From accurate atomic data to elaborate stellar modeling: Structure and collisional data, opacities, radiative accelerations.

Transcription

1 From accurate atomic data to elaborate stellar modeling: Structure and collisional data, opacities, radiative accelerations. DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University By Franck Delahaye ***** The Ohio State University 2005 Dissertation Committee: Approved by Prof. Anil K. Pradhan, Adviser Prof. Marc H. Pinsonneault Prof. Donald M. Terndrup Adviser Astronomy Graduate Program Dr. Claude J. Zeippen

2 ABSTRACT The Atomic physics field has served astronomy for a long time and it has also been stimulated by it. Despite the maturity of the field, the new requirements for accuracy and completeness from astronomers demand more challenging calculations. Current problems require the relaxation of approximations and a detailed study of different effects like relativistic effects, radiation damping and target expansions. We carry out elaborate relativistic atomic calculations using the Breit-Pauli R-matrix method to generate radiative and collisional data for atoms and ions of astrophysical interest. Electron impact excitation of He-like ions (N, O, Ne, Mg, Al, Si, S and Ca) have been calculated including relativistic effects, the levels up to the complex n = 4 and radiation damping. A detailed study of their effects is presented here and the comparison with previous works shows differences of up to 30% in the effective collision strength for the z-line. This line with 3 other transitions between the complexes n = 1 and n = 2 constitute an important tool for plasma diagnostics and such differences will have strong consequences in the analysis of X-Ray spectra of astronomical objects. ii

3 We present the relativistic calculation of Kα resonances for oxygen ions that are responsible for X-Ray absorption features observed in the spectrum of the Seyfert 1 galaxy MCG and other AGNs. The resonance oscillator strengths have been computed from the photoionization cross sections and appear to be strong. Comparison with the recent experimental and theoretical study of Kα photoionization of O II shows good agreement. These data should allow a more complete analysis of X-Ray spectra from AGNs and refine the general picture of such complex objects. We report the first large relativistic calculation of the photoionization cross section of Fe II. The detailed fine structure in the present work agrees well with the experimental results from Kjeldsen et al. (2002) and represents an improvement over the previous non-relativistic results based on the LS coupling approximation. As a complement to the X-Ray analysis of AGNs, these data should be useful to the theoretical template of Fe II emissions crucial in the analysis of the UV/O/IR counterpart of the spectrum. The new OP opacities are presented, outlining the importance of the inner-shell transitions for opacities at high temperature - high density regimes. Despite the overall good agreement with another source of data (OPAL), the differences are noticeable. This new set of data is used to determine constraints on the solar composition and to shed some light on the Solar Convection Problem. While the discrepancies between the two sources are minor for the Rosseland mean opacities, iii

4 they affect more severely the radiative accelerations. The comparison between the accelerations computed with OPAL data and those obtained with OP data outlines how sensitive they are to details in monochromatic opacities. The consequences for stellar models are discussed. iv

5 To Jeanne-Claire, Hubert, Solène and Mael. v

6 ACKNOWLEDGMENTS I am greatly indebted and grateful to Claude J. Zeippen who introduced me to the work of a researcher long ago. He allowed me to discover a new, exciting and stimulating world. He never stopped supporting me at all levels. He encouraged me in the right direction even when I had lost all hope. I thank him for his competence, his patience and his humor. I would like to thank my adviser, Anil K. Pradhan, who warmly welcomed me in his group at OSU. I thank him for his guidance and fruitful discussions, for his precious help and constant support during these difficult - but rewarding - years of a PhD student. I would like to thank Marc H. Pinsonneault who introduced and hooked me to this exciting topic of stellar theory. His rigorous and knowledgeable approach taught me a lot and serves me immensely. I am greatful for his endless patience and help. I thank Mike J. Seaton for his advice and constructive discussions on opacity and radiative accelerations. He gave me generously of his time and knowledge. I would like to thank Werner Eissner for his help on running and debugging the codes, and conversations on theoretical questions. I am grateful to Georges Alecian, vi

7 Keith A. Berrington, Emile Biémont, Claudio Mendoza and Peter J. Storey for their helpful comments and suggestions. I thank my collaborators Anil K. Pradhan, Sultana N. Nahar, Guo-Xin Chen and Justin Oelgoetz for allowing me to present our joint work in the present thesis. It has been very important for me to be part of the international Iron Project team. I am grateful to all the project members for excellent collaboration. I would like to thank all the people in the Department of Astronomy at OSU who always diligently solved all the problems I have encountered in sometimes very tricky situations. They always did their best and even more. I will not endeavour to give here a list of names for fear of forgetting someone among all those who helped me and make this department a stimulating and peaceful place. Their care allowed me to pursue my PhD endeavor in the best conditions. I would like to thank the CNRS and the Observatoire de Paris for the financial and logistic support provided during my stays in Meudon, France. In particular, I am grateful to Fabienne Casoli, former Deputy Director of the INSU, for her help. While in France, I have been an associate member of two laboratories at the Observatoire de Paris: first the DAEC and more recently the LUTh, with the much appreciated support of their respective heads, Georges Alecian and Jean-Michel Alimi. vii

8 I thank my parents heartily for insisting on my going to the university, always leaving open for me as many options as possible. Both Jeanne Claire and I want to thank our families and friends for their warm support during this extraordinary journey. Spending six years in the United States led us to meet many interesting people and I want to thank all of them for helping us to make this adventure so wonderful. viii

9 VITA November 26, Born Versailles, France B.S. Physics, Université Paris 7, France M.S. in Geophysics, I.P.G.P. - Université Paris 7, France Graduate Teaching and Research Associate, The Ohio State University PUBLICATIONS Research Publications 1. F. Delahaye, M. H. Pinsonneault, Comparison of radiative accelerations obtained with atomic data from OP and OPAL, ApJ 625, pp (2005). 2. N. R. Badnell, M. A. Bautista, K. Butler, F. Delahaye, C. Mendoza, P. Palmeri, C. J. Zeippen and M. J. Seaton, Up-dated opacities from the Opacitiy Project, MNRAS 360, pp (2005). ix

10 3. F. Delahaye, Sultana N. Nahar, Anil K. Pradhan and Hong Lin Zhang, Resolution and accuracy of the resonances in R-matrix cross section, J. Phys. B: At. Mol. Opt. Phys., Volume 37, Issue 12, pp (2004). 4. A.K. Pradhan, G.X. Chen, F. Delahaye, S. Nahar, J. Oelgoetz, X-ray absorption via K-alpha resonance complexes in oxygen ions, Monthly Notices of Roy. Astro. Soc. 341, 1268 (2003). 5. F. Delahaye and A.K. Pradhan, Electron impact excitation of Heliumlike oxygen up to n=4 levels including radiation damping, J.Phys. B: At. Mol. Opt. Phys., Volume 35, pp (2002). 6. S.N. Nahar, F. Delahaye, A.K. Pradhan, C.J. Zeippen, Atomic data from the Iron Project XLIII. Transition probabilities for FeV, Astron. Astrophys. Suppl. Ser., 144, (2000). 7. E. Biémont, F. Delahaye, C.J. Zeippen, Transition rates for the doubletquadruplet intersystem lines in C II and N III, J. Phys. B: At. Mol. Opt. Phys., Volume 27, Issue 24, pp (1994). x

11 FIELDS OF STUDY Major Field: Astronomy xi

12 Table of Contents Abstract Dedication Acknowledgments ii v vi Vita ix List of Tables List of Figures xv xvii 1 Introduction X-Ray Fe data Opacities Radiative Accelerations The Opacity Project and The Iron Project My Contribution Scope of the Dissertation General Method for atomic data calculations Introduction The Close Coupling theory and the R Matrix method xii

13 2.2.1 The Target The Coupled Integro-Differential equations The R-matrix method The BPRM suite of programs An Electron Impact Excitation calculation: He-like ions X-Ray diagnostics and He-like ions Helium-like Oxygen: O VII Details of calculations Results and Discussion Helium isoelectronic sequence: N, Ne, Mg, Al, Si, S, Ca Inner-shell excitation for X-Ray transitions: Oxygen ions Details of calculations Results and Discussion A Photoionization calculation: Fe II Introduction Details of calculations Results and Discussion Opacities for stellar structure and evolution calculations Introduction General definitions Some Details on the opacity calculations: New OP data xiii

14 6.3.1 Equation of state Inner-shell transitions Frequency mesh and opacity sampling Results Inner-shell transitions, monochromatic opacities and Rosseland mean Comparison OP-OPAL Opacities and Radiative accelerations Introduction Method Results Comparison between New OP and OP OP and OPAL: Direct comparison for C OP vs OPAL: Other elements Discussion Stellar Model differences Further discussion Conclusion Opacities and solar model: Helioseismology constraints on the solar composition Introduction Method: On the error analysis of the input physics Initial solar model xiv

15 8.2.2 Opacity tables Uncertainties and error budget Constraints on the composition Conclusion General Conclusion Accuracy of Atomic data Future Work A Atomic Units 187 B Reduced collision strengths and asymptotic behavior. 188 Bibliography 190 xv

16 List of Tables 3.1 Energies for the 31 levels of O VII compared to values compiled at NIST Transitions probabilities and oscillator strengths for OVII compared to values compiled at NIST Kα resonance oscillator strengths f r for oxygen ions Energies for the 108 levels of Fe II compared to the values compiled at NIST and Nahar & Pradhan (1994, NP94) Energies for the 108 levels of Fe II compared to the values compiled at NIST and Nahar & Pradhan (1994, NP94) Energies for the 108 levels of Fe II compared to the values compiled at NIST and Nahar & Pradhan (1994, NP94) (ρ-t) points where direct comparison is possible Composition from Grevesse & Sauval κ R : rms of the percentage difference between OP and OPAL OPAL and OP κ R and γ C Comparison of γ(c) obtained with OPAL and OP data without mte effect Comparison of γ(c) obtained with OPAL and OP data including mte effect Comparison of the C acceleration obtained with OPAL and OP data Comparison of g rad with mte included Composition xvi

17 8.2 Different sources of error in the determination of R CZ Best solution xvii

18 List of Figures 1.1 XMM RGS spectra of 5 stellar system (Güdel et al. 2001) Artistic representation of a black hole and its surrounding (NASA/CXC/SAO) Chandra HETG spectrum of MCG (Lee et al. 2001) FeII pseudo-continuum emission in I Zw 1 UV spectrum (Vestergaard & Wilkes, 2001) Abundance anomalies in M15 (Behr et al. 2000) Iron convection zone predicted by stellar models including radiative levitation (Richard et al. 2001) Isochrones for M92 (Vandenberg et al. 2002) Configuration space in the R-Matrix method The RMATRX flow chart Simplified level scheme for He-like ions Collision strengths for the principal X-Ray line transitions Collision strengths for transitions from the ground state to the n = 3 complex Partial collision strengths from the ground state to the n = 2 complex Partial collision strengths from the ground state to the n = 3 complex Total averaged collision strengths from the ground state to the n = 2 complex xviii

19 3.7 Effective collision strengths for the principal lines (z, x, y, w) Effective collision strengths for transitions within the n = 2 complex Effective collision strengths for transitions between different complexes Detail of collision strengths for transition 1 1 S S Detail of collision strengths for transition 1 1 S P o Detail of collision strengths for transition 1 1 S P o Detail of collision strengths for transition 1 1 S P o Reduced collision strengths for He-like ions Details of reduced effective collision strengths for the intercombination transition Comparison of effective collision strengths for He-like ions with previous work Photoionization cross sections of Kα resonance complexes in oxygen ions, OI OV I Fe II photoionization cross sections: LSJ vs LS Fe II photoionization cross sections: Comparison with experimental results The weighting function f(u) for the calculation of κ R Monochromatic opacities for C and Fe and total from OP and OPAL Rosseland mean opacity from OP with and without inner-shell transitions Rosseland mean opacity from OP and OPAL Comparison of the Rosseland mean opacity: OP vs OPAL at high densities Comparison of the Rosseland mean opacity: OP vs OPAL at low densities113 xix

20 6.7 Comparison of the Rosseland mean opacity: OP vs OPAL for the solar model Monochromatic opacities for C and Fe Differences in the Rosseland mean opacities between OP and OPAL Differences in radiative accelerations for the solar model Difference in accelerations for T eff = 10000K logr = Difference in accelerations for M = 1.3M Difference in accelerations for M = 1.5M Theoretical and Observational value of R CZ and Y surf Difference (Theory - Observation) for R CZ and Y surf Oxygen and Iron abundances needed to fit R Sun CZ and Y Sun surf Predicted values of R CZ and Y surf when the composition is modified Photoionization cross sections of O II Percentage differences in photoionization rates of O II Monochromatic bound-free opacity of O II xx

21 Chapter 1 Introduction For a variety of reasons, philosophical, metaphysical or due to scientific curiosity humans have always looked at other worlds outside ours and tried to understand outer space. This quest has had to rely on the capacity of our eyes to gather valuable information from what we see above our head. Even today, the extraterrestrial material that we have in hand is limited to a few meteorites found at the surface of our planet and samples from the Moon brought back during the Apollo missions. Besides the work of sky surveys and the description of the motion of celestial objects, present astronomical research is for a large part based on the interpretation of the spectra emitted or absorbed by various objects. The analysis of lines (in spectroscopy) over a broad range of wavelengths, from gamma ray to radio, allows astronomers to deduce characteristics like the chemical composition, densities, temperatures, distances or velocities of a wide variety of objects. From our local Sun to the most distant quasars, from the intergalactic medium to massive black holes at the center of host galaxies the radiation arises essentially from atomic processes involving the interaction of atoms, ions or molecules and electrons and/or photons. Among the most important processes, electron impact excitation, photoionization, 1

22 recombination, photo-excitation and radiative decays have to be studied in order to interpret these observations. Astronomy, in that sense, has stimulated the study of these quantum mechanical phenomena. The astrophysical community has to rely on theoretical calculations to provide the large sets of atomic data needed to interpret their observations. Because the theory and methods used to produce these data date from the thirties to sixties and seventies it often seems that this is routine work. However the more demanding astronomical problems have challenged the atomic and molecular physics field by requiring more data for more complex systems with greater accuracy and this at all wavelengths. High accuracy and completeness require elaborate physical methods, efficient numerical techniques, optimized computer codes and knowledgeable data producers X-Ray The high quality and wealth of spectra obtained by Chandra and XMM since their launch (July 29, 1999 and December 10, 1999 respectively) have made the X-Ray astronomy one of the most active topics of astronomy. From the study of young stellar objects, cataclysmic variables and neutron stars to the observation of accretion by black holes living at the center of galaxies all domains of astronomy have been affected. 2

23 Figure 1.1 (Güdel et al. 2001) shows the soft X-Ray spectra of five stellar systems obtained with the XMM Reflection Grating Spectrometers (RGS). We clearly see prominent lines from N, O, Ne, Mg and Si. These lines arise from transitions between low-lying energy levels of ions many times ionized (only 2 electrons left). These lines from He-like ions, as presented in chapter 3, are powerful tools which may be used to determine the electron density and temperature of the emitting plasma as well as abundances for the different species. It is important to revise the available data including the important relativistic effects as well as higher complex levels and radiation damping that affect significantly the results for these transitions and therefore the quality of the diagnostic as presented in chapter 3. The paradigm of the black hole is represented in Figure 1.2. Active Galactic Nuclei (AGNs) are the most luminous objects observed. The radiation emitted cannot be attributed to stars. The paradigm is that a massive black hole, sitting at the center of a galaxy, accretes matter. As the gas converges to the center via an accretion disk, the gravitational energy is converted into radiation. As the radiation escapes the central region, it interacts with the surrounding matter, forming the Broad Line Region (BLR), the closest to the central engine and the Narrow Line Region (NLR). The resulting radiation gives rise to broad emission lines, coming from the BLR and narrow lines from the NLR superimposed on a continuum. Presently, the mechanisms occurring in the center of AGNs are poorly known, but detailed analysis of the spectra of the BLR of AGNs can constrain the model. 3

24 With the spectra of the active galactic nuclei obtained with Chandra and XMM, this general model is being tested and details of the structures are probed. A spectral representation of the Seyfert 1 galaxy MCG is shown in (Fig. 1.3). Data of this quality allows unprecedented refinement in the description of the region surrounding the black hole, giving evidence of the presence of warm absorbers characterized by the inner-shell excitation absorption feature from O VII near E = 0.7 kev. However this is challenged by another interpretation proposed by Branduardi-Raymont et al. (2001) who attribute the same feature to a highly red-shifted emission lines from H-like ions probing a region much closer to the central black hole. Describing the other features present in this spectrum will help to shed some light on this very complicated region. For example the large deep around E = 0.55keV is unidentified but suggestions lead to absorption features from oxygen ions. Atomic calculations of inner-shell transitions of all oxygen ionization states presented in chapter 4 will clearly represent good candidates Fe data The role of iron in astrophysics is of prime importance. It is an abundant element with many lines appearing in the spectra of most astrophysical objects. Iron ions can be used as a diagnostic tool to probe the region close to black hole, thought to sit in the center of AGNs, and to determine the physical conditions of 4

25 the emitting region. Iron ions also have a direct physical impact on the structure of stars via their contributions to the opacity. They directly influence the aspect as well as the evolution of stars. Since their identification for the first time in quasar spectra by Greenstein and Schmidt (Greenstein et al. 1964), iron emission lines have been observed in many other AGNs and are found to often dominate their spectra. It has been shown (Boroson et al. 1992) also that the intensity of Fe II lines increases as the broad lines get narrower. The complexity of the atomic structure of Fe ions combined with the uncertainties about the mechanism of the central engine of AGNs make the interpretation difficult. Moreover, the large number of lines for iron ions associated with the high velocity of the Broad Line Region (BLR) can generate, in certain wavelength intervals (UV, Optical), a pseudo-continuum (see Fig. 1.4, Vestergaard & Wilkes, 2001). This makes it impossible to use iron lines to derive the physical conditions and the mechanisms responsible for this radiation. Also any diagnostic from non-iron lines, due to the alteration of the continuum by the Fe pseudo-continuum or blending with other lines, becomes unreliable. The state of the actual theoretical model of Fe emission is not accurate enough to allow a meaningful analysis of AGN spectra. The limitation comes from two sources of inaccuracy. The actual radiative transfer treatment in most theoretical work, uses the escape probability approximation. This is not an adequate formulation for an inhomogeneous medium like BLR (Collin et al. 2000). Moreover, 5

26 the atomic data used to solve the transfer equation are not accurate or are simply missing. We need to provide the most accurate atomic data possible by taking into account the important excitation mechanisms of Fe. This will allow us to generate theoretical Fe spectra templates and use them to analyze Fe spectra of AGNs and constrain the physical conditions of the emitting region Opacities The Sun is the closet star that we can easily observe. The quantity of data associated to their high quality make it a favored laboratory to test stellar theory. The obserations of the different pulsation modes from the sun (helioseismology) provides a useful probe of the interior of our star and complements the information obtained for the surface by other techniques. The excellent agreement between solar models and helioseismology validated our knowledge on stellar structure and evolution and the solar parameters were considered robust and reliable. The density and temperature profiles, the chemical compositions, the nuclear rates, the opacities were therefore supposed to be known with great precision until recently. However, the results depends on the composition adopted. A new analysis of the atmospheric composition of the Sun (Asplünd et al. 2005) argue that the standard solar content of C, N, O and Ne has been largely overestimated (by more than 30% for C and O). Changing the metal content of a mixture affects directly its properties, especially 6

27 its behavior when light goes trough, in other words its opacity. The Rosseland mean opacity (κ R ) defines how the transfer of radiation takes place from the center of the star, where it is produced by nuclear reactions, towards the surface, and therefore is a key ingredient in stellar theory. Small changes in κ R modify drastically the structure of the stellar models. Applying the new composition in solar models leads to a serious conflict between the helioseismic data and the predictions of solar interior models. Different solutions have been proposed among which a revision of the opacities (see, for example, Bahcall & Pinsonneault 2004; Bahcall, Serenelli, & Pinsonneault 2004; Bahcall, Serenelli, & Basu 2005; Basu & Antia 2004; Antia & Basu 2005; Turck-Chi eze et al. 2004; Seaton & Badnell 2004). For stellar interiors only one source of opacities, OPAL opacities, have been available to stellar modelers. Proposing an independent alternative for the opacity is essential as a scientific check of the quality of data. The OP-IP team has extended the range of validity of the OP opacities to broader physical conditions including those typical of stellar interiors to provide an independent test on opacities (Badnell et al. 2005; see chapter 7). The results raise some questions on our ability to understand stellar physics and stellar atmospheres to which we try to answer with the new OP opacities in Chapter 9. 7

28 1.4. Radiative Accelerations It has long been established that certain stars show abundance anomalies (see Fig. 1.5, Behr et al. 2000) at their surface and that these peculiarities are due to micro-diffusion processes and more precisely to radiative levitation (Michaud 1970; Watson 1970). Each species interacts with the radiation differently depending on its atomic structure and its capability to absorb photons. During the process, absorbtion or scattering, the momentum from the photon is transfered to the atom (ion). This is the radiative acceleration. The combination of this process that makes elements rise and the gravitational force which tends to make them sink modify the local composition, the local mean opacity and therefore the structure. Modeling these phenomena is challenging and at present, the comparison between the predictions and the observations reveals the limitation of the theory or of the data used to define the radiative accelerations (i.e. the monochromatic opacities). As discussed above the study of pulsation modes of the Sun with the space based observatory SOHO (Solar and Heliospheric Observatory) and the ground based GONG project (Global Oscillation Network Group) has demonstrated the power of helioseismic studies as a tool to provide valuable information on the solar interior. As a logical extension similar studies on other stars (asteroseismology) should be as fruitful and would allow precise tests of the theoretical models under a wider range of physical conditions. With the first observations from MOST (Microvariability and 8

29 Oscillations of STars, Canadian observatory launched on June 30, 2003) and the perspective of those from COROT (COnvection ROtation and planetary Transit, launch scheduled for 2006) the Asteroseismology is well under way. This is necessary, for example, to estimate more accurately the age of the stars and particularly of old metal-poor stars that would give an independent way to estimate the age of the universe (see Fig. 1.7, Vandenberg et al. 2002; Chaboyer et al. 1992; Chapter 8 for more references). The accumulation of specific elements in certain layers should occur in any type of stars for which radiative acceleration is important. Besides sdb, Am-Fm, Ap stars other types are also affected. Richard et al. (2001) showed that levitation may create an extra convection zone due to the diffusion of iron peak elements (Fig. 1.6, Richard et al. 2001). For all the above reasons, accurate radiative acceleration calculations directly coupled to stellar codes are of crucial importance to stellar theory. In chapter 8 we present the results of my implementation of radiative accelerations using OP opacities in YREC The Opacity Project and The Iron Project The Opacity Project (OP) has been initiated by Mike J. Seaton (1987) to compute accurate atomic data needed in the calculation of opacities. The 9

30 OP team was composed of researchers from Europe, North America and South America who decided to cooperate to develop and/or improve the atomic physics methods and codes and to produce the required data. In 1993 the data were made available to the general public via a database named TOPbase (Cunto et al. 1993, at the Centre de données de Strasbourg (CDS) and all the work on opacities for stellar atmospheres has been published in two books (The Opacity Project team, 1995 and 1997). Then members of the OP team decided to pursue the collaboration under the Iron Project (IP) and used all the experience acquired under OP to compute collisional data of iron ions and other ions essential to astrophysics. Besides this primary target, the IP aimed at improving certain radiative data obtained under OP and at computing some other missing radiative data. Particular attention has been given to the inclusion of relativistic effects. Given the extent of the task there is still a large amount of the existing data that have been calculated in LS coupling (non relativistic). My work in the IP consists in contributing to the extensive new relativistic calculations. Now the OP-IP team provides 2 databases (TOPbase and TIPbase) and an opacity server allowing the on-line calculation of Rosseland mean opacities for mixtures containing H, He, C, N, O, Ne, Na, Mg, Al, Si, S, Ar, Ca, Mn, Cr, Fe and Ni in any proportion. It provides the radiative accelerations for a given set of ρ T points. The new opacity package is also available for the public to download. It 10

31 contains all the monochromatic opacities and codes to calculate Rosseland means and radiative accelerations My Contribution I have worked on the detailed study of the relativistic effects and radiation damping on electron impact excitation of He-like ions that are essential in X-Ray astronomy as described previously. Concerning the Fe ions, I have carried out the first large relativistic calculations of photoionization cross sections for the low-lying levels of Fe II which compare favorably with the observed results from laboratory experiment (Kjeldsen et al. 2002). One of the most recent development concerns the extension of the opacity work to include inner-shell transitions. This extends the validity of the OP opacities to a broader range of temperatures and densities (especially toward high ρ T regimes). They are now applicable to stellar interiors (more details are given in Chapter 8). I have participated in the calculation of these inner-shell transitions and have done a large part of the tests on these new data and monochromatic opacities. Finally, as a direct application of this work on opacity, and outside of the OP-IP work, my interest for stellar structure and evolution theory led me to develop and include the routines necessary to the calculation of radiative accelerations in the stellar code Yale Rotational Evolutionary Code or YREC (Chapter 9). 11

32 1.6. Scope of the Dissertation In this first chapter we have just described the motivation for the calculations of atomic data of astronomical importance. We have presented different topics in which new atomic data are crucial to solve different problems with a focus on X-Ray astronomy, AGN Fe-spectra and stellar applications for opacities. The present work is divided in two parts. The first one concerns the atomic calculations specifically and is organized as follows: In the following chapter we describe the theory and methods used to compute the required data. Chapter 3 and 4 present the results of calculations concerning X-Ray applications. The former includes the results of electron impact excitation of He-like ions, the later details the calculations of inner-shell transition (K α) for Oxygen ions. In Chapter 5 the first results of the relativistic computations of photoionization cross sections of Fe II are reported with a comparison with experimental data. Chapter 6 closes this part by giving some remarks on the accuracy and completeness of the atomic data. The second part focuses on specific stellar applications of opacities. More precisely in chapter 7 we present the definition of opacities, the impact of improved atomic data onto it and a comparison with another source (OPAL). We conclude by exposing the effects on stellar models due to the differences in the Rosseland mean. In Chapter 8 more details highlight more acutely the discrepancies between the two sources of opacities. Finally in Chapter 9 the application of the new opacities allows to give some answers to the Solar Composition Problem. 12

33 Fig XMM RGS spectra of 5 stellar systems (Güdel et al. 2001). The prominent lines are generated by transitions between the low lying levels form He-like ions (N, O, Ne, Mg and Si). The relative intensities of the different lines for each ion reveal the characteristics of the plasma in coronal equilibrium. The variation from the top to bottom panel indicates the differences in temperature from one system to another. 13

34 Fig Artistic representation of a black hole (NASA/CXC/SAO). The black hole at the center is surrounded by a disk of hot gas, and a large dough-nut or torus of cooler gas and dust. 14

35 Fig (a) MCG spectrum(deredshifted) superposed on a dusty warm absorber model (orange dashed line) and Galactic-absorbed power law (purple dashdotted line). (b) Inset: the spectrum of Cyg X-1 (purple-green) and the OVII resonance series for n 3 (orange) over-plotted on MCG in the OVII edge and Fe L region (Lee et al. 2001).. 15

36 Fig Non-iron emission model (solid line) overplotted on the continuumsubstracted I Zw 1 spectrum (dotted line). The detected emission lines are labelled (Vestergaard & Wilkes, 2001). 16

37 Fig Fe and Mg Abundance anomalies and rotation velocities in M15 (Behr et al. 2000). 17

38 Fig Iron convection zone predicted by stellar models including radiative levitation (Richard et al. 2001). (a) M = 1.5M, (b) M = 1.7M and (c) M = 2.5M. γ corresponds to the iron convection zone (Richard et al. 2001). 18

39 Fig Fit of the M92 fiducial (Steson & Harris 1988) to the field subgiant HD Non-diffusive and diffusive isochrones have been overlaid onto the observed CMDs (Vandenberg et al. 2002). 19

40 PART I: ATOMIC DATA 20

41 Chapter 2 General Method for atomic data calculations Introduction The main atomic processes that give rise to the line spectra of astronomical objects are of two sorts, radiative or collisionnal. In the first type the atom or ion interacts with a photon while the second type involves collisions between an atom (ion) and an electron. The most important processes are: Photo-excitation and radiative decay: A i (N) + hν A f (N) (2.1) Photo-ionization: A i (N + 1) + hν A f (N) + e (2.2) Recombination: A i (N) + e A f (N + 1) + hν (2.3) 21

42 Electron impact excitation: A i (N) + e A f (N) + e (2.4) (2.5) where A(N) corresponds to the target ion, A(N + 1) indicates that the ion has N + 1 electrons. The subscripts i and f are for initial and final states. In order to describe such processes and derive atomic data of use for applications, we have to represent a system made of a nucleus of nuclear charge Z and many electrons. The full description can be achieved by solving the time independent Shrödinger equation, H Ψ = E Ψ (2.6) where H is the Hamiltonian, Ψ is the eigenfunction associated to the eigen energy E. In the present thesis we used the atomic unit system in which e = m = h = 1. All corresponding units of length, time, energy etc. are given in appendix A. In order to be complete the Hamiltonian should account for all the interactions between the different particles present in the system. This includes the kinetic energy of the electrons, their potential energy generated by the nucleus, the repulsive interaction between electrons, the spin-orbit for each electron, the spin-spin, spin-other orbit 22

43 and orbit-orbit interactions between electrons. In the case of a light element of low charge, we can use the non-relativistic Hamiltonian, N Hnr N = ( 1 i=1 2 2i Z ) N N (2.7) ri j=i+1 i=1 r ij where 2 i = 1 r i d 2 dr 2 i r i L2 i r 2 i (2.8) and r i is the radius of the i-th electron from the nucleus considered as infinitely heavy and reduced to a point and r ij the inter-electron distance. In the case of intermediate Z (Z <30) elements, important relativistic effects must be included, for example using the Breit-Pauli approximation to give the Breit-Pauli Hamiltonian, H N BP = H N NR + H N mass + H N D 1 + H N so (2.9) where H N nr is the non-relativistic Hamiltonian defined above, H N mass is the mass correction term, H N D 1 is the Darwin term and H N so is the spin-orbit term which are defined as follows: the mass correction of the electron kinetic energy Hmass N = 1 N 4 α2 4 i (2.10) i=1 the one body Darwin term H N D 1 = 1 4 Zα2 N 2 i i=1 1 r i (2.11) 23

44 the spin-orbit interaction H N so = N i=1 Zα 2 (l ri 3 i.s i ) (2.12) For species with higher Z, other relativistic effects must be included but will not be treated here because it is out of the scope of the present work. One important approximation is to solve the problem in the framework of the Hartree-Fock theory in which one considers each electron as an independent particle moving in a Coulombian potential due to the nucleus and corrected in order to include the screening effect to this potential due to the repulsive interaction with the other electrons and the indiscernability between electrons (Pauli exclusion principle). In the case of collisional calculations, one uses the variational principle to minimize the energies which gives a system of coupled integro-differential equations (Seaton 1953; Burke & Smith 1962; Eissner & Seaton 1972). One method used to solve this system of equations is called the R-Matrix method and is described in the following section. It is the formalism used by the Opacity Project - Iron Project The Close Coupling theory and the R Matrix method In the case of collisional work, we attempt to find the wave function solution to the Schrödinger equation (eq. 2.5) for a system with (N+1) electrons describing 24

45 the interaction of a free electron with what is called the target (a nucleus with N electrons). The eigenstates Ψ, solutions of this equation, are usually developed in terms of the target and colliding electron wave functions Ψ Γ = i χ Γ i (x 1,..., x N )θ Γ i (x N+1 ) (2.13) where Γ SLM L M S π, x l rσ represents the spin (σ) and spacial (r (r, ϑ, ϕ)) coordinates for electron l, χ i is the target ion wave function in a specific state S i L i π i or level J i π i, and θ i is the wave function for the (N+1) th electron in a channel labeled as S i L i (J i )π i k 2 i l i (SLπ) [Jπ] and k 2 i is the electron energy associated to the channel i. The sum includes all the possible combinations of the target states and colliding electron wave function that give the required symmetry Γ for Ψ Γ (x 1,..., x N+1 ) with the eigen energy E Γ representing the total energy. For convenience we will omit the index Γ. E = E i + ki 2 where E i is the target energy. Depending on the value of ki 2, we have open channels when ki 2 0 and closed channels when ki 2 0. In the coupled channel or close coupling (CC) approximation the wave function expansion Ψ is restricted to include only the channel i with strong coupling and limits the infinite sum to the number of channel n cc which depends on the number of target states included in the calculation. The expansion becomes n cc Ψ(x 1,..., x N+1 ) = A χ i (x 1,.., x N )θ i (x N+1 ) + c j Φ j (x 1,..., x N+1 ), (2.14) n bc i j 25

46 where A is the antisymetrisation operator and where the second sum is introduced (a) to compensate for the orthogonality conditions between the continuum and the bound orbitals of the target, and (b) to represent additional short-range correlations that are often of crucial importance in scattering and radiative CC calculations for each symmetry. The Φ j s are correlation wave functions also referred to as bound channels as opposed to the free channels n cc. The notion of free channel refers to the radial function of the colliding electron that can freely vary in order to solve the Schrödinger equation for the system with (N+1) electron. We can combine the angular and spin part of the wave function θ i (eq. 2.13) for the (N+1)th electron to the target wave function χ i which give χ i and n cc 1 Ψ(X) = A χ i ( X) F i (r N+1 ) + c i Φ j (X) (2.15) r N+1 i n bc i where X stands for (x 1,..., x N+1 ) and X corresponds to (x 1,..., x N, ϑ N+1, ϕ N+1, σ N+1 ), F i (r N+1 ) is the reduced radial function of the colliding electron. In the relativistic BPRM calculations the set of SLπ are re-coupled in an intermediate (pair) coupling scheme to obtain (e + ion) states with total Jπ, followed by diagonalisation of the (N+1)-electron Hamiltonian. 26

47 The Target The initial step in a collisional calculation, is to generate the target representation. The target wave function for a state i is developed in a configuration interaction (CI) expansion. χ i (x 1... x N ) = j c ij φ j (x 1... x N ) (2.16) The mixing coefficients c ij are determined by diagonalization of the Hamiltonian (2.5) and each configuration state is constructed from the Slater orbitals. Using the Slater determinant, we have φ j (x 1... x N ) = 1 N! u 1 (x 1 ) u 1 (x 2 )... u 1 (x N ) u 2 (x 1 ) u 2 (x 2 )... u 2 (x N ). u N (x 1 ) u N (x 2 )... u N (x N ). (2.17) where the Slater orbital u i in the central field approximation can be constructed as the product u i (x) = u i (r, σ) = 1 r P nl(r)y m l l (ϑ, ϕ)χ ms (σ), (2.18) of a radial function 1P r nl(r), an orbital function (spherical harmonics) Y m l l (ϑ, ϕ) and a spin function χ ms (σ). 27

48 Because the orbital and spin functions are known mathematical functions, the problem consists in solving the radial Schrödinger equation for the P nl (r) { } d 2 l(l + 1) + 2V (λ dr2 r 2 ls, r) + ε nl P nl (r) = 0 (2.19) where V (λ l, r) is the Thomas-Fermi-Dirac potential and the λ ls are used as the variational parameters to optimize the potential in order to minimize a set of chosen energies. The structure code used in the present work is SUPERSTRUCTURE (Eissner et al. 1974) in a version due to Nussbaumer and Storey (1978). The code SUPERSTRUCTURE includes relativistic corrections not only like those presented in equations (2.9,2.10 and 2.11) but also some two-body operators such as the spin-other-orbit term, the spin-spin term, the 2-body Darwin term and the spin-spin-contact term. They are generally included in radiative calculations using SUPERSTRUCTURE, but in the case of collisional work, in order to remain consistent only the 3 terms from the Breit-Pauli approximation present in the RMATRX code are taken into account. Once generated, the radial functions of the target are used in the RMATRX package to rebuilt the target and generate the (N+1) bound states (second sum in eq. 2.13). 28

49 The Coupled Integro-Differential equations Given a set of the target and bound channel wave functions the problem consists in solving the Schrödinger equations for the unknown radial functions F i of the colliding electron and the coefficients c j (eq. 2.14). These functions have to obey the boundary conditions which define the scattering matrix S lim F ij(r) = r l i+1 r 0 (2.20) and lim F ij(r) = k 1/2 r i (e iθi(r) δ ij e iθi(r) S ij ) for ki 2 > 0, i = 1,.., n a (2.21) 0 for k 2 i < 0, i = n a + 1,.., n cc (2.22) where n a is the number of open channels, j = 1,..., n a and the Coulomb phase θ i is θ i = k i r 1 2 l iπ + Z N (ln(2k i r) + arg(γ)(l i + 1 Z N ı) (2.23) k i k i with l i being the angular momentum of the scattered electron in channel i, Z being the nuclear charge and N being the number of electrons in the target. For numerical applications one defines the K-matrix such that real wave functions instead of complex ones are used. They satisfy the real boundary conditions lim F ij(r) = k 1/2 r i (sinθ i (r)δ ij + cosθ i (r)k ij ) for ki 2 > 0 (2.24) 29 0 for k 2 i < 0 (2.25)

50 with the relation between the n a n a matrices S and K given by S = 1 ık 1 + ık (2.26) The transmission matrix is then defined as T = 1 S = 2ıK 1 ık (2.27) Using the Kohn variational principle (W. Kohn 1948) one derives the system of equations (Eissner & Seaton 1974) that the functions F i must satisfy { d 2 dr l i(l i + 1) 2 r 2 + 2z } r + k2 i F i (r) = 2 n j=1 (V ij (r) + W ij (r)) F j (r)+ nl Λ i nl P nl (r)δ lli (2.28) which represents the so-called Coupled Integro-Differential equations. The potential V is the target potential including the screening and the long distance component and W represents the exchange interaction. The last sum is included to impose the orthogonality between the radial functions of the target P nl and of the colliding electron F i. Finally the physical quantities of interest, the cross section σ tot and the collision strength Ω for a transition between an initial state i and final state j are defined in LS coupling as σ tot (γ i L i S i γ j L j S j ) = πa2 o k 2 i LSπ (2L + 1)(2S + 1) LSπ Tij 2 (2.29) l i l j 2(2L i + 1)(2S i + 1) 30

51 and Ω(i, j) = (2L i + 1)(2S i + 1)k 2 i π σ tot (i j) (2.30) The R-matrix method The use of the R-matrix method for collisional calculations between an electron and an atom started in the early 70 s (Burke et al. 1971; Burke & Seaton 1971) and is the method adopted by OP-IP team (Seaton 1987, Hummer et al. 1993) in a version that is constantly improved upon from the initial non-relativistic code (Berrington et al. 1974) to the intermediate stage with the one-body Breit-Pauli terms (Scott & Burke 1980; Scott & Taylor 1982) to the present relativistic form BPRM (Seaton 1987, Hummer et al. 1993) and to future development with the inclusion of the relativistic 2-body terms (Chen & Eissner 2005, code in the testing phase). The specificity of this method (Burke et al. 1971) lies in the division of the configuration space in two parts called the inner region and the outer (also called asymptotic) region (see Fig. 2.1). The inner region consists of a sphere of radius a within which the interactions between the target and the free electron are considered strong. The boundary is chosen such that the charge distribution of the target is contained within the radius a. The exchange and correlation reactions are vanishing in the outer region and the problem can be treated with a simpler approximation 31

52 for a free electron moving in the long range potential of the target. Both solutions (for the inner region and for the outer region) have to be consistent at the boundary. This is provided by a condition on the wave functions and their first derivatives which should be continuous at the boundary. The inner region In the inner region the total wave function can be expanded on a basis set ψ k independent of the energy such that Ψ(E) = k A Ek ψ k (2.31) with ψ k of the form ψ k = A ij n bc 1 χ i ( X) F i (r N+1 ) + c ik Φ j (X) (2.32) r N+1 i where F i can be expanded on the continuum orbital basis set v i such that n cc Ψ k (X) = A i nrang2 j n bc 1 χ i ( X) v ij (r N+1 )a ijk + c ik Φ j (X). (2.33) r N+1 i with A Ek regrouping the coefficients a ijk and c ik. The radial basis functions v ij are solutions of { d 2 dr l i(l i + 1) 2 r 2 + V o (r) + k 2 ij } v ij (r) = 32 n max(l i ) n=l i +1 Λ ijn P nli (r) (2.34)