1.5 Zeeman Doppler Imaging

Research Collection

Doctoral Thesis

Multi-line diagnostics of magnetized stellar atmospheres

Author(s): Sennhauser, Christian

Publication Date: 2010

Permanent Link: https://doi.org/10.3929/ethz-a-006298517

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ETH Library Diss ETH No. 19210

Multi-line diagnostics of magnetized stellar atmospheres

Christian Sennhauser

Diss ETH No. 19210

Multi-line diagnostics of magnetized stellar atmospheres

Diss ETH No. 19210

Multi-line diagnostics of magnetized stellar atmospheres

A dissertation submitted to

ETH Zurich

for the degree of Doctor of Sciences

Presented by

Christian Sennhauser

Dipl. Phys. ETH born January 22, 1980 from Kirchberg, SG

accepted on the recommendation of

Prof. Dr. M. R. Meyer, examiner Prof. Dr. S. V. Berdyugina, Prof. Dr. S. K. Solanki, Prof. Dr. A. O. Benz, co-examiners

Zurich, 2010

— To my dear family —

Abstract

This thesis is devoted to the polarization spectra emerging from stellar photospheres, and the methods to recover the physical conditions under which they formed. Stars are the most prominent constituents of our universe. They convert nuclear binding energy into electromagnetic radiation, an energy source necessary to maintain thermodynamical order, which is equivalent to information, or life in the broadest sense. It is natural that they amazed humanity ever since, and that, driven by their relentless curiosity, people put unreasonable effort into the accumulation of knowledge about stars. Understanding our Sun as an only very nearby member of these nocturnal skyresidents, however, has justified the investigation of stellar activity phenomena in modern astronomy. Stellar magnetism is of particular importance, not only because it plays a growing role in stellar evolution models, from the formation phase to the ex- pulsion of planetary nebulae, but it is also believed to be the origin of energy released in flares, which in the case of the Sun can affect life on earth. Ac- cordingly, this thesis is partly concerned with the detection of stellar magnetic fields. Especially for late-type stars, magnetic activity is often observed to go along with other surface inhomogeneities, such as chemical abundance, temperature and density divergence. The complete physical properties of stellar photospheres are imprinted in the spectral lines. The limited amount of flux arriving at the earth, during a time interval short enough that the observable stellar disk can be assumed as unchanged, often results in insufficient photon x statistics. In order to account for this, the combination of multiple absorption lines shall unsheathe information about the photospheric region in which they form. First the concepts of line formation in magnetized stellar atmospheres will be introduced, with emphasis on the Zeeman effect. The importance of considering the polarization state of light in the Stokes formalism when measuring magnetic fields will become apparent. With the aim to outperform existing multi-line techniques, we develop in this thesis two methods, each based on an independent novel approach to disentangle blended line profiles. In a first attempt we account for the nonlinearity of blends in intensity spectra, de- parting from the the weak line approximation (WLA). Based on a heuristical formula, we derive an analytical function for line summation. The main achievement in the course of this thesis is the development of the Zeeman component decomposition (ZCD) and its numerical implementation. With the assumption of a common line-to-continuum opacity line profile, it is an inversion code for the simultaneous fitting of thousands of spectral lines for a given set of polarized spectra. ZCD sets a lower threshold for the detection of weak magnetic fields (< 1 G), at the same time increasing the reliability of intermediate fields, not being restricted to the weak field approximation. As an application of ZCD, we report the first detection of a weak magnetic field on the K giant Arcturus (αBoo) in Chapter 4. The reproduction and study of research objects in a laboratory is an impos- sibility in astrophysics. We fully depend on the observation of light, emitted by an unstoppable experiment, that has found its way onto the primary mirror of a telescope. The effort put in the development of observing instruments and their performance should cope with the advantages of modeling theories. The last part of this work is dedicated to the calibration and upgrade of an existing polarimeter using liquid-crystal variable retarders (LCVRs), which greatly reduces the systematic error, and enables fast-switching polarization modulation for a broad spectral region including many lines. The aim to increase the informative value of spectral line profiles by building high-precision instruments and improving our analyzing techniques will allow us in the near future to apply ever more sophisticated diagnostic tools to stellar observations, which are currently only applicable for the Sun. Zusammenfassung

Die vorliegende Doktorarbeit befasst sich mit dem polarisierten Lichtspek- trum, das von Sternatmospharen¨ abgestrahlt wird, und Methoden die es erlauben, die physikalischen Bedingungen bei der Enstehung dieses Lichts zu rekonstruieren. Sterne sind augenscheinlich die auffalligsten¨ Bestandteile unseres Univer- sums. Sie formen nukleare Bindungsenergie in elektromagnetische Strahlung um, eine notwendige Energiequelle um die thermodynamische Ordnung, die Information aufrechtzuerhalten, die wir Leben nennen. Verstandlicherweise¨ haben Sterne die Menschheit seit jeher verwundert. Getrieben von ihrer unban-¨ digen Neugier haben Menschen scheinbar unvernunftig¨ viel Aufwand be- trieben, ihr Wissen uber¨ die Sterne zu vergrossern.¨ Die Einsicht, dass unsere Sonne auch nur ein Reprasentant¨ jener nachtlichen¨ Himmelsbewohner ist, gibt der modernen Astronomie einen naheliegenden Grund die Mechanismen fur¨ Sternaktivitat¨ zu studieren. Dem Sternmagnetismus kommt dabei eine besondere Bedeutung zu. Er spielt nicht nur eine zunehmend entscheidende Rolle in Sternentstehungs- modellen, von der kollabierenden Molekulwolke¨ bis zum Ausstoss planetarer Nebel, sondern ist vermutlich die treibende Kraft bei Sonneneruptionen, die das Leben auf der Erde nicht bestimmen, aber beeinflussen konnen.¨ De- mentsprechend befasst sich die vorliegende Arbeit mit dem Messen von Stern- magnetfeldern. Speziell auf kuhlen¨ Sternen mit einer effektiven Temper- atur . 5500 K geht magnetische Aktivitat¨ oft einher mit Inhomogenitaten¨ auf xii der Oberflache:¨ chemische Haufigkeit,¨ Temperatur- und Dichteunterschiede. Spektrallinien sind die Fingerabdrucke¨ von Sternatmospharen.¨ Sie erlauben prazise¨ Ruck-schl¨ usse¨ auf die physikalischen Eigenschaften. Die beschrankte¨ Anzahl Photonen, die auf der Erde warend¨ eines Zeitintervalls gemessen werden, in dem der Stern als unverandert¨ angenommen werden kann, verur- sacht jedoch Rau-schen im Profil der Spektrallinien. Das Kombinieren vieler solcher Linien soll helfen, die darin enthaltenen Informationen besser zu ent- schlusseln.¨ Im einfuhrenden¨ Kapitel werden die Konzepte zur Enstehung von spek- tralen Linienprofilen in magnetischen Sternatmospharen¨ erlautert,¨ wobei dem Zeeman Effekt eine entscheidende Rolle zukommt. Die Berucksichtigung¨ der Polarisationseigenschaften von Licht werden im sogenannten Stokes For- malismus verdeutlicht. Im weiteren Verlauf werden zwei Methoden zum Kombinieren mehrerer Spektrallinien mit dem Ziel entwickelt, bestehende Viel-Linien-Methoden zu ubertre¨ ffen. Der gegenseitigen Beeinflussung von Linienprofilen wird in zwei unterschiedlichen Herangehensweisen Rechnung getragen. In Kapitel 2 wird, ausgehend von einem heuristischen Ansatz, eine analytische Formel fur¨ das Aufeinanderlegen von beliebig starken Linien hergeleitet. Die Entwicklung und numerische Implementierung der Methode ‘Zeeman component decomposition’ (ZCD, Kapitel 3) steht im Zentrum dieser Dok- torarbeit. ZCD ist ein Inversionscode, der Tausende von Spektrallinien einer Messung unter der Annahme eines gleichbleibenden Profils der Linien-zu- Kontinuum Opazitat¨ rekonstruiert. Damit wird die Limite fur¨ detektierbare longitudinale Magnetfelder entscheidend heruntergesetzt (< 1 G), wahrend¨ auf der anderen Seite auch die Genauigkeit fur¨ starke Felder gesteigert wird. Die Anwendung von ZCD auf eine einzelne Messung des prominenten roten Riesen Arcturus (αBoo) lieferte die erste direkte Detektion eines Magnet- feldes auf diesem Stern, vorgestellt in Kapitel 4. Die Reproduktion und Messung von Studienobjekten im Labor ist in der Astronomie nicht moglich.¨ Wir sind vollstandig¨ abhangig¨ vom Licht, das aus- gesandt von einem fortlaufenden Experiment den Weg auf den Hauptspiegel eines unserer Teleskope gefunden hat. Es ist daher wichtig, die Entwick- lung von Messinstrumenten auf die Fortschritte von theoretischen Modellen abzustimmen. Der letzte Teil dieser Arbeit ist deshalb der Instrumentation gewidmet. Kapitel 5 beschreibt die Kalibration und Umsetzung eines auf Flussigkristallen¨ (LCVR) basierenden Polarimeters, wodurch eine hochfre- xiii quente Polarisationsmodulation und eine wesentliche Reduktion des system- atischen Fehlers erreicht wird. Das Ziel, hochprazise¨ Teleskope und Instrumente zu bauen, bestehende Modelle und Techniken zu verbessern und neu zu erfinden, um noch mehr Informationen aus polarisierten Spektrallinien herauszuschalen,¨ wird es uns in naher Zukunft ermoglichen,¨ noch genauere Diagnose-Techniken auf Stern- spektren anzuwenden, wie sie erst in der Sonnenphysik zum Einsatz kommen.

Contents

Abstract ...... ix

Zusammenfassung ...... xi

1. Introduction ...... 1 1.1 Stellarmagneticfields...... 1 1.1.1 Origins of stellar magnetism ...... 1 1.1.2 Distribution across the HR diagram ...... 2 1.2 Measuring stellar magnetic fields ...... 8 1.2.1 Light through a keyhole ...... 8 1.2.2 Interaction of light with magnetized matter ...... 10 1.2.3 Polarization: the Stokes formalism ...... 15 1.2.4 Spectropolarimeters ...... 17 1.3 Atomic spectropolarimetry ...... 22 1.3.1 Polarized radiative transfer ...... 22 1.3.2 The Unno-Rachkovsky solution ...... 24 1.3.3 The line absorption profile ...... 26 1.4 Multi-line techniques ...... 35 1.4.1 State-of-the-art methods ...... 35 1.4.2 Methods developed in this thesis: a new benchmark . . 38 1.5 Zeeman Doppler Imaging ...... 41 1.6 Outline of the thesis ...... 42 xvi Contents

Bibliography ...... 43

2. NDD: A new analyzing technique for spectroscopy ...... 53 2.1 Introduction...... 54 2.2 Weak line decorrelation and deblending ...... 56 2.3 Optically thin versus optically thick lines ...... 58 2.4 Numerical implementation ...... 63 2.4.1 Critical blends ...... 64 2.5 Results...... 66 2.5.1 Atomic proﬁles: Stokes I ...... 66 2.5.2 Molecular proﬁles: Stokes I ...... 68 2.6 Conclusions...... 69 Appendix A: Inverse interpolation ...... 71 Appendix B: Solving algorithms ...... 73 Appendix C: Derivative of the extended interpolation formula.... 74 Bibliography ...... 77

3. Zeeman component decomposition ...... 79 3.1 Introduction...... 80 3.2 PrinciplesoftheZCD...... 82 3.3 Nonmagnetic line blending ...... 84 3.3.1 Blended line profile and opacity ...... 85 3.3.2 The scaling function ...... 86 3.4 Zeeman component blending ...... 87 3.4.1 An illustrative example ...... 88 3.4.2 Anomalous splitting versus triplet approximation . . . 89 3.5 Explicit formulas ...... 95 3.5.1 Zeeman component opacities ...... 96 3.5.2 Final expressions for Stokes IQUV ...... 97 3.5.3 Combining two Zeeman components for Stokes I . . . 98 3.6 Numerical implementation ...... 99 3.7 Application to simulated data ...... 102 3.7.1 Longitudinal magnetic fields: Stokes I, V ...... 102 3.7.2 Inclined magnetic fields: Stokes IQUV ...... 104 3.8 Conclusions...... 108 Bibliography ...... 111 Contents xvii

4. Detection of a weak magnetic ﬁeld on Arcturus ...... 113 4.1 Introduction ...... 114 4.2 Observation and analysis ...... 115 4.2.1 Analysis of additional spectra ...... 117 4.3 Results...... 118 4.4 Discussion...... 119 4.4.1 Possible magnetic ﬁeld generators ...... 119 4.4.2 Sources for circular polarization cross-talk ...... 122 4.5 Conclusions...... 123 Bibliography ...... 125

5. Achromatizing a liquid-crystal spectropolarimeter ...... 129 5.1 Introduction ...... 130 5.2 Retardance Fitting and Optical Characterization ...... 134 5.2.1 Achromatic Wave-Plate Characterization ...... 136 5.3 Achromatizing an LCVR Polarimeter ...... 141 5.3.1 Observing with an LCVR Spectropolarimeter . . . . . 144 5.3.2 Retardance Fitting Deprojection ...... 150 5.3.3 Stokes-Based Deprojection ...... 153 5.3.4 Beneﬁts of Stokes-Based Deprojection ...... 155 5.4 HiVIS LCVR Spectropolarimeter ...... 156 5.4.1 HiVIS LCVR Observations ...... 157 5.4.2 HiVIS LCVR Stokes-Based Deprojection ...... 159 5.4.3 HiVIS LCVR Optimization ...... 160 5.5 Polarization Properties of the HiVIS Spectrograph ...... 164 5.5.1 Polarization With Image Rotator ...... 164 5.5.2 Polarization Without Image Rotator ...... 166 5.6 AEOS Telescope Polarization ...... 169 5.7 Discussion And Conclusions ...... 171 Bibliography ...... 173

6. Summary ...... 177

Curriculum Vitae ...... 179

List of Publications ...... 181

CHAPTER 1

Introduction

1.1 Stellar magnetic ﬁelds

1.1.1 Origins of stellar magnetism There is a wealth of quantities used to parametrize different stars. Among them are mass, radius, age, rotation rate, chemical composition, luminosity and temperature. Strictly speaking, none of these parameters are independent of each other. As a prominent example, the Hertzsprung-Russell (HR) diagram classifies stars according to their temperature, luminosity, and evolutionary stage. Stellar magnetism is a comparably young phenomenon, first reported by Babcock (1947), but since then plays a growing role in the field of star formation and evolution. A thorough study of magnetic activity, revealing the relation to other fundamental parameters, will provide an advanced picture of stellar physics. It may even be of overriding importance, shaping up as a remnant directly linked to the physical conditions during the protostellar cloud collapse, determining other quantities to a large extent. Two kinds of origins for stellar magnetic activity have been suggested; the continuous generation by dynamo processes, and the conservation of magnetic flux from their natal clouds, known as so-called frozen fields. Elaborate review papers on stellar magnetism are given by Berdyugina (2009) and Do- 2 Chapter 1. Introduction nati & Landstreet (2009). The topic of starspots as particular tracers of the internal dynamo activity is covered by Berdyugina (2005) and Strassmeier (2009). In late-type stars, dynamo processes driven by inner convection and differential rotation, parametrized by ∆Ω, are believed to act as a generator for magnetic activity. The Ω-effect transforms a poloidal magnetic field into a toroidal magnetic field. In addition, the so-called α-effect is responsible for the conversion of a torodial field into a poloidal field due to the Coriolis force. This αΩ-dynamo was proposed by Parker (1955). Recent examples of reviews on stellar (and solar, in particular) dynamo theory are given by Brandenburg & Subramanian (2005) and Charbonneau (2005). The radiative envelopes of upper main-sequence (MS) stars (A and earlier) cannot support these dynamos. The lack of classical activity phenomena such as non-thermal radio and X-ray emission from chromospheres and coronae, thermal winds, flares, starspots, solar-cycle-type variability etc. are due to the diminishing of strong envelope convection in stars hotter than 8 000 K (Land- street 1998), until outer convection disappears altogether at Teff 9 000 K (Schrijver & Zwaan 2000). Their magnetic fields are the remnants≈ of inter- stellar or dynamo fields which existed at the pre-main sequence evolutionary stage (Moss 1989), or remain conserved directly from cloud-collapse. For high-luminosity massive stars, for which Kelvin-Helmholtz timescales are in the order of the duration of infall, pre-MS evolution may be completely pre- vented (Hartmann 2009).

1.1.2 Distribution across the HR diagram

With the onset of advanced magnetic field studies in the eighties and nineties, most of the effort was put into measuring A, B, and active solar-type stars. More advanced instruments and analyzing techniques in the past decades have enabled the detection of magnetic fields in fainter, yet much more abundant late K and M-dwarfs, as well as in more massive B and O stars, which were previously assumed to be non-magnetic. The following short overview tries to classify magnetic activity by mass, and evolutionary stage (pre-MS, MS, post-MS). 1.1. Stellar magnetic fields 3

High-mass stars

While massive pre-MS (proto-) stars of masses M > 10M , where M is the solar mass, are very rare objects, young intermediate-mass⊙ stars from⊙ 2 to 10M embedded in high-rate accreting disk systems called Herbig Ae∼ /Be ∼systems⊙ are commonly observed. Besides features in the spectral energy distribution (infrared excess, 3 µ peak) attributed to dusty envelopes and evaporation at the inner edge of the disk (Dullemond & Dominik 2004), evidence of high-velocity infall for some targets suggests magnetospheric accretion (Muzerolle et al. 2004). Due to their high luminosities, the dust destruction radii are comparably large (0.3 1 AU), and their apparent brightness is also high, making them good targets− for spectropolarimetric studies as well. Sur- veys among HAe/Be stars and individual studies (e.g., Alecian et al. 2008) so far have revealed a handful of magnetic active objects, whose fractional bulk incidence is 10% (Wade et al. 2007, 2009), which corresponds to a total of 9 stars with fields∼ exhibiting predominant dipole components of kG strength. Among MS A and B stars, the chemically peculiar Ap/Bp stars are most probable candidates for photospheric magnetism. They exhibit intense lines of some ionized metals and rare-earths, pointing to a vast overabundance (up to 106 solar values) of these elements in the star’s surface layers. The first attempt by Preston (1971) to construct a surface magnetic field distribution, and later studies, for example by Mathys et al. (1997), led to the conclusion that the field strength distribution of magnetic Ap stars has a maximum at 3 to 5 kG, with all surface field strengths ranging from a couple of kG to a few tens of kG. On the other hand, Borra & Landstreet (1980) and Landstreet (1992) found a continuous distribution without a lower limit. The most complete cat- alog of magnetic stars by Romanyuk (2000) includes 211 chemically peculiar upper MS stars. This list is extended by many groups (e.g., Kochukhov & Bagnulo 2006) While Egret & Jaschek (1981) assumed 11% among B7-A3 stars to be Ap/Bp stars, recent studies suggest half of this fraction (5% reported by Mathys 2009). Considering this to be an upper limit of the frequency of magnetic stars, this number is consistent with the fraction of magnetically active HAe/Be objects, as pointed out earlier. However, not all Ap/Bp are magnetic (72 out of 96 reported by Kudryavtsev et al. 2006). A small fraction of magnetic HAe/Be stars evolve into weakly active Ae, Am, A (resp. B) stars. Currently, weak longitudinal fields (up to 300 G) are reported for more massive O and 4 Chapter 1. Introduction early B-type stars (Hubrig et al. 2009), and even weaker for Be stars (Yudin et al. 2009). Presumably, the structuring of these fields remains dipole-like (e.g., Kochukhov et al. 2004a,b) when arriving on the MS, which suggests that they might be fossil. Originally, abundance peculiarities of stars which are, to some extent, un- characteristic of their occurrence in the part of the galaxy and their formation epoch, are thought to result from a complex interplay between gravitational settling, radiative levitation, mass-loss through stellar winds, meridional cir- culation, convection and turbulence within the stellar envelope (Wade et al. 2000). The fact that in the atmospheres of magnetic stars, abundances can be outstandingly peculiar, and more remarkably appear to be distributed ir- regularly within a given star’s atmosphere, indicates that a magnetic field can be a precondition in the formation of such peculiarities and, furthermore, a determining factor of the physical state of the whole envelope. Detailed measurements of both the magnetic field structure of a given star and its chemical abundance peculiarities therefore will allow us to test theoretical models of stellar atmospheres and envelopes.

Intermediate-mass stars

Dynamo-driven magnetic activity of stars of about one solar mass is believed to evolve during their lifetimes, from the early T Tauri phase along the MS and into the giant branches. Being related to stellar rotation, non-homogeneous kG fields are recovered on fast rotating and supposedly fully convective young T Tauri stars, using various observational techniques (e.g., Guenther et al. 1999; Johns-Krull 2007). Along the main sequence, the decrease of magnetic activity and change of the topology goes along with loss of angular momentum and other surface activity phenomena. Early studies on global mean longitudinal magnetic fields of late-type stars include ten F5-M3 stars reported by Borra et al. (1984), Hubrig et al. (1994), Hubrig & Mathys (1997), Plachinda & Tarasova (1999), one being a MS star, the others late-type giants or subgiants. Latest results suggest that at least in special cases, also post-MS red giants may still possess structured magnetic fields of the order of a few Gauss (Konstantinova-Antova et al. 2008, 2009), comparable to the global field strengths of MS-stars. Below we follow the evolutionary stages. 1.1. Stellar magnetic fields 5

T Tauri stars are young pre-MS stars characterized by enhanced lithium absorption, exhibiting X-ray emission and flaring activity which are related to surface magnetism. Although photospheric fields, as well as large starspots have also been directly measured (Hartmann 2009, and references therein, p. 158), their field topology is largely unknown. Classical T Tau stars (CTTS) have accretion disks with infalling material onto the star, where one important difference compared to HAe/Be stars is that the central dust depletion radii, defined by the dust evaporation temperature of 1500 K, are much smaller (for a solar-mass star 0.1 AU). On the other hand,∼ the lack of near- infrared excess or optical continuum∼ emission, characteristic of disks (and infall) of some pre-MS stars gave rise to the term “weak” emission T Tauri stars (WTTS). These show high levels of chromospheric and coronal X-ray emission (e.g., Gudel¨ et al. 2007). The differences between CTTS and WTTS allows us to determine which properties are due to disk accretion and which to magnetic activity. In contrast to what we have learned from HAe/Be stars, Smirnov et al. (2004) suggested a variable photospheric field for T Tau based on Stokes IV measurements (cf. Sect. 1.2.2). Zeeman Doppler imaging (ZDI) maps obtained from LSD profiles (see Sects. 1.4.1 and 1.5) also indicate a more complex field structure (Donati et al. 2007, 2008a). The attempt to describe them in terms of a multipole expansion may, however, seem ques- tionable.

Solar-type stars are MS stars with spectral type ranging from F7 to K2, where convection is responsible for the energy transport from half the stellar radius to the outer layers. These stars and are of special interest,∼ as the Sun is one of them. A detailed study of magnetic surface structures of solar-like stars of ages comparable to the Sun’s is not possible with present techniques, due to the Sun being a relatively slow rotator. Solar progenitors exhibit higher activity levels in all domains, as supported by faster rotation (e.g., ξ Boo A observed by Petit et al. 2005; HD 190771 by Petit et al. 2009), which makes them possible targets for ZDI studies. These stars may even exhibit polarity reversals, which is in agreement with current dynamo theory. The signs of nonaxisymmetric fields are supported by theoretical models (Kitchatinov 2001). Recent results (e.g., Santos et al. 2010) confirm that standard solar activity indicators (Ca ii, Hα) can be used to trace the stellar magnetic cycle in solar-type stars just as well, as proposed by Wilson (1968). However, trac- ing the progression of the topology of magnetic fields will impose far more 6 Chapter 1. Introduction distinct constraints on evolutionary models. First indications suggest that the large-scale toroidal component dominating the surface field structures of rapid rotators is replaced by a mostly poloidal field for low rotation rates. Petit et al. (2008) found for their sample a critical period of 12 d where the two regimes merge. However, this effect may be caused by≈ decreasing spatial resolution in slower rotators. While the connection of surface activity with rotation, both declining with age (Skumanich 1972) is well established, the techniques presented in this thesis shall denote a further step towards a better interpretation of local and global magnetic fields on slowly and intermediately rotating solar-type stars, increasing our understanding of the physics responsible for solar activity and all phenomena related to it.

Post-MS red giants are stars of masses 0.8 . M . 10M that evolved away from the MS into the red giant branch (RGB) and asymptotic⊙ giant branch (AGB). The formation of a helium core and onset of CNO-cycle and H-shell burning causes an excess pressure for the hydrogen envelope which then ex- pands. This process leaves both solar-mass and higher-mass stars in a phase when their surface temperatures are comparable, and both types exhibit hydrogen convection zones extending deeply into the stellar interior (cool bottom processing, cf. Herwig 2005, and references therein). Blackman et al. (2001) showed that a dynamo at the interface between the rapidly rotating core and the more slowly rotating envelope of the star can generate moderate photospheric magnetic fields ( 100 G). Such a magnetic field may rule the mass loss geometry and the global∼ shaping of ejected (bipolar) planetary nebulae. Nordhaus et al. (2008) could produce kG toroidal fields at the base of the convection zone for both RGB and AGB stars via a shear-driven αΩ- dynamo, in order to model isotopic abundances where convection resupplies shear. Possible observational evidence of remnants of such fields in the photosphere are provided by Konstantinova-Antova et al. (2008, 2009) and Lebre` et al. (2009), although their sample stars rotate faster than the bulk of red giants. Stronger surface fields (> 100 G) have been detected on peculiar evolved giants, such as the tidally locked binaries of type RS CVn (e.g., ZDI maps of II Peg by Carroll et al. 2007), or the very rapidly rotating FK Com stars (Korhonen et al. 2009). They are also assumed to originate from a dynamo. Descendants of strongly magnetic Ap/Bp stars seem to keep their presumably fossil fields (e.g., Auriere` et al. 2008). Detections of very weak magnetic 1.1. Stellar magnetic fields 7

fields in the order of 1 G on the solar neighbor K-giant Pollux (Auriere` et al. 2009) and the first M-supergiant Betelgeuse (Auriere` et al. 2010), where giant convection cells could sustain a local dynamo, give new insight in the diverse field of evolved stars, and raise further questions about the origins of magnetism on these objects.

Low-mass stars

Stars with masses < 0.8M and nuclear lifetimes beyond 10 Gyr are called red dwarfs while on the MS,⊙ and constitute at least 80% of the stellar population in the galaxy. Stellar evolution models predict that stars later than M4 are fully convective. It is believed that the solar-type shell dynamo is gradually replaced by a turbulent, distributed dynamo, which generates magnetic fields of a different topology. Such late-type stars are very faint (usually >8 mag) even in the solar neighborhood, and polarimetric measurements have proven to be difficult. Using the Hα line as a tracer for magnetic activity, the increase for stars later than M5, with a peak activity near type M8, from a survey of 8000 late-type dwarfs by West et al. (2004) indicates the change in behaviour when passing from the partially to the fully convective regime. A corresponding change in the structure of the magnetic field from spectropolarimetric surveys is suggested (e.g., Berdyugina et al. 2008; Donati et al. 2008b). Regarding the radius and mass of a star as most fundamental parameters, Lopez-Morales´ (2007) describes how the deviations from predicted linked values are either due to magnetic activity and/or metallicity. Similar to solar-type stars, the connection between (magnetic) surface activity and rotation rate is found for M dwarfs. Reiners & Basri (2007) measured the Zeeman broadening of molecular FeH lines for a set of more than 20 stars ranging from M2 to M9, and found the strongest fields (> 3 kG) on the fastest rotators. Being the coolest stars in the sample, this result is in agreement with the findings from Hα surveys mentioned above. 3-D simulations of fully convective stars by Browning (2008) show that indeed rotation may play a crucial role in setting the field strength and morphology, with increasing fields reducing the solar-like differential rotation, up to quenching it entirely. Stokes V measurements analyzed in the weak field approximation by Reiners & Basri (2009) further support observational evidence for a change in magnetic topology at the boundary to full convection. However, they correctly remark that truthful determination of field strengths from molecular circular 8 Chapter 1. Introduction polarization patterns has to involve the Paschen Back effect, as treated by Berdyugina et al. (2005). Finally, measurements of magnetic activity indicators such as radio, X- ray, and Hα emission extend our knowledge steadily down the HR diagram. Examples are the recently reported 1.7 kG L dwarf (< 0.1M ) by Berger et al. (2009) and other investigations≈ into the magnetic field properties⊙ of ultracool dwarfs (e.g., Berger et al. 2010).

1.2 Measuring stellar magnetic ﬁelds

1.2.1 Light through a keyhole In general, we can only use the information imprinted in the light coming from astronomical objects to study their properties. Spectroscopy, the interaction of light with matter, is the diagnostic tool for inferring the physical properties of the material where light is absorbed (cool material in front of a hot light source), emitted (kinetic energy is transformed into radiation), or scattered (by atoms, ions, and/or electrons). For stellar photospheres, the first processes is usually of importance, while stellar chromospheres and other hot circumstellar matter is studied through emission lines. Also the effect of scattered light has started to play a role as a diagnostic tool for circumstellar disks and the detection of planets. Particles other than photons as information carri- ers are used only in a very limited number of special astronomical objects. In the following two paragraphs, we will quickly review the importance of other observational tools for astroparticle physics, while electromagnetic radiation is exclusively used as an exploring tool in the rest of this thesis. Neutrinos are produced in very high numbers during atomic fusion in stellar cores. Due to their extremely small scattering cross-section, they escape the stellar interior unaffected by the outer shells and are therefore the only means to directly examine their formation region. Having the disadvantage that they also are extremely hard to detect, the only objects from which we receive ‘enough’ neutrino flux are the Sun, gamma-ray bursts (e.g., Achter- berg et al. 2008), and nearby supernovae. Most of the energy released during a supernova explosion is assumed to be carried away by neutrinos, enabling the star to fully collapse (e.g., Woosley & Janka 2005; Kotake et al. 2006). From a cosmological point of view, a so far unrealistic increase in detector sensitivity to extremely low-energy neutrinos would enable us to discover the 1.2. Measuring stellar magnetic fields 9 cosmic neutrino background, originating from earlier times in the galactic history than the cosmic microwave background (CMB). On the other hand, the cosmic ray showers in the earth’s atmosphere are more of an annoyance to a stellar astronomer, since they cause undesired imprints on our detectors. They are induced by highly energetic protons and heavier charged ions (He, C, Fe, etc.), by electrons at a smaller rate and rarely + by gamma rays (85% p , 12% He, 2% e−, 1% others). The average energy of cosmic rays is E 2 109 eV. h i ≈ × Should star travel and very efficient heat-shielding remain science fiction, we will never be able to physically probe the ingredients of stars, with the exception of the solar wind. We are literally locked out from the place where astronomy happens. However, a lock is only a lock. Modern astronomy was born with the first measurements of the spectral distribution of the intensity of light, i.e., spectroscopy1. It was the first glance through the keyhole of the barring door. The detection and interpretation of missing light in the solar Fraunhofer spectrum initialized the characterization of stellar atmospheres in terms of element abundances, density, surface gravity, etc. via detailed analysis of atomic absorption/emission lines (see Sect. 1.3). Magnetic fields also play an important role as energy storage in all areas of astrophysics. Unfortunately, their imprints on light at typical strengths in most astrophysical sources are almost hidden. The modifications of spectral lines caused by a magnetic field in the formation region barely affects the intensity spectrum, but they do appear when we look at the polarization spectrum of light. The comprehension of intensity as just one of four polarization states of light, and the development of corresponding detectors is an ongoing process and will provide astronomers with a much more profound understanding of the Universe. Currently, the door is being opened, but only to the width of a chink. As a necessary tool, the polarization properties of light as electro- magnetical waves, and their characterization by the Stokes formalism will be introduced in Sect. 1.2.3. However, the underlying effect that causes polarization in a magnetic field, enabling us to detect them in the first place, is the Zeeman effect, discussed in the following.

1 There is an ambiguity in the word spectroscopy. It either denotes the physics describing the interaction of light with matter. Or, it simply means that the intensity of light is recorded as a function of wavelength (or frequency). 10 Chapter 1. Introduction

1.2.2 Interaction of light with magnetized matter

First described by Zeeman (1897), the Zeeman effect (ZE) imprints a polarization signal in the spectrum, which enables to recover the full magnetic field vector. As a matter of fact, it is only sensitive to large-scale magnetic fields. Fields in spatially unresolved line forming regions will cause cancellation of the polarization signals. In stellar astronomy, lacking spatial resolution, it is nonetheless the most important diagnostic tool for detection of magnetic fields. For the Sun, being well spatially resolved, other origins for polarized radiation can be taken into account: scattering polarization, and the Hanle effect. Both effects originate from a local symmetry braking, caused by the anisotropy of the radiation field (e.g., Stenflo & Nagendra 1996; Trujillo Bueno 2001). The orientation of the emergent linear polarization depends on the scattering plane and thus on the location on the solar disk. Integrated observations of a spherical object at all angular positions would results in zero net linear polarization, since the symmetry braking is again removed to first order (for Hanle effect on stars, e.g., see Ignace et al. 1995; Nordsieck & Ig- nace 2005). Therefore, both effects are largely irrelevant for stars other than the Sun, unless they are not spherically symmetric due to rotational platten- ing. The mathematical derivation of the Zeeman resp. Pascheneffect follows Sakurai (1994) and Landi Degl’Innocenti & Landolfi (2004).

The Zeeman regime

Consider an atom with an electronic state of total angular momentum J = L+S with quantum numbers J, L, S . We assume Russell-Saunders coupling, where the orbital angular momentum vectors li of the individual electrons superpose to form L, and so do the spin vectors si to form S, instead of ﬁrst coupling to each other ( j j coupling). This state is (2J+1)-fold degenerate, i.e., there exist magnetic sublevels M = J,... J with the same energy. − The Hamiltonian of a n-electron atom in presence of a magnetic potential 1.2. Measuring stellar magnetic ﬁelds 11

A(r) in the Hartree-Fock approximation is given by

n 2 2 n pj 1 e e H = +V r j + pj A r j + A r j pj 2me  2 ri r j − 2mec · · Xj= 1   X i , j | − | X j = 1    H0  Hee HBL n 2 2 | e{zA r j } | {z } | {z } + (1.1) 2m c2 Xj=1 e where V(r) is the spherically symmetric potential field of the nucleus at position r. Hee describes the Coulomb interactions between the electrons. The term quadratic in A starts to play a role for strong fields (quadratic Zeeman effect at field strengths > 10 kG) and will be omitted for the rest of the calculation. If the magnetic field is uniform in the z-direction (B=Bzˆ), we have

1 A = (Byxˆ Bxyˆ) (1.2) −2 − Inserting into Eq. (1.1) yields for the fourth term (interaction of B with the orbital motions of the electrons): e HBL = BLz (1.3) −2mec with Lz = lz,1 + . . . +lz,n the orbital angular momentum operator in z-direction for all electrons. On the other hand, the electrons interact with the external magnetic ﬁeld due to their magnetic moment, giving rise to an additional term e e H = µ B = − s B = − BS . (1.4) BS − · m c j · m c z Xj e e Furthermore, the magnetic moment of the electrons interacts with the effective magnetic ﬁeld caused by their orbital motion. The Hamilton operator of the spin-orbit interaction is given by

H = ζ r l s . (1.5) SO j j · j Xj

In the case of Russell-Saunders coupling (Hee > HSO), the spin-orbit Hamil- tonian is labeled LS-term, leading to the ﬁne-structure. The total Hamiltonian then consists of the following terms: 12 Chapter 1. Introduction

2 n pj H0 = j=1 2m +V r j e P 1 e2 Hee = , 2 i j ri r j | − | HLS = ζP(L S) e B · HB = − | | (Lz + 2S z) (1.6) 2mec

(the factor 2 before S z arises from the electron spin gyromagnetic g-ratio). According to Condon & Shortley (1935), ζ is the spin-orbit coupling constant with the dimensions of energy. Introducing γ = ~ωL/ζ, with ωL the Larmor frequency, if γ 1, S precesses much faster around J than around B. In other ≪ words, HB is a small perturbation to H0+HLS . This is the Zeeman regime. The ﬁrst order energy shifts are then given by H . h Bi To compute the energy shifts H in the Zeeman regime, we note that h Bi

2 Jz, H0 +HLS = 0 and J , H0 +HLS = 0, (1.7) h i since H0+HLS is spherically symmetric. So the total angular momentum J and its projection M along an arbitrary z-axis are good quantum numbers, and we denote the eigenstates by α, JM , where α summarizes the electronic con- | i ﬁguration quantum numbers. To compute the expectation value of Jz and S z, we have to recall an important identity for tensor operators, a simpliﬁcation of the Wigner-Eckart theorem, known as the projection theorem. It applies in the case where the tensor operator does not change J, J~ being the maximum eigenvalue of Jz, or, in other terms, the operator does not change the total angular momentum:

α′, JM J V α, JM α′, JM′ V α, JM = h | · | i JM′ J JM . (1.8) h | q | i ~2 J(J+1) h | q | i

Applying Jz to the projection theorem immediately gives, of course, M~. For S we ﬁndh i h zi 1.2. Measuring stellar magnetic ﬁelds 13

S J S = h · i J h zi ~2 j( j+1) h zi J2 L2 + S2 = − J D 2~2 j( j+1) E h zi ~2 [J(J+1) L(L+1) + S (S +1)] = − J 2~2 J(J+1) h zi 1 S (S +1) L(L+1) = + − J (1.9) "2 2J(J+1) # h zi

The Zeeman energy shifts are ﬁnally given by

∆EB = gM~ ωL, (1.10) where M = J, J+1,..., J, thus they are equally spaced between J and J, proportional− to the− Land´efactor g of the electronic state. − With these results the dipole transition frequency between a lower and upper energy level are

ω = ω + ω (g M g M ) , (1.11) l,u 0 L u u − l l where ω0 is the frequency of the unperturbed transition. From the Wigner- Eckart theorem we ﬁnd the M- and J-selection rules (conditions for which the overlap integral of the initial and the ﬁnal state is non-zero and angular momentum conservation):

∆J = 0, 1, J = 0 9 0 and ∆M = 0, 1 (1.12) ± ± Transitions with ∆M = q = 0 are called π transitions, those with ∆M = M M = q= 1 are called σ resp. σ transitions. For J =0 1, there is one u− l − ± b r → transition for each σb, π, σr (normal Zeeman triplet case). In general, both the lower and the upper level split into (J+1) sublevels, and multiple transitions with a given ∆M are possible, giving rise to different Zeeman splitting patterns (anomalous ZE). Assuming no atomic coherences and complete redistribution of the populations among the Zeeman sublevels, the normalized transition strengths S q (Ml, Mu) are listed in Table 1.2.2. For a precise description and classification see Landi Degl’Innocenti & Landolfi (2004). 14 Chapter 1. Introduction

Transition π σb (M M 1) σr (M M+1) → − → J J M2 1 (J+M)(J+1 M) 1 (J M)(J+1+M) → 4 − 4 − J J 1 J2 M2 1 (J+M)(J 1+M) 1 (J M)(J 1 M) → − − 4 − 4 − − − J J+1 (J+1)2 M2 1 (J+1 M)(J+2 M) 1 (J+1+M)(J+2+M) → − 4 − − 4 Tab. 1.1: Relative strengths of transverse Zeeman components

The two extremes for the orientation of the magnetic vector with respect to the observer are called transverse, reps. longitudinal Zeeman eﬀect. In the former, the ﬁeld B is perpendicular to the line of sight, and both the π and σ components are linearly polarized, and no circular polarization is observed (Stokes V = 0). In the case where B is parallel to the line of sight, the σ components are contrariwise circularly polarized, while the π component becomes invisible (Stokes Q, U = 0). For the implications on the Stokes parameters in the general case, see Sects. 1.3.2, 1.3.1

The Paschen-Back regime The case in which γ 1 is called the complete Paschen-Back regime. The internal energy due to≫ the magnetic ﬁeld is much larger the LS-coupling (HB HLS ), and S precesses around B almost independently of the pre- cession≫ of J around B. We are left with cylindrical symmetry, and the good quantum numbers are mL and mS, the projections of L resp. S along the magnetic ﬁeld. HB is diagonal in the representation of such a basis, and the eigenvalues (energy shifts due to B) are given by

α (l, s) , m , m H α (l, s) , m , m = ω (m + 2m ) (1.13) h L S | B| L Si L L S In contrast to Eq. (1.10), there is only a level shift proportional to B, but no level splitting (both mL and mS have a well-defined value), i.e., we are left in the case of the normal Zeeman effect. In the intermediate case (incomplete Paschen-Back regime), where γ 1, ∼ i.e., HB is comparable to HLS , the energy eigenvectors gradually evolve from α (L, S ) , M, J to α (L, S ) , M , M . Analytical solutions exist for doublet | i | L Si terms only (S=1/2). In all other cases, the eigenvalues of HLS + HB can be found only by numerical methods. 1.2. Measuring stellar magnetic fields 15

1.2.3 Polarization: the Stokes formalism For the description of polarized light, usually a formalism introduced by G.G. Stokes in 1852 is applied. The advantage of the so called Stokes vectors lies in the separation of light into linearly and circularly polarized parts on one hand, and in allowing its direct measurement on the other hand. For the derivation, let us start with the Maxwell equations, by which all phenomena of electro- dynamics can be described2.

1 ∂B B = 0 E + = 0 ∇ · ∇× c ∂t ρ 1 ∂E j E = B = (1.14) ∇ · ǫ0 ∇× − c ∂t c ǫ0 In vacuum (ρ=0, j=0), we obtain directly the wave equations

E = 0 B = 0, (1.15)

= ∆ 1 ∂2 where c2 ∂t2 stands for the D’Alambert operator. If we reduce to one dimension, this− is u(z, t)=0. For a spherical-symmetric ansatz

1 u(z, t) = f (r, t) , r we get f (r, t) = g(r ct) + h(r+ct) , − where g and f are arbitrary continuously diﬀerentiable functions of argument (r ct). Since the wave equations (1.14) require f (0, t) = 0, the general solution− is 1 u(z, t) = g(r ct) g(r+ct) , r − − describing a spherical wave including time inversion. If we request periodicity and call λ the spatial period after which the function g has the same value,

g(ct r λ) = g(ct r) , (1.16) − − − 2 In fact, for a complete description of electrodynamic phenomena, we have to involve the Lorentz force, which arises form the Lorentz transformation of the Coulomb force, i.e. the relativistic aspect has to be incorporated. Note that Maxwell’s equations are already invariant under Lorentz transformation. 16 Chapter 1. Introduction we obtain for the electric ﬁeld E for a monochromatic plain wave in z-direction: E = E exp [ik (z ct)] (1.17) 0 · − From the periodicity condition (1.16) follows for the constant k (called wave number): 2π k λ = 2π k = (1.18) · ⇒ λ Therewith, we write for the angular frequency ω: 2πc k ct = t = ωt, (1.19) · λ which allows to rewrite Eq. (1.17) as: E = E exp [ikr iωt] . (1.20) 0 · − In general, E0 is a complex number. Going back to only the z-direction, where equation (1.20) describes a plain wave, this means that E0 = x, y, 0 , which follows from nE E o ∂2E 1 ∂2E ∂ = , E =0 Ez = 0 = const. ∂z2 c2 ∂t2 ∇· ⇒ ∂z ⇒ Ez By choosing adequate boundary conditions, we then achieve = 0. If both Ez components of E0 are real, i.e. E0 = xex + yey, E is called linearly polarized, meaning that both componentsE of theE wave oscillate in phase. For iπ/2 x = y (= E0 ) , but x = y e± , i.e. phase-shifted by 90 ◦, the wave is called circularlyE E | polarized| ,E meaningE that E(0, t) circulates with oscillation frequency ω. In general, if x , y (= E0 ) , and the phase shift is not exactly 90 ◦, we iφ E E | | have x = y e with an arbitrary angle φ, and we call the wave elliptically polarized.E E

The Stokes vectors are now given as:

2 2 I = x + y DE E E 2 2 Q = x y DE − E E U = 2 x y cos φ D E E E V = 2 x y sin φ D E E E 1.2. Measuring stellar magnetic ﬁelds 17

I is the sum of polarized and unpolarized parts of the light. Q and U represent the fraction of linearly polarized radiation orthogonal to each other, where the basis vectors for the definition of Q and U can be defined by the observer. V is the fraction of circularly polarized light. The brackets are supposed to indicate that in reality, we never deal with perfectly monochromatic light, so that we use time-averages in the definition (1.21) for the Stokes parameters I, Q, U and V. In general, a light beam consists of mutually uncorrelated photons with different polarization states, because the electronic transition processes by which they are created are stochastically independent of each other. This statistical ensemble will then be partially polarized.

On the contrary, a totally polarized plane wave would yield a constant relation of x and y, due to a ﬁx phase shift φ. Therefore, for totally polarized light appliesE E I2 = Q2 + U2 + V2. (1.21) In general though, for partially polarized light we have

I2 Q2 + U2 + V2. (1.22) ≥ The degree of polarization can ﬁnally be given by

Q2 + U2 + V2 P = . (1.23) r I2

1.2.4 Spectropolarimeters The usefulness of the Stokes formalism becomes apparent when the polarization state of light has to be actually measured. In the instrument’s frame, the angle θ is deﬁned to measure the angle in the plane perpendicular to the incoming light beam. We adopt a coordinate system in which Stokes Q and U measure the amounts of intensity linearly polarized along position angles θ = 90◦ and θ = 135◦, respectively, whereas positive Stokes V is the right- handed circularly polarized intensity. Clearly, a light beam with linear polarization along θ = 0◦ superposed to a beam which is polarized along θ = 90◦ will decrease the degree of polarization of the total beam. Therefore, to ﬁnd Stokes Q we measure linear polarization along θ = 90◦ minus the amount of polarization perpendicular to it. Denoting the intensity of light with linear 18 Chapter 1. Introduction

polarization along θ as Iθ, and right/left-handed circular polarization intensity = as Ir/l, the Stokes vector can be written as (note that Iθ Iθ+180◦ ):

I I  Q   I90 I0  I =   =  −  . (1.24)  U   I135 I45     −   V   Ir Il     −      A polarimeter therefore has to convert the polarization information into brightness modulations. A spectrometer using some disperser (prism, diﬀraction grating) then separates the incoming wave into a wavelength spectrum, measured by an electronic detector. The combination of the two instruments (polarimeter, spectrograph) is usually referred to as spectropolarimeter. Using the vector representation of polarized light, all transformations of a default Stokes vector Iin into an outcoming Iout within an optical system can be written as a linear system of equations using a 4 4 set of transfer coeﬃcients, the Mueller matrix: ×

Iout = MIin (1.25)

The simplest case of linear polarization conversion into brightness modulation is obtained using a linear polarizer (optical element transmitting only light which is linearly polarized along some angle ψ). Four subsequent measurements with the polarizer at angles ψ = 0◦, 45◦, 90◦, 135◦ yields Stokes Q and U, according to Eqs. (1.24). Instead of rotating the linear polarizer (or the spectrograph + detector, see Sect. 5.1 for examples), retarders are used to alter the polarization state of the incoming light wave (Fig. 1.1). A typical wave plate is a birefringent crystal. Light polarized along the ordinary axis of the material travels at a diﬀerent speed than light polarized perpendicular to it, causing a phase shift φ=δo δe of the two beams3. The Mueller matrix M of a retarder with its principal axis− (fast axis) at an angle θ and retardance φ is given by

3 The subscripts ”o”, resp. ”e” refer to the so-called ordinary and extraordinary axis of the retarder. If the refractive index of the ordinary axis no is larger than ne, as in calcite, the terms ”fast axis” and ”slow axis” apply in this order. 1.2. Measuring stellar magnetic ﬁelds 19

Pol. Modulator Analyzer Disperser Detector (Retarder) Disperser:

λ λ+dλ dθ Spatial direction Spatial

Fig. 1.1: A schematic of a spectropolarimeter. The polarization state of incident light from the left is modulated in the retarder section (rotatable wave plate or liquid- crystal variable retarder LCVR). An analyzer (polarizer) projects the beam onto a defined linear polarization state (+Q). The beam is spectrally dispersed (prism, diffrac- tion grating) and imaged onto the detector (CCD). A polarizing beamsplitter gives displaced, orthogonally polarized Q exit beams (dashed lines), which are located at ± different positions on the detector.

M(θ,φ) = R( θ) M(φ) R(θ) − 10 00 100 0 10 00 0 cos 2θ sin 2θ 0 010 0 0 cos 2θ sin 2θ 0 (1.26) =      −   0 sin 2θ cos 2θ 0   0 0 cos φ sin φ   0 sin 2θ cos 2θ 0   −   −     00 01   0 0 sin φ cos φ   00 01              where the rotation matrix R is applied twice, once to rotate to the system in which M(φ) is the matrix representation of the retarder, and then back into the instrument’s coordinate system. The polarizer (oriented at ψ = 90◦ with the Mueller matrix Mpol) is then assigned the task of an analyzer, in the sense that it determines how much of which polarization states have been transformed into +Q (in the case of θ = 45◦ and φ = 90◦, right handed circular polarization +V has been fully mapped onto +Q). The output intensity Iout of the polarimeter is then 20 Chapter 1. Introduction

1000 10 0 0 Iin 2 2 1  0100   0 C2 +S 2 cos φ S 2C2(1 cos φ) S 2 sin φ   Qin  Iout =  − −  2  0000   0 S C (1 cos φ) S 2 +C2 cos φ C sin φ   U     2 2 2 2 2   in   0000   −   V     0 S 2 sin φ C2 sin φ cos φ   in     −    1  2  2    = 2 Iin + C2 +S 2 cos φ Qin + S 2C2(1 cos φ)Uin S 2 sin φ Vin h − − i T = [o1, o2, o3, o4][Iin, Qin, Uin, Vin] (1.27)

where C2 =cos 2θ and S 2 =sin 2θ. Thus, in general the measured intensity is a linear combination of the incident Stokes vector, further referred to as Stokes component. A minimum of four measurements i=1,..., 4 is needed with sets of angles θ,φ such that the modulation matrix O = (oi j) is regular. The input 1 T Stokes vector can then be recovered by applying D = O− to [I1, I2, I3, I4]out. However, to map any pure polarization state to Stokes +Q (defined by the orientation of the analyzer), the retarder section in Fig. 1.1 consists of two elements. The modulation scheme that allows to retrieve the input Stokes parameters by simple difference of two exposures (normal sequence) then consists of six measurements defined by Eq. (5.16) in Sect. 5.3.1.

Dual-Beam-Exchange Polarimetry

If a polarizing beamsplitter is used as an analyzer instead of a linear polarizer, it is possible to measure two complementary linear polarization components simultaneously. In addition, the photon efficiency is increased by a factor of 2, since the light that was blocked by the linear polarizer is not lost anymore. On the other hand, the optical paths for the two beams are different, and they are mapped on different areas of the detector. This causes differential instrumental effects, which can be accounted for by calibrating the two channels relative to each other (flat-field and zero point). In order to minimize aberra- tion of the dual-beam setup, it is necessary to measure each Stokes component twice by exchanging the two complementary polarized beams. The use of a polarizing beamsplitter in combination with the standard modulation scheme is the common way to achieve this (discussed in Sect. 5.3.4). 1.2. Measuring stellar magnetic fields 21

Sensitivity-tunable demodulation using liquid-crystal variable retarders

All optical components in the polarimeter are sources of systematic errors in the demodulation process of the Stokes parameters. So far, we used idealized representations of all functional elements of our polarimeter. Most important, the retarders were assumed to be achromatic (wavelength independent). Many astronomical polarimeters (for references see Chapter 5) use rotating retarders of birefringent material, called achromats. In Chapter 5 we deal with the High-resolution Visible and Infrared Spectrograph (HiVIS) at the 3.67 m Advanced Electro-Optical System (AEOS) telescope located on mount Haleakala, Maui. It was recently upgraded with a λ/4 (quarter) wave plate in addition to the existing half-wave and Savart plate. However, these wave plates are not fully achromatic, which greatly affects the modulation matrix O = (oi j), i = 1,..., n, j = 1, 2, 3, 4, where the oi j’s are given in Eq. (1.27) and n the number of measurements. The characterization and retardance-fitting of two wave plates will be described in Sect. 5.2.1 to investigate the effects on the demodulation. Liquid-crystal variable retarders (LCVRs) exhibit both certain advantages and drawbacks compared to achromats. Using wave plates, the retardance variation is achieved by moving (rotating) them in the optical path. This causes a systematic error during or between exposures. In addition, the process is fairly slow (in the order of seconds). LCVRs allow for fast-switching (milliseconds) polarimetric modulation without moving optics, enabling to remove time-dependent systematic effects (e.g., telescope guiding errors). As a trade-off, the retardance of liquid crystals varies with wavelength and temperature. While this is negligible when observing in a small spectral window, typical for solar applications (TIP & LPSP, Mart´ınez Pillet et al. 1999), the effect in broad-band observations is immense, rendering the demodulation efficiency highly chromatic. By calibrating the two LCVRs in the laboratory we are able to perform a retardance fitting demodulation, which works sufficiently well. However, a Stokes-based deprojection determining the Mueller matrix of the optical system by measuring known input states incorporates all effect of the optical path. Applying a set of voltages to the LCVRs, corresponding to the normal sequence at one single wavelength, it is possible to recover the input states in all wavelengths. We trace the efficiency of the inversion process with the condition number of the modulation matrix. Plain calibration of the system will allow to choose a set of voltages which optimize 22 Chapter 1. Introduction sensitivity to all (‘efficiency balance’) or one specific Stokes parameter in a desired wavelength region, offering an unknown form of versatility.

1.3 Atomic spectropolarimetry

1.3.1 Polarized radiative transfer Radiative transfer (RT) describes the propagation of light through matter. We will in the following only consider 1-dimensional radiative transfer, i.e., the propagation of light along a path s. Considerable contributions to the theory of RT in a magnetic field were made by Unno (1956), Rachkovsky (1962b), Beckers (1969a,b), Landi Degl’Innocenti (1983), and Stenflo (1994). We will restrict ourselves to absorption effects (in the presence of a magnetic field) as the fundamental issue of polarization spectrum formation. Our aim is to provide an insight to the role of spectrally local deviations κL from the continuum absorption coefficient κc of the material, giving rise to spectral lines, i.e., distinct signatures in the Stokes parameters. A unpolarized light beam of intensity Iλ, passing along the path element ds can be subject to absorption by interaction with the matter, where the absorption is proportional to Iλ:

dI = κ I ds (1.28) λ − λ λ At the same time, there is also emissivity ǫλ, increasing the intensity:

dIλ = ǫλds (1.29)

We introduce the monochromatic optical depth τc,λ (s) of a medium as the optical path length along the viewing direction, measured against the direction of propagation of the photons:

dτ (s) = κ (s) ds (1.30) c,λ − λ Dividing Eqs. (1.28), (1.29) by κλds, we obtain in the limit ds 0 the radiative transfer equation in the plane parallel case: →

dIλ ǫλ = Iλ = Iλ S λ, (1.31) dτc,λ − κλ − 1.3. Atomic spectropolarimetry 23

where S λ is called the source function. On its way through the medium (opposite τc-direction), Iλ will increase if S λ > Iλ, or it will decrease if S λ < Iλ. T In polarized radiative transfer, we use the 4-vector I = (Iλ, Qλ, Uλ, Vλ) instead of Iλ, and Eq. (1.31) becomes a matrix equation dI = (ML + Mc) I ( jL + jc) (1.32) dτc,λ − where M is the Mueller absorption matrix, both for the continuum and for local (in the sense of wavelength) variations of the absorption coefficient caused by spectral lines. Similarly, jL is the line emission vector, and jc the emission contribution from the continuum. M has a symmetric component responsible for absorption effects, whereas the antisymmetric part of M contains the anomalous dispersion terms. The latter originates from the real part of the refractive index of the material, and is responsible for the transformation of polarization states during propagation. Therefore, the effect is also called Faraday rotation. Explicitly

ηI ηQ ηU ηV  ηQ ηI 2ρV 2ρU  ML + Mc =  −  + E, (1.33)  ηU 2ρV ηI 2ρQ   −   ηV 2ρU 2ρQ ηI   −  where E the 4 4 identity matrix, and  ×

ηI S  ηQ   0  jL + jc = S   +   (1.34)  ηU   0       ηV   0      where we disregard continuum polarization  by letting the continuum contribute only to the intensity spectrum. The components of the line absorption matrix ML are given by:

1 2 1 1 2 ηI = φ0 sin γ 1 sin γ (φ+ + φ ) , 2 2 2 − 1 1 − − 2 ηQ = φ0 (φ+ + φ ) sin γcos 2χ, 2 − 2 − (1.35) 1 h 1 i 2 ηU = 2 φ0 2 (φ+ + φ ) sin γ sin 2χ, 1 − − ηV = (hφ+ φ ) cos γ, i 2 − − with 24 Chapter 1. Introduction

2Ju+1 φ0, (υ) = κ0/κc S q(Ml, Mu) Z(υ qυn) , q = 1. (1.36) ± − ± Xn=1 Here Z is the proﬁle of the line absorption coeﬃcient with values between 0 and 1, and

υ = c (λ λ ) /λ and υ = λ ∆E / (2π~) (1.37) − 0 0 n 0 n are the distance from the undisturbed transition wavelength and the individual shifts of the Zeeman subtransitions in velocity space (c is the speed of light in [km/s]). The corresponding expressions for ρQ,U,V are obtained by replacing φ0, in ± Eq. (1.36) by ρ0, , and Z by Zρ, which would then be the shape of the line ± dispersion. The absorption coeﬃcient in the line center κ0 denotes the scaling factor for Z (and Zρ):

κ = κ Z(υ qυ ) (1.38) L 0 − n Finally, the orientation of the magnetic field vector B is defined by two angles γ (inclination) and χ (azimuth): γ measures the angle between B and the line of sight (z-direction), while χ is the angle between the x-axis and the projection of B onto the x-y-plane. Polarized radiative transfer according to Eq. (1.32) neglects coherent scattering or atomic polarization (imbalance between magnetic sublevels). The special case for local thermal equilibrium (LTE) can be obtained by setting S = Bλ(T), the Planck function. The important thing to note is that the emitted Stokes vector depends only on the profiles Z, Zρ, the magnetic field B, the line-to-continuum opacity κ0/κc, and the source function S .

1.3.2 The Unno-Rachkovsky solution For an arbitrary medium all quantities in Eq. (1.32) are depth dependent. For- mal solutions (cf. Landi Degl’Innocenti & Landi Degl’Innocenti 1985; Miha- las 1978) require a model atmosphere listing all necessary parameters as functions of optical depth (or log τ, where the variations are generally smoother). While applicable for a single line, or a very limited number of lines profiles, these sophisticated methods are very computer intensive and the simulation 1.3. Atomic spectropolarimetry 25 of thousands of spectral lines in a stellar atmosphere is not feasible. Fur- thermore, if measuring an integrated physical quantity is presumably difficult (e.g., due to small statistics), inferring the distribution of the quantity seems impossible. The idealization of a Milne-Eddington (ME) atmosphere for the photospheric layer is a widely used simplification, in both stellar and solar physics. The basic assumption is that the line absorption matrix ML is independent of depth. This implies that the magnetic field and the shapes of the absorption- dispersion profiles are constant in the line formation layers. With the aim to further combine the information about these quantities from multiple spectral lines, the former assumptions seem acceptable. In the ME approximation, the line formation region is fully characterized by the condition that the source function has a linear τ dependence. If we introduce the optical depth scale as measured parallel to the normal vector of the local plane-parallel atmosphere instead along the line of sight, and if θ denotes the angle between the two directions, we replace τc by τ/µ, where µ=cos θ. For the source function we define:

B(τ) = B0 + B1τ. (1.39) Introducing the parameter β as the limb-darkening factor,

µB /B β = 1 0 , (1.40) 1 + µB1/B0 we write the analytic solution of the polarized radiative transfer Eq. (1.32) for the emergent (τ=0) Stokes vector from an atmospheric element with a normal inclined by θ to the line of sight (Rachkovsky 1962a,b):

2 2 2 2 (I/Ic)λ = 1 β + β (1 + ηI) (1 + ηI) + ρQ + ρU + ρV /∆, − h i (Q/I ) = β[(1 + η )2 η + (1 + η ) (η ρ η ρ ) c λ − I Q I V U − U V +ρQ ηQρQ + ηUρU + ηV ρV ]/∆, (1.41) (U/I ) = β[(1 + η )2 η + (1 + η ) η ρ η ρ c λ − I U I Q V − V Q +ρU ηQρQ + ηU ρU + ηV ρV ]/∆, (V/I ) = β[(1 + η )2 η + ρ η ρ + η ρ + η ρ ]/∆, c λ − I V V Q Q U U V V 26 Chapter 1. Introduction

where

∆= (1 + η )2 (1 + η )2 η2 + η2 + η2 + ρ2 + ρ2 + ρ2 I I − Q U V Q U V (1.42) η ρ +h η ρ + η ρ 2 , i − Q Q U U V V the solution known as the Unno-Rachkovsky solution.

Disk-integrated limb darkening In night-time astronomy, the stellar disks cannot be resolved, and the observation is recorded at all possible angles 0<θ<π/2 simultaneously. Therefore, the limb-darkening factor from Eq. (1.40) is not deﬁned. We have to inte- grate over the whole visible disk to get an average parameter β¯. Deﬁning β0 = B1/B0,

1 1 µβ0 β¯ = dµ = ln(1 + β0) ln(1 + µβ0) dµ (1.43) Z0 1 + µβ0 − Z0 through integrating by parts. With the substitution x=1+µβ0 we obtain

1 1+β0 ln(1 + µβ0) dµ = [x lnx x]1 /β0 = ln(1 + β0) (1 + 1/β0) 1, (1.44) Z0 · − − and finally β¯ = 1 ln(1 + β ) /β . (1.45) − 0 0 Equations (1.41), (1.42), in combination with Eq. (1.45) provide the foundation on which we base our further consideration on the formation of polarization spectra, given the input profile of the local deviations of the continuum absorption coefficient, κL/κc, which we describe in the next section.

1.3.3 The line absorption profile Line broadening is one of the main ingredients for the formation of Stokes profiles. Here we shall review the processes responsible for the shape of the line opacity profile for an atomic dipole transition. We will first consider radiation damping, giving rise to the so-called natural width of the line. The derivation is inspired by quantum mechanical time evolution operators and 1.3. Atomic spectropolarimetry 27 perturbation theory. The extensive carrying out is due to the author’s amaze- ment at this fundamental effect, in combination with Heisenberg’s uncertainty principle. Since the natural total half width of a spectral line in stellar astronomy is by about 4 orders of magnitude smaller than the most prominent line broadening mechanisms, we will in a second step take into account important additional effects, in order to point out the wealth of observable quantities which are imprinted in a spectral line profile, and represents both the physical conditions and the abundances of chemical elements in the stellar atmosphere. This thesis is devoted to the analysis of spectral line profiles and extracting physical information from them. Therefore, our aim is to understand and to determine these quantities, and to strengthen the justification of recovering the line profile from spectra.

Natural broadening In the following we will disclose the effects responsible for a finite lifetime of the atomic levels set by their decay via the radiation process itself, according to Sakurai (1994). We consider a total Hamiltonian H of an atomic system such that it can be split into a part H0, which is time-independent (note that this definition of H0 does not correspond to the definition in Eq. 1.6), and a time-dependent part V(t), which is called a perturbation of the system:

H = H0 + V(t) (1.46) We assume that we know the energy eigenstates n and the energy eigenvalues | i En of the unperturbed problem H0 n = En n . Since n represents a complete set of orthonormal eigenstates, the| keti 4 state| i α of the| i system in an arbitrary state at time t can be written as a lines combination| i of the basis states:

iH0t/~ α, t = c (t) e− n (1.47) | i n | i Xn All state kets were so far assumed to be in the Schrodinger¨ picture, where time evolution acts solely on the state kets and is determined by the total Hamiltonian. With Eq. (1.47) in the current form, we have introduced the

4 The term ket goes back to Dirac. The state ket is postulated to contain complete information about the physical state. 28 Chapter 1. Introduction interaction picture, where the evolution of the state ket is determined by the perturbation alone, through the time-dependent probability amplitudes cn(t) (cf. Eq. 1.55). Therefore, the state ket in the interaction picture (subscript I) is given by

α, t = eiH0t/~ α, t , (1.48) | iI | iS while for operators, the observables are given by

iH0t/~ iH0t/~ VI = e Ve− . (1.49)

On the other hand, we deﬁne a time evolution operator UI (t) from time t0 to t in the interaction picture

α, t = U (t) α, t , (1.50) | iI I | 0iI for which holds the Schrodinger-like¨ equation d i~ U (t) = V (t) U (t) (1.51) dt I I I From Eq. (1.51) can be seen that UI (t) was ﬁxed in time if VI was absent. With the initial condition UI (t0) = 1, an approximate solution to Eq. (1.51) called the Dyson series can be obtained by iteration:

i t i 2 t t′ UI (t) = 1 dt′VI t′ + − dt′ dt′′VI t′ VI t′′ + .... (1.52) − ~ Z ~ Z Z t0 t0 t0 If the system is the eigenstate i at t = t , we deﬁne a perturbation expan- | i 0 sion for the amplitude cn(t):

n U (t) i = n i, t = c (t) = c(0)(t) + c(1)(t) + c(2)(t) + . . . (1.53) h | I | i h | iI n n n n Comparing the right-hand side of this equation to terms with corresponding orders in VI (t), we ﬁnd (using Eq. 1.49):

(0) cn (t) = δni t t (1) i i iωnit′ cn (t) = −~ n VI (t′) i dt′ = −~ e Vni (t′) (1.54) t0 h | | i t0 2 t t′ (2) iR iωRnmt′ iωmit′′ cn (t) = − m dt′ dt′′e Vnm (t′) e Vmi (t′′) ~ t0 t0 P R R 1.3. Atomic spectropolarimetry 29

where we have used that ωni =(En Ei/~). The probability P(i n) of finding i after time t t in state n for n−, i is → | i − 0 | i P(i n) = c(1)(t) + c(2)(t) + . . . 2. (1.55) → | n n | On the other hand, what happens to ci(t) itself (the probability of finding i, t = i )? Assume that the perturbation was turned on infinitely slowly from |zeroi in| thei remote past (t ), reaching a constant value V at t = 0 (to avoid the effect of a sudden→ change −∞ in the Hamiltonian). Setting V(t) = eηtV, we may let η 0 at the end of the calculation to converge to perturbation that (was) is constant→ for all times t. Inserting V(t) into Eqs. (1.49) and (1.54) yields (t ) 0 → −∞

i i 2 e2ηt i V 2e2ηt c (t) = 1 V eηt + − V 2 + | mi| (1.56) i − η~ ii ~ | ii| 2η2 ~ 2η (E E + i~η) Xm,i i − m up to second order. Letting now η 0 we ﬁnd → c˙ (t) i i V 2 i i − V + − | mi| = − ∆ . (1.57) c (t) ≃ ~ ii ~ E E + i~η ~ i i Xm,i i − m The last equality was made because Eq. (1.57) is time-independent. There- i∆it/~ fore, the ansatz ci(t) = e− is justiﬁed. The time evolution of i is given by | i

i∆it/~ i∆it/~ iEit/~ i, t = c (t) i = e− i = e− e− i , (1.58) | iI i | iI | iI | iS i.e., due to the perturbation V, we observe a shift of the eigenenergies Ei (1→) Ei +∆i. According to the right-hand side of Eq. (1.53), we expand ∆i = ∆i + (2) (1) ∆i +. . . and compare with Eq. (1.57) to get to ﬁrst order ∆i =Vii = i V i , as expected from time-independent perturbation theory. h | | i 2 To interpret ∆i we need the residual theorem to ﬁnd 1 1 limǫ 0 = iπδ(x) , (1.59) → x + iǫ x − and eventually 30 Chapter 1. Introduction

 2  i  Vmi  π 2 c (t) = exp[ −  (V + | |  t V δ(E E ) t] (1.60) i ~  ii E E  − ~ | mi| i − m  Xm,i i m  Xm,i  ∆(1) −   i   (2)  (2) |{z} Re∆  Im∆ /~=:Γi  i  − i  | {z } | {z } The probability of ﬁnding state i at time t is (Eq. 1.55) | i

2 Γit/2 c (t) = e− , (1.61) | i | where Γi characterizes the rate at which state i decays. The key thing to note is that the ﬁnite lifetime of state i is solely due| i to the imaginary part of the | i energy shift ∆i, which only appears in second order perturbation theory. From Fourier inversion, we ﬁnd for the decay power spectrum (energy delivered per unit time, in contrast to time momentary energy spectrum) Γ 1 I(ω) = i (1.62) 2π (ω ω )2 +Γ2/4 − i i Note that 2/Γ = τ is the mean lifetime of state i . We can identify ~Γ as the i i | i i uncertainty in energy ∆E, and τi as ∆t to get

~Γ τ = ∆E ∆t ~ (1.63) i i ≃ the time-energy uncertainty relation. Since Eq. (1.62), which was obtained by perturbing the Hamiltonian of the system, is equivalent to the solution of a damped oscillator with damping constant γ in classical electromagnetic theory, which occurs even for a completely isolated system, we conclude that the damping is in fact due to an additional energy term. The Hamiltonian H0 in Eq. (1.46), although including all interactions of the system, was still not complete. There is obviously a coupling to a hidden energy reservoir.

Additional line broadening While the natural broadening described in the previous section is determined by the (atomic) system alone, the dominant broadening effects depend on the interaction of the system with its surroundings (collisional or pressure broadening), and the multitude of reference frames of the absorbing particles 1.3. Atomic spectropolarimetry 31 with respect to the observer’s, each causing a Doppler shift of the center of line. The term Doppler broadening refers to the effect caused by the velocity dispersion in a gas, while rotational broadening comes from the simultaneous observation of stellar surface elements with different apparent velocities.

Doppler broadening. For purely thermal motions of particles, their velocity distribution is given by a Maxwellian, describing the probability of ﬁnding a particle with a line of sight velocity in the interval (ξ, ξ+dξ)

n(ξ) 1 ξ2/ξ2 dξ = e− 0 dξ, (1.64) N ξ0 √π i.e., a Gaussian distribution with the variance ξ0 = √2kT/ma, where ma is 23 the mass of the atom involved in the transition, k = 1.38 10− J/K, and the Doppler width ·

∆λD = λ0ξ0/c, (1.65) with λ0 being the line central wavelength in the atom’s rest frame. It is common to deﬁne

υ = (λ λ )∆λ , − 0 D y = ξ/ξ0, (1.66) Γλ2 a = 0 , 4πc∆λD given that other broadening effects are Lorentzian (as in the case of natural broadening), with Γ its full width at half maximum. It is important to note that the lower (final) state of a transition is in general also an excited state of the atom, characterized by a frequency width Γ f . Therefore, we exchange the damping width in Eq. (1.62) by Γ=Γi +Γ f . If we understand that the total absorption coefficient is given by

2 ∞ πe κλdλ = f, (1.67) Z0 mac where f is the oscillator strength (proportional to the square of the dipole transition matrix element), the absorption coeﬃcient per atom at wavelength λ is given by the convolution integral 32 Chapter 1. Introduction

∞ n(ξ) κλ = κ(λ ξλ/c) dξ, (1.68) Z0 − N which is usually written as

λ2 √πe2 f κ = ( , υ) λ 2 H a (1.69) mac ∆λD with

y2 a ∞ e H(a, υ) = − dy (1.70) π Z (v y) + a2 −∞ − being the Voigt function. For stellar applications a 0.1. For a = 0.01 the change between Gaussian and Lorentzian character lines≤ near v=2.7 and has 3 H(0.01, 2.7) 10− (Unsoeld 1968). To account≈ for local, small-scale inhomogeneities, it is possible to introduce the so-called microturbulence and macroturbulence parameters. The former is usually applied to account for eﬀects in the line formation region causing additional Gaussian broadening via

2 ξ0 = 2kT/ma + ξmicro, (1.71) q whereas macroturbulence causes Gaussian broadening of the Stokes parameter S directly:

1 2 2 ξ /ξmacro S λ = [S λ]old e− (1.72) ξmacro √π (In the following, S always denotes one element of the Stokes vector).

Collision broadening. In the classical impact theory, a single perturber comes by at large speed and causes a momentary disruption of the wave train of the relaxating atom. Explicitly, suppose that in the time period T between two perturbations, the atom emits a wavetrain

g(t) = eiω0t. (1.73)

Although we know from Sect. 1.3.3 that the spectral distribution of this radiation should be Lorentzian, we assume for the moment a monochromatic wave, hence the use of ω0. The energy spectrum is then given by 1.3. Atomic spectropolarimetry 33

T T 1 1 cos ((ω ω0) T) = i(ω0 ω)t i(ω0 ω)t = E (ω, T) e − dt e− − dt − − 2 . 2π Z0 ! Z0 ! π (ω ω0) − (1.74) However, collisions do not occur periodically. Rather, we have to average over all collision times T. If we denote the mean time between two collisions by τc (the subscript c stands for collisions), and assume the probability for the interval time T to decrease exponentially, W(t) dT = exp ( T/τ) dT/τ, the mean energy spectrum is −

∞ Γ 1 E (ω) = E (ω, T) W(t) dT = . (1.75) 2 2 Z0 2π (ω ω ) +Γ /4 − 0 As for the natural linewidth Γnat, the impact collisional broadening is again Lorentzian, with a width Γcol =1/τcol. With a Lorentzian in Eq. (1.73) instead of a monochromatic wave, the resulting energy spectrum is a convolution of two Lorentz profiles, which is again a Lorentzian with full width Γtot = Γnat +Γcol. In contrast to the impact theory, the quasi-static approximation applies in cases where the perturber passes slowly enough to disturb the term structure of the de-exciting atom by an external electric field. This is typically true for ions (in contrast to e−, which are better described by the impact theory due to their low mass). In the nearest-neighbor approximation, only the strongest electric field component (usually from the nearest perturber) is taken into account. Following Weisskopf (1932) the interaction between atom and perturber is classified by

p ∆ω = Cp/r , (1.76) with Cp the interaction constant (determined by calculation or measurement), and r the distance at the moment of closest encounter (for the impact theory), or the distance to the next ion (quasi-static nearest-neighbor approximation). The power index n defines the type of the interaction. For the linear Stark effect (p = 2), the resulting broadening profile has a 5/2 Holtzmark shape (dip in the line center and ∆ω− in the far wings). It applies to hydrogen(-ic) atoms, whose high spatial∼ sensitivity to perturbing electric fields is due to their permanent dipole moment. In the special case of the Balmer lines, assuming that Stark broadening from interactions with 34 Chapter 1. Introduction electrons dominates the line shape (hot atmospheres), the Inglis-Teller estimate gives a relation between the electron density Ne and the highest upper Balmer Balmer level nmax for which the transition n = 2 nmax is still separable from Balmer → transitions with n > nmax (Inglis & Teller 1939):

log N = 23.2 7.5 log nBalmer. (1.77) e − max Resonance broadening (p=3) is mainly important for collisions of hydrogen atoms with one another, and is a noticeable broadening effect of Balmer lines in solar-type atmospheres, where the temperature is high enough to have hydrogen in the n = 2 state, but still electrons take over the role as major perturbers. For non-hydrogenic atoms and ions (systems without dipole moments), collisional broadening arises from the quadratic Stark effect (p=4) in hot atmospheres, where e− and ions are largely abundant, and from Van der Waals interactions (p = 6) with neutral hydrogen in cool stellar atmospheres. The short scale lengths of both interaction types validate the impact approximation, resulting in Lorentzian profile shapes.

Rotational broadening A macroscopic broadening of spectral lines in the observable Stokes parameters is caused by the rotation of the star. In the following we always choose the coordinate system such that z-axis coincides with the LOS and that the stellar rotation axis is inclined by an angle i to the y-axis, lying in the y-z-plane. If φ,ϑ measure the angles between the LOS and the x-axis, resp. y-axis, the velocity component vz in the direction of the observer is given by

vz = vr sin i sin φ sin ϑ, (1.78) where vr is the equatorial rotational velocity of the star. On the stellar surface, vz is constant for all point for which sin φ sin ϑ=const. In the given coordinate system, this holds for x = const, i.e., for stripes parallel to the y-axis across the stellar disk, if we neglect diﬀerential rotation. If we further assume that the emergent Stokes proﬁles do not vary across the stellar disk, the observable Stokes parameters S obs may be written as a convolution

∞ S obs = S (λ ∆λ) G(∆λ) d∆λ = S (λ) G(λ) . (1.79) Z − ∗ −∞ 1.4. Multi-line techniques 35 where G(λ) has the shape of a half ellipse. Gray (1992) also gives an explicit expression for G(λ) for limb-darkening of the form β = 1 ǫ+ǫ cos θ. − Rotational broadening does obviously equally affect all lines in the spectrum. For non-hydrogenic metal lines present in cool and solar-like atmospheres, essentially for transition metals with atomic number 22 Z 29, the absorption coefficient broadening effects are also self-similar (especially≤ ≤ if the differences in mean Doppler velocity of different species are blurred by a global microscopic turbulence parameter). Multi-line techniques, discussed in the next section, take advantage of the fact that many spectral lines are affected by broadening effects in a similar way.

1.4 Multi-line techniques

To study temperature inhomogeneities on the stellar surface (see also Sect. 1.5) one requires a precise shape of absorption lines, whereas for the detection of very weak magnetic fields, high sensitivity towards polarization signals is needed. In general, the signal-to-noise (SNR) ratio in single line profiles is not sufficient to deduce the desired amount of information. Multi-line techniques (MLT) encounter this problem by assuming that many lines in the spectrum contain the same information. Combining them will eventually increase the SNR.

1.4.1 State-of-the-art methods

Stating that different lines contain the same information means that they have something in common which is measurable. Answering the question what exactly is assumed to be similar among different spectral lines will be the basic idea behind any MLT. So far, there has been a generally accepted view, even though not very much questioned, that Stokes profiles of different lines should be combined. MLTs that are based on this principle are the line addition (LA) technique, least squares deconvolution (LSD), and principal component analysis (PCA). Each of them is presented in the following. In Sect. 1.4.2, we introduce two new methods developed in this thesis: nonlinear deconvolution with deblending (NDD, Chapter 2) and Zeeman component decomposition (ZCD, Chapter 3). 36 Chapter 1. Introduction

Line adding The first attempt to find a common Stokes profile from circular polarization spectra was based on adding a number of selected spectral lines (Semel 1989; Semel & Li 1996). Each local line profile with line central wavelength λi can be transformed to the Doppler coordinate υ, with ∆υ = c (λ λi) /λi, where LA − c is the speed of light in km/s. The weight wi for each line is given by LA wi =ωi geff,i, where ωi is the equivalent width, and geff,i the effective Lande´ factor. Note· that for other Stokes parameters than Stokes V, other weights would have to be used. The mean Stokes V profile, calculated independently for each Doppler coordinate υ j of the velocity grid, is then given by (Semel et al. 2009):

LA i wi V υ j Υi ZLA x = − , (1.80) j P wLA i i P where Υi is the Doppler coordinate of the center of line i and υ j is the Doppler coordinate axis. Using the residual intensity RI = 1 I/Ic, Eq. (1.80) can be applied equally to Stokes I, Q, U or V. Although− it is often stated that no assumptions are made to compute ZLA, this is not true. If no correlation between different datasets is assumed, there would be no reason to combine them to find a mean signature in the first place. These correlations constitute the set of adopted assumptions.

Least squares deconvolution The LSD technique was introduced for the detection of Zeeman signatures in stellar polarized spectral by Donati et al. (1997). Their considerations are based on the fact that in the weak ﬁeld approximation (WFA), i.e., when the Zeeman splitting in units of the Doppler width υH 1, we ﬁnd for the polarized Stokes parameters (χ=0): ≪

V υ cos γ ∂I/∂υ (1.81) ∝− H Q = U υ2 sin2γ ∂2I/∂2υ (1.82) ∝− H From Eq. (1.10) we recall that υ g ﬀλ. In combination with the assumption H ∝ e that the shape of the local Stokes proﬁle S (υi) is similar for all spectral lines (weak line approximation WFA), for each Stokes parameter S = RI, Q, U and 1.4. Multi-line techniques 37

V, we have in particular that RI (υi) simply scales with the local line central 2 2 depth parameter dc,i, and ∂RI/∂υ = dc,i k1, as well as ∂ RI/∂ υ = dc,i k2, where k1, k2 are just proportionality functions (equal for all lines). The weighting factors for Q, U and V depending on the properties of the local line i can be extracted as:

LS D LS D LS D 2 2 wV,i = geﬀ,i λi dc,i and wQ,i = wU,i = geﬀ,i λi dc,i. (1.83) With the use of a line mask M = wLS Dδ υ Υ (1.84) S S,n j − i Xi the spectrum for a given Stokes parameter S can be written in terms of a convolution expression S = M ZLS D. Including the diagonal error matrix W, S ∗ S with Wi,i = 1/σ υ j Υi being the inverse standard deviation of the expected noise at each velocity − point, the least squares solution to Eq. (1.84) is:

MT W2 S ZLS D = S · · (1.85) S MT W2 M S · · S LS D LA In comparison with the LA method, we note that wi and wi for Stokes V are in fact equal under the given assumptions. For weak lines, the equivalent width of the line is directly proportional to the wavelength and the number of absorbers present, ωi λi dc,i. This is known as the linear part of the curve of growth (Unsoeld 1968;∝ Mihalas 1978).

Principal component analysis In stellar astronomy, the term PCA refers to the method of reconstructing noise-reduced individual line profiles using a multi-line principal component analysis. Introduced by Semel et al. (2006) and Mart´ınez Gonzales´ et al. (2008), the basic idea is that each Stokes profile can be represented by a truncated basis of orthogonal eigenprofiles obtained from a dataset of local profiles. This is assumed to be true due to the fact that the imprints of local variations on the stellar surface are similar in all spectral lines. We denote the local profiles for the Stokes parameter S as xi = S υ j Υi , each consisting of m velocity points (the cardinal numeral of j is equal − to m for each line i=1,..., n). With x¯ the n-element vector containing the mean x¯i of each data vector x , the m m covariance matrix C is given by i × 38 Chapter 1. Introduction

C = (x x¯) (x x¯)T (1.86) i − i − Xi

The set of eigenvalues λl and eigenvectors sl of the matrix C are deﬁned by Csl = λl sl. If we demand the eigenvalues to be in descending order, λ1 . . . λ , we write each local Stokes proﬁle as ≤ ≤ m m

xi = αi,l sl, (1.87) Xl=1 where αi,l = xi sl is the projection (scalar product) of the i-th data vector onto the l-th eigenvector of C. This expansion states that s1 contains the most coherent features in the dataset, whereas the information carried by sl is in- creasingly incoherent with increasing l. Therefore, the denoising takes place by truncating the number of eigenvectors used in the representation of xi in Eq. (1.87) to m′ principal components (m′ < m). If m′ is equal to 1, each Stokes profile is represented by the first principal component, which yields basically the same result as LSD. The idea of PCA is to overcome the limitations of identical lineshapes, WFA and WLA, by letting m′ > 1, arguing that subsequent eigenvectors contain information inherent to specific spectral lines. Thus the aim of PCA is not to find a common line pattern, but to reconstruct individual line profiles with less statistical noise.

1.4.2 Methods developed in this thesis: a new benchmark There is a problem common to all MLTs described in the previous section, namely the issue of blends. Individual line profiles are not completely independent of each other, and their information content can get mixed up: a feature of a line i blended with another line appears at different Doppler velocities as seen from each line. PCA would identify such a distortion as an incoherent feature. The reasoning of LA and LSD that the contributions from blends are “arbitrarily” situated and randomized (Semel et al. 2009) is incon- clusive for a limited number of lines, or for blends that are systematically ordered, such as in a molecular band. The first new method of nonlinear deconvolution with deblending (NDD), described in detail in Chapter 2 of this thesis, deals with the issue of blends. It allows a pixel at Doppler coordinate υ j to be affected by any number of lines i with υ Υ below a certain limit, usually a few times the full width at half j − i

1.4. Multi-line techniques 39 maximum of a single absorption line profile. In Sect. 2.2 we describe how Eq. (1.85) can be improved into a true deconvolving approach for blended local profiles, but with the restriction of linearly added line profiles. However, when disentangling the contributions from individual spectral lines, the assumption of linear line summation is clearly wrong except for a very limited number of weak lines. The NDD uses a heuristical approach, based on the so-called interpolation formula, first introduced by Minnaert (1935) for the calculation of equivalent widths. In Sect. 2.3 we derive an algorithm that accounts for the nonlinearity in blended profiles, while considering the individual saturation depths in different regimes. Using inverse interpolation formulas described in Sect. 2.6, the quality of the inferred common intensity pattern can be increased. We also take them into account when forming analytical expressions for the Jaboci and Hessian matrices used in the nonlinear solving algorithms (Gauss-Newton or Levenberg-Marquardt). An issue inherent to all MLTs, including NDD, is that they compare Stokes profiles directly. In fact, even if all spectral lines well separated in wavelength, their profiles would not be similar (e.g., Kurucz 1993). Five simulated profiles (using a ME atmosphere) of the same multiplet transition, but with different relative strengths are shown with dashed lines in the upper three panels and the lower left and middle panel of Fig. 1.2. Dividing each by their central depths, we obtain the solid-line profiles. In the lower right panel of Fig. 1.2, these ‘normalized’ profiles are plotted on top of each other. Their shapes are obviously not similar. In fact, even in the most simple case of line formation theory, Stokes line profiles are only similar for weak magnetic fields and equal line depths. In Chapter 3 we approach the problem from a different perspective. We argue that it is not the observable Stokes profiles, but a single underlying line- to-continuum opacity profile that is similar among different spectral lines. In general, this profile is assumed to be a Voigt-function. By allowing it to take an arbitrary shape, we account for both turbulence in the line forming region, and inhomogeneities across the stellar surface. The use of weak and stronger lines at the same time, which form at different heights in the atmosphere, requires that κL/κc is independent of depth in the line formation region. This is the characteristic of a Milne-Eddington atmosphere, which in turn allows us to use the Unno-Rachkovsky solution to produce different Stokes parameters. Using all the benefits elaborated in Chapter 2 concerning blends, the Zee- man component decomposition (ZCD) is the first inversion method using ra- 40 Chapter 1. Introduction

Fig. 1.2: Line profiles with different relative strengths (dashed curves), increasing from left to right in top and lower panels. The solid curves represent scaled profiles with central depth equal to unity. Lower right panel: The assembly of all rescaled line profiles.

diative transfer (ME-model) to simultaneously fit thousands of spectral lines in all Stokes parameters. The assumption that one arbitrary κL/κc profile is responsible for the formation of all Stokes parameters bears the possibility to extract an average magnetic field vector (provided all Stokes parameters are available). Therefore, ZCD is currently averaging polarized Zeeman signatures that are caused by complicated field structures. In Sect. 3.5 we will derive algorithms for the interactions of the three Zeeman components σ , π0 that are independent of the WFA, as well as the WLA. Another powe±rful feature of the ZCD is that it treats the line strength proportionality factors β¯ η0 (cf. Eq. 1.45) as free parameters for each line, so that it does not rely on· pre-calculated line depths. Moreover, the wealth of relative line strengths enables the ZCD to extract the temperature stratification and element abundances, which will be explored in more detail in the future. 1.5. Zeeman Doppler Imaging 41

1.5 Zeeman Doppler Imaging

All diagnostics discussed so far are restrained to the disk integrated light from the star. Local structures, such as temperature and magnetic field inhomogeneities (e.g., starspots) are averaged over the whole visible disk, since direct surface resolving observations are not common for stars other than the Sun. However, in the case when broadening of line profiles is dominated by stellar rotation (the projected rotational velocity v sin i is larger than the local Doppler width), a variety of local Stokes profiles emerging from inhomogeneities in the line formation region will be imprinted in the observable line profile. A possibility to map the surface structure from rotationally broadened profiles was first formulated by Deutsch (1958). The name Doppler Imaging (DI) was introduced by Vogt & Penrod (1983), but further developed into Zeeman Doppler Imaging (ZDI) by utilizing polarization profiles to recover magnetic field maps by Semel (1989). The resulting image reconstruction algorithms, first implemented by Brown et al. (1991) and others by now, rely on the precise knowledge of spectral line profiles to recover temperature, abundance, or magnetic field variations. This is the future field of application for the methods described in this thesis. Line profiles have to be combined to obtain a SNR sufficient for the analysis, but systematic errors in the applied multi-line technique causing noticeable artifacts in the recovery of the free parameters are to be avoided. For instance, assume a magnetically active star, with a spot that has a lower temperature than the rest of the photosphere and a radial magnetic field. Depending on the rotational phase, linear polarization (the spot is near the stellar limb, transverse Zeeman effect) or circular polarization signals (the spot is near the disk center, longitudinal Zeeman effect) can arise. In addition, the temperature difference causes a change of the local continuum level, relative to the mean continuum of the visible disk, and a different local line- to-continuum opacity itself. As a result, we observe traveling distortions in the normalized Stokes profiles as the star rotates. These distortions will be visible at specific Doppler shifts in the line profile, corresponding to the apparent LOS velocity vz of the spot. In other words, there is a one-to-one correlation between the wavelength position of the distortion in the line profile and positions on the stellar surface which correspond to that apparent velocity. Positions within an interval (vz, vz +dvz) appear as vz-stripes parallel to the y-axis. 42 Chapter 1. Introduction

From a single snapshot, it is therefore possible to assign a local profile distortion to a vz-stripe on the stellar disk. However, the permitted positions in y-direction increase with cos vz (a distortion in the line center can correspond 5 to a spot anywhere on the rotational axis, since vz = 0 everywhere along it) . Therefore, reliable reconstructions of 2-dimensional stellar surface maps are based on time resolved observations with full phase coverage. Considering what has been said, the requirements for observations include high SNR, and high spectral resolution at the same time. While a good sample of magnetic Ap/Bp stars can be analyzed using individual line profiles due to sufficient apparent stellar brightness, modeling of typically fainter late-type stars mostly relies on multi-line techniques. They became the overwhelming approach to precise stellar magnetic field diagnosis. The present work shall contribute to further development of ZDI by providing more realistic common line profiles, overcoming the weak field approximation and accounting blended profiles by introducing realistic physics into the multi-line technique analysis.

1.6 Outline of the thesis

The conscientious recording and analysis of polarized spectra are the key ingredients towards a better understanding of stellar activity. In this introduc- tory Chapter we reviewed some of the achievements and open questions of stellar magnetism, and introduced diagnostic tools which enable us to tackle them. The extraction of information from spectral line profiles is of main interest. It is important to distinguish and understand the different effects that contribute to the formation of spectral lines. Only then can we venture to combine multiple lines to increase the SNR in search of specific parameters. Oversimplified assumptions may lead to delusive results. The previous discussion should have disclosed the strengths and weaknesses of existing methods. In the remaining part of this thesis we develop sophisticated techniques in an attempt to extract a line shape that holds specific properties of the stellar atmosphere from a multitude of spectral atomic and molecular lines. In Chapter 2 we will first concentrate on the deconvolution and deblending of

5 Theoretically, limb-darkening effects may help to further constrain the position near the limb. 1.6. Outline of the thesis 43 intensity spectra. As pointed out in Sect. 1.5, special features in the line profile can be attributed to surface peculiarities at a radial velocity on the rotating star. In a blended line profile, such distortions appear at different velocities for different individual lines. Quite general, we want to know how much one spectral line contributes to a given position in a blended line profile. Instead of treating the spectrum as a convolution of dirac functions located at the central wavelengths with a common line profile, it is crucial to disentangle the contributions of separate lines. We will show that with our approach it is possible to find the single common line profile from a limited number of heavily blended lines, such as in a molecular band. Chapter 3 provides an advanced algorithm, taking into account that the observable Stokes profiles of weak and stronger (not saturated!) lines are intrinsically different. With the use of the analytic (Unno-Rachkovsky) solutions of polarized radiative transfer, we are able to infer the magnetic field strength (and orientation, if Stokes Q and U are available) on the assumption of a common line-to-continuum opacity profile and a uniform field. Fitting thousands of spectral lines in a spectrum, we abandon the dependence on pre-calculated line depths. We have established a numerical code that dynam- ically assesses the utility of each line to obtain a best fit, allowing us not to restrict ourselves to lines with central depths above a certain threshold inherent to other methods. ZCD enables us to detect longitudinal fields beyond the limit of state-of-the-art techniques. The detection of a weak magnetic field presented in Chapter 4 is the first in a series of studies we plan to carry out using ZCD in the future. In the course of this thesis, nighttime observations at several telescopes around the world (CFHT, Hawaii; TNG, Canary Islands; TBL, France) were carried out. The author experienced how much influence the performance of an instrument has on the outcoming data. The ongoing improvement of instrumental techniques, especially in polarimetry, is vital to the diagnostic progress. In Chapter 5 we document the upgrade of the HiVIS instrument, installed at the AEOS telescope on mount Haleakala, Maui, to a full Stokes polarimeter using LCVRs. These non-moving retarders allow for fast-switching polarization modulation and tunable sensitivity towards different Stokes parameters in the desired spectral region, using a Stokes-based demodulation technique of the whole instrument. 44 Chapter 1. Introduction Bibliography

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Nonlinear deconvolution with deblending: A new analyzing technique for spectroscopy†

C. Sennhauser1, S. V. Berdyugina2, D. M. Fluri1

Abstract

Spectroscopy data in general often deals with an entanglement of spectral line properties, especially in the case of blended line profiles, independently of how high the quality of the data may be. In stellar spectroscopy and spectropolarimetry, where atomic transition parameters are usually known, the use of multi-line techniques to increase the signal-to-noise ratio of observations has become common practice. These methods extract an average line profile by means of either least squares deconvolution (LSD) or principle component analysis (PCA). However, only a few methods account for the blending of line profiles, and when they do, they assume that line profiles add linearly. We abandon the simplification of linear line-adding for Stokes I and present a

† This chapter is published in Astronomy & Astrophysics 507, 1711 (2009) 1 Institute for Astronomy, ETH Zurich, 8093 Zurich, Switzerland 2 Kiepenheuer Institut fur¨ Sonnenphysik, 79104 Freiburg, Germany 54 Chapter 2. NDD: A new analyzing technique for spectroscopy novel approach that accounts for the nonlinearity in blended profiles, also il- luminating the process of a reasonable deconvolution of a spectrum. Only the combination of those two enables us to treat spectral line variables independently, constituting our method of nonlinear deconvolution with deblending (NDD). The improved interpretation of a common line profile achieved com- pensates for the additional expense in calculation time, especially when it comes to the application to (Zeeman) doppler imaging (ZDI). By examining how absorption lines of different depths blend with each other and describing the effects of line-adding in a mathematically simple, yet physically meaningful way, we discover how it is possible to express a total line depth in terms of a (nonlinear) combination of contributing individual components. Thus, we disentangle blended line profiles and underlying parameters in a truthful manner and strongly increase the reliability of the common line patterns retrieved. By comparing different versions of LSD with our NDD technique applied to simulated atomic and molecular intensity spectra, we are able to illustrate the improvements provided by our method to the interpretation of the recovered mean line profiles. As a consequence, it is possible for the first time to retrieve an intrinsic line pattern from a molecular band, offering the opportunity to fully include them in a NDD-based ZDI. However, we also show that strong line broadening deters the existence of a unique solution for heavily blended lines such as in molecular bandheads.

2.1 Introduction

A multi-line approach for increasing the signal-to-noise (S/N) ratio of measured line profiles was first introduced by Semel (1989) and Semel & Li (1996). It was developed into the LSD method by Donati et al. (1997). The aim of these methods is to determine the common Stokes profiles of polarized spectra when assuming the weak-field regime. It is assumed that the line profiles are the same for all lines in the spectrum except for a factor, which is given by the properties of the transition i f , i.e., wavelength, effective Lande´ factor, and oscillator strength,| andi → by| thei stellar atmosphere in which the lines form, i.e., shape and depth of the (intensity) line. These mean Stokes profiles are called common line patterns, or Zeeman patterns Z(v) of a spectrum, which are presented in velocity space. The spectrum is then a correlation of this common line pattern with a linemask, which defines the 2.1. Introduction 55 line position (wavelength) and scaling in intensity. Another approach is that of principal component analysis (PCA), where denoising is achieved by di- agonalizing a cross-product matrix of individual spectral lines to reconstruct the data with a truncated basis of eigenvectors (e.g., Mart´ınez Gonzales´ et al. 2008). Methods such as LSD and PCA can be assumed to be filtering methods, which extract common line patterns of Stokes I, Q, U, and V in more detail, thus enabling a more precise description of the stellar atmosphere in which the lines formed, including the temperature and both the strength and orientation of the magnetic fields. For the task of Zeeman doppler imaging (ZDI), that reconstructs the brightness and magnetic field distribution on a stellar surface from input multiple exposures at different rotational phases, such a multi- line technique seems to be an appropriate tool for providing highly resolved information about the star merged in one single line profile. To retrieve this common line pattern, it is insufficient to represent a spectrum as an accumulation of separate spectral lines, but it is also necessary to deconvolve blended line profiles, i.e., to account for the contributions of closely neighboring lines (in all Stokes parameters). So far, LSD methods assume that contributions from different lines add up linearly for all Stokes parameters, i.e., that the spectrum is a true convolution in a mathematical sense. We show that this is only valid for weak absorption lines, and we present a self-consistent way of analyzing arbitrary line depths. The main goal was to find a formalism that describes how line depths ‘add’ in a physical sense, which we found in terms of the interpolation formula by Minnaert (1935), introduced for quite a different purpose. In this paper, we consider intensity spectra only, while a consistent procedure for polarized spectra will be developed in a forthcoming paper. The paper is structured as follows. In Sect. 2.2, we consider the basics of multi-line techniques, introducing the principles of deblending. We then compare optically thin and thick cases in Sect. 2.3, to develop a general formalism from which we derive our nonlinear deconvolution with deblending (NDD) technique. The numerical implementation of our new method is given in Sect. 2.4, which is supplemented by Appendices A,B, and C, where we provide details about inverse interpolation, solving algorithms, and defining the Ja- cobi matrix explicitly. A comparison of linear and nonlinear deconvolution, applied to simulated atomic and molecular lines is presented in Sect. 2.5. Finally, in Sect. 2.6, we summarize our conclusions. 56 Chapter 2. NDD: A new analyzing technique for spectroscopy

2.2 Weak line decorrelation and deblending

We first consider how to extract a proportionality function (Zeeman pattern) that is constant for all lines in a given intensity spectrum. It is obvious that if we assume similar line profiles, i.e. no saturation effects, each line profile is given by the characteristic Zeeman pattern, which we choose to have a central line depth of 1, “scaled” by the central line depth dc,i of the current line i. For the surrounding local line profile Iloc, however, it is more appropriate to consider (1 I (v)) = d (1 Z (v)) , (2.1) − loc c,i − I rather than Iloc(v) = dc,i ZI(v). The line depths dc,i of each line constitute the linemask matrix MI (cf. Sect. 2.2) with elements mi,I

M (v) = m δ (v v ) , (2.2) I I,i − i Xi where vi and mI,i are respectively the position in velocity space and weight of each spectral line. To be able to write Eq. (2.1) in terms of a linemask, we complete the transformations

I˜ = 1 I and Z˜ = 1 Z . (2.3) − I − I In the weak line approximation, a spectrum can be written in terms of a correlation expression of the so-called line pattern function, or linemask M, with the sought-after common line pattern Z,

I = M Z . (2.4) I ∗ I As proposed by Donati et al. (1997), this is equivalent to a (linear) system of equations I=M Z . If we know the error σ of each spectral pixel j, we can weight each I· I j of those equations with the inverse error 1/σ j by multiplying both sides with a square diagonal matrix S, where S j j = 1/σ j. The least squares solution is then given by MT S2 I Z = I · · . (2.5) I MT S2 M I · · I If the pixel errors are unknown, S can be chosen to be the identity matrix and therefore omitted. In our discussion, we do not consider the error matrix, but 2.2. Weak line decorrelation and deblending 57 do remember that if provided, it is important to include it during all further steps. It should be emphasized here that for intensity spectra, the transformations given by Eq. (2.3) must be made prior to applying Eq. (2.5), because for Stokes I the normalized continuum equals 1, instead of zero as for V, Q, and U. It is common to interpret Eq. (2.5) in terms of evaluating the common line pattern only for one single bin v′ of a given velocity grid at a time, i.e., by transforming local line proﬁles Iloc,i for each line i = 1, ..., n into velocity space with their line centers as origin and deﬁning a system of n equations

I1 v′ = m1 Z(v′) . .

In v′ = mn Z(v′) (2.6) In this approach, M and I are one-dimensional arrays, and inserting them into Eq. (2.5) yields the least squares solution of one element of ZI at a given velocity grid point. Equations (2.6) assume that there are no blended lines or more precisely that there is only a contribution to Z from one line at a time. The eﬀect of blends is assumed to be averaged statistically. However, in general a measurement at wavelength λ′ corresponding to velocity v′, theoretically is aﬀected by all lines in the spectrum. An explicit equation for I at v′ is then

I v′ = m1 Z v1′ + m2 Z v2′ + m3 Z v3′ + ..., (2.7) where vi′ is the velocity distance measured from the line center λ0,i. In practice, most of the contributions are of course negligible. Only terms for which vi′ lies within the grid of Z are accounted for. Thus, if we assume the ﬁrst line in our linelist to be blended with the second one, the equation would have to look like

I v′ = m1 Z v1′ + m2 Z v2′ . (2.8)

The equation for the next pixel at λ′′ ( v′′) is then →

I v′′ = m1 Z v1′′ + m2 Z v2′′ . (2.9) For some additional pixel atλ˜, the contribution from a third line could start to play a role, whereas the ﬁrst line is already too far away, yielding

I(v˜) = m2 Z(v˜2) + m3 Z(v˜3) . (2.10) 58 Chapter 2. NDD: A new analyzing technique for spectroscopy

This approach not only deconvolves the spectrum for separate lines, but does so it in an analytical way for blended line proﬁles. The only reason why the previous method yields reasonable results is due to stressed statistics, while here we solve Eq. (2.5) simultaneously for all components of Z. Hence, M is a matrix of dimensions p times z, where p is the number of pixels in the spectrum (p can be on the order of 105), and z the number of components for the common line pattern Z. However, if two spectral lines are separated by more than two times the velocity limit of Z, there will be no equations in the intermediate interval, so the number of equations can be smaller than the number of pixels. The (overdetermined) system of equations then resem- bles the following (remembering that the indices of m denote the lines in the linemask): I1 m1 [0] . .  .   [0]T ..   .    Z1  .   m2 [0] m1   .  =   . (2.11)    T .. ..   .   .   [0] . .  ·    .     Zz     m [0] m [0] m       3 2 1     ·   . .. ..     Ip   . . .      The diagonal alignment  of the weighting factors is clearly evident, since they indicate the pixel at which a speciﬁc line begins to contribute to the Z-velocity grid, and how it migrates through the bins as we go through our measurement points. We note that the alignment is diagonal strictly only if the distance between two measurement points precisely equals the binsize of our velocity grid, which is not always the case. If the binsize is larger, some weighting factors are allocated one below the other, whereas for too small a binsize, they can be shifted by more than one column.

2.3 Optically thin versus optically thick lines

In Eq. (2.7), we assume that each datapoint is a linear combination of diﬀerent contributions from our sought-after common line pattern. According to the Eddington-Barbier relation in LTE for the optically thin case, we write for the line depth (Boehm-Vitense 1989)

2 κl d ln Bλ Rλ = , (2.12) 3 κc dτc τc=2/3

2.3. Optically thin versus optically thick lines 59

where Bλ (T) is the Planck function, κc the continuous absorption coefficient, and κl the line absorption coefficient. Equation (2.12) mainly expresses that for weak lines, the line depth at every point within the line increases in pro- portion to κl/κc, or proportionally to the number of atoms Ns absorbing in the line. For a blended line profile Rλ consisting of two lines R1,λ and R2,λ, we may then write

Rλ = R1,λ + R2,λ. (2.13)

Fig. 2.1: Changes in the line proﬁle with increasing κl/κc in the case of optically thin lines (κl κc, upper panel) and optically thick lines (κl/κc 1, lower panel). ≪ ∼ Thus, it seems obvious that the assumption about linearly added line pro- ﬁles only holds for weak lines. For optically thick lines, where κl/κc 1 or is greater than 1, Eq. (2.12) is no longer a valid approximation, since∼ it can easily produce total line depths > 1. In general, we search for a line-summing rule for Rtot,λ that meets the following criteria:

i) Rtot (R1, R2) = Rtot (R2, R1) ;

ii) R (R , R ) = R +R , for R , R 1; tot 1 2 1 2 1 2 ≪

iii) Rtot (1, R2) = 1, for any R2; 60 Chapter 2. NDD: A new analyzing technique for spectroscopy

iv) Rtot (R1, 0) = R1;

v) Rtot (R1, Rtot (R2, R3)) = Rtot (Rtot (R1, R2) , R3) . A simple function exists that satisfies the above conditions, i.e., that (i) is symmetric in its arguments; (ii) adds line depths linearly for optically thin lines; (iii) if any of its arguments approaches 1, it does so, too; and according to (iv) and (v) can handle an arbitrary number of lines, even if that number is 1. This function is as follows: R (R , R ) = R +R R R . (2.14) tot 1 2 1 2 − 1 · 2 Although this formula was derived occurred in a totally heuristic manner, it is obvious that it is able to produce far more reliable results than a simple linear approach, especially when Ri ≮ 0.15 (Unsoeld 1968), because of its second term, which accounts for the effects of line saturation and makes it impossible to reach line depths greater than 1. Nevertheless, saturation levels in general vary from line to line, and are <1. For optically very thick lines for which κl/κc at the line center goes to infinity, the residual is approximated by the source function Sλ(τc = 0), and the limiting central depth is given by Rλ0 in terms of Ic Sλ(0) Rλ0 = − , (2.15) Ic as illustrated in Fig. 1. In the case of NLTE, Sλ(0) may approach zero, for instance in resonance lines, and Rλ0 becomes 1, although this is an exceptional case. Attempts to modify Eq. (2.14) by means of tuning the function towards a more realistic saturation level <1 are destined to fail, since any parameter inserted in the second (nonlinear) term would prevent the function from meeting criterion (v). We instead need another formula that interpolates Rλ between Eq. (2.12), which is for both weak lines and the wings of strong lines, and Eq. (2.15), which is for the optically thick parts of the line. Introducing the quantity

2 κl(λ) d ln Bλ X(λ) = τc=2/3, (2.16) 3 κc dτc | the approximation of the line proﬁle is given by the interpolation formula according to Minnaert (1935): 1 R = . (2.17) λ 1 + 1 X(λ) Rλ0 2.3. Optically thin versus optically thick lines 61

For strong lines, where κl/κc is high, X(λ) is much larger than 1, and the term 1/X (λ) can be neglected, yielding R R . On the other hand, for optically λ ≈ λ0 thin lines, 1/X(λ) 1/Rλ0 and Rλ X(λ). We propose a new≫ application of≈ this formula, which has so far only been employed to calculate equivalent widths and curves of growth. Following our derivation given above, we state that the quantity X(λ) can be regarded as the sum of different terms that contribute to the absorption in the line, as an extreme case for instance, the sum over the absorbing atoms. It therefore seems reasonable that the combination of Xi(λ) for different spectral lines is given by the sum Xtot(λ) = i Xi(λ) of all the contributing quantities. For a blended line profile, we thenP write 1 R = . (2.18) λ,tot 1 + 1 i Xi(λ) Rλ0,tot P To find Xi(λ) as a function of Ri,λ, we simply rearrange Eq. (2.17) for X(λ), yielding 1 = Xi(λ) 1 1 . (2.19) R R i,λ − i,λ0 Combining Eqs. (2.18) and (2.19) provides a formula that enables us to calculate the line depth at a given wavelength for an arbitrary number of contributing lines, given their individual local line depths and their saturation levels. For a blended line profile, the saturation depth for each contributing line is obtained by interpolating the surfaces as shown in Fig. 2. The quantity Rλ0,tot in Eq. (2.18) represents the saturation depth of the blended line, and can be calculated similarly by setting Ri,λ = Ri,λ0 , and in turn Ri,λ0 = 1. By expe- rience, Rλ0,tot is always very close to one for a sufficiently large number of blended lines. To calculate line profiles, we must determine in practice the source function. Only in the case of LTE Sλ is known (namely Sλ = Bλ). For the Milne- Eddington Model of line transfer, Mihalas (1978) provides a formula for the saturation depth of a line that depends only on temperature and wavelength, and neglects scattering in both the continuum and line (assuming LTE in the line). In our case, we often deal with strong lines that do not form in LTE and scattering is present. This approach then seems unsatisfactory also when we take into account that level population depends on the excitation energy of the ground state for a given electronic transition. Therefore, to estimate 62 Chapter 2. NDD: A new analyzing technique for spectroscopy

Fig. 2.2: Saturation levels for Fe I lines with lower excitation energies of 4.6 eV calculated by the code STOPRO.

saturation depths as accurately as possible, we employed the code STOPRO (Solanki 1987; Frutiger et al. 2000; Berdyugina et al. 2003), solving the polarized radiative transfer equations for 33 diﬀerent elements and ions, 6 stellar atmospheric models (Kurucz 1993), 31 wavelengths covering the region from 4000 to 10 000 Å, and every possible lower excitation energy starting from 0.1 eV and increasing in 0.3 eV steps (the maximal lower excitation energy for an ion is of course lower than the next ionization energy level).

To combine all formulas together, we recall our empirical line-summing formula in Eq. (2.14), where we had to assume that all saturation depths equal one, and compare this with the interpolation formulas Eqs. (2.18) and (2.19). We expand Eq. (2.18) into a Taylor series for two lines [R1(λ) , R2(λ)] := R, 2.4. Numerical implementation 63

by setting R1,λ0 and R2,λ0 equal to 1 and obtain 1 R +R 2 R R R = = 1 2 − · 1 · 2 tot 1 + 1 R R 1 + 1 1 1 2 1/R 1 1/R 1 − · 1− 2− α D R (R) = = tot |[R 0] Rα = R +R R R + . . . (2.20) α! · 1 2 − 1 · 2 Xα | | We can immediately see that Eq. (2.14) represents the ﬁrst two non-zero terms of the Taylor expansion of the interpolation formula. Our estimation was therefore quite accurate, and can be helpful in practice, if saturation levels are unknown, and one beneﬁts from the simplicity of Eq. (2.14).

2.4 Numerical implementation

A common problem in spectral analysis is optimal binning. Spectral lines in the blue part of a spectrum are narrower than those at red wavelengths, since (non-transitional) broadening effects depend on wavelength. After transforming the wavelength scale into the velocity domain centered on the line wavelength, we can compare line profiles from different wavelength regions. How- ever, any predefined velocity grid (usually at equidistant points) will never match measured data points transformed into the velocity space of a given line. The easiest way is to simply assign the measured velocity to the closest gridpoint. In such a case, however, one disregards the very accurate wavelength information of a spectrum (accurate to a fraction of an Angstrom). Here we propose to employ inverse interpolation, allowing multiple gridpoints to incorporate information of one datapoint. We implemented quadratic inverse interpolation for NDD and the methods used for comparison, yielding a noticeable improvement in retrieved common line patterns. A detailed description is given in Appendix A: ‘Inverse interpolation’, where we also present a practical algorithm for inverse spline interpolation. Since Eqs. (2.14) and (2.18) are nonlinear functions, there is no matrix representation that would enable us to employ Eq. (2.5) to find a solution in Z. Thus, we have to apply iterative methods to approach a least squares solution in Z. We intentionally do not write the solution, because we show 64 Chapter 2. NDD: A new analyzing technique for spectroscopy later that, depending on how densely lines are blended with each other, multiple minima may appear in the topology of solutions. In Appendix B: ‘Solv- ing algorithms’, we present two algorithms implemented to find a minimum; the Gauss-Newton with Minimization (GNwM) method and the Levenberg- Marquardt (LM) method. Although the latter is in general preferable because it minimizes the χ2 merit function far more efficiently, the former should not be underestimated because of its simplicity. Another problem, which is atypical in the optimization of functions, is that we do not have a “model” function that depends on our z parameters, but rather it is the true form of our function depending on the number of parameters that is unknown. Therefore, the derivatives necessary for the Jacobi and Hessian matrices in the solving algorithms are also unknown, and need to be determined for each equation separately. Iterative evaluation of derivatives would cause our code to be extremely slow since we consider 104 equations and more, depending on as many as several hundred parameters. Therefore, we resolved this problem analytically, by determining all partial derivatives when we defined our system of nonlinear equations, depending on the number of contributing blends and the different gridpoints that they contribute to (see Appendix C: ‘Derivative of the extended interpolation formula’). We note that these two numbers are not necessarily equal, which is the case for only two line centers lying within the velocity bin, and causes additional problems in finding the derivatives.

2.4.1 Critical blends

As we have mentioned, the ability of a deconvolution method to identify a common line pattern in blended line profiles depends on the proximity of the line centers. For instance, if we have dozens of lines that are broadened significantly and separated by only fractions of an Angstrom, they would form a broad “valley” with many degenerate solutions for the intrinsic line profile. We approximate the Doppler broadening of a line by

λ 2 2 2 ∆λD = ξ + ξ + (v sin i) , (2.21) c · q th turb where ξth = 2RT/µ is the mean thermal velocity of an atom, ξturb accounts for the turbulentp motions in the stellar atmosphere, and v sin i represents the 2.4. Numerical implementation 65

Fig. 2.3: Performance of our NDD method in recovering common line patterns from a blended line proﬁle with line center separation ∆λc1,2 and eﬀective broadening ∆λD,instr.. stellar rotation. To allow for instrumental broadening, we adopt

2 2 2 ∆λD,instr = (∆λD) + (∆λinstr) . (2.22) For a separation of ∆λc1,2 between the line centers of two spectral lines, in- ﬂuenced by the above broadening eﬀects, we claim that the performance of a deconvolution code theoretically depends on the ratio

= ∆λ /∆λ . (2.23) Q1,2 D,instr c1,2 Figure 3 illustrates the ability of our deconvolving code to recover the original line proﬁles. We blend two identical line patterns, apply our code and compare the retrieved proﬁle to the original single input pattern. We plot 66 Chapter 2. NDD: A new analyzing technique for spectroscopy

2 the sum of the variance ( σi ) as a function of total line broadening in units of line separation (Eq. 2.23).P For a given 1,2, the ability to recover the original single line pattern is roughly constant, butQ errors then begin to increase rapidly if 1,2 1. We emphasize that this limit does not depend on the method, but thatQ the≥ absolute value of the error does! In practice, however, results depend on the distance of line centers. For a given ratio and spectral resolution Q1,2 R, fewer pixels carry information about the line, if ∆λc1,2 is small compared to the situation where ∆λc1,2 is large. That is, the (intensity) gradient in the pixel frame is far steeper, imposing diﬃculties for the applied LSD method in recovering details from the common line proﬁle and resulting in larger errors per pixel.

2.5 Results

2.5.1 Atomic profiles: Stokes I We simulated an intensity spectrum containing 7 blended spectral lines (Fig. 4), among them two iron ions, as well as vanadium and nickel, each with very different lower excitation energies (to test the reliability of the Minnaert approach and its dependence on individual saturation levels, see Eq. (2.18)), to probe the functionality of both the different approaches and the nonlinear solving methods. This does not represent a true spectrum since the real line centers were shifted, to produce a challenging blended profile. Therefore, the abscissa marks are omitted, both v sin i and instrumental broadening were assumed to have particular values, and we assumed a noise level of 0.5%. The ∼ line central depths dc,i range from 0.38 to 0.66, and they were ‘measured’ by simulating single line profiles. We emphasize that the deconcolving performance relies on precise knowledge of the parameters defining the weighting factors mi. For Stokes I, the line central depth is the only determining parameter, and, as can be seen in Eq. (2.1), if they are concordant with the (well-normalized!) spectrum, the recovered Zeeman pattern should have normalized wings and a central line depth dc,Z exactly equal to 1, regardless of the strength of the original line depths. Only such a common line pattern, scaled to individual line depths, and treated according to Eqs. (2.18) and (2.19) for blended profiles, will be able to reproduce the original spectrum. Therefore, the true central line depth of the retrieved Zeeman pattern is a strong indica- tion of the effectiveness of a multi-line method. 2.5. Results 67

Fig. 2.4: Simulated intensity spectrum of 7 blended lines. The dashed vertical lines mark the line centers; element names and ionization levels are labeled at the bottom.

We applied the conventional LSD (‘normal LSD’), given by Eq. (2.6), and the deblending, yet linear method, corresponding to Eq. (2.11). Both methods can be solved using Eq. (2.5). We compare them to our NDD method according to our realization of the Minnaert interpolation formula, applying both the Gauss-Newton with Minimization and the Levenberg-Marquardt method for the iterative solving of the nonlinear sets of equations (see Fig. 5). ‘Normal LSD’ (dash-dotted curve at the bottom), which is not a true deconvolution method, fails in this attempt, when all lines are heavily blended, resulting in strong wiggles in the wings, a continuum level dependent on the number of blended lines, if there is a noticeable continuum at all, and a central line depth significantly smaller than 1. The other linear, but deconvolution approach (dashed line) is far more successful, but the wings are (asymmetri- cally) broadened, and the line center does not reach 0. Both of these results infer that the linear approach for this spectrum is unsuccessful, because the absorbing region is entirely optically thick. When the Minnaert formula and inverse quadratic interpolation are applied, as presented in Sect. 2.4, the result is a noticeably smoother profile compared to a ‘nearest bin’ attempt. The nonlinear (NDD) method reproduces a fairly perfect profile within the line, showing a central line depth 1. The differences between the two solving numerical methods are marginal,≃ so the two profiles almost coincide. However, 68 Chapter 2. NDD: A new analyzing technique for spectroscopy in the far wings of the Zeeman pattern, which are arguably already in the continuum, the nonlinear solving causes errors, because of the handling of small numbers (if we recall that we operate with 1 I). This problem should be solved by applying boundary conditions to the− continuum part of the profile.

Fig. 2.5: Recovered intensity pattern from diﬀerent methods employed to the atomic spectrum given in Fig. 4.

2.5.2 Molecular profiles: Stokes I In molecular bands and especially in the band head, a large number of blends contribute to the measured line profile, and serious deconvolution and provi- sion for nonlinearity become mandatory. To illustrate the capabilities of our new method, we applied the method to a simulated molecular TiO γ(0,1) band at 7591 Å, as shown in Fig. 6. Our NDD technique recovers a common line pattern that almost coincides with the input Z-profile except for the right wing of the line. This is due to the band head, where the bulk of blends interfere with each other in a way that basically removes the individual profiles, pre- cluding a unique solution. As mentioned before, this problem increases for greater broadening and more significant blending. Nevertheless, our method 2.6. Conclusions 69 is capable of retrieving the necessary details of the line profile (e.g., width, see Fig. 7), which features a line central depth of virtually equal 1, and being easily able to identify possible irregularities within the entire profile. A comparison with the results for the linear attempt shows that the effects of nonlinearity become noticeable in terms of their conspicuously asymmetric wings and shallow central profile. Again, ‘normal LSD’ fails, because it lacks deconvolving ability.

Fig. 2.6: Simulated molecular TiO γ(0,1) band, including Doppler broadening and ∼ 1% noise level. The dashed-dotted vertical lines indicate the line centers. The hashed region of the band head is where many blends are packed together and i, j 1 Q ≫

2.6 Conclusions

Our nonlinear approach to deconvolving blended line profiles by applying the Minneart interpolation formula, combined with the advantages provided by inverse interpolation, represents a significant revision of least squares deconvolution techniques, mainly because the latter do not incorporate a deconvolution of blended lines. Given a linemask that includes all relevant atomic or molecular data, we are able to disentangle arbitrarily heavily blended spectra in a physically consistent manner, fully abandoning any preliminary assumptions about the intrinsic single line profile, apart from its existence. This 70 Chapter 2. NDD: A new analyzing technique for spectroscopy

Fig. 2.7: Recovered intensity pattern from different methods employed to study the molecular band given in Fig. 6. allows us to retrieve the true common line patterns even from a limited number of blended lines and will enable (Zeeman) Doppler Imaging to be applied even to noisy datasets of narrow spectral intervals. Our method is also able to properly disentangle molecular bands consisting of many tens of blended line profiles within a narrow spectral range. Heav- ily broadened molecular spectra do not offer, however, a unique solution for the recovery of an intrinsic line profile: results always show depressed right wings. For the time being, it is nonetheless the tool most capable of identify- ing weak features in molecular absorption lines. We place emphasis on the importance of a proper deconvolution of the spectrum to be analyzed, highlighting the need to disentangle blends. As shown in the derived common line patterns in Sect. 2.5, any non-deconvolving method imposes additional line broadening by neglecting the influence of close-by blends. In a forthcoming paper, we discuss whether ZDI codes based on these methods infer stellar spots at systematically higher latitudes, due to a technical artifact. The method described in the present paper offers a general and direct for- Appendix A: Inverse interpolation 71 malism concerning how spectral line parameters, as well as ambient and instrumental effects affect an observed intensity profile. It can be used to ad- dress a variety of problems in stellar spectrum analysis, and it is likely that it will find more general applications in many types of spectroscopy, where the quality of data is insufficient to apply direct fitting of spectra.

Appendix A: Inverse interpolation

Each datapoint in a spectrum has a distinct distance in wavelength to the center of line λ0,i for the spectral line i, which in turn corresponds to a distance in velocity space ∆vi. The grid for which we wish to retrieve our common line pattern Z consists of equidistant points, in other words, bins of constant binsize ∆bin. If we consider ∆bin 3 km/s and a pixel with ∆vi = 5.1 km/s, one is willing to argue that it corresponds≡ to the gridpoint Z(v=6 km/s), simply because it is the closest one. However, one could instead inversely interpolate, so as to express the datapoint in terms of multiple Z-gridpoints. Applying quadratic interpolation, we obtain

Z(v=5.1) = p Z(v=3) + p Z(v=6) + p Z(v=9) , (2.24) 0 · 1 · 2 · where the constants p0, p1, p2 can be calculated with the Lagrange-formula for interpolation:

(v v0) . . . (v vk 1) (v vk+1) . . . (v vn) pk(v) = − · · − − − · · − , (2.25) (vk v0) . . . (vk vk 1) (vk vk+1) . . . (vk vn) − · · − − − · · − yielding

Z(v=5.1) = 0.195 Z(v=3) + 0.91 Z(v=6) 0.105 Z(v=9) . (2.26) − Another method that was implemented was that of spline interpolation, since we expect our Zeeman pattern to be smooth and continuous. The diﬃ- culty lies in analytically solving the linear system of equations with its typical tridiagonal, strictly diagonally dominant (z 2) (z 2) matrix (again, assum- − × − ing that ∆bin =constant): 6 + + = + = = pi 1 4pi pi+1 2 (Zi+1 Zi 1) Ai, i 1,..., n 1, (2.27) − (∆bin) − − 72 Chapter 2. NDD: A new analyzing technique for spectroscopy

and p0 = pn =0, for n+1=z knots (gridpoints). The cubic polynomials deﬁning the spline are known to be given as

3 3 pi+1(v vi) +pi(vi+1 v) Zi+1 ∆bin S i(v) = − − + pi+1 (v vi) 6 ∆bin ∆bin − 6 − Zi ∆bin + pi (vi+1 v) . (2.28) ∆bin 6 − − After Gauss elimination, Eq. (2.27) can be written as

0 A1 4 1 1 0 1 − . p A2 A1  1 ..  1  − 4  0 4 4 1 .  4 1   −  . = A3 15 A2 4 A1  (2.29)  4 .   .   − −   0 4 ..     15 4 1   15     A4 A3 A2 A1   −. .   pn 1   −56 −15 −4   .. ..   −   .       .          To express the coeﬃcients pi after back-substitution of this linear system of equations, we introduce the recursive sequences

qi+1 = 4 qi qi 1, q0 =0, q1 =1 (2.30) − − qi ri+1 = Ai+1 ri, r1 = A1 (2.31) − qi+1

Using this, we can write for pi, i=2,..., n:

ri 1 pi = − pi 1 −qi 2 . (2.32) − 4 − qi 1 − − or, without the use of the ri’s:

i 1 i 1 k qi 1 pi + k−=1 ( 1) − − qk Ak pi 1 = − − − (2.33) − P qi 1 − We have presented the sought-after coeﬃcients in Eq. (2.28) in terms of the quantities Ai, where the Ai’s in turn are linearly dependent on Zi, as given in Eq. (2.27). Even if a datapoint is aﬀected by only one spectral line, it will then be represented as a linear combination of all elements of Z, where in the case of quadratic interpolation, it will always be represented by three of them. Appendix B: Solving algorithms 73

Appendix B: Solving algorithms

For the Gauss-Newton method, following Nipp (2002), we represent our set of equations by fi(Z) yi − = ri, (2.34) σi where yi are the measurement values and ri the residuals. We search for a Z Rz for which r 2 =χ2(Z) is minimal. We complete a Taylor expansion around∈ k k2 a starting value Zcur (second and higher order terms (h.o.t.) are neglected) to obtain

f(Z) = f(Zcur + ξ) = fcur + Acurξ + h.o.t., (2.35)

p z where Acur R × denotes the Jacobi matrix at Zcur, and inserting it into Eq. (2.34) yields∈ the linearized residual equations

S (A ξ + f y) = ρ , (2.36) · cur cur − cur where S j j = 1/σ j is the diagonal matrix containing the inverse pixel errors. This set of linear equations can be solved by applying Eq. (2.5). We then derive ξnext and thus an improved approximation Znext = Zcur +t ξnext, where we deﬁne the parameter t so that t ξnext represents the direction of the steepest 1 1 descent. That is, for t=1, 2 , 4 ,... we test whether

2 2 χ (Zcur +t ξnext) < χ (Zcur) . (2.37)

After k iterations, we then force Zk to converge and assume that ρk rk. The disadvantage of the steepest descent methods is that the new gradient≃ at the minimum point of any line minimization is perpendicular to the direction just traversed. If not, the new gradient would have a nonzero component along the previous direction, which we claimed to have traversed until the minimum, and so the new direction would not be the steepest gradient. Therefore, one must make a right angle turn, which does not, in general, lead to the minimum. The Levenberg-Marquardt method is more eﬃcient, because of its favor- able direction of descent. Again, we deﬁne the χ2 merit function in terms of least squares to be p y f (Z) 2 χ2(Z) = i − i (2.38) " σ # Xi=1 i 74 Chapter 2. NDD: A new analyzing technique for spectroscopy

Suﬃciently close to the minimum, we expect the χ2 function to be well approximated by a quadratic form, according to Press (1992) 1 χ2(Z) γ d Z + Z D Z, (2.39) ≈ − · 2 · · where d is an z-vector and D is an z z matrix. If Eq. (2.39) is a good × approximation, then one leap will take us from the current values Zcur to the minimizing ones Zmin, i.e., 1 2 Z = Z + D− χ (Z ) . (2.40) min cur · −∇ cur h i2 However, if (2.39) does not represent the shape of χ at Zcur, i.e., when we are far from the minimum, then all we can do is apply the steepest descent method, as we did for the Gauss-Newton minimization Z = Z constant χ2(Z ) . (2.41) next cur − · ∇ cur The Levenberg-Marquardt method adopts both extremes, by varying smoothly between the inverse-Hessian method (2.40) and the steepest descent method (2.41), depending on how far we are from the minimum. In addition, it provides an estimated covariance matrix of the standard errors in the ﬁtted values Z.

Appendix C: Derivative of the extended interpolation formula

For our solving algorithms, we develop the partial derivatives for each equation with respect to each element of our sought-after common line pattern Z(v) to produce the Jacobi matrix and the Hessian matrix, which can also be approximated by the ﬁrst derivatives. If we deﬁne d = 1/R for 1 j z, j j,λ0 ≤ ≤ where z is the number of parameters in Z, and dtot = 1/Rλ0,tot, then we rewrite Eq. (2.19) with the weights m j as m Z X = j j (2.42) j 1 d m Z − j j j and Eq. (2.18), for 1 i p, where p equals the number of equations, as ≤ ≤ si fi = , with si = X j (2.43) 1 + dtot si jXj ∈[ ]i Appendix C: Derivatives 75

th where j i is the list of line-indices that contribute to the i equation. The partial derivatives are then given by ∂ f (1 + d s ) d s 1 i = tot i tot i = − 2 2 . (2.44) ∂si (1 + dtot si) (1 + dtot si)

∂s m j 1 d jm jZ j + m jd jm jZ j m i = − = j 2 2 (2.45) ∂Z j 1 d jm jZ j 1 d jm jZ j − − Therefore, for the elements of the Jacobi matrix (∂ f /∂Z)i j we ﬁnd that ∂ f ∂ f ∂s m i = i i = j 2 (2.46) ∂Z j ∂si ∂Z j 2 (1 + dtot si) 1 d jm jZ j − Acknowledgements. This work is supported by the EURYI (European Young Investigator) Award provided by the European Science Foundation (see www.esf.org/euryi http://www.esf.org/euryi ) and SNF grant PE002-104552.

Bibliography

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Astron. Astrophys. 225, 456 Semel, M., Li, J., 1996, “Zeeman-Doppler Imaging of Solar-Type Stars: Multi Line Technique”, Sol.Phys. 164, 417 Solanki, S. K., 1987, “Photospheric layers of solar magnetic ﬂuxtubes”, Ph.D. thesis, ETH, Zurich, Switzerland Unsoeld, A., 1968, “Physik der Sternatmosph¨aren”, Springer-Verlag, Berlin CHAPTER 3

Zeeman component decomposition for recovering common proﬁles and magnetic ﬁelds†

C. Sennhauser1, S. V. Berdyugina2

Abstract

High resolution spectropolarimetric data contain information about the region where atomic and/or molecular lines form. Existing multi-line techniques assuming similarities in shapes of line profiles can extract generalized Stokes signatures from noisy spectra. However, the interpretability of these signatures is limited by the commonly employed weak-field and weak-line approximations. On the other hand, inversion techniques based on realistic polarized radiative transfer can interpret complicated individual line profiles but still unable to handle the informative wealth of broad-band spectra. We present a new method, Zeeman component decomposition (ZCD), which combines the versatility of an unconstrained line profile resulting from a multi-line analysis with the radiative transfer physics implying that one profile constitutes all

† This chapter is published in Astronomy & Astrophysics 522, A57 (2010) 1 Institute for Astronomy, ETH Zurich, 8093 Zurich, Switzerland 2 Kiepenheuer Institut fur¨ Sonnenphysik, 79104 Freiburg, Germany 80 Chapter 3. Zeeman component decomposition

Stokes parameters. We show that the ZCD is capable of inferring a common Zeeman component profile as well as a reliable magnetic field vector from noisy broad-band spectra. We employ an analytic polarized radiative transfer solution describing formation of polarized line profiles in a Milne-Eddington atmosphere. The ZCD is built as a nonlinear inversion procedure with a number of free parameters, namely an unconstrained line profile, the line central depths, and the magnetic field parameters B , γ and χ. The procedure is applied to all Stokes parameters simultaneously.| | We carefully analyze blending of line profiles and Zeeman components and obtain practical analytical expressions. By comparing the anomalous Zeeman splitting with the commonly used triplet approximation, we obtain an estimate of the error, helping us to identify the cases where the simplification is not applicable. We demonstrate the capabilities of the ZCD by applying it to simulated Stokes I, V, and full I, Q, U, V spectra. The first test shows that the ZCD outperforms standard multi-line techniques in finding common line profiles for noisy polarization spectra and, in addition, consistently recovers the line-of-sight magnetic field. Trials with I, Q, U, V spectra demonstrate the ability of the ZCD to work with noisy linear polarization spectra and recover the magnetic field parameters in realistic scenarios.

3.1 Introduction

Multi-line techniques assuming similar line profiles have become a standard feature of stellar astronomy for increasing the signal-to-noise ratio (SNR) of spectropolarimetric measurements. However, their benefits are limited due to the widely-used weak field approximation (WFA), which is incompatible with a large variety of Stokes profiles emerging from local stellar and solar magnetic fields. Precise measurements of these fields would provide major constraints for both large-scale generation mechanisms such as the magnetic dynamo and for more localized phenomena such as magnetic flux emergence. Even for the Sun, recent reports on the internetwork magnetic field differ by an order of magnitude, from roughly 10 gauss up to several hundreds of gauss (e.g., Lites et al. 2008, Orozco Suarez´ et al. 2007). The limitations in accuracy and/or spatial resolution of current spectropolarimetric observations inhibit non-ambiguous deductions of magnetic field strengths and configura- tions. 3.1. Introduction 81

The techniques commonly applied to non-solar broad-band high-resolution spectropolarimetric observations are based on various rough assumptions. First, the local profile is represented only by one single spectral line at a time, while it is indispensable to account for blends line profiles in all Stokes parameters. Second, they oversimplify the extraction of common line profiles by assuming that contributions from overlapping lines add up linearly for the continuum-normalized Stokes I (intensity), Q, U, (linear polarization) and V (circular polarization) spectra. Third, the reconstruction of the magnetic field vector from the mean polarization signal neglects both saturation effects and deviations of the Zeeman splitting patterns from a pure triplet. The latter two assumptions are only valid in the case of weak fields and weak lines, as shown in Sect. 3.4. We have developed a new method of Zeeman component decomposition (ZCD), which represents an improvement over existing multi-line techniques in various aspects. We carry out precise deconvolution and account for nonlinearity in blends using the Milne-Eddington model. This makes it possible to incorporate various heavily blended profiles, such as found in cool stars and molecular bands. Further, we analyze and take into account individual saturation levels of Zeeman components. This permits us to include lines with arbitrary splitting and treat Zeeman multiplets adequately. A detailed consideration of the formation process of polarization profiles allows us to simultaneously deploy Stokes I, Q, U, and V spectra to determine a single common line profile. The latter provides a strong additional constraint to our problem and allows us to extract field strengths for each constituent absorber. The key principle of our new method is to solve for separate components in a Zeeman multiplet instead of the entire profile signature. This basically leads to the derivation from three polarization spectra Q, U, and V of no more than three parameters B, γ, and χ, i.e., the length and orientation of the magnetic field vector. This method represents an upgrade to our recent technique of nonlinear deconvolution with deblending (NDD, Sennhauser et al. 2009), which accounts for the nonlinearity in blended profiles. This paper is structured as follows. In Sect. 3.2, we explain the idea and operation principles of the ZCD. While discussing the used radiative transfer solution, we derive a line-adding formula in Sect. 3.3 for blended spectral lines. In the subsequent Sect. 3.4, we investigate blending of individual Zee- man components and consider the validity of an effective Lande´ factor geff for different types of sublevel splitting. The final expressions used for the inver- 82 Chapter 3. Zeeman component decomposition sions are given in Sect. 3.5. There we also discuss in detail deviations of the summation of Zeeman components from a linear model. Some details on the numerical implementation of the method are provided in Sect. 3.6. In order to test the performance of the ZCD, we apply it to simulated data in Sect. 3.7. Finally, in Sect. 3.8, we summarize our conclusions.

3.2 Principles of the ZCD

The main objective of the ZCD is to find a single common line pattern Z which is able to reproduce all Stokes parameters for a given full set of measurements. This pattern is the Zeeman component profile, which constitutes all four Stokes parameters simultaneously. Along with this, an orientation and a strength of a magnetic field present in the line forming region are obtained. Allowing the profile to take an arbitrary shape, our new method combines advantages of a multi-line technique with the polarized radiative transfer. Among current multi-line techniques, the least squares deconvolution (LSD), first introduced by Semel (1989) and Semel & Li (1996) and further developed by Donati et al. (1997), and the principal component analysis (PCA, e.g. Mart´ınez Gonzales´ et al. 2008) are well-established and frequently used methods. The LSD extracts a weighted mean Stokes profile, while the PCA diagonalizes a cross-product matrix of individual spectral lines to reconstruct the data with a truncated basis of eigenvectors. If only the first eigenvector (corresponding to the largest eigenvalue) is taken into account, the result is equivalent to that of the LSD. Subsequent eigenvectors contain information about individual line properties, quickly turning into pure noise of measurements. The main drawback of the LSD mean polarization profile is its reduced interpretability due to the WFA and the weak line approximation (WLA). Al- though Semel et al. (2009) pointed out that, being independent of the WLA, the center of gravity method applied to a mean Zeeman signature (MZS) can infer realistic field strengths up to 3 kG, the shape of the MZS, chiefly in the case of Stokes I, is still distorted due to the WLA. Second, both the LSD and the PCA are applied to one Stokes parameter at a time, i.e., neglect their common origin. In contrast to the assumptions of the LSD and the PCA, the polarized radiative transfer assembles the full Stokes vector using only one intrinsic line profile Z, which is often assumed to be the Voigt function (in the absence of 3.2. Principles of the ZCD 83 the magnetooptical effect). To account for inhomogeneities in the line forming regions, more sophisticated inversion codes, such as SPINOR (Frutiger et al. 2000; Berdyugina et al. 2003) or SIR (Ruiz Cobo & del Toro Iniesta 1992), introduce multiple model atmospheres resulting in a more complex line profile. In general, in the presence of a magnetic field B the line can be represented by three so-called Zeeman profiles denoted by σ+, π0, and σ . For a normal Zeeman triplet, their relation to an arbitrary intrinsic profile− Z is straightforward:

eλ2 π (υ)= (υ) , σ (υ) = π υ c , = , 0 Z 0 q 2 geff B q 1 (3.1) ± − 4πmec ∆λD ! ± where geff is the effective Lande´ factor and

υ = (λ λ ) /∆λ , (3.2) − c D

λc being the central wavelength, and λD the Doppler width. The following equations show how the σ+, π0, and σ profiles constitute the four Stokes parameters in the simple case of weak lines− formed by pure absorption (e.g. Stenflo 1994). The Stokes vector is then composed of linear combinations of the Zeeman components weighted by trigonometric functions of γ and χ, the two angles defining the orientation of the magnetic vector:

1 2 1 1 2 Iλ 1 π0 sin γ 1 sin γ (σ+ + σ ) , (3.3) − 2 − 2 − 2 − 1 1 2 Qλ π0 (σ+ + σ ) sin γ cos 2χ, 2 − 2 − 1 1 2 Uλ π0 (σ+ + σ ) sin γ sin 2χ, 2 − 2 − 1 Vλ (σ+ σ ) cos γ, 2 − −

Knowing the value of the line central depth dc, the maximum of Z can be rescaled to unity1. In an idealized case, when absorption takes place only in a thin, homogeneous layer of the star, the common Zeeman component proﬁle

1 In the Paschen-Back regime, individual strengths of the Zeeman components are different. Knowing the relative values of the strengths (e.g., Landi Degl’Innocenti & Landolfi 2004, Berdyugina et al. 2005) helps to rescale the components. 84 Chapter 3. Zeeman component decomposition turns out to be a Voigt-function. In general, however, the emerging line profile has contributions from different layers, and its shape is a priori unknown. In addition, spatially unresolved structures on the stellar/solar surface further complicate the issue. In spite of that, the ZCD is able to solve for the magnetic vector and an arbitrary Zeeman component profile Z, containing information on atmospheric inhomogeneities. Note that Eqs. (3.3) are shown here only for illustrating the basic principle of the ZCD. In practice, we use the analytic solutions of the polarized radiative transfer problem first found by Unno (1956) and Rachkovsky (1962b,a). Their capability will be discussed in detail in the subsequent sections. Most important is that they are not confined to weak magnetic fields. As long as the Zeeman splitting is much smaller than the fine-structure splitting (LS- coupling), the Zeeman shift is given as in Eq. (3.1). In order to also allow for stronger fields, i.e., in the Paschen-Back regime, we modify the displacement term. For instance, Berdyugina et al. (2005) introduced new, individual effec- + 0 tive Lande´ factors g , g , and g− for the σ+, π0, and σ components, respectively, which can be used to derive the corresponding− Zeeman shifts. Thus, our approach allows us to combine spectral lines of any magnetic sensitivity forming in various magnetic environments.

3.3 Nonmagnetic line blending

Solving for the Zeeman components of individual lines requires first of all the disentanglement of blended line profiles. Standard cross-correlation techniques treat a given spectrum as a convolution of the common line profile with a linemask containing the strength (depth) and positions of all contributing lines in the spectrum. This implies that contributions from different lines add up linearly. Such an assumption is invalid in all cases but for a very limited amount of extremely weak lines. In fact, the linemasks used in LSD even exclude weak lines where the WLA would apply, because they introduce an unreasonable amount of noise. In this section we investigate blending of unsplit lines (B = 0) and derive an explicit formula for nonlinear adding spectral lines. Similar to Sennhauser et al. (2009), this enables us to disentangle a blended line profile into individual profiles, useful in many applications where the separate contributions into a blend have to be known. 3.3. Nonmagnetic line blending 85

3.3.1 Blended line proﬁle and opacity In the Milne-Eddington model, the wavelength dependent ratio of the line opacity in the transition i f , κ , to the absorption coeﬃcient in the | i → | i i f continuum, κc, is independent of (optical) depth:

ηλ = κi f (λ) /κc = const in τ. (3.4)

If we assume the line to be formed by absorption under the local thermodynamical equilibrium (LTE), and if we adopt also the source (Planck) function to be linear in optical depth Bλ (τ) = a + bτ, the ﬂux in the line can be computed as (Mihalas 1978)

2 F = a + b/(1 + η ), (3.5) λ 3 λ which contains the Eddington-Barbier relation that the emergent ﬂux in the continuum (ηλ = 0) is the source function at the optical depth 2/3, Fc = B (τ = 2 ). For the residual depth R =1 I = F /F we obtain λ 3 λ − λ λ c

Rλ = R0 ηλ/(1 + ηλ), (3.6) where R0 is the central depth of a theoretical line for which ηλ . This parameter is also called saturation depth of the line. → ∞ The problem of disentangling a proﬁle of two blended lines is equivalent to forming a total depth Rtot consisting of two individual absorption depths R1 and R2 at a given wavelength. For weak lines, where the line opacity is small compared to the absorption in the continuum (η 1), λ ≪

∂Rλ Rλ ηλ = R0 ηλ. (3.7) ≈ ∂η = λ ηλ 0

For two separate absorbers, the total opacity is just the sum of the two individual opacities,

ηλ,tot = κi, f,1 + κi, f,2 /κc = ηλ,1 + ηλ,2, (3.8) and therefore for optically thin lines we obtain

Rλ,tot = Rλ,1 + Rλ,2. (3.9) 86 Chapter 3. Zeeman component decomposition

Equation (3.7) does not hold for optically thick lines, where κi f /κc & 1. How- ever, as stated above, the quantity ηλ is linear in all terms responsible for the absorption in the line, e.g. transition probability or number density. Adding another line (i.e. increasing the absorption) is therefore equivalent to increasing one of those numbers. For a blended line proﬁle consisting of n lines we thus write as a generalization of Eq. (3.6):

n n R = R η / 1 + η . (3.10) λ,tot s,i λ,i  λ,i Xi Xi     To find ηλ,i as a function of Rλ,i, following Sennhauser et al. (2009), we rearrange Eq. (3.6) for ηλ, yielding η = R /(R R ). (3.11) λ λ 0 − λ In combination, Eqs. (3.10) and (3.11) provide a powerful tool to find the total absorption depth if the individual line depths are known, or, on the other hand, to disentangle a blended profile into separate contributions. While Eq. (3.8) is not limited to assumptions of LTE or the Milne-Eddington model, one may argue that Eq. (3.5) depends on the choice of ai, bi, i.e., on the optical depth scale τi of a given transition i. However, the validity to express the residual depth directly in terms of ηλ does not depend on the actual values of ai, bi. They only affect R0, a parameter which we tabulate employing the code STOPRO (Solanki 1987; Frutiger et al. 2000; Berdyugina et al. 2003), solving the polarized radiative transfer equations for a large number of wavelengths, atmospheric models, elements and ions. Therefore, the only requirement for this model is that the source function is linear, whereas the actual behavior (offset, steepness) is irrelevant.

3.3.2 The scaling function As an interesting side result from our previous discussion, we can ﬁnd a wavelength dependent scaling function fλ that transforms a given line proﬁle Rλ with strength η into one with relative strength s. The latter can be a measure of the oscillator strength, or level population, or element abundance. Using Eq. (3.6), we can write R (s) R η s 1 + η s (1 + η ) λ = 0 λ λ = λ . (3.12) Rλ 1 + ηλ s · R0ηλ 1 + ηλ s 3.4. Zeeman component blending 87

Applying Eq. (3.11) then yields

Rλ(s) = Rλ fλ(s) , (3.13) where s R f (s) = · 0 . (3.14) λ R +R (s 1) 0 λ · − The development of a line proﬁle with initial central depth dc and saturation level R0 depending on s is illustrated in Fig. 3.1.

Fig. 3.1: Behavior of a given line Rλ (thick line) with a varying strength. The scaling function fλ is given by Eq. (3.14).

3.4 Zeeman component blending

The line adding algorithm derived in the previous section neglects polarization caused by the presence of a magnetic field in the medium interacting with the incident radiation. Therefore, it is only valid for unsplit lines (B = 0). In this section, we discuss how the Zeeman splitting affects the observed profile, and, in particular, the way the Unno-Rachkovsky solution handles blending of individual Zeeman components originating from one or multiple spectral lines. Understanding the mechanism of blending different Zeeman (sub- )components is important for interpreting the results of the ZCD. To clarify 88 Chapter 3. Zeeman component decomposition the underlying idea, in Sect. 3.4.1 we present an example illustrating the difference in blending of Zeeman components of the same kind and those of orthogonal polarization. This difference is essential for accounting for blending in the case of the anomalous Zeeman effect, which we discuss in Sect. 3.4.2. There we also further assess deviations from linear multi-line techniques. The final explicit solutions for all Stokes parameters are provided in Sect. 3.5.

3.4.1 An illustrative example We start by characterizing individual Zeeman components as different absorbers. In the classical picture, the Zeeman components σ+, σ and π0 are − described as three linear oscillators ox, oy and oz, wherez ˆ is the direction of the magnetic field vector2. In the general case, when all three components are visible, the observed motions (projected on the plane perpendicular to the LOS) of the oscillators are not linearly independent, and their effect on the respective Stokes vector is subject to the angle γ betweenz ˆ and the LOS. However, for γ=0, only ox and oy are visible and orthogonal. Thus, in this case, we can identify σ+ and σ with two independently absorbing media, i.e., − light absorbed by σ+ does not affect σ , and vice versa. This can be illustrated by two equal boxes, which can be filled− with different absorbing materials of amounts m1 and m2, having however an identical central wavelength and line broadening. The boxes are evenly illuminated by a plane-parallel light source of total intensity Ic, while the emerging light is gathered at a detector with no spatial resolution (Fig. 3.2). For a plane-parallel atmosphere perpendicular to the LOS and containing only oscillators ox and oy, the content of the box 1 represents the amount of σ+ absorbers, and the content of the other box that of σ . An empty box implies an absence of oscillators in that direction. − First, let the box 1 contain either m1 or m2 amount of absorbers while the box 2 remain empty (this most simple case is not illustrated in the figure). tot Then the corresponding observed spectra Ri (i = 1, 2) are obviously given by

tot tot Ii 1 Ii 1 Ic 1 1 Ii 1 Ri = 1 = 1 + = = Ri. (3.15) − Ic − 2 Ic 2 Ic ! 2 − 2 Ic 2

2 Although σ+ and σ denote right- and left-handed circularly polarized light, caused by − circular motions of the oscillators o+ and o , each of them can be represented as a linear − combination of two linear oscillators ox and oy with a phase diﬀerence of 90◦ and 90◦, respectively. − 3.4. Zeeman component blending 89

So, we observe exactly half the amount of the absorption as compared to the proﬁles Ii/Ic that would emerge directly from the box 1, because we also observe Ic passing through the other, empty box. Now, if the box 1 is ﬁlled with m1 absorbers and the box 2 with m2, further referred to as case a), the total residual absorption depth is simply the sum of the individual values (Fig. 3.2, left panel): 1 Rtot = (R + R ) . (3.16) a 2 1 2

In the case b), the box 1 contains both m1 and m2 while the box 2 is again empty (Fig. 3.2, right panel). The contributing profiles form a total residual depth Rtot according to Eqs. (3.10) and (3.11), yielding 1 R + R 2R R Rtot = 1 2 − 1 2 , (3.17) b 2 1 R R − 1 2 where for simplicity we assumed the saturation depths R0,i to be equal 1. When we assume m1 + m2 of ox, such as in case b), the atmosphere is transparent for light with the electric field vector alongy ˆ. For an unpolarized light source Ic, this means that only half the amount of light will be subject to absorption, with the other half simply transmitted. In case a), our model atmosphere contained the same total amount of oscillators as in case b), but divided among ox, oy with multiplicity m1, m2. Absorption now takes place “independently” in two different media, resulting in a different residual depth, as seen in Eqs. (3.17), (3.16). The above experiment shows that it is essential to distinguish between the cases of combining components of the same type and of different types. Explicit formulas for calculating the total residual depth as function of γ and the three components will be given in Sect. 3.5.

3.4.2 Anomalous splitting versus triplet approximation So far we assumed that all spectral lines are Zeeman triplets, i.e., a transition splits into three magnetic components in the presence of a magnetic field. This is true for transitions J : 0 1, where J is the total angular momentum quantum number, and this case→ is called normal Zeeman effect. More often, for J > 1, transitions consist of multiple Zeeman subcomponents, and this case is called anomalous Zeeman effect. In principle, the latter can be approximated by the triplet case when the energy shifts within one kind of Zeeman 90 Chapter 3. Zeeman component decomposition

Ic Ic

I1 I2 I1I2 Ic

Ic Ic Ic(I1+I2)-I1I2 Ic

Det ect or Det ect or

1 1 c c /I /I tot tot I I

0 l 0 l Case a) Case b)

Fig. 3.2: Incident plane-parallel light Ic (dotted lines) is passing through two boxes, that can be filled with (identical) absorbing material (triangles, squares) and is detected by a spectrometer. The observed spectrum is plotted as solid line at the bottom of each panel. Left panel: The total line profile is the sum of the two individual profiles caused by only material 1 (I1/Ic, dashed line) and 2 (I2/Ic, dot-dashed line), respectively. Right panel: If box 1 contains both absorbing materials, the residual depth is calculated using Eqs. (3.10), (3.11), and cannot exceed a value 0.5, since half of Ic always passes through box 2. Saturation effects are visible in the center of the line. components are replaced by a corresponding center of gravity. Here we assess the difference in Stokes profiles arising due to the triplet approximation as compared to the true transition splitting taking into account blending of the Zeeman subcomponents. We show that substantial errors can be made even for field strengths . 1 kG, depending on the line splitting. Therefore we further require our ZCD technique to be able to handle arbitrary Zeeman patterns, where we assume that the subcomponent profiles of each line are equal (hypothesis of complete redistribution, e.g. Landi Degl’Innocenti 1976). 3.4. Zeeman component blending 91

We will proceed by recalling the anomalous Zeeman effect and deriving an analytical expression for the location of the maximum Stokes V as a function of splitting and the line broadening parameter a in the case of a Voigt profile. Knowing the location of the maximum, we then compare the Stokes V amplitudes in the triplet and anomalous cases at this position for three different Zeeman patterns. In the case of the anomalous Zeeman effect, the (2J + 1)-fold degenerate energy levels in the presence of a magnetic field split onto magnetic sublevels corresponding to M = J,..., J. The selection rules (conditions for which the overlap integral does− not vanish) require ∆J = 0, 1, where J : 0 0 is forbidden, and ∆M = 0, 1. For a given transition, the± ensembles of subtran-→ sitions with ∆M = 1 are± denoted as σ , while those with ∆M =0 are referred ∓ ± to as π0. In the Zeeman regime, both the splitting pattern and the individual strengths S n of the subcomponents are symmetric around the central wavelength and depend on ∆J and M as well as on the electron spin and orbital momentum (see, e.g., for atoms Sobelman 1972 and for molecules Berdyug- ina & Solanki 2002). In this case, the usual renormalization for the strengths S n within each ∆M subset is

S n = 1. (3.18)

Xn∆M

A mean energy shift of the σ-components is characterized by the effective Lande´ factor geff, i.e., the mean splitting weighted with the corresponding strengths. This is a way to describe arbitrary splittings in terms of a normal triplet. In the Paschen-Back regime, the symmetry around the central wavelength vanishes, and the sums of strengths for each ∆M are no longer equal (e.g., Berdyugina et al. 2005). Here the line profile can be approximated by a + 0 triplet with the three different Lande´ factors g , g , and g−. Also, the violation of the selection rule ∆J =0, 1 has to be taken into account. ± In stellar spectropolarimetry it is common to assume that the maximum Stokes V amplitude of a spectral line increases linearly with the product of the line of sight magnetic field component and geff of the line (WFA). Here we evaluate when this assumption loses its validity due to noticeable splitting. We partially adopt the notation of Stenflo (1994). For the magnetic quantum numbers Ml, Mu and the Lande´ factors gl, gu of the upper and lower levels, respectively, the individual Zeeman shifts for q=1 (Ml = Mu+1, and Mn = Mu) 92 Chapter 3. Zeeman component decomposition are given by

∆E = (g M g M ) ω ~ = (g (M + 1) g M ) ω ~, (3.19) n l l − u u L l n − u n L with ωL =eB/2me the Larmor frequency, and the anomalous splitting in units of the Doppler broadening ∆En/~ υn = . (3.20) ∆ωD To investigate the deviations from the anomalous to the normal Zeeman effect, let us assume here that the line profile is given by a Voigt function H (a, υ) with a =Γ/4π∆νD, and a line central depth dc, instead of our unconstrained profile Z(υ), yielding

2Ju+1 σ (υ) = S n dc H(a, υ qυn) , q = 1. (3.21) ± − ± Xn=1 In Eq. (3.21), we made use of the symmetry of the splitting pattern for the σ components. Note that we use dc in contrast to the usual definition using the line-to-continuum opacity ratio η. For the Zeeman triplet case (Eq. 3.1), we define in accordance with Eq. (3.20) the Zeeman splitting in units of the Doppler broadening ωL υH = geff . (3.22) ∆ωD To be able to assess the difference between multiplet splitting and the triplet approximation in terms of maximum amplitude in circular polarization, we need to know where Stokes V has its maximum. For weak magnetic fields and weak lines, V ∂H/∂υ (Stenflo 1994), i.e., the location υmax where its derivative is largest∝ is constant for υ 1. For a=0, H is a Gaussian, and H ≪ 1/2 υmax,0 = 2− . (3.23)

We then approximate the dependence on the parameter a for B 1 by a linear function ≪ υ (a, B 0) υ +m a. (3.24) max → ≈ max,0 a From numerical simulations for various 0 < a 0.1 we find ma = 0.2855 0.0003 (Fig. 3.3, left panel). Our calculated points≤ show a constant offset of± 3 6.4 10− , which does not influence the determination of m . ∼ · a 3.4. Zeeman component blending 93

The locations of the maximum Stokes V as a function of component shifts 3 2 1 υH for a = 10− , 10− , 10− are shown with dashed lines in the right panel of Fig. 3.3. The values for υ 0 are given by Eq. (3.24). As υ increases H → H the maximum location υmax moves away from the line center. When the line is fully split (yet in the Zeeman regime), υH & 1 and υmax (a, υH) reaches an asymptotic value υH (dotted line). To obtain an analytical expression for υmax in the case when a=0, we search for the extrema of Stokes V and solve

2 2 (υ υH) (υ+υH) dV (υ, υ ) = (υ υ ) e− − (υ + υ ) e− = 0. (3.25) H − H − H Since there is no analytical solution to this equation, we make a Taylor expansion up to the second order in υ around the point υ0 =1/√2, which yields

2 b +ǫb+ c+ +ǫ c 2ǫδ − − − − − υmax (υH) = p , (3.26) d+ + ǫd − where √ 2 √ 3 b = 1 + 2υH 4υH + 2υH ± ± ∓ √ 2 √ 3 4 c = 1 2υH υH 2 2υH υH ± ± − ∓ − √ √ 2 3 d = 2 3 2υH 2υH (3.27) ± ± ∓ − δ = 1 5υ2 + 3υ4 − H H ǫ = e2 √2υH .

As shown in Fig. 3.3, the function given by Eq. (3.26) (bold solid line) is a good approximation for υH . 1. For a magnetic field of . 1 kG, we have υH < 0.5, and the curves for a,0 (dashed lines) differ from that of a=0bya constant shift (cf. right panel scale-up) given by Eq. (3.24). Thus, combining these two equations, we finally obtain

2 b ǫb+ + c+ +ǫ c 2ǫδ υmax (a, υH) − − − − − +ma a. (3.28) ≈ d+p+ ǫd − Using Eq. (3.28), we can now investigate the Stokes V amplitude in the anomalous splitting case at the position where the corresponding triplet approximation curve has its maximum. We consider three diﬀerent splitting patterns as shown on the left panel of Fig. 3.4. The relative errors δV/V for these patterns as a function of (triplet) component shifts are shown in on the right 94 Chapter 3. Zeeman component decomposition

Fig. 3.3: Behavior of the location where Stokes V is the largest as a function of the Voigt profile parameter a and the σ-component shift υH in the Zeeman triplet. Left panel: Diamonds show the maximum positions υmax numerically calculated for different values of a and very small B (υH 0). A linear fit (solid line) is sufficient → to describe the dependence of υmax on a. Right panel: Dashed lines are numerical 3 2 1 simulations of υmax (a, υH) for a = 10− , 10− , 10− . The dotted line describes its asymptotic behavior when the line is fully split. The bold solid line is an analytical solution given by Eq. (3.26) for a=0.

panel of Fig. 3.4. In cases where the sublevel splitting is small (∆υn υH, 2 2 ≪ e.g., in the transition F5/2 D3/2), the center of gravity approximation works very well (dashed curve). The intermediate case matches the results from Semel et al. (2009). However, if ∆υn . υH, the error increases rapidly with υ , reaching 10% (e.g., in 2D 2P ) already at υ 0.3. For example, a H 3/2 3/2 H ∼ line with λ0 =6000 Å, ∆λD =0.1 Å, and geﬀ =2, will reach this value at ∆λ 4πm c = D e B υH 2 0.9 kG. (3.29) geﬀλ0 e ≈

Since ∆λD λ, all curves in Fig. 3.4 scale with λ for a given Doppler velocity, rendering∝ the triplet approximation worse at longer wavelengths. Note that δV/V depends on the the relative shifts and strengths of individual σ- components within a splitting pattern, and not on the absolute shift of the gravity center. The eﬀect is therefore independent of geﬀ. 3.5. Explicit formulas 95

Fig. 3.4: Left panel: Diﬀerent types of anomalous Zeeman splitting patterns. The lengths and shifts of the bars from the center are proportional to the relative strengths and energy shifts, respectively. From Sobelman (1972). Right panel: Deviation of the maximum Stokes V amplitude in the anomalous case with respect to that in the triplet approximation as a function of the triplet splitting in Doppler units.

3.5 Explicit formulas

In this section we aim to provide analytic expressions for Stokes I, Q, U, and V as functions of the three common Zeeman profiles σ , and π0. The shape of each corresponds to the observable profile at B=0 with± an average line depth of all contributing lines, and not to the underlying intrinsic profile appearing the Unno-Rachkovsky solutions (cf. the use of dc in Eq. (3.21)). Similar to the line adding algorithm derived in Sect. 3.3, we will now explicitly see how different Zeeman components add up. In other words, we will be able to write each Stokes parameter in terms of the three Zeeman profiles, which differ only by a proportionality factor depending on central depths of the contributing lines. So, Z(υ) , dc, B,γ,χ is our set of free parameters. Provided has to be a list of wavelengths and quantum numbers of possible spectral lines for the input spectrum. To keep the expressions at a comprehensive level, we skip the magnetooptical effect throughout this section, for which the calculations work similarly.

This offers us a valuable tool to disentangle the individual σ , and π0 profiles of a (blended) line and to demonstrate the difference between± blending of σ+ with σ and of σ with π0 components. Also, we can compare the result with that− of linear models± and emphasize the difficulties arising when the 96 Chapter 3. Zeeman component decomposition

LSD or PCA combine Stokes profiles. First, in Sect. 3.5.1 we show how the Zeeman component opacities can be expressed in terms of σ , π0, γ and χ. Then, we substitute them into the radiative transfer solutions± and obtain final expressions for Stokes I, Q, U, and V in Sect. 3.5.2 without a detailed discussion. Finally, in Sect. 3.5.3, we illustrate in detail the behavior of Stokes I when two Zeeman component profiles are added up and discuss errors for a linear model.

3.5.1 Zeeman component opacities

The analytic radiative transfer solutions RI, RQ, RU , RV are functions of γ, χ, and the local line opacities in units of the continuum opacity η+,0, , i.e., − RI = RI (η+,η0,η ,γ,χ). Explicit expressions are provided by, e.g., Stenflo (1971), and Arena− et al. (1990). To express the Zeeman component profiles via opacities, we first consider σ+ in the longitudinal case, i.e., when the electron in the classical picture oscillates in the x-y-plane perpendicular to the LOS. As shown in Sect. 3.4.1, σ+ then represents exactly half of the absorption capability of the medium. Therefore we have, using Eq. (3.6)

1 Iσ+ η+/2 σ+ = − = , (3.30) 2 1+η+ when assuming R0 to be equal to 1. To account for arbitrary orientations, we characterize the circular motion with frequency ω of the σ+ electron component by two linear oscillators, σx, σy: σ+ = σ cos (ωt) + σ sin (ωt) . (3.31) x · y · If we incline the plane of oscillations of σ+ by the angle γ around the x-axis, the observed motion of σ+ becomes

σ+ = σ cos (ωt) + σ sin (ωt) cos γ. (3.32) x · y · · The absorption eﬃciency is equal to the squared time-average amplitude. If we also note that the observable motion of the π-electron is proportional to sin γ, we get the orientation factors s(γ) for the σ and π components, respectively:

2π/ω ω 2 1 2 sσ(γ) = cos (ωt) + sin (ωt) cos γ = 1 sin γ, (3.33) 2π Z − 2 0 3.5. Explicit formulas 97

2 sπ(γ) = sin γ. (3.34) Using Eqs. (3.33) and (3.34), we obtain 2ρ 1 2ρ sρ(γ) ηρ = , or ηρ = , ρ = σ , π0, (3.35) 1 2ρ s (γ) · 1 2ρ ± − ρ − which simpliﬁes to Eq. (3.30) for the σ components in the case of γ=0. Another way to see this is to insert, for instance, σ+ into RI of the Unno- Rachkovsky equations, and to solve the result for η+:

8σ+ η+ = , (3.36) (2σ+ 1) (3+cos(2γ)) − which after some algebra turns out to be equivalent to Eq. (3.35).

3.5.2 Final expressions for Stokes IQUV

Knowing now the expressions for η+, η0 and η in terms of the Zeeman com- − ponent profiles σ+, π0, σ , γ and χ, we can insert them into the transfer equation solutions and obtain− the final expressions for Stokes parameters. In the general case, all three components contribute to the line profile:

RI (σ+, π0, σ , γ) = 1 + c1 [1 σ+ π0 σ +4σ+π0σ ] /∆ − − − − − −

RQ (σ+, π0, σ ,γ,χ) = sσ(γ) cos (2χ) Σ/ [2∆] − (3.37) RU (σ+, π0, σ ,γ,χ) = sσ(γ) sin (2χ) Σ/ [2∆] −

RV (σ+, π0, σ , γ) = sσ(γ) cos (γ) (2π0 1) (σ+ σ ) /∆, − − − − where 2 c1 = sσ(γ)

c2 = 2sσ(γ) (1 sπ(γ)) 2 − c3 = sπ(γ) c = 4s (γ) (1 s (γ)) (3.38) 4 π − π ∆ = c1 +c2π0 (σ+ +σ )+(c3 +c4π0) σ+σ − − − Σ = 2sσ(γ) π0 sπ(γ) (σ+ +σ ) 4 (1 sπ(γ)) π0 (σ+ +σ ) − − − − − + 4sπ(γ) σ+σ + 2 (3cos (2γ)+1) σ+π0σ , − − and with sσ(γ), sπ(γ) as given in Eqs. (3.34). 98 Chapter 3. Zeeman component decomposition

3.5.3 Combining two Zeeman components for Stokes I In this section we discuss in detail how two arbitrary Zeeman component profiles blend together in Stokes I within our model. More specifically, we consider two pairs: σ+ and σ , and σ+ and π0. Further, we derive and discuss expressions for errors arising− when the components are added linearly. Since Stokes I does not depend on χ, we set this angle equal to 0 for the moment. From the radiative transfer solutions given by Eqs. (3.37), we obtain for the pair of the σ+ and σ profiles: − 2 2 sσ(γ) (σ+ +σ ) sπ(γ) σ+σ = − − RI, (σ+, 0, σ , γ) 2 − 2 , (3.39) ± − sσ(γ) sπ(γ) σ+σ − − and for the pair of the σ+ and π0 profiles:

sσ(γ) (σ+ +π0) + 2 (sπ(γ) 1) σ+π0 RI,+0 (σ+, π0, 0, γ) = − . (3.40) s (γ) + 2 (s (γ) 1) σ+π σ π − 0 Fig. 3.5 contains six panels, three rows of 2 panels each. In the following we refer to the three rows of panels in Fig. 3.5 as top, middle and lower panels (note that the middle panels share the abscissa values with the lower panels). The two functions given by Eqs. (3.39) and (3.40) are plotted in the top panels of Fig. 3.5 for γ = π/4, 3π/8, 7π/16, 15π/32, π/2 (solid lines, from top to bottom). For simplicity, we plot RI, and RI,+0 assuming that in the ﬁrst case ± σ+ = σ and in the second case σ+ = π0. The result of the linear summation of the components− is shown by dashed line. For small values of γ (< π/4), deviations from the linear case are small for both RI, and RI,+0. However, combining σ components for π/4 < γ . π/2 results± in noticeably smaller total depths, for± all σ 0.5. Note that the two special cases γ=0 and γ=π/2 correspond to the left± and≤ right panels of Fig. 3.2, respectively. To ﬁnd the values of σ and π0 for which the relative error introduced by ± the linear approximation is largest, we solve for σ+(= σ ) −

∂ δRI, ± = 0 (3.41) ∂σ+ RI, ± and an analogous equation for σ+(= π0), yielding 19+12 cos (2γ) 4 (7cos γ+cos (3γ))+cos (4γ) σmax, (γ) = − (3.42) ± 16sπ(γ) 3.6. Numerical implementation 99 and sσ(γ) sσ(γ) /2 σ (γ) = − . (3.43) max,+0 1 ps (γ) − π The middle panels of Fig. 3.5 show the functions given by Eqs. (3.42) and (3.43), respectively. For a pair of σ components, the relative error at small γ is largest for intermediate component± depths (σ = 0.25), whereas for larger ± γ, the error is largest for stronger components. The relative error for RI,+0 is always largest for 0.25 σ+ (=π0) < 0.3, i.e., when σ- and π-proﬁles are of similar strengths in the Zeeman≤ triplet. The values of the maximum errors are given by

δRI, 4 max ± = tan (γ/2) , RI, ± δR 1+2sσ(γ) 2 √2 √sσ(γ) max I,+0 = − . RI,+0 1 sπ(γ) − Obviously, these occur at σmax, and σmax,+0, respectively. The two functions ± are shown in the lower panels of Fig. 3.5. For RI, , the relative error is small for small γ, reaching 3% at γ = π/4, but it increases± rapidly up to 100% for γ π/2. When adding up σ+ and π0, the relative error is largest ( 17%) for γ near→ 0 and decreases for larger inclination angles. ∼

3.6 Numerical implementation

The ZCD algorithm is realized as an inversion procedure with the minimization of the discrepancy between the data y and the model f . Representing our set of Eqs. (3.37) for each pixel i in terms of the measurements yi with individual errors σi by fi yi − = ri, (3.44) σi 2 we search for the solution which minimizes ri by applying a Gauss-Newton minimization method, as described by SennhauserP et al. (2009). In contrast to other codes, instead of using response functions we ﬁnd the matrix of derivatives analytically for our parameters B , γ, χ, d , Z, which takes 5 times MOD c ∼ the computing time of one evaluation step of the function fi. For the evaluation of fi at wavelength λi for the Zeeman subcomponent k of the transition q (q = 1, 0, 1) of the contributing line j, we perform spline − 100 Chapter 3. Zeeman component decomposition

Fig. 3.5: Behavior of RI, (σ++σ , left panels), and RI,+0 (σ++π0, right panels). Top ± − panels: The resulting blend depth when both components have equal depth (abscissa values) for diﬀerent angles γ ranging from π/4 to π/2 (top to bottom solid lines). The dashed line represents the linear sum of the two components. Middle panels: Component depth for which the relative error with respect to the linear approximation δRI/RI is largest, depending on the angle γ. Lower panels: Maximum relative errors δRI/RI as a function of γ. See text for detailed discussion.

interpolation of our common line proﬁle Z(v) at the velocity vi jqk, whereas for the Jacobi matrix, we use quadratic inverse interpolation. There exists an ambiguity between the amplitude of Z and the line depths parameters dc. For example, consider that the spectral line j has a “true” central depression of 0.5 at its laboratory wavelength center λ0, j. Let ξ be the current solution to the current linearized residual equations and Z(v=0) = 1, and dc, j = 0.4 be the current parameter guesses. Many variations ξZ0 , ξdc, j of 3.6. Numerical implementation 101

these two parameters may satisfy the condition Z0 dc, j =0.5. Since we require the amplitude of Z to be equal to 1 during each step· of inversion, while the location of the maximum may well vary with respect to the initial guess, we try to inﬂict any undesired amplitude change ∆Z as additional alteration ∆dc, j on the current ξdc, j for all lines j. Thus, assuming that f Z + ξZ, dc + ξdc is a good solution, we look for a ∆dc which satisﬁes the condition

˜ Z + ∆Z (dc + ∆dc) = A′ dc + ξdc , (3.45) where

- A′ = max(Z + ξZ), the amplitude of the trial Z′ =Z + ξZ located atv ˜;

- Z˜ = Z(v˜), the value of the current Z at the location where Z′ is max;

- ∆Z = A′ 1, the diﬀerence in overall amplitude (the amplitude of the current Z−equals 1).

Rearranging Eq. (3.45) we ﬁnd

∆d = A′ d + ξ / Z˜ + ∆Z d . (3.46) c c dc − c If the current Z and dc were good parameters already, Eq. (3.45) simpliﬁes to

Z˜ + ∆Z (dc + ∆dc) = Z˜ dc, (3.47) · and

∆dc = dc ∆Z/ Z˜ + ∆Z . (3.48) − The new parameters will then be

d′ = d + ξ ∆d c c dc − c Z′ = (Z + ξZ)/A′, (3.49) where normalization by A′ ensures the new amplitude to be equal to 1. 102 Chapter 3. Zeeman component decomposition

3.7 Application to simulated data

Here we demonstrate the performance of the ZCD method using simulated data. The input spectra have to be normalized to the continuum, which is the only requirement (ordering or spectral interval between pixels is not required). We first apply it to Stokes I, V data and explore its capability to recover a longitudinal magnetic field BLOS from circular polarization signals fully embedded in noise (Sect. 3.7.1). Then, we employ another set of I, Q, U, V simulated data, allowing us to retrieve the full magnetic field vector (Sect. 3.7.2). We compare the inferred common line profiles (and the corresponding mean Stokes V profiles) with those obtained by the LSD technique with and without deblending (see Sennhauser et al. 2009). We have simulated a local solar-type spectrum (temperature, element abundance) using the full polarized radiative transfer code STOPRO (see Sect. 3.3.1) for references with a variety of noise levels and magnetic field vectors, con- volved with gaussian to account for instrumental broadening. The spectrum includes 35 atomic lines in the wavelength interval between 521.5 and 529.8 nm, and all of them are used to retrieve unknown parameters. However, for better visibility, we show in the figures only 15 lines in the window from 526.2 to 526.65 nm in the figures. Note that the line central depths parameters were assumed to be altogether unknown. The spectral resolution of the simulated spectra was of 30 mÅ per pixel, i.e. typical for high-resolution solar spectropolarimetry.∼ The chosen binsize for the ZCD common Zeeman profile was 1.14 km/s.

3.7.1 Longitudinal magnetic ﬁelds: Stokes I, V

To test the capability of the ZCD to retrieve only BLOS, Stokes I, V data with B = 20 G and γ = 0, and a continuum noise of 1 % were generated. These are shown in Fig. 3.6 with dotted lines in the upper two panels. Best fits obtained with the ZCD are drawn with solid lines. For comparison, the noise-free input spectrum for Stokes V is shown as dash-dotted line. It hardly deviates from the ZCD fitted spectrum. The lower left panel shows the inferred common Stokes I profile from our ZCD method (solid line) corresponding to the recovered BLOS=18 G. The common profiles from the LSD with deblending (dashed) and the standard LSD (dash-dotted) are superposed. Note that in order to compare them to the ZCD, they had to be renormalized. 3.7. Application to simulated data 103

Fig. 3.6: The result of the ZCD deployed to Stokes I, V simulated data. Upper panels: Simulated (dotted) and recovered (solid line) spectra for Stokes I and V, as well as the noise-free input spectrum (dash-dotted line) for Stokes V. Lower panels: Retrieved common line proﬁles from the ZCD (solid) and from the LSD with/without deblending (dashed/dash-dotted lines) for Stokes I (left panel) and V (right panel).

The striking difference between the common line profiles obtained with the LSD and our ZCD method is a substantial broadening of the former. This is because of the following two reasons: First, line strengths influence the shape of the LSD profile, as stronger lines are broader than optically thin lines, while the ZCD profile is not affected by the strengths of the contributing lines. Secondly, contributions from blends, which are either not accounted, as in the standard LSD, or improperly (linearly) added, as in the deblending LSD, cause a noticeable broadening, especially in the far wings of the profile 104 Chapter 3. Zeeman component decomposition

(see dash-dotted curve at v 10 km/s in Fig. 3.6, lower left panel). ∼ For a comparison, we also created a common Stokes V profile from the ZCD common line profile, using averaged line parameters, as in other multi- line techniques. The lower right panel of Fig. 3.6 shows the obtained circular polarization signatures. Clearly, the noise is too high for the LSD to recover a reasonable signal from the Stokes V only (dashed and dash-dotted lines). For the ZCD, applied to the Stokes I and V simultaneously, the Zeeman component profiles are well constrained by the intensity spectrum. This allows for a much more precise determination of the component separation and results in a reasonable average Stokes V profile. We further tested the reliability of our ZCD method by varying an im- posed noise. We produced 125 simulated spectra with the same BLOS of 20 G for each noise level between 1/6% and 1%. Fig. 3.7 shows the recovered BLOS for two sets of 25 spectra with the noise levels of 0.33 % (stars) and 1 % (triangles). The average magnetic field strength values of 20.1 3.2 G (stars) and 18.4 9.1 G (triangles) are close to the true value of 20 G.± The same is valid for other± noise levels: 19.9 G, 19.8 G, 20.4 G, 19.7 G, 20.0 G, while the uncertainty increases proportionally to the SNR of the input spectra (Fig. 3.7, lower panel). We also varied initial magnetic field strength values Binit between 50 and 50 G to demonstrate the independence of the solution on the initial guess.− We conclude that for this spectrum, the ZCD is robust up to the noise level a factor of 5 larger than the mean Stokes V amplitude. This gain can be further increased∼ by employing a larger number of spectral lines.

3.7.2 Inclined magnetic fields: Stokes IQUV To test the capability of the ZCD to deal with full Stokes data, we simulated spectra with BMOD = 200 G, γ = 70◦, and χ = 30◦. Figure 3.8 shows results obtained from the ZCD when 0.2 % noise is added to the spectra. The top four panels display the data (dotted) and the fits (solid lines) for Stokes I, V, Q, and U, as well as the noise-free input spectra (dash-dotted lines) for Stokes QUV. The lower left panel shows the inferred ZCD common Zeeman profile (solid line), as well as the retrieved magnetic field parameters. The corresponding profiles obtained using the LSD with (dashed) and without deblending (dash- dotted) are plotted too. The Stokes I profiles reveal the same issues for the LSD profiles as in the case of a longitudinal field (Sect. 3.7.1). But with a circular polarization signal-to-noise 7 times higher in this case than in the ∼ 3.7. Application to simulated data 105

Fig. 3.7: Top panel: Retrieved magnetic field strengths obtained by the ZCD deployed to 20 G Stokes I, V data with the noise levels of 0.33 % (stars) and 1 % (triangles). The results are shown for two sets of 25 spectra with different initial values for BLOS. Lower panel: Standard deviations of BLOS from 125 runs each at different noise levels. previous one, the LSD also detects a reasonable Stokes V signature (lower right panel). The ZCD is also capable of extracting meaningful linear polarization signals despite the high noise level (S/N<1) and to determine the orientation of the magnetic field. While the fit from ZCD for Stokes V coincides with the noise-free input spectrum, Stokes Q and U show the limitations of Eqs. (3.37). The remaining low noise ripples in the fitted linear polarization spectra in Fig. 3.8 are due to the fact that our Zeeman component profile is not constrained to be neither symmetric nor monotonic in the wings. We have again tested the robustness of the ZCD solution by adding different amounts of noise to the same spectrum and simulating 100 different realizations for each noise level. The results are shown in Fig. 3.9, where three panels present expectation values with error bars for the magnetic field 106 Chapter 3. Zeeman component decomposition

Fig. 3.8: The result of the ZCD deployed to Stokes IQUV simulated data. Top four panels: Simulated (dotted) and ﬁtted (solid line) spectra for Stokes I, V, Q, and U, as well as the noise-free input spectra (dash-dotted lines) for Stokes QUV. Lower panels: Retrieved common line proﬁles from the ZCD (solid), and from the LSD with/without deblending (dashed/dash-dotted lines) for Stokes I (left panel) and V (right panel). 3.7. Application to simulated data 107

parameters BMOD, γ and χ.

The best-fit parameters BMOD and γ showed distributions close to normal, and we calculated their expectation values as the means for each noise level. The distribution of χ values showed however three peaks, i.e. far from a normal distribution. The highest peak was at 30◦, the second one at 60◦, and a (smaller) third one at 150◦. The explanation for this is as follows. As can be seen in Eqs. (3.3), Stokes Q and U have a (different) trigonometric dependence on χ. Only determination of Q/U selects the correct values for χ. Therefore, if the noise distribution allows the code to identify both Stokes Q and U signals, the ratio is well defined, and χ is found correctly (within a normal distribution). However, if Stokes Q is too much embedded in noise (which is more probable than for U for our choice of χ), the code picks one 1 √ of two equally probable angles: 2 sin 3/2 =30◦ or 60◦. If Q is detectable but 1 U is not, then it finds 2 cos (1/2)=30◦ or 150◦. We attempted to account for this special kind of distribution and consid- ered values of 60◦ and 150◦ as partially true. We applied the following formula for the expectation value of χ

1 1 χ = cos− cos (χ) + sin− sin (χ) /2, (3.50) h i where cos (χ), sin (χ) denote the mean value. The lowest panel of Fig. 3.9 shows these weighted mean values χ . One can see that they are within 10 % of the true value for all tested noiseh i levels, whereas the standard deviations are all larger than 10 %, reaching 100 % for the 0.5 % noise.

For BMOD and γ there is a clear trend to increase for higher noise levels. However, the LOS component of the field is quite stable. In addition, the functions for the mean recovered values saturate for high noise levels, not ex- ceeding 350 G and 80◦, respectively. We explain this behavior as follows. The inversion procedure has two ways of fitting stochastic features created by the noise in the linear polarization spectra: by increasing BMOD and/or γ. Since our considerable Stokes V signal constrains in first order only the LOS component of B, both BMOD and γ increase. This effect saturates chiefly because of the constraints by Stokes I (line broadening, or splitting, respectively). 108 Chapter 3. Zeeman component decomposition

Fig. 3.9: Statistical analysis of the results from the ZCD deployed to full Stokes spectra with (BMOD = 200 G, γ = 70◦, χ = 30◦). Expectation values and standard deviations for the magnetic ﬁeld parameters were obtained from 100 runs at each noise level.

3.8 Conclusions

We have developed an efficient method, the Zeeman component decomposition, which is able to extract a common Zeeman component profile from high-resolution broad-band Stokes I, Q, U, V spectra. The shape of this profile is fully unconstrained. This makes the ZCD useful for studying stellar (and solar) inhomogeneous atmospheres. We consider that all Stokes parameters comprise this common profile, assuming that each line forms locally in a Milne-Eddington atmosphere. Being based on the analytical radiative transfer solutions, the ZCD overcomes limitations of the weak-field and weak-line approximations, which are typical for other multi-line techniques. The advantage of the ZCD is that it is applied simultaneously to all Stokes parameters, assuming a uniform magnetic field. The intensity spectrum strongly constrains the profile, while polarized spectra provide essential information 3.8. Conclusions 109 on the magnetic vector. In other words, we allocate the z + n + 3 (common profile + n lines + magnetic vector) free parameters to the individual Stokes parameters according to their information content (with the magnetic parameters allocated to all of them with different weights). The signal-to-noise ratio for Stokes I is normally two orders of magnitude larger than for Stokes V, which in turn is 10 times larger than for Q and U. Therefore, Stokes I ∼ defines the common profile and line central depths, with only BMOD, γ and χ being left for the polarized spectra. BMOD is responsible for the individual shifts of the component profiles, while γ and χ basically determine the ratios of Q, U and V. Other multi-line techniques applied to individual Stokes spectra operate with 4 z free parameters, relying on the precise a priori knowledge of line central depths,× so that the interpretability of 3 z of them is question- able and strongly affected by noise. From this point of× view the advantage of the ZCD can be easily seen. Thus, solving for the Zeeman component profile we obtain at the same time parameters of the full magnetic field vector in the line forming region. We showed that the ZCD is capable to retrieve reliable values from very challenging, i.e., noisy spectra. Furthermore, the fact that the recovered line central depths on average deviate by no more than 0.024 from the predetermined values strongly reduces the dependency on knowledge of stellar parameters. The only fixed parameters left are the two intrinsic transition parameters energy shift (wavelength), and electronic configuration. In this paper, we presented and illustrated with examples main principles of the ZCD. While not constraining the common line profile allows for diverse atmospheric conditions, the inferred magnetic field strength and orientation are assumed to be uniform. Further improvements can be done to account for strong rotational and/or instrumental broadening. These posterior convolution effects on spectra are known to degrade the functionality of deconvolution techniques, because the process of line formation and convolution with a non- delta kernel are noncommuting mathematical operations. A solution to this problem shall be discussed in a forthcoming paper. At the same time, we will further develop ZCD into a code for Zeeman Doppler imaging. In contrast to existing methods, our approach shall dynam- ically assess the number of resolution elements necessary to reproduce the observations, expanding the overall profile into a series of local sub-profiles. We shall thereby overcome the limitation of a uniform magnetic field to model arbitrary polarization profiles, even from single snapshots. Despite the spatial mapping inherent to every Doppler imaging technique, this approach should 110 Chapter 3. Zeeman component decomposition maintain the high sensitivity to very weak magnetic fields, as well as the ability to model strong magnetic fields, of the current ZCD method. Acknowledgements. We thank the referee, Prof. G. A. Wade, for many valuable comments that helped improve the paper. This work is supported by the EURYI (European Young Investigator) Award provided by the European Science Foundation (see http://www.esf.org/euryi) and SNF grant PE002-104552. Bibliography

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First detection of a weak magnetic ﬁeld on the giant Arcturus. Remnants of a solar dynamo?†

C. Sennhauser1, S. V. Berdyugina2

Abstract

Arcturus is the second closest K giant and among the brightest stars in the sky. No magnetic field has been directly detected so far, while Ca ii H&K lines as activity indicators suggest Arcturus to be magnetically active. From three observations we infer the mean longitudinal magnetic field strengths and put them into the context of an intraseasonal activity modulation. We apply our new Zeeman component decomposition (ZCD) technique to single sets of Stokes I and V spectra, measuring a longitudinal component of the magnetic field responsible for tiny Zeeman signatures in spectral line profiles. For two of the spectra we report the detection of the Zeeman signature of a weak longitudinal magnetic field of 0.65 0.26 G and 0.43 0.16 G. The third measurement is less significant, but all± the measurements± fit well a rotationally

† This chapter has been submitted for publication in Astronomy & Astrophysics 1 Institute for Astronomy, ETH Zurich, 8093 Zurich, Switzerland 2 Kiepenheuer Institut fur¨ Sonnenphysik, 79104 Freiburg, Germany 114 Chapter 4. Detection of a weak magnetic field on Arcturus modulated activity cycle with four active longitudes. For the first time a magnetic field on Arcturus was directly detected. This field can be attributed to a diminishing solar-type αΩ-dynamo acting in the deepening convection zone of Arcturus. This work demonstrates that our new method ZCD is lowering the detection limit for very weak magnetic fields from spectropolarimetric measurements.

4.1 Introduction

Arcturus (α Boo) is a K1.5 III star ascending the red giant branch. Being the second brightest star in the northern hemisphere, Arcturus has been the object of many studies. While observations in Doppler velocity suggested long-period variability on time scales of a few hundred days (Gray & Brown 2006, and references therein), its short-term variability has been quantitatively characterized by the identification of individual modes of oscillation (Tarrant et al. 2007). The properties of granulation was measured by the same group, with an estimated time-scale of 0.5 d. Being a single star, its mass is relatively poorly known but comparable∼ to that of the Sun (0.8 0.3M by Bonnel & Bell 1993). The heterogeneity of the effective temperature± (Gri⊙ ffin 1996) has abated and it is nowadays generally accepted to be 4300 K. Already Ayres et al. (2003) concluded that Arcturus may sustain a modest level of magnetic activity responsible for the heating of the coronal structures.The long-term study of Ca ii H&K lines as classical activity indicators by Brown et al. (2008) reveals a range of variability periods between the years 1984 and 2007, exhibiting an apparent magnetic cycle with an estimated duration of .14 yr. Considering the known correlation of magnetic activity with surface temperature as observed for the Sun (Gray & Livingston 1997) and other late-type MS stars (Gray et al. 1996b,a, and references therein), periodic temperature changes of 20 K inferred from line-depths ratio variations support this theory. Assuming≈ a rotational period of 730 d (Peterson et al. 1993), or about 2 yr (Gray & Brown 2006), with v sin i = 1.5 0.3 km/s, the seasonal variability (200–250 days) can be attributed to four active± longitudes. Observed periods in H&K line bisectors can be explained by differential rotation and latitudinal migration of active regions. In this context, the direct detection (via Zeeman effect) of a magnetic field on Arcturus is a long time coming. The first magnetic field measurement of 4.2. Observation and analysis 115

2.9 1.8 G reported by Hubrig et al. (1994) was taken right at the activity maximum,± but error bars are too high for an unambiguous detection. We have recently developed a new multi-line analyzing technique for polarized spectra, called Zeeman component decomposition (ZCD). Encouraged by the performance on simulated Stokes I and V spectra and its ability to recover very weak longitudinal field strengths whose Zeeman signatures are completely embedded in noise (Sennhauser & Berdyugina 2010), we applied ZCD to three datasets taken at the Canada-France-Hawaii telescope (CFHT). We report here the detection of a supposedly varying, very weak magnetic field of about half a Gauss at the photosphere of Arcturus. Section 4.2 describes our observation and the principles of ZCD used for the data analysis. In Sect. ?? we present our results in the context of a seasonal variability caused by four active longitudes. We discuss possible origins for the magnetic field on Arc- turus in Sect. ??, and we investigate potential sources of systematic errors responsible for spurious signatures in circular polarization. In Sect. ?? we summarize our conclusions.

4.2 Observation and analysis

The three observations of Arcturus were obtained at the CFHT using ES- PaDOnS (Donati et al. 2006) on August 2 2006, August 23 2008 and De- cember 6 2008. Data reduction was made with the package Libre-ESpRIT installed at CFHT (Donati et al. 1997). We used ESPaDOnS in circular polarization mode, recording 4 sub-exposures to obtain the continuum-normalized Stokes I (intensity) and Stokes V (circular polarization) parameters. The total integration time were 16 s in 2006 and 8 s in 2008, resulting in a peak signal- to-noise ratio (SNR) well above 1000 in the wavelength region 650-850 nm. However, the SNR gradually decreases toward shorter wavelengths, containing most of the spectral lines processed, dropping below a value of 300 (2006) and 450 (2008) at λ 430 nm. ∼ To recover the mean longitudinal magnetic field, we applied the Zeeman component decomposition analysis, described in Sennhauser & Berdyugina (2010). ZCD is an inversion technique based on Milne-Eddington assumptions. Treating the strength of each line as a free parameter, ZCD does not rely on pre-calculated line masks, which depend on a given set of stellar parameters. Necessary are only the central wavelengths and atomic configura- 116 Chapter 4. Detection of a weak magnetic field on Arcturus

Fig. 4.1: An interval of Stokes I, V spectra of Arcturus, taken at the CFHT/ESPaDOnS on Aug 2, 2006 (solid lines), and best ﬁts from ZCD (dash-dotted line). ZCD automatically omits spectral features/lines that cannot be ﬁtted well. Lower panel: the recovered Stokes V signal is fully embedded in noise of the original measurement.

tions of all possible lines typically observed for a spectral class. During the inversion, ZCD extracts a single line-to-continuum opacity profile common to both Stokes I and V from thousands of lines, and infers a mean longitudinal magnetic field strength at the effective heights of formation of these lines. In addition, ZCD fits the line strengths to the observations simultaneously for Stokes I and V, rejecting lines that cannot be fitted well enough, or whose strengths are below a certain threshold value (usually 1% residual depth, depending on the type of spectra). To estimate the error bars for the recovered magnetic field strength a Monte Carlo simulation was used for each individual spectrum as the most reliable way of error assessement.

In Fig. 4.1 we show a fragment of the observations (solid lines) for Stokes I (top panel) and Stokes V (bottom panel). The ﬁtted spectra obtained from ZCD are superimposed as dash-dotted lines. Note that the scaling for V/Ic is smaller than the noise level of the observations, in order to make the ﬁt from ZCD visible. 4.2. Observation and analysis 117

4.2.1 Analysis of additional spectra We have earlier proved the ZCD functionality with multiple numerical tests (Sennhauser & Berdyugina 2010). Here, as a further test, we present results of applying the ZCD to two late-type supergiants 32 Cyg (HD 192909) and λ Vel (HD 78647). They were previously analysed by Grunhut et al. (2010) who employed the LSD technique (Donati et al. 1997) and inferred ﬁrst-order moments of the line-of-sight (LOS) component of the magnetic ﬁeld. Table 4.2.1 lists the results from Grunhut et al. (2010), compared to the values recovered by our ZCD.

Tab. 4.1: Comparison of recovered BLOS from LSD and ZCD for two different late- type giants Name Spec. type (B σ) [G] (B σ) [G] LOS ± LSD LOS ± ZCD 32 Cyg K3Ib+ 1.16 0.49 0.53 0.16 λ Vel K4.5Ib-II 1.72 ± 0.33 0.90 ± 0.13 ± ± First, we note that whenever there is a magnetic field detection from LSD, there is also one from ZCD (true not just for these two stars). However, while tests have shown that the inferred magnetic field strengths from noise-free, simulated spectra of unblended lines are, of course, identical for both methods, the values for these two test objects (exhibiting to some degree balanced antisymmetric Stokes V LSD signatures) from LSD are a factor of two larger than from ZCD. There are several possible reasons to cause this discrepancy. The mean longitudinal magnetic field BLOS derived from the LSD Stokes V profile ZV given in velocity space v is proportional to vZV dv. This quantity is very sensitive to small errors in the far wings of ZV , whichR can be introduced by the following effects:

- The LSD ZV profile suffers from artificial broadening. Neglecting intrinsic blends (i.e., blends that are not caused by rotational or instrumental broadening) leads to broadening of the recovered mean Stokes V profile. This effect strongly depends on the type of spectra and is more severe for late-type stars.

- The wings of ZV are affected by noise, which contributes to the integral more as we go farther into the wings. The choice of integration limits can therefore strongly affect the retrieved BLOS. 118 Chapter 4. Detection of a weak magnetic field on Arcturus

- Part of the signature is not caused by the star in question, but by a secondary object (e.g. binary, as for 32 Cyg, especially near conjunctions).

The uncertainty introduced by the choice of the measurement window for a noise-free spectrum is in the order of several percent (Kochukhov et al. 2010). This error can be drastically increased by all effects mentioned above, especially in their combination. In addition, known continuum depression of the LSD Stokes I profile and subsequent renormalization also affects the determination of BLOS. All these effects may lead to an overestimation of the magnetic field strength inferred from the first-order moment method for the two example stars. We conclude that the different assumptions underlying LSD and ZCD may lead to different results. For slowly rotating late-type stars, the more sophisticated treatment of line blending of ZCD, accounting for the different shapes of weak and stronger lines, may render ZCD the more appropriate choice.

4.3 Results

The recovered longitudinal magnetic ﬁeld strength and corresponding uncertainty values for Arcturus are collected in Table 4.3. The second column indicates the number of lines used by ZCD. The initial number of lines of over 5000 in our linelist was reduced by the code during inversion, individually for each spectrum, to increase the goodness-of-ﬁt.

Tab. 4.2: Recovered magnetic ﬁeld strength for three diﬀerent spectra of Arcturus

Obs. date # lines BLOS [G] σB [G] 2 Aug 2006 4289 0.65 0.26 24 Aug 2008 3919 -0.23 0.20 6 Dec 2008 3758 0.43 0.16

Using the average of the retrieved line strengths, the inferred line-to- continuum opacity profile κL/κc from ZCD was transformed into mean Stokes I and V profiles, which are presented with solid lines in Fig. 4.2 for one of the observations. Stokes I was renormalized to a line central depth equal to 1. For comparison, the results from the least-square deconvolution (LSD) analysis, 4.4. Discussion 119 described in Donati et al. (1997), were overplotted as dotted lines. In addition we analysed a sum of orthogonally polarized spectra, a so-called “null spectrum”, which should contain zero circular polarization. We conclude that the influence of blends on one hand, and the combination of weak and strong lines on the other, caused the LSD profiles to appear slightly broader than the Stokes I signature obtained from ZCD. For Stokes V, there is no significant signal in the LSD Zeeman signature, as seen in the middle panel of Fig. 4.2. However, the Zeeman profile for Stokes V obtained from κL/κc, BLOS, for a line at λ = 500 nm and with a Lande´ factor of 1.23 (the average of all lines processed) shows an amplitude at the 2.6 σ level, while the profile for the null spectrum is practically zero (corresponding to 0.01 G, lower panel of Fig. 4.2). We want to emphasize that those are not mean− Zeeman signatures in the terminology of Semel et al. (2009), since ZCD works on the assumptions of a uniform magnetic field. This constraint enables ZCD to lower the detection limit for the mean longitudinal component of magnetic fields.

4.4 Discussion

As can be seen from Table 4.3 the recovered photospheric longitudinal magnetic field of Arcturus is perhaps not constant in time. Taking into account the gaps between the different observations this had to be expected. The H+K S II index observations by Brown et al. (2008) show intraseasonal periods of 253 days between 1984 and 1986 and 207 days in 1986–1989. Assuming 0.65 G to be the amplitude of an intraseasonal magnetic field variability, our three data- points can be fitted by a 208 day period. According to the seasonal variability with an estimated minimum cycle duration of 14 yr found by the authors mentioned above, the earliest possible next maximum in H+K is 2009. Placing our observations in this context, this means that they were taken either during a period of increasing activity, or during an extended activity minimum.

4.4.1 Possible magnetic field generators The first of two possible origins for magnetic fields is a local dynamo from giant convection cells, which is suggested to generate the recently discovered field of 1 G on the M supergiant Betelgeuse, reported by Auriere` et al. (2010). ∼ 120 Chapter 4. Detection of a weak magnetic field on Arcturus

Fig. 4.2: The recovered ZCD line-to-continuum opacity profile, transformed into a mean Stokes I profile with central depth equal to unity (upper panel), mean Stokes V profile (middle panel), and the diagnostic null spectrum (lower panel), drawn as solid lines. The error bars were estimated with Monte Carlo simulations. Superposed as dotted lines are the corresponding mean profiles obtained with the LSD technique.

Due to the large apparent diameter of Arcturus (21.05 0.21 mas), Lacour et al. (2008) attempted to directly image its surface via int±erferometry, using a technique similar to that successfully applied to Altair (α Aql) by Monnier et al. (2007) with a resolution of < 1 mas. They failed to reveal surface inhomogeneities, and conclude that the most probable brightness distribution is that of a simple limb darkened disk. However, due to the limited number of resolution elements, the existence of large convection cells, modeled by e.g. Kiss et al. (2006), analogous to solar granulation, cannot be ruled out. 4.4. Discussion 121

For Betelgeuse, Haubois et al. (2009) explain their imaging and interferomet- ric observations, and their variability, by the occurrence of such giant granules. These large convective cells could sustain local small-scale dynamos even without stellar rotation (Freytag 2003; Dorch & Freytag 2003). For Arc- turus, however, the predicted number of 500 cells, using the scaling relation for the size of stellar granules from Freytag∼ et al. (2002), is too large to explain the power spectrum of the stochastic noise in the data of Brown et al. (2008). In addition, the velocity span of the λ6252.56 Fe i line lacks a correlation with the predicted 14 yr magnetic cycle. The observed properties mentioned above indicate that such a dynamo is not expected to operate on Arcturus, where granules are smaller than on Belelgeuse and show no long-term variability, while magnetic activity concentrates on active longitudes. In the case of Arcturus, a solar-like αΩ-dynamo driven by convection and differential rotation is more probable. As outlined by Brown et al. (2008), there is a similarity in behavior of the Ca ii H&K activity indicator on the Sun and on Arcturus, revealed by line-depth ratio variations with a 2 yr time lag as observed in late-type dwarfs (e.g., Gray 1994; Gray et al. 1996a). To- gether with the suspected presence of four active longitudes exhibiting latitudinal migration, these are strong indicators that the photospheric and chromospheric characteristics are closely related to the magnetic surface activity of a solar-type MS star. Active longitudes are common among cool, magnetically active stars, ranging from zero-MS stars (Berdyugina & Jarvinen¨ 2005) to solar-type stars (Berdyugina & Usoskin 2003; Lanza et al. 2009) and to rapidly rotating red giants (Berdyugina & Tuominen 1998; Korhonen et al. 2002). Being attributed to a non-axisymmetric component of a large-scale magnetic field (Berdyugina et al. 2002; Moss 2005), they normally coexist with an axisymmetric component responsible for overall cyclic variations of stellar activity. The loss of angular momentum during both the MS and red giant stages diminishes the overall activity and also changes the stellar structure. Modeling shows that a weaker differential rotation supports the dominance of active longitudes, while a stronger differential rotation leads to a dominant axisymmetric mode. Thus, active longitudes and a possible 14-yr cycle on Arcturus suggest a solar-like dynamo with a weaker differential rotation. The magnetic fields detected in single red giants by Konstantinova-Antova et al. (2008, 2009) are likely due to a solar-type dynamo. While all stars in their sample were relatively fast rotators (9.4 < v sin i < 29 km/s), with magnetic fields ranging from 1 to 10 G, Arcturus represents the first slowly-rotating K 122 Chapter 4. Detection of a weak magnetic field on Arcturus giant for which a magnetic field possibly generated by a solar-type dynamo has been detected.

4.4.2 Sources for circular polarization cross-talk

5 When dealing with circular polarization signals at the 10− level, potential sources of spurious Stokes V signals need to be inspected. The first is the issue of contamination of circular polarization by other Stokes parameters. For the ESPaDOnS polarimeter, such a cross-talk is caused by stress birefringence in a triplet lens and in the atmospheric dispersion corrector (Donati 2006). The triplet lens used in August 2006 had a linear to circular cross- talk of about 2 3%. The effect of optical pumping (Happer 1972) can cause gas clouds intervening− the LOS to produce absorption lines with fractions of linear polarization of the order of 1%, if the absortptive region is within 2 stellar radii (Kuhn et al. 2007). With∼ such a mechanism and a linear to circular cross-talk, magnetic fields are not required explain Stokes V signals of 4 the order of 10− in the case of partially obscured stars (e.g., Herbig Ae/Be or AGB). Not having reached the AGB yet, this model does not apply for Arcturus, but it may not be neglected in the case of highly evolved giants, or giants with circumstellar envelopes, for which very weak circular polarization signals are being reported (e.g., Auriere` et al. 2009, 2010). Another possible source for misinterpretation of Stokes V signals is stellar pulsation. Two subsequent exposures measuring Stokes I+V and I V by rotating a quarter wave plate in front of an analyzer can be shifted in wavelength− due to a change in radial velocity of the star between the exposures and cause a spurious polarization signal. Hatzes & Cochran (1993) found day-to-day variations in radial velocities of up to 100 m/s for Arcturus. Can this source of cross talk into Stokes V be responsible for the detected Zeeman signatures? At a wavelength of λ=500 nm and a Lande´ factor of 1.23, our half a gauss for the longitudinal magnetic field B corresponds to a Zeeman velocity shift of e ∆v = λ B 0.35 m/s. (4.1) H 4πm ≈ Assuming a time lag of 1 min between two subexposures, the upper limit for the radial velocity modulation is ∆vrad . 0.07 m/s (the exposure times themselves are negligibly small, 4 s for our observations). This effect is diminished for dual beam polarimeters with beam exchange (e.g., Tinbergen & Rutten 4.5. Conclusions 123

1997), which are designed to eliminate such systematic errors and sources of spurious signals to ﬁrst order (Donati et al. 1997). However, a combination of slow polarization modulation and large pulsation amplitudes can cause spurious signals and should be veriﬁed for pulsating stars.

4.5 Conclusions

For the first time, a weak magnetic field on Arcturus was detected. The maximum mean longitudinal component measured via Zeeman effect has a strength of 0.65 0.26 G. We suggest that the difference between our three measurements is± consistent with a 208 day period due to the intraseasonal variability of the activity revealed by other indicators. Combined with the observations of Brown et al. (2008), we suggest a diminishing solar-like dynamo as origin of magnetic activity, excluding the possibilities of a remnant fossil field as in EK Eri (Dall et al. 2010, and references therein), or the local small scale dynamos driven by large convection cells, as suggested in the case of the supergiant Betelgeuse (Auriere` et al. 2010). The deepening of the convection zone and further rotational slowdown during the RGB phase may alter the characteristics of an active dynamo mode into a turbulent, or shear- driven αΩ-dynamo acting between the rapidly rotating (helium-)core and the outer shells (Nordhaus et al. 2008). Long-term spectropolarimetric monitoring of Arcturus is needed to confirm its magnetic activity cycle in terms of Zeeman-detection, along with the traditional activity indicators. The performance of the multi-line analysis method ZCD enables us to detect very weak stellar magnetic fields, and makes ZCD a powerful tool for such studies. The present work is the first in a series of analysis of polarimetric measurements of active MS stars, early and late-type giants. Acknowledgements. We thank Prof. Gregg Wade for providing the two Arcturus Stokes IV spectra recorded in 2008 and for his valuable comments on the paper. This work is supported by the EURYI (European Young Investi- gator) Award provided by the European Science Foundation (see www.esf.org/euryi http://www.esf.org/euryi ) and SNF grant PE002-104552. We acknowledge the use of the VALD atomic database.

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Achromatizing a liquid-crystal spectropolarimeter: Retardance vs Stokes-based calibration of HiVIS†

D. M. Harrington1, J. R. Kuhn1, C. Sennhauser2, E.J. Messersmith1 & R. J. Thornton3

Abstract

Astronomical spectropolarimeters can be subject to many sources of systematic error which limit the precision and accuracy of the instrument. We present a calibration method for observing high-resolution polarized spectra using chromatic liquid-crystal variable retarders (LCVRs). These LCVRs allow for polarimetric modulation of the incident light without any moving optics at frequencies 10Hz. We demonstrate a calibration method using pure Stokes input states that≥ enables an achromatization of the system. This Stokes-based deprojection method reproduces input polarization even though highly chromatic instrument eﬀects exist. This process is ﬁrst demonstrated in a labora-

† This chapter is published in Publ. Astron. Soc. Pac. 122, 420 (2010) 1 Institute for Astronomy, University of Hawaii, Honolulu-HI-96822 2 Institute for Astronomy, ETH Zurich, 8093 Zurich, Switzerland 3 Department of Physics, West Chester University, West Chester, PA 130 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter tory spectropolarimeter where we characterize the LCVRs and show example deprojections. The process is then implemented the a newly upgraded HiVIS spectropolarimeter on the 3.67m AEOS telescope. The HiVIS spectropolarimeter has also been expanded to include broad-band full-Stokes spectropolarimetry using achromatic wave-plates in addition to the tunable full- Stokes polarimetric mode using LCVRs. These two new polarimetric modes in combination with a new polarimetric calibration unit provide a much more sensitive polarimetric package with greatly reduced systematic error.

5.1 Introduction

Stellar high resolution spectropolarimetry is an underutilized and often complex technique, in part because polarimetric signatures are often less than 1% and in many cases less than 0.1%. Nevertheless, when it can be reliably applied it is a powerful remote sensing diagnostic. In this small-signal regime, where high signal-to-noise is essential, there are important instrumental limits to the derived accuracy and precision. There are several methodologies used for making precision measurements. The most obvious but practically most difficult method is to build a specialized instrument (and telescope) specifically optimized for polarimetry. One minimizes the number of optical elements, keeps all optical folds to near- normal incidence, minimizes scattered light and has some form of modulation or beam-swapping to remove systematic effects. A spectropolarimeter has many functional components essential for sensitive linear or circular polarization measurements. For example, polarizing beamsplitters, fixed and variable waveplates, and specialized synchronous detectors are common sources of polarimetry-specific systematic errors. These errors include detector variations (pixel-to-pixel efficiency and field illumination effects), changes in illumination during the exposures comprising a measurement (telescope guiding error, beam wander from moving optics, atmospheric transparency variations and seeing) and any polarimetric or chromatic effects induced by the entire optical system. Some of these are easily correctable and some are not. Typically, one or two retarders are inserted in the beam to modulate the incident polarization for detection. These can be several types of achromatic retarders, Fresnel rhombs, liquid crystal variable retarders (either ferro-electric or pneumatic) or piezo-elastic modulators. 5.1. Introduction 131

In some cases, the instrument, retarder or analyzer rotates to accomplish the modulation. Various choices of retardance have been used for various applications to optimize sensitivity to different types of incident polarization or for different expected signatures. Since typical polarimetric detections at these small levels are often limited by instrument systematic effects, careful designs must mitigate as many error sources as possible. Notable examples are the ESPaDOnS fiber-fed spectrograph on the 3.6m Canada France Hawaii Telescope (CFHT) and it’s copy Narval on the 2m Telescope Bernard Lyot (TBL) telescope (cf. Semel et al. 1993; Donati et al. 1999; Manset & Donati 2003). These instruments work at a resolution R 65,000 covering 3700-10500Å and are in active use by many (cf. MAPP1 or∼ MiMeS2 ). The two∼ telescopes are equatorial and the polarimetric modulation is done immediately after a collimating lens and an atmospheric dispersion corrector (ADC). The instrument is fiber-fed and has a time-variable continuum polarization at the several percent level (ESPaDOnS Instrument Website 3). A similar continuum polarization is seen in the William Wehlau Spectropolarimeter unit with a similar design to the ESPaDOnS package (Ev- ersberg et al. 1998). The transmissive optics (lens and ADC) currently induce a time-variable cross-talk of a few percent, but for most spectropolarimeters this is quite benign. Lower spectral resolution instruments, such as HPOL and ISIS, typically have more stable polarimetric properties that have yet to be seen at higher spectral resolution (Wolff et al. 1996, ISIS Spectropolarime- try Manual Tinbergen & Rutten 19974). Ultimately it is calibration of the telescope and instrument that achieves the highest possible system polarimetric accuracy. Even though it is difficult to keep instrumental polarization below 1%, the effort in building a low polarization system isn’t lost, as the ultimate calibrated polarization performance improves multiplicatively. Thus, sensitive techniques which determine the Mueller matrix of the telescope/polarimeter (Beck et al. 2005b,a; Kuhn et al. 1994; Joos et al. 2008; Patat & Romaniello 2006; Tinbergen 2007) will achieve a lower overall system noise when the raw instrumental errors are minimized. High-resolution astronomical spectropolarimeters have common design

1 http : //lamwws.oamp. fr/magics/mapp/FrontPage 2 www.physics.queens.ca/ wade/mimes/ 3 www.cfht.hawaii.edu/Instruments∼ /Spectroscopy/Espadons/ 4 www.ing.iac.es/Astronomy/observing/manuals/ 132 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter elements. Typically, rotating achromatic retarders are placed before a calcite- based dual-beam analyzer - a Wollaston prism in collimated space or a Savart plate at an image plane. ESPaDOnS and Narval use two half-wave and one quarter-wave Fresnel rhombs before a Wollaston prism. Two other dual-beam instruments, PEPSI on the 8.4m LBT at R 310,000 and HARPS on the ESO 3.6m telescope at R 115,000 are in various∼ stages of construction (Strass- meier et al. 2003 & 2008;∼ Snik et al. 2008). The HARPS polarimetric package was required to fit inside an already- existing space. This required the analyzer to be a Foster prism with a cylindrical lens on one beam and a CaF2 prism for the second beam. The design includes separate quarter-wave and half-wave super-achromatic plates for circular and linear polarization that cannot be used simultaneously. The PEPSI design is similar to ESPaDOnS in that a lens collimates the telescope focus and retarders with a Wollaston perform the modulation. An- other lens in combination with an atmospheric dispersion corrector form an image on fibers which feed the spectrograph. Instead of Fresnel rhombs, a super-achromatic quarter-wave plate is chosen. Linear polarization sensitivity is achieved by physically rotating the entire polarization package. The High-resolution Visible and Infrared Spectrograph (HiVIS) we discuss here was an instrument initially built for spectroscopy but modified for linear spectropolarimetry (Thornton et al. 2003; Harrington et al. 2006; Har- rington & Kuhn 2008). A Savart plate and a rotating achromatic half-wave plate were installed at the entrance slit. One of the main complications of this system is the eight oblique reflections between the sky and the polarization analyzer. Since the 3.67m Advanced Electro-Optical System (AEOS) telescope is altitude-azimuth and HiVIS is in a coude´ path, the many oblique reflections cause pointing-variable cross-talk that can completely swap linear and circular polarization from telescope input to output at some wavelengths and pointings. Nevertheless, the instrument has been successfully used for sensitive linear spectropolarimetric studies of stars as well as other studies (cf. Harrington & Kuhn 2007, 2009a, 2009b). Rapid modulation of the incident beam polarization, synchronous with the detector, is the one technique for removing time-dependent systematic effects. If this modulation exceeds a kilohertz, then even atmospheric seeing errors can be minimized. This technique has been developed by the solar com- munity. For instance, the various incarnations of the ZIMPOL I and II solar imaging polarimeter have used piezo-elastic modulators or ferro-electric liq- 5.1. Introduction 133 uid crystals in combination with charge shuffling on a masked CCD to remove seeing induced systematic errors (Povel 2001; Stenflo et al. 1997; Gandorfer et al. 2004; Stenflo 2007). The Advanced Stokes Polarimeter (ASP) and La Palma Stokes Polarimeter (LPSP) are other notable examples (c.f. Elmore et al. 1992; Lites 1996; Mart´ınez Pillet et al. 1999). This technique has been adapted for night-time spectropolarimetric use at the Dominion Astrophysical Observatory using ferro-electric liquid crystals and a fast-shuffling unmasked CCD. Though this instrument can only record a small spectral region at lower spectral resolution, fast modulation removes several systematic errors (Monin et al. in Prep). We present here a spectropolarimeter using retardance-varying liquid crystals. These liquid crystals add additional complexity as the retardance is chromatic and varies as a function of temperature. However, these chromatic liquid crystals can be quite useful when considering their performance characteristics. The retardance varies without physical motion of the liquid crystals. This removes systematic errors caused by moving an optical element during or between exposures. Another advantage is that this type of liquid crystal can switch from zero to half-wave retardance in several milliseconds. These retarders can be used in any type of fast-switching application. Another trade-off lies in detector cost and complexity. In cross-dispersed echelle spectropolarimeters, the detector area must be used efficiently. If one can encode polarimetric information in a minimum number of spectral orders and minimize the required number of CCD reads, one can optimize a system for faster, more efficient operation. These trade-offs motivate the use of LCVR systems. We show here that a Stokes-based deprojection method allows reconstruction of the incident polarization even when using LCVRs. In order to demonstrate this deprojection and improve the utility of HiVIS, we have implemented two different polarimetric modes. The first is a standard full-Stokes polarimetric mode using two rotating achromatic wave plates. The second is a fast-switching polarimetric mode using the liquid-crystal variable retarders. With this new full-Stokes capability we were able to thoroughly calibrate and compare the polarization properties of HiVIS and quantify the sources of systematic error and polarimetric cross-talk caused by the various optical elements. In this paper we will demonstrate the calibration of the new retarders, performance characteristics of the upgraded HiVIS, and the Stokes-based deprojection routines we have developed. First, the performance and calibration 134 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter of the retarders using a laboratory spectropolarimeter will be presented in section 2. The calibration of the LCVRs as well as the Stokes-based deprojection methods will be presented in section 3. The upgraded HiVIS spectropolarimeter properties will be presented in section 4. Section 5 will present the HiVIS spectrograph polarization properties and section 6 will present the AEOS telescope polarization properties.

5.2 Retardance Fitting and Optical Characterization

In the Stokes formalism, the polarization state of light is denoted as a 4-vector:

I Q I =   (5.1)  U     V    In this formalism, I represents the total intensity, Q and U the linearly polarized intensity along position angles 0◦ and 45◦ in the plane perpendicular to the light beam, and V the right-handed circularly polarized intensity. Note that according to this deﬁnition, linear polarization along angles 90◦ and 135◦ will be denoted as Q and U, respectively. Furthermore, Q and U and V are said to be orthogonal− states− of polarization. The degree of polarization can be deﬁned as a ratio of polarized light to the total intensity of the beam:

Q2 + U2 + V2 P = (5.2) p I The intensity of unpolarized light will be represented in this paper as:

I = (1 P)I (5.3) u − The normalized Stokes parameters are denoted with lower case and are deﬁned as:

q Q 1 u = U (5.4)   I    v   V          5.2. Retardance Fitting and Optical Characterization 135

For details on polarization of light, see an excellent text: Collett 1992.

The upgrade to full Stokes polarimetry on HiVIS required the addition of an achromatic quarter-wave plate to the standard polarimetric mode and the installation of a completely new mode using two LCVRs. New calibration optics were also installed to deliver broad-band linear and circular polarization to the HiVIS spectrograph. We acquired and tested two ’compensated’ LRC-200 IR-1 Meadowlark LCVRs. These retarders were designed for 0 to over half-wave retardance over the 6500Å to 9500Å range. We also acquired a Boulder Vision Optic quarter-wave plate (QWP) and half-wave plate (HWP) designed to be achromatic in the 4000Å to 7000Å range. Calibration polarizers consisted of a pair of Codixx wire-grid polarizers (1” round), a pair of Edmund High-Contrast polarizers, a pair of Edmund wire-grid polarizers (50mm square) and a pair of simple sheet linear polarizers (2” round). We will denote these as Codixx, Edmund, wire-grid’ and sheet polarizers respectively.

We describe below the characterization of these new optics using our laboratory spectropolarimeter. The spectropolarimeter is essentially a collimated halogen light source with a pair of polarizers to generate and analyze polarized light. Light is collected in a fiber-fed Ocean Optics USB 200 spectrograph. The polarimetric efficiency for all the polarizer pairs used in this paper were measured with the laboratory spectropolarimeter. The sheet, Edmund, Codixx and wire-grid polarizer pairs were first measured both crossed (minimum transmission) and parallel (maximum transmission) to characterize their degree of polarization. The wavelengths showing 95% polarization for each polarizer pair are shown in Table 1. ≥

The sheet polarizer has a high degree of polarization from roughly 4800Å to 8200Å. The sheet polarizer was >80% polarizing from the near-UV limit of the spectrograph at 3800Å, above 95% in the 4200Å to 8200Å region and falling sharply with wavelength after 8200Å. Both the Edmund and Codixx polarizers show high degree of polarization longward of 5200Å. The wire- grid polarizers repeatably showed between 99.0% and 99.8% polarization for the full spectral range of 3800Å to 10000Å. All of the measurements were reproducible to better than 1%. Our four well-calibrated polarizer pairs are used to test the optics in the laboratory and in the HiVIS spectrograph. The sheet polarizers were used for short wavelengths and the Edmund polarizers were used for long wavelengths. 136 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

Tab. 5.1: Optic Wavelength Ranges Name Wavelength (Å) Sheet 4200-8200 Edmunds 5200-10000 Codixx 5200-10000 Wire-grid 3800-10000 LCVR 6500-9500 Wave-plate 4000-7000

This Table shows the name of each optic used in calibrations. The top section describes the polarizers used and the measured range where the measured polarization was 95%. The bottom section shows the design range for the retarders used. ≥

5.2.1 Achromatic Wave-Plate Characterization To calculate retardance, we take the ratio of transmission spectra with the retarder in between polarizers. The maximum transmission spectrum I is recorded with parallel fast axes of both polarizers and wave plate. This yieldsk the transmission through all optics of the system. Next, the fast axes of the retarder and second polarizer are rotated 45◦ and 90◦ respectively. This spectrum is labeled I , to denote the crossed polarizers. We then deﬁne the wavelength dependent⊥ transmission spectrum as

I (λ) T(λ) = 1 ⊥ (5.5) − I (λ) k Thus the transmission spectrum is always between 0% and 100% and is independent of the spectrum of the light source. Any polarization-independent absorption caused by the retarder or polarizers is included in both I and I and does not inﬂuence the transmission spectrum. Since the fast axis⊥ align-k ment is independent of wavelength and the polarizers are used only in their highly polarizing wavelength ranges, the transmission can be directly related to the LCVR retardance. To describe how polarized light propagates through any optical system, a Mueller matrix is constructed which describes how every incident state is transferred to an outgoing state. The Mueller matrix is a 4x4 set of transfer coeﬃcients which multiplies the input Stokes vector to create the output 5.2. Retardance Fitting and Optical Characterization 137

Fig. 5.1: This Figure shows the measured transmission spectra T(λ) calculated as in Equation 5.5 for the QWP and HWP (dot-dash lines). The corresponding relative intensity spectra I and I are shown as solid lines. The HWP is shown in the top two k ⊥ panels and QWP is shown in the bottom two panels. The left hand y-axis is relative intensity. The right hand y-axis shows the transmission ratio in percent. The relative intensities for the HWP are recorded through crossed Edmund polarizers on the top left panel and crossed sheet polarizers on the top right panel. The QWP intensities are recorded through parallel Edmund polarizers on the bottom left panel and parallel sheet polarizers on the bottom right panel. There are four individual measurements repeated at 0◦, 90◦, 180◦ and 270◦ rotation for the polarizer pairs giving rise to the many curves in each panel.

Stokes vector:

Iout = MIin (5.6)

If the Mueller matrix for a system is known, then one inverts the matrix and deprojects a set of measurements to recover the inputs: 138 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

1 Iin = M− Iout (5.7) The inverse Mueller matrix for the system we call the deprojection matrix. One can represent the individual Mueller matrix terms as describing how one incident state transfers to another:

II QI UI VI IQ QQ UQ VQ M =   (5.8)  IU QU UU VU     IV QV UV VV    In our lab spectropolarimeter θ measures the angle in the plane perpendicular to the light beam. If we deﬁne θ = 90◦ to be the direction of Stokes Q, we can write the standard Mueller matrix for a wave plate oriented at θ = 45◦ with its won transmission function Tret and a variable retardance φ as:

1 00 0  0 cos φ 0 sin φ  Mret(45◦,φ) = Tret − . (5.9)  0 01 0     0 sin φ 0 cos φ    The Mueller matrix of an ideal linear polarizer looks as follows:

1 1 0 0 ∓ 1  1 100  Mpol = T pol ∓ . (5.10) 2  0 000     0 000      Where the negative sign corresponds to θ = 0◦ while the positive sign is θ = 90◦. Since the light of the calibration lamp is unpolarized, Ilamp = T Tlamp [1, 0, 0, 0] , the Stokes vector of the light incident on the retarder can be written as 1 I = T T [1, 1, 0, 0]T (5.11) in 2 lamp pol1 − Our analyzer (the second polarizer) has a transmission function T pol2 and is a polarizer oriented vertically. With this Mueller matrix and input Stokes vector, we solve the transfer equation to recover the phase variations:

Iout = MpolMretIin (5.12) 5.2. Retardance Fitting and Optical Characterization 139

1 If we note that 2 IlampT pol1TretT pol2 = I we write the output Stokes vector as: k

1 + cos(φ)  1 cos(φ)  Iout = I  − −  (5.13) k  0     0    This represents a purely polarized output beam with I = Q. Since we are measuring this intensity through crossed polarizers, the output− Stokes vector is simply I as denoted above. The only variable is a wavelength dependent retardance⊥ (φ). The measured output intensity becomes:

I (λ) = I (λ)(1 + cos(φ)). (5.14) ⊥ k In terms of the transmission function deﬁned in Equation 5.5 we get an equation for the retardance:

1 φ(λ) = cos− [T(λ)] (5.15) This equation applied to the measured transmission curves gives us the absolute retardance variation with wavelength (φ(λ)). The measurements of I , I and the corresponding retardances are shown in the four panels of Figurek 5.1.⊥ The HWP measurements are shown in the top two panels. The QWP measurements are shown in the bottom two panels. Both wave plates were rotated through 360 degrees for both polarizer sets giving four independent measurements. In order to additionally test the repeatability of the mounting and aligning procedure, the QWP was unmounted and remounted for an additional set of measurements. This remounted measurement was indistinguishable from the others. An ideal HWP oriented with the fast axis at 45◦ to the input polarization rotates the input state to the orientation of the ﬁnal polarizer, giving 100% transmission through crossed polarizers. An ideal QWP would only give 50% transmission through both parallel and crossed polarizers. Thus, the bottom two panels have the right-hand y axis going to 50%. Since the transmission spectra of the wave plates are only useful when the polarizers have a high degree of polarization, these transmission spectra are only plotted for long wavelengths in the left-hand boxes and are only plotted for short wavelengths in the right-hand boxes. 140 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

Fig. 5.2: The average deviation in retardance (in degrees phase) calculated for both 1 quarter and half wave plates through polarizer sets. The equation φ = cos− [T(λ)] is implemented for each observation. An arbitrary oﬀset has been applied for clarity.

Since the QWP gives 50% transmission through both parallel and crossed polarizers, we decided to leave the polarizers parallel for the QWP measurement as it involves one less rotation of an optical element and gives less chance for systematic error. Thus the top two panels of Figure 5.1 show measured relative intensities either near 0 or 25 while the bottom two panels show relative intensities near 12 or 25. One can also see the degree of polarization of the polarizers decreasing outside their designed wavelength ranges since the measured intensities show only small diﬀerences between crossed and parallel conﬁgurations.

The corresponding retardance variation, computed as in Equation 5.15, can be seen in Figure 5.2. Both QWP and HWP phase deviations measured through both sheet and Edmund polarizer pairs are shown. The HWP shows less than 10 degrees retardance deviation for the 4000Å to 8000Å region. The QWP is slightly less chromatic at longer wavelengths but more chromatic short of 5000Å. 5.3. Achromatizing an LCVR Polarimeter 141

Fig. 5.3: The throughput (transmission) of the two liquid crystal variable retarders measured by the Varian Cray spectrophotometer. Each LCVR was measured at three diﬀerent rotation angles and they are nearly indistinguishable on this plot. The Cray internal light source is highly polarized at most wavelengths but only a mild ’ripple’ is seen at the diﬀerent rotation angles.

5.3 Achromatizing an LCVR Polarimeter

In order to use the LCVRs in a spectropolarimeter, one must know how the retardance varies with applied voltage and wavelength. The transmission function is highly chromatic but independent of polarization. Figure 5.3 shows the measured LCVR transmission function. A Varian Cary 5000 spectrophotometer was used to measure the transmission of the two LCVRs we label A and B. The Cary spectrophotometer was calibrated for dark and unobscured transmission. The LCVR transmission was measured at three separate rotation angles. Since we measured the Cary light source to be highly polarized, and the LCVRs are highly birefringent and chromatic with zero applied voltage, any rotational modulation of the LCVRs could come from the polarization properties of the incident light, detector or of the LCVRs themselves. There were significant differences between the two LCVRs at short wavelengths but there was no significant dependence on LCVR orientation. Both 142 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

LCVRs transmitted nearly perfectly longward of 6000Å. LCVR A had a significantly lower overall transmission around 4000Å. Both LCVRs showed small-amplitude ripples that were orientation dependent. LCVR A transmits over 50% more light at very short wavelengths, but LCVR B transmits significantly more from 4000Å to 6000Å. Using the lab spectropolarimeter we measured the transmission spectra, I , with the LCVR fast axis aligned with both the front and rear polarizers. Thek transmission spectra, I , was then measured for various LCVR voltages ⊥ between crossed polarizers with the fast axis of the LCVR rotated by 45◦. The retardance values were recovered using Equation 5.15 for each voltage in the same manner as the achromatic wave plates. It should be stressed that when the LCVR fast axis is aligned with the axis of either polarizer, there is no measurable effect with applied voltage. The LCVRs show no measurable rotation of the fast axis with voltage or wavelength. These compensated Meadowlark LCVRs are designed to have zero retardance around 5-7 volts with over half-wave retardance at low voltages. With these specifications, we convert the measured transmission spectra in to retardance spectra. We find the half and full transmission points through crossed polarizers as a function of wavelength and voltage then assign the retardance 1 as cos− (T). To illustrate this process, another set of measurements of both LCVR A and B was taken with 56 separate voltages from 0V to 10V. Sampling of 0.25- 0.50V per step was performed around regions of rapid change with wider sampling at low and high voltages. These measurements were in agreement with the ones presented previously. Several hundred spectra were again recorded and normalized for each voltage and averaged to create a single precise transmission spectrum. Figure 5.4 shows these transmission spectra interpolated to a regular voltage and wavelength grid. The contours for half and full transmission are overlaid representing 1/4, 1/2 and 3/4 wave retardances. The structure of Figure 5.4 arises from the liquid crystal material properties and design of the LCVR. The liquid crystal itself is a layer of uni-axial birefringent material. The molecular axes are naturally aligned and the material is fixed between two rubbed glass surfaces to orient the material. On the inside of the glass surfaces is a thin transparent layer of indium-tin oxide that allows a voltage to be applied across the material. The material has maximal birefringence under low voltage as all the material is aligned parallel to the rubbed glass surfaces. As the voltage across the liquid crystal increases, 5.3. Achromatizing an LCVR Polarimeter 143

Fig. 5.4: The LCVR-A transmission spectra through both the Edmund and sheet polarizers interpolated onto a regular voltage grid. Contours representing maximum transmission (and 1/2 wave retardance) as well as half-maximum transmission (and 1/4 & 3/4 retardance) are overlaid. Red corresponds to maximum transmission (1) and black corresponds to zero transmission.

the liquid crystal material in the center of the layer is torqued in response to the electric field. This central material aligns itself with the field while the material attached to the glass surfaces remains fixed and supplies a restoring torque. As voltage is increased the birefringence is reduced as the total degree of molecular alignment in the liquid crystal is reduced. The molecules in the center become aligned with the electric field while those at the surface remain aligned with the surface. Thus the electric field scrambles the bulk alignment of the material. Since these liquid crystals cannot sustain a DC voltage for a long time without damage, an AC square wave at 2kHz is applied which simply alternates the sign of the field while preserving the amplitude. One can see in Figure 5.4 that there is a region between 0 and 1 volts where there is essentially no change with voltage. This corresponds to the region where the electric field across the liquid crystal is insufficient to cause any molecular rotation in the material. There is a strong change seen from 1V to 4V as the liquid crystal molecules begin to move. Above 4 volts, the 144 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter material is mostly saturated and there is essentially no birefringence and very low chromatism. The two LCVRs have very similar retardance at a fixed voltage. However, two-decimal place precision in voltage is required to accurately reproduce retardances and an independent calibration must be done for each LCVR. These retardance measurements are repeatable to better than a few percent. The results of Figure 5.4 were reproduced even after a complete rebuild of the lab spectropolarimeter. There is a significant temperature dependence for the LCVRs. Retardance measurements were taken at 20◦ and 40◦. The computed retardance when heated was subtracted from the retardance measured when cool to form the temperature coefficients (in radians per degree temperature change) shown in Figure 5.5. There is a decrease in retardance with an increase of temperature ranging from 0.02 to 0.08 radians per degree. This range of voltages corresponds to the region of maximal sensitivity to LCVR input voltage, only spanning 1.2-3V. There is generally low retardance variation with temperature at higher voltages and longer wavelengths as the physical retardance values are actually lower. There is generally higher temperature sensitivity at shorter wavelengths and lower voltages as these settings correspond to higher birefringence and stronger implicit voltage dependence. This shows that temperature is an important variable that must be controlled when using LCVRs.

5.3.1 Observing with an LCVR Spectropolarimeter As a way of illustrating the deprojection process we will illustrate the extraction of the Stokes parameters from a typical observing sequence for a hypo- thetical spectropolarimetric observation of the 7590Å TiO band. In a typical dual beam spectropolarimeter sequence, one exposure will correspond to a measurement of one Stokes parameter. There are other modulation schemes that record linear combinations of the Stokes parameters in other ways, but many night-time high-resolution spectropolarimeters function in this manner. In a typical sequence, a series of 6 exposures is taken corresponding to positive and negative Stokes parameters. The dual-beam analyzer gives two orthogonally polarized spectra per exposure. The orthogonally polarized exit beams become the top and bottom spectra imaged on the detector. Since the analyzer sends I + Q to the top spectra and I Q to the bottom spectra and all other states are split equally between top and− bottom spectra, one can 5.3. Achromatizing an LCVR Polarimeter 145

Fig. 5.5: The liquid crystal temperature dependence shown in radians induced phase change per degree temperature increase. All values are negative showing that retardance drops with increasing temperature.

simply subtract the top spectra from the bottom spectrum to obtain a polarized spectrum in one Stokes parameter. The retarders act to modulate the incoming polarization so that the analyzer will produce two orthogonally polarized beams which are sensitive to other Stokes parameters in the incident beam. The typical setup with a Savart plate analyzer, shown in Figure 5.6, is to have the first LCVR with its fast axis aligned with the Savart plate axis and a retardance φ1. The second LCVR is rotated by 45◦ and has a retardance φ2. The typical procedure is to use combinations 0, 90, 180 and 270 degrees phase to achieve the modulation. The first step in this example is calibrating the liquid crystals. From measurements like those of Figure 5.4 for 7590Å one finds that voltages of 6.50, 2.80, 2.10 and 1.65 correspond to the retardances of 0, 90, 180 and 270 degrees phase respectively. These voltages then become the settings for a normal sequence. The LCVRs are set to their respective phases to measure one particular Stokes parameter, say (φ1, φ2)=(0,90) for +U and the exposure is taken recording the corresponding spectra. The LCVR phases for each setting can be denoted as: 146 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

00 000 Exp1 φ1 , φ2 +Q Exp2 φ00, φ180 Q    1 2   −   Exp3   φ90, φ090   +U    =>  1 2  =>   (5.16)  Exp4   φ90, φ270   U     1 2     Exp5   φ00, φ090   −+V     1 2     Exp6   φ00, φ270   V     1 2   −       

α β This can be more compactly written as φ1 and φ2 where α and β denote the retardance angle at the nominal wavelength. The phases are denoted as α=(0,0,90,90,0,0) and β=(0,180,90,270,90,270). Each orthogonally polarized beam exiting the Savart plate is a pure polarization state represented as a scalar intensity coeﬃcient multiplying a Stokes vector for pure +Q:

1 +1 I = I   (5.17) top top  0     0     

What we are interested in is how the incident polarization state maps to the intensity coefficients Itop and Ibot. This can be calculated by simply multiplying the Stokes vector of the incident light by the combined Mueller matrix for the two LCVRs at their respective orientations and retardances and a linear polarizer which represents the Savart plate analyzer. The main result is that unpolarized light is always recorded with equal intensity in both top and bottom beams. For every LCVR retardance setting of the normal sequence, the desired Stokes parameter is always recorded with a coefficient of 1 while the other Stokes parameters are recorded equally with coefficients of± 0.5. The prescription for where each Stokes parameter falls in each spectrum and the two corresponding LCVR phases (shown parenthetically in degrees) is given by: 5.3. Achromatizing an LCVR Polarimeter 147

LCVR1 LCVR2 Savart Plate

oe Fast Fast Axis Axis

Fig. 5.6: The schematic for the HiVIS spectropolarimeter. Light incident from the left hits the ﬁrst LCVR, oriented with the fast axis vertical. The second LCVR is rotated by 45◦. The Savart plate is two calcite crystals bonded such that the extraordinary beam for the ﬁrst crystal becomes the ordinary beam in the second crystal. The Savart plate axis is oriented vertically, giving orthogonally polarized Q exit beams. ±

I1top =+Q + (Iu + U + V)/2 (φ1,φ2) = (00, 000) I = Q + (I + U + V)/2 (φ ,φ ) = (00, 000) 1bot − u 1 2 I = Q + (I + U + V)/2 (φ ,φ ) = (00, 180) 2top − u 1 2 I2bot =+Q + (Iu + U + V)/2 (φ1,φ2) = (00, 180)

I3top =+U + (Iu + Q + V)/2 (φ1,φ2) = (90, 090) I = U + (I + Q + V)/2 (φ ,φ ) = (90, 090) 3bot − u 1 2 I = U + (I + Q + V)/2 (φ ,φ ) = (90, 270) 4top − u 1 2 I4bot =+U + (Iu + Q + V)/2 (φ1,φ2) = (90, 270)

I5top =+V + (Iu + Q + U)/2 (φ1,φ2) = (00, 090) I = V + (I + Q + U)/2 (φ ,φ ) = (00, 090) 5bot − u 1 2 I = V + (I + Q + U)/2 (φ ,φ ) = (00, 270) 6top − u 1 2 I6bot =+V + (Iu + Q + U)/2 (φ1,φ2) = (00, 270)

This ’normal sequence’ records one Stokes parameter in one exposure by encoding that parameter in a single top or bottom spectra with no sensitivity to the other Stokes parameters. For instance, Stokes Q is entirely contained in the ﬁrst exposure (I1top and I1bot) at the optimum wavelength and the other states are equally recorded in both beams. The I1top spectrum minus the I1bot 148 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

Fig. 5.7: The retardance versus wavelength for four selected LCVR voltages. The solid line shows LCVR A and the dashed line shows LCVR B. The vertical line shows the TiO wavelength of 7590Å where the LCVR retardances were optimized to be 0.00, 0.25, 0.50 and 0.75 waves.

spectrum gives only Stokes Q sensitivity. The sign of each state is reversed in even versus odd exposure numbers. This sign reversal is the key to the dual beam concept as it enables systematic error subtraction and independence from detector effects. Each Stokes parameter is recorded on the same set of pixels with opposite sign. Subtrac- tion of the two spectra removes detector noise but adds the Stokes parameters. Once this normal sequence is observed, we have recorded 12 spectra in 6 exposures where simple subtraction of individual spectra is enough to disentangle each Stokes parameter. The three exposures recording the negative Stokes parameters are used to remove CCD cosmetic effects and any pixel-to-pixel sensitivity variations. Even though the LCVRs are chromatic, the polarization information at all other wavelengths is still present. Extracting the polarization spectra is possible but requires a more complex deprojection process. This gives rise to wavelength-dependent sensitivities for each Stokes parameter. In order to deproject the observations, measurements of the LCVR retardance must be used. Figure 5.7 shows the retardance for the four voltages used in this sequence. There is minimal difference between individual LCVRs as the solid 5.3. Achromatizing an LCVR Polarimeter 149 and dashed lines overlay quite nicely. In order to represent our two LCVR polarimeter for all wavelengths, we calculate the Mueller matrix for a 2- LCVR system with arbitrary phase. We will denote the cosine of a phase as cφ and the sine of a phase as sφ. The Mueller matrix of the two LCVR system (M2LCVR) is just Mret(45,φ1) multiplied by Mret(0,φ2):

1 00 0 10 0 0  0 cφ2 0 sφ2   01 0 0  M2LCVR =  −  .   (5.18)  0 01 0   0 0 cφ1 sφ1       0 sφ2 0 cφ2   0 0 sφ1 cφ1     −      This can be multiplied out to create the full Mueller matrix for the liquid crystals:

10 0 0  0 cφ2 sφ1 sφ2 cφ1 sφ2  M2LCVR =  −  (5.19)  0 0 cφ1 sφ1     0 sφ2 sφ1cφ2 cφ1cφ2   −  The dual-beam spectropolarimeter essentially has two sepa rate analyzers. The top and bottom spectra can be represented as having passed through a linear polarizer oriented vertical (0◦) and a second linear polarizer oriented horizontal (90◦). The spectra recorded through each beam can be calculated by multiplying the 2-LCVR Mueller matrix by the Mueller matrix for the linear polarizers which select the individual top and bottom beams. If the beam recorded through the vertical polarizer is the ’top’ spectra and the horizontal polarization is the ’bottom’ spectra, the equation for the intensity coeﬃcients of the recorded spectra becomes:

I 1 I cφ Q sφ sφ U + sφ cφ V top = u − 2 − 2 1 2 1 (5.20) Ibot ! 2 Iu + cφ2Q + sφ2 sφ1U sφ2cφ1V ! − With the equations for the coefficients multiplying each Stokes parameter in each of the exposures, typically called the efficiencies, one can determine the chromatic effects of the LCVRs and how to deproject each of the Stokes parameters. Figure 5.8 illustrates the wavelength dependence of these efficiencies for each Stokes parameter optimized for 7590Å. In the normal sequence, the top and bottom spectra record one Stokes parameter multiplied by an efficiency 1. The other Stokes parameters are all equally present in ± 150 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter both top and bottom spectra with a coefficient of 0.5. Each panel of the Fig- ure shows the efficiencies for an individual Stokes parameter as a function of wavelength. During the first exposure, shown in the top box, Stokes Q is correctly multiplied 1 at 7590Å as both LCVRs are at zero retardance and show low chromatism± at high voltage. The measurement of Q (dashed curve) puts the second LCVR at half-wave retardance and lower voltage.− The chromatic effects are strong enough to make both efficiencies +1 by 4500Å. Stronger chromatic effects are seen in the other Stokes parameters because of the general higher retardances (and lower voltages) present on both LCVRs.

Fig. 5.8: The LCVR eﬃciencies from Equation 5.20 as functions of wavelength using the ﬁrst three retardance settings of the ’normal’ sequence. The top box shows the standard ’Q’ sequence, the middle box the ’U’ sequence and the bottom box the ’V’ sequence. The solid lines are the + Stokes parameters while the dashed lines are the - Stokes parameters. Only for a narrow wavelength range around 7590Å does the normal sequence properly map incident polarization to the required QUV as noted ± by the vertical lines.

5.3.2 Retardance Fitting Deprojection Given that we have now measured the retardance for the LCVRs for all possible applied voltages, an easy deprojection to attempt is simply to calculate the Mueller matrix of the LCVR polarimeter for all wavelengths using Equa- 5.3. Achromatizing an LCVR Polarimeter 151 tion 5.18. We will investigate a sample observing sequence for input linear polarization as an example of this deprojection via retardance ﬁtting. In order to properly deproject a series of 12 spectra from 6 exposures into a single polarization measurement, one must create a ’deprojection matrix’. Typically, one calculates the retardance of the LCVRs and then solves the system of equations which represent the polarized radiative transfer for the spectropolarimeter. We will illustrate this process for 7590Å-optimized measurements. The observing sequence gives a set of redundant equations for the polarized spectra:

Imeas = DIin (5.21) These equations depend on the scalar intensity coeﬃcients and can be expressed in terms of sines and cosines of the LCVR retardances. The retardances for each LCVR (φ1,φ2) were expressed in Equation 5.16. To simplify the notation, we will number the spectra by exposure (1 to 6) and note their detector location (top or bot). We note that the retardances are implicitly functions of wavelength. The coeﬃcients are:

1 1 1 1 1 I1top 1 cφ2 sφ2sφ1 +sφ2cφ1 − 1 − 1 1 1 1 I1bot 1 +cφ2 +sφ2sφ1 sφ2cφ1    2 2 2 − 2 2   I2top   1 cφ sφ sφ +sφ cφ     − 2 − 2 1 2 1   I   1 +cφ2 +sφ2sφ2 sφ2cφ2   2bot   2 2 1 2 1     3 3 3 − 3 3   I3top   1 cφ2 sφ2sφ1 +sφ2cφ1  Iu    − 3 − 3 3 3 3   I3bot   1 +cφ +sφ sφ sφ cφ  Q   =  2 2 1 − 2 1    (5.22)  I   1 cφ4 sφ4sφ4 +sφ4cφ4  U  4top   2 2 1 2 1       +− 4 +− 4 4 4 4     I4bot   1 cφ2 sφ2sφ1 sφ2cφ1   V     5 5 5 − 5 5     I5top   1 cφ sφ sφ +sφ cφ       − 2 − 2 1 2 1     I   1 +cφ5 +sφ5sφ5 sφ5cφ5   5bot   2 2 1 2 1     6 6 6 +− 6 6   I6top   1 cφ2 sφ2sφ1 sφ2cφ1     − 6 − 6 6 6 6   I6bot   1 +cφ +sφ sφ sφ cφ     2 2 1 − 2 1      In order to solve this system we simply invert the equations:

DT I I = meas (5.23) in DT D This is easily implemented for all wavelengths using the retardances we have already calculated in the previous section. An example of the deprojection process using the lab spectropolarimeter will be shown here. The LCVR 152 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

fast axes were positioned as a polarimeter with LCVR A at 0◦ and LCVR B at 45◦. Eight complete linear polarization states were input by rotating the front polarizer through 360◦ in 45◦ increments. A full polarimetric sequence was taken for each input state using the voltages chosen to optimize the polarimeter for 7590Å. The overall transmission used to normalize each measurement was recorded through parallel polarizers with both LCVRs oriented vertically. Since the retardance with wavelength functions are known, the complete polarimetric properties of the input can be deprojected using Equation 5.23. Figure 5.9 shows the deprojection for all eight input states. The input linear polarization was extracted well for all wavelengths. The input states were recovered and the cross-talk states were generally around 0.1 or less though with some chromatism and some values as high as 0.2. One can see from this simple example of a normal observation sequence that there are still significant systematic errors present from phase fitting. The LCVR phases are measured in a different optical setup than when the LCVRs are used as a polarimeter. This leads to a wide range of potential errors. As we shall see in the next section, a method where phase fitting is not required performs much better.

Fig. 5.9: The deprojection of pure linear polarization input using the standard deprojection of Equation 5.23. Though chromatic, the deprojected outputs shown here reproduce the pure input states with values near 1. The cross-talk Stokes parameters are the curves that are below 0.3. The discontinuity around 7500Å is the wavelength transition between sheet and Edmund polarizers. 5.3. Achromatizing an LCVR Polarimeter 153

5.3.3 Stokes-Based Deprojection

A deprojection method which uses measurements of known input states is straightforward and direct. It incorporates all effects of the optical path and not just those of the LCVRs. There is no need to calculate the phase of the LCVRs and no interpolation is necessary. Using our lab spectropolarimeter, this is illustrated in the following example. We inserted fully polarized light into our lab spectropolarimeter as pure inputs +Q, +U, +V (as well as Q, U, V to show consistency). For each input state, we took the six measurements− − − applying the ‘normal sequence’ of voltages to our two LCVRs. In the lab spectropolarimeter we have a polarizer as an analyzer so we only record the ’top’ spectra from Equation 5.22. Since our input space is three-dimensional, our output (prime) space will have (at most) three independent vectors, which we created by defining Q′ = I 2top − I1top, U′ = I4top I3top, and V′ = I6top I5top. Note that we could have chosen completely− different combinations− of measurements to build up our three prime space vectors, but our choice was influenced by our knowing that at 7590Å, these three vectors represent the (unnormalized) values of Q, U, and V from our retardance measurements. As orthonormal input basis vectors we use fully polarized light - all three pure Stokes parameters define the basis vectors. The two front linear polarizer states defining +Q and +U were used as well as a quarter wave plate oriented to feed in +V. These are our pure input states used for the Stokes-based deprojection. Then the columns of the Mueller matrix for our setup are directly given by (Q′,U′,V′) for each of the input basis vectors. If we now want to retrieve the input state of any measurement we again build up the quantities I2top-I1top, I4top-I3top, I6top-I5top, and we apply the inverse of our Mueller matrix to them. The problem turns out to be always well-posed, as one can see by looking at the (wavelength dependent) condition number of the Mueller matrix. The condition number, defined as the absolute value ratio of the highest to lowest eigenvalues of the mueller matrix, is a measure of the invertability of a linear system. In optical systems this definition is the most useful for system optimization and error propagation calculations (cf. Compain et al. 1999; de Martino et al. 2003). The condition number is well below 10, and increasing only for wavelength short-ward of 6000Å, where the polarization performance in the system starts to drop. As two examples, Figure 5.10 shows the Stokes-based deprojection method with the front polarizer input states orthogonal to the calibrations ( Q − 154 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter and U respectively). The recovered Stokes parameters are the solid top lines near− 1.0. The dotted and dashed lines show linear and circular cross-talk.

Fig. 5.10: The deprojected Stokes parameters are shown here. The input state is reproduced at a value near 1. The cross-linear and circular states are quite small, being roughly 0 for linear and 0.1 for circular. Compare this improved method with that of Figure 5.9.

The circular cross-talk is near zero with the linear cross-talk value being around 0.05. This represents roughly a factor of 4 improvement from the method utilizing LCVR phases presented previously. It should be pointed out that the linear cross-talk is essentially achromatic. This can be easily explained as simply an error in the rotation angle of the linear polarizer when set a negative input state. With a ratio of linear Stokes parameters of 0.5 to 1 1.0, one can compute an angular error as 0.5 tan− (q/u) of less than 1.5◦. This corresponds to the error what one would expect for a manual rotation of the linear polarizer. The simple fact that the cross-talk is achromatic suggests that it is the improper rotation angle of the polarizer used to create the pure input Stokes parameters that is the cause of the error, not the improper removal of chromatic LCVR retardance. 5.3. Achromatizing an LCVR Polarimeter 155

5.3.4 Benefits of Stokes-Based Deprojection The main benefit of Stokes-based deprojection can be seen when considering how errors effect measured Stokes parameters. In dual-beam spectropolarimeters, the standard observing sequence is followed in order to accomplish two independent goals. One goal is to create polarization spectra that are independent of pixel-to-pixel efficiencies. Since these pixel-to-pixel variations are usually not known to better than a percent, precision spectropolarimetry requires observations to be independent of the gain. The other main goal is to choose retardances that map each Stokes parameter into these normalized double differences. For the standard sequence used with the achromatic wave plates this can be implemented as follows:

Q 1 I1top I2top I1bot I2bot 1 q = = ( − − ) = (q1 + q2) (5.24) I 2 I1top + I2top − I1bot + I2bot 2 One can show that this method is independent of pixel-to-pixel gains by propagating gain coefficients through Equation 5.24. Suppose that the incident radiation intensity in a top pixel is multiplied by a gain factor gt. Ona bottom pixel the gain is gb. In the standard choice of retardances, the intensity of the incident radiation switches beams so that the I1top intensity before detection is the same as I2bot. I1bot is identical to I2top. Thus there are only two incident intensities denoted as It and Ib which switch places between exposure. The incident radiation is modulated so that the gains cancel and the normalized double difference (Equation 5.24) becomes: 1 g I g I g I g I I I q = ( t t − t b b b − b t ) = t − b (5.25) 2 gtIt + gtIb − gbIb + gbIt It + Ib This is exactly as desired - the pixel-to-pixel gain variations have been eliminated and all that remains is a normalized difference which records one Stokes parameter. There is no systematic-error induced by pixel-dependent variations. The difference ratio method has the added benefit of removing the optical mis-alignment errors. The slit tilt and ’derivative error’ caused by wavelength misalignments between ’top’ and ’bottom’ spectra are removed because the same pixels are used for each normalized difference (q1 and q2). While the normalized double differences always cancel the pixel-to-pixel gains, the chromatic effects of the LCVRs require a deprojection from input to measured QUV. The Stokes-based deprojection utilizes calibration observations taken with as similar an optical configuration to the measurements as 156 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter possible. One can see the quality of the Stokes-based deprojection by comparing the retardance-fitting deprojection of Figure 5.9 with Stokes-based deprojection of Figure 5.10. Aside from the small ripples inherent in the LCVR transmission function, the Stokes-based method is much more achromatic, less prone to optical sources of error and easier to implement.

5.4 HiVIS LCVR Spectropolarimeter

After demonstrating Stokes-based deprojection with the lab spectropolarimeter, we installed an LCVR spectropolarimeter unit and the associated calibration optics on the HiVIS spectrograph. We are presently upgrading HiVIS to include better optics, optomechanics and a fast-switching mode utilizing these LCVRs. This upgrade involves using a new charge-shifting detector. The original detectors, two 2k by 4k CCID-20 were replaced with new CCID-20s and electronics that allow charge to shuffle in one direction. If the modulation is performed faster than telescope guiding errors, a major source of systematic error can be removed. In order to do this fast modulation, the charge will be shifted on the CCD without reading out an exposure. This charge shifting must also be synchro- nized with the polarimetric modulation. The spectral orders will not be al- lowed to cross in the charge shifting process as there is presently no shuttering during an exposure. In order to maximize signal-to-noise and make efficient use of the telescope, the number of imaged spectra must be minimized and this optimization procedure implemented. We do not need to have one exposure per Stokes parameter. As long as the polarization information is encoded in the recorded spectra in an extractable manner, the number of exposures can be reduced. The new HiVIS spectropolarimeter has several upgrades. The additions include a second rotating Boulder-Vision-Optic (BVO) achromatic quarter- wave plate, a remounting of all the polarizing optics with full xyz control, a new dekkar with full xyz control, calibration optics, a new Savart plate which has a larger clear aperture, more displacement and is more accurately bonded and a new mount which allows for easy switching between the standard wave plate mode and the fast-switching liquid-crystal mode. The new Savart plate has a 17% larger displacement and no detectable leak into the undesired e-e and o-o beams (compared to the 1% measured for 5.4. HiVIS LCVR Spectropolarimeter 157 the old Savart plate shown in Harrington & Kuhn 2008). The new dekkar and Savart plate allows for more sharply defined and wider spectral orders on the focal plane. The dekkar stage allows for calibration optics to be mounted just upstream of the slit and all pure input states (QUV) are reproduced with the achromatic wave-plates above the 95% level for all wavelengths observed (5500Å-7000Å). Once this accurate reproduction of incident polarization at the slit was verified using the wave-plates the new 2-LCVR spectropolarimeter fast switching system was installed. We first show the properties of the HiVIS liquid-crystal system and then illustrate the Stokes-based deprojection for the system. The variations in signal- to-noise ratio with wavelength in the deprojected output will be discussed along with implications for the design and optimization of an achromatized spectropolarimeter.

5.4.1 HiVIS LCVR Observations In order to make the HiVIS LCVR system very robust, empirical calibration hardware and software has been developed. A new calibration unit consisting of an Edmund high-contrast polarizer and a BVO quarter-wave plate creates pure input states ( Q, U, V) at the spectrograph slit mirror. These pure inputs are used to derive± ± the± retardances and Mueller matrix coefficients of the LCVRs exactly as mounted and run during the night. As an example of this technique, we will use a sample set of Hα observations and calibrations. These observations were obtained using the Apogee 3k2 detector. The Apogee has 13 spectral orders in the 5500Å to 7000Å range each with 3056 independent polarization measurements per order. We will define Stokes +Q as parallel to the slit. The LCVR voltages were selected to be only roughly close to the ’normal sequence’ for Hα. Deliberate rounding errors in voltage were introduced to show that, even with significant retardance selection error, the Stokes-based deprojection method works well. Voltages of 7.25, 2.95, 2.10 and 1.90 volts were chosen corresponding to retardances of -0.004, 0.215, 0.425 and 0.596 waves when mounted in the lab spectropolarimeter. This lab spectropolarimeter differs from the HiVIS setup in temperature and the two LCVRs were remounted introducing potential alignment variations. These values are rough and the deprojection does not require a derivation of the actual retardances. For each pure input state, the standard set of QUV observations listed in ± 158 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

Fig. 5.11: This Figure shows the 6 measured polarization spectra of the normal sequence. The six individual exposures produce 12 spectra which are diﬀerenced as in Equation 5.24 representing QUV. The Stokes parameters have been normalized. ± The solid lines show q, the dashed line shows u and the dot-dash line shows v. To ± ± ± illustrate the repeatability of the measurement, two complete observation sequences were taken with pure +Q and Q inputs. The sign has been reversed for all measured − output states for the Q input. Thus there are 24 spectra plotted here. See text for − details.

Equation 5.16 were taken. The data was processed using the IDL reduction scripts described in Harrington & Kuhn (2008). We found that each echelle order required using a dark subtraction from the non-illuminated regions ad- jacent to each spectral order. A spatially and temporally variable dark current at the 100-ADU level is present in the Apogee detector and is removed independently for each exposure. If the usual dual-beam polarization analysis is performed using these observations then the input pure polarization states are poorly reproduced, as expected. The LCVR chromatism and our deliberate introduction of imperfect retardances causes significant changes in how the Stokes parameters are recorded on the detector. This is the perfect situation to illustrate the power of this deprojection technique to correct an imperfect polarimeter. Figure 5.11 shows the measured Stokes parameters derived from the six exposures for two 5.4. HiVIS LCVR Spectropolarimeter 159 full observing sequences. Pure +Q was input for one sequence and pure Q was input for another sequence. The sign of the measured Stokes parameters− has been reversed for the Q input sequence for ease of plotting. The high voltage setting corresponding− to zero retardance (7.25, 7.25) shows a near- perfect reproduction of q for all wavelengths (solid line on top) whereas the half-wave retardance± on the second LCVR induces a fairly chromatic dependence with only 70% reproduction of q. The u states are nearly zero at ± Hα but are highly chromatic in that the dashed lines cross zero but reach -.4 and +.8 at short wavelengths. The v states are very similar to u but with a significant offset - there is no wavelength where both + and - v are zero.

5.4.2 HiVIS LCVR Stokes-Based Deprojection Pure input states [+Q, +U, +V] were observed with a full polarization sequence described above (6 exposures, 12 spectra). As each set of pure input states are observed twice, one set with lower retardances and one set with higher retardances, we actually get two independent measurements of each input state and hence two independent deprojection matrices. The measurements of each pure input state at low and high retardance form a transfer matrix at low and high retardance respectively. We then input pure negative states for testing the calibration routines [ Q, U, V]. These calibration observations also give us 6 exposures and 12− spectra− − which correspond to QUV measurements at both low and high LCVR retardance values. The transfer matrix calculated using the positive states were inverted and used to deproject each input negative state. The resulting empirical calibration, shown in Fig- ure 5.12, shows the residuals are less than 0.1. The three input states (negative pure states) map to essentially pure measured output states (the curves near +1). The sign was reversed for clarity. The curves near zero correspond to the circular ’cross-talk’ that has not been removed by this deprojection procedure. These cross-talk values are quite small, being typically around 0.05 and show little chromatic variation. Considering that the calibration optics were manually rotated and are on simple post-mounts, we consider this correction to be excellent given the chromaticity of the LCVRs. As noted earlier, achromatic oﬀsets in cross-talk can result from imprecise rotation of the calibration optics. It is the chromatic variation of the deprojected Stokes parameters that is directly caused by errors in the deprojection process. The chromatic variation in Figure 5.12 is much smaller than the achromatic oﬀsets from zero. 160 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

Fig. 5.12: This Figure illustrates the empirical inversion process with LCVRs on HiVIS. Input states +Q, +U and +V were observed to create the deprojection matrix. The calibration optics were rotated to input negative states. These measured and deprojected negative Stokes parameters are shown here. The input state is reproduced at a value near 1 with the cross-linear and circular states are quite small.

5.4.3 HiVIS LCVR Optimization

Since deprojection removes the constraint that one Stokes parameter correspond to one exposure, we are free to choose retardances that give the desired precision for each Stokes parameter. One can optimize the observing sequence and retardances for a given requirement by controlling the properties of the deprojection matrix. For instance, the condition number of the deprojection matrix gives the ratio of the maximum and minimum eigenvalues (cf. Compain et al. 1999; de Martino et al. 2003). This number is an upper limit to the maximum relative variance of the corresponding deprojected Stokes parameters. Said another way: Given equal exposure times for the measured spectra, the statistical noise in the deprojected spectra will have a ratio of maximum to minimum variances that follows the condition number. We calculate the condition number by first applying the IDL single-value- decomposition routine SVDC to our deprojection matrix then taking the ratio of maximum to minimum eigenvalues (ξmin, ξmax). The condition number, c, 5.4. HiVIS LCVR Spectropolarimeter 161 is defined as: ξ c = max (5.26) | ξmin | We have investigated this relationship using our HiVIS observations. Us- ing the deprojection matrix we have calculated for the LCVRs, we computed the condition numbers as in Equation 5.26. The condition number for the low- retardance setting (high voltages) was always below 3. The condition number for the high-retardance setting (low voltages) was generally higher but always less than 4. This implies that, for a constant exposure time and statistical noise per recorded spectrum, the signal-to-noise of the resulting deprojected spectra will vary by no more than √3 and √4 respectively for the two settings. We then compared the variance of our polarimetric measurements both pre- and post- deprojection. Since the Apogee detector has 13 spectral orders each with 3056 independent polarization measurements, we get 13 variance measurements in the 5500Å to 7000Å range. The variance σpre,i were computed for each order i before deprojection. Since the exposure times were constant, σpre,i = σpre. Accordingly, the variances for each order j post- de- projection σpost, j were computed. Since the condition number gives the ratio of maximum to minimum variance, we define the maximum measured variance change as

σpost, j δvmeas = MAX j( ) (5.27) σpre In addition, we also simulated error propagation through our calculations with simulated input Stokes parameters. In the IDL reduction package, we simply replaced the measured Stokes parameters (13 orders, 3056 wavelengths per order) with noise about zero with a variance of 1 (σpre=1). The deprojec- sim tion was performed and the variances σpost, j were determined. We can compute the variances of this simulated data in analog to Equation 5.27:

sim δvsim = MAX j(σpost, j) (5.28)

Figure 5.13 shows the condition number, measured variance change (δvmeas) and simulated variance change (δvsim) which track each other to better than 0.2 across the recorded wavelength range. The ﬁnal test performed was to verify that the coeﬃcients in the deprojection matrices agree with the relative error values. The deprojection pro- 162 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

1 cess multiplies the inverse Mueller matrix (M− ) by the measured polarization spectra. Each deprojected spectrum is a sum of three input spectra multiplied by the corresponding deprojection matrix element. The variance of statistical noise will increase as the square of the deprojection matrix element and the variance will also increase as the sum of these squared values. The variance should go as n=a2+b2+c2 where a, b and c represent the row of the deprojec- 1 tion matrix (M− ). This we term the ’noise amplification coefficient’. This coefficient also agrees with the simulated variance and the condition number to better than 0.2 as seen in Figure 5.13. If we define each term of the depro- 1 th th jection matrix as Mi− j for the i row and j column, we can define these noise amplification coefficients ni as:

1 2 ni = (Mi− j ) (5.29) Xj Since these noise ampliﬁcation coeﬃcients are a direct measure of the signal-to-noise computed for each deprojected Stokes parameter, they can be used to optimize an observing sequence for any desired outcome.

Fig. 5.13: This Figure illustrates the relationship between the condition number, measured variance, simulated variance, and ’noise-ampliﬁcation-coeﬃcient’. Since the variances are computed as ratios, all four curves should agree.

There are many schemes for tuning polarimetric measurements where the 5.4. HiVIS LCVR Spectropolarimeter 163 properties of the retarders (fast-axis orientation and retardance) are varied in order to give some speciﬁed performance of the condition numbers. There are eﬃciency-balanced schemes, which maintain condition-numbers of 1, optimize for equal sensitivity between linear (QU) and circular (V) or optimize on other constraints. (c.f. Nagaraju et al. 2007; Compain et al. 1999; de Mar- tino et al. 2003).

The ultimate limitations of this system lie in the measurement of the Mueller matrix terms. In this example, even where manual rotation of two different optics induced a substantial systematic error, the achromatized performance was acceptable. The measured cross-talk error in the output Stokes parameters was satisfactorily achromatic. Uncertainties in voltages or calculated LCVR phases do not influence the performance of the Stokes-based deprojection. There were systematic offsets that can be explained as 1-2◦ errors in rotation of the calibration optics. The chromatic error in the recovered polarization are significantly lower. A motorized stage for the calibration optics should improve the achromatization even further.

This system is designed to be used for line spectropolarimetry which is an inherently differential technique. The continuum polarization accuracy of this system is limited by these deprojection uncertainties, as well as the polarization induced by the optics of the system. However, this calibration is quite sufficient to create a stable polarimetric reference frame for measurements of polarization across individual spectral lines. One of the most widely used high-resolution spectropolarimeters, ESPaDOnS, has a cross-talk of 2% and a wavelength-dependent, time-variable continuum polarization (Donati 2006, ESPaDOnS Instrument Website 5). Absolute polarization measurements suf- fer from many systematic errors but the change in instrument properties over an individual spectral line are negligible. Hence, achromatization of the instrument performance to an accuracy of a few percent is quite sufficient to allow for accurate differential measurements.

5 www.cfht.hawaii.edu/Instruments/Spectroscopy/Espadons/ 164 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter

5.5 Polarization Properties of the HiVIS Spectrograph

In order to perform accurate polarimetry of astronomical sources, a polarization calibration of the telescope and spectrograph optics is also required. The deprojection procedure we have just outlined is quite general and can also be applied to undo cross-talk induced by other optical elements in an instrument. The AEOS telescope with 5 oblique fold mirrors and the HiVIS spectrograph with its image rotator is a suitable example. The AEOS telescope induces pointing-dependent cross talk. The image rotator in HiVIS induces rotation- dependent cross-talk. The spectrograph itself also induces cross-talk even without the image rotator. Since the Savart plate analyzer is just after the slit, only the optics before the slit induce polarimetric effects. In this section, we will use the achromatic wave-plate spectropolarimeter to measure the polarization properties of the HiVIS spectrograph optics. The image rotator mount was aligned and remounted to allow repeatable removal and re-insertion of the image rotator with negligible influence on the spectrograph beam path. The optical path from the last coude´ mirror through the spectrograph fore-optics to the analyzer has many elements. There are three near-normal incidence fold mirrors and a 5.5m focal length collimator before coming to the image rotator (3 oblique reflections). Another fold, the tip-tilt mirror, a 1.1m focal length sphere and another fold are used to create the stellar image at the slit. To calibrate with pure linear polarization, the wire grid polarizer was mounted at the calibration stage inside a mask to polarize the diffusion screen and to block unpolarized light. This serves as a polarized flat field creating pure QU inputs to the spectrograph fore-optics.

5.5.1 Polarization With Image Rotator The polarization response of the spectrograph fore-optics was first measured with the image rotator in the beam and aligned vertically. Pure Q and U states were input. The cross-talk effects can be best illustrated± by separat-± ing the output measurements into three components - the reproduction of the input state, the linear cross-talk and the circular-cross-talk. The measured output separated in this manner is shown in Figure 5.14. Only input U shows significant wavelength-dependent cross-talk. The overall degree of polarization, p, calculated as p2=q2 + u2 + v2, is very high for both Q and U inputs. 5.5. Polarization Properties of the HiVIS Spectrograph 165

Fig. 5.14: This Figure shows the output Stokes parameters for four inputs (+Q, +U, Q, U) with the image rotator in the nominal orientation. Solid lines are − − for Q inputs, dashed lines are for U inputs. The top box shows the measured out- ± ± put state for the same input state (eg. +Q out for +Q in). The middle panel shows the non-input linear Stokes parameter showing linear cross-talk of up to 0.2. The bottom panel shows circular polarization and is entirely cross-talk.

The values range from 96-100% and thus the ’depolarization’ terms are small. The negative and positive input states give results that are opposite in sign but very similar in magnitude and morphology. The linear cross-talk of the middle panel shows symmetry about zero and a very small 0.1-0.2 amplitude. There is significant linear cross-talk for input U and a small amount for input Q at longer wavelengths. The third panel shows high UV cross-talk for input U at both short and long wavelengths with essentially a linear dependence with wavelength. The UV cross-talk fortuitously goes to zero around 6500Å. The induced polarization is measured to be quite small, being under 5% to 6% in these wavelengths. Since the induced polarization and depolarization terms are small with nearly 100% reproduction of the input degree of polarization, the Mueller matrix for the HiVIS system is essentially a 3x3 matrix 166 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter involving QUV terms in rotation and linear-circular cross-talk. To investigate the image rotator further, the polarization properties of the spectrograph are measured with the image rotator at angles of -90,-45,0,+45 and +90 degrees. Induced polarization (unpolarized flat field measurements) as well as purely polarized inputs were measured at four polarizer orientations (+Q, +U, Q, U) for each of the 5 image rotator orientations. This set of 20 observations− − allows us to characterize the wavelength-dependent changes with rotator angle. The first thing to note about the flat field is that there is significant linear polarization for all the image rotator orientations. The q terms all generally decrease to longer wavelengths while the u terms are more generally flat with wavelength. The 90◦ observations show a very significant u of 0.1-0.15 ± though with low v. The 45◦ observations show significant v at short and long wavelengths, with all observations± showing zero induced circular polarization around 6200Å, 300Å shorter than other orientations. This shows that the induced polarization is at the 2% to 15% level and is highly dependent on the image rotator. The overall polarization properties of Figure 5.14 are generally reproduced at the varying image rotator orientations. The polarization reference frame rotates with the image rotator as expected. There is some change in the exact form of the linear to circular cross-talk but the relative amplitude does not change by much. The linear cross-talk is roughly double and there is some additional QV circular cross-talk when the rotator is oriented 45◦. We should note that some calibrations were repeated with the polarizer at± different spatial locations and the resulting observations did not change significantly.

5.5.2 Polarization Without Image Rotator Given the strong polarization variation when using the image rotator as well as the high fold angles of the mirrors, we decided to implement a mode with the image rotator removed from the beam. This removes our control of the projected slit orientation on the sky. However, we gain some very positive beneﬁts. The removal of moving oblique-fold mirrors greatly reduces the complexity, magnitude and wavelength-dependence of the cross-talk. We also gain in throughput and get reduced static wavefront errors caused by the high- angle folds. The AEOS telescope itself has ﬁve 45◦ folds before reaching our spectrograph, but now there are no high-angle folds in the spectrograph optics. 5.5. Polarization Properties of the HiVIS Spectrograph 167

Fig. 5.15: This Figure shows the measured polarization properties with the image rotator removed. Pure input Q is shown with a solid line and was measured twice. Pure input U is the dashed line. The top panel shows the reproduced linear polarization input (Qin to Qout, Uin to Uout). The middle panel shows linear cross-talk. The bottom panel shows the measured Stokes V running from 0 to about 0.4.

Though the rotator adds roughly 15cm to the optical path after the collimator, this is a small change given the 550cm path length from the collimator to the tip tilt pupil image. The flat field illumination pattern did not significantly change and the imaged spectral orders are essentially identical with and without the image rotator. The alignment of the system is good enough that the image rotator can simply be removed and replaced as needed without changing any other mirror orientations. The polarization measurements showed that the fore-optics induced polarization is typically 5% to 6% percent depending on the illumination pattern. The flat field screen and tip-tilt guiding piezo were moved to measure the response to a change in the illumination pattern. There are differences caused by the varying illumination, but a coherent pattern is easily recognized. The flat field is dominated by Stokes q being around 0.05 to 0.06. Stokes u is 5 to 10 times smaller, being 0.006 to 0.012. The induced circular polarization, v, is another order of magnitude smaller than this. The total degree of polar- 168 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter ization is essentially the degree of linear polarization of 5% to 6% and this is essentially all Stokes q. This is quite a small variation given the 2% to 15% that can be induced with the image rotator at various orientations. The major improvements can be seen in the reproduction of pure input linear polarization. Figure 5.15 shows the measured polarization properties for input Q (repeated twice, solid lines) and U (dashed lines). The top panel shows the reproduced linear polarization input at a roughly constant 98% for input Q and a 98% falling to 90% for an input U. The middle panel shows the linear cross-talk. The chromatic variation is very small and corresponds to less than 1◦. The bottom panel shows the measured Stokes V which is nearly 0 for input Q (just as with the image rotator in place) but only 0 rising to 0.4 for an input U (as opposed to 1 changing to 1 with the image rotator in place). The general depolarization± is much improved.∓ This can be seen as the pure linear inputs being reproduced at the 90% to 98% level. The total degree of polarization is always above the 98% level. The mirror-induced rotation of the plane of polarization is nearly negligible, in contrast to the 10◦ seen with the image rotator. The induced V was very strong and very chromatic with the image rotator, but the induced V now has a maximum of 0.4 only for input U and only at long wavelengths. There is essentially no circular cross-talk at 5500Å. As we had seen earlier, both the induced polarization and the depolarization are small. This alone suggests that one can assume that any linear to circular cross-talk has a corresponding circular to linear component. In order to verify this assumption, we remounted the slit calibration optics (high-contrast polarizer and quarter-wave plate) just down stream of the diffusion screen in order to create a Stokes V input. We obtained several full-Stokes observations with both +V and V inputs at various combinations of polarizer and quarter- wave plate orientations.− The input V had U cross-talk in direct response to the UV cross-talk seen in Figure 5.15. We also measured linear polarization as a consistency check and found excellent agreement. As expected, the image rotator is the most significant source of induced polarization and cross-talk in our polarimeter. Removing the image rotator created several polarimetric advantages. The circular polarization cross-talk in the system is 1 linearly changing to 1 from 5500Å to 7000Å for linear ± ∓ polarization input at 45◦. This induced circular polarization becomes much smaller without the image rotator, showing only UV cross-talk rising to 0.4 at long wavelengths. The chromatic rotation of the plane of polarization was 5.6. AEOS Telescope Polarization 169

Fig. 5.16: This Figure shows the measured twilight polarization properties using HiVIS without the image rotator. Each panel shows Stokes quv for twilight pointing at the zenith. The left panel shows cardinal pointings: north, east, south and west. The right panel shows non-cardinal pointings: north-east, south-east, south-west, north- west. The degree of polarization for all measurements was always 70% to 80%. At non-cardinal pointings, the linear-circular cross-talk was severe and reached 100% at 6500Å.

up to 10◦ with the rotator but is less than 1◦ without the rotator. In general the depolarization is low both with and without the image rotator. The induced polarization without the rotator is almost entirely Stokes q at 5% to 6% with negligible induced circular polarization and with only a moderate variation with changing illumination patterns. This is to be expected as Stokes Q is deﬁned parallel to the slit which is parallel to the image rotator axis. The spectrograph is more easily deprojected without the rotator as the condition number of the deprojection matrix is much smaller. This allows us to recover an accurate polarimetric reference frame at the entrance to the spectrograph and to separate the telescope polarization properties from the spectrograph.

5.6 AEOS Telescope Polarization

Knowing the polarization properties of the HiVIS spectrograph with and without the image rotator, we can now quantify the polarization properties of the AEOS coude´ path. A simple experiment to illustrate the cross-talk is to look at 170 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter twilight at the zenith. The linear polarization properties of twilight observed with HiVIS were investigated in Harrington & Kuhn (2008). They adopt a simple Rayleigh scattering model (cf. Liu & Voss 1997; Lee 1998; Cronin et al. 2006). With the sun on the horizon in the west the zenith is highly polarized with the polarization angle pointing north-south. This is a simple approximation that holds quite generally (cf. Pomozi et al. 2001; Suhai & Horvath´ 2004; Horvath´ et al. 2002).

The AEOS coude´ path has ﬁve 45◦ reﬂections and there is a pair of mirrors corresponding to the azimuth axis and a pair of mirrors corresponding to the altitude axis. If we point the telescope at the zenith and simply change azimuth through 360◦, we can measure the polarization properties of this coude´ path as two mirrors cross and uncross. In Harrington & Kuhn (2008), the linear polarization of the zenith was found to decrease from 70% to 80% at some pointings to 15% at other pointings. We now know that without the image rotator there is nearly no linear-circular cross-talk around 5500Å but there is 0.4 at 7000Å.

We repeated the twilight experiments without the image rotator using the new full-Stokes capabilities with achromatic wave plates. Figure 5.16 shows the measured Stokes parameters of twilight at the zenith on September 5th 2009. On this night the sun set at 280◦ azimuth. The left hand panel shows the Stokes parameters at azimuths of north, east, south and west (10, 100, 190 and 280). The right panel shows north-east, south-east, south-west and north-west (55, 145, 235 and 325). One can see that for the cardinal direction pointings of the left panel, the linear polarization is almost all Stokes q with v varying from -0.2 to +0.2 from 5500Å to 7000Å. Conversely, at the non-cardinal pointings of the right panel, the observed polarization is substantially more chromatic and is entirely Stokes v at some wavelengths. The degree of polarization is again observed to be always 70% to 80% for all wavelengths. With these new observations we can conclude that the loss of linear polarization and the rotation of the plane of polarization seen in Harrington & Kuhn at 6500Å is due to linear-circular cross talk. We are in the process of measuring the telescope polarization properties at all pointings. 5.7. Discussion And Conclusions 171

5.7 Discussion And Conclusions

We have presented a Stokes-based deprojection method applied under various circumstances to highly chromatic astronomical systems. These methods are an easy and robust way to restore a reasonable polarization reference frame to a system with very significant chromatism and cross-talk. Since it is often more cost-effective to add spectropolarimetric capabilities to existing instruments, one typically has to deal with substantial systematic effects. Stokes-based deprojection is a straightforward and repeatable method to remove some of these errors. A simple laboratory spectropolarimeter was built to characterize and test our new optics as well as to demonstrate the Stokes-based deprojection methods in the lab. Retardances of our achromatic wave plates as well as LCVRs were measured. We verified the achromatic performance of the wave plates over their designed wavelength range. The LCVRs were characterized by creating a grid of retardances versus wavelength and voltage. An LCVR spectropolarimeter was constructed and a standard sequence of observations were taken. The measurements were first deprojected using standard retardance- fitting. This retardance based deprojection was then compared to the Stokes- based deprojection method. The Stokes-based method was both easier to implement, less chromatic and more accurate. The Stokes-based deprojection methods were then demonstrated with a new HiVIS LCVR spectropolarimeter. HiVIS now includes several optical / mechanical improvements, a polarization calibration stage, a new full-Stokes achromatic wave plate mode and a liquid-crystal mode. This LCVR mode is now fully characterized to allow the choice of polarization sensitivities optimized for a wide range of applications. We have shown that one need not map individual Stokes parameters to individual exposures. For instance, we are implementing a mode where our new fast shuffling detector can inter- leave orders on the focal plane. With this information, we can implement a mode where the condition number of the deprojection matrix is optimized for different sensitivities within each exposure. Linear, circular or full-Stokes sensitivity can be specified within each frame to minimize systematic error in measurements of each Stokes parameter. We then used the achromatic wave-plate mode to fully calibrate the HiVIS spectrograph and show, as expected, that the main polarimetric effect of the spectrograph mirrors is linear-circular cross-talk caused by the image rota- 172 Chapter 5. Achromatizing a liquid-crystal spectropolarimeter tor. With this image rotator removed, the spectrograph still has linear-circular cross talk at longer wavelengths at the 0.4 level. We then used this spectrograph calibration information to show that the AEOS telescope also induces 100% linear-circular cross-talk at the zenith when at non-cardinal azimuths. The image rotator and telescope can both induce 100% linear-circular cross-talk. These polarimetric results are repeatable and as such, may be calibrated. Accurate absolute polarimetry from any modern alt-az telescope requires careful cross-talk calibration as we describe here. Stokes-based deprojection applied to highly chromatic systems will significantly improve the repeatability and accuracy of spectropolarimetric measurements. The LCVR system, though nominally chromatic, can be used to obtain broad-band spectropolarimetry with good accuracy and repeatability. This achromatization allows one to realize the performance advantages of a fast-switching system with no moving parts. We outlined how systematic errors resulting from rotating wave-plates and slow modulation may now be eliminated from the HiVIS spectropolarimeter. Acknowledgements. This program was partially supported by the NSF AST-0123390 grant, the University of Hawaii, the AirForce Research Labs (AFRL) and an SNF grant PE002-104552. Bibliography

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Summary

The wealth of information imprinted on the polarization state of light is a se- cret well hidden in photon noise for most astronomical targets. The present work illustrates both analysis and instrumental techniques to trace the signatures of spectral lines to a degree beyond current limits. The big question of “the role of magnetic fields in the universe” and in stars in particular cannot be answered without measuring these fields in the context of other parameters. In this thesis we have developed, tested and applied new diagnostic tools that will help to shed light on the (magnetic) surface activity not only for stars at the extreme of parameter space, but also for more quiet Sun-like main-sequence objects in particular: The shape of absorption line profiles is crucial for inferring properties of the (unresolved) region of line formation. Addressing the important question of the reliability of other, commonly used multi-line techniques, we have come up with a new technique called Nonlinear deconvolution with deblending (NDD). For the first time, a multi-line technique based on the premise of an unconstrained common Stokes profile takes into account the nonlinearity of intrinsically blended line profiles, enabling us to unravel even a molecular band as an example of the most challenging of entangled profiles. Expanding this approach to the polarized Stokes parameters, Zeeman Com- ponent Decomposition (ZCD) is a code for the simultaneous inversion of 178 Chapter 6. Summary thousands of spectral lines. Within the current assumption of a uniform field distribution, it sets a new threshold for the detectability of very weak magnetic fields, while at the same time increasing the interpretability of the common absorption line profile. The confirmation, or rather refutation, of astronomical models, is ultimately ascertained by observations. The ongoing effort to build advanced instruments is a vital ingredient to our knowledge. Aiming for a high-precision spectropolarimeter with various efficiency schemes toward different Stokes parameters, we calibrated, implemented and tested a new polarimetric unit for the HiVIS instrument using liquid-crystal variable retarders. We demonstrated how a Stokes-based deprojection method can be applied for dual-beam broad-band spectropolarimetry to reduce systematic effects and to be state-of- the-art in terms of both accuracy and versatility. The detection of a weak longitudinal magnetic field on the evolved giant Arcturus from a supposedly diminishing solar-type dynamo denotes a further step toward a better understanding of the underlying stellar activity generators. The gain in polarimetric sensitivity in our application involves a loss in spatial resolution. In general, whenever noise dominates over possible circular polarization signals, it seems expedient to increase the sensitivity toward fewer, simpler magnetic structures at the expense of stellar disk-resolving power. Using the ratio of instrumental spectral resolution to the projected rotational velocity of the star at the same time may serve to estimate the optimum number of surface resolution elements for each observation. This will greatly reduce the numerical artifacts seen in many (Zeeman-) Doppler imaging maps. Although established for decades, the full potential and application possibilities of multi-line techniques have so far only been skimmed at the surface, with ZCD trying to bring about less doubted renown in the near future. The simultaneous use of multiple spectral lines is one of the few diagnostic tools contrived by stellar astronomers, that may eventually find its application in solar physics, and not vice versa. Curriculum Vitae

of Christian Sennhauser, born January 22, 1980 in Zurich, Switzerland since 2007 Research assistant at the Institute for Astronomy, ETH Zurich (ETHZ), Switzerland March 2007 Diploma thesis on Measurements of stellar magnetic ﬁelds at IRSOL at the Institute for Astronomy of the ETHZ under the guidance of Prof. Dr. S. V. Berdyugina 2001 – 2007 Studies of Physics at ETHZ January 2000 Matura type D (languages) 1993 - 2000 High School in Zurich, Switzerland 1987 - 1993 Primary School in Adliswil, Switzerland

Member of the Swiss Society of Astrophysics and Astronomy (SGAA) 180 Curriculum Vitae List of Publications

Publications in refereed journals

Sennhauser, C.; Berdyugina, S. V.; Fluri, D. M., “Nonlinear deconvo- • lution with deblending: a new analyzing technique for spectroscopy”, 2009, Astron. Astrophys., 507, 1711

Sennhauser, C.; Berdyugina, S. V., “Zeeman component decomposition • for recovering common proﬁles and magnetic ﬁelds”, 2010, Astron. Astrophys., 522, A57

Sennhauser, C.; Berdyugina, S. V., “First detection of a weak magnetic • ﬁeld on the giant Arcturus. Remnants of a solar dynamo?”, 2010, As- tron. Astrophys., submitted

Harrington, D. M.; Kuhn, J. R.; Sennhauser, C.; Messersmith, E. J.; • Thornton, R. J., “Achromatizing a Liquid-Crystal Spectropolarimeter: Retardance vs. Stokes-Based Calibration of HiVIS”, 2010, Publ. As- tron. Soc. Pac., 122, 420

Publications in conference proceedings

Sennhauser, C.; Berdyugina, S. V.; Fluri, D. M., 2009, “Zeeman-Doppler • Imaging of Stellar Magnetic Fields with Atomic and Molecular Lines”, in S. V. Berdyugina, K. N. Nagendra, R. Ramelli (eds.), “Solar Polar- 182 List of Publications

ization 5: In Honor of Jan Stenﬂo”, vol. 405 of “Astron. Soc. Pac. Conf. Ser.”, 543

Sennhauser, C.; Berdyugina, S. V.; Fluri, D. M., 2009, “LSD-a non- • linear approach”, in “15th Cambridge Workshop on Cool Stars, Stellar Systems and the Sun”, vol. 1094 of “AIP. Conf. Ser.”, 732

Sennhauser, C.; Berdyugina, S. V., 2010, “Zeeman Component Decom- • position (ZCD) of Polarized Spectra: Application for the Quiet Sun In- ternetwork Magnetic Field”, in S. R. Cranmer, J. T. Hoeksema, J. L. Kohl (eds.), “SOHO-23: Understanding a Peculiar Solar Minimum”, vol. 428 of “ASP Conf. Ser.”, 113