maurice/thesis/thesis2.pdf · ABSTRACT High Resolution X-ray Spectroscopyof Massive Stars Maurice...
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High Resolution X-ray Spectroscopy of Massive Stars Maurice Andrew Leutenegger Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Graduate School of Arts and Sciences COLUMBIA UNIVERSITY 2006
Transcript of maurice/thesis/thesis2.pdf · ABSTRACT High Resolution X-ray Spectroscopyof Massive Stars Maurice...
High Resolution X-ray Spectroscopy of Massive Stars
Maurice Andrew Leutenegger
Submitted in partial fulfillment of the
requirements for the degree
of Doctor of Philosophy
in the Graduate School of Arts and Sciences
COLUMBIA UNIVERSITY
2006
c©2006
Maurice Andrew LeuteneggerAll rights reserved
ABSTRACT
High Resolution X-ray Spectroscopy of Massive Stars
Maurice Andrew Leutenegger
This thesis presents studies of high-resolution X-ray spectra of massive stars.
Diffraction grating spectrometers onboard the XMM-Newton and Chandra satellite
X-ray observatories have revolutionized our understanding of X-ray emission from
massive stars, allowing the resolution of individual spectral lines and the study of
their Doppler profiles. I discuss the use of line strengths and ratios to constrain
temperature distributions, elemental abundances, and distribution of X-ray emit-
ting plasma near a star. I also discuss the interpretation of Doppler profiles in light
of their unexpected lack of asymmetry.
Contents
1 Introduction 1
1.1 Background and motivation . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Line-driven winds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Radiative transfer in stellar winds . . . . . . . . . . . . . . . . 4
1.2.2 The CAK model of a line-driven wind . . . . . . . . . . . . . . 14
1.2.3 Measurement of wind parameters in steady state models . . . 18
1.2.4 Observational evidence for variability and instability in
line-driven winds . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.2.5 Theory of instabilities in line-driven winds and generation of
X-ray emitting shocks . . . . . . . . . . . . . . . . . . . . . . . 20
1.3 X-ray emitting plasmas in the coronal approximation . . . . . . . . . 23
1.3.1 Atomic processes in plasmas . . . . . . . . . . . . . . . . . . . 24
1.3.2 Rates, rate coefficients, and cross sections . . . . . . . . . . . . 26
1.3.3 Ionization balance . . . . . . . . . . . . . . . . . . . . . . . . . 27
1.3.4 Discrete line emission . . . . . . . . . . . . . . . . . . . . . . . 29
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1.3.5 Bremsstrahlung . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
1.3.6 Model parameters for a coronal plasma . . . . . . . . . . . . . 31
1.3.7 Helium-like triplet ratios . . . . . . . . . . . . . . . . . . . . . 32
1.4 Emission line Doppler profile models . . . . . . . . . . . . . . . . . . 35
1.4.1 The Doppler profile model . . . . . . . . . . . . . . . . . . . . 36
1.4.2 Doppler profiles with resonance scattering . . . . . . . . . . . 41
1.5 X-ray spectroscopic instrumentation . . . . . . . . . . . . . . . . . . . 43
1.5.1 The XMM-Newton Reflection Grating Spectrometer . . . . . . 45
1.5.2 The Chandra High Energy Transmission Grating Spectrometer 48
2 High resolution X-ray spectroscopy of ζ Puppis with the XMM-Newton
Reflection Grating Spectrometer 51
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.2 Observations and Data Analysis . . . . . . . . . . . . . . . . . . . . . 55
2.2.1 Light Curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2.2 Emission Line Intensities and Emission Measure Analysis . . 57
2.2.3 Continuum emission analysis . . . . . . . . . . . . . . . . . . . 58
2.2.4 He-like triplet ratios . . . . . . . . . . . . . . . . . . . . . . . . 62
2.2.5 Line profile analysis . . . . . . . . . . . . . . . . . . . . . . . . 64
2.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3 X-ray spectroscopy of η Carinae with XMM-Newton 69
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3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Observation and Data Analysis . . . . . . . . . . . . . . . . . . . . . . 72
3.2.1 EPIC spectral and imaging analysis . . . . . . . . . . . . . . . 73
3.2.2 RGS spectral analysis . . . . . . . . . . . . . . . . . . . . . . . . 75
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
3.3.1 Temperature distribution . . . . . . . . . . . . . . . . . . . . . 88
3.3.2 Abundance measurements . . . . . . . . . . . . . . . . . . . . 90
3.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4 Measurements and analysis of helium-like triplet ratios in the X-ray spec-
tra of O-type stars 97
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.2.1 Radial dependence of the f/i ratio . . . . . . . . . . . . . . . . 102
4.2.2 The effect of photospheric absorption lines . . . . . . . . . . . 106
4.2.3 The integrated ratio . . . . . . . . . . . . . . . . . . . . . . . . 111
4.2.4 He-like line profiles . . . . . . . . . . . . . . . . . . . . . . . . 120
4.3 Data reduction and analysis . . . . . . . . . . . . . . . . . . . . . . . . 122
4.3.1 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.3.2 Fitting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
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4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5 Evidence for the importance of resonance scattering in X-ray emission
line profiles of the O star ζ Puppis 151
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.2 Data reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
5.3 Best fit He-like profile model . . . . . . . . . . . . . . . . . . . . . . . 158
5.3.1 The profile model . . . . . . . . . . . . . . . . . . . . . . . . . . 158
5.3.2 Best fit model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
5.4 Best fit model including the effects of resonance scattering . . . . . . 164
5.4.1 Incorporating resonance scattering into OC01 . . . . . . . . . 164
5.4.2 Best fit model including resonance scattering . . . . . . . . . 167
5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
5.5.1 Comparison of results . . . . . . . . . . . . . . . . . . . . . . . 177
5.5.2 Plausibility of the importance of resonance scattering . . . . . 178
5.5.3 Impact of resonance scattering on Doppler profile model pa-
rameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
5.5.4 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
5.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
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List of Figures
1.1 Illustration of the definition of φ(x) and Φ(x). . . . . . . . . . . . . . . 10
1.2 Ion fractions q for iron as a function of temperature. . . . . . . . . . . 28
1.3 Diagram of the interaction of the n = 2 triplet levels of He-like ions . 33
1.4 Model Doppler profiles . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.1 RGS spectrum of ζ Puppis . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.2 RGS light curve of ζ Pup . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.3 Inferred emission measure distribution of ζ Pup . . . . . . . . . . . . 59
2.4 The Ne IX triplet of ζ Pup . . . . . . . . . . . . . . . . . . . . . . . . . 66
2.5 Doppler profiles of Ly α lines of ζ Pup . . . . . . . . . . . . . . . . . . 66
3.1 EPIC-MOS image of the field around η Carinae . . . . . . . . . . . . . 74
3.2 EPIC-MOS2 spectrum of η Car . . . . . . . . . . . . . . . . . . . . . . 79
3.3 First order RGS spectrum of η Car . . . . . . . . . . . . . . . . . . . . 79
3.4 RGS source and background spectra of η Car . . . . . . . . . . . . . . 81
3.5 Cross-dispersion profile of N VII Ly α . . . . . . . . . . . . . . . . . . 81
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3.6 Cross-dispersion profile of O VIII Ly α. . . . . . . . . . . . . . . . . . . 82
3.7 Cross-dispersion profile of O VII He α. . . . . . . . . . . . . . . . . . . 82
3.8 Inferred emission measure distribution of η Car . . . . . . . . . . . . 86
4.1 Model UV flux for ζ Ori . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2 f/i ratio for the Mg XI triplet of ζ Ori . . . . . . . . . . . . . . . . . . . 110
4.3 f/i ratio for six He-like triplets observed in ζ Pup . . . . . . . . . . . 115
4.4 f/i ratio for five He-like triplets observed in ζ Ori . . . . . . . . . . . 116
4.5 f/i ratio for five He-like triplets observed in ι Ori . . . . . . . . . . . . 117
4.6 f/i ratio for five He-like triplets observed in δ Ori . . . . . . . . . . . 118
4.7 MEG data and best-fit model for S XV in ζ Pup . . . . . . . . . . . . . 128
4.8 HEG data and best-fit model for S XV in ζ Pup . . . . . . . . . . . . . 129
4.9 MEG data and best-fit model for Si XIII in ζ Pup . . . . . . . . . . . . 129
4.10 MEG data and best-fit model for Mg XI in ζ Pup . . . . . . . . . . . . 130
4.11 MEG data and best-fit model for Si XIII in ζ Ori . . . . . . . . . . . . . 130
4.12 HEG data and best-fit model for Si XIII in ζ Ori . . . . . . . . . . . . . 131
4.13 MEG positive first order data and best-fit model for Si XIII in ζ Ori . 131
4.14 MEG negative first order data and best-fit model for Si XIII in ζ Ori . 132
4.15 MEG data and best-fit model for Mg XI in ζ Ori . . . . . . . . . . . . . 132
4.16 MEG data and best-fit model for Si XIII in ι Ori . . . . . . . . . . . . . 133
4.17 HEG data and best-fit model for Si XIII in ι Ori . . . . . . . . . . . . . 133
4.18 MEG data and best-fit model for Mg XI in ι Ori . . . . . . . . . . . . . 134
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4.19 MEG data and best-fit model for Si XIII in δ Ori . . . . . . . . . . . . . 134
4.20 HEG data and best-fit model for Si XIII in δ Ori . . . . . . . . . . . . . 135
4.21 MEG data and best-fit model for Mg XI in δ Ori . . . . . . . . . . . . . 135
4.22 Two dimensional plots of confidence intervals for fit parameters for
Mg XI in ζ Pup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.23 Comparison of measurements and calculations for Si XIII in ζ Ori . . 145
4.24 Comparison of measurements and calculations for S XV in ζ Pup . . 147
5.1 O VII triplet with best fit OC01 He-like triplet model . . . . . . . . . 162
5.2 N VI triplet with best fit OC01 He-like triplet model . . . . . . . . . . 163
5.3 Influence of βSob on Doppler profile shape . . . . . . . . . . . . . . . . 168
5.4 Influence of τ0,∗ on Doppler profile shape. . . . . . . . . . . . . . . . . 169
5.5 O VII triplet with best fit model assuming resonance scattering with
βSob = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
5.6 O VII triplet with best fit model assuming resonance scattering with
βSob = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
5.7 N VI triplet with best fit model assuming resonance scattering with
βSob = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
5.8 N VI triplet with best fit model assuming resonance scattering with
βSob = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
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List of Tables
2.1 Measured fluxes for prominent emission lines in the spectrum of ζ Pup 60
2.2 Upper limits on the strengths of prominent K edges in the X-ray
spectrum of ζ Pup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
2.3 Comparison of photoexcitation and decay rates of the 2 3S state for
ζ Pup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4 Velocity widths and shifts of the Lyman α lines observed in the
spectrum of ζ Pup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.1 Best fit parameters for the EPIC-MOS2 spectrum of η Car . . . . . . . 76
3.2 Measured fluxes for prominent emission line complexes in the RGS
spectrum of η Car . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1 Parameters adopted for He-like triplets . . . . . . . . . . . . . . . . . 104
4.2 Adopted stellar parameters . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3 Comparison of He-like ratio calculations . . . . . . . . . . . . . . . . . 119
4.4 Parameters for He-like profile fits . . . . . . . . . . . . . . . . . . . . 126
4.5 Parameters for He-like Gaussian fits . . . . . . . . . . . . . . . . . . . 127
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4.6 Comparison of fit parameters with previous work . . . . . . . . . . . 139
4.7 Comparison of R1 to R0 . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
5.1 List of observations of ζ Pup with net exposure times . . . . . . . . . 157
5.2 Model fit parameters for O VII in the spectrum of ζ Pup . . . . . . . . 171
5.3 Model fit parameters for N VI in the spectrum of ζ Pup . . . . . . . . 172
5.4 Expected characteristic Sobolev optical depth for strong resonance
lines observed in the spectrum of ζ Pup . . . . . . . . . . . . . . . . . 180
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ACKNOWLEDGMENTS
I thank my advisors, Steve Kahn and Frits Paerels for their support, advice,
and for providing the impetus to get me started in X-ray spectroscopy. I thank
David Cohen and Stan Owocki, my collaborators in my work on massive stars and
the X-ray emission from their winds. I thank my colleagues: Marc Audard, Ehud
Behar, Jean Cottam, Ming Feng Gu, Ali Kinkhabwala, Thierry Lanz, Kaya Mori,
John Peterson, Andy Rasmussen, Doug Reynolds, Masao Sako, Dave Spiegel, and
Jacco Vink. I thank the members of my thesis committee for their careful reading
of this manuscript: Chuck Hailey, Szabi Marka, and Amber Miller. Finally, I thank
Lalla Grimes for her administratitive support.
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Chapter 1
Introduction
1.1 Background and motivation
Massive stars greatly influence the evolution of baryonic matter in the universe.
They interact strongly with their environment through their winds and radiation
fields, influencing star formation and the state of the ISM, and the supernovae that
occur at the end of their lives, together with Type Ia supernovae, are the source of
all metals in the universe (e.g. Arnett 1996).
Aside from their direct consequences for the ISM, the radiatively driven winds
of massive stars influence their evolution; they are also intrinsically interesting as
tests of our understanding of radiation hydrodynamics and stellar atmospheres.
Massive stars lose a significant fraction of their initial mass through their
wind over their main sequence lifetime, and continue to lose mass through the
luminous blue variable and Wolf-Rayet stages of evolution, leading to significant
evolutionary consequences (Chiosi & Maeder 1986).
1
2
The X-ray emission from massive stars arises from a diverse array of mecha-
nisms that are all thought to involve their winds in some way. Massive star binaries
often are bright X-ray sources. A system consisting of a massive star and a compact
object (a high-mass X-ray binary) is typically very luminous and emits X-rays that
are created by accretion of the wind onto the companion and then reprocessed in
the wind of the massive star (e.g. Vela X-1, Cyg X-3). A system consisting of two
massive stars may have a strong wind-wind collision (a colliding wind binary),
leading to a high temperature shock front (e.g. γ2 Vel, η Car). Most single massive
stars are not very bright X-ray sources in comparison with HMXBs or CWBs. The
X-rays from these stars are thought to originate in shocks in their winds caused
by instabilities in the radiative acceleration mechanism (e.g. ζ Pup, ζ Ori). A
significant number of single massive stars show anomalously bright or hot X-ray
emission; these are thought to be special cases of winds influenced by strong sur-
face magnetic fields (e.g. θ1 Ori C, τ Sco) or the presence of a circumstellar disk
(e.g. γ Cas). However, the vast majority of OB stars are thought to emit X-rays
because of the wind-instability mechanism.
Spectroscopy of the X-ray emission from shocks resulting from the wind in-
stability is the focus of much of this thesis. X-ray spectroscopy allows us to test
our understanding of the physics of stellar winds and instabilities, and also to
measure key quantities of the winds including the mass-loss rates, elemental abun-
dances, temperatures of shocked material, and spatial distribution of X-ray emitting
plasma; these are all goals of this thesis.
In this thesis, I first give a description of the theory of stellar winds and in-
stabilities (§ 1.2); and of X-ray spectroscopy of coronal plasmas (§ 1.3); I review
recent work on formation of X-ray emission line Doppler profiles (§ 1.4); and I
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describe the new generation of high-resolution diffraction grating X-ray spectrom-
eters (§ 1.5). In Chapter 2, I present a preliminary analysis of the XMM-Newton
Reflection Grating Spectrometer (RGS) X-ray spectrum of the bright O star ζ Pup.
In Chapter 3, I analyze the RGS spectrum of the X-ray emitting nebula ejected by
the anomalous star η Car. In Chapter 4, I discuss the measurement of line ratios in
the emission of helium-like ions in the winds of O stars and their use to constrain
the location of X-ray emitting plasma. In Chapter 5, I present evidence for the
hypothesis that resonance scattering may be important in the formation of Doppler
profiles of emission lines in the X-ray spectrum of ζ Pup.
1.2 Line-driven winds
Massive stars have been known to have outflows since the observation of blue-
shifted absorption in their UV spectra by spectrometers flown on rockets in the
late 1960s (e.g. Morton 1967b,a; Carruthers 1968; Morton et al. 1969). The solar
coronal wind theory was obviously inadequate to explain the high velocities of
the outflows, since the ionization of the winds is quite moderate. It was quickly
realized that radiation pressure resulting from scattering in spectral lines could
efficiently accelerate the winds (Lucy & Solomon 1970).
In this section I review our knowledge of line-driven winds from massive
stars. I first review radiative transfer theory for moving atmospheres in § 1.2.1.
In § 1.2.2 I discuss the formulation of a steady state model of the wind in the
tradition of Castor, Abbott, & Klein (1975). In § 1.2.3 I review measurements of
fundamental wind parameters in the context of steady state models. In § 1.2.4 I
discuss observational evidence for variability in line-driven winds, and in § 1.2.5
4
I review the theory of instabilities in line-driven winds, numerical hydrodynamic
simulations of the winds, and the theoretical work on X-ray production.
1.2.1 Radiative transfer in stellar winds
In this section I discuss the basic physics of the line-driving force. I derive the
radiative acceleration for an absorption line in a stellar wind, and the angular
dependence of the local escape probability for scattered light.
1.2.1.1 Physical picture of line-driven winds
There are a few crucial conceptual ingredients to line-driven winds: momentum
transfer in spectral lines, reduction of self-shadowing due to Doppler shifts, and the
simplification of the radiative transfer problem by the approximation that scattering
is localized, known as the Sobolev approximation (Sobolev 1960).
The idea that momentum transfer in a stellar atmosphere may be made more
efficient by the reduction of self-shadowing resulting from Doppler shifts is due
to Milne (1926). The proposal of a wind driven by this effect followed soon on
the discovery of evidence for outflows in the UV spectra of massive stars (Lucy &
Solomon 1970).
The momentum carried by a photon is p = E/c. The average momentum
transferred in a scattering event should be the same as for pure absorption, since
the reemission of the scattered photon is in a random direction in the atom’s frame.
The opacity in lines is much greater than in the continuum; however, most
of the potential opacity is “wasted” in self-shadowing. The part of the spectrum
5
absorbed by an optically thick line in a static atmosphere is set by the thermal
Doppler width, vth. The momentum transfer rate is
dp
dt=
Lν∆ν
c=
Lνν0vth
c2 . (1.1)
When the atmosphere is accelerated into a wind, however, the atoms are able to
absorb successively shorter wavelengths as they are accelerated. If the line remains
optically thick over the flow, the momentum transfer is
dp
dt=
Lνν0v∞c2 , (1.2)
where v∞ is the terminal velocity of the wind. For typical thermal and terminal
velocities, this is a factor of ∼ 100 enhancement.
The maximum momentum transfer in the single scattering limit would occur if
the whole spectrum was blocked by optically thick lines, in which case dp/dt = L/c.
This is a factor of c/v∞ ∼ 100 enhancement over the momentum transfer due to
scattering in a single line. Alternatively, for a given momentum transfer we could
write
Neff =dp
dt
c2
Lv∞, (1.3)
where Neff is the effective number of optically thick lines.
1.2.1.2 Coulomb coupling
Because hydrogen is completely ionized in the photospheres and winds of O stars,
momentum must be transferred from the driven ions by collisions. If this momen-
tum transfer is not effective, a wind might develop where metals are preferentially
lost. A good discussion of this topic is given in Lamers & Cassinelli (1999).
6
The characteristic timescale for momentum transfer through collisions scales
approximately with the inverse of the electron density:
tc ∝ n−1e ∝
r2v(r)M. (1.4)
The characteristic timescale for acceleration of ions scales with the inverse of the
line acceleration:
td ∝ g−1i ∝
r2
L∗. (1.5)
The condition for effective coupling is roughly that the timescale for acceleration
should exceed that for momentum loss through collisions:
td > tc, (1.6)
or, putting in numbers appropriate to stellar winds of massive stars,
L∗v(r)M
< 5.9 × 1016, (1.7)
where the luminosity is given in solar units, the velocity in km s−1, and the mass-loss
rate in M⊙ yr−1. For ζPup, L ∼ 106 L⊙, M ∼ 2−5×10−6 M⊙ yr−1, and v∞ ∼ 2500 km s−1.
Thus, the condition for Coulomb coupling is easily satisfied for ζ Pup; however,
the winds of early B-type stars have much lower mass-loss rates, and Coulomb
coupling may not occur in their winds.
1.2.1.3 Geometry and coordinates of stellar winds
Stellar winds are typically described in spherical coordinates or in ray/cylindrical
coordinates. In spherical coordinates, azimuthal symmetry with respect to the line
of sight is usually assumed. Thus, we may consider two coordinates, radius r and
7
line-of-sight projection µ = cos θ. In ray coordinates, we describe a point by the
distance z = µr along a ray with impact parameter p =√
r2 − z2.
Consider a point in a wind with projected velocity vz = µv(r). It is useful to
define the frequency shift with respect to the comoving frame in thermal Doppler
units:
x =∆ν
∆νD=
(
ν − ν0vz
c
) 1∆νD
(1.8)
where ∆νD = ν0 vth/c and ν0 is the rest frequency of the transition. Thus, −1 < x < 1
gives the characteristic range of frequencies scattered at that point in the wind.
Let us also define:
Q ≡ −dx
dz=
1vth
dvz
dz=
1vth
v
r(1 + σµ2) (1.9)
where
σ ≡ r
v
dv
dr− 1. (1.10)
Because the wind terminal velocity is much larger than the thermal velocity
of ions, the scattering of light of a given rest frequency traveling along a ray from
the photosphere is confined to a small region of physical space and of the range of
velocity space occupied by the wind. In the Sobolev approximation, this region is
approximated as a point (the Sobolev point).
The physical length scale over which photons of a given frequency may scatter
is called the Sobolev length:
Lµ ≡ vth
(
dvz
dz
)−1
=1Q. (1.11)
For the Sobolev approximation to be valid, physical conditions must not change
significantly on approximately this length scale.
8
We define the Sobolev optical depth
τµ =χ
Q=
τ0
1 + σµ2 (1.12)
where χ ≡ κρ is the absorption coefficient (cm−1), κ is the opacity per unit mass
(cm2 g−1), ρ is the density (g cm−3), and
τ0 =χ r vth
v(r)(1.13)
is the lateral (µ = 0) Sobolev optical depth. Alternatively,
1τµ=
1 − µ2
τ0+µ2
τ1(1.14)
where
τ1 =χ vth
dv/dr(1.15)
is the radial (µ = 1) Sobolev optical depth. The Sobolev optical depth gives the
optical depth to scattering across the entire Sobolev region along a given line of
sight.
Note that the Sobolev optical depth does not depend on the thermal velocity;
this is because the value of the absorption coefficient χ depends on the definition of
x and thus the thermal velocity through its appearance in the line profile function.
Also, τ1 ∝ (dv/dr)−1, while τ0 ∝ (v/r)−1; this shows that photon escape in the radial
direction is facilitated by the radial velocity gradient, while photon escape in the
lateral direction occurs because of the spherical divergence of the wind. Finally,
when dv/dr = v/r, σ = 0 and τµ = τ0 = τ1; this is the point in the wind of (local)
constant expansion, so that there is no angular dependence to the Sobolev optical
depth.
9
1.2.1.4 Solution of the transfer equation
Let us consider the radiative transfer of a single, isolated spectral line in an ideal-
ized stellar wind that is smooth and spherically symmetric, with a monotonically
increasing velocity that is directed radially outward and is much greater than the
speed of sound. For readers unfamiliar with radiative transfer formalism, Mihalas
(1978) and Rybicki & Lightman (1979) contain treatments of this subject.
The appropriate transfer equation is
∂I
∂z−Q∂I
∂x= χφ(x)[S − I]. (1.16)
Here I is the specific intensity, S is the source function, χ = κLρ (cm−1) is the
absorption coefficient of the line integrated over frequency, andφ(x) is a normalized
line profile function giving the frequency dependence of the absorption. x and Q
are defined in Equations 1.8 and 1.9.
We assume that the spatial derivative term in Equation 1.16 may be neglected
in comparison with the frequency derivative (Sobolev 1960). We may thus write
the transfer equation as∂I
∂x= τµ φ(x)[I − S]. (1.17)
Define
Φ(x) =∫ ∞
x
φ(x′) dx′. (1.18)
Note that dΦ(x) = −φ(x) dx. In the Sobolev approximation, Φ(x) is the Heaviside
step function H(−x). A graphical representation of φ(x) and Φ(x) is shown in
Figure 1.1.
Using the integrating factor eτµΦ(x), we may solve the transfer equation
∂
∂x(I eτµΦ(x)) = −τµ φ(x) SeτµΦ(x), (1.19)
10
Figure 1.1 Illustration of the definition of φ(x) and Φ(x).
11
by applying appropriate boundary conditions. To solve this, note that
−∫
dxφ(x)eΦ(x) =
∫
dΦ(x)eΦ(x). (1.20)
and use the Sobolev approximation to pull the source function out of the integral
on the right-had side of the equation:
I(x1) eτµΦ(x1) − I(x2) eτµΦ(x2) = S(xSob)(eτµΦ(x1) − eτµΦ(x2)) (1.21)
where xSob is the Sobolev point. xSob = 0 in the comoving frame.
1.2.1.5 The line-driving force
Let us consider the scattering of radiation from the stellar core. The boundary
condition for unscattered stellar emission is
I = Ic D(µ). (1.22)
Here Ic refers to the specific intensity of the stellar core, and D(µ) is unity for µ > µ∗
and zero otherwise under the assumption of no limb darkening. (µ∗ ≡√
1 − (R∗/r)2
is the value of µ at the edge of the stellar disk as observed from a point outside the
star). This boundary condition applies at x → ∞, where there is by definition no
scattering. We can evaluate Equation 1.21 at x and∞, giving
I(x) = Ic D(µ) e−τµΦ(x) + S (1 − e−τµΦ(x)). (1.23)
If we average this over frequency, we get
I ≡∫ ∞
−∞dxφ(x) I(x) = Ic D(µ) p(µ) + S (1 − p(µ)) (1.24)
12
where
p(µ) =∫ ∞
−∞dxφ(x) e−τµΦ(x) =
1 − e−τµ
τµ(1.25)
is the angle-dependent Sobolev escape probability.
The flux is given by
H ≡ 〈µ I(µ)〉 = Ic 〈µD(µ) p(µ)〉. (1.26)
Terms proportional to S vanish because it is an odd function of µ. In other words,
scattering does not affect the line-driving force (although it does affect its stability;
see the discussion in § 1.2.5).
If we approximate the emission from the star as coming from a point source
(µ = 1),
H ≈ Ic D p1 = Ic W(r)1 − e−τ1
τ1(1.27)
where D = 〈D(µ)〉 =W(r) is the geometrical dilution and p1 ≡ p(µ = 1).
The radiative acceleration is proportional to the flux:
g =4πκ∆νD
cH = gthin
1 − e−τ1
τ1= gthin p1 (1.28)
where
gthin =4πκ∆νD
cIcW(r) (1.29)
is the radiative force due to an optically thin line. The radiative force for an optically
thick line is
gthick =gthin
τ1=
gthin
κvth
1ρ
dv
dr=
4πc2 IcW(r)
1ρ
dv
dr. (1.30)
Thus, the optically thin force scales with the strength of the driving flux, while the
optically thick force scales with the velocity gradient.
13
1.2.1.6 The scattered radiation field
Consider the solution of the transfer equation given in Equations 1.19, 1.23, and
1.24. We would like to solve for the source function; it has a scattering term and a
thermal emission term:
S = J(1 − ǫ) + Bǫ = Ssc + Sth (1.31)
where
J ≡ 〈I〉. (1.32)
ǫ is the collisional destruction parameter; it gives the probability of thermalization
of a scattered photon per scattering event. Thermalization typically occurs when
an ion in an excited state is collisionally de-excited by electron impact. For most of
the situations we are interested in, it can be taken to be small.
Using the solution to the transfer equation for radiation from the stellar core
from Equation 1.24, we get
J = Icβc + S(1 − β) (1.33)
where
βc = 〈D(µ)p(µ)〉 (1.34)
is the probability for a photon from the photosphere to penetrate to this point (the
“core penetration probability”) and
β = 〈p(µ)〉 (1.35)
is the angle-averaged escape probability.
The source function is then given by
S =βcIc(1 − ǫ) + Bǫ
β(1 − ǫ) + ǫ . (1.36)
14
In the limit of negligible thermalization (ǫ → 0), the scattering of the stellar
radiation field is given by
Ssc =βcIc
β. (1.37)
This means that the scattered radiation field is given by the amount of light pene-
trating from the star (βcIc), enhanced by a factor of 1/β because of locally trapped
photons.
Again, in the limit of small ǫ, the thermal emission term is given by
Sth =Bǫ
β=
Sth,0
β. (1.38)
In other words, the thermal emission is enhanced by a factor of 1/β as well, also
because of trapping.
1.2.2 The CAK model of a line-driven wind
Castor, Abbott, & Klein (1975, CAK) computed the properties of a line-driven wind
under a number of simplifying assumptions. One of their key contributions was to
develop a method for analytical treatment of the effect of an ensemble of lines. This
model is the basis for all contemporary steady-state models of line-driven winds
(e.g. Abbott 1980, 1982; Friend & Abbott 1986; Pauldrach et al. 1986; Pauldrach 1987;
Puls 1987; Pauldrach et al. 1994; Taresch et al. 1997; Haser et al. 1998; Pauldrach et al.
2001). Here we discuss a simple CAK-like model and the relation of its properties
to those of a real wind. A significantly improved formalism is presented in Gayley
(1995), and the reader is advised to refer to this work as well.
To calculate the radiative acceleration due to a number of non-overlapping
spectral lines, we must sum over all lines. Contemporary models perform detailed
15
NLTE calculations of level populations and calculate radiative transfer in the actual
lines; however, it is of significant value in understanding the behavior of line-driven
winds (and much simpler) to consider an analytic expression for the line force.
CAK assumed that the behavior of the radiative force is somewhere in between
the optically thin and optically thick cases:
g ∝ L∗r2
(
1ρ
dv
dr
)α
(1.39)
with 0 < α < 1. A value of α = 0 represents a wind composed entirely of optically
thin lines, while a value of α = 1 corresponds to the completely optically thick case.
The radial dependence of the flux is given in the point source approximation.
This can be recovered if we assume a truncated power-law opacity distribution
N(κ) ∝ κα−2e−κ/κ0 . (1.40)
and integrate the line force over this distribution:
gCAK =
∫ ∞
0dκN(κ)g(κ) (1.41)
The assumption of a power-law opacity distribution must be discarded in favor
of a line list in a more realistic stellar wind model, but it is a very useful tool in
constructing and understanding a simple model.
Now let us consider the structure of a wind with line force given by Equa-
tion 1.39. The momentum equation is
vdv
dr+
GM∗
r2 − 1ρ
dp
dr− gL − ge = 0 (1.42)
where the terms are, in order, net acceleration, gravity, the gas pressure gradient,
radiative acceleration in lines, and radiative acceleration in the continuum.
16
Define
w ≡ v2
2GM∗, (1.43)
u ≡ −1/r, and
w′ ≡ dw
du=
r2v
GM∗
dv
dr(1.44)
Neglecting gas pressure and the radiative force due to continuum opacity, we
can write
w′ + 1 − C(w′)α = 0 (1.45)
where
C =KL∗GM∗
(4πGM∗
M
)α
. (1.46)
We have used the continuity equation in deriving this result. K is the constant of
proportionality in Equation 1.39.
This equation admits 0, 1, or 2 solutions; in the case with one solution (the
“critical” solution), its derivative with respect to w′ is zero. This gives the solution
w′ =α
1 − α (1.47)
and
C =α−α
(1 − α)1−α . (1.48)
Solving for the velocity law and the mass loss rate, we get
v(r) = v∞
(
1 − R∗r
)0.5
(1.49)
with
v∞ =
√
α
1 − αvesc (1.50)
17
and
M = 4πα(
GM∗
1 − α
)1−1/α
(KL∗)1/α. (1.51)
We have assumed that the “critical” solution is the correct solution, but there
is no reason to do so on the basis of this analysis. If we follow CAK and include
gas pressure in our analysis, however, the critical solution is the only one. It is not
at all clear that real winds correspond to this solution, though.
The mass-loss rate in a line-driven wind is set by the conditions around the
critical point; because the wind is highly supersonic above the critical point, the
flow can have no effect on the conditions at the base of the wind. On the other hand,
radiation pressure continues to accelerate the wind far above the critical point, so
the velocity law depends on the conditions throughout the wind.
Although the critical point in a line-driven wind is not the sonic point, the
wind velocity at the critical point is of order the sound speed, so the Sobolev
approximation is not necessarily valid. Furthermore, there is a large body of
observational and theoretical evidence for variability and inhomogeneity in winds.
For a more detailed discussion of solution topologies and the validity of steady-
state models, see e.g. Poe, Owocki, & Castor (1990), Owocki (1990) and Owocki &
Zank (1991).
The assumption of radially directed photons (µ = 1) in Equation 1.27 results
in larger accelerations close to the star, and hence a steeper velocity law. Friend &
Abbott (1986) and Pauldrach et al. (1986) have independently found the finite-disk
correction for the CAK model. They find the velocity-law exponent is approxi-
mately 0.8 (compared to 0.5 for CAK). They also find that the mass-loss rate is
lower than CAK by about a factor of two, which is also attributable to the lower
18
radial momentum transfer near the wind base.
1.2.3 Measurement of wind parameters in steady state models
The two most important quantities to measure in a stellar wind are the mass-loss
rate and the terminal velocity. The mass-loss rate is much more difficult to measure
accurately. A good review of measurements of properties of stellar winds is given
in Kudritzki & Puls (2000).
Mass-loss rate diagnostics generally fall into one of two categories: those re-
sulting from photon emission and scaling with density squared, and those resulting
from absorption and scaling with density.
The two main emission diagnostics are H α emission and radio free-free emis-
sion. Because these scale with the emission measure, dEM = nenidV, they are both
susceptible to overestimating the mass-loss rate if the wind is clumped.
Consider a clumped wind in which all the mass occupies a fraction f of the
wind volume. The density will be larger by a factor 1/ f over a smooth wind,
and the emission measure will also be larger by a factor f . The mass-loss rate is
proportional to the density, so in a smooth wind EM ∝ M2, and in a clumpy wind
EM ∝ M2 f .
Early measurements of M found good agreement between radio and H α (e.g.
Lamers & Leitherer 1993). It was argued that this presented evidence that clumping
was not important, since the H α emission comes from very close to the star and
the radio emission comes from far out in the wind, making coincidental agreement
in clumping factors unlikely. More recent measurements using these techniques
have indeed found discrepancies (Fullerton et al. 2006; Puls et al. 2006).
19
Unsaturated UV absorption profiles can give very accurate measurements of
the ion column density as a function of Doppler shift, and thus the wind density.
Deriving the wind density from the ion column density, however, depends on
having an accurate calculation of the ionization balance, a nontrivial problem. In
fact, calculation of the ionization balance is also subject to errors if the wind is
clumped and this clumping is not taken into account, due to the dependence of
the recombination rate on the density. One way around this difficulty is to use the
absorption line of an ion that is the dominant ionization state, but this is not always
possible. Some recent work has focused on measurements of absorption due to
sodium-like phosphorus (P V), which is thought to be the dominant ionization stage
of phosphorus in the winds of at least some spectral types of O stars; downward
revisions of O star mass loss rates at least of order a few appear to be called for
(Massa et al. 2003; Hillier et al. 2003; Bouret et al. 2005; Fullerton et al. 2006).
1.2.4 Observational evidence for variability and instability in
line-driven winds
There are several lines of evidence for inhomogeneity and instability in radiatively
driven winds. In addition to this, the winds are expected to be unstable based on
theoretical arguments and numerical hydrodynamic simulations.
The most direct observational evidence for instability is discrete absorption
components (DACs; e.g. Lamers et al. 1982; Howarth & Prinja 1989; Prinja et al.
1992; Eversberg et al. 1998; Kaper et al. 1999). They are narrow absorption lines
superimposed on broad P Cygni absorption troughs; time series observations show
that they move from lower to higher velocities.
20
Extended black troughs in absorption profiles are another line of evidence for
inhomogeneity. In a smooth outflow, the width of the totally black part of a satu-
rated P Cygni profile (which occurs at the terminal velocity) should be the thermal
velocity width of the ions; the rest of the profile is partially filled in with scattered
light. However, a non-monotonic velocity structure can preferentially backscatter
light, leading to the observed black troughs (Lucy 1982a, 1983). This effect has been
reproduced in synthetic profiles generated from numerical hydrodynamic simula-
tions with non-monotonic velocity structure arising from the line-driven instability
(Puls et al. 1994; Owocki 1994).
Evidence for shocks in the winds of O stars comes from their soft X-ray emis-
sion and their non-thermal radio excess. The soft X-rays are produced thermally
in the several MK plasma arising from the strongest shocks in the wind (Harnden
et al. 1979; Seward et al. 1979; Cassinelli & Swank 1983); the non-thermal radio
excess is thought to be from synchrotron emission from particles that have under-
gone Fermi acceleration in the shocks (White 1985; Bieging et al. 1989). The shocks
themselves are thought to be a result of nonmonotonic velocity fields produced
by the instability in the line driving force (Lucy & White 1980; Owocki et al. 1988;
Feldmeier et al. 1997b).
1.2.5 Theory of instabilities in line-driven winds and generation
of X-ray emitting shocks
The idea that the line driving force might be unstable was recognized by Lucy &
Solomon (1970). The basic physical idea is that if the line driving force depends on
deshadowing, then perturbatively deshadowing a small amount of gas can lead to
21
runaway acceleration through further deshadowing.
Under more detailed analysis, some authors found that line-driven winds
should indeed be unstable in the approximation of optically thin perturbations
(e.g. Carlberg 1980), while Abbott (1980) found that in the Sobolev approximation
perturbations give rise to acoustic waves and do not become unstable. Owocki
& Rybicki (1984) reconciled these approaches by dropping these approximations;
they found that perturbations on length scales greater than the Sobolev length
behave as predicted by Abbott, but that perturbations on smaller length scales are
indeed unstable, leading to growth of order 100 e-folds by 1.5 stellar radii.
Lucy (1984) showed that the scattered radiation field can reduce or eliminate
the instability. From the point of view of an unperturbed parcel of gas, the wind is
moving away from it in all directions, and the scattered radiation field is symmetric,
resulting in no net force; for a perturbed parcel of gas, the scattered radiation in
the direction of the perturbation is Doppler shifted into resonance with it, causing
a force to develop in opposite direction of the perturbation.
Owocki & Rybicki (1985) incorporated scattering in their earlier stability anal-
ysis and showed that, while this damping is effective in reducing the instability
near the base of the wind, by the time the wind has reached 1.5 stellar radii, the
instability is already half as strong as in the absence of scattered radiation.
Since the initial results analyzing the instabilities of winds, numerical hydro-
dynamic simulations have confirmed their existence and given much insight into
their nature. The simulations of Owocki, Castor, & Rybicki (1988) ignored the ef-
fects of scattering in order to make the computations tractable. They found a wind
structure with dense clumps and a highly rarefied interclump medium. The inter-
22
clump medium is subject to runaway acceleration until it runs into a clump, where
there is a reverse shock, heating the rarefied gas to X-ray emitting temperatures.
This is in contrast to earlier phenomenological models (Lucy & White 1980; Lucy
1982b), which assumed forward shocks caused by dense clumps plowing through
an ambient medium as an origin for X-rays.
Cooper (1994, Ph.D. thesis) and Cooper & Owocki (1994) developed models
of the X-ray emission using hydrodynamic codes. They found that thick and thin
winds have different behavior: thin winds can be modelled assuming adiabatic
cooling of shocked material dominates over radiative cooling; thick winds are
subject to numerical instability when radiative cooling is included, but the shock
cooling may be modelled by assuming it occurs rapidly in comparison with the
flow timescale of the wind, leading to steady-state cooling. The thin wind model
generally underpredicts the X-ray emission of late O stars by a factor of 10 or
greater, while the thick wind model overpredicts the emission of early O stars by
up to a factor of 10.
Feldmeier et al. (1997b) performed numerical hydrodynamic simulations with
seed perturbations at the base of the wind. They found that these models have
clump-clump collisions, with small, fast cloudlets overtaking larger clumps. This
can lead to substantially enhanced X-ray emission in comparison with a tenuous
interclump medium piling into a reverse shock on a clump. This model predicts
an X-ray flux a factor of a few lower than is observed, which is significantly closer
than the adiabatic model of Cooper (1994). The X-ray emission at any given time
comes from only one or a few shocks, which would lead one to expect the model to
predict significant variability. However, in three dimensions the wind is likely to
behave incoherently on some (unknown) lateral scale, smoothing out the observed
23
X-ray flux.
1.3 X-ray emitting plasmas in the coronal approxima-
tion
Many astrophysical X-ray emitting plasmas can be described by a simple model
known as the coronal approximation, so called because it was first used in describ-
ing the Sun’s corona (Elwert 1952). This model makes a number of simplifying
assumptions.
First, it is assumed that X-ray photons created in the plasma escape without
further interaction. Second, the electrons and ions are assumed to have Maxwellian
velocity distributions, and to have the same temperature as each other; the ion-
ization balance in the plasma is also assumed to be in equilibrium. For a sudden
change in temperature, the timescale for achieving ionization equilibrium is in-
versely proportional to the electron density. Finally the excited state populations
of ions are taken to be negligible compared to the ground state populations; col-
lisional interactions only occur with ions in the ground state. This condition is
satisfied if the electron density is sufficiently low.
All of these assumptions may be partially relaxed to obtain special cases of
the coronal approximation. If strong resonance lines are optically thick, resonance
scattering may be important (this has been observed in the elliptical galaxy NGC
4636 by Xu et al. 2002); populations of metastable states may be high enough
that collisional- or photoexcitation from those states is not negligible (e.g. Gabriel
& Jordan 1969; Blumenthal, Drake, & Tucker 1972; Mauche, Liedahl, & Fournier
24
2001); and in a non-ionization-equilibrium plasma (NIE), ionization states may not
be in equilibrium. NIE plasmas are found in young supernova remnants that have
not had enough time to reach ionization equilibrium.
There are two other very different plasma models which are frequently in-
voked by X-ray astronomers: photoionized plasmas, where the ionization balance
is set by an external ionizing source and the plasma is overionized compared to
the electron temperature; and stellar atmospheres, where the optical depth is not
negligible and collisional de-excitation of excited states is significant.
Examples of plasmas which are well-explained by the coronal approximation
or a modified version thereof include stellar coronae, accretion shocks (e.g. in the
accretion columns of some cataclysmic variables), supernova remnants, the hot
intracluster medium, the hot, shocked plasma in O star winds, and colliding winds
in massive-star binaries. Photoionized plasmas are typically observed near intense
sources of ionizing radiation, which can be created by the accretion of material onto
a compact object; examples include the outflows from the nuclei of active galaxies,
material around accretion disks in low-mass X-ray binaries, and stellar winds in
high-mass X-ray binaries. Stellar atmospheres of normal stars are much too cool to
emit X-rays, but the atmospheres of young, cooling neutron stars and white dwarfs
are quite hot and can emit significant soft X-rays.
1.3.1 Atomic processes in plasmas
The important events we need to keep track of in a coronal plasma are changes
in ionization stage and emission of photons. These are both ultimately caused by
electron-ion collisions. The possible outcomes of such a collision are direct exci-
25
tation, direct ionization, radiative recombination (RR), and dielectronic (resonant)
capture.
Assuming the excited electron is from the valence shell, the result of direct ex-
citation in a coronal plasma is always radiative decay. In both the case of ionization
and excitation, it is possible to ionize or excite a core electron rather than a valence
electron. This is usually not a very important process to consider in collisional
ionization equilibrium, since electrons with high enough energies to do this do not
coexist with the appropriate ionization states.
Resonant capture refers to the simultaneous capture of a free electron and
excitation of a bound electron, resulting in a doubly excited state. If the excited
electron comes from the core, double-autoionization or auto-double-ionization may
occur (a net single ionization), but this is again unlikely in CIE. If the doubly excited
state decays radiatively twice, the process is called dielectronic recombination (DR).
If the doubly excited state autoionizes but leaves one electron in an excited state
the process is called resonant excitation (RE). RE is factored into rate coefficients for
electron impact excitation, and DR is combined with RR to get total recombination
rate coefficients. Both RE and DR can make significant and sometimes dominant
contributions to rate coefficients.
Good overviews of these processes are given in the Ph.D. theses of Gu (2000,
Chapter 1, § 3) and Sako (2001, Chapter 1, diagrams on pp. 53-54), as well as in
Mewe (1999), and Kahn (2005).
26
1.3.2 Rates, rate coefficients, and cross sections
Consider an infinitesimal amount of plasma with monoenergetic electrons of veloc-
ity v in which events caused by electron-ion collisions are occurring. In the coronal
approximation, the ions are always in the ground state when collisions occur. The
number of events of process j occurring from the ground state of ion z of element
Z per unit time per unit volume is given by
dN j(v) = ne nz v σ j(v) dV dt, (1.52)
where σ j(v) is the cross section for process j, and ne and nz are the densities of
electrons and ions, respectively.
For a plasma at temperature T, the rate is given by averaging Equation 1.52
over a Maxwellian velocity distribution:
dN j(T) = 〈dN j(v)〉 = C j(T) ne nz dV dt (1.53)
where we have defined the rate coefficient (in units cm3 s−1) for process j
C j(T) ≡ 〈vσ j(v)〉 =∫ ∞
v0
dv fM(v,T) v σ j(v). (1.54)
Here v0 is the threshold velocity for process j to occur.
For a radiative decay j from state i, the number of events occurring per unit
time per unit volume is given by
dN j(T) = A j ni dV dt (1.55)
where A j is the (spontaneous) rate of process j and ni is the density of ions in state
i.
27
1.3.3 Ionization balance
In a coronal plasma, the ionization balance is determined by the competition be-
tween ionization and recombination. Using Equation 1.53, we have
1ne
dnz
dt= nz−1 CI
z−1(T) − nz(CIz(T) + CR
z (T)) + nz+1 CRz+1(T) (1.56)
where CIz(T) is the rate coefficient for collisional processes resulting in ionization
from charge state z to z + 1 and CRz (T) is the rate coefficient for recombination
processes from charge state z to z − 1. We have neglected processes involving
multiple ionizations, but it is possible to include them in a more sophisticated
treatment.
In equilibrium the time derivative is zero, and Equation 1.56 reduces to
nz+1
nz=
CIz(T)
CRz+1(T)
. (1.57)
This set of equations can be solved to obtain the ionization fractions qz(T), as
described in Mewe (1999).
In Figure 1.2 we show a plot of the ionization balance of iron from the APED
database (Smith et al. 2001). The stable He-like and Ne-like configurations dom-
inate over a wide range of temperatures. The rapid changes in ionization state
from a few MK to a few tens of MK allows us to use Fe L-shell spectroscopy as
a sensitive thermometer for hot plasmas. For further useful illustrations of the
ionization balance, see the Ph.D. theses of Sako (2001) and Peterson (2003).
28
Figure 1.2 Ion fractions q for iron as a function of temperature.
29
1.3.4 Discrete line emission
Let us consider the population of state j in an ion of atomic number Z and charge
z. The rate of change in the population of state j can be written
dn j
dt= ne(nzCzj + nz−1C(z−1) j + nz+1C(z+1)i) +
∑
k
nk Akj − n j
∑
k
A jk. (1.58)
The first three terms give the population of state j through direct excitation, inner-
shell ionization, and recombination. The fourth terms gives the population of state
j by radiative decay from higher energy states states k, and the last term gives the
radiative decay rate to lower energy states k. We have neglected all terms involving
multiple ionizations or recombinations.
Let us assume the plasma is in a steady-state so that the time derivative is
zero. The rate of transition ji per unit volume is given by
n j A ji = B ji
ne(nzCzj + nz−1C(z−1) j + nz+1C(z+1)i) +∑
k
nk Akj
(1.59)
where
B ji = A ji/∑
k
A jk (1.60)
is the branching ratio for decay from state j to state i.
Substituting Equation 1.59 into itself to eliminate the dependence on the pop-
ulations of other states, we get
n j A ji = B ji ne nZ
qz
∑
k
Czk B′kj + qz−1
∑
k
C(z−1)k B′kj + qz+1
∑
k
C(z+1)k B′kj
(1.61)
where B′kj
refers to the effective branching ratio from state k to state j over all
possible intermediate states. B′j j
is defined to be unity.
30
Define the line power (photons cm3 s−1)
P ji ≡ B jinH
ne
AZ
qz
∑
k
CzkB′kj + qz−1
∑
k
C(z−1)kB′kj + qz+1
∑
k
C(z+1)kB′kj
(1.62)
and the emission measure
dEM ≡ n2e dV. (1.63)
The rate of photon emission can then be written
n j A jidV = P ji(T)dEM, (1.64)
which has the advantage of separating the temperature and density dependence of
the emission. Physically, the line power is the product of the rate coefficient for the
population of state j by all processes multiplied by the branching ratio for decay to
state i. The product of an emission measure and a rate coefficient gives a rate.
1.3.5 Bremsstrahlung
Bremsstrahlung can occur when an electron collides with an ion. In temperatures
typical of shocks in O star winds (0.1-0.5 keV), line emission is dominant in radiative
cooling. In higher temperature plasmas (> 1 keV), such as are found in the shock of
a colliding-wind binary, bremsstrahlung is dominant in radiative cooling, although
line emission is still of enormous diagnostic value.
The spectrum of bremsstrahlung emission from an isothermal plasma is nor-
mally given as the product of the classical spectrum times a (slowly varying and
often of order unity) quantum mechanical correction, the Gaunt factor. This is
because both formulations have the same scaling with physical parameters. The
31
emission spectrum for a plasma with a Maxwellian distribution of electrons at
temperature T is given in Rybicki & Lightman (1979):
dPν(T) = Pν,0∑
i
z2i gi(ν,T) T−1/2 e−hν/kT ne ni dV (1.65)
where Pν,0 = 6.8 × 10−38erg cm3 s−1 Hz−1 K1/2 and the summation is over all ion
species. gi(ν,T) is the velocity averaged Gaunt factor. This spectrum is flat for
low frequencies, with an exponential cutoff at photon energies comparable to the
plasma temperature.
The total power per unit volume is
dP(T) = P0
∑
i
z2i gi(T) T1/2 ne ni dV (1.66)
where gi is the frequency average of the velocity averaged Gaunt factor, and P0 =
1.4 × 10−27erg cm3 s−1 K1/2.
1.3.6 Model parameters for a coronal plasma
Consider an X-ray emitting plasma which is well-described by the coronal model
and by a single temperature. The emission spectrum is described by the sum of all
discrete and continuum processes:
dLλ(T) =
Pλ,brems(T)/n2e +
∑
i, j
E ji φ ji(λ) P ji(T)
dEM. (1.67)
Here φ ji(λ) is the line profile of transition ji.
Real astrophysical plasmas are never described by one temperature; they are
always multiphase. We can rewrite Equation 1.67 to reflect this:
dLλ =
Pλ,brems(T)/n2e +
∑
i, j
E ji φ ji(λ) P ji(T)
dEM
dTdT (1.68)
32
where we have defined the distribution of plasma over temperature by the differ-
ential emission measure dEM/dT.
Combined with the elemental abundances, which enter into the line powers,
the differential emission measure distribution uniquely specifies the emission from
a coronal plasma.
1.3.7 Helium-like triplet ratios
The n = 2 → 1 emission of helium-like ions is observed to consist of a triplet: the
resonance line (1s 2p 1P1→ 1s2 1S0), the intercombination line (a blend of 1s 2p 3P1,2
→ 1s2 1S0), and the forbidden line (1s 2s 3S1 → 1s2 1S0). Strictly speaking, this is
not a triplet, but because the intercombination line is blended for astrophysically
observable He-like emission, it is referred to as one. The line ratios are set by the
collisional rate equations described in § 1.3.4; the ratio R ≡ f/i has a very weak
temperature dependence. The interaction of the triplet states and the ground state
is depicted in Figure 1.3. The transition wavelengths for 1s 2s 3S1→ 1s 2p 3PJ are in
the UV for astrophysically interesting ions.
The 1s 2s 3S1 state in He-like ions is metastable. Decay to ground is forbidden
in the dipole approximation, but proceeds via magnetic dipole interaction. Because
the decay rate is slow, the rate for collisional- or photoexcitation of this state to the
1s 2p 3P0,1,2 states may become comparable. If this is the case, the forbidden line is
weaker and the intercombination line stronger.
Solving the rate equations gives this expression (Gabriel & Jordan 1969; Blu-
33
i
P3
J
2
1
0
S1
S3
1
0
X−ray
UV
f
Figure 1.3 Diagram of the interaction of the n = 2 triplet levels of He-like ions.
Solid lines show electron impact excitation from 1s 2s 3S1 at high density. Dashed
lines show photoexcitation and radiative decay.
34
menthal, Drake, & Tucker 1972):
R = R01
1 + φ/φc + ne/nc. (1.69)
R0 gives the “unperturbed” limit of the ratio in the limit of no excitations to higher
levels; it is a function of the rate coefficients populating the n = 2 levels and the
branching ratio of decays from 3PJ to ground. φ gives the photoexcitation rate from
3S1 to 3PJ, and ne is the electron density, while φc and nc are the critical rate and
density where R = R0/2. For calculations of R0, φc, and nc, see e.g. Blumenthal,
Drake, & Tucker (1972) or Porquet et al. (2001).
The f/i ratio of an individual He-like triplet only provides information about a
limited range of densities or photoexcitation rates; ifφ or ne is lower than the critical
value by more than a factor of ten, it will be difficult to observe a perturbation to
the ratio, and if φ or ne is greater than the critical value by more than factor or ten,
it will be difficult to measure the forbidden line at all. However, the decay rate
of the 1s 2s 3S1 state has a strong dependence on the atomic number Z. If we can
observe f/i for a range of Z, this allows us to probe a broad range of densities or
exciting fluxes. For example, the critical density for O VII is a few times 1010 cm−3,
while for Si XIII it is a few times 1013 cm−3.
As will be shown in Chapter 2 and in more detail in Chapter 4, in X-ray spectra
of O stars the strong UV flux from the star causes many of the forbidden lines to
almost disappear and others to diminish. Because of the geometrical dependence
of the UV flux seen by ions in the wind, the observed line ratios, in conjunction
with the (well-known) UV spectrum of the star, constrain the location of the X-ray
emitting plasma. This is an important diagnostic because it is independent of other
35
diagnostics of the location of the X-ray emitting plasma such as Doppler profiles of
emission lines, the interpretation of which depends on assumptions regarding the
line-of-sight velocity of the X-ray emitting plasma.
The wavelengths of the 1s 2s 3S1→ 1s 2p 3PJ transitions decrease with increasing
Z. For Z ≥ 14 (silicon), the transition wavelengths are shortward of the Lyman edge.
Both the measured and calculated UV fluxes of O stars are much more uncertain
shortward of the Lyman edge, making any inferences of plasma location based on
line ratios of Z ≥ 14 correspondingly uncertain.
1.4 Emission line Doppler profile models
Because of the high velocities of O-star stellar winds (∼ 1500 − 2500 km s−1), it is
possible to spectrally resolve the Doppler profiles of X-ray emission lines, which
are formed in shocked plasma distributed throughout the wind. In the absence
of photoelectric absorption by the unshocked bulk of the wind, the profile shape
contains information about the radial distribution of the X-ray emitting plasma.
This is because the velocity of the wind is a function of radius. Plasma located
close to the star has a low velocity, and thus only emits at lower Doppler shifts,
while emission at high Doppler shifts must come from plasma far away from the
star. When absorption is added into the model, the profile becomes asymmetric;
this is because X-rays emitted on the far side of the star must pass through a greater
column of absorbing material, so that the profile has a blueward skew. One can
thus measure the radial distribution of the X-ray emitting plasma from the width
of the profile and the thickness of the wind from the profile asymmetry.
36
Owocki & Cohen (2001, hereafter OC01) have suggested a simple parametrized
model for stellar wind X-ray emission line profiles, which I recapitulate here briefly.
I also discuss the inclusion of resonance scattering in the model, as derived by
Ignace & Gayley (2002, hereafter IG02).
1.4.1 The Doppler profile model
In the OC01 model, the wind is taken to consist of a smooth, spherically symmetric,
two-component fluid. The main component is the cold, unshocked bulk of the
wind, which has a temperature of order the photospheric temperature. The other
component is a hot, shocked, X-ray emitting plasma. The second component emits
the X-rays, while the first absorbs them. In reality, the wind may be clumpy; indeed,
it must be inhomogeneous at some level if there is to be any X-ray emission at all.
However, the approximation of smoothness is valid as long as the number of X-ray
emitting blobs is large and porosity effects are not important (e.g. Oskinova et al.
2006; Owocki & Cohen 2006).
The X-ray luminosity as a function of wavelength is given by the volume
integral of the emissivity, attenuated by absorption along the line of sight:
Lλ = 4π∫
dV ηλ(µ, r) e−τ(µ,r) (1.70)
First, we show how to evaluate the profile assuming we know how to calculate
the optical depth.
OC01 assume an emissivity that scales with density squared times a volume
filling factor fX(r):
ηλ = Cρ2(r) fX(r)δ(
λ − λ0
1 −µv(r)
c
)
. (1.71)
37
Here the wavelength dependence of the emission is contained in a line profile
function that is a δ function in the Sobolev approximation.
Define the scaled wavelength
x ≡(
λ
λ0− 1
)
c
v∞(1.72)
such that x = ±1 corresponds to the red/blueshift when moving away from the
observer at the terminal velocity. Note that this is different than the convention in
Sobolev theory, where x is defined as a frequency shift scaled in units of the thermal
velocity.
Using Lxdx = Lλdλ, the luminosity is
Lx = 8π2C
∫ 1
−1dµ
∫ ∞
R∗
dr r2 fX(r)ρ2(r) e−τ(µ,r) δ(x + µw(r)) (1.73)
where w(r) = v(r)/v∞ = (1 − R∗/r)β is the scaled velocity.
Using the continuity equation, ρ = M/(4πr2v(r)), and integrating over dµusing
the delta function, we have
Lx =CM2
2v2∞
∫ ∞
rx
drfX(r)
r2w3(r)e−τ(µ,r)
∣
∣
∣
∣
∣
∣
µ=−x/w
. (1.74)
The lower limit of integration rx = R∗/(1−|x|1/β) is chosen to enforce the condition of
the delta function. The third factor of w(r) in the denominator comes from changing
variables in the δ function.
For numerical quadrature, it is preferable to change to inverse radial coordi-
nates u = R∗/r:
Lx =CM2
2v2∞R∗
∫ ux
0du
fX(u)w3(u)
e−τ(µ,r)
∣
∣
∣
∣
∣
∣
µ=−x/w
(1.75)
where ux = 1 − |x|1/β.
38
OC01 take the volume filling factor of the X-ray emitting plasma to be zero
below a radius R0, reflecting the fact that the instabilities do not develop near the
base of the wind, and a power law in radius above that, fX(r) ∝ r−q or fX(u) = fX,0 uq.
This gives
Lx =CM2 fX,0
2v2∞R∗
∫ umax
0du
uq
(1 − u)3β e−τ(µ,r)
∣
∣
∣
∣
∣
∣
µ=−x/w
. (1.76)
We have defined umax ≡ min(u0, ux), where u0 = R∗/R0.
The optical depth is evaluated in ray coordinates. The impact parameter is
p ≡ r√
1 − µ2, and the distance along the ray is z ≡ µr. The optical depth is then
given by
τ(p, z) =∫ ∞
z
dz′κρ(r′) (1.77)
Again using the continuity equation, we have
τ(p, z) =κM
4πv∞R∗
∫ ∞
z
dz′R∗r′2w(r′)
= τ∗ t(p, z) (1.78)
where
τ∗ =κM
4πv∞R∗(1.79)
is the characteristic optical depth and
t(p, z) =∫ ∞
z
dz′R∗r′2w(r′)
. (1.80)
If the wind velocity were constant (β = 0), then the optical depth along a radial ray
is t(p = 0, z) = t(r) = R∗/r; thus, τ∗ would be the optical depth from the observer to
the surface of the star.
The integral t(p, z) has analytic solutions for integer values of β, but must be
numerically evaluated otherwise. Because the optical depth integral is nested in
the line profile integral, it is more convenient to assume an integer value of β and
39
use the analytic solution. Since most O-star winds have β ∼ 1, we take β = 1 as the
canonical case. We give the analytic expression here such that it can be evaluated
without using complex numbers.
t(p, z) =
∞, p ≤ 1, z ≤√
1 − p2
1+√
1+z2
z− 1, p = 1
1y
(
π2 + arctan
(
1y
)
− arctan(
zy
)
− arctan(
zyr
))
, p > 1
log(
zz−1
)
, p = 0
12z∗
log[
(z+z∗)(µ+z∗)(1−z∗)(z−z∗)(µ−z∗)(1+z∗)
]
, p < 1
(1.81)
In these expressions, y =√
p2 − 1, z∗ =√
1 − p2, and all quantities with dimensions
of length are in units of R∗. The first case listed is for obscuration by the stellar core.
In Figure 1.4 we show model profiles for a wind with a constant filling factor,
an onset radius for X-ray emission of 1.5 stellar radii, and a range of characteristic
optical depths.
Several modifications of or extensions to this model are possible. Clumping of
the cold absorbing component of the wind may cause it to become porous, resulting
in a reduction of the effective optical depth (e.g. Oskinova, Feldmeier, & Hamann
2004, 2006; Owocki & Cohen 2006). In Chapter 4, we investigate the effect of the
radial dependence of line ratios of He-like triplets on the line profiles. In Chapter 5
we show that resonance scattering may be important in Doppler profile formation.
The derivation of this effect, originally described in IG02, is the subject of the next
section.
40
Figure 1.4 Model profiles with q = 0, R0 = 1.5R∗, and τ∗ = 0, 1, 3, 5, 10, 100. The
profiles go from symmetric to blueward skewed in increasing order of τ∗.
41
1.4.2 Doppler profiles with resonance scattering
In this section I recapitulate the derivation of the effects of resonance line scattering
on the Doppler profile of an emission line in the X-ray spectrum of an O star. This
was originally discussed in IG02, although not in the specific context of the model
of OC01.
IG02 suggested resonance scattering as a possible explanation for the unex-
pected lack of strong asymmetry observed in Doppler profiles of X-ray emission
lines in O stars. This can be effective if the lateral (azimuthal) escape probability
is much higher than the radial escape probability, so that we only observe photons
with a low projected velocity. In Sobolev theory, radial escape results from the
local radial velocity gradient (dv/dr), while lateral escape results from the spherical
divergence of the wind (v/r), so this effect could be important further out in the
wind, where the radial velocity gradient is small and the spherical divergence is
large.
We use the definition of x in Equation 1.72, where it is defined to be the
wavelength shift in the observer’s frame. We further define
x′ = x + µw(r), (1.82)
the wavelength shift in the comoving frame.
With this definition of x′ we set
Φ(x′) =∫ x′
−∞dyφ(y), (1.83)
where y is a dummy variable for integration. Note that now dΦ(x′) = φ(x′)dx′.
We consider the transfer equation including continuum absorption:
∂I
∂z= χφ(x)(S − I) − χcI. (1.84)
42
Using the integrating factor e−τce−τµΦ(x) we get
∂
∂z[Ie−τce−τµΦ(x)] = χφ(x)Se−τce−τµΦ(x). (1.85)
Here we have used
τc(p, z) ≡∫ ∞
z
dz′χc(z′), (1.86)
where dτc = −χc dz.
We integrate Equation 1.85 along a ray with impact parameter p. Inserting
Equation 1.38 (which gives the enhancement to thermal emission due to trapping)
and using the definition S ≡ η/χ, we find the observed specific intensity:
I(p) =∫ ∞
−∞dzηth
βe−τc φ(x′) e−τµΦ(x′), (1.87)
where ηth = C fX(r)ρ(r)2, as in OC01.
The X-ray line profile can be calculated by integrating over all rays p:
Lx = 4π∫ 2π
0dα
∫ ∞
0p dp I(p) (1.88)
which can be rewritten in spherical coordinates:
Lx = 8π2∫ ∞
R∗
r2 dr
∫ 1
−1dµηth(r)β
e−τc(µ,r)φ(x′) e−τµΦ(x′). (1.89)
In the Sobolev approximation, the line profile can be treated as a δ function.
We use it to evaluate the µ integral, as in OC01. Using Equation 1.25, we get
Lx = 8π2∫ ∞
rx
r2 drηth(r)w(r)
e−τc(µ,r) p(µ)β
∣
∣
∣
∣
∣
∣
µ=−x/w
. (1.90)
The expression derived is identical to Equation 1.74 (Equation 8 of OC01), with
the modification that the integrand is multiplied by the normalized escape proba-
bility p(µ)/β. The calculation of model profiles from this expression is performed
in Chapter 5.
43
1.5 X-ray spectroscopic instrumentation
The design of modern X-ray observatories is quite different from conventional op-
tical observatories for a number of reasons. First, because the earth’s atmosphere
is opaque to X-rays, all observations must be made from space, and X-ray obser-
vatories must be satellite-based. Due to the high cost of launching satellites into
orbit, weight must be minimized, placing strong constraints on optics and instru-
ment design. Second, it is not possible to reflect X-rays at normal incidence, with
the exception of multilayer materials optimized for a narrow wavelength band.
Because of this, broadband X-ray telescopes reflect X-rays at grazing incidence.
Reflection angles of order a few degrees are typically needed to obtain sufficient re-
flectivity. X-ray telescopes generally use the Wolter Type I design, in which X-rays
are reflected twice, with the first surface being a paraboloid of revolution and the
second a hyperboloid. Because grazing incidence mirrors have a small geometrical
area compared to their diameter, mirror shells are nested inside one another to
create a mirror module with sufficient geometrical area. Also, the mirrors must
be long to intercept a significant amount of light at grazing incidence, and are
thus proportionally heavier than an optical telescope of similar geometrical area.
Finally, because the fluxes of typical astrophysical X-ray sources are very small, it
is crucial to maximize the effective area of the observatory. The stringent weight re-
quirements of a satellite-based observatory work against the design goal of a large
effective area, so it is necessary to optimize mirror and instrument construction to
obtain the maximum possible performance with the minimum possible weight.
For the first twenty years after the discovery of X-ray emission from O stars, it
was not possible to obtain high-resolution, high-throughput X-ray spectra of them.
44
Although significant information was extracted from the proportional counter and
CCD spectra of O stars which were available, notably the rough scaling of the
X-ray luminosity with the bolometric luminosity (Long & White 1980; Pallavicini
et al. 1981; Chlebowski et al. 1989; Berghoefer et al. 1996b) and the lack of strong
absorption of soft X-rays which implied the location of X-ray emitting plasma far
out in the winds of O-stars (Cassinelli & Swank 1983), ultimately these spectra were
of limited value in understanding the physical nature of sources which emit most
of their X-rays in discrete emission lines.
With the launch of the XMM-Newton and Chandra observatories, both includ-
ing high-resolution, high-throughput diffraction grating spectrometers, our un-
derstanding of O stars (and almost every other class of astrophysical object) has
been greatly advanced. It is now possible to resolve individual emission lines and
blends, as well as their Doppler profiles, where before it was only possible to see
the crudest features. The most important spectral diagnostic now available is the
shape of Doppler profiles of emission lines, which contain information about the
radial distribution of X-ray emitting plasma and the amount of X-ray absorption
by the bulk cold portion of the wind. This is also the diagnostic placing the most
stringent requirements on the instrument, especially in terms of effective area. For
a comparison of the effective area and resolving power of various spectrometers
and the requirements of spectral diagnostics, see Paerels (1999) and the Ph.D. thesis
of Cottam (2001).
45
1.5.1 The XMM-Newton Reflection Grating Spectrometer
The XMM-Newton1 observatory (Jansen et al. 2001) provides high-throughput,
high-resolution X-ray spectroscopy in the soft X-ray band. It has three coaligned
X-ray telescopes, each of which is composed of 58 nested mirror shells. There
are two sets of X-ray spectrometers on XMM: the moderate-resolution European
Photon Imaging Cameras (EPIC) and the high-resolution Reflection Grating Spec-
trometers (RGS). In the focal plane of one telescope is the EPIC-pn CCD imaging
spectrometer (Struder et al. 2001). In the other two telescopes, a reflection grating
array (RGA) is mounted behind the mirror module, intercepting about half the
light and dispersing it to the RGS focal plane camera (den Herder et al. 2001). The
other half of the light goes to the EPIC-MOS CCD imaging spectrometers (Turner
et al. 2001). There is also a coaligned optical/UV telescope, the Optical Monitor
(OM) (Mason et al. 2001).
The design of the RGS is motivated by the desire to obtain high resolution X-ray
spectra using lightweight, high-throughput, moderate angular resolution mirrors.
This requires high dispersion, or equivalently a diffraction grating with a high
effective ruling density. Because the gratings are mounted so that reflection occurs
at grazing incidence, the effective ruling density is the projected ruling density,
which is much higher than the actual ruling density of the gratings. Exploiting this
fact is a design choice; the consequence of this choice is that the gratings must be
very flat and precisely aligned.
The RGS gratings are mounted in an array which is placed behind the telescope
in the path of the focused light. The gratings are placed on an inverted Rowland
1http://xmm.vilspa.esa.es
46
circle (see e.g. Paerels 1999), a configuration designed to minimize aberrations due
to arraying. The grating period has a slight variation, which is equivalent to using
a curved grating to refocus the dispersed light.
The focal plane of each RGS has a strip of nine CCDs arrayed along the dis-
persion direction. There is no detector at the zero order reflection point; because of
this, the wavelength scale calibration is dependent on accurate position informa-
tion from the optical star tracker and accurate calibration of the internal geometry
of XMM.
The CCDs are similar to other CCD imaging spectrometers used in X-ray
observatories. They are back-illuminated, which increases their sensitivity at low
energies. The intrinsic spectral resolution of the detectors is used to filter out a
large portion of the background events (those whose energies as determined by the
CCD pulse height are too different from the nominal wavelength assigned by the
detector position to be real photon events), and also to distinguish emission from
different spectral orders. This is illustrated in the bottom panel of Figure 9 in den
Herder et al. (2001) .
Two of the CCDs - one from each RGS - have failed in flight. As a result, RGS
2 has no coverage around 20.0-24.0 Å, and RGS 1 has no coverage around 10.5-13.7
Å. The most important spectral feature in the 20-24 Å range is O VII He α. The
range 10.5-13.7 Å contains the α transitions of Ne X and Ne IX, as well as numerous
L-shell lines from various charge states of iron.
The CCDs have a number of hot pixels and hot columns - pixels and columns
of pixels which generate spurious events. These are reliably detected and removed
by the RGS data processing routines. Of course, any data near a hot column or
47
pixel is lost. Many hot pixels and columns became functional again after the RFCs
were cooled to a lower operating temperature around revolution 530.
The background in the CCDs mostly comes from low-energy protons from the
solar wind which are focused by the telescopes and reflected from the gratings.
Because the gratings do not disperse the protons, the background is strongest at
smaller reflection angles, or on the portions of the detector corresponding to short
wavelengths. The proton background is uniform in the cross-dispersion coordinate,
so that it is possible to measure a background spectrum from the non-source region
in cross-dispersion coordinates and subtract it from the source region spectrum.
Because the RGS has high dispersion, it is well-suited to extended source
spectroscopy, even though it is a slitless spectrometer. In the limit that the spatial
extent of a source is substantially larger than the point spread function of the
telescope, the dispersion is the only relevant instrumental quantity in determining
the spectral resolution. Another way of looking at this is that RGS achieves high
spectral resolution even with a telescope of only moderate spatial resolution, so
that it is less sensitive to source extent than a grating spectrometer which derives
its high spectral resolution partly from a very high resolution telescope, such as the
Chandra HETGS.
An expression for the spectral resolution of RGS for an extended source can be
derived from the grating equation, mλ = d(cos β − cosα), where α is the incoming
reflection angle and β is the outgoing angle. If we take the derivative of λ with
respect to α, we have
dλ =d sinα
mdα. (1.91)
This gives a first order spectral resolution of approximately ∆λ = 0.1 Å ∆α, where
48
∆α is the source extent in arcminutes.
The typical terminal velocity of an O star is v∞ ∼ 2000km s−1. The spectral
resolution of RGS for a monochromatic line is 50 mÅ (FWHM), but the ability of
RGS to measure a wavelength shift or to distinguish a profile shape is significantly
better than this; for example, for a line with many counts, the central wavelength
can easily be measured to a few mÅ relative accuracy. The absolute accuracy is set
by the knowledge of the wavelength scale, which is dependent in part on attitude
information. For comparison, a resolution of 50 mÅ corresponds to about 750 km/s
at 20 Å, while a resolution of 5 mÅ corresponds to 75 km/s.
The RGS has a bandpass of 6-38 Å. This covers most of the X-ray emission
observed from typical O stars such as ζ Pup, although a few relatively unabsorbed
sources show significant EUV emission (e.g. Cohen et al. 1996). The emission is
dominated by lines, although there is significant continuum due to bremsstrahlung
even at the low temperatures observed in O stars. The most prominent lines are
K-shell lines from Ne, O and N (depending on the amount of CNO processed
material observed in the photosphere and wind), as well as L-shell lines from Fe.
The RGS has high effective area over the wavelength range where these transitions
occur, making it well-suited to study the X-ray spectra of O stars.
1.5.2 The Chandra High Energy Transmission Grating Spectrome-
ter
The Chandra2 X-ray observatory has an X-ray telescope, two different transmission
grating arrays which can be inserted behind the telescope (the High and Low2http://cxc.harvard.edu
49
Energy Transmission Gratings, or HETG and LETG), and CCD and microchannel
plate focal plane detectors. The High Energy Transmission Grating Spectrometer
(HETGS) is formed when the HETG array is inserted and the resulting dispersed
spectrum is read out onto a strip of six CCDs, ACIS-S.
The HETG array has two different types of grating, the high energy grating
(HEG) and medium energy grating (MEG). The HEG has half the grating period of
the MEG. The two grating types have their dispersion direction rotated away from
each other, but still close to lengthwise along the detector strip. The two dispersed
spectra form an “X” in the detector plane. This prevents confusion of events from
the two gratings.
The X-ray telescope on Chandra has a much higher spatial resolution than the
XMM telescopes; 0.5′′ for Chandra compared to 5′′ for XMM. Thus, even though
the projected ruling density of RGS is higher than the ruling density of the HEG
and MEG, the spectral resolution of the HETGS is a factor of a few higher than that
of RGS. However, because it depends on the spatial resolution of its telescope to
achieve its high spectral resolution, even moderately extended sources noticeably
degrade the spectral resolution of the HETGS. On the other hand, in many cases the
spatial resolution of Chandra makes a difference in separating different point sources
in the same field; young O stars in star forming regions are often surrounded by
moderately bright point sources at the few arcsecond scale. This is the case with
the Trapezium, which contains the anomalous bright young O star θ1 Ori C (Schulz
et al. 2001).
The effective area of the HETGS is high from about 1.5-15 Å, but it falls off
quite rapidly at long wavelengths. At these wavelengths, RGS has much more
50
effective area.
The HETGS has substantially lower background due to soft protons than
RGS. This is partly because the dispersion is lower, so that the detector area per
unit wavelength is smaller, and partly because the higher spatial resolution of the
telescope allows a smaller cross-dispersion region to be used for the source region.
In contrast to XMM, Chandra is dithered in order to smooth over gaps between
CCDs. As a result, although hot pixels, hot columns, and CCD gaps exist in the
HETGS, there are no gaps in the spectral coverage.
Because of its factor of three better resolution than RGS, spectra from the MEG
are more than adequate to study spectra of O stars in terms of spectral resolution.
However, because of its low effective area at long wavelengths, it does not obtain
spectra of as high statistical quality in the same integration time.
Chapter 2
High resolution X-ray spectroscopy of
ζ Puppis with the XMM-Newton
Reflection Grating Spectrometer1
We present the first high resolution X-ray spectrum of the bright O4Ief supergiant
star ζ Puppis, obtained with the Reflection Grating Spectrometer on-board XMM-
Newton. The spectrum exhibits bright emission lines of hydrogen-like and helium-
like ions of nitrogen, oxygen, neon, magnesium, and silicon, as well as neon-like
ions of iron. The lines are all significantly resolved, with characteristic velocity
widths of order 1000 − 1500 km s−1. The nitrogen lines are especially strong, and
indicate that the shocked gas in the wind is mixed with CNO-burned material, as
has been previously inferred for the atmosphere of this star from ultraviolet spectra.
1Published in Astronomy & Astrophysics Vol. 365 as “High resolution X-ray spectroscopy of
ζ Puppis with the XMM-Newton Reflection Grating Spectrometer” by S. M. Kahn, M. A. Leuteneg-
ger, J. Cottam, G. Rauw, J.-M. Vreux, A. J. F. den Boggende, R. Mewe, & M. Gudel
51
52
We find that the forbidden to intercombination line ratios within the helium-like
triplets are anomalously low for N VI, O VII, and Ne IX. While this is sometimes
indicative of high electron density, we show that in this case, it is instead caused
by the intense ultraviolet radiation field of the star. We use this interpretation to
derive constraints on the location of the X-ray emitting shocks within the wind that
are consistent with current theoretical models for this system.
53
2.1 Introduction
The initial discovery of X-ray emission from O stars with the Einstein Observa-
tory in the late 1970s (Harnden et al. 1979), sparked a vigorous field of research
aimed at better understanding the production of hot gas in such systems. Most
current models invoke hydrodynamic shocks resulting from intrinsic instabilities
in the massive, radiatively driven winds from these stars (for a detailed review see
Feldmeier et al. 1997a). Support for this picture comes from the fact that the X-ray
flux is not highly absorbed at low energies (Cassinelli & Swank 1983; Corcoran
et al. 1993), which is expected if the X-ray emitting gas is distributed throughout
the wind rather than in some form of hot corona in the outer atmosphere of the
star. However, the X-ray observations to date have not been especially constrain-
ing for stellar wind models. This is primarily due to the low spectral resolution of
the available nondispersive detectors, which has precluded the study of individual
atomic features so crucial to the unambiguous determination of physical conditions
in the shocked gas.
In this Letter, we present one of the first high resolution X-ray spectra of an
early-type star, the O4Ief supergiant ζ Puppis, which we have obtained with the
Reflection Grating Spectrometer (RGS) experiment on the XMM-Newton Observa-
tory. ζ Pup is an excellent target for such work since it is the brightest O star in
the sky, and has consequently been very well studied at longer wavelengths. In
particular, Pauldrach et al. (1994) have developed a very detailed NLTE model
of the atmosphere and wind of this star, constrained by high quality ultraviolet
spectra from Copernicus and IUE. They find a mass loss rate of ∼ 5.1×10−6 M⊙ yr−1
and an effective temperature ∼ 42 000 K. The derived abundances indicate that the
54
atmosphere is mixed with CNO-processed material, consistent with the theoretical
picture of a highly evolved star near the end of core hydrogen burning. Hillier
et al. (1993) found that the Pauldrach et al. model is consistent with the ROSAT
X-ray spectrum of ζ Pup, but only if the X-ray emission arises in shocks distributed
throughout the wind. At the very lowest energies (≤ 200 eV), they find that the
emitting material must be ≥ 100R∗ away from the star. This is a consequence of
the fact that in the Pauldrach et al. model, helium is mostly only singly ionized in
the outer regions of the wind, so that the photoelectric opacity is very high at soft
X-ray energies.
Our XMM-Newton RGS spectrum is dominated by broad emission lines of
mostly hydrogen-like and helium-like charge states of nitrogen, oxygen, neon,
magnesium, and silicon, and neon-like ions of iron. The data provide a number
of important constraints on the nature and location of the X-ray emitting material
in the ζ Pup wind. Of particular interest is the fact that we see a suppression of
the forbidden line and enhancement of the intercombination line in the helium-like
triplets of nitrogen, oxygen, and neon. We show that this is a natural consequence
of the intense ultraviolet radiation field in the wind, and that it allows us to place
constraints on the location of the X-ray emitting shocks relative to the star. Addi-
tional constraints come from the emission line velocity profiles. We show that the
data are consistent with the predictions of the Pauldrach et al. model.
In section 2, we describe the details of the RGS observation, the nature of our
data reduction and analysis, and the key observational features of the spectrum.
In section 3, we consider the implications of our results for our understanding of
ζ Pup, and of the X-ray emission from O stars in general.
55
2.2 Observations and Data Analysis
The RGS covers the wavelength range of 5 to 35 Å with a resolution of 0.05 Å,
and a peak effective area of about 140 cm2 at 15 Å. ζ Pup was observed for 57.4 ks
on 2000 June 8. The data were processed with the XMM-Newton Science Analysis
Software (SAS). Filters were applied in dispersion channel versus CCD pulse height
space to separate the spectral orders, and the source region was separated with a
1′ spatial filter. The background spectrum was obtained by taking events from a
region spatially offset from the source. The wavelengths assigned to the dispersion
channels are based on the pointing and geometry of the telescope and are accurate
to∼ 0.008 Å (den Herder et al. 2001). The effective area was simulated with a Monte
Carlo technique using the response matrix of the instrument and the exposure maps
produced by the SAS. Based on ground calibration, we expect the uncertainty in the
effective area to be less than 10% above 9 Å and at most 20% for shorter wavelengths
(den Herder et al. 2001). A fluxed spectrum, corrected for effective area, with the
two first order spectra added together to maximize statistics, is presented in Fig. 2.1.
There is a small discontinuity in the spectrum near the nitrogen Ly β line at 20.91
Å caused by a gap between two CCD chips. The spectrum was also extracted for
analysis in XSPEC using standard SAS routines.
Due to a large solar flare event that occurred close to the time of the ζ Pup
observation, all three EPIC detectors and the Optical Monitor on XMM-Newton
were switched off while this source was observed. Thus, only data from RGS are
available for analysis.
56
Figure 2.1 First order background subtracted spectrum. It has been corrected for
the effective area of the instrument.
2.2.1 Light Curve
Previous ROSAT observations of ζ Pup (Berghoefer et al. 1996a) show evidence for
a 16.67 h period of variability in the X-ray emission between 0.9 and 2.0 keV. We
have extracted a light curve (Fig. 2.2) of the events in this energy range in the RGS
data set. There is no significant observed variation on this time scale apparent to
the eye, and the fit to a constant intensity yields χ2ν = 0.9. However, our upper limit
to the percentage variation at this period is not inconsistent with the Berghoefer
et al. (1996a) detection.
57
2.2.2 Emission Line Intensities and Emission Measure Analysis
The spectrum displayed in Fig. 2.1 is composed almost entirely of emission lines,
with a weak underlying continuum. The lines are all broadened, with characteristic
velocity widths of order 1000 km s−1. In nitrogen, oxygen, and neon the He-like
forbidden lines are very weak and the intercombination lines are bright. The line
fluxes (Table 2.1) were measured by taking the integrated flux of the spectrum over
the line and subtracting a corresponding amount of continuum flux. Lines that
originate from the same ion and that are blended were evaluated as one complex
(for example, the He-like triplets or the Fe XVII emission around 15 or 17 Å).
The continuum strength was determined by taking the flux of a spectral region
free of lines but near the line in question. In cases where this was not possible,
the continuum strength was interpolated from the strength in other regions of
the spectrum. Some of the measurements were complicated by the presence of
overlapping lines. When possible, the flux of these other lines was estimated from
the flux of lines originating from the same ion by comparing with line power ratios.
We calculated the emission measure (see Fig. 2.3) for each ion assuming solar
abundances (Anders & Grevesse 1989) and a temperature given by the temperature
of formation for the dominant lines from that charge state. The emission measure
is
EM = Fline4πd2
PlineA fi(2.1)
where Fline is the observed flux in the line, d is the distance to the source, Pline is
the line power, A is the elemental abundance, and fi is the ion fraction evaluated at
the temperature of formation. We used line powers from the APEC code (Smith &
Brickhouse 2000), which includes ion fractions from Mazzotta et al. (1998). We take
58
d = 450 pc (Schaerer et al. 1997). The line fluxes must be corrected for interstellar
absorption, although this is a minor effect. We take NH = 1020 cm−2 (Chlebowski
et al. 1989) and we use cross sections from Morrison & McCammon (1983).
We find that the emission measures derived from the nitrogen emission lines
are at least an order of magnitude greater than those for carbon and oxygen, which
both have temperature of formation ranges that overlap with that of nitrogen.
This indicates that the ratios AN/AO and AN/AC are substantially higher than solar,
even allowing for a factor of two uncertainty due to the crudeness of the emission
measure analysis. This result is consistent with the inference of atmospheric abun-
dances by Pauldrach et al. (1994), based on their analysis of UV spectra. It indicates
that the wind material in ζ Puphas been significantly mixed with matter that has
undergone CNO burning in the stellar interior.
The emission measure expected for a smooth, spherically symmetric wind
with parameters appropriate to ζ Pup is EM = 6.5× 1060cm−3. The fact that we find
EMs that are lower by 4 to 5 orders of magnitude implies that only a small fraction
of the wind material is heated to X-ray emitting temperatures.
2.2.3 Continuum emission analysis
The continuum emission has a very low intensity relative to the bright line emission.
To ensure that the continuum that we see in the spectrum is real, and not, for
example, an artifact of faulty background subtraction, we plotted the first order
events in a region free of emission lines as a function of their spatial distribution in
the cross-dispersion direction, without subtracting the background. The peak at the
location of the source spectrum is clearly visible, and indicates the presence of true
59
Figure 2.2 Light curve of first order events in the range 6.2 to 13.8 Å, corresponding
to 0.9 to 2.0 keV. Events from both instruments have been binned together. The time
dependence is plotted as a function of the 16.67 hr period reported in Berghoefer
et al. (1996a).
Figure 2.3 Emission measure calculated from each line as a function of Tform. The
horizontal error bars indicate the temperature range over which Pline fi is at least
half its maximum value. Solar abundances are assumed.
60
Table 2.1 Measured fluxes for prominent emission lines.
Linea Fluxb
Si XIII (6.65, 6.69, 6.74) (1.1 ± 0.4) × 10−4
Mg XII (8.42) (4.1 ± 2.8) × 10−5
Mg XI (9.17, 9.23, 9.31) (2.0 ± 0.7) × 10−4
Ne X (12.13) (1.8 ± 0.7) × 10−4
Ne IX (13.45, 13.55, 13.70) (5.0 ± 1.0) × 10−4
Fe XVII (15.01, 15.26) (7.4 ± 0.7) × 10−4
Fe XVII (16.78, 17.05, 17.10) (4.7 ± 0.8) × 10−4
O VIII (18.97) (3.50 ± 0.22) × 10−4
O VII (21.60, 21.80, 22.10) (5.1 ± 0.4) × 10−4
N VII (24.78) (6.4 ± 0.5) × 10−4
N VI (28.78, 29.08, 29.53) (1.13 ± 0.11) × 10−3
C VI (33.74) (6.4 ± 2.8) × 10−5
aRest wavelengths are given in Å.
bFlux is in units of photons cm−2 s−1
61
continuum, well above background. The detected continuum also significantly
exceeds the scattered light contribution from the ensemble of detected emission
lines.
We looked for evidence of discrete photoelectric absorption edges in the con-
tinuum. Although there are no edges evident, the constraints are weak due to the
low strength of the continuum emission. The upper limits on the optical depths
are given in Table 2.2. These upper limits are incompatible with the detection of
a strong edge feature near 0.6 keV reported by Corcoran et al. (1993), but that is
perhaps not surprising, considering the complexity of the spectrum and the low
spectral resolution of their measurements.
The continuum is so weak that it is difficult to ascertain its overall shape,
especially in the region from 8 - 17 Å where there are many lines. We tried fitting
with a bremsstrahlung model in XSPEC, but the results were inconclusive.
Table 2.2 Upper limits on the strengths of the K-edges of Ne, O, and N.
Ion λ τ0
(Å)
Ne X 9.10 < 0.5
Ne IX 10.37 < 0.5
O VIII 14.23 < 0.2
O VII 16.78 < 0.3
N VII 18.59 < 0.5
N VI 22.46 < 0.5
62
2.2.4 He-like triplet ratios
The forbidden to intercombination line ratios, R = f/i, for helium-like oxygen and
neon were obtained by fitting in XSPEC. In each case, we fit a Gaussian to the Ly α
line of the respective element, and used that profile for each of the three components
in the He-like triplet. We also take into account both the low level continuum and
lower flux lines in the near vicinity. For Ne IX we find R = 0.34± 0.11 and for O VII
we find R = 0.19 ± 0.08. The fit to Ne IX is shown in Fig. 2.4. These values are
well below the expected values for low density plasmas in collisional equilibrium
(Porquet et al. 2001). A similar analysis did not work very well for the N VI He-like
triplet because of the complexity of both the Ly α and He-like profiles for that
element. Nevertheless, it is clear from the data that the forbidden line is strongly
suppressed for N VI as well. The Si XIII and Mg XI triplets are too blended to allow
the forbidden line and intercombination line to be quantitatively separated. Again,
it is clear from the data that the Mg XI forbidden line is suppressed, although not
as strongly as for Ne IX.
The conversion of forbidden line to intercombination line emission in high
density plasmas is a well known effect. However, this can also occur at much
lower densities if the plasma is exposed to a strong UV radiation field (Mewe &
Schrijver 1978a). To produce a low R ratio, electrons populating the 2 3S1 state
must be excited into the 2 3P state. The state 2 3S1 is metastable, but it is easily
populated at collisional equilibrium temperatures, and the corresponding emission
line intensity is normally strong. When the excitation rate to the 2 3P levels becomes
comparable to the decay rate to the ground state, the forbidden line is suppressed.
In a high density plasma, this occurs because the collisional excitation rate from 2 3S1
63
to 2 3P competes effectively with radiative decay. However, since this is a dipole
transition, it can also be photoexcited if the ambient UV flux at the appropriate
wavelength is sufficiently high. To calculate the photoexcitation rate, we estimated
the emission from ζ Pup for the frequencies of the 2 3S1 to 2 3P transitions for N, O,
Ne, Mg, and Si. We assumed a blackbody spectrum with Teff = 42 000K (Pauldrach
et al. 1994). The rate of photoexcitation is given by RPE = Fνπe2
mcf , where f is the
oscillator strength. We used oscillator strengths from Cann & Thakkar (1992) and
Sanders & Knight (1989). The flux is given by Fν = 2π(1−√
1 − (R∗R
)2)Iν, where Iν is
the specific intensity for the blackbody.
We used the archived IUE UV spectrum of ζ Pup in addition to the Copernicus
UV spectrum (Morton & Underhill 1977) to assess the validity of our blackbody
model for the UV emission from the photosphere. We compared the measured
flux at the wavelengths of the 2 3S1 to 2 3P transitions for N VI, O VII, and Ne IX
to the flux predicted by the blackbody model. After correcting the measured flux
for absorption, these agree to within a factor of two. Since the critical radius
where RPE = Rdecay depends approximately on√
Fν, the radii we calculate are valid
to within a factor of√
2. The measured UV spectra also indicate that there is
negligible optical depth in these lines due to the wind. This is expected since these
are excited state transitions, and since the helium-like plasma only represents a
very small fraction of the wind.
In Table 2.3 we list the decay rates of 2 3S1 and the photoexcitation rates
from 2 3S1 to 2 3P evaluated at the photosphere. It is clear from this calculation
that the forbidden line suppression we observe is in fact due to photoexcitation
from ζ Pup’s high UV flux, and not due to collisional excitation at high densities,
as long as the emitting regions are close enough that the UV is not sufficiently
64
diluted. We can thus place constraints on the location of shock formation for each
of the observed lines. Since the forbidden lines in N, O, and Ne are strongly
suppressed, the emission must occur at radii smaller than the radii at which the
photoexcitation and decay rates are equal (the critical radius). Since Mg XI has a
somewhat suppressed forbidden line, its emission probably occurs near the critical
radius. The emission from Si must occur farther out than the critical radius.
Table 2.3 Comparison of photoexcitation and decay rates of the 2 3S state.
Ion A a φ∗ b Rcc
(s−1) (s−1) (R∗)
Si XIII 3.56 × 105 4.83 × 106 2.7
Mg XI 7.24 × 104 7.36 × 106 7.1
Ne IX 1.09 × 104 1.11 × 107 22.6
O VII 1.04 × 103 1.66 × 107 89
N VI 2.53 × 102 1.99 × 107 198
aDecay rates for 2 3S→ 1 1S are from Drake (1971).
bPhotoexcitation rates for 2 3S→ 2 3P near the stellar photosphere.
cRadius at which A = φ.
2.2.5 Line profile analysis
The projected velocity (vp) profile for a thin spherical shell (with a single radial
velocity) is flat, or dIdvp= (const.). We expect the emission line profile to appear as a
convolution of the radial emission intensity with this flat projected velocity profile.
65
The velocity of each shell is given as a function of radius by the conventional β-
model: v(r) ≈ v∞(1 − r0/r)β (Lamers & Cassinelli 1999) where β ≈ 0.8 and r0 ≈ R∗.
Lines formed at larger radii will therefore appear broader than lines formed close
to the star. Furthermore, lines originating from larger radii are formed over a
region with a small velocity gradient. Therefore, we expect these lines to appear
more flat-topped than lines originating closer to the star. This is apparent in the
observed Ly α lines, which are plotted in velocity space in Fig. 2.5. The N VII
peak is noticeably broader and has a substantially different shape than the other
peaks in the plot. It also shows evidence for resolved, discrete structure, given the
resolution of the instrument. Note that other lines overlap with the Ne X line; it
does not have a red shoulder.
In Table 2.4 we list the shifts in the line centroids and the line widths in
velocity space for Ne X, O VIII, and N VII. The fit to N VII is poor due to the
complex structure in the line profile.
Table 2.4 Velocity widths and shifts of the Lyman α lines.
Ion Velocity Shift a Velocity Width a
(km s−1) (km s−1)
Ne X 250 ± 125 940 ± 150
O VIII 400 ± 80 1230 ± 80
N VII 0 ± 60 1370 ± 100
aPositive shifts are blueshifts.
66
Figure 2.4 The Ne IX triplet with best fit model. Only RGS 2 data are shown for
clarity, but both were used in the fit. There is also a Fe XVII line at 13.825 Å. The
rest wavelengths of the three lines are at 13.45 Å (r), 13.55 Å (i), and 13.70 Å (f).
Figure 2.5 Ly α lines in velocity space. Nitrogen is the solid line, oxygen is dashed,
and neon is dotted. The intensities have been renormalized for comparative pur-
poses. The error bar is representative for the nitrogen line. Note the discrete
structure in the nitrogen peak. The shoulder on the Ne X line is not a velocity
feature; it is an emission line from Fe XVII.
67
2.3 Discussion
The RGS spectrum we have presented provides very strong confirmation for a
number of key aspects of conventional models of X-ray emission from early-type
stars in general, and for ζ Pup in particular. The emission is dominated by broad
emission lines indicative of hot plasma that is flowing outward with the wind. We
do not see evidence for strong attenuation at low energies, which confirms that the
X-ray emitting regions are not concentrated at the base of the wind, but instead
are distributed out to large radii. These very simple results agree well with the
predictions of wind instability models.
As we have shown, the suppression of the forbidden lines in the He-like
triplets of low Z elements allows us to derive upper limits to the radii of the
emitting shocks in each case. It is interesting to compare these values with Fig. 1
of Hillier et al. (1993), which is a plot of the radius at which optical depth unity
is reached in the wind as a function of X-ray energy for the NLTE model of ζ Pup
calculated by Pauldrach et al. (1994). At the energies of the N VI, O VII, Ne IX,
Mg XI, and Si XIII lines, unit optical depth is achieved at 22, 23, 9, 3.5, and 2.5 stellar
radii, respectively, in this model. These values are quite compatible with both our
derived upper limits for N VI, O VII, Ne IX, and Mg XI, and our derived lower limit
for Si XIII. Since the density in the wind drops off like r−2, and the emissivity is
proportional to n2, we expect radii characteristic of the smallest radius at which the
overlying wind is still transparent to the respective line, in each case. Our results
are consistent with this expectation.
Further support for this picture comes from the observed velocity profiles.
The higher Z lines have characteristic widths ∼ 1000 km s−1, whereas the N VII
68
line is distinctly broader. The terminal velocity in the ζ Pup wind is 2260 km s−1
(Groenewegen & Lamers 1989), so the higher Z lines are most likely emitted at only
a few stellar radii, whereas N VII can come from considerably further out.
We have shown that the respective line intensities are suggestive of significant
enhancements of the nitrogen abundance relative to carbon and oxygen, as one
would expect for CNO-processed material. This, again, agrees well with the Paul-
drach et al. model. Meynet & Maeder (2000) recently presented new evolutionary
models for rotating single stars. They found that rotational mixing produces a sig-
nificant surface helium and nitrogen enhancement. Meynet & Maeder suggested
that stars with enhanced He-abundances and large projected rotational velocities
are natural descendants of very fast rotating main-sequence stars. In these stars,
the chemical enrichment at the surface is very fast and as a result of the strong
rotational mixing the chemical structure of these stars could be near homogeneity.
With its large projected rotational velocity of v sin i ≃ 203 km s−1 (Penny 1996),
ζ Pup most probably falls into this category.
Chapter 3
X-ray spectroscopy of η Carinae with
XMM-Newton1
We present XMM-Newton observations of the luminous star η Carinae, including a
high resolution soft X-ray spectrum of the surrounding nebula obtained with the
Reflection Grating Spectrometer. The EPIC image of the field around η Car shows
many early-type stars and diffuse emission from hot, shocked gas. The EPIC
spectrum of the star is similar to that observed in previous X-ray observations,
and requires two temperature components. The RGS spectrum of the surrounding
nebula shows K-shell emission lines from hydrogen- and helium-like nitrogen
and neon and L-shell lines from iron, but little or no emission from oxygen. The
observed emission lines are not consistent with a single temperature, but the range
of temperatures observed is not large, spanning ∼ 0.15 − 0.6 keV. We obtain upper
limits for oxygen line emission and derive a lower limit of N/O > 9. This is1Published in the Astrophysical Journal Vol. 585 as “X-ray spectroscopy of η Carinae with
XMM-Newton” by M. A. Leutenegger, S. M. Kahn, & G. Ramsay
69
70
consistent with previous abundance determinations for the ejecta of η Car, and
with theoretical models for the evolution of massive, rotating stars.
71
3.1 Introduction
The massive, luminous star ηCarinae is famous for an extended outburst beginning
in 1843, during which it temporarily became the second brightest star in the sky.
This outburst gave rise to a bipolar optical nebula, obscuring the star from direct
observation. η Car is thought to be very massive (M ∼ 100 M⊙), and to lose mass at
a rate of M ∼ 10−3 M⊙ yr−1. For a general review of its history and properties, see
Davidson & Humphreys (1997).
Einstein observations of ηCar showed it to be a complex X-ray source (Seward
et al. 1979; Seward & Chlebowski 1982; Chlebowski et al. 1984). ηCar has two X-ray
emission components: hard, absorbed (NH ∼ 5 × 1022 cm−2), spatially unresolved
emission coming from the star, and soft, extended emission coming from the nebula
around the star. The Einstein observations also showed that there are many other
X-ray sources in the field around η Car, and that there is diffuse X-ray emission
with an extent of about a degree.
Ginga observations found evidence for iron K-shell emission from ηCar consis-
tent with Fe XXV, indicating a thermal origin for the hard X-ray emission (Koyama
et al. 1990). Corcoran et al. (1995) used ROSAT PSPC observations to show that the
hard X-ray emission is variable.
ASCA observations obtained much higher quality spectra, and found evidence
for a very strong N VII Lyα feature, which was thought to result from the supersolar
abundance of nitrogen in the ejecta (Tsuboi et al. 1997; Corcoran et al. 1998). This
also was consistent with previous optical and UV spectroscopic observations of the
ejecta around η Car (Davidson et al. 1982, 1986).
72
Recent Chandra ACIS-I imaging observations have resolved η Car spatially at
the subarcsecond scale (Seward et al. 2001). The soft X-ray nebula shows complex
structure with several knots of X-ray emission. Chandra HETGS observations have
given us the first high resolution X-ray spectrum of the star, showing that the hard
emission is non-isothermal, with emission lines from H- and He-like iron, calcium,
argon, sulfur, and silicon (Corcoran et al. 2001).
In this paper, we report the results of XMM-Newton observations of η Car,
including the high resolution soft X-ray spectrum obtained with the Reflection
Grating Spectrometer (RGS) (den Herder et al. 2001). Until now, no X-ray observa-
tory has been able to obtain high resolution soft X-ray spectra of extended sources.
RGS has a spectral resolution of about 0.1 Å for the ∼ 1’ nebula of ηCar, or λ∆λ∼ 200
at 20 Å. This is important in the case of η Car, because we can study the physical
state of the X-ray nebula in detail, and obtain much more accurate elemental abun-
dance measurements than with a CCD spectrometer. We also present the EPIC
image of the field and the CCD spectrum of η Car.
3.2 Observation and Data Analysis
η Car was observed with XMM on 2000 July 27-28 for a total of 50 ks, split into two
nearly consecutive observations. The EPIC-MOS1 (Turner et al. 2001) and EPIC-pn
(Struder et al. 2001) cameras were operated in full-frame mode, and MOS2 was
operated in small window mode. All of the EPIC cameras used the thick filter. Due
to the optical brightness of η Car, the Optical Monitor was blocked.
The EPIC data were processed with SAS version 5.3.0, and the RGS data were
73
processed with a development version of the SAS (xmmsas 20011104 0842-no-aka).
Standard event filtering procedures were followed for RGS and EPIC. Times with
high particle background levels were filtered out, leaving 45.9 ks of usable RGS
exposure and 39.5 ks of EPIC exposure.
ηCar is somewhat piled up in MOS1, but not in MOS2 or pn. The 0.2 − 10 keV
count rate of η Car in MOS2 was 2.5 cts s−1. Because it has moderately higher
spectral resolution than pn, we use only MOS2 for spectroscopy. The canned
MOS2 response matrix appropriate for standard MOS event grade selection was
used (e.g. PATTERN = 0 − 12). A background region as free as possible of sources
and diffuse X-ray emission was chosen for the MOS2 spectral analysis.
3.2.1 EPIC spectral and imaging analysis
The smoothed, exposure corrected EPIC-MOS image in the 0.3-2.5 keV band is
shown in Figure 3.1. It includes data from both MOS cameras. η Car is clearly
visible in the center, and several bright stars are also present, including WR 25
and several O-type stars. There is also substantial emission from either unresolved
point sources or diffuse gas. This emission is a substantial X-ray (non-particle)
background contaminant to the RGS spectrum, as discussed below.
The EPIC-MOS2 spectrum of η Car is shown in Figure 3.2. It is similar to the
Chandra HETGS spectrum (Corcoran et al. 2001). There are a number of strong
emission lines, including K-shell lines of hydrogenic and helium-like Si, S, Ar, Ca
and Fe. There is also an iron K fluorescence line from neutral iron in the optical
nebula. As found using the Chandra HETGS spectrum, a two temperature model
gives a significantly better fit to the EPIC MOS spectrum than a one temperature
74
Figure 3.1 Combined, exposure corrected EPIC-MOS image of the field aroundηCar
in the energy range 0.3-2.5 keV. The contrast scale is logarithmic. There is substantial
emission from diffuse gas and/or unresolved point sources over the entire field of
view.
75
model. We only fit the data from 2 − 10 keV in order to include only emission
from the star and to exclude emission from the nebula. The results of the fit are
shown in Table 3.1; abundances are quoted relative to solar (Anders & Grevesse
1989). We assume a distance of 2.3 kpc in determining the luminosity (Davidson &
Humphreys 1997).
Table 3.1 also shows that Si, S, Ar and Fe are marginally underabundant,
although the relative ratios are close to solar. Corcoran et al. (2001) found that S is
marginally overabundant, while Si and Fe were solar within the errors. However,
the inferred abundances may have systematic errors as a result of fitting a two
temperature model to what is likely a continuous distribution of temperatures (for
the both the Chandra and the EPIC data).
The unabsorbed luminosity is about the same compared to the epoch of the
Chandra observation. The equivalent width of the neutral iron K fluorescence
feature is 64 eV, compared to 39 eV at the time of the Chandra observation, but still
lower than the lowest EW observed with ASCA (Corcoran et al. 2000).
3.2.2 RGS spectral analysis
The RGS spectrum is shown in Figure 3.3. It is background subtracted and corrected
for effective area, as described below. It shows emission lines from helium-like
and hydrogen-like neon and nitrogen, and also Fe XVII and XVIII. These lines all
originate from the nebula, in contrast with the higher energy emission from the
star. There is no obvious emission from oxygen, from which one would expect
prominent emission lines, given the range of temperatures implied by the presence
of the other emission lines. There is also no obvious emission from charge states of
76
Table 3.1 Best fit parameters for the EPIC-MOS2 spectrum.
Comp. 1 a Comp. 2 a
NHb 5.7 ± 0.1 15.2+0.73
−1.8
kT c 1.52+0.05−0.07 4.64+0.13
−0.08
LXd 2.5 × 1034 1.2 × 1035
Si e 0.6 ± 0.1
S e 0.57 ± 0.06
Ar e 0.61+0.12−0.14
Ca e 1.1 ± 0.2
Fe e 0.59+0.02−0.03
EW Fe K f 64+30−7
χ2ν
g 1.11
aThe best-fit model had two components, with NH, kT, and LX given by the two columns of this table.
bColumn densities are in units of 1022 cm−2.
cTemperatures are in keV.
dLX for the 2-10 keV band in ergs s−1. A distance of 2.3 kpc is assumed (Davidson & Humphreys
1997). Unabsorbed luminosities are reported.
eAbundances are relative to solar.
f Equivalent width of the neutral iron fluorescence line at 6.4 keV is reported in eV.
gFor 351 degrees of freedom.
77
iron higher than Fe XVIII. Fe XIX would be harder to see, as the brightest emission
line would lie at ∼ 13.5 Å, which would be indistinguishable from Ne IX at this
resolution; however, if a substantial amount of Fe XX emission was present, a
strong emission line would be present at ∼ 12.8 Å. There is also no evidence for
detectable continuum emission. Thermal bremsstrahlung is less prominent relative
to line emission at temperatures around 0.5 keV than at higher temperatures. Thus,
from inspection it is clear that although the plasma is not isothermal, the range
of temperatures present is limited, and also that the abundance of nitrogen is
supersolar while that of oxygen is subsolar.
There are also emission lines from helium-like and hydrogen-like magnesium
and silicon, but these originate from the point source, with the possible exception
of Mg XI. This is known from the Chandra HETGS spectrum (Corcoran et al. 2001),
which does not include emission from the nebula and shows emission from all of
these lines. The RGS cross-dispersion profiles of these emission lines are consistent
with point-like emission. The cross-dispersion profile of Mg XI is also consistent
with extended emission, so its origin is unclear. Those emission lines are physically
associated with emission extending to much higher energies than are accessible
with RGS, so we do no attempt to model them in the analysis presented below.
Two main complications are encountered in the analysis of these RGS data.
First, the X-rays come from an extended source, and second, there is a substan-
tial background flux of soft X-rays which are diffuse or unresolved, and which
originates over essentially the entire spatial field of view of RGS. These dif-
fuse/unresolved X-rays presumably come from the many OB stars in the field
of view, and from truly diffuse, hot, shocked gas. The diffuse emission affects
all wavelengths, but the most severe confusion occurs in the wavelength range
78
∼ 10 − 20 Å. This is because the diffuse gas and unresolved point sources have
substantial iron L-shell and O VIII emission, and also because the column density
is high enough to absorb most of the diffuse emission at long wavelengths.
Because the soft X-ray emission is extended, the spectral and spatial infor-
mation are inherently coupled. A Monte Carlo simulation can be a useful data
analysis technique in this case (Peterson et al. 2004). This is not a feasible approach
for this observation because there is no way to reproduce the diffuse/unresolved
background component with a simple model. Background subtraction is also
an issue, because the background point sources and diffuse emission could have
different spectra, and are not uniformly distributed on the sky. In practice, the
background spectrum produced by a segment of the data offset from η Car in
cross-dispersion coordinates is fairly constant. The RGS 1 count rate spectra from
the source and background cross-dispersion regions are shown in Figure 3.4 to
demonstrate this, although there are differences in the crucial region around O VIII
Ly α. The adopted procedure is to produce a background subtracted spectrum
which includes an assessment of the systematic error caused by the background
subtraction. We estimate the potential systematic error to be 25% of the background
strength in a given wavelength bin. This systematic error is most important in low
flux lines with high background, e.g. O VII and VIII.
Using this method, we then measure emission line strengths by taking the
total flux in the neighborhood of the line, taken to be within 0.3 Å of the rest
wavelength. Because the bremsstrahlung continuum emission is weak compared to
line emission in the inferred temperature range, this does not produce a substantial
overestimate of the line fluxes. For a few important emission lines, one expects
strong emission from other ions at about the same wavelength, which would lead
79
Figure 3.2 EPIC-MOS2 spectrum of η Car with best fit model.
Figure 3.3 First order RGS spectrum of η Car. It has been background subtracted
and corrected for effective area.
80
to an overestimate of the line strength. The most important correction is to Ne X
Ly α, which is at roughly the same wavelength as the strongest 4d-2p transitions
in Fe XVII. A correction is made based on the observed 3-2 transitions in Fe XVII.
A Monte Carlo simulation of the RGS effective area is used together with the
measured source counts to obtain the emission line fluxes. Archival Chandra ACIS-I
imaging observations of η Car are used to provide a spatial distribution for soft
photons. The sky coordinates of photons with energy below 1.2 keV are used as
a spatial event list for the Monte Carlo simulation. We assume that the spatial
distribution does not vary as a function of energy, and that the exposure map is
approximately constant over the ∼ 1′ size of the nebula. Actual exposure variations
are at or below the 1% level.
Figure 3.5 shows the RGS 2 cross-dispersion image of the N VII Ly α line,
including all photons within 0.3 Å of the rest wavelength, plotted together with
the Monte Carlo cross-dispersion image using the Chandra data. The profiles are
slightly different, which indicates that there is some variation in the image of the
nebula at different temperatures. For comparison, Figures 3.6 and 3.7 show the
cross-dispersion images of O VIII Ly α and O VII He α respectively. A more
detailed discussion of the significance of these features follows below.
Table 3.2 gives the measured line fluxes for the major emission line complexes
in the RGS spectrum of η Car. It also gives the intrinsic line fluxes, which correct
for interstellar absorption, and in the case of Ne X Ly α for the Fe XVII 4d-2p lines.
Because the continuum flux in the soft X-ray spectrum is negligible, it is not
possible to measure neutral edge strengths or the equivalent neutral hydrogen
column density with RGS. The column density used in the X-ray literature for
81
Figure 3.4 Spectra of the source (black) and background (blue and red) regions
in RGS1. The largest discrepancies between the two background spectra occur
around 18 − 19 Å.
Figure 3.5 Cross-dispersion profile of N VII Ly α. The black line is the data, and the
red line is the Monte Carlo using the Chandra image.
82
Figure 3.6 Cross-dispersion profile of O VIII Ly α.
Figure 3.7 Cross-dispersion profile of O VII He α.
83
η Car is NH = 2 × 1021 cm−2 (Seward et al. 1979). This value comes from three
sources. The first is a fit to the low resolution IPC spectrum, giving a value of
2 × 1021 cm−2. The second is the Savage et al. (1977) measurement of UV H I and
H2 absorption to a somewhat nearby star, HD 92740 (=WR 22), for which N(H I +
H2) = 1.8 × 1021 cm−2. The last is the conversion of the optical extinction to neutral
hydrogen column density using the relations of Gorenstein (1975) and Ryter et al.
(1975). Stars near to ηCar have EB−V = 0.4 mag (Feinstein et al. 1973), so the column
density obtained is 2.7 × 1021 cm−2. Other measurements of EB−V to nearby stars
yield similar results to Feinstein et al. (1973) (Herbst 1976; Forte 1978).
Of these methods, we cannot rely on the first, as the fitting procedure is
degenerate even for the high resolution RGS spectrum, and the second is also
unsatisfactory, as HD 92740 is too far away from ηCar to expect the column density
to be the same. The third method is also problematic, as the extinction to Tr 16 is
anomalous and variable. The extent to which it is anomalous is controversial, with
different works obtaining values of R = AV/EB−V ranging from 3.2 to 5.0 for different
stars in Tr 16 and in the vicinity of η Car (Feinstein et al. 1973; Herbst 1976; Forte
1978; Turner & Moffat 1980; The & Groot 1983; Tapia et al. 1988; The & Graafland
1995). This implies that the relations of Gorenstein (1975) and Ryter et al. (1975)
underestimate the total column density, since they take R ∼ 3.1. However, these
relations should at least provide a lower limit to NH. In their review, Davidson &
Humphreys (1997) adopt a value of AV = 1.7 for the interstellar (non-circumstellar)
extinction to η Car. Using the Gorenstein (1975) relation, we obtain NH = 3.7 ×
1021 cm−2. Also, as an alternative to the Savage et al. (1977) measurement, we can
use the Diplas & Savage (1994) measurement of the H I column towards HD 303308,
which is much nearer to ηCar than HD 92740. They find NH = 2.8×1021 cm−2. This
84
can also be considered a lower limit, as a substantial fraction of the hydrogen may
be ionized. For simplicity we will take the equivalent hydrogen column density to
be NH = 3.0× 1021 cm−2, and assess the systematic effects of a higher column on the
temperature distribution and abundance measurements.
There is no evidence for emission from O VII He α. Although there is no
strong emission line corresponding to O VIII Ly α, there is a marginal detection
of flux at this wavelength, and a small line-like feature. Because O VIII Ly α is so
weak in this spectrum, and because it is one of the stronger features in spectra of
typical O-type stars and collisionally ionized plasmas in general, the question of
contamination by nearby sources is important to address. It is possible that some
or all of the observed O VIII Ly α flux is attributable to a nearby star. HD 303308
is the brightest object close enough to cause confusion. If this star was the source
of the apparent O VIII feature, it would have an apparent wavelength of 18.9 Å
(slightly blueshifted), and it would be about 1′ east of the bright knot seen in the
southwest of the Chandra image. The observed RGS feature is inconsistent with this
requirement. In addition, the O VIII flux obtained from a two temperature thermal
plasma model fit to the EPIC-MOS spectrum is 10−5 photons cm−2 s−1, compared
with 4 ± 2 × 10−5 for η Car. Thus, it seems unlikely that any point source can
account for the O VIII feature in η Car. However, O VIII Ly α is only detected at the
2σ level, so the detection is not very secure.
We do not use the data to obtain an upper limit to the C VI emission. The
high column density would make any upper limit a weak one, and the fact that
we expect a very low carbon abundance means that the more easily measurable
oxygen abundance provides a stronger physical constraint on the nucleosynthetic
signatures of the CNO cycle.
85
In Figure 3.8 we plot the emission measure inferred for individual emission
line complexes assuming a single temperature. For each ion, the error bar is
marked at the temperature of maximum line strength. The abundances are taken
to be solar (Anders & Grevesse 1989), except nitrogen and oxygen. The abundance
of nitrogen is assumed to be 11 times solar, which is approximately the sum of
the solar abundances of carbon, nitrogen, and oxygen. The abundance of oxygen
is chosen to have a value such that O VII and N VII are consistent, as discussed
below. It is clear that a single temperature cannot account for the different species
observed. Furthermore, the lack of emission from Fe XX shows that there cannot be
a substantial amount of plasma at temperatures above 0.6 keV (7 MK). The presence
of N VI emission shows that there must be emission from temperatures down to at
least about 0.15 keV (1.7 MK), but emission at lower temperatures is unconstrained;
there are no spectral features we would expect to see if lower temperatures were
present, given the probable low carbon abundance and high column density.
Because the strength of the N VII and O VII lines have very similar temperature
dependence, especially near the temperature of maximum line strength (Tm), one
may derive an upper limit to the abundance ratio by setting the oxygen abundance
such that the curves in the plot are consistent near Tm. This is possible because we
know that there is no emission from high temperatures, where the relative flux of
N VII and O VII has a substantial temperature dependence. The same is true for
Fe XVII and Ne X. We measure a lower limit N/O > 9, while Fe/Ne is consistent
with the solar ratio (Anders & Grevesse 1989) to 0.1 dex. The error introduced by
directly comparing ion emission measures assuming a single temperature is less
than 0.1 dex. If we allow that the column density assumed could be 4 × 1021 cm−2,
the lower limit to N/O becomes N/O > 8. Although it is possible that the column
86
Figure 3.8 Inferred emission measure distribution. The abundance of nitrogen is
set to the sum of solar C+N+O, while oxygen is set so that O VII is consistent with
N VII. Other elements have solar abundances.
87
density is higher than that, this uncertainty clearly cannot affect the N/O lower
limit very strongly.
Alternatively, if we take the O VIII detection at face value and fix the oxygen
abundance by requiring that the differential emission measure should not have a
dip at log T = 6.5, the N/O ratio is increased by about 0.2 dex from our lower limit,
to N/O = 14.
The appearance of the ion emission measure plot is moderately affected by the
set of reference abundances chosen. The most recent work on solar abundances
(Grevesse & Sauval 1998; Holweger 2001; Allende Prieto et al. 2001, 2002) have
substantial variations relative to each other and Anders & Grevesse (1989) in the
abundances of CNO and Fe. These variations are of order 0.1 dex. Given the levels
of uncertainty in the measurements themselves and in the abundances, the plot is
consistent with a differential emission measure which is flat from log T = 6.0− 6.6,
declining substantially to log T = 6.8. In any case, the measurement of the ratio
N/O is unaffected by the choice of reference abundances.
The shape of the emission measure distribution would be affected if the column
density were substantially higher than 4 ± 1 × 1021 cm−2. Rather than looking
flat, with a dropoff at high temperatures, it would be decreasing with increasing
temperature.
Mg XI has also been included in the plot. For this point to be consistent with
the rest of the plot, we would have to take the abundance of magnesium to be at
least 0.3 dex higher than iron and neon. While this is not out of the question, there
is no good reason to have an overabundance of only magnesium (which would
have no relation to CNO cycle abundance changes if it were real). The most likely
88
explanation is that Mg XI emission comes from the star rather from the nebula.
There is also evidence for a feature at about 7.85 Å, the wavelength of Mg XI He β.
This feature is strong compared to Mg XI He α, as would be expected for the high
column density observed in the spectrum of the star. This is also similar to the
Mg XII Ly α to β ratio observed with Chandra (Corcoran et al. 2001).
3.3 Discussion
There are two main results from the analysis of the RGS spectrum of η Car. The
first is a constraint on the range of temperatures in the nebula (0.15 − 0.6 keV),
which allows us to infer shock velocities for the expansion of the ejecta into the
surrounding medium. The second is a lower limit on the nitrogen to oxygen
abundance ratio (N/O > 9). This allows us to constrain the evolution of η Car.
3.3.1 Temperature distribution
The upper end of the temperature distribution is strongly constrained by the Fe
L-shell spectrum. The lack of measurable emission from charge states higher
than Fe XVIII rules out the presence of appreciable quantities of gas above ∼
0.6 keV. The lower end of the distribution appears to be flat, based on the emission
from Ne IX, N VII and N VI. However, there are no other potentially observable
spectral lines originating from ions that exist at lower temperatures than N VI,
so the emission measure distribution cannot be constrained below about 0.2 keV.
Davidson, Walborn, & Gull (1982) and Davidson et al. (1986) find UV emission
lines from N I through N V in the spectra of the ejecta, so there is certainly a range
89
Table 3.2 Measured fluxes for prominent emission line complexes.
Line Flux a Intrinsic Flux a
Mg XI He α 0.59 ± 0.13 0.84 ± 0.18
Ne X Ly α 0.71 ± 0.14 0.73 ± 0.31 b
Ne IX He α 1.76 ± 0.28 4.47 ± 0.71
Fe XVIII 3d-2p 0.51 ± 0.17 1.23 ± 0.41
Fe XVIII 3s-2p 0.43 ± 0.19 1.47 ± 0.65
Fe XVII 3d-2p 1.84 ± 0.35 5.32 ± 1.01
Fe XVII 3s-2p 1.95 ± 0.32 8.15 ± 1.34
O VIII Ly α 0.4 ± 0.2 2.2 ± 1.1
O VII He α < 0.4 < 5.6
N VII Ly α 4.02 ± 0.25 36.7 ± 2.3
N VI He α 1.55 ± 0.24 52.4 ± 8.1
aFluxes are in units of 10−4 photons cm−2 s−1
bThe intrinsic flux of Ne X Ly α has been corrected for blending with 4-2 transitions of Fe XVII.
90
of temperatures present. As noted in the previous section, a substantial difference
between the assumed absorption and the real absorption could change the overall
shape of the emission measure distribution, especially at low temperatures.
Weis, Duschl, & Bomans (2001) attempt to correlate the observed projected
velocity of optical blobs which are spatially coincident with X-ray emission using
Hubble Space Telescope and ROSAT HRI images. The velocities they find in
the brightest X-ray regions would produce plasma at temperatures an order of
magnitude higher than observed, assuming that the ejecta was colliding with a
stationary ISM. Of course, η Car should be surrounded by a wind blown bubble
out to much larger radii than 0.3 pc (the radius of the X-ray nebula), and the
material inside the bubble should be streaming outward. It seems likely that
the observed shock temperature reflects the velocity at which the X-ray emitting
ejecta are overtaking the previously emitted stellar wind. The temperature range
0.15−0.6 keV implies a shock velocity range of 300−700 km s−1. If the X-ray emitting
ejecta date from the great eruption of 1843, then the rough expansion velocity for a
free expansion is ∼ 0.3 pc / 150 yr = 2000 km s−1, so the velocity of the stellar wind
before the great eruption was ∼ 1500 km s−1.
3.3.2 Abundance measurements
The N/O ratio observed in the ejecta has implications for the evolution of η Car. It
is clearly a signature of CNO processing, and the degree of conversion of oxygen
to nitrogen observed in the ejecta is high.
All massive main-sequence stars burn hydrogen on the CNO cycle. Its nucle-
osynthetic signatures are the conversion of most of the catalytic carbon and oxygen
91
to nitrogen, and the burning of H into He. For CNO processed material to be ob-
served on the surface of these stars, or in their ejecta, it must be transported there
from the core.
Previous measurements of N/O in η Car give similar but generally less con-
straining results than our RGS measurements. Optical and UV spectroscopy of
the S condensation (corresponding spatially roughly to the brightest X-ray knot
(Seward et al. 2001)) shows that most CNO is nitrogen and that the helium mass
fraction is 0.40 ± 0.03 (Davidson et al. 1982, 1986). A quantitative measurement of
the CNO abundance ratios is not made because of their dependence on ionization
and thermal structure, and also because some oxygen and carbon may be in solid
grains. It should be noted that the measured value of the helium mass fraction
may be systematically too low if the ionization balance of helium was not properly
modelled.
More recent measurements of the abundances in the S condensation have
been made by Dufour et al. (1997) with HST-FOS. They report CNO and He abun-
dances for the S2 and S3 “sub-condensations”, respectively, of [N/O] > 1.72, 1.75,
[N/C] > 1.95, 1.85, and Y = 0.39, 0.42. They did detect weak oxygen and carbon
lines, but treated them as upper limits due to potential contamination from the
foreground H II region. However, they also find that preliminary analysis of the
S1 and S4 sub-condensation spectra show much lower N and He enrichment, with
correspondingly lower N/O and N/C ratios.
Previous X-ray observations (Tsuboi et al. 1997; Corcoran et al. 1998; Seward
et al. 2001; Weis et al. 2002) have shown the presence of a strong N VII Ly α feature
in the spectrum, but the CCD spectra lacked the resolution to strongly constrain
92
the O VII and O VIII features. Our measurement of N/O is not limited by the
spectral resolution of RGS, but rather by source/background contamination, and
the observed line strength is not influenced by the formation of dust grains or large
uncertainties in the temperature distribution of the plasma.
Recent HST-STIS long-slit spectroscopy of the central star have obtained a
lower limit of N/O >∼ 1 (Hillier et al. 2001). This is a conservative interpretation
of the data; the lower limit could easily be taken to be an order of magnitude
higher. On the other hand, UV spectra taken with HST-GHRS show evidence for
moderate carbon depletion which may be inconsistent with the level of depletion
found in the ejecta (Lamers et al. 1998). In light of recent work indicating that
η Car may be a binary system (Damineli 1996; Damineli et al. 2000), the apparent
contradiction in the stellar and nebular abundances is taken to be an indication that
the star producing the carbon features is actually the secondary (assuming the star
that produced the nebula is the primary). Walborn (1999) points out that there are
several difficulties with this conclusion, the most obvious being the concealment
of the luminous blue variable (LBV) primary.
Spectroscopic measurements of the abundances of the central star cannot in-
validate the RGS abundance measurements, but the binary scenario requires us to
treat the nebular abundances with some care. It is unlikely that both members of
a binary system could contribute substantially to the ejecta around η Car, but it is
possible, in principle, that the ejecta from the primary could mix with the wind
of the secondary. If both stars had a high N/O ratio, but substantially different
helium abundances, then it would be possible to misinterpret the significance of
the nebular abundances. However, this is not a likely scenario, so we make the
simplest assumption, which is that the observed nebular abundances reflect the
93
current surface abundances of the primary.
The signatures of CNO processing have been observed in various types of hot
stars, including OBN stars, blue supergiants, and LBVs (Maeder 1995). However,
CNO processed material is not observed on the surface of all hot stars, and the
amount of processed material observed spans a wide range. The fact that N/O is
so high in the ejecta of η Car is strongly constraining, regardless of the mechanism
responsible for mixing.
The two most plausible mechanisms which could have resulted in the mea-
sured abundances in the ejecta of η Car are : 1.) η Car is on the main sequence
and is rotating. This rotation has caused very thorough mixing. 2.) η Car is in a
post-red-supergiant blue supergiant phase, and the CNO abundance ratios are a
result of the onset of convection in the envelope during the red supergiant phase.
We refer in particular to the discussion in Lamers et al. (2001), which deals with
the same question in the case of other LBV nebulae. We can use the measured N/O
ratio in conjunction with the He abundance of Davidson et al. (1986) to assess the
plausibility of these two mechanisms.
Using Figure 3 of Lamers et al. (2001) for the case of an 85 M⊙ star with
Z = 0.02, we find that for log (N/O) > 1.0, log (He/H) > −0.3, or Y > 0.67. This
simply reflects the fact that although a high surface ratio of N/O can be obtained
in the red supergiant phase, this can only happen if the star lost enough of its
envelope on the main sequence to allow core processed material to dominate the
resulting composition. This value of Y is not consistent with the Davidson et al.
(1986) measurement of Y = 0.4, although a conservative assessment of the possible
errors, particularly in measuring the helium mass fraction, does not allow us to
94
rule out that η Car could be a post-red-supergiant object.
Meynet & Maeder (2000) make predictions for the abundances of rotating
massive stars. Their Z = 0.02 model with an initial rotation velocity of 300 km s−1
and a mass of 120 M⊙ predicts Ys = 0.89 and N/O = 45.4 at the end of H-burning.
While this value of Y is also not consistent with the observed value, in this case Y
will clearly be lower earlier in the life of the star, whereas if the mixing is efficient
enough, N/O will already be high enough to be consistent with the measured
lower limit. This is an important point; the conversion of oxygen to nitrogen in
CNO burning is considerably slower than the conversion of carbon to nitrogen. If
rotational mixing is responsible for the observed abundances, the mixing timescale
must be short compared to the evolutionary timescale. As pointed out in Maeder
(1987), the ratio of the mixing timescale to the main sequence lifetime in rapidly
rotating stars is indeed expected to decrease with increasing mass.
3.3.3 Summary
We have analyzed XMM-Newton X-ray spectra of η Car. The EPIC spectral data
from the star are consistent with past observations by ASCA and Chandra. The data
are not consistent with an isothermal plasma, but require at least two temperatures.
The RGS spectra show that the nebula is nonisothermal and has strongly non-
solar CNO abundances. The temperature range in the nebula is 0.15 − 0.6 keV.
If this is interpreted as a shock velocity, it corresponds to 300 − 700 km s−1. We
find a lower limit of N/O > 9, which is indicative of very thorough mixing in
the envelope of η Car. Taken with previous measurements of the surface helium
abundance Y = 0.4, this implies that η Car is a main-sequence object with some
95
strong mixing mechanism at work, although it does not decisively rule out the
possibility that it is a post-red-supergiant object.
96
Chapter 4
Measurements and analysis of
helium-like triplet ratios in the X-ray
spectra of O-type stars1
We discuss new methods of measuring and interpreting the forbidden-to-inter-
combination line ratios of helium-like triplets in the X-ray spectra of O-type stars,
including accounting for the spatial distribution of the X-ray emitting plasma and
using the detailed photospheric UV spectrum. Measurements are made for four
O stars using archival Chandra HETGS data. We assume an X-ray emitting plasma
spatially distributed in the wind above some minimum radius R0. We find min-
imum radii of formation typically in the range of 1.25 < R0/R∗ < 1.67, which is
consistent with results obtained independently from line profile fits. We find no
1Accepted for publication in the Astrophysical Journal as “Measurements and analysis of helium-
like triplet ratios in the X-ray spectra of O-type stars” by M. A. Leutenegger, F. B. S. Paerels, S. M.
Kahn, & D. Cohen
97
98
evidence for anomalously low f/i ratios and we do not require the existence of
X-ray emitting plasmas at radii that are too small to generate sufficiently strong
shocks.
99
4.1 Introduction
Since the discovery of X-ray emission from OB stars by Einstein (Harnden et al. 1979;
Seward et al. 1979), the exact mechanism for X-ray production has been something
of a mystery. X-ray emission from OB stars had been predicted by Cassinelli
& Olson (1979), who proposed that an X-ray emitting corona could explain the
observation of superionized O VI through Auger ionization of O IV. However,
subsequent observations showing less attenuation of soft X-rays than would be
expected from a corona lying below a dense stellar wind made a purely coronal
origin seem unlikely (Cassinelli & Swank 1983). Macfarlane et al. (1993) also found
that a distributed X-ray source was necessary to explain the observed O VI UV P
Cygni profile in ζ Pup. Furthermore, with no expectation of a solar-type α − Ω
dynamo in OB stars with radiative envelopes, the coronal model fell out of favor.
Subsequently, several scenarios in which magnetic field generation and dynamos
could exist in OB stars have been proposed (Charbonneau & MacGregor 2001;
MacGregor & Cassinelli 2003; Mullan & MacDonald 2005). Since these models have
been proposed, the primary observational evidence invoked by their proponents
is anomalously low f/i ratios in the X-ray emission of a few He-like ions in several
stars. Re-examining these line ratios and determining whether they require a
coronal model to explain them is one of the main goals of this paper.
Shocks arising from instabilities in the star’s radiatively driven wind have
been considered to provide a more likely origin for the observed X-ray emission, as
they are expected to be present, given the line-driven nature of these winds (Lucy
& White 1980; Lucy 1982b; Krolik & Raymond 1985; Owocki, Castor, & Rybicki
1988; Feldmeier 1995). However, there have been difficulties in reproducing the
100
observed X-ray properties of O stars, such as the overall X-ray luminosity and
the spectral energy distribution, from stellar wind instability models (Hillier et al.
1993; Feldmeier 1995; Feldmeier et al. 1997a,b). Until recently, the quality of the
available spectral data provided little insight into these problems, since the CCD
and proportional counter spectra could not resolve individual spectral lines.
Recent high resolution X-ray spectroscopy of OB stars by the XMM-Newton
Reflection Grating Spectrometer (RGS) (Kahn et al. 2001; Mewe et al. 2003; Raassen
et al. 2005) and the Chandra High Energy Transmission Grating Spectrometer
(HETGS) (Schulz et al. 2000; Waldron & Cassinelli 2001; Cassinelli et al. 2001;
Miller et al. 2002; Cohen et al. 2003; Kramer et al. 2003; Gagne et al. 2005; Cohen
et al. 2006) have answered some questions while raising new ones. Some stars have
X-ray spectra that appear consistent with emission from shocks in the wind, but the
detailed comparisons to predicted spectral models are still problematic. Both Wal-
dron & Cassinelli (2001, hereafter WC01) and Cassinelli et al. (2001, hereafter C01)
have found low forbidden-to-intercombination line ratios in one set of helium-like
triplets each in the X-ray spectra of ζOri and ζ Pup. They infer from this that some
of the X-ray emitting plasma is too close to the star to allow shocks of sufficient
velocity to develop.
Other stars (θ1 Ori C and τ Sco) have X-ray spectra that are unusually hard
and have relatively small line widths. While these stars might be considered prime
candidates for a coronal model of X-ray emission - especially after having magnetic
fields detected via Zeeman splitting (Donati et al. 2002, 2006) - their behavior is
better understood in terms of the magnetically channeled wind shock model, rather
than a model of magnetic heating (Schulz et al. 2000; Cohen et al. 2003; Schulz et al.
2003; Gagne et al. 2005; Donati et al. 2006).
101
Finally, we note that for all of the O giants and supergiants observed, the
line profiles are less asymmetric than predicted, given the high mass-loss rates
measured for these stars using radio free-free emission, H α emission, and UV
absorption lines (Waldron & Cassinelli 2001; Kahn et al. 2001; Cassinelli et al. 2001;
Miller et al. 2002; Kramer et al. 2003; Cohen et al. 2006). This implies either a lower
effective opacity to X-rays in their winds (e.g. due to clumping or porosity effects
(Feldmeier et al. 2003; Oskinova et al. 2004, 2006; Owocki & Cohen 2006)), or lower
mass-loss rates (Crowther et al. 2002; Massa et al. 2003; Hillier et al. 2003; Bouret
et al. 2005; Fullerton et al. 2006).
One of the key diagnostic measurements available to us in understanding the
nature of X-ray emission in OB stars is the forbidden-to-intercombination line ratio
in the emission from ions that are isoelectronic with helium. This ratio is sensitive
to the UV flux, and thus to the proximity to the stellar surface. This allows us to
constrain the location of the X-ray emitting plasma independently of other spectral
data, such as emission line profile shapes.
In this paper we discuss methods for using the f/i ratio to constrain the
location of X-ray emitting plasma in O star winds. In particular, we explore the
effects of a spatially distributed source motivated by the broad line profiles. We
discuss the effects of photospheric absorption lines, as well as the f/i ratio expected
for a plasma emitted over a range of radii, taking account of detailed line shapes
when signal-to-noise allows. We find that accounting in detail for photospheric
absorption lines is not important, as long as the X-ray emission originates over a
range of radii.
These methods are then applied to He-like triplet emission in a set of archival
102
Chandra observations of O stars. Our primary result is that good fits can be achieved
for most lines with models having emission distributed over the wind, with mini-
mum radii of about 1.5 stellar radii. We find that none of the data require the X-ray
emitting plasma to be formed very close to the photosphere.
This paper is organized as follows: In § 4.2 we review the physics of line
formation in He-like species (§ 4.2.1), explore the effects of spectral structure in the
photoexciting UV field (§ 4.2.2), and of spatial distribution of the X-ray emitting
plasma (§ 4.2.3), while incorporating the line-ratio modeling into a self-consistent
line-profile model (§ 4.2.4). In § 4.3 we discuss the reduction and analysis of
archival O star X-ray spectra. In § 4.4 we give the results of this analysis, fitting high
signal-to-noise complexes with the self-consistent line-profile model described in
§ 4.2.4 and fitting the lower signal-to-noise complexes with multiple Gaussians and
interpreting these results according to the spatially distributed picture described
in § 4.2.3. In § 4.5 we discuss the implications of these results, and in § 4.6 we give
our conclusions.
4.2 Model
4.2.1 Radial dependence of the f/i ratio
The physics of helium-like ions in coronal plasmas has been investigated in numer-
ous papers (Gabriel & Jordan 1969; Blumenthal et al. 1972; Gabriel & Jordan 1973;
Mewe & Schrijver 1975, 1978a,b,c; Pradhan & Shull 1981; Pradhan 1982; Porquet
et al. 2001). The principal diagnostic is the ratio of the strengths of the forbidden to
intercombination lines, R ≡ f/i. We will use the calligraphic R to refer to this ratio,
103
and the italic R to refer to distances comparable to the stellar radius.
The upper level of the forbidden line (2 3S1) is metastable and relatively long-
lived. When the excitation rate from 2 3S1 to the upper levels of the intercombination
line (2 3P1,2) becomes comparable to the decay rate of the forbidden transition, the
line ratio is altered.2 The excitations may be due to electron impacts in a high
density plasma, or due to an external UV radiation source.
Gabriel & Jordan (1969) (hereafter GJ69) and Blumenthal, Drake, & Tucker
(1972) (hereafter BDT72) derive the expression
R = R01
1 + φ/φc + ne/nc(4.1)
where φ is the photoexcitation rate from 2 3S to 2 3P, and φc is the critical rate at
whichR is reduced toR0/2. Similarly, ne is the electron density, and nc is the critical
density.
In Table 4.1 we give our adopted values for the atomic parameters necessary
for calculation of He-like triplet ratios. We adopt the BDT72 values for φc because
they have calculated it for all the ions we are interested in, and because more recent
calculations are not substantially different. However, we use the more recent values
for R0 from Porquet et al. (2001); their calculations of R0 are slightly lower than
those of BDT72. Porquet et al. (2001) also give values for G ≡ ( f + i)/r, evaluated at
Tmax, the temperature at which emission from that He-like ion is the strongest. We
cite G(Tmax) for comparison with our measurements.
Because densities high enough to cause a change in the line ratios exist only
2The 2 3S1 state may also be excited to the 2 3P0 state, but this state does not decay to ground,
so we omit it from our discussion. However, in Gabriel & Jordan (1969) and Blumenthal, Drake, &
Tucker (1972), the formal treatment involves all states.
104
Table 4.1 Parameters adopted for He-like triplets.
Ion f a λ1a λ2
a φcb R0
c G(Tmax) c
(Å) (Å) (s−1)
S XV 0.0507 738.32 673.40 9.16E5 2.0 · · ·
Si XIII 0.0562 865.14 814.69 2.39E5 2.3 0.68
Mg XI 0.0647 1034.31 997.46 4.86E4 2.7 0.71
Ne IX 0.0700 1272.81 1248.28 7.73E3 3.1 0.74
O VII 0.0975 1638.28 1623.61 7.32E2 3.7 0.90
N VI 0.1136 1907.26 1896.74 1.83E2 5.3 0.88
aOscillator strengths and transition wavelengths are from CHIANTI (Dere et al. 1997; Young et al.
2003). Oscillator strength is for the sum of all three transitions 2 3S1 → 2 3PJ. λ1,2 are the transition
wavelengths for 2 3S1 → 2 3P1,2 respectively.
bφc are from BDT72.
cR0and G (Tmax) are from Porquet et al. (2001), except S XV, which is from BDT72.
105
very close to the star, we consider only the photoexcitation term. If there are O stars
with f/i ratios that are measured to be too low to be explained by photoexcitation,
it is appropriate to consider the effects of high density; this is not the case for any
of our measurements.
The expression for φmay be evaluated as follows, given a model stellar atmo-
sphere Eddington flux Hν:
φ =16π2e2
mecf
Hνhν
W(r) (4.2)
where W(r) = 12(1 −
√
1 − (R∗/r)2) is the geometrical dilution.
The expression for the R ratio derived by GJ69 is written such that f is the
sum of the oscillator strengths for 2 3S1 to all three of 2 3PJ, despite the fact that 2 3P0
does not decay to ground, and 2 3P2 only contributes for high Z. For low Z (Ne IX
and lower) Hν should be evaluated for 2 3S1 → 2 3P1. For Mg XI and higher Z ions
it is more accurate to evaluate Hν for both 2 3S1 → 2 3P1 and 2 3P2 and weight the
average by the relative contributions to the effective branching ratio. Of course,
this is only necessary if Hν is substantially different for the two transitions.
Since the flux of UV radiation seen by ions in a stellar wind decreases in
proportion to the geometrical dilution factor W(r), the R ratio is also a function of
radius. It is helpful to express it in this form:
R(r) = R01
1 + 2 P W(r)(4.3)
with P = φ∗/φc and
φ∗ = 8ππe2
mecf
Hνhν. (4.4)
The value of the R ratio near the photosphere is then Rph = R0/(1 + P).
106
In this paper we perform calculations and make measurements for a sample
of four O stars observed by Chandra: ζ Pup, ζ Ori, ι Ori, and δ Ori. The relevant
properties of these stars are given in Table 4.2. The effective temperatures and
gravities of the stars are taken from Lamers & Leitherer (1993) and then rounded
off to the closest values calculated on the TLUSTY O star grid (Lanz & Hubeny
2003).
4.2.2 The effect of photospheric absorption lines
The expression for R(r) written in the last paragraph involves an approximation
that must be explored further. We assumed a photospheric UV flux that would
be diluted by geometry, but we neglected the Doppler shift of the absorbing ions.
Over the range of Doppler shifts seen in a stellar wind, there can be many photo-
spheric absorption lines. This introduces an additional radial dependence to the
photoexcitation rate, and thus the R ratio:
φ(r) ∝ Hν(r)W(r) (4.5)
with the Doppler shifted frequency as seen by an ion at radius r:
ν(r) = ν0
(
1 +v(r)
c
)
(4.6)
In this expression a positive velocity represents a blue shift.
In Figure 4.1 we show a plot of the photospheric UV flux for a model repre-
senting ζ Ori near the 2 3S1→ 2 3P1,2 transitions of Mg XI. The model is taken from
the TLUSTY O star model grid (Lanz & Hubeny 2003). Note that for Mg XI, most
of the intercombination line strength still arises from the 2 3P1 to ground transition.
107
Figure 4.1 Model UV flux for ζOri near the 2 3S1→ 2 3P1,2 transitions of Mg XI (J = 1
is on the top, J = 2 on the bottom), plotted as a function of wavelength (bottom
axis) and scaled stellar wind velocity, w(u) = v(u)/v∞ (top axis). The flux is given
in units of 1020 photons cm−2 s−1 Å−1
. The dashed line shows the rest wavelength
of the O VI line at 1031.91 Å. For comparison, the average continuum flux we use
for this ion and this star is 1.67, in the same units. The model flux is taken from the
TLUSTY O star grid (Lanz & Hubeny 2003).
108
Table 4.2 Adopted stellar parameters.
Star Spectral type a Teffb log g b v∞
c
(kK) (cm s−2) (km s−1)
ζ Pup O4 I 42.5 3.75 2485
ζ Ori O9.5 I 30.0 3.25 1860
ι Ori O9 III 35.0 3.50 2195
δ Ori O9.5 II 32.5 3.25 1995
aSpectral types are given for reference and are taken from the Garmany values reported in Table 1
of Lamers & Leitherer (1993).
bEffective temperatures and surface gravities are the values on the TLUSTY O star grid that are the
closest approximations to the values used in Lamers & Leitherer (1993).
cTerminal velocities are taken from Prinja, Barlow, & Howarth (1990).
109
We also compute theR ratio using an averaged value of Hν, which we compare
to the R ratio calculated using the non-averaged (radially dependent) Hν. We do
this to understand whether it is important to explicitly account for photospheric
absorption lines, or whether it is sufficient to calculate R using an averaged value
of the photospheric UV flux. We use the average value of Hν over the range where
0.1 < R/R0 < 0.9, or 9 > 2 P W(r) > 0.111. There are two reasons for this: when the
photoexcitation rate is much less than the critical rate, the effect of photospheric
lines on R is small; and when the photoexcitation rate is so high that the forbidden
line is very weak, we can’t measure variations in the forbidden line strength. We
estimate this range using the continuum UV flux. In cases where R does not ever
get reduced to 0.1R0 (even at the photosphere) because the UV flux is not strong
enough, we average from the rest frequency to the frequency at which R = 0.9R0.
In Figure 4.2 the red lines show R(r) for averaged (dashed) and non-averaged
(solid) Hν for Mg XI for the star ζ Ori. There are substantial fluctuations in R(r) for
the non-averaged case. The solid lines in the figure are discussed in the following
section; they represent the effects of averaging the emission over a range of radii,
as opposed to simply over a range of frequencies.
In making this figure we have ignored all additional Doppler shifts, as the
purpose of the plot is mainly to illustrate qualitatively the effect of photospheric
absorption lines on the R ratio. Examples of potentially relevant Doppler shifts are
the thermal velocities of the ions (of order 100 km s−1 for neon at 0.4 keV), stellar
rotation (typically 100−200 km s−1 for O-type stars, although the wind also rotates),
and the non-monotonicity of the stellar wind due to shocks (e.g. Feldmeier 1995, of
order a few 100 km s−1). We have also treated the star as a point source rather than
a finite disk, which would change the projected velocity as a function of position
110
Figure 4.2 The f/i ratio for the Mg XI triplet of ζ Ori plotted as a function of
the inverse radial coordinate u ≡ R∗/r. The solid lines are for the actual model
photospheric UV flux, while the dashed lines are for an averaged value. The
bottom pair of lines (red) show the local radial dependence of R(u); the top pair of
lines (black) show the integrated ratio R(u0) observed for the whole star (see text).
Note that u and u0 are not comparable physical quantities, since u corresponds
to a single radius, which could be interpreted as a characteristic radius, while u0
corresponds to the minimum radius for the onset of X-ray emission. The solid lines
include the effects of the photospheric UV flux for transitions to both the 2 3P1 and
2 3P2 states, although the 2 3P1 state is far more important for Mg XI. The peaks in
the red solid line correspond to the absorption lines in the top panel of Figure 4.1.
111
on the stellar disk. All of these effects are small compared to the wind terminal
velocity, but they could diminish the impact of photospheric lines on the f/i ratio
by smearing out the photospheric spectrum.
One possibly important effect we neglect is scattering by resonance lines of
ions in the wind. This is probably relevant only for Mg XI. In this case the O VI
line at 1031.91 Å is on the blue side of the 2 3S1 → 2 3P1 transition at 1034.31 Å,
which means that it could scatter the UV light from the photosphere to a different
wavelength. However, it is not clear that this will greatly affect the line ratio, as
the scattering process does not generally destroy photons. The detailed effects of
scattering by this transition could be assessed by modelling the radiative transfer
in the wind at this wavelength range, but this is beyond the scope of this work.
4.2.3 The integrated ratio
In the preceding two subsections we calculated the radial dependence of the f/i
ratio. Here we will calculate the f/i ratio integrated over an emitting volume that
may span a wide range of radii. After all, for any realistic model of a stellar wind,
we expect the X-ray emitting plasma to be distributed over a large range of radii
(although it could be a small range of radii for a coronal model). We cannot directly
observe the ratio as a function of radius, but only the overall ratio, or the ratio as a
function of the observed Doppler shift.
We make the simple assumptions that the emissivity of the X-ray emitting
plasma scales as the wind density squared above some onset radius. This is the
same set of assumptions as the model of Owocki & Cohen (2001), with the two ad-
ditional simplifications that there is no continuum absorption and that there is no
112
radial variation in the X-ray filling factor. These approximations are not unreason-
able, considering the low characteristic optical depths and the radial dependence
of the filling factor reported by Kramer et al. (2003) for fits to line profiles in the
Chandra HETGS spectrum of ζ Pup, especially for high Z, where the optical depths
are expected to be smallest.
To calculate the integrated strength of the forbidden and intercombination
lines, we weight the integrand with the normalized (radially dependent) strength
of each line.3 The weights are
f (u) = GR(u)
1 +R(u)(4.7)
and
i(u) = G1
1 + R(u). (4.8)
Here u ≡ R∗/r is the inverse radial coordinate. We have introduced G ≡ (for +
int)/res to ensure that the weighting factors are properly normalized relative to the
resonance line; we will discuss this in more detail in the next section. The radial
dependence of R was discussed in the previous sections (cf. Equation 4.3).
The integrated ratio is then
R(u0) =
∫
dV η f∫
dV ηi
(4.9)
where η f,i are the emissivities of the forbidden and intercombination lines. The
3We could instead express the integrated ratio as a single volume integral of the f-to-i ratio with
a weighting term for the overall emissivity of the complex, but we feel that formalism we use here,
of a ratio of two separate emissivity integrals, is more intuitive. However, the two methods are
formally equivalent.
113
integrals are∫
dV η f,i ∝∫ ∞
R0
Ω(r)r2drρ2(r) f , i(r) ∝∫ u0
0
duΩ(u)w2(u)
f , i(u) (4.10)
where we have usedρ(u) ∝ u2/w(u). Ω(u) = 2π(1+√
1 − u2) is the solid angle visible
by the observer (i.e. not obscured by the stellar core). w(u) = v(u)/v∞ = (1 − u)β is
the scaled velocity; we take β = 1 as a convenient approximation, as discussed in
the following section. R0 is the onset radius for X-ray emission, and u0 = R∗/R0 is
its inverse. The He-like line strength weights f , i(u) are given by Equations 4.7 and
4.8, respectively.
In Figure 4.2, the solid lines show the integrated f/i ratio as a function of u0
for Mg XI in ζ Ori. The integrated ratio is very similar for both the averaged flux
(black dashed line) and unaveraged (black solid line) flux cases. Since it is much
simpler to consider only a single value of photospheric UV flux and because it
agrees well with the more detailed treatment, we do so in the rest of this paper.
However, it should be noted that if one modeled the X-ray emission as arising near
a single radius or Doppler shift, as might be appropriate for a coronal model, the
actual photospheric flux (including absorption lines) would have to be included in
the modeling.
It is important to note that there are two separate physical effects being consid-
ered here: the first is the effect of using the actual photospheric spectrum instead of
a wavelength average, and the second is the averaging of the R ratio over a range
of radii. What Figure 4.2 shows is that the first effect is not important if we include
the second. However, when comparing the radius inferred from a localized model
to the minimum radius inferred from the distributed model, it is crucial to realize
that they are physically different quantities. The radius in a localized model can be
114
taken literally as the characteristic location of the X-ray emitting plasma, but in the
distributed model, the minimum radius is the smallest radius where there is X-ray
emission; it can be interpreted physically as the shock onset radius.
In Figures 4.3,4.4,4.5, and 4.6, we show R(u) and R(u0) for all He-like ions
observed in the four O stars we consider in this paper. These plots all assume an
averaged value of the photospheric UV flux. For a given measured value ofR, there
are substantial differences between the value of u0 derived assuming a distributed
plasma and the value of u derived assuming a plasma dominated by one radius -
that is, u0 is always larger than u for a single radius, as one would expect.
In Table 4.3 we compare our calculations using TLUSTY model stellar atmo-
sphere fluxes to the same calculations using Kurucz (1979) fluxes, as in WC01 and
C01. We make the comparison for one key ion for each paper, both of which have
their 2 3S1 → 2 3PJ transition wavelengths in the Lyman continuum. We use R0
values taken directly from the plots of WC01 and C01. For most ions in these two
papers, R0 is taken from BDT72, but for Si XIII, WC01 use R0 = 2.85, while the
BDT72 value is 2.51. The values of R0 given in BDT72 are systematically higher
than those in Porquet et al. (2001).
There are substantial differences between our calculations of Rph (the value
of R at the photosphere) and those of WC01, C01, and Miller et al. (2002). These
differences mainly arise from differences in the continuum flux of the photospheric
models shortward of the Lyman edge; the TLUSTY models generally predict a
factor of 2-3 more than the Kurucz models.
The combination of the different Lyman continua and R0 values lead to sub-
stantially higher values of Rph for Si XIII and S XV in WC01, C01, and Miller
115
Figure 4.3 The f/i ratio for six He-like triplets observed in ζ Pup. The dashed lines
show the radial dependence of R, while the solid lines show the dependence of the
integrated ratio R on the inverse minimum radius u0 = R∗/R0. The curves fall in
increasing order of Z from left to right. The colors are black for N VI, red for O VII,
orange for Ne IX, green for Mg XI, blue for Si XIII, and purple for S XV. S XV is
omitted from the subsequent three figures because it is not observed in the spectra
of those stars.
116
Figure 4.4 The f/i ratio for five He-like triplets observed in ζ Ori. Scheme is as in
Figure 4.3.
117
Figure 4.5 The f/i ratio for five He-like triplets observed in ι Ori. Scheme is as in
Figure 4.3.
118
Figure 4.6 The f/i ratio for five He-like triplets observed in δ Ori. Scheme is as in
Figure 4.3.
119
Table 4.3 Comparison of He-like ratio calculations.
HνE
b P c Rph/R0 R0d Rph
e
ζ Ori Si XIII this work 1.97 3.09 0.244 2.3 0.56
WC01 a 0.633 0.993 0.502 2.85 1.43
ζ Pup S XV this work 8.71 3.21 0.238 2.0 0.48
C01 a 4.95 1.82 0.355 2.04 0.72
aWe used Kurucz model atmospheres to reproduce these authors’ calculations. We assumed that
ζ Pup was represented by a model with Teff = 40kK and log g = 4.0 and ζ Ori by a model with
Teff = 30kK and log g = 3.5.
bThe photospheric UV flux, HνE , is given in units of 107 photons cm−2 s−1 Hz−1.
cP ≡ φ∗/φc is discussed in Equations 4.3 and 4.4.
dFor WC01 and C01 we used the R0 values shown on their plots.
eOur calculations for Rph (which is the value of R near the photosphere) using the Kurucz model
atmospheres agree with the figures of WC01 and C01.
Note. — In this table we compare the adopted photospheric UV flux and the He-like triplet ratio
calculations of WC01 and C01 to those in this work.
120
et al. (2002). This means that we would infer systematically larger radii than these
authors, given the same measured value of R.
Regardless of the differences between TLUSTY and Kurucz model atmo-
spheres, there are substantial uncertainties in the Lyman flux of any model at-
mosphere; this part of the spectrum is generally inaccessible to observation, and
the models’ Lyman continua have not been directly verified experimentally. In
the two cases where early B stars have been directly observed in the Lyman con-
tinuum with EUVE, the fluxes have been roughly an order of magnitude above
models (Cassinelli et al. 1995, 1996); however, it should be pointed out that these
stars are significantly cooler than the O stars we are studying, so that their Lyman
fluxes are more sensitive to changes in the temperature structure in the outer at-
mosphere. Furthermore, the effective temperature scale used for O stars in the past
may be systematically too high (Martins, Schaerer, & Hillier 2002), which would
also have more of an effect on the part of the spectrum shortward of the Lyman
break. However, the effect of the uncertainty in the model Lyman continuum flux
is significantly larger than the effect of the correction to the effective temperature
scale.
4.2.4 He-like line profiles
Although it may sometimes be easier to measure the f/i ratio directly and compare
it to a calculation for the ratio as a function of distance from the star, it is potentially
much more powerful to calculate line profiles including the radial dependence of
the line ratio and compare these to the data. The expression for the line profile
121
derived in OC is
Lx = C
∫ ux
0du
fX(u)w3(u)
e−τ(u,x) (4.11)
In this expression, the volume filling factor of X-ray emitting plasma is fX(u) ∝ uq,
while x refers to the velocity-scaled dimensionless Doppler-shift parameter. τ(u, x)
is the optical depth along the line of sight to the observer, which is usually written
as the product of a geometrical integral, t(u, x), and a dimensionless constant,
τ∗ =κM
4πv∞R∗, the characteristic optical depth. It should be noted that the expression
for the optical depth is only analytic for integral values of the velocity law index
β; otherwise it must be evaluated numerically. Because the expression for Lx must
also be evaluated numerically, it is preferable to take β to be an integer in order
to avoid a multidimensional integral. β = 1 is the best integer approximation for
most O stars (see, e.g., Puls et al. (2006) for models that include clumping).
To account for the relative line strengths of the triplet, we simply multiply
the integrand with the weighting factors f (u) = G R(u)1+R(u) or i(u) = G 1
1+R(u) . This
normalizes the forbidden and intercombination lines to the resonance line, which
may be calculated using the above expression with no modification. If it is desirable
to normalize the sum of all three weighting factors to unity, one may divide them
by 1+G. In this work we have assumed that G does not vary with radius. Although
G does depend on temperature, the variation is not strong, and the X-ray emitting
plasma is likely multiphase. If there is any variation in the line profile shapes
caused by a radial dependence in G, it is not likely to be detectable except with
data of very high statistical quality.
In comparison with the integrated plots presented in the previous section, a
line profile with τ∗ > 0 has a higher R ratio than one with no absorption, given
122
the same value of u0. This is because the forbidden line is only formed farther out
where absorption is less, while the intercombination line is mainly formed close
to the star, where absorption is greater. Nonzero positive values of q cause R to
go down, because relatively more emission comes from close to the star, while
negative values cause R to go up.
In comparison with normal line profiles, the intercombination line has weaker
wings, as it becomes much weaker far away from the star. On the other hand,
the forbidden line is relatively flat topped; because of photoexcitation, the profile
appears as if it has a larger effective value of R0 than the resonance line.
The addition of the radial dependence of f/i ratio to the OC profile model
has the appealing property of enforcing self-consistency between the radial de-
pendences of the Doppler profile and the f/i ratio. Also, although it does make
the quite reasonable assumption that the X-ray emitting plasma follows the same
β-velocity law as the wind, it is not tied to any particular heating mechanism. In
§4.3, we use this model to fit Chandra HETGS spectra of four O stars.
4.3 Data reduction and analysis
In this section we fit He-like triplets in the Chandra HETGS data of four O stars:
ζ Pup, ζ Ori, ι Ori, and δOri. We only fit Mg XI, Si XIII, and for ζ Pup S XV. This is
because Ne IX and lower Z He-like species generally have R < 0.2 in O stars and
therefore do not contain significant information in the line ratio.
123
4.3.1 Data processing
Primary data products were obtained from the Chandra data archive and processed
using standard CIAO routines outlined in the CIAO grating spectroscopy threads.4
The versions used were CIAO 3.1 and CALDB 2.28. The spectral fitting was done
with XSPEC 11.3.1. The C statistic (Cash 1979) is used instead of χ2 because of
the low number of counts per bin. For ζ Ori and ι Ori the data were split into
two observations each, which were fit simultaneously. Emission lines were fit over
a wavelength range of [λr(1 − v∞/c) − ∆λ, λ f (1 + v∞/c) + ∆λ], where ∆λ is the
resolution of MEG at that wavelength. This range was chosen to include the entire
emission line, but at the same time to prevent the quality of the continuum fit from
influencing the fit statistic for the line. To get the continuum strength for a given
line, we first fit it outside this range, but near the wavelength of the line.
Because the MEG has substantially more effective area than the HEG at longer
wavelengths, we used only the MEG ±1 order data for Si XIII in ζ Pup and for
Mg XI for all stars. For the S XV complex in ζ Pup and the Si XIII complex in the
other stars, the statistics are poorer, and the contribution of the HEG is significant,
so we simultaneously fit both the HEG and MEG±1 order data. The MEG±1 order
data for Si XIII in ζ Ori are inconsistent, so we fit each of them separately; this
inconsistency is discussed in more detail in the results subsection.
4.3.2 Fitting procedure
We use two different fitting procedures, depending on the number of counts in the
triplet. For triplets with many counts (Mg XI for all stars, and Si XIII for ζ Pup),4http://cxc.harvard.edu/ciao/threads/gspec.html
124
we fit them with the He-like OC profile described in § 4.2.4. The fixed model
parameters are the line rest wavelengths, the terminal velocity of the wind, the
velocity law index β = 1, the unaltered f/i ratio R0, and the averaged photospheric
UV strength. The fit parameters from the profile model are q, τ∗, and u0, in addition
to the G ratio and the overall normalization. The four fit parameters other than
normalization are fit on a grid with spacing 0.2 for q and τ∗, and spacing 0.05 for u0
and G.
For lines with few counts (S XV for ζ Pup and Si XIII for the other stars), we
fit a three Gaussian model to prevent overinterpretation. Rather than using three
individual Gaussians, which would have three separate normalizations, we use a
model with parameters G, R, the overall normalization, and the velocity width,
which is taken to be the same for all the lines in a given complex. This avoids
fitting problems due to covariance in individual line normalizations, which can be
a problem in blended line complexes. It also allows us to directly measure the line
ratios and their errors, which are the quantities of interest. We fit the parameters
on a grid with spacing 2×10−3 for σv, 0.2 forR, and 0.1 or 0.2 for G. We interpret the
results of these multi-Gaussian fits using the integrated ratio formalism described
in § 4.2.3 and shown in Figures 4.3-4.6.
In all cases we add a continuum component to approximate bremsstrahlung
emission. This is represented by a power law of index 2 with normalization chosen
to fit the continuum near the line. Care is taken to avoid including moderately
weak spectral lines in the continuum fit. A power law of index 2 is not necessarily
appropriate for the continuum in general, but over a sufficiently short range in
wavelength, any reasonable continuum shape is statistically indistinguishable. An
index of 2 is chosen because this gives a flat continuum when Fλ is plotted versus
125
wavelength.
We do not expect any other strong lines to contaminate our line fits. Mg XII
Ly γ is at approximately the same wavelength as the Si XIII forbidden line, but
even in ζ Ori, where Si XIII is relatively weak, the strength of Mg XII Ly γ expected
based on the strength of Mg XII Ly α is not enough to affect our measurements
significantly.
4.4 Results
The results of the fits are summarized in Tables 4.4 and 4.5. The fits are plotted
with the data in Figures 4.7-4.21. The data have been rebinned for presentation
purposes in some of the plots, but in all cases the data were fit without rebinning.
We show two-parameter confidence interval plots for the profile fit to Mg XI
for ζ Pup in Figure 4.22. These confidence intervals are qualitatively representative
of our results for all the line complexes; they demonstrate that there is a moderate
correlation of the parameters q and u0 in the profile fits, and that the other param-
eters are not strongly correlated. The correlation in q and u0 is expected, as both
parameters influence the radial distribution of plasma, and therefore both the f/i
ratio and the profile width.
The goodness of fit is tested by comparing the fit statistic to that obtained
from Monte Carlo simulations from the model. The percentage of 1000 realizations
having C less than the data is given in the tables of results. These percentages can
be thought of as rejection probabilities.
The helium-like line profile fits generally are adequate to explain the data; they
126Table 4.4. Parameters for He-like profile fits
Star Ion q τ∗ u0 R0a G Rb Flux c C Bins MC d
ζ Pup Mg XI 0.0+0.4−0.2 1.0+0.4
−0.4 0.70+0.05−0.05 1.43 0.70+0.15
−0.10 0.41 17.7+0.9−0.9 135.3 136 39.1
Si XIII 0.0+0.6−0.4 0.6+0.4
−0.2 0.70+0.05−0.10 1.43 1.05+0.15
−0.15 0.90 11.9+0.7−0.7 116.2 98 84.7
ζ Ori Mg XI −0.4+1.0−0.2 0.2+0.2
−0.2 0.6+0.10−0.1∗ 1.67 1.05+0.05∗
−0.2 0.82 6.5+0.5−0.6 267.2 240 73.4
ι Ori Mg XI −0.8+0.2−0.0 0.0+0.2
−0.0 0.75+0.05−0.10 1.33 0.90+0.20∗
−0.25 0.72 3.5+0.5−0.5 266.6 256 80.2
δ Ori Mg XI −0.8+0.4−0.0 0.0+0.2
−0.0 0.80+0.05−0.10 1.25 0.60+0.25
−0.10∗ 0.75 4.0+0.5−0.5 123.9 124 18.2
aR0 is given in units of the stellar radius; it is calculated from R0 = 1/u0, and retains an extra digit to avoid
rounding error.
bR is reported without an uncertainty because it is the value of the f/i ratio calculated by the best fit model.
cFlux is given in units of 10−5 photons cm−2 s−1.
dMC is the percentage of Monte Carlo realizations of the model having C less than the data does for that model.
Note. — Errors are 2σ, or ∆C = 4 for one degree of freedom. Asterisks indicate parameters that were still within
2σ at the edge of the fit range.
127
Table 4.5 Parameters for He-like Gaussian fits.
Star Ion σv/c R G Flux a C Bins MC b R0
(10−3)
ζ Pup S XV 2.4+0.4−0.4 1.0+0.4
−0.4 0.9+0.2−0.2 3.1+0.3
−0.3 191.2 216 78.4 1.1+0.4−0.1
ζ Ori Si XIII c 2.4+0.2−0.0 2.8+0.8
−0.8 1.2+0.2−0.1 2.45+0.15
−0.15 432.1 496 97.9 ≥ 2.1
Si XIII d 2.0+0.2−0.2 ≥ 2.8 0.9+0.2
−0.2 2.4+0.4−0.4 57.6 59 35.3 ∞
Si XIII e 3.0+0.8−0.6 ≥ 1.6 2.0+1.0∗
−0.6 2.4+0.6−0.5 85.6 59 98.8 ≥ 1.4
ι Ori Si XIII 2.8+0.4−0.4 2.8+0.6
−0.8 1.6+0.4−0.2 1.54+0.24
−0.24 305.1 532 84.9 ≥ 3.2
δ Ori Si XIII 1.2+0.0−0.2 2.2+1.0
−0.4 0.7+0.1−0.1 1.88+0.16
−0.16 214.0 258 62.9 ≥ 2.2
aFlux is in units of 10−5 photons cm−2 s−1.
bMC is the percentage of Monte Carlo realizations of the model having C less than the data does for
that model.
cCombined fit to positive and negative first order HEG and MEG data.
dFit to positive first order MEG data only.
eFit to negative first order MEG data only.
Note. — Errors are 1σ, or ∆C = 1. Asterisks indicate parameters that were still within 1σ at the
edge of the fit range.
128
Figure 4.7 MEG data and best-fit model for S XV in ζ Pup. The positive and
negative first order data have been coadded. The data are shown with error bars,
and the model is shown as a solid line. The rest wavelengths of the resonance,
intercombination, and forbidden lines are shown with dotted lines. This scheme is
used in all subsequent figures presenting the data. Except where stated explicitly,
the plots of Gaussian fits show the joint best fit to both the HEG and MEG data,
even though data from only one grating are presented at a time.
129
Figure 4.8 HEG data and best-fit model for S XV in ζ Pup. The positive and negative
first order data have been coadded.
Figure 4.9 MEG data and best-fit model for Si XIII in ζ Pup. The positive and
negative first order data have been coadded.
130
Figure 4.10 MEG data and best-fit model for Mg XI in ζ Pup. The positive and
negative first order data have been coadded.
Figure 4.11 MEG data and best-fit model for Si XIII in ζ Ori. The positive and
negative first order data have been coadded. Figures 4.13 and 4.14 show the
positive and negative first order MEG data separately.
131
Figure 4.12 HEG data and best-fit model for Si XIII in ζ Ori. The positive and
negative first order data have been coadded.
Figure 4.13 MEG positive first order data and best-fit model for Si XIII in ζOri. The
model is the best fit to only the positive first order data.
132
Figure 4.14 MEG negative first order data and best-fit model for Si XIII in ζ Ori.
The model is the best fit to only the negative first order data.
Figure 4.15 MEG data and best-fit model for Mg XI in ζ Ori. The positive and
negative first order data have been coadded.
133
Figure 4.16 MEG data and best-fit model for Si XIII in ι Ori. The positive and
negative first order data have been coadded.
Figure 4.17 HEG data and best-fit model for Si XIII in ι Ori. The positive and
negative first order data have been coadded.
134
Figure 4.18 MEG data and best-fit model for Mg XI in ι Ori. The positive and
negative first order data have been coadded.
Figure 4.19 MEG data and best-fit model for Si XIII in δ Ori. The positive and
negative first order data have been coadded.
135
Figure 4.20 HEG data and best-fit model for Si XIII in δ Ori. The positive and
negative first order data have been coadded.
Figure 4.21 MEG data and best-fit model for Mg XI in δ Ori. The positive and
negative first order data have been coadded.
136
Figure 4.22 Two dimensional plots of confidence intervals for fit parameters for
Mg XI in ζ Pup. The shades of grey represent 1, 2, and 3 σ, or ∆C < 2.3, 6.17, 11.8
(as appropriate for two degrees of freedom), and the cross represents the best fit.
There is a moderate correlation of the fit parameters q and u0, as one would expect.
We have made similar plots for the other He-like profile fits (not shown) to look
for correlations in fit parameters. These plots also show a moderate correlation
between q and u0.
137
are all formally statistically acceptable. The fact that the fits can simultaneously
account for the profile shape and the f/i ratio indicates that the values of u0 obtained
are not an artifact of the profile model. In other words, we can explain both the line
ratios and profile shapes with a single model for the radial distribution of X-ray
emitting plasma.
The fit parameters obtained for the He-like profile fits are generally consistent
with those obtained in Kramer et al. (2003) and Cohen et al. (2006) from non-
helium-like line profile fits. The R ratios for the helium-like line profile fits are
also consistent with those measured in Kahn et al. (2001), WC01, C01, and Miller
et al. (2002). The values of u0 for all four stars fall in the range 0.6 < u0 < 0.8, or
1.25 < R0 < 1.67. This is substantially closer to the star than the values of u inferred
in Kahn et al. (2001), WC01, and C01 from f/i ratios. This reflects the difference
between assuming a single radius of formation as opposed to a distribution of radii.
We now consider the lower signal-to-noise complexes, which we fit with Gaus-
sians. For the Si XIII lines in ιOri and δOri, the R ratio is not strongly constrained,
and in both cases the data are consistent with R = R0. The goodness of fit is for-
mally acceptable in both cases. If anything, it is surprising that the R ratio is not
slightly lower in both cases, considering the values of u0 measured for the Mg XI
lines.
The R ratio measured in S XV in ζ Pup is equivalent to a value of R0 = 1.1+0.4−0.1,
based on Figure 4.3. The 1σ upper limit to R0 is consistent with what is seen in
other lines and with the expectations of hydrodynamic models of wind shocks
(Feldmeier et al. 1997b; Runacres & Owocki 2002). The fit to these lines is formally
acceptable.
138
The fit to the Si XIII complex of ζOri is poor. Because the positive and negative
first order MEG data look very different, we fit them separately in addition to the
joint fit. These additional fits are shown in Figures 4.13 and 4.14. Part of the
difference in appearance is a result of the Si XIII complex falling on a chip gap
in the negative first order, which reduces the effective area and makes it uneven.
However, even accounting for this there is a substantial difference in the fit results
for the two orders, both for R and for G. It is possible to get a satisfactory fit using
only the positive first order MEG data, but fitting the negative first order by itself
gives a poor fit. Because the negative first order data for this complex falls on a
chip gap, cannot be fit well by a three Gaussian model, and has substantially fewer
counts than the positive first order, we consider it to be unreliable.
In Table 4.6 we compare our fits for Si XIII in ζ Ori and S XV in ζ Pup to those
of WC01 and C01, respectively. There is not enough information in their original
work to directly compare their best fit model to ours; they do not give the velocity
broadenings or overall normalizations. We use their published values of R and G
and find the best fit parameters for velocity broadening and normalization. WC01
do not present their measurements of G, but we infer from the temperature range
they claim is allowed that they measure G in the range 0.8-2.0. We assume the
best fit was in the middle of this range, or G = 1.4. In both cases, we also tried
letting G be a free parameter, in order to test the validity of their R measurements
independently of any claims about G. For S XV in ζ Pup, we found that the best fit
occurs with a substantially different value of G than that reported by C01.
Our measurement of the R ratio of Si XIII in ζ Ori is significantly different
than that of WC01. Statistically, their best-fit measurement has a C value that is
11.7 greater than our best fit. For one interesting parameter, this is excluded at
139
Table 4.6 Comparison of fit parameters.
Star Ion σv/c (10−3) R G C bins R0, R d
This work ζ Pup S XV 2.4 1.0 0.9 191.2 216 1.1+0.4−0.1
C01aa 2.8 0.61 2.06 200.7 < 1.2
C01bb 2.4 0.61 0.9 192.0
This work ζ Ori Si XIII 2.4 2.8 1.2 432.1 496 ≥ 2.1
WC01 c 2.2 1.2 1.4 443.8 < 1.08
aFor C01a we used the published value of R and G.
bFor C01b we used the published value of R and the best fit value for G.
cFor WC01 the published value of G was the best fit, assuming their value of R.
dFor our work this column gives the inferred minimum radius of formationR0, while for the previous
work this column gives the inferred radius of formation R.
Note. — In this table we compare our fit parameters to those of C01 and WC01 for two line
complexes. In all cases, we used the best fit σv and normalization.
140
more than 3σ. The reported value of R = 1.2 ± 0.5 is also very different than our
measured value of 2.8 ± 0.8. Two points should be reiterated: first, when fitting all
the data, we do not get a statistically acceptable fit, but the positive first order MEG
data can be well-fit, and this fit has an R ratio which is comparable to the value we
measure using all the data; furthermore, whether we use all the data or exclude
the questionable negative first order MEG data from the fit, we get essentially the
same result. Second, we are using essentially the same model as WC01, but merely
measure very different parameter values, even when fitting exactly the same data.
This may stem from the fact that WC01 used very early versions of the CIAO tools
(J. P. Cassinelli, private communication).
Our measurement of the R ratio of S XV in ζ Pup is somewhat different than
that of C01. We also found that the if we fix R to the value they reported, the best
fit value of G is substantially different than their measurement. Although their best
fit model with R = 0.61 is not excluded at the 1σ level, the model based on their
measured value of G has a value of C which is greater than that of our best fit by
9.5, despite the fact that we both fit a three Gaussian model. Although we do not
exclude their best-fit value of R at 1σ, it is also puzzling that our range of fit values
should be significantly different from that of the previous work.
4.5 Discussion
We have used the f/i ratios of He-like triplets in conjunction with their line profiles
to constrain the radial distribution of X-ray emitting plasma in O stars. Our results
are consistent with the results of Kramer et al. (2003) and Cohen et al. (2006) in the
sense that the spatial distribution we infer from f/i ratios (additionally constrained
141
in some cases by line profile fitting) is consistent with these authors’ results from
fitting line profiles to high signal-to-noise individual lines.
Our results for Si XIII in ζ Ori are different from the initial analysis which
claimed that the location of the emitting plasma was extremely close to the star.
These differences are due both to our assessment of the relative line fluxes and to
our modeling of the line formation. Table 4.6 shows a comparison of our mea-
surements and inferred radii of formation; we find that the Si XIII is at least 1.1
stellar radii above the photosphere (R0/R∗ = 2.1). Part of the difference in inferred
radii originates in our different calculations of the radial dependence of R. This
is illustrated in Figure 4.23, where we plot our calculations and measurements of
R(u0) and compare them to the calculations and measurements ofR(u) from WC01.
(It is important to note that the range of radii indicated on the plot by the thickened
lines refers to that allowed by the statistical error in the measurement of R, and not
to a physical extent of the X-ray emitting plasma). We also show what our inferred
radius of formation would be if we inferred a single radius from our measured
value of R instead of an onset radius R0 in a distributed model (assuming that
an averaged value of the photospheric UV flux could be used). This is intended
to make it clear that the major sources of disagreement are the actual R measure-
ments and the UV fluxes of the adopted model atmospheres. In fact, even taking
the reported upper limit on R (of 1.7) from WC01, and assuming a single radius of
formation (dash-dot curve in Figure 4.23), our analysis shows that the formation
radius is consistent with values larger than 2R∗. Although we also assume a spatial
distribution of X-ray emitting plasma, this does not contribute to the new, larger
formation radii.
We make a similar comparison with the earlier results C01 for S XV for ζ Pup
142
in Table 4.6 and Figure 4.24. In this case the measured range of allowed values
of R is different but overlapping. The different measured range of R combined
with a somewhat higher model photospheric UV flux leads us to infer a minimum
radius of formation as large as 1.5R∗; however the allowed range of minimum radii
extends down to nearly the photosphere, in agreement with the results of C01.
The upper range of allowed minimum radii is reasonable in the context of stellar
wind models for X-ray-emitting plasma formation, but the lower range is certainly
not. While the difference between our measurements and calculations and those
of C01 is not great, it is enough to allow that the S XV emission could reasonably
be produced in a wind shock model.
These results obviate the need for any kind of two-component model for the
origin of X-ray emission in O stars, as suggested by WC01, C01, and Mullan &
Waldron (2006). For the Si XIII line in ζOri the range of acceptable minimum radii
of formation we infer are quite reasonable in the wind-shock paradigm. For the
S XV line in ζ Pup, the upper end of the range of acceptable minimum radii of
formation we infer is acceptable in the wind-shock paradigm, although the lower
end of the range is not. Taken together, we can say that the wind-shock paradigm
is consistent with these data; we do not exclude the possibility that S XV in ζ Pup
is formed very close to the star or is formed in a process outside of the wind-shock
paradigm, but we do not require this. We note that numerical simulations of the
line-driven instability show that large shock velocities, and therefore hot plasma,
occur quite deep in the wind, almost as soon as the damping effects of the diffuse
radiation field are overcome by the onset of the instability growth (Runacres &
Owocki 2002). While hybrid wind-coronal mechanisms are not excluded by the
data, there is nothing in the X-ray spectral data that requires such complex models,
143
and the principle of Occam’s razor leads us to suggest that it is more reasonable
to assume a wind shock origin for all X-rays from the O stars we are studying,
if it is possible to explain the data this way. Another argument against inferring
extremely small formation radii for these two ions is the lack of any evidence,
either from line profiles or from f/i ratios, of the presence of emission from lower
ion stages at these very small radii, as would be expected from the rapid radiative
cooling of plasma containing S XV or Si XIII at the densities expected this far down
in the wind.
It should be noted that there is one other claim in the literature of an anoma-
lously low f/i ratio measurement requiring an X-ray production mechanism out-
side of the standard wind shock paradigm. Waldron et al. (2004) find evidence
for this in their analysis of the X-ray spectrum of Cyg OB2 8A; in this case the
basis for their claim is emission from S XV and Ar XVII. However, these ions’ 2 3S1
→ 2 3PJ transitions are in the Lyman continuum, where results are very sensitive
to model atmosphere uncertainties, so the inferred radii are subject to substantial
uncertainties. Furthermore, the data have very low signal-to-noise.
The characteristic optical depths we measure from profile fits are substantially
smaller than one would expect, given the published mass-loss rates. Detailed
calculations of the expected values of τ∗ are beyond the scope of this paper, but
it is safe to say that we would expect to see characteristic optical depths at least
of order a few at 9 Å; our measurements for Mg XI give τ∗ = 1 for ζ Pup, and
less for the other stars. However, Si XIII and Mg XI give poorer constraints on
the optical-depth/mass-loss-rate discrepancy than longer wavelength lines, such as
O VIII and N VII Ly α, where the photoelectric absorption cross-section per unit
mass is higher, so that τ∗ is larger and produces a more asymmetric profile.
144
WC01, C01, and Miller et al. (2002) compare the radii they infer from mea-
surements of f/i ratios in He-like triplets to the radii of optical depth unity, R1, for
the wavelength at which that He-like ion emits. These values of R1 were calculated
using mass-loss rates from the literature and assumed a smooth wind density. They
claim that the inferred radii correspond roughly to R1, so that we are observing
plasma at the closest point to the star where we can see it. Kahn et al. (2001) make
a similar conjecture. Table 4.7 compares the values of R1 from these papers to
those derived using the methodology of this paper. Several lines show evidence
for emission from inside the predicted R1. This is in agreement with the low values
of τ∗ we have measured, as well as the measurements of Kramer et al. (2003) and
Cohen et al. (2006). There is now mounting evidence from analysis of unsaturated
UV line profiles that the literature mass-loss rates of O stars may be too high by
at least a factor of a few (Massa et al. 2003; Hillier et al. 2003; Bouret et al. 2005;
Fullerton et al. 2006). In addition, porosity may reduce the effective X-ray optical
depths of O star winds (Feldmeier et al. 2003; Oskinova et al. 2004, 2006; Owocki
& Cohen 2006).
4.6 Conclusions
We have investigated the effect of a radially distributed plasma on the forbidden-
to-intercombination line ratio in helium-like triplets, as well as variations in the
exciting photospheric flux as a function of Doppler shift throughout the wind. We
find that the fact that the plasma is likely distributed over a range of radii and
Doppler shifts allows us to use an averaged value of the photospheric continuum
instead of accounting for it in detail. We also find that the value of R0 derived
145
Figure 4.23 Comparison of measurements and calculations for Si XIII in ζ Ori.
Calculations are thin lines, and measurements are thickened over the allowed
range of R. The solid line shows R(u0) from this work; the dash-dot line shows the
R(u) for a single radius, but using an averaged value of the TLUSTY UV flux; and
the dashed line shows the calculations and measurements of WC01. Note that the
range shown by the thickened lines represents the allowed range of measured R or
R values, and does not represent the physical extent of the X-ray emitting plasma.
In the case of R the model assumes a single radius of formation, while for R the
value of u0 inferred corresponds to the minimum radius for X-ray emission. The
fact that the allowed range of R graphically mimics the distribution of plasma radii
for the upper limit value to u0 is a coincidence.
146
Table 4.7 Comparison of R1 to R0.
Star Ion R1 R0a
ζ Pup Mg XI 2.5 1.43 ± 0.10
Ne IX 5 < 2.5
O VII 4 < 4
ζ Ori Mg XI 1.5 1.67+0.33−0.24
Ne IX 2.8 < 2.22
O VII 2.2 < 2.85
δ Ori Mg XI 1.2 1.25+0.18−0.07
Ne IX 2.1 < 2.22
O VII 2.8 < 3.33
aR0 is measured using an He-like line profile fit for Mg XI (see Table 4.4). Upper limits for O VII
and Ne IX are derived from upper limits to the f/i ratio of R < 0.1 and R < 0.2, respectively, which
are taken to be representative for all three stars.
Note. — In this table we compare the radius of optical depth unity R1 calculated by WC01, C01,
and Miller et al. (2002) to measurements of R0, the minimum onset radius for X-ray emission.
147
Figure 4.24 Same as Figure 4.23, but for S XV for ζ Pup, and we are comparing our
work to C01.
148
assuming a distribution of radii is substantially smaller than the value of R derived
assuming a single radius.
We have used the f/i ratio of helium-like triplets to constrain the radial distri-
bution of X-ray emitting plasma in four O-type stars. We find that the minimum
radius of emission is typically 0.6 < u0 < 0.8, or 1.25 < R0/R∗ < 1.67 with the
emission extending beyond this initial radius with either a constant filling factor
or one that increases slightly with radius. This is consistent with the results of line
profile fits using the model of Owocki & Cohen (2001) (Kramer et al. 2003; Cohen
et al. 2006). However, some of the minimum radii of formation are well inside the
radius of optical depth unity calculated using the mass-loss rates in the literature,
implying that either the effective opacities are lower (e.g. due to porosity effects
Feldmeier et al. 2003; Oskinova et al. 2004, 2006; Owocki & Cohen 2006) or the
mass-loss rates are lower than the literature values (Massa et al. 2003; Hillier et al.
2003; Bouret et al. 2005; Fullerton et al. 2006) or both. We also measure low values
of the characteristic optical depth τ∗ compared to what one would expect based on
the literature mass-loss rates, which is consistent with the same conclusions.
We find that there is no evidence for anomalously low f/i ratios in high-Z
species. Our measurements do not require X-ray emission originating from too
close to the star to have sufficiently strong shocks, nor do we need to posit the
existence of a magnetically confined corona. This conclusion is based partly on
different measured values of f/i ratios and partly on higher photospheric UV fluxes
on the blue side of the Lyman edge in the more recent TLUSTY model spectra.
We have fit He-like emission line complexes with profile models that simul-
taneously account for profile shapes and line ratios. These models constrain the
149
radial distribution of plasma both through the line ratio and the profile parameters
u0 and q. We find that they are capable of producing good fits to the data, showing
that the information contained in the line ratios and profile shapes are mutually
consistent.
150
Chapter 5
Evidence for the importance of
resonance scattering in X-ray emission
line profiles of the O star ζ Puppis1
We fit the Doppler profiles of the He-like triplet complexes of O VII and N VI in
the X-ray spectrum of the O star ζ Pup, using XMM-Newton RGS data collected
over ∼ 400 ks of exposure. We find that they cannot be well fit if the resonance
and intercombination lines are constrained to have the same profile shape. How-
ever, a significantly better fit is achieved with a model incorporating the effects of
resonance scattering, which causes the resonance line to become more symmetric
than the intercombination line for a given characteristic continuum optical depth
τ∗. We discuss the plausibility of this hypothesis, as well as its significance for our
1Submitted to the Astrophysical Journal as “Evidence for the importance of resonance scattering
in X-ray emission line profiles of the O star ζ Puppis” by M. A. Leutenegger, S. P. Owocki, S. M.
Kahn, & F. B. S. Paerels
151
152
understanding of Doppler profiles of X-ray emission lines in O stars.
153
5.1 Introduction
High resolution X-ray spectra obtained with diffraction grating spectrometers on
the Chandra and XMM-Newton X-ray observatories have revolutionized our under-
standing of the X-ray emission of O stars in the last five years. In the canonical
picture, the X-rays are emitted in plasmas heated by shocks distributed throughout
the wind (Cassinelli & Swank 1983; Corcoran et al. 1993; Hillier et al. 1993; Corcoran
et al. 1994); the shocks are created by instabilities in the radiative driving force (e.g.
Lucy & White 1980; Owocki, Castor, & Rybicki 1988; Cooper 1994; Feldmeier, Puls,
& Pauldrach 1997b). Although some stars show anomalous X-ray emission that
can be explained by a hybrid mechanism involving winds channelled by magnetic
fields (e.g. τ Sco andθ1 Ori C, Donati et al. 2002; Cohen et al. 2003; Gagne et al. 2005;
Donati et al. 2006), a number of “normal” O stars have X-ray spectra that are mostly
consistent with the wind-shock paradigm (e.g. ζ Pup, ζ Ori, δ Ori). The works
describing the first few high resolution spectra of normal O stars obtained found
some inconsistencies with expectations (Waldron & Cassinelli 2001; Kahn et al.
2001; Cassinelli et al. 2001; Miller et al. 2002; Waldron et al. 2004), but more recent
quantitative work based on the simple empirical Doppler profile model of Owocki
& Cohen (2001, hereafter OC01) has resolved many of these problems (Kramer,
Cohen, & Owocki 2003; Cohen et al. 2006; Leutenegger et al. 2006). The main
outstanding problem is the relative lack of asymmetry in emission line Doppler
profiles, which, if taken at face value, would imply reductions in the literature
mass-loss rates of an order of magnitude (Kramer et al. 2003; Cohen et al. 2006;
Owocki & Cohen 2006).
Although there is mounting evidence from other lines of inquiry suggesting
154
that the literature mass-loss rates may be systematically too high (Massa et al.
2003; Hillier et al. 2003; Bouret, Lanz, & Hillier 2005; Fullerton, Massa, & Prinja
2006), there are also subtle radiative transfer effects that could cause emission
line profiles to be more symmetric than one might naively expect. Two effects
that have been investigated in the literature are porosity (Feldmeier, Oskinova, &
Hamann 2003; Oskinova, Feldmeier, & Hamann 2004, 2006; Owocki & Cohen 2006)
and resonance scattering (Ignace & Gayley 2002, hereafter IG02). Porosity could
lower the effective opacity of the wind to X-rays, thus symmetrizing emission lines.
However, Oskinova et al. (2006) and Owocki & Cohen (2006) have found that the
characteristic separation scale of clumps must be very large to show an appreciable
effect on line profile shapes, which makes it difficult to achieve a significant porosity
effect. Resonance scattering can symmetrize Doppler profiles by favoring lateral
over radial escape of photons; it is an intriguing possibility, but to date it has not
been tested experimentally.
In this paper, we present evidence for the importance of resonance scattering in
some of the X-ray emission lines in the spectrum of the O star ζ Pup. We show that
the blend of resonance and intercombination lines of two helium-like triplets in the
very high signal-to-noise XMM Reflection Grating Spectrometer (RGS) spectrum
of ζ Pup cannot be well fit assuming that both lines have the same profile, but can
be much better fit assuming the profile of the resonance line is symmetrized by
resonance scattering.
This paper is organized as follows: in § 5.2 we discuss the reduction of over
400 ks of XMM RGS exposure on ζ Pup; in § 5.3 we briefly recapitulate the results
of OC01 and Leutenegger et al. (2006) for Doppler profile modelling (§ 5.3.1), and
we show that the He-like OC01 profile model does not give a good fit to the O VII
155
and N VI triplets of ζ Pup (§ 5.3.2); in § 5.4 we generalize the results of OC01 to
include the effects of resonance scattering as derived in IG02 (§ 5.4.1), and we fit
this model to the data (§ 5.4.2); in § 5.5 we discuss our results; and in § 5.6 we give
our conclusions.
5.2 Data reduction
The data were acquired in eleven separate pointings. The first two observations
were Performance Verification, while the rest were Calibration; they are all available
in the public archive. The ODFs were processed with SAS 7.0.0 using standard
procedures; periods of high background were filtered out. Only RGS (den Herder
et al. 2001) data were used in this paper, but EPIC data are available for most of the
observations. Processing resulted in a coadded total of 415 ks of exposure in RGS
1 and 412 ks in RGS 2. The observation IDs used and net exposure times are given
in Table 5.1.
RGS has random systematic wavelength scale errors with a 1σ value of ±7 mÅ
(den Herder et al. 2001). A 7 mÅ shift could lead to significant systematic errors in
the model parameters measured from a line profile. Because of this, we coadd all
observations using the SAS task rgscombine. Assuming the systematic shifts are
randomly distributed, coadding the data will result in a spectrum that is almost
unshifted (depending on the particular distribution of shifts of the individual ob-
servations), but that is broadened by 7 mÅ; this effect is much easier to account for
in our analysis. We have assumed that the data do not vary intrinsically. We have
not formally verified that the data show no significant intrinsic variation, but upon
visual inspection the data do not appear to vary more than expected from statistical
156
fluctuations combined with the aforementioned random systematic errors in the
wavelength scale.
Spectral fitting was done with XSPEC 12.2.1; the line profile models are imple-
mented as local models. The C statistic (Cash 1979) is used instead of χ2 because
of the low number of counts per bin in the wings of the profiles.
Because of the failed CCD on RGS2, we only have RGS1 data for O VII He α.
We only fit RGS2 data for N VI He α because the complex falls on a chip gap for
RGS1.
For each complex we fit, we first measured a local continuum strength from
a nearby part of the spectrum uncontaminated by spectral features. We modeled
this continuum as a power-law with an index of two, which is flat when plotted
against wavelength. When fitting a line profile, we fit a combination of the local
continuum (fixed to the measured value) plus the line profile model to the data.
For the N VI He α complex, we also included emission from C VI Ly β at
28.4656 Å, since the red wing of this line overlaps the blue wing of the resonance
line of N VI He α. The model parameters for C VI Ly β are assumed to be the same
as for N VI, and it is assumed to be optically thin to resonance scattering.
Emission lines and complexes were fit over a wavelength range ofλ− < λ < λ+.
Here λ± = λ0(1± v∞/c)±∆λ, where ∆λ is the resolution of RGS at that wavelength.
λ0 is the shortest wavelength in the complex for λ− and the longest wavelength for
λ+.
157
Table 5.1 List of observations with net exposure times.
obsid a texp,R1b texp,R2
b
0095810301 30.6 29.8
0095810401 39.7 38.3
0157160401 41.5 40.2
0157160501 32.8 32.8
0157160901 43.4 43.4
0157161101 27.0 27.0
0159360101 59.2 59.2
0159360301 22.0 22.0
0159360501 31.5 31.5
0159360901 46.6 46.6
0159361101 41.1 41.0
aXMM Observation ID.
bNet exposure time in ks.
158
5.3 Best fit He-like profile model
5.3.1 The profile model
In this section we briefly recapitulate the results of OC01 for the Doppler profile of
an X-ray emission line from an O-star wind, and the extension of these results to a
He-like triplet complex by Leutenegger et al. (2006).
In the physical picture of the OC01 model, the wind is a two-component
fluid; the bulk of the wind is relatively cool material of order the photospheric
temperature, while a small fraction of the wind is at temperatures of order 1-5 MK,
so that it emits X-rays. The cool part of the wind has some continuum opacity to
X-rays and can absorb them as they leave the wind.
The OC01 formalism casts the line profile in terms of a volume integral over
the emissivity, attenuated by continuum absorption:
Lλ = 4π∫
dVηλ(µ, r)e−τ(µ,r) (5.1)
where ηλ(µ, r) is the emissivity at the observed wavelength λ and τ(µ, r) is the
continuum optical depth to X-rays of the wind.
The line profile can be expressed in terms of the scaled wavelength x ≡ (λ/λ0−
1) c/v∞ = −vz/v∞; this gives the shift from line center in the observer’s frame in
units of the wind terminal velocity. The sign convention is such that positive x
corresponds to a redshift.
OC01 derive an expression for the line profile in terms of an integral over the
inverse radial coordinate u = R∗/r:
Lx = L0
∫ ux
0du
fX(u)w3(u)
e−τ(x,u). (5.2)
159
In this equation we have used the following expressions: w(u) ≡ v(u)/v∞ = (1−u)βv
is the scaled velocity; τ(x, u) is the (continuum) optical depth to X-rays emitted
along a ray to the observer; fX(u) ∝ uq is the filling factor of X-ray emitting plasma;
and ux ≡ min(u0, 1 − |x|1/βv) is the upper limit to the integral. u0 = R∗/R0 is the
inverse of the minimum radius of X-ray emission R0, and 1− |x|1/βv is a geometrical
cutoff for the minimum radius that emits for a given value of x. We have used βv
as the exponent of the velocity law rather than the customary β to avoid confusion
later in the paper. The integral for Lx can be evaluated numerically.
The optical depth in this expression is derived in OC01. It is written as the
product of the characteristic optical depth τ∗ = κM/4πR∗v∞ times a dimensionless
integral containing only terms depending on the geometry. It can be evaluated
analytically for integer values of βv. For non-integer values of βv, the optical
depth must be evaluated numerically, which is computationally costly, and thus
not convenient in conjunction with the radial integral of the line profile. Because
of this we assume βv = 1 throughout this paper, which is a good approximation for
ζ Pup, and also for O stars in general.
The interesting free parameters of this model are the exponent of the radial
dependence of the X-ray filling factor, q; the characteristic optical depth to X-rays
of the cold plasma, τ∗; and the minimum radius for the onset of X-ray emission R0.
Leutenegger et al. (2006) extend this analysis to a He-like triplet complex.
The only difference is that the forbidden-to-intercombination line ratio has a radial
dependence due to photoexcitation of the metastable upper level of the forbidden
line:
R ≡f
i= R0
11 + φ/φc
= R01
1 + 2PW(r). (5.3)
160
Here φ is the photoexcitation rate from the upper level of the forbidden line; it
depends on the photospheric UV flux and scales with the geometrical dilution
W(r); φ∗ is the photoexcitation rate near the photosphere, so that φ = 2φ∗W(r);
φc is the critical photoexcitation rate, which is a parameter of the ion; and P =
φ∗/φc is a convenient dimensionless parameter that gives the relative strength of
photoexcitation and decay to ground near the star such that R(R∗) = R0/(1 + P). In
this paper, we use values of P calculated from TLUSTY stellar atmosphere models
(Lanz & Hubeny 2003) as described in Leutenegger et al. (2006). Values of R0 are
taken from Porquet et al. (2001).
To modify the expressions for the forbidden and intercombination line profiles
to account for this effect, the emissivity is multiplied by the normalized line ratio:
η f (r) = η(r)R(r)
1 + R(r)(5.4)
and
ηi(r) = η(r)1
1 + R(r). (5.5)
5.3.2 Best fit model
In this section we model the Doppler profiles of the O VII and N VI He-like
triplets with the He-like profile of Leutenegger et al. (2006) described in § 5.3.1.
The forbidden line is very weak for these two ions, and the intercombination line
profile predicted by the model is not very different from the resonance line profile.
The main difference in the profile of the resonance and intercombination lines is
that the extremes of the wings are somewhat weaker. This is because the f/i ratio
reverts to the low-UV-flux limit at very large radii (> 100R∗ for O VIII for ζ Pup), so
161
that the intercombination line strength is reduced by a factor of a few. However,
this has only a small effect on the profile shape.
Although it is weak, the predicted strength of the forbidden line is a good check
on the consistency of the profile model. The value of the characteristic optical depth
τ∗ can have a strong effect on the observed f/i ratio by setting the value of R1, the
radius of optical depth unity. However, this effect is degenerate with the value of
q, the exponent of the radial dependence of the X-ray filling factor.
In Figures 5.1 and 5.2, we show the Doppler profiles of the O VII and N VI
He-like triplets, together with the best fit models. The best-fit parameters are given
in Table 5.2 and 5.3. There are significant residuals in both fits. The N VI triplet
shows stronger residuals than O VII. The residuals have a systematic shape: the
model predicts a greater flux than the data on the blue wing of the resonance line
and the red wing of the intercombination line, while it underpredicts the data in
the center of the blend.
The systematic nature of the residuals implies that there is something different
about the shapes of the Doppler profiles of the resonance and intercombination
lines. Qualitatively, the residuals are consistent with the model resonance line
being too blue and therefore too asymmetric, and the model intercombination line
being too red and therefore too symmetric.
Resonance scattering has been proposed by IG02 as an explanation for the
properties of O-star X-ray emission line Doppler profiles. If it is important, it can
cause significant symmetrization of profiles of strong resonance lines. Because this
is in qualitative agreement with our observations, we explore this idea further.
162
Figure 5.1 O VII triplet with best fit OC01 He-like triplet model (not including the
effects of resonance scattering). The top panel shows the data in black with error
bars and the model in red. The flat red line shows the assumed continuum strength.
The bottom panel shows the fit residuals.
163
Figure 5.2 N VI triplet with best fit OC01 He-like triplet model (not including the
effects of resonance scattering). Scheme is as in Figure 5.1. The C VI Ly α line at
28.4656 Å is also included in the fit, as well as the other fits to the N VI triplet.
164
5.4 Best fit model including the effects of resonance
scattering
5.4.1 Incorporating resonance scattering into OC01
In this section we discuss the modifications to the model of OC01 needed to include
the effects of resonance scattering. The calculation of a Doppler profile including
the effects of resonance scattering was performed in IG02; however, they used two
simplifying assumptions that we relax here. The first assumption we relax is that of
very optically thick resonance lines, and the second is that of a constant expansion
velocity.
It is also desirable to recast the results of IG02 in terms of the formalism of
OC01 in order to facilitate comparison of results from different model parameters.
To include the effects of resonance scattering in the OC01 formalism, we mul-
tiply the integrand of Equation 8 of OC01 by the normalized escape probability
p(µ)/β, giving
Lx =CM2
2v2∞
∫ ∞
R∗
drH[w(r) − |x|]f (r)
r2w3(r)
e−τc(µ,r) p(µ)β
µ=−x/w(r)
. (5.6)
Here
p(µ) =1 − e−τµ
τµ(5.7)
is the angle-dependent Sobolev escape probability and
β =12
∫ 1
−1dµ p(µ) (5.8)
is the angle-averaged Sobolev escape probability. The physical motivation for this
term comes from the Sobolev theory developed in IG02; the escape probability p(µ)
165
describes the angular distribution of escaping photons, while the factor β gives the
increased emission over thermal resulting from the trapping of scattered photons.
Another way to look at the factor β is that it normalizes the emission to be the
same as for the case of no resonance scattering, which should be the case as long
as photons are not trapped long enough to be thermalized.
In these equations,
τµ =τ0
1 + σµ2 (5.9)
is the Sobolev optical depth, where
σ =r
v
∂v
∂r− 1 =
βvu
1 − u− 1. (5.10)
Here we have used the inverse radial coordinate u ≡ R∗/r; we have also used βv
to denote the exponent of the velocity law, v(r) = v∞(1 − R∗/r)βv , in order to avoid
confusion with the angle-averaged Sobolev escape probability.
The factor τ0 gives the Sobolev optical depth in the lateral direction µ = 0; it is
given by
τ0 =38λ
re
c
vfi ni σT r. (5.11)
Here re is the classical electron radius; σT is the Thomson cross-section; fi is the
oscillator strength of the transition; and ni is the ion density. To explicitly put in all
dependence on the radial coordinate, we use the continuity equation, M = 4πρr2v,
giving
τ0 =λrecM
4R∗v2∞
(
fini
ρ
)
u
w2(u)= τ0,∗
u
w2(u)(5.12)
where we have defined the parameter
τ0,∗ =λrecM
4R∗v2∞
(
fini
ρ
)
(5.13)
166
as a characteristic Sobolev optical depth. The factor
ni
ρ=
ni
ne
ne
ρ=
Ai qi fX
µN mp
(5.14)
gives the ratio of the ion number density to the mass density. Here Ai is the
abundance of the element relative to hydrogen; qi is the ion fraction; fX is the filling
factor of X-ray emitting plasma; and µNmp is the mean mass per particle. We take
this ratio to be a constant with radius, although in principle the ion fraction and
filling factor could vary.
In this paper, we take τ0,∗ as a free parameter. τµ then has the radial and
angular dependence given by Equations 5.9 and 5.12.
In order to evaluate β, it is necessary to perform the integral over µ. In the
approximation that the Sobolev optical depth is very large, the integral is analytic,
and we getp(µ)β=
1 + σµ2
1 + σ/3. (5.15)
If we cannot make this approximation, the integral over µ cannot be evaluated
analytically. However, there is an analytic approximation given in Castor (2004, pp.
128-129, attributed to Rybicki) that is accurate to∼ 1.5%. We use this approximation
to calculate β for finite values of τ0,∗ .
IG02 assume a constant wind expansion velocity (βv = 0) and that the Sobolev
optical depth is large. Under these assumptions, we recover the expression p(µ)/β =
(3/2)(1 − µ2), which has the same µ dependence as the result derived in IG02.
The approximation of constant expansion is not unreasonable at large radii,
but βv = 1 is closer to the actual velocity law of ζ Pup, and it is no more difficult
to implement in our model. However, we wish to consider the possibility that the
167
effective βv for the purposes of resonance scattering could be different than for the
wind as a whole. For example, since the X-ray emitting plasma is too ionized to
have much effective line opacity in the UV, it should not be driven, and thus the
local velocity gradient might be better described in terms of a βv = 0 model without
radial acceleration, even while the overall mean velocity of the wind is described
well by a velocity law with βv = 1.
Let us thus define βSob to be the value of βv used in calculating σ. We consider
two cases in this work: βSob = 0 corresponds to no local velocity gradient for X-ray
emitting plasma, and βSob = 1 (= βv) means that the local X-ray and global bulk
wind velocity gradients are equal.
We have implemented this as a local model in XSPEC. The Sobolev optical
depth has angular and radial dependence as given by Equations 5.9 and 5.12.
The additional parameters added to the OC01 model are a switch to turn on or
off completely optically thick scattering; the characteristic Sobolev optical depth
τ0,∗ (used when the completely optically thick switch is off); and the value of the
velocity law exponent used in calculating σ, βSob.
In Figures 5.3 and 5.4 we compare the effects of various values of τ0,∗ and
βSob. The trend is for higher values of τ0,∗ and lower values of βSob to give more
symmetric profiles.
5.4.2 Best fit model including resonance scattering
In this section we fit He-like profile models including resonance scattering to the
O VII and N VI complexes. We fit each complex twice: once assuming βSob = 1 and
168
Figure 5.3 Comparison of the influence of different values of βSob on Doppler profile
shape. All models have q = 0, u0 = 2/3, and τ∗ = 5. The most asymmetric model is
optically thin. Both of the other models use the approximation that τ0,∗ is infinite;
the more asymmetric of the two has βSob = 1, while the least asymmetric has
βSob = 0.
169
Figure 5.4 Comparison of the influence of various values of the characteristic
Sobolev optical depth τ0,∗ on Doppler profile shape. All models have q = 0,
u0 = 2/3, τ∗ = 5, and βSob = 0. In order from most asymmetric to least the models
have τ0,∗ = 0, 1, 10,∞.
170
once assuming βSob = 0. The best-fit models are shown in Figures 5.5-5.8. The best
fit parameters are given in Tables 5.2 and 5.3.
The O VII profile is well fit by either value of βSob. We tested goodness of fit
by comparing the fit statistic of 1000 Monte Carlo realizations of the model to the
fit statistic of the data; both models are formally acceptable. The fit with βSob = 1
is better than that with βSob = 0 , but only by ∆C = 3.8, which is about 2σ for one
interesting parameter. The fit with βSob = 0 has a significantly smaller value of τ0,∗
than the fit with βSob = 1, as would be expected. The fit with βSob = 1 is statistically
consistent with the approximation that the Sobolev optical depth becomes infinite.
The N VI profile is much better fit by either model including resonance scat-
tering than it is by the original model. Furthermore, the model with βSob = 0 gives
a significantly better fit than the model with βSob = 1. However, neither model
is formally acceptable, and even the βSob = 0 model shows residuals of the same
qualitative form as the original model, albeit of a much lower strength. For both
models including resonance scattering, the optically thick approximation gives a
better fit than a profile with finite Sobolev optical depth.
To test the significance of profile broadening introduced by coadding data
with random systematic errors in the wavelength scale, we have also fit each best-
fit model with an additional 7 mÅ Gaussian broadening. In all cases, the best-fit
parameters did not change significantly and the fit statistics were not significantly
worse. Thus we conclude that our analysis is not strongly affected by this broad-
ening.
171
Table 5.2 Model fit parameters for O VII.
βSob q τ∗ u0 τ0,∗ G a n b C c MC d
· · · -0.21 1.6 0.62 · · · 0.91 6.91 152.2 · · ·
1 0.15+0.06−0.07 4.1+0.3
−0.4 > 0.68 > 50 1.11+0.03−0.04 6.88 ± 0.07 85.3 0.578
0 0.15+0.07−0.07 4.1 ± 0.4 > 0.63 5.9+3.2
−1.8 1.02+0.04−0.03 6.88 ± 0.07 89.1 0.730
aG = ( f + i)/r is assumed not to vary with radius.
bNormalization of entire complex (r + i + f ) in units of 10−4 photons cm−2 s−1.
cFor 83 bins.
dFraction of 1000 Monte Carlo realizations of model having C less than the data.
Note. — The first row gives the best fit for a model not including resonance scattering (i.e. the
model of OC01 and Leutenegger et al.). The second row gives the best fit for a model including
resonance scattering with βSob = 1, and the last row has βSob = 0. We used a value of P = 1.67 × 104
for all O VII profile models (Leutenegger et al. 2006).
172
Table 5.3 Model fit parameters for N VI.
βSob q τ∗ u0 τ0,∗ G a n b nβc C d
· · · -0.34 0.5 0.58 · · · 0.87 1.562 0 510.5
1 -0.09 2.1 0.50 thick 1.10 1.559 0.87 292.2
0 0.06 3.0 0.48 thick 1.15 1.552 1.25 188.4
aG = ( f + i)/r is assumed not to vary with radius.
bNormalization of entire N VI complex in units of 10−3 photons cm−2 s−1.
cNormalization of C VI Lyman β in units of 10−5 photons cm−2 s−1.
dFor 117 bins.
Note. — The first row gives the best fit for a model not including resonance scattering (i.e. the
model of OC01 and Leutenegger et al.). The second row gives the best fit for a model including
resonance scattering with βSob = 1, and the last row has βSob = 0. The C VI Lyman β line is assumed
to have the same values of q, τ∗, and u0 as the N VI triplet, and is assumed not to be affected by
resonance scattering. We used a value of P = 1.01 × 105 for all N VI profile models (Leutenegger
et al. 2006).
173
Figure 5.5 O VII triplet with best fit model assuming resonance scattering with
βSob = 1. Scheme is as in Figure 5.1.
174
Figure 5.6 O VII triplet with best fit model assuming resonance scattering with
βSob = 0. Scheme is as in Figure 5.1.
175
Figure 5.7 N VI triplet with best fit model assuming resonance scattering with
βSob = 1. Scheme is as in Figure 5.1.
176
Figure 5.8 N VI triplet with best fit model assuming resonance scattering with
βSob = 0. Scheme is as in Figure 5.1.
177
5.5 Discussion
5.5.1 Comparison of results
The profile fits presented in § 5.3.2 clearly show that the O VII and N VI He-like
triplet complexes in ζ Pup cannot be fit by models that assume the same profile
shapes for the resonance and intercombination lines. The profile fits presented in
§ 5.4.2 show that these complexes can be much better fit by a model including the
effects of resonance scattering.
However, although the O VII complex is well fit by a model including the
effects of resonance scattering, the N VI complex shows differences in profile shape
between the resonance and intercombination line that are greater than our model
can reproduce, even under the most generous conditions (τ0,∗ → ∞, βSob = 0).
Furthermore, one would expect the two complexes to show relatively similar pa-
rameters; for example, since the elemental abundance of nitrogen appears to be
roughly twice that of oxygen, one would expect the parameter τ0,∗ to be about
twice as large for the fit to N VI as it is for O VII. But a fit to the N VI profile with
βSob = 0 and τ0,∗ ≈ 10 (roughly twice the value measured for O VII) would give
a substantially worse fit than a model with infinite Sobolev optical depth, which
itself has significant residuals.
The fact that the apparent discrepancy between the shapes of the resonance
and intercombination line profiles is much greater for N VI than for O VII implies
that whatever the symmetrizing mechanism for the resonance line is, it is signifi-
cantly stronger for N VI. There is no obvious explanation for this in the resonance
scattering paradigm.
178
5.5.2 Plausibility of the importance of resonance scattering
It is worth revisiting the plausibility arguments of IG02 to confirm that one would
expect resonance scattering to be important for these ions in the wind of ζ Pup.
The relevant quantities to estimate are the Sobolev optical depth and the ratio of
the Sobolev length to the cooling length.
The Sobolev length is given by
Lµ =1 + σ
1 + σµ2
vth
dv/dr=
vth
v/r
11 + σµ2 (5.16)
(e.g. Gayley 1995). The cooling length is given by
52
k∆T
neλv, (5.17)
as derived in IG02.
Taking the ratio,
Lc
Lµ=
52
k∆T
neΛ
v
vth
v
r(1 + σµ2) (5.18)
=52
k∆T
Λ
4πµNmp
M
v∞vth
v2∞R∗
w3(u) fX
u(1 + σµ2) (5.19)
where we have used M = 4πµNmpner2v for a smooth wind, and added a filling
factor fX to correct for the ratio of the density of the X-ray emitting plasma to the
mean density expected for a smooth wind.
Putting in some representative numbers appropriate to ζ Pup, we have
Lc
Lµ= 10 (1 + σµ2)
w3(u) fX
u
1M6
(5.20)
where M6 is the mass-loss rate in units of 10−6 M⊙ yr−1. We have used Λ = 6 ×
1023 erg s−1 cm3, ∆T = 2MK, µN = 0.6, vth = 50 km s−1, v∞ = 2500 km s−1, and
R∗ = 1.4 × 1012 cm.
179
This expression is greater than unity for lateral escape except at small radii
(r < 2R∗) if the filling factor is of order unity. However, if the filling factor is
significantly less than unity, the Sobolev approximation may not be valid.
We now consider the expected values of the characteristic Sobolev optical
depth,
τ0,∗ =λ re c M
4R∗ v2∞
(
fini
ρ
)
=λ re c M
4µN mp R∗ v2∞
fi Ai qi fX. (5.21)
Putting in appropriate values, we get
τ0,∗ = 120(
fiAi
10−3
λ
20 Å
)
qi fX M6 (5.22)
We give calculations of τ0,∗/qi fX for important lines in O star spectra in Table 5.4.
We have assumed solar abundances for all elements except C, N, and O (Anders &
Grevesse 1989). We assumed that the sum of CNO is equal to the solar value, with
carbon being negligible and with nitrogen having twice the abundance of oxygen;
this is an estimate based on the observed X-ray emission line strengths. Note that
the Sobolev optical depth scales with the wavelength of the transition; this means
that the Sobolev optical depths are significantly smaller for an X-ray transition than
they are for a comparable UV transition.
Again, if the X-ray filling factors are of order unity, the characteristic Sobolev
optical depths for the resonance lines of N VI and O VII are large, but X-ray filling
factors of order 10−3 or less are sufficient to cause the lines to become optically
thin. However, the requirement that the Sobolev length in the lateral direction be
smaller than the cooling length is about as stringent, so that if resonance scattering
is important for strong lines, the Sobolev approximation should also be valid.
The high filling factors required are at odds with the simple two-component
180
Table 5.4 Expected characteristic Sobolev optical depth.
λ fia Ai
b τ0,∗/qi fXc
(Å)
N VI r 28.78 0.6599 0.9 103
β 24.90 0.1478 20
N VII Ly α 24.78 0.1387, 0.2775 d 19, 37
O VII r 21.60 0.6798 0.45 40
β 18.63 0.1461 7
O VIII Ly α 18.97 0.1387, 0.2775 d 7, 14
Fe XVII 15.01 2.517 0.047 11
15.26 0.5970 2.5
16.78 0.1064 0.5
17.05 0.1229 0.6
Ne IX r 13.45 0.7210 0.12 7.0
β 11.55 0.1490 1.2
Ne X Ly α 12.13 0.1382, 0.2761 d 1.2, 2.4
Mg XI r 9.17 0.7450 0.038 1.6
Mg XII Ly α 8.42 0.1386, 0.2776 d 0.27, 0.53
Si XIII r 6.65 0.7422 0.036 1.1
Si XIV Ly α 6.18 0.1386, 0.2776 d 0.19, 0.37
aOscillator strengths are from CHIANTI (Dere et al. 1997; Landi et al. 2006).
bAssumed abundance relative to hydrogen in units of 10−3.
cThis number is calculated using Equation 5.22 assuming a mass-loss rate of 10−6 M⊙ yr−1.
d Lyman α is a doublet.
181
fluid picture of the OC01 model, since the X-ray filling factors are known to be very
low. However, if we take the wind to be resolved on scales of order the Sobolev
length into the two components, the filling factor would just be ratio of the local
density to the mean density at that radius. This filling factor would still likely be
less than unity for the X-ray emitting plasma, but not as low as the X-ray filling
factor for the whole wind. This conjecture is a significantly stronger assumption
than is made in OC01.
5.5.3 Impact of resonance scattering on Doppler profile model
parameters
If resonance scattering is important in Doppler profile formation in the X-ray spectra
of O stars, it may lead to a partial reconciliation with the literature mass-loss rates.
The best fit models for O VII have τ∗ = 4.1, and the best fit model for N VI has
τ∗ = 3.0. If we speculate that somehow the resonance line of N VI is even further
symmetrized than predicted by our model, as the residuals in our best-fit model
imply, the value of τ∗ demanded by the intercombination line profile residuals
should be somewhat higher; a reasonable guess would be τ∗ ∼ 4 − 5.
These characteristic optical depths are higher than those measured by Kramer
et al. (2003) for ζ Pup by applying the model of OC01 to Doppler profiles observed
with the Chandra HETGS; the lines studied in that paper were mostly resonance lines
as well. They are still somewhat lower than one would expect given the literature
mass-loss rates; however, a detailed comparison with opacity calculations and
mass-loss rates remains to be done. New, sophisticated analyses of UV absorption
line profiles indicate that the published mass-loss rates of O-star winds are too high
182
(Massa et al. 2003; Hillier et al. 2003; Bouret et al. 2005); the most recent systematic
analysis of galactic O stars finds that for at least some spectral types, the published
mass-loss rates must be at least an order of magnitude too great (Fullerton et al.
2006). This leads to the curious possibility that measurements of the mass-loss rates
of O stars using Doppler profiles of X-ray emission lines could be higher than the
most recent UV line profile measurements, which is the opposite of the problem
currently being addressed by the community.
Our measurements provide suggestive constraints on the radial dependence
of the X-ray emission. The O VII fit has q ∼ 0, and the onset radius for X-ray
emission is not well constrained apart from being significantly inside the radius of
optical depth unity. Although our model does not provide a good fit to N VI, a
model that accounts for the symmetrization of the resonance line may also show
a similar radial distribution of X-ray emitting plasma. If the characteristic optical
depths to X-rays are of order a few at longer wavelengths in the wind of ζ Pup,
these constraints on the radial dependence may help to break possible profile
fitting degeneracies. A profile model with q ≡ 0 and u0 obscured by absorption
would have two fitting parameters for optically thick lines (τ∗ and τ0,∗) and one for
optically thin lines (τ∗). Thus, high signal-to-noise, optically thin Doppler profiles
with significant continuum absorption may provide robust measurements of the
mass-loss rates of O stars. A good candidate for this is the 16.78 Å line of Fe XVII,
which is likely not to be optically thick, and which is not blended with other lines.
183
5.5.4 Future work
Here we give a list of questions raised by this analysis that should be addressed in
future work.
1. The discrepancies in the fits in this paper must be resolved. The fact that
we cannot fit the N VI profile well is unsatisfactory. The difference between the
appearance of the N VI complex and the O VII complex requires explanation.
2. The effect of resonance scattering on other resonance lines in the X-ray
spectrum should be considered. Furthermore, unless we can make concrete pre-
dictions for the importance of resonance scattering for these lines, there may be
significant fitting degeneracies between resonance scattering and low characteristic
continuum optical depths.
3. The effect of multiple lines on resonance scattering should be explored.
Of special importance is the calculation of the profile of a close doublet, such as
Lyman α. In that case, the splitting between the two lines is of order the thermal
velocity of the ions.
5.6 Conclusions
We have fit Doppler profile models based on the parametrized model of OC01 to
the He-like triplet complexes of O VII and N VI in the high signal-to-noise XMM-
Newton RGS X-ray spectrum of ζ Pup. We find that the complexes cannot be well
fit by models assuming the same shape for the resonance and intercombination
lines; the predicted resonance lines are too blue and the predicted resonance lines
184
are too red. This effect is what is predicted qualitatively if resonance scattering is
important.
We find that models including the effects of resonance scattering give signif-
icantly better fits. However, there is significant disagreement between the O VII
and N VI profiles in the degree of resonance line symmetrization that is difficult to
understand in the framework of the resonance scattering model. Nevertheless, the
general trend of the resonance scattering model to give more symmetrized profiles
provides an interesting alternative (or supplement) to models that assume reduced
wind attenuation due to reduced mass-loss rates and/or porosity.
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