Download Magnetic fields and the variable wind of the β Ori early-type supergiant

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. It would of course be necessary for the spectroscopic data
reduction to be accomplished relatively quickly (within a few days), so that once an
HVA is observed the necessary spectropolarimetric observations could then be collected using a larger telescope. Of course, in order to ensure that spectropolarimetry
of an HVA could be collected, observatories in both hemispheres would ideally participate; with the installation of HARPSPol at the European Southern Observatory’s
3.6 m La Silla telescope (Piskunov et al., 2011), such an observing program may be
feasible.
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