suggests that the variability we see in Rigel’s wind is not due to magnetic confinement. Upper limits established for spotted geometries indicate that local ordered magnetic fields of a few hundred up to thousands of gauss might go undetected, if the spots subtend only a few degrees of the stellar disk. A weak High Velocity Absorption event seems to have occurred around the same time as the most densely sampled epoch of observations in this study; additionally, shortly after the end of the spectropolarimetric time series, interferometric observations (as yet unpublished) witnessed a true CHAPTER 6. DISCUSSION AND CONCLUSIONS 133 HVA, the third such event observed for Rigel. If localized surface fields are in fact a key element of HVA events, models of the phenomenon must be able to do so within these limits. While localized magnetic structures have yet to be detected in early-type stellar photospheres, they have long been invoked as an explanation for transient phenomena such as HVAs, which bear the signature of both simultaneous outflows and infalls of matter (Israelian et al., 1997) and rigid co-rotation (Kaufer et al., 1996b), both characteristics of magnetically supported plasmoids. The hypothetical FeCZ predicts the existence of plasmoids generated in a convective dynamo embedded within the radiative zone, a possibly unifying explanation for microturbulent broadening, nonradial pulsations, DACs, wind clumping, and possibly X-ray emission (Cantiello et al., 2009, 2011), all near-ubiquitous features of OB stars. DACs have previously been ascribed to CIRs driven in part by bright spots on the stellar surface (e.g. Cranmer & Owocki, 1996); interestingly, such bright spots are a natural consequence of the FeCZ model, in which the spots arise due to plasmoids generated within the FeCZ dynamo rising to the surface due to magnetic buoyancy (Cantiello et al., 2011). Lower limits for the surface field generated by this mechanism provided by Cantiello & Braithwaite (2011) indicate that, for a star with Rigel’s fundamental parameters (see Table 1.1), the surface field should be at least of order 5–10 G (see Fig. 6.2). While there are no predictions for the filling factor, even the upper limits obtained for the largest spots considered here cannot rule out such magnetic bright spots. Under the theory that a large perturbation requires a large disturbance, small spots separated by a small angular distance would be unlikely to be able to support the enormous loops implied in the magnetic interpretation of the HVA phenomenon, CHAPTER 6. DISCUSSION AND CONCLUSIONS 134 Figure 6.2: The figure has been slightly modified from the original in Cantiello & Braithwaite (2011) to show the position of β Ori (black dot). The blue contours show lower limits for magnetic fields generated by a FeCZ. CHAPTER 6. DISCUSSION AND CONCLUSIONS 135 making the upper limits for large spots perhaps of more relevance to HVAs per se. As to the FeCZ model, there are as yet no predictions as to the numbers or sizes of such spots, but as the FeCZ is rather thin on the scale of the star they are unlikely to be very large. Non-radial pulsations have also been suggested to have a connection to HVAs, and indeed during the period of the weak HVA a pulsational mode appears to have been visible in the photospheric lines. The pulsational variability also seems to have influenced the core of the Hα line, although the correlation in the dynamic spectra is imperfect. Nonradial modes with relatively long periods have been detected in the concurrent MOST photometry and AST radial velocities by Moravveji et al. (accepted). It has also been suggested that pulsations and macroturbulence are the same phenomenon (e.g. Aerts et al., 2009); as the macroturbulent broadening in Rigel’s photosphere is comparable to the rotational broadening, if true pulsations would have to be incorporated at the base of any rigorous theoretical treatment of the processes affecting variable mass loss in BA SGs. In addition to providing an avenue for exploring the mechanisms influencing mass loss, a model able to reproduce the observed photospheric variability due to NRPs would enable more precise characterization of the wind environment itself, since the contribution of pulsations to line broadening could be removed on a spectrum by spectrum basis and a relatively pure wind spectrum obtained. The relation between non-radial pulsations and photospheric magnetic fields, if any, is not well understood. It has been suggested (see e.g. Cantiello et al., 2009) that the phenomena might share a common source in the FeCZ, with low-density metallic convection zones driving both non-radial pulsational modes and a dynamo, leading CHAPTER 6. DISCUSSION AND CONCLUSIONS 136 to magnetic flux tubes that rise and generate magnetic spots on the stellar surface. This is similar to the physics presumed to be behind sunspot formation although – due to the much higher luminosities involved with early-type stars – the spots are much brighter than the surrounding photosphere. Such magnetic hot spots have been suggested to generate CIRs, leading to the DACs that are a near-ubiquitous feature of OB wind lines. The crude spot model used here considers only the contribution of localized photospheric magnetism to the Stokes V profile. While this is sufficient for the establishment of upper limits, a more realistic treatment would use the magnetic bright spots as the source surface of an MHD wind simulation to test if fields between the lower limits given by Cantiello & Braithwaite (2011) and the upper limits established here are capable of sustaining the plasmoids hypothesized to generate HVAs, whilst also solving radiative transfer equations to determine if such formations are indeed capable of generating the dramatic LPV characterizing the phenomenon. The spectral analysis of wind variability would also benefit from the application of software such as fastwind (Puls et al., 2005) to the Balmer lines, in order to generate a model of the underlying static wind to combine with non-LTE model atmosphere synthetic profiles; the residual variation might then be used to study the properties of the wind in detail at high spectral resolution. Extremely weak, complex fields of the sort detected in Vega and Sirius by Petit et al. (2010, 2011) cannot be ruled out. Such fields appear to be of an entirely different order from those of magnetic OB stars: extremely weak (order of 0.1–1 G), showing no signs of evolution over multiple epochs, with highly complex topologies as determined through Zeeman Doppler Imaging (Alina et al., 2010; Petit et al., 2010). In order to CHAPTER 6. DISCUSSION AND CONCLUSIONS 137 explore the possibility of such fields, future spectropolarimetric observations would need to achieve an LSD SNR of 200,000–2,000,000, as determined through artifically tuning the noise parameter σV in the model profiles. This would require around 200– 30,000 observations to be collected over a short period of time – ideally a single night – assuming the LSD SNRs of individual observations are comparable to those of the present data. Under such conditions a dipole of order 1 G might be detected with a few hundred observations achieving an LSD SNR of around 200,000, however probing much beneath this limit for a star of Rigel’s spectral type is impractical with current instrumentation. If magnetic spots are a factor, observational verification of the magnetic loop hypothesis would be even more challenging than detecting a weak dipole. Depending on the filling factor of the spots, detection of localized magnetic fields of the order of 10 G would necessitate an extraordinarily high LSD SNR (around 106 assuming a spot diamter of ∼ 10◦ ), requiring once again hundreds to thousands of spectropolarimetric obsevations to be collected in a short time. Further complicating matters is that HVAs are currently impossible to predict; those that have been observed so far were captured by happenstance. With no way to schedule observations in advance, it is difficult to justify the significant investment in telescope time such a project would require. A possible solution would be to combine a spectroscopic monitoring campaign on smaller telescopes (which need not be able to observe anything but Hα) with a single night of observations with a large telescope. As HVAs are not particularly subtle, low-resolution spectrographs should be capable of capturing them; as such instrumentation is now relatively affordable to amateur astronomers, this could CHAPTER 6. DISCUSSION AND CONCLUSIONS 138 represent an opportunity for collaboration between the professional and amateur communities. Depending on the locations of participants, the spectroscopic monitoring could potentially be able to observe Rigel throughout the year, generating a valuable long-baseline time series. 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