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On the Feynman path into the sun
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On the Feynman path into the sun
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ON THE FEYNMAN PATH INTO THE SUN by Yung-Ching Liang A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (PHYSICS) August 2013 Copyright 2013 Yung-Ching Liang Dedication To my family ii Acknowledgments First of all, I would like to express my heartfelt gratitude to Werner D¨ appen who is my advisor and also my mentor. Thank you for being greatly helpful and generous to me. I would like to thank Professors Nelson Eugene Bickers, Stephan Haas, Aiichiro Nakano, and Joseph Wang, for serving my PhD advisory committee and being friendly and supportive. At the end, I would also like to thank my family and my wife, Hsiao- Hsuan Lin. Your support is essential to the completion of my Ph.D. study. iii TableofContents Dedication ii Acknowledgments iii ListofFigures vi Abstract xi Chapter1: Introduction 1 1.1 Basic Structure of the Sun . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Looking through the Sun . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.1 Introduction to Helioseismology . . . . . . . . . . . . . . . 9 1.2.2 Observing the Solar Oscillation . . . . . . . . . . . . . . . . 10 1.2.3 Mathematical Description of Solar Pulsations . . . . . . . . 12 1.2.4 Asymptotic Oscillations . . . . . . . . . . . . . . . . . . . 15 Chapter2: TheoryaboutSolarandStellarmodeling 17 2.1 Relevant Time Scales of Stars . . . . . . . . . . . . . . . . . . . . . 17 2.2 Mathematical Model of Stellar Structure . . . . . . . . . . . . . . . 19 Chapter3: EquationofStateforSolarandStellarModeling 23 3.1 Chemical Picture Approach . . . . . . . . . . . . . . . . . . . . . . 24 3.1.1 Free Energy Minimization Method . . . . . . . . . . . . . . 24 3.1.2 MHD Equation of State . . . . . . . . . . . . . . . . . . . . 27 3.2 Physical Picture Approach . . . . . . . . . . . . . . . . . . . . . . 28 3.2.1 Activity Expansion Method . . . . . . . . . . . . . . . . . . 29 3.2.2 OPAL Equation of State . . . . . . . . . . . . . . . . . . . 33 3.2.3 Virial Equation of State . . . . . . . . . . . . . . . . . . . . 38 Chapter4: VirialEquationofStateviaFeynman-Kacpathintegral 41 4.1 Statistical Mechanics of the Grand Partition Function . . . . . . . . 43 4.2 Feynman-Kac Path Integral Approach . . . . . . . . . . . . . . . . 43 4.2.1 Single Particle System . . . . . . . . . . . . . . . . . . . . 44 iv 4.2.2 Maxwell-Boltzmann Many-Particle System . . . . . . . . . 45 4.2.3 Virail Expansion in Feynman-Kac Representation . . . . . . 48 4.2.4 Corrections from Quantum Statistics . . . . . . . . . . . . . 50 Chapter5: ProbingtheSunwiththeVirialEquationofState 56 5.1 Domain of Applicability of Virial Equation of State . . . . . . . . . 57 5.1.1 Coulomb Coupling Parameter . . . . . . . . . . . . . . . . 58 5.1.2 Degeneracy Parameter . . . . . . . . . . . . . . . . . . . . 59 5.1.3 Ionization Degree . . . . . . . . . . . . . . . . . . . . . . . 60 5.1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Preliminary Discussions . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2.1 Solar Data and Parameters . . . . . . . . . . . . . . . . . . 62 5.2.2 Issues of Computing Q and E functions . . . . . . . . . . . 65 5.3 Results and Discussions . . . . . . . . . . . . . . . . . . . . . . . . 71 5.3.1 Physics of Pressure . . . . . . . . . . . . . . . . . . . . . . 72 5.3.2 Physics of Thermodynamic Quantities . . . . . . . . . . . . 86 5.3.3 Physics of the heavy elements . . . . . . . . . . . . . . . . 93 5.3.4 Effects of the Linear Term in theQ Function . . . . . . . . 96 5.3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 98 Chapter6: Low-TemperatureExtensionsoftheVirialEquationofState forSolarModeling 100 6.1 Saha Equation of State . . . . . . . . . . . . . . . . . . . . . . . . 101 6.1.1 Introduction to Theory . . . . . . . . . . . . . . . . . . . . 101 6.1.2 Results and Discussions . . . . . . . . . . . . . . . . . . . 103 6.2 Scaled Low Temperature Expansions . . . . . . . . . . . . . . . . . 108 6.2.1 Introduction to Theory . . . . . . . . . . . . . . . . . . . . 108 6.2.2 Results and Discussions . . . . . . . . . . . . . . . . . . . 118 6.3 Conlcusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 Chapter 7: Comparisons with the OPAL and CEFF Equations of States 127 7.1 Compare SLT&FK Virial with OPAL and CEFF forH System . . . 128 7.2 Compare FK Virial with OPAL and CEFF forH&He System . . . . 133 7.3 Compare FK Virial with CEFF forH&He&O System . . . . . . . . 135 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 Chapter8: Conclusion 141 Bibliography 143 v ListofFigures 1.1 The interior structure of the Sun. (Figure provided by J. W. Leibacher) 5 1.2 The difference between blue-band intensity and red-band intensity is proportional to the oscillation velocity,I(+4)I(4) 24 @I() @ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 Doppler images of solar oscillations measured by using MDI (Michel- son Doppler Imager) instrument [54] on the SOHO. The left is the original image, and the right is the average image after removing the velocity of rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4 -l diagram of power spectrum from helioseismology. (Figure pro- vided by A. G. Kosovichev) . . . . . . . . . . . . . . . . . . . . . . 14 1.5 Sound waves propagate between the solar surface and the turning points. (Figure provided by A. G. Kosovichev) . . . . . . . . . . . . 16 4.1 Difference between the squared sound speed obtained from the inver- sion methodology with two different solar models. Figure provided by S Basu. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5.1 Coupling parameters throughout the Sun for different charged par- ticles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 5.2 The electron degeneracy parameter is below the critical value 1 through- out the Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 5.3 The ionization degrees of different states ofH andHe are estimated by using the Saha equation. . . . . . . . . . . . . . . . . . . . . . . 60 5.4 The temperature-radius map expresses the relation between the tem- perature and the relative radius throughout the Sun. r R = 1 is at the solar surface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 vi 5.5 The abundances of different elements involved in the solar data. . . 63 5.6 The ranges ofx for different particles in the Sun. . . . . . . . . . 65 5.7 The behaviors of the Q and E functions for smallerx values. . . . . 66 5.8 The behaviors of the Q function for largeerjxj values. . . . . . . . . 66 5.9 Difference between the Pad´ e approximant in [45] and the exact func- tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.10 Compare the Pad´ e approximant in [58], the asymptotic functions, and the exact function. . . . . . . . . . . . . . . . . . . . . . . . . 68 5.11 The behavior of the zeta function(x). . . . . . . . . . . . . . . . . 69 5.12 The left graph shows the behaviors of Q with the linear term and without the linear term. The right graph expresses the relative dif- ference between them. . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.13 Compare the normalized total pressure and some normalized partial pressure for theH system throughout the Sun.P id andP DH express the ideal gas pressure and Debye-H¨ uckle pressure respectively. . . . 73 5.14 The behaviors of different normalized partial pressure defined in Eq. (5.22) of theH system. The curve number represents the order of the term calculated on the right hand side of Eq. (5.22). . . . . . . . 75 5.15 Absolute values of the different normalized partial pressure of the H system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.16 Compare the normalized total pressure and some normalized partial pressure for theH&He system throughout the Sun. . . . . . . . . . 79 5.17 The behaviors of different normalized partial pressure defined in Eq. (5.22) of theH&He system. . . . . . . . . . . . . . . . . . . . . . 81 5.18 Compare the normalized total pressure and some normalized partial pressure for theH&He&O system throughout the Sun. . . . . . . . 83 5.19 The behaviors of different normalized partial pressure defined in Eq. (5.22) of theH&He&O system. . . . . . . . . . . . . . . . . . . . 84 5.20 The power exponents and T are calculated by using the FK- virial EOS for theH system, and both contain the singularities. . . . 88 vii 5.21 The behavior of the the power exponent of theH system in the solar interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.22 The behavior of the power exponent T of theH system in the solar interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 5.23 The behavior of the specific heat c v of the H system in the solar interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.24 The behavior of the adiabatic exponent 1 of the H system in the solar interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.25 The behavior of the sound speed v s of the H system in the solar interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 5.26 Compare the behaviors of the the power exponent for the systems with different constituents. . . . . . . . . . . . . . . . . . . . . . . 93 5.27 Compare the behaviors of the the power exponent T for the systems with different constituents. . . . . . . . . . . . . . . . . . . . . . . 94 5.28 Compare the behaviors of the the power exponentc v for the systems with different constituents. . . . . . . . . . . . . . . . . . . . . . . 94 5.29 Compare the behaviors of the the adiabatic exponent 1 for the sys- tems with different constituents. . . . . . . . . . . . . . . . . . . . 94 5.30 Compare the behaviors of the the sound speed v s for the systems with different constituents. . . . . . . . . . . . . . . . . . . . . . . 95 5.31 TheO atoms in the system are substituted by theHe atoms. The left graph represents the absolute values of relative discrepancies of the thermodynamic quantities between the substituted system and the original system. The right graph expresses the relative discrepancy of the squared sound speed between the substituted system and the original system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.32 The O atoms in the system are substituted by C atoms. The left graph represents the absolute values of relative discrepancies of the thermodynamic quantities between the substituted system and the original system. The right graph expresses the relative discrepancy of the squared sound speed between the substituted system and the original system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 viii 5.33 The influences of the linear term for the thermodynamic quantities of theH&He&O system. . . . . . . . . . . . . . . . . . . . . . . . 98 6.1 Compare the normalized total pressure calculated from different EOS for theH system (the dot line represents P P id = 1). . . . . . . . . . . 103 6.2 The behavior of ratio throughout the Sun. . . . . . . . . . . . . . 105 6.3 Compare calculated by using the Saha EOS with the FK virial EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.4 Compare T calculated by using the Saha EOS with the FK virial EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 6.5 Compare c v calculated by using the Saha EOS with the FK virial EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.6 Compare 1 calculated by using the Saha EOS with the FK virial EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.7 Compare the sound speedsv s calculated by using the Saha EOS with the FK virial EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.8 Qualitative phase diagram of hydrogen gas (cited from [4]). . . . . . 118 6.9 The behavior of s throughout the Sun. . . . . . . . . . . . . . . . . 119 6.10 The behaviors of theb k functions throughout the Sun. . . . . . . . . 120 6.11 The behaviors of theh k functions throughout the Sun. . . . . . . . . 121 6.12 The corrections to the Saha pressureP Saha throughout the Sun. . . . 121 6.13 The absolute values of the corrections to the Saha pressure P Saha throughout the Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.14 The absolute values of the corrections to the Saha pressure P Saha throughout the Sun. . . . . . . . . . . . . . . . . . . . . . . . . . . 123 6.15 Compare the normalized total pressure calculated by using different EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 6.16 Compare the thermodynamic quantities calculated by using differ- ent EOS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 ix 7.1 Compare the thermodynamic quantities computed by using the SLT EOS, the FK virial EOS, and OPAL (the approximations Eq. (6.31) and Eq. (6.32) are used for the curveSLT 1). . . . . . . . . . . . . 129 7.2 Compare the thermodynamic quantities computed by using the SLT EOS, the FK virial EOS, and OPAL (the approximations Eq. (6.33) and Eq. (6.34) are used for the curveSLT 2). . . . . . . . . . . . . 130 7.3 The absolute values of the relative discrepancy of the thermody- namic quantities between different EOS and OPAL. . . . . . . . . . 131 7.4 The absolute values of the relative discrepancy of the thermody- namic quantities between different EOS and CEFF. . . . . . . . . . 132 7.5 Compare the thermodynamic quantities computed by using the FK virial EOS with CEFF and OPAL. . . . . . . . . . . . . . . . . . . 137 7.6 The absolute values of the relative discrepancy between the thermo- dynamic quantities of the H&He System from the FK virial EOS and OPAL/CEFF. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 7.7 The discrepancy between the squared sound speed of the H&He System from the FK virial EOS and OPAL/CEFF. . . . . . . . . . . 138 7.8 Compare the thermodynamic quantities of the H&He&O System computed by using different EOS. . . . . . . . . . . . . . . . . . . 139 7.9 The absolute values of the relative discrepancy between the ther- modynamic quantities of theH&He&O System from the FK virial EOS and CEFF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 7.10 The discrepancy between the squared sound speed of theH&He&O System from the FK virial EOS and CEFF. . . . . . . . . . . . . . . 140 x Abstract This study deals with solar-physics applications of a recent equation-of-state formalism based on the formulation of the so-called “Feynman-Kac (FK) representation”. This formalism leads to an exact virial expansion of the thermodynamic functions in powers of the particle densities of a Coulomb plasma (“exact” here refers to the accuracy of the low-order virial coefficients of the expansion). By taking advantage of the exact and analytic form of this virial expansion, we can probe the thermodynamic properties of the solar interior, both in detail and to an accuracy that has so far not been achieved with currently available equation-of-state formalisms. For reacting plasmas, virial-expansion equation of state have an intrinsic problem when the plasma is less than fully ionized. Fortunately, in most parts of the Sun’s interior, the plasma is almost completely ion- ized. Therefore, the FK virial equation of state can be applicable, but only in the deeper (and hotter) solar interior. The precise boundary of the domain of validity of the virial expansion depends on the elements included in the formalism. Since the computational effort increases tremendously with an increasing number of chemical elements, here, we choose to represent the heavier elements by a single one, oxygen. This approximation is reasonable for the following reason. In the Sun, the major heavy elements (that is, the elements other thanH andHe which together comprise more than 98% of the mass frac- tion) areC,N andO. They all have a similar effect in the equation of state. Then, the other elements, such asFe andNe, are so little abundant that they have altogether a very xi small effect in the equation of state (although they are very important in spectroscopy). Assuming the relevant constituents of the Sun to beH,He, andO, we have examined the effects on thermodynamic quantities from each of these components. With the aid of the FK formalism, we have studied the influences of the contributions to the respective different partial pressure. This not only helps us to understand the intrinsic challenge of applying the virial expansion, but also to analyze the various effects quantitatively. We have also tackled the aforementioned main limitation of the virial expansion, that is, its breakdown for relatively low temperatures, where the plasma is clearly less than fully ionized. For this, we have replaced the virial equation for these outer regions for the Sun by another, similar equation of state, the so-called “scaled low temperature (SLT)” expansion. Current versions of SLT are so far only for one chemical element, there- fore restricted to the hydrogen part of the solar plasma. However, since hydrogen is the most abundant element in the Sun (more than 90 % by number), SLT in its current form is already useful, if one adds the contributions of the remaining elements (He and heavy elements) using conventional equation-of-state formalisms. In such a procedure, an improved H part and a conventional treatment of the other elements will none the less lead to a net improvement of the overall equation of state compared to entirely con- ventional formalisms. Our results show (i) that the SLT EOS is not only consistent with the conventional reacting ideal-gas equation of state (the so-called “Saha equation” ) at low temperature and low densities, but (ii) that there is also a smooth matching with the FK virial expansion EOS up to the order of density 2 . Comparing our resulting combined equation of state (FK plus SLT) with currently popular equations of state, such as OPAL (developed at Livermore), the relative discrepancies of the relevant ther- modynamic quantities are about 10 3 to 10 4 . Since these differences are of the same order as the accuracy of helioseimic inversions for thermodynamic quantities, we have demonstrated that the FK equation of state extended by SLT will be a serious player xii in solar modeling. Further technical steps still have to be taken before the FK equa- tion of state can be used in a fully-fledged helioseismic study. Our result demonstrates the feasibility of such an application, which will then tell whether the FK virial EOS is more accurate than the so-far best equation of state OPAL. With the exact nature of FK (devoid of several approximations used in OPAL), there is a promise of a successful outcome. However, at the moment, a final comparison is not possible because of the current state of the SLT EOS. Once it will be available for a multi-element plasmas, our product consisting of the SLT EOS for the solar exterior, and the FK virial EOS for the solar interior, will likely become the most accurate EOS for solar and stellar modeling. xiii Chapter1 Introduction Without any doubt, the Sun is the most important star to us. Myriad effects in our daily lives rely on the energy from the Sun. In addition to its vital importance, among the numerous stars, the Sun also plays a crucial role in physical research, especially in astrophysics. The reason is simply its proximity. Except for the Sun, the nearest star, Proxima Centauri, is about 4 light years away, while the distance to the Sun is just about 8 light minutes. Thus the Sun is 10 5 times closer. The Sun provides us therefore with the unique opportunity to observe and study it with highly accurate and profound details. Not only does this help us to study the Sun itself, but it also enhances our general knowl- edge about how the Sun’s behavior is governed by the laws of physics. This, in turn, can be extrapolated to other stars, enhancing our understanding of their structure and evolu- tion. From a physics point of view, in stellar interiors we encounter extreme conditions, such as high density, high temperature, high pressure, and strong magnetic fields, all of which cannot be realized in terrestrial laboratories. As a consequence, solar and stellar modeling depends crucially on theoretically computed material properties, most impor- tantly the equation of state, opacity, and the rate of nuclear-energy production. Thanks to solar observations, we obtain very accurate constraints on these material properties, making the Sun a formidable laboratory for plasma physics. Of course we have to live with the limitation that the specific physical conditions of the Sun are given and cannot be varied, but the resulting information still have the power to challenge our theoretical understanding of stars. 1 In this chapter, we first briefly introduce the basic properties and the structure of the Sun, including the most important observational technique for solar oscillations, the so- called “helioseismology”. In Chapter 2, we introduce the fundamental technique of solar and stellar modeling. Among its necessary ingredients from basic physics, the equation of state plays the most crucial role. It is the focus of the present study. Because of the aforementioned extreme conditions, only theory can provide the equation of state. Dif- ferent equations of state will lead to different solar models, which are then up for helio- seismic tests. In Chapter 3, two different approaches for obtaining a theoretical equation of state are discussed, and the two most popular equations of states for solar modeling are presented. One of them is the so-called “MHD” (Mihalas-Hummer-D¨ appen) equa- tion of state, which is realized by a free-energy-minimization technique. Its approach is therefore called the “chemical picture”. The other is the so-called “OPAL” (Opacity at Livermore) equation of state, which is realized by a grand-canonical partition function technique. Its approach is called the “physical picture”. In the last 25 years, these two different equations of state have been quite successful, but they are each not without significant drawbacks. The present study is therefore dedicated to yet another equation of state that has the potential to overcome limitations of both MHD and OPAL. This alternative approach is also derived in the framework of the physical picture: it leads to a virial expansion for the equation of state. In the following two chapters, it is presented as part of a complete theory. In Chapter 4, we start to explain the derivations of the virial equation of state for the quantum plasma. Specifically, we utilize a theory based on the Feynman-Kac path integral in order to be consistent with the derivation of the other important theory: the scaled low-temperature expansion. The final result of this chapter will serve as one of the main formalism of my study. In Chapter 5, we will use the equation of state derived in Chapter 4 to probe the thermodynamic properties of the Sun. To make use of this equation, we are able to discuss the various effects of the 2 Property Value Mass (M ) (1.98910.0004)10 33 g Radius (R ) (6.96260.0007)10 10 cm Surface temperature (T ) (57772.5) K Luminosity (L ) (3.8460:01) 10 33 erg/s Table 1.1: Physical characteristics of the Sun [13, 15, 57]. thermodynamic properties of the Sun in details. This also shows one of the advantages of this exact virial equation of state. In addition, in order to have a similarly exact equa- tion of state in the outer region of the Sun where the plasma is only partially ionized, we introduce the theory of the scaled low-temperature expansion in Chapter 6. We then apply the theory to a hydrogen plasma under the solar conditions. The results are also compared with the virial equation of state and simple analytic equations of state (the so-called “Saha equation”) in order to see how the scaled low-temperature expansion method can go beyond the Saha equation and approach the results of the virial equation from the deeper end of the Sun. In Chapter 7, we compare the results obtained from the previous discussions with the results from two the well studied equation of states. These comparisons can serve as the benchmark to understand the quality of data computed by using the virial expansion or the scaled low-temperature expansion. Finally, we will conclude our study in Chapter 8. 1.1 BasicStructureoftheSun Some fundamental properties of the Sun are summarized in the Tab. 1.1. In general, the shape of a star is very close to a sphere. The distance between the Sun and the Earth can be measured directly with the technology of radar echoes to the nearest planets, and the radius of the Sun can be inferred from measuring its angular size of the Sun on earth. The mass of the Sun can be obtained from the Kepler’s third law by measuring the period 3 of the planetary motion. By measuring the brightness, which is the radiation flux, with the space-based radiance measurements, the luminosity of the Sun can be determined from the relation, L = 4R 2 B; (1.1) where the L is the luminosity, R is the distance to the Sun, and B is the measured brightness. Since the radiation form the surface of a star can be well approximated as black body radiation, the effective surface temperature can be also determined from the luminosity by using the Stefan-Boltzmann law, L = 4R 2 T 4 ; (1.2) where is the Stefan-Boltzmann constant. Besides, the chemical composition on the surface of the Sun can be measured from the spectroscopy. The relative abundances ratio of the heavy elements to hydrogen can be determined from the strengths of their spectra except the helium, and the value in mass is 0:0165 0:0017 [10]. Because the spectrum of helium is too faint and complex, the abundance of helium cannot be analyzed by this means. Now, we understand the structure of the Sun is as Fig. 1.1. The interior structure of the Sun is like an onion, has a stratified structure because the Sun has spherical sym- metry and its gravity changes along the radius. In different regions, even though the composition doesn’t change significantly, they have very different physical properties and therefore can be classified into different shell structures. In general, all stars have this similar structure, but the stars with different mass or age might comprise different types of layers. For example, a star with mass above 1.5 solar mass will have the con- vection layer below the radiation layer instead. The distinct stratified structures of the Sun are briefly summarized as the following: 4 Figure 1.1: The interior structure of the Sun. (Figure provided by J. W. Leibacher) Core: 0.r. 0:1R This is the place where the energy is produced to maintain the life of the Sun and keep it shining steadily. In this region, the intense heat makes atoms fully ionize into nuclei, protons, electrons, and neutrons. The neutrons can quickly escape from the core because of its neutral charge. However the protons and electrons remain in the core as neural plasma. The core has very high temperature and density, and thus can ignite the nuclear fusion processes. It turns out, most of the power in the Sun is generated via the p-p (proton-proton) chain reaction, and only about 0:8% of the power is from the CNO (carbon-nitrogen-oxygen) cycle. The basic reactions in the p-p chain are 1 H + 1 H!e + + e + 2 D 2 D + 1 H! + 3 He: 5 The produced 3 He can fuse further to produce 4 He via three different paths. These three different branches of the p-p chain are pp-I chain 3 He + 3 He! 2 1 H + 4 He; pp-II chain 3 He + 4 He! + 7 Be 7 Be +e ! e + 7 Li 7 Li + 1 H! 4 He + 4 He; pp-III chain 7 Be + 1 H! + 8 B 8 B!e + + e + 8 Be 8 Be! 4 He + 4 He: Different branch can produce different amount of energy and products. Their reac- tion rates are also different due to their different scattering cross sections. For the temperature of the solar core, the pp-I chain is the dominate reaction to produce the energy. During the whole process of fusion, the hydrogen is consumed and turned into helium; meanwhile, about 26:73Mev of energy is released from the reactions. Most of this energy is carried by the gamma rays and the other small part (about 2%) is carried by the neutrinos. Because the interaction with neutrinos is very week, the neutrinos can easily escape the Sun without losing their energy. The heat from the core is mostly produced by the absorptions of the gamma rays. This prevents the Sun from gravity collapse and maintains the stability of the Sun. 6 Radiation Zone: 0:1R .r. 0:7R Outside of the core is the radiation zone. In contrast to the core, the nuclear fusions cannot occur here due to its lower temperature. Even though the temperature in this region is lower than the core and some nuclei and electrons can combine into atoms, the plasma here is still almost fully ionized. As the radiations move out- ward from the core, they keep being absorbed, remitted, and also scattered by the plasma. These phenomena cause the radiations transport diffusively; meanwhile, the plasma thermally equilibrates with certain temperature gradient. However, the plasma is still not opaque enough to cause great temperature gradient, hence the plasma is still mechanically stable and does not move convectively. The main characteristic of this region is that the dominate mechanism of energy transfer is still via the radiation transfer. Convection zone: 0:7R .r. 0:999R The temperature decreases as the position is farther from the center of the Sun. When the temperature is low enough so that more atoms are formed and the plasma becomes more opaque, the equilibrium between the radiation and plasma induces large temperature gradient. As the temperature gradient is so large that the plasma becomes mechanically unstable, the convective motion starts to occur. A pack of hotter plasma floats outward and becomes cool when it touches the top of the convection zone, and then sinks to the bottom to be heated again. This cyclic motion is stirred by the buoyancy due the density difference between the plasma pack and its environment. In the meanwhile, the convective motion carries the energy from the top of the radiation zone to the top of the convection zone. It turns out that this method can transfer more energy than the radiation in this region, which means the convection is much more efficient than the radiation to 7 transfer the energy. This feature makes the convection zone physically different from the radiation zone. Photosphere: 0:999R .r.R On the top of the convection zone, there is a thin layer (about 100 km thick) which is opaque to the light called photosphere. This layer is similar to the solid surface of the Earth; we cannot look through the depth of the photosphere. In this region, the temperature is already very low and many atoms can form to absorb the radiations from the interior completely, hence the sunlight we see is almost emitted from this thin photosphere. The sunlight is thermal radiation and close to a black body radiation with temperature about 5800 degrees Kelvin. In addi- tion, on the top of the Sun’s photosphere, there is also an atmosphere consists of chromosphere, and corona. 1.2 LookingthroughtheSun The core, the radiation zone, and the convection zone make up the interior of the Sun. Although, as mentioned in the previous section, many fundamental properties of the Sun can be measured by means of traditional methods, like observing the planetary motions, and analyzing the spectra, those methods cannot reveal much information about the inte- rior of the Sun to us. The reason is quite fundamental. The photons we observed are actually all from the Sun’s photosphere. Even though the energy is originally gener- ated from the nuclear reactions in the solar core and transfers outward in the form of radiation, the information about the interior of the Sun was already washed out before the energy reaches the surface. The opaque interior has made the photons be absorbed and re-emitted for many times before they reach the surface. That is also why we can treat the surface radiation from the Sun as black body radiation. It might sound puzzled. 8 However, it turns out that we still can understand the interior structure by utilizing the Sun’s light. This is due to the ability of performing high spacial resolution measure- ments of the oscillations on the solar surface. 1.2.1 IntroductiontoHelioseismology Around 1960, the regular oscillating phenomena with about 5 minute periods were first observed among the complex and chaotic oscillations on the solar surface by Leighton, Noyes, and Simon [38]. At beginning, people thought those oscillations were just atmo- spheric phenomena at the photosphere and didn’t look deeper. It was only until around 1970, Ulrich, Leibacher and Stein [59, 37] proposed a model to explain the origin of those oscillations and then was later confirmed by Deubner [21]. Since then, people understand that the regular oscillations at the photosphere are originated from the prop- agating waves in the resonant cavity beneath the solar photosphere. The waves are first agitated stochastically by the turbulent motions in the outermost part of the convection zone. They propagate within the resonant cavity and interfere with each other to form standing waves with many different resonant modes. As a result, the spatial and tem- poral coherent oscillations are formed on the solar surface. This discovery created a new branch of the solar physics called helioseismology; a name modified from another similar field, seismology. It gives a way to measure the interior of the Sun, even though indirectly, but is still rigorous enough. In practice, it offers a high-precision diagnostic of the thermodynamic quantities in the solar interior. Another way to study the interior of the Sun is to use the solar neutrinos. The neutrinos are produced from the nuclear fusions in the solar core and transport almost freely to reach the Earth due to the very weak interaction for neutrinos. People used to be puzzled by the missing two-third of the amount of the neutrinos: the so called solar neutrino problem. They only mea- sured about the one-third of the amount predicted by the solar model and observations 9 from helioseismology. They thought the solar model might need to be modified. It is only after people had confirmed the phenomenon of neutrino oscillation, this puzzle was completely resolved. It turns out that there are only one-third of neutrinos can remain as electron neutrinos when they travel to the Earth and can be measured by the equipments [2]. In spite of its ability to carry the interior information,compare to helioseismology, the neutrino measurement is too difficult and still has big errors. Besides, it can only reveal the information about the region of the solar core. In short, helioseismology offers a unique tool can study the whole interior region of the Sun locally and precisely. 1.2.2 ObservingtheSolarOscillation The oscillations on the solar photosphere can be clearly detected by using the Doppler spectroscopy. Due to the Doppler effect, when the light sources have relative motions to the observers, the observed frequencies will be shifted according to the relative velocity. The frequencies decrease (also called redshift) as the sources leave away from observer, and increase (called blueshift) for the other way around. By observing a spectrum line of some specific element, like hydrogen or calcium, the velocity of the light emitting source can be inferred from comparing the observed frequency with its standard frequency. In practice, the measurements are made by using the intensity of two narrow bands on both sides of the peak of the spectrum line. Since the intensity profile of the spectrum line on the frequency axis is shifted with the velocity of the emitting source, the intensities of those two bands will change oppositely, as illustrated in Fig. 1.2. The reason for choosing this instead of the center frequency is because it is more sensitive due to their nonzero slope. To combine this trick with a sensitive CCD can offer a two-dimensional image of the velocity map on the solar surface with very high spatial resolution. The difference between the red-band intensity image and blue-band intensity image is equivalent to a 10 Figure 1.2: The difference between blue-band intensity and red-band intensity is pro- portional to the oscillation velocity,I( +4)I(4) 24 @I() @ . velocity-level image, but it also mixes with the unwanted global velocity due to the rota- tion of the Sun. This can be corrected by subtracting the mean intensity image which only contains the velocity from the rotation (see Fig. 1.3). Although the quantities mea- sured are the velocities instead of oscillating amplitudes, the oscillation frequencies still can be extracted because they share the same frequencies and just have different phases. Taking the Doppler images for long time sequences can serve as the raw data from helio- seismology. In order to obtain the useful information, the data need to be converted to reciprocal images in the frequency-momentum space by using the Fourier transforma- tion. However, the integration of the Fourier transformation requires long enough time for taking the data in order to obtain the power spectrum with high enough signal-to- noise ratio. For this reason, people have to be able to observe the Sun continuously. This can be achieved by three different methods. One is to use the satellite located at the Lagrangian points, the second is to use the observatories at north or south poles, and the third is to rely on the international cooperation of the observatories located around the world. In the first way, there are already some satellites have been launched and operate for a while, e.g. SOHO (Solar and Heliospheric Observatory). Even though this method is very expensive, inflexible, and has short life time, its data has higher quality. On the other hand, the third method is the easiest option and can also offer very reliable 11 Figure 1.3: Doppler images of solar oscillations measured by using MDI (Michelson Doppler Imager) instrument [54] on the SOHO. The left is the original image, and the right is the average image after removing the velocity of rotation. data. For example, the TON (Taiwan Oscillation Network), GONG (Global Oscillation Network Group), are this kind of international cooperative projects. 1.2.3 MathematicalDescriptionofSolarPulsations Even though the pulsations of the Sun are violent, their amplitudes are still much smaller compare to the size of the Sun ( r R 10 4 ). Thus they can be treated as small pertur- bations on the top of equilibrium Sun. This looks naive approximation can actually simplifies the theory greatly as a linear theory. In addition, because the periods of the oscillations are much shorter than the time for thermal energy to dissipate, the adiabatic approximation is sufficient to discuss the dynamics of the solar oscillations. By using hydrodynamics, the whole process can be described by four partial differential equations with four unknowns: 0 =r ( 0 r); (1.3) 12 0 @v @t =rp 0 + 0 r + 0 r 0 ; (1.4) r 2 0 =4G 0 ; (1.5) p 0 +rrp 0 = 1 p 0 0 ( 0 +rr 0 ): (1.6) The 0 is the perturbation of the local density, 0 is the static local density, r is the displacement, v is the velocity, p 0 is the perturbation of local pressure, is the local gravitational potential, 0 is the perturbation of the local gravitational potential, G is the gravity constant, and finally the 1 = ( @lnp @ln ) ad is the first adiabatic exponent. In the spherical coordinate, by using the spherical harmonic representation, the perturbation quantities and the displacements can be decomposed into different spherical harmonic normal modes: r = r a r + a + a = r a r + h a h ; (1.7) wherea r ,a anda are unit vectors in ther, and directions respectively, anda h is along the direction perpendicular to the a r . By means of this, the dynamic equations ((1.3)-(1.6)) can be further simplified into the following set of the ordinary differential equations: d r dr =( 2 r + 1 1 p dp dr ) r + 1 c 2 ( S 2 l ! 2 1)p 0 l(l + 1) r 2 ! 2 0 ; (1.8) dp 0 dr =(! 2 N 2 ) r + 1 1 p dp dr p 0 + d 0 dr ; (1.9) 13 Figure 1.4: -l diagram of power spectrum from helioseismology. (Figure provided by A. G. Kosovichev) 1 r 2 d dr (r 2 d 0 dr ) =4G( p 0 c 2 + r g N 2 ) + l(l + 1) r 2 0 ; (1.10) where S l (S 2 l = l(l+1)c 2 r 2 ) is the Lamb frequency, N (N 2 = g( 1 1 p dp dr 1 d dr )) is the buoyancy frequency, andc (c 2 = 1 p ) is the speed of the sound in adiabatic conditions. Those equations ((1.3)-(1.6)) together with appropriate boundary conditions at r = 0 andr =R constitute an eigenvalue problem, and only exist some particular eigenfre- quencies . Each mode is described completely by three parameters: the radial ordern, the harmonic degreel, and the azimuthal orderm [14]. If we ignore the rotation of the Sun, it will have rotational symmetry. The modes will show 2l + 1 degeneracy for each l harmonic degree and the radial eigenfunctions and eigenfrequencies are independent ofm. The results of the eigenfrequencies are usually drawn as a-l diagram and can be compared with the data from helioseismology in Fig. 1.4. 14 1.2.4 AsymptoticOscillations Those oscillations not only reveals the solar interior structure, but also can probe it locally. To understand why the waves are trapped within some ranges under the photo- sphere, the geometric wave approach provides a simple but still rigorous approximation. First of all, the validity of using the ray theory relies on the relatively much longer paths for the propagation of the waves compare to the wave lengthes. This is satisfied in gen- eral. Therefore, we can approximate the wave fronts as rays. The wave vector can be decomposed into two perpendicular components; one is in the radial direction and the other is perpendicular to it, and satisfies the dispersion relation ! 2 =c 2 jkj 2 =c 2 (k 2 r +k 2 h ): (1.11) After using the definition ofk h (k 2 h = l(l+1) r 2 ), Eq. 1.11 can be expressed as k 2 r = ! 2 c 2 l(l + 1) r 2 = ! 2 c 2 (1 S 2 l ! 2 ): (1.12) This equation simply shows that the wave propagation is controlled by the sound speed which depends on the temperature of the medium. Because the temperature becomes higher in the deeper region of the Sun, the sound speed also becomes higher. Thus, k r decreases with the depth, until k r = 0 at somewhere r = r t , the wave propagates horizontally and Eq. 1.12 becomes c(r t ) r t = ! p l(l + 1) : (1.13) It is similar to the phenomenon of refraction of light as its phase velocity changes; the wave is bent away from its propagating direction. At some point r t , called turning point, the radial component of the wave vector will decrease to zero and then become 15 2.3.3 Asymptotic behavior of stellar oscillations Figure 2.3: Reflection, refraction, and turning points of sound waves inside the sun. (Fig- ure provided by A. G. Kosovichev) If we ignored the fact that our sun is a little ellipsoidal instead of a perfect sphere, the spherical harmonic is still a good approximation to represent normal mode. (True normal modes of the ellipsoid would have a main spherical analog with a little in-mixing from a few adjacent spherical modes). The three characteristic wave-numbers that follow this choice would ben, which is the radial order;l, which is the degree; andm, which is the azimuthal order[10]. The displacement of a normal mode can be written as, 14 Figure 1.5: Sound waves propagate between the solar surface and the turning points. (Figure provided by A. G. Kosovichev) negative. This means the wave starts to propagate backward from the turning point and move outward from the center. On the other side, because there is no gas outside of the Sun, the wave cannot propagate beyond the solar radius (c decreases rapidly at the surface and becomes zero outside the surface). The outgoing wave must be reflected and propagates inward. Thus, the waves are trapped and travel back and force within the region between the turning pointr t and the surface as shown in Fig. 1.5. The depth of the turning point depends on the angular order of the vibration mode, and a wave with lower angular order can penetrate deeper into the Sun. The acoustic mode withl = 0 can penetrate the center, and the highest mode been observed (l w 1000) is trapped above about 99:8% of the radius. This important behavior can be used to get information about local structures by analyzing different wave modes. This gives helioseismology the power to measure the interior of the Sun locally. 16 Chapter2 TheoryaboutSolarandStellar modeling A star, by definition, is a celestial body that is bound by its self-gravity and radiates energy supplied by an internal energy source, e.g. an nuclear reaction or gravity col- lapse. The Sun is a typical example of a star. It emits light to sustain our daily life on the Earth, and the energy is generated from the fusion reactions in its core. A very useful tool called R-H (Hertzsprung-Russell) diagram can classify most of the stars into different main categories. It turns out that almost 90% of the stars lie on the diagonal region of the diagram, and called main sequence stars. Our Sun is also one of them. 2.1 RelevantTimeScalesofStars A quick understanding about a star can be obtained from analyzing its characteristic time scales. Even the simple dimensional analysis can reveal important information about the system. In practice, by comparing the time scales of different mechanisms, we can have qualitative glimpse about the physical properties of the stars, especially, our Sun. Moreover, these time scales can be used to criticize if some approximations are made properly. In general, there are three important time scales. Dynamical time scale This time scale is a measure of how long it takes to restore the mechanical equi- librium. We can consider the extreme condition. If the internal pressure of a star 17 is totally removed, it will rapidly contract due to its own gravity. For this case, this time should be about the free-fall time from the surface of the star to reach its center with the gravitational acceleration same as on the surface, dynamical = r 2R 3 GM 1 p G ; (2.1) Though this is a rough estimation, the order of magnitude would still be reason- able. Actually, compare to the exact calculation, the error turns out to be just about 10%. For the example of the Sun, the dynamical time is about 35 minutes. Thus, if there is any non-equilibrium event happens, a star should be able to settle to equilibrium on this time scale. This offers the criterion for a star can be treated as in hydrostatics equilibrium. Thermal time scale Since the energy of a star is continuously depleted by emitting radiations outward, we can estimate the life of a star if we assume the consumed energy is totally from its gravitational energy, th = GM 2 RL ; (2.2) For the Sun, this time is about 1:6 10 7 years, which is much shorter than the Sun’s age, about 4:5 10 9 years. This confirms that the energy source of the Sun is not the gravitational energy. In addition, this time scale will be about the time for a star to recover thermal equilibrium if it is perturbed out of thermal equilibrium. 18 Nuclear energy time scale The meaning of this quantity is similar to the previous one except the energy source is nuclear fusion. nuc = f"Mc 2 L (2.3) This estimates the time about how long a star will evolve: the life of a star. For the Sun, it is about 10 billion years. To conclude for the Sun, the order of its characteristic time scales is dynamical th nuc . This simple relation can already reveal some important information about the Sun to us. Because the Sun is evolving with the time scale much longer than the others and even much longer than any of our observation time, we can regard the Sun as in hydrostatic and thermal equilibrium at each time step during its evolution. Actually, this statement is true for all the main sequence stars. That is why we can still discuss the static structures of stars since we cannot really see how it would evolve during our short lives. 2.2 MathematicalModelofStellarStructure In order to be able to handle the theory to obtain useful information, some assumptions are necessary in order to make the theory as simple as possible but still rigorous enough. The simplest case is an isolated star, and the influence of rotation and magnetic field are neglected. These imply a star with spherical symmetry. Furthermore, in order to be able to write a few equations to model the stellar structure, an approximation is needed; a star satisfies the local thermodynamic equilibrium (LTE). This is more than just an approximation. LTE offers us the situation to understand and study it theoretically with great deal of our knowledge, the thermodynamics. This is a very good approximation as long as we are focusing on the regions with size much larger than microscopic scale but 19 still much smaller than the temperature changing distance, and the time scale is much longer than the mean free times for the thermal equilibration. Base on those hypotheses, a mathematical model of the stellar structure can be writ- ten down by using a set of equations according to the mechanisms with different physics. On the other hand, if any neglected effect is needed we can always reconsider it after to modify the theory, or treat it perturbatively if the effect is small enough. Hydrostatic balance equation The mechanical equilibrium between the pressure and the gravity can be expressed under this equation. The gravitation force acting downward on the spherical shell is balanced with the upward net pressure due to the pressure differ- ence between the upper and lower boundaries of the shell: dp dr = Gm r 2 ; (2.4) wherep is the pressure and is the density at position of radius r, andm is the mass within the sphere of radiusr. The density and the mass are related by the continuity equation, dm dr = 4r 2 : (2.5) Energy conservation equation The energy is lost by the net energy flux flowing outward, and eventually radiates out of the Sun. This energy lose is compensated by the energy generation in the core in order to keep the star stationary. The balance of the energy in the star can be written by the energy conservation law as following: dL dr = 8 < : 4r 2 " nuc , within the core 0 , otherwise, (2.6) 20 where theL is the local luminosity at r, and the nuc is the nuclear energy genera- tion rate per unit mass. Energy transport equation In the different regions of the stellar interior, the energy is transferred by different mechanisms. The temperature gradients can be described by dT dr = 8 < : 3L 64r 2 T 3 , for radiative transport ( dT dr ) c , for convective transport, (2.7) where T is the local temperature, is opacity, and is the Stefan-Boltzmann constant. These four differential equations describe how the functions of local quantities change with the radius from the center up to the surface. Since these four equations are obtained from the fundamental physics, they should be universal for different stars. Nevertheless, they are not enough for solving seven unknowns (p,,T ,m,L,, and). To make the equations close, we need three extra conditions. They are the equation of state (EOS), the opacity, and the nuclear energy generation rate. The equation of state describes how the local thermal quantities: the pressure, the density, and the temperature are related under LTE. The opacity is the quantity represents how the radiation is absorbed in the material and describes the efficiency about the radiation transport. The energy generated from the nuclear fusion is described by the nuclear reaction rate. As they do not have any universal function forms to present, they would be different for various stars. Those three ingredients describe the material properties of a star. From the other point of view, these three equations should be discussed with their microscopic physics. On the other hand, the previous four equations are described by the macroscopic physics; they do not relate to the microscopic details of a star and thus are universal. This distinction 21 also presents the difficulty of calculating those three material equations because the microphysics is always more complicated than the macrophysics. To conclude, this set of stellar equations should be able to describe any normal star from the fundamental ground. In order to get a solution, e.g. a unique star, a set of ade- quate boundary conditions would be required. Among them, some should be satisfied with the observations on the solar surface (see Tab. 1.1), Solar mass : M Solar luminosity : L Solar radius : R Uniform initial composition : X 0 and the others at stellar center, M(r = 0) = 0 L(0) = 0; (2.8) One of main complexity of solving the problem comes from the calculation of the opac- ity. It entangled with the EOS in complicated way and cannot be isolated in general. However, the EOS has the advantage to be isolated from the opacity and nuclear reac- tion rate in the convection zone where the equations ((2.4)-(2.7)) do not contain both of them. This offers a chance to study the EOS of the Sun independently. In addition, this can be probed by using the local power of helioseismology. This is one reason why we concentrate on the study of the EOS to improve our understanding of solar structure. 22 Chapter3 EquationofStateforSolarandStellar Modeling The equation of state is a thermodynamic potential that relates the thermodynamic quan- tities of a system to a pair of independent variables (e.g. temperature and density). Thermodynamic equilibrium is assumed. The simplest case is for the one component classical ideal gas in which there is no interaction between any particles and the parti- cles are points without any structures. First of all, in thermodynamic equilibrium, all the thermodynamic quantities can be described by three thermodynamic parameters: pres- surep, density, and temperatureT . Nevertheless, they are not totally independent but related by an EOSp = p(T;), which in this simple case is ( being the mass of the particles,k B the Boltzmann constant) p = k B T : (3.1) Even though this is highly simplified, it still can play a good approximation for most applications in astrophysics as long as the effects of interactions are weak enough and the classical physics is still valid to describe the system. Despite the classical ideal gas law combined with the Saha equation have been successfully employed to have great understanding about the solar structure and evolution, the uncertainty compared to the high-precision helioseismology data is still unsatisfied. This is due to the absence of two effects in classical ideal gas model: the interaction and the quantum effects. This 23 means, even though the solar plasma is slightly non-idea and non-classical, we still need to consider their corrections under the constraint of helioseismology. It turned out that the relative precision from helioseismology has been reached as high as 10 5 , but the relative uncertainties among various proposed EOS are still higher than 10 3 . This gives us the room to improve the EOS to have better match with experiment observations. During the development of the EOS for a solar modeling, there are two major EOS have being extensively used in the last two decades due to their own advantages: the MHD (Mihalas-Hummer-Dappen) EOS [30, 19, 43, 44] and the OPAL (Opacity Project at Livermore) EOS [48, 52, 50, 51]. They based on two different approaches; the first one is in chemical picture, and the second one is in physical picture. 3.1 ChemicalPictureApproach The idea is to consider a system with fixed total particles, and all the possible species of particles (e.g. electrons, ions, atoms, and molecules) are introduced at once. The numbers of each species can vary but constrained by the corresponding reaction laws (e.g. ionization, association, and recombination) and different conservation laws (e.g. the charge conservation, and the mass conservation). When the system reaches thermo- dynamic equilibrium, those particle numbers will be determined by minimizing the total free energy. 3.1.1 FreeEnergyMinimizationMethod For a system with Hamiltonian H and volume V , its canonical partition function at temperatureT is Z =Tr[e H k B T ]; (3.2) 24 and the corresponding free energy is F (T;V;N i ) =k B T lnZ; (3.3) whereN i represent the particle numbers of different species. At thermodynamic equi- librium, the values ofN i (i.e.N eq i ) should satisfy the relation F (T;V;N eq i )F (T;V;N i ) (3.4) for all possibleN i values. This is the condition of minimization of the free energy, also known as the free energy minimization method (FEMM) [26, 28, 27, 25, 17]. Following this, the EOS can be derived: p(T;V;N eq i ) = @ @V F (T;V;N eq i ): (3.5) In general, the Hamiltonian has the form, H(~ p 1 ;~ p 2 ;:::; ~ p N ;~ r 1 ;~ r 2 ;::: ~ r N ) = N X i=1 p 2 i 2m +U(~ r 1 ;~ r 2 ;:::; ~ r N ): (3.6) In order to simplify the calculation, the Hamiltonian is first postulated to be classical. Thus the partition function can be separated and written as a factorized form: Z = 1 h 3N Z ::: Z dp N exp( N X i=1 p 2 i 2m ) 1 N! Z ::: Z dre U N X i=1 g i e E i =Z trans Z config Z int ; (3.7) 25 where the translation term: Z trans = 1 h 3N Z ::: Z dp N exp N X i=1 p 2 i 2m ; (3.8) only contains the kinetic energy, and the configurational term: Z config = 1 N! Z ::: Z d~ r N exp[U(~ r 1 ;~ r 2 ;::: ~ r N )]; (3.9) only contains the external potential energy between the particles. However, the inter- nal energy of bound particles needs to be considered in order to complete the partition function. This is written as the internal term, Z int = N X i=1 g i exp(E i ); (3.10) where the E i is the i-th bound energy level and g i is the corresponding degeneracy. Furthermore, the quantum effect correction can be included in the electron translational part since the electron has much lighter mass than the other particles. So we can extract it from the originalZ trans and the final partition function can be expressed as Z =Z e Z trans Z config Z int : (3.11) The benefit of writing in the factorized form is that its free energy can be written as an additive form: F =K B T ln(Z e Z trans Z config Z int ) (3.12) =F e +F trans +F config +F int : (3.13) 26 This makes the analysis and modification easy. If any effect needs to be considered, we can just modify the relevant term without changing the others. Besides, in general, F e +F trans should dominate the free energy (i.e. the ideal classical gas with degenerate electrons), and also has analytical form. In contrast,F config +F int has rather minor con- tribution. Hence, any other physical effects can be easily considered by adding higher order term to it as corrections. 3.1.2 MHDEquationofState Among those terms in the free energy,F int usually contains infinite number of states and is therefore divergent. The simplest example is the internal structure of the hydrogen atom. Its internal term is Z int = X nl 2(2l + 1) exp(E nl =k B T ) (3.14) = 1 X n=0 2n 2 exp(E nl =k B T ); (3.15) whereE nl = 13:6eV n 2 is the hydrogen energy level. This infinite sum is obviously diver- gent, and would cause the total free energy divergent. However, in practice, this problem should not exist since the situation that a single bond particle does not interact with its environment is unreal. The actual F int should be finite. This problem turns to be the main part of obtaining an EOS by using the chemical picture:how to deal with Z int . There are many different ways had been proposed to go around this difficulty. The different approaches all rely on how to cut the infinite sum properly. For example, Saha equation: only consider the ground state energy. Confined atom(CA): assume the bound electrons are confined in a sphere with the average volume of ions. 27 Statically screened Coulomb potential (SSCP): use the screened Coulomb inter- action to limit the number of states. Those methods use the hard-cutoff technique to deal with the internal term which means the infinite sum is truncated into some finite sum. On the other hand, the MHD method uses another approach: the soft-cutoff. This method uses the occupation weightsw is to model the distribution of bound states of the composite particles by continuous and monotonically decreasing occupation probability. This method turns to be a better approximation for modeling the internal structure. The MHD was originally devel- oped by D. Mihalas, D. G. Hummer, and W. D¨ appen as part of the international Opacity Project. In their method, w is represents the probability of finding species s at its i-th ionization state. Thus the corresponding internal partition function, Z int = X i;s w is g is e E is ; (3.16) is also convergent. Following this, the MHD EOS can be obtained. 3.2 PhysicalPictureApproach In contrast to the chemical picture, the physical picture starts from more fundamental ground rather than phenomenological approaches. The theory only considers a sys- tem with the electrons and different species of the nuclei. In principle, all the possible species of composite particles (e.g. ions, atoms, and molecules) should be able to be found in it as bound states. The virtue of it is the theory does not need to include any undetermined phenomenological parameters, but only involves the fundamental electro- magnetic interaction. This makes the theory unbiased and clear. In addition, the higher order corrections can be made in a systematical way, instead of heuristic way as in the 28 chemical picture method. However, the theory is usually too complex to be solved. Besides, it cannot be easily modified to improve the approximation like the chemical picture did. 3.2.1 ActivityExpansionMethod Consider the example of an one component plasma (OCP) system with the Hamiltonian H, the volume V , and the temperature T . From the definition, its grand canonical partition function is (z;N;T ) = 1 X N=0 (e ) N Q N (V;T ); (3.17) where Q N (V;T ) is the N-particle canonical partition function. For the classical and quantum systems,Q N (V;T ) are different and expressed separately as Q N (N;T ) = 1 N!h 3N Z d 3N pd 3N r exp( X i p 2 i 2m X i<j u ij ); (3.18) and Q N (V;T ) =Tre H = Z d 3N r X (1;:::;N)e H (1;:::;N): (3.19) Furthermore, the expression of the grand canonical partition function can be further written as a more elegant form, (z;N;T ) = 1 X N=0 (z) N Z N (V;T ) N! ; (3.20) 29 where z = e 3 is called activity, and = p 2~ 2 =mkT is the thermal de Broglie wavelength. For a classical system or a quantum system,Z N is Z N (V;T ) = Z d 3N r exp( X i<j u ij ); (3.21) or Z N (V;T ) = Z d 3N r(N! 3 X (1;:::;N)e H (1;:::;N)) = Z d 3N rW N (1;:::;N); (3.22) whereW N (1;:::;N) is defined as W N (1;:::;N)N! 3N X (1;:::;N)e H (1;:::;N) (3.23) In this expression, the classical and quantum grand canonical partition functions share a similar form, and can be treated in a unified way. The EOS can be derived from the grand canonical partition function by using the thermodynamic relations P kT = 1 V log (z;V;T ) 1 v =z @ @z ( 1 V log (z;V;T )): (3.24) In principle, they can be expended in power ofz (the activity expansion) as P kT = 1 X l=1 b l z l 1 v = 1 X l=1 lb l z l (3.25) 30 since the logarithmical function is an analytical function. Furthermore, the relation between b l and Z N (V;T ) is like the cumulants and moments in statistics, and can be easily obtained. For a quantum system, b l (V;T ) 1 l! 3l3 V Z d 3 r 1 :::d 3 r l U l (1;:::;l); (3.26) where U l (1;:::;l) represent the ”cumulants” and are related to the ”moments” W l (1;:::;l) as U 1 (1) =W 1 (1) = 1 (3.27) U 2 (1; 2) =W 2 (1; 2)W 1 (1)W 1 (2) (3.28) U 3 (1; 2; 3) =W 3 (1; 2; 3)W 1 (1)W 2 (2; 3) (3.29) W 1 (2)W 2 (3; 1)W 1 (3)W 2 (1; 2) + 2W 1 (1)W 1 (2)W 1 (3) . . . (3.30) On the other hand, the classical system has similar relations but can be further simpli- fied by introducing a new function f(r ij ) = e P i<j u ij 1. The advantage of it is thatf(r ij ) plays a better perturbative function thane P i<j u ij , especially for a dilute system. Besides, in this expression, b l (V;T ) only contains connected l-particle clus- ter integrals. Therefore, the relation for b l (V;T ) can be derived by using the cluster expansion theory (all the details can be found in [29]) as b l (V;T ) 1 l! 3l3 V (sum of all possible connected l clusters) 1 l! 3l3 V Z ::: Z X 1i<jl Y all bonds f ij dr 3 1 :::dr 3 l : (3.31) 31 For example, b 1 = 1 V Z d 3 r 1 = 1; (3.32) b 2 = 1 2! 3 V Z d 3 r 1 d 3 r 2 f 12 = 1 2 3 Z d 3 r 12 f 12 ; (3.33) b 3 = 1 3! 6 V Z (f 31 f 21 +f 32 f 31 +f 32 f 21 +f 32 f 31 f 21 )d 3 r 1 d 3 r 2 d 3 r 3 ; (3.34) etc. Although these activity expansion formulas for the EOS are rigorous, the coeffi- cients b l (V;T ) are plagued by the divergence (except b 1 (V;T ) = 1). Thus, the diver- gence needs to be removed before to make any application from it. It turns out that there are two different sources of the divergence: the long-range (also called infrared) divergence and short-range (also called ultraviolet) divergence. The first one comes from the long range nature of the Coulomb interaction, and the second one is due to the singularity of the attractive Coulomb interaction. For a classical system, Mayer [41] and Abe [1] proposed the re-summation method to remove the long-range divergence. On the other hand, the short-range divergence is ambiguously solved by putting some short-range cut-off. In order to explain the re-summation method clearly, the activity expansion EOS can be rewritten as p k B T = X l=1 b l z l =z +S + 1 X m=2 z m! ( @ @z z) m2 ( @S @z ) m 1 v =z @ @z ( p k B T ); (3.35) whereS is defined as S 1 X l=1 l l + 1 z l+1 = 1 X l=2 a l l 1 z l+1 : (3.36) 32 l is the irreduciblel-cluster integral and is defined as l 1 l!V Z ::: Z X 1i<j(l+1) Y irreducible bonds f ij dr 3 1 :::dr 3 l : (3.37) Follow Mayer’s and Abe’s ideas, the divergence in S function can be removed by regrouping the terms in the summation and summing the ring diagrams first. Then the S can be rewritten as S =S R + 1 X m=2 S m ; whereS R = 1 12 3 D ; D = r k B T 4z ; (3.38) S 2 = 2z 2 Z 1 0 [e q(r)=k B T 1]r 2 dr +z 2 (2 2 D 2 2 D );q(r) = 1 r e r= D : (3.39) After performing the re-summation, the long-range divergence has been cured by replac- ing the long-range Coulomb potential with the short-range screened potential. From this result, the activity expansion of the pressure can be reordered in order of activityz, p k B T =z + R + 1 X m=2 (C m +z @ R @z @C m @z ) (3.40) R =S R + 1 X m=2 z m! ( @ @z z) m2 ( @S R @z ) m ; C 2 =S 2 ; C 3 =S 3 + z 2! ( @S 2 @z ) 2 (3.41) Even though the expressions above are for a system of one component plasma (OCP), the results can be easily generalized to more than one component. 3.2.2 OPALEquationofState In the short distance (shorter than the de Broglie wavelength), the classical physics becomes invalid and the quantum theory is needed. In addition, the quantum mechanical effect:the uncertainty principle, naturally smears out the short-range divergence of the 33 Coulomb potential. Therefore, the quantum plasma offers the complete theory about the systems of plasma. The OPAL was developed as part of Livermores OPAL opacity recalculation project [55, 31]. It follows the idea of activity expansion of a classical system; first works in the classical framework and uses the re-summation method, and then includes quantum corrections to obtain the theory of quantum plasma. Similar to (3.39), for two component plasma (TCP) gas, it becomes S 2;ij =a 2;ij +z i z j h 2(e i e j ) 2 D 2 (e i e j ) 2 D i ; (3.42) where a 2;ij =2z i z j Z 1 0 e q ij (r)=k B T 1 r 2 dr; q ij (r) = e i e j r e r= D ; (3.43) and we can define s ij S 2;ij z i z j : (3.44) For quantum plasma, this becomes [49] s ij =b ij + 1 + 2 ; (3.45) b ij = a 2;ij z i z j = 3 2 Tr(e H ij H 0 ij ); (3.46) 1 = 2(eie j ) 2 D ; (3.47) 2 = 2 (eie j ) 2 D w( ij ); (3.48) 34 wherew ij is a quantum correction term, and becomes 1 under classical limit. ij is the thermal de Broglie wave length ij = h= p 2 ij k B T with the reduced mass ij . In addition, the EOS up to second order activity expansion is (pp 0 ) = X ij z i z j b ij : (3.49) When the two components in TCP have opposite charge, like electrons and protons, the b ep contains two parts: one is the bound state part b b ep and the other is scattering state partb c ep , and b ep =b b ep +b c ep ; (3.50) b b ep = 3 ep 2 X nl (2l + 1)e E nl ; (3.51) b c ep = 3 ep 2 X nl (2l + 1) Z 1 0 e E nl d dE l (E)dE: (3.52) The bound state part is divergent as explained in the section of the chemical picture, however both parts can be reorganized into a convergent forms by doing the integration by parts twice, and using the Levison theorem twice [48, 36]. The results are b b ep = 3 ep 2 X nl (2l + 1) e E nl 1 +E nl ; (3.53) b c ep = 3 ep 2 " 1 X l (2l + 1) Z 1 0 dE d l dE e E nl + X nl (2l + 1) X nl (2l + 1)E nl : # (3.54) The compensation between the scattering state part and the bound state part cures the divergence of the original bound state part and gives an convergent effective internal 35 partition function. This form is usually referred as Planck-Larkin partition function (PLPF): PLPF = X nl (2l + 1) e E nl 1 +E nl : (3.55) PLPF is not a real partition function, but rather a part of the second cluster coefficient, which reveals the results of the compensation between the scattering and the bound states. However, it can be used to define the activity of new composite particles. For example, the hydrogen atom is not assumed at the beginning in the system of e-p TCP, but the activity of this new compound can be recognized as z H 2z e z p b p ep =z e z p 3 ep X nl (2l + 1) e E nl =k B T 1 + E nl k B T : (3.56) More general, the activity of the composite particle composed of two speciesa,b can be defined as z ab 2z a z b b b ab =z a z b 3 ab X nl (2l + 1) e E nl =k B T 1 + E nl k B T (3.57) Therefore, the originalS 2 of activityfz e ;z p g can be rewritten as of activityfz e ;z p ;z H g S 2 =z H +S 2 =z H + (z 2 e s ee + 2z e z p s ep z 2 p s pp ); (3.58) wheres ep is the continuum state part ofs ep s ep =b c ep + 2(eie j ) 2 D 2 (eie j ) 2 D w( ij ): (3.59) 36 This new activity can be further used to include heavier elements, such as helium. By considering all possible augmented set of activities in the second order,fz e ;z p ;z H :z eH g, the equation of state becomes [46, 47] p k B T = X i z i +S R + 1 2! X i z i ( @S R @z i ) + X ij z i z j s c ij ; (3.60) wherei;j runs over the augmented set of activities: S R = 1 (12 D ) 3 ; D = k B T 4[z e e 2 e +z p e 2 p +z H e 2 H +z e e 2 eH +:::] 1 2 (3.61) z H = X nl z nl H (3.62) z nl H =z e z p 3 ep (2l + 1) e E nl =k B T 1 + E nl k B T (3.63) z eH = X nln 0 l 0 z nln 0 l 0 eH (3.64) z nln 0 l 0 eH =z e z nl H 3 eH (2l 0 + 1) e E nln 0 l 0=k B T 1 + E nln 0 l 0 k B T (3.65) s c ij =b c ij + 1 + 2 ; 1 = 2(e i e j ) 2 D ; 2 = 2 (eie j ) D w( ij ) (3.66) b c ij = 3 ij 2 " X l (2l + 1) Z 1 0 dE d l dE e E=k B T + X nl (2l + 1) X nl (2l + 1) E nl k B T # (3.67) whereb c ij is the continuum state part ofb ij . For the pairs such asee orpp they would not have the compensated parts. The last two terms other thanb c ij ins c ij are from the re-summation method to remove the divergences. 37 3.2.3 VirialEquationofState The activity expansion EOS has both the pressure and the density expanded in terms of power ofz. By assuming the pressure can be expanded as a power series in the density, those two equations can be easily combined into one EOS as a density expansion of pressure by eliminating all the power of activityz. The result is called virial equation: Pv kT = 1 X l=1 a l (T )( 1 v ) l1 ; (3.68) where thea l (T ) is thel-th virial coefficient, and the relations between thea l andb l are a 1 = b 1 = 1 a 2 = b 2 a 3 = 4b 2 2 2b 3 a 4 = 20b 3 2 + 18b 2 b 3 3b 4 ::: (3.69) It can be proved that thea l (T ) only contains the integrals with the graphes of irreducible l-cluster. In principle, both the virial EOS and the activity EOS are rigorous. However, because the higher order terms are more complicated to calculate, both the infinite series need to be truncated up to some power in practical calculation. This introduced error surely brings the discrepancy between them. Moreover, although the virial EOS is more nature since it does not contain the activity which is not a directly measurable quantity, the activity expansion EOS is superior to the virial EOS for studying a system with pro- cesses of chemical reactions or ionizations due to the invalidity of density expansion in this regime. This point can be explained more clear by using one example. Consider 38 a system of hydrogen atoms, and the the hydrogen molecule can form via the chemical reaction process, H +H$H 2 : (3.70) The activity expansion EOS of this system is p k B T =z H +z 2 H b HH H =z H @p=k b T @z H =z H + 2z 2 H b HH ; (3.71) where the expansion is truncated up to second order in order to make the calculation simpler but still include the effect of formation ofH 2 molecules. Besides, we only need to concentrate on two extreme temperature since theH 2 molecules can only form when the temperature is low enough. First, at very high temperature, we can get the result b HH ! 0 from the definition ofb l . Hence we obtain H =z H from the activity expan- sion of the density because thez H is still some finite number. Then we can substitute thez H in the pressure expansion with H and obtain a relation which is exactly the same as the virial expansion up to the second order, p k B T = H 2 b HH = H + 2 a HH : (3.72) This demonstration shows the high temperature limit of the truncated activity expansion can recover the the result of virial expansion up to the same order. For the very low temperature,b l will be exponentially large. From the activity expansion of density, we can getz H ! p H =2b HH H and use it in the activity expansion of the pressure to obtain p k B T =z 2 H b HH = H 2b HH b HH = H 2 = H 2 : (3.73) 39 This equation correctly describe the pressure for a H 2 system with half density since every two H atoms combine into one H 2 molecule. However the same conclusion cannot be obtained by using the virial expansion. When the temperature decreases to very low temperature, the second virial coefficient a 2 = b 2 simply increases to large negative number. At some low temperature, the pressure would become zero and then become negative when the temperature decreases further. This nonphysical result does not only come from the truncation of the virial expansion, but also the assumption of density expansion breaks down in this situation. To conclude, this simple example shows the activity expansion is more capable than the virial expansion to deal with the processes of formation of composite particles. However ,the virial expansion can be still useful and more nature as long as it is valid, which means, for this example, it is under the condition of almost full dissociation. 40 Chapter4 VirialEquationofStatevia Feynman-Kacpathintegral Among the competing equation-of-state formalisms for solar modeling, both MHD and OPAL EOSs are very successful in describing the structure in the Sun [18]. So far, the best current equation of state is OPAL, an activity-expansion based theory (see Fig. 4.1), yet the virial expansions of the equation of state have the potential to become the best EOS for solar modeling due to multiple reasons. In principle, each order of density expansions of the virial EOS can be calculated exactly, systematically, and analytically. Figure 4.1: Difference between the squared sound speed obtained from the inversion methodology with two different solar models. Figure provided by S Basu. 41 All the non-ideal contributions are included rigorously. This makes it likely to be the most accurate EOS. Moreover, the code of calculating OPAL EOS is not an open source and the results are only offered as pre-calculated tables for various applications. On the other hand, the analytical formulas of the virial EOS can be utilized easily, flex- ibly, and efficiently. During the development of exact virial expansions for quantum Coulomb systems, different methods had been adopted by different people to obtain the high orders of virial coefficients. For some main contributions, Ebeling and his collaborators [22] computed the virial coefficients for a Coulomb potential with a long- distance cutoff by using ladder approximation to obtain results. Alastuey and Perez [9] used the path integral [24] in the framework of the Feynman-Kac (FK) representation [56] to derive the virial EOS. Brown and Yaffe [12] also obtained the similar results by utilized the effective field theory. Up to now, the highest order of virial coefficients has been completely calculated is up to 5/2-th power of the density. Both results from Alastuey’s and Brown’s works are consistent up to that order. Each method has its own value. The approach based on the formalism of the FK path integral maps a quantum Coulomb system of charged particles into a classical system of loops with modified Coulomb interactions. Its advantage is that the tools developed for a classical system can be thus utilized to simplify the calculations. Furthermore, this method is also con- venient for calculating the scaled low-temperature expansion of the EOS: a theory goes beyond Saha equation in both low temperature and low density regions [5]. Under this approach, the exact low-temperature asymptotic expansions can be naturally obtained within the screen cluster representation (more details will be discussed in Ch.6). Base on these reasons, our work will follow the formulas from Alastuey’s and Perez’s results [7, 8, 9]. 42 4.1 Statistical Mechanics of the Grand Partition Func- tion The grand partition function of a quantum system can be obtained from the grand canon- ical ensemble =Tr exp[(H N X N )]; (4.1) where the index represents the volume of the system, and indicates the species of particles. The trace Tr is taken over all states that satisfy the boundary conditions and the required symmetry according to their spin statistics. The partition function of a system almost contains all the information about its thermodynamics under thermal equilibrium. Many thermodynamic quantities can be derived form the partition function by taking log function and/or derivatives of it. For example, pressure and density can be obtained from the grand partition function by using the relations, P = lim TL 1 ln ; (4.2) =z @ @z h lim TL 1 ln i j ; (4.3) where lim TL means thermodynamic limit, andz = exp( ) is the fugacity of element . 4.2 Feynman-KacPathIntegralApproach The method of path integral does not only provide an alternative of solving the quan- tum mechanical problems but sometimes can also make the calculation easy, clear, and elegant. The physical picture obtained under this approach is more profound than the 43 traditional one. Besides, it is usually more suitable for performing the perturbation cal- culation. 4.2.1 SingleParticleSystem To explain the method of the FK path integral, we start with a system containing only one particle with massm subjected to an external potentialV (~ r). Its Hamiltonian reads H = ~ 2 2m 4 +V (~ r): (4.4) Under the method of path integral, the diagonal elements of its density matrix <~ rj exp(H)j~ r> at temperatureT can be expressed as <~ rj exp([ ~ 2 2m +V (r)])j~ r>= X paths exp( S(~ r;~ r) ~ ); (4.5) where S(~ r;~ r) = ~ Z 0 dt[ m 2 ( d~ r(t) dt ) 2 +V (~ r(t))] (4.6) is the corresponding action with imaginary timet. In the formal FK representation, this can be further expressed as <~ rj exp([ ~ 2 2m +V (r)])j~ r> = 1 (2 2 ) 3=2 Z D( ~ ) exp[V ? (~ r; ~ )]; (4.7) V ? (~ r; ~ ) = Z 1 0 dsV (~ r + ~ (s)); (4.8) where = (~ 2 =m) 1=2 is the thermal de Broglie Wavelength. In this expression, the time and path have been re-parameterized ast =s~ and~ r (t) =~ r + ~ with the condi- tion ~ (0) = ~ (1) = ~ 0. Thus the dimensionless path, called Brownian bridge ~ (s), starts 44 from the origin at times = 0 and returns back at times = 1. Meanwhile, a Gaussian factor from the kinetic energy part of the Hamiltonian has already been absorbed into the functional measure. Therefore, the integral measure becomes a normalized Gaussian measure and is entirely defined by its covariance Z D( ~ ) (s) (t) = 8 > > < > > : s(1t); st t(1s); ts: (4.9) The equation (4.7) gives another physical picture hidden in the quantum density matrix. Under this expression, the original system of a quantum particle is mapped to a system of a classical extended object, a loop, subjected to an average potential V ? (~ r; ~ ), and exp[V ( ~ )] is the corresponding classical Boltzman factor for each random shape of the loop. The advantage of making the quantum system like a classical system enables us to deal the calculations with tools developed in classical physics. At the end, the partition function of this system can be further expressed as = Z d~ r<~ rj exp([ ~ 2 2m +V (r)])j~ r>= Z d~ r 1 (2 2 ) 3=2 Z D( ~ ) exp[V ? (~ r; ~ )]: (4.10) 4.2.2 Maxwell-BoltzmannMany-ParticleSystem The result from the previous section can be extended to many particle system straightfor- ward. To begin with, we only study a system with Maxwell-Boltzmann statistics here. This means all the particles are distinguishable, and therefore no resulting exchange effects. We can consider a system consisting different particles of element . Each particle has massm , chargee , and spin . In here we only consider the Coulomb 45 interaction as the only interaction between the particles. This makes a good approxi- mation as long as the relativistic effects are not important and can be neglected. This is generally suitable for most applications in astronomy, especially for solar modeling. The further advantage under this approximation is that we can also ignore the retarda- tion effects. Therefore our discussion will be restricted to the Coulomb systems. The Hamiltonian of a Coulomb system is H N; = X i ~ 2 2m i i + 1 2 X i6=j e i e j j~ r i ~ r j j ; (4.11) where the indexi = [ k ] is a double index,k runs from 1 to the number of particlesN of element, and runs from 1 to the number of elements. Its grand partition function can be written as MB = 1 X N=0 Y z N N ! (2 + 1) N Z N Y i d~ r i D ~ R N j exp(H N )j ~ R N E ; (4.12) where j ~ R N >= i j~ r i >; (4.13) and < ~ R N j exp(H N )j ~ R N >= Y 1 (2 2 ) 3N=2 Z Y i D( ~ i )exp " 2 X i6=j e i e j v(E;E 0 ) # : (4.14) In this expression, the pathE actually denotes (;~ r; ~ ). The form of the potential is v(E;E 0 ) = Z 1 0 dsv c (j~ r + ~ (s)~ r 0 0 ~ 0 (s)j); (4.15) 46 where thev c is the pure Coulomb potential. Unlike the electrostatic potential between two classical charged loops, the potential v(E;E 0 ) represents an average potential between the positions on two loops interacting at the same ”time” s. Under the clas- sical limit (j~ r ~ r 0 j ), which means the sizes of the loops are much smaller than the separations between the loops, the Coulomb form v(E;E 0 ) 1=j~ r~ r 0 j will be revived. The grand partition function can be further written as a more compact form by redefining a new phase-space measure dE=dd~ rD( ~ ) and a general fugacity z(E) = (2 + 1)z (2 2 ) 3=2 . It becomes MB = 1 X N=0 1 N! Z N Y k=1 dE k z(E k ) Y k<l [exp[e k e l v(E k ;E l )]]: (4.16) Due to its similarity to a classical grand partition function, the ideas from Mayer, Abe, and Meeron [1, 41, 42] can be reused here. We can introduce the similarf bonds as in the Mayer’s cluster expansion theory: f(E k ;E l ) = exp[e k e l v(E k ;E l )] 1; (4.17) The grand partition function of the quantum particle system ends up to be MB = 1 X N=0 1 N! Z N Y k=1 dE k z(E k ) Y k<l [1 +f(E k ;E l )]: (4.18) In this classical like picture, the quantum mechanical aspect (the uncertainty principle) is hidden in the complex shapes of the loops. The extended shapes of the loops repre- sent the original quantum fluctuations of quantum particles. The method of path integral clearly describes the quantum effects pictorially. In contrast, even though the effective potential approaches can also provide a classical like picture of a quantum system: map 47 a system of quantum particles to a system of classical particles, they introduces com- plicated many-body interactions between the classical particles. On the other hand, the interactions between the loops are strictly two-body interactions. 4.2.3 VirailExpansioninFeynman-KacRepresentation The form of the equation (4.18) is identical to the classical grand partition function with similar Mayer’sf bonds. In this analogy, the normal spatial positions of classical particles are replaced by the generalized coordinatesE. The mathematical structures of its thermodynamic functions and correlations in terms of fugacity expansion or density expansion with the Mayer’s cluster diagrams still remain the same: the topology of Mayer’s diagrams and the symmetry counting are unchanged. Thus we can just borrow the results from Mayer’s classical cluster expansion. However, each Mayer-like diagram is divergent because the two-body potential of loops are long-ranged, like the Coulomb potential of two charged particles. This difficult can be again removed by applying the re-summation method introduced by Abe and Meeron as described previously. First of all, thef bond is expanded into the following from, f(E;E 0 ) = f C (j~ r~ r 0 j) + 1 2 f 2 C (j~ r~ r 0 j) + Z 1 0 ds[ ~ (s): ~ r + 0 ~ 0 (s): ~ r 0 ]f C (j~ r~ r 0 j) +f T (E;E 0 ); (4.19) wheref C is the shape-independent Coulomb bondf C (j~ r~ r 0 j) =e e 0v C (j~ r~ r 0 j), and the truncated bondf T contains all the remaining terms. This useful decomposition separates the integrable bondf T in which it decays as 1=j~ r~ r 0 j whenj~ r~ r 0 j!1 from the other nonintegrable bonds. The diagrams including those nonintegrable bonds can be rearranged and the corresponding convolution chains can be re-summed up as Debye 48 screened potential D (r) = exp(r)=r with = (4 X e 2 MB ) 1=2 [20]. After this resummation procedure, part of nonintegrable bonds are replaced by the re-summed integrable bonds in which its potential is the short-range screened potential D (r). The other unchanged long-range bonds are either convoluted with short-range bond or mul- tiplied with one or more chains to be integrable. These algebraic decay bonds make the difference from the classical Coulomb systems, which contains an exponential decay of the correlation with its effective potential. The pressure can be obtained by using the thermodynamic relations, F MB = F MB id + 2 X ; Z 1 0 dg Z d~ r MB T;g ( ~ 0;~ r)e e v C (r); (4.20) P MB = X MB @ @ MB lim TL h F MB i j lim TL h F MB i : (4.21) First, the free energy F MB can be calculated from the idea free energy F MB id and the two-body correlation function MB T;g which can be calculated from the identity MB T ( a ~ r a ; b ~ r b ) = Z D( ~ a )D( ~ b )(E a )(E b )h(E a ;E b ): (4.22) In this expression, the Ursell functionh(E a ;E b ) is defined as (E a )(E b )h(E a ;E b ) =z(E a )z(E b ) lim TL 2 ln ? z(E a )z(E b ) ; (4.23) and the loop density(E) with shapeE is (E) =z(E) lim TL ln ? z(E) : (4.24) 49 In order to obtain the virial expansion as a particle density expansion, the loop den- sity can be further expanded by the particle density which is reduced to double integer series in ( MB ) 1=2 and ln MB . Then, the diagrammatic representation of the pressure with Maxwell-Boltzmann statistics can be obtained by substituting the particle density expansion of(E) in the free energy equation (4.20) and taking the derivative of particle density. 4.2.4 CorrectionsfromQuantumStatistics In order to have complete quantum effects, the quantum statistics, either Bose-Einstein statistics or Fermi-Dira statistics, needs to be included. First of all, the allowed states must be either symmetric or antisymmetric under exchange of particles and both situa- tions can be expressed as Slater sums j ~ R N z N > S = 1 Q (N !) 1=2 X P Y (P) i j~ r P(i) z P(i) > S : (4.25) P is a permutation of (1:::N ),P (i) = (P (k);), and (P) is either 1 if the parti- cles are bosons (with integer spin ) or the signature (1) if the particles are fermions (with half integer spin ). According to these states, the corresponding grand partition function will become = 1 X N=0 Q z N [ Q (N !)] 2 X P;P 0 Y (P ) (P 0 ) X f z i g Y i < z P 0 (i)j z P (i)> Z N Y i d~ r i <~ r P 0 (i)j i j exp(H N; )j i j~ r P (i)>: (4.26) 50 In this expression, the summation under double permutation P P;P 0 can be decomposed into two parts: the sum of diagonal matrix elements and the sum of off-diagonal matrix elements. The sum of diagonal part obviously corresponds to Maxwell-Boltzmann statistics because there is no exchange of particles between two states. On the other hand, the terms in the off-diagonal sum represent exchange of some particles between two states. In the FK representation, the path for a particle exchanging the position from ~ r k to position~ r l can be again parameterized by using the Brownian bridge ~ (s) as ~ ! kl (s) = (1s)~ r k +s~ r l + ~ (s); (4.27) and satisfies the conditions ~ ! kl (0) = ~ r k and ~ ! kl (1) = ~ r l . Therefore, the exchanged particle in the FK representation is equivalent to an open loop where one end of it is at position~ r k and the other end is at~ r l . For the terms in the off-diagonal sum, they also include various unchanged particles between two states, and those particles are still expressed as closed loops, like the particles with Maxwell-Boltzmann statistics. Under this picture, the off-diagonal sum can be arranged as the order of how many particles are exchanged, or in the other word, the order of number of open loops in the classical like systems. This approach behaves like a perturbative expansion with the Maxwell- Boltzmann system discussed in the previous section as the reference system. Besides, the term with the order ofn exchanged particles contributes the power ofz n in the sum of grand partition function. Thus, the grand partition function can be expanded as = MB f1 1 2 X (1) 2+1 (2 + 1) z 2 (2 2 ) 3 Z d~ r 12 Z D( ~ 1 )D( ~ 2 ) exp h v(F 12 ;F 21 ) Z 1 0 dg Z dE a (E a jF 12 ;F 21 ) v(E a ;F 12 ) +v(E a ;F 21 ) i +O(z 3 )g;(4.28) 51 where(E a jF 12 ;F 21 ) is the one-body density in the presence of the two open loopsF 12 andF 21 interacting with the closed loopE a . Follow this construction, the total pressureP also can be expressed as an expansion around theP MB in power of the fugacityz P =P MB + 1 X n=2 E n ; (4.29) where theE n contains the contribution ofn exchanged particles with the power ofz n . As discussed previously,P MB can be presented as diagrammatic series in terms of par- ticle density MB . The fugacity z and the Maxwell-Boltzmann particle density MB can be further eliminated by applying the relations (4.2)and (4.3) to and MB and the results only contain the real density . After that, the desired particle density expansion of the pressure, the virial expansion, can be obtained by collecting the terms with the same power of particle density . This systematic scheme has been utilized to obtain 52 the virial expansion of pressure exactly up to the order 5=2 (this equation will is be called ”FK virial EOS” or ”virial EOS” thereafter): P = X 3 D 24 + 6 (ln 2 1) X ; 3 e 3 e 3 p 2 X ; 3 Q(x ) 3 3 X ; e 3 e 3 ln ( D ) + p 2 X (1) 2+1 (2 + 1) 3 2 E(x ) 3 2 p 2 X ; e e D 3 Q(x ) 2 4 X ; e 4 e 4 D ln ( D ) + 3 2 p 2 X (1) 2+1 (2 + 1) 3 2 e 2 D E(x ) + 1 16 X 2 ~ 2 e 2 m 3 D +( 1 3 3 4 ln 2 + 1 2 ln 3) X ; 4 e 4 e 4 D +C 1 X ;; 5 e 3 e 4 e 3 1 D +C 2 X ;; ; 6 e 3 e 3 e 3 e 3 3 D +O( 3 ln) (4.30) with D = (4 X e 2 ) 1=2 , l = e e , m = m m =(m + m ), = (~ 2 =m ) 1=2 ,x = p 2 l ,C 1 = 15:205 :001, andC 2 =14:733 :001. The function Q(x ) refers to the so-called quantum second-virial coefficient first introduced by Ebeling and co-workers [22]: Q(x ) = 1 ( p 2 3 ) lim R!1 ( Z r<R d~ r h (2 2 ) 3=2 <~ rje h j~ r>1 + e e r 2 e 2 e 2 2r 2 i + 2 3 3 e 3 e 3 [ln( 3 p 2R ) +C] ) ; (4.31) 53 where the Euler-Mascheroni constant isC = 0:577216. This represents the direct quan- tum mechanical contribution. The function E(x ) describes the quantum exchange contribution and is defined as the two-body exchange integral E(x ) = (2 p ) lim R!1 Z r<R d~ r<~ rje h j~ r>: (4.32) In bothQ andE functions,h represents the one-body Hamiltonian with reduced mass m subjected to the Coulomb potentiale e =r. The functionsQ andE only depend on the temperature via the dimensionless Born parameter (x ), and can be expressed as useful converging series [34, 32, 60] Q(x ) = x 6 p 8 (x ) 2 1 6 ( 1 2 C E + ln 3 1 2 )(x ) 3 + 1 X n=4 p (n 2) (1 + n 2 ) ( x 2 ) n ; (4.33) E(x ) = p 4 + x 2 + ln 2 p 4 (x ) 2 + 1 X n=3 p (1 2 (2n) )(n 1) (1 + n 2 ) ( x 2 ) n (4.34) with the Riemann zeta function(n) and the gamma function (n) defined separately as (n) = 1 X x=1 1 x n ; (4.35) (n) = Z 1 0 x n1 e x dx: (4.36) For the integern, the function can be further simplified as (n) = (n 1)!; (4.37) (n + 1 2 ) = (2n)! 4 n n! p : (4.38) 54 In conclusion, the first term in the FK virial EOS is obviously the same as the clas- sical idea gas pressure. All the other non-ideal or non-classical contributions are cor- rections to it. Noticeably, the second term is the so called Debye-H¨ uckel term for the correction from the screened Coulomb interaction and also can be obtained by using the mean field approximation. Both of first two terms are purely classical. The next higher order terms start to contain the quantum effects, e.g. quantum diffraction, recombina- tion, scattering, and exchange. In this FK path integral approach, the physical pictures for each terms can be clearly understood. In addition, the higher correction terms can be derived rigorously and systematically if needed. 55 Chapter5 ProbingtheSunwiththeVirial EquationofState With the analytical expression of the FK virial EOS ready, we can begin the research on the applications to the Sun: to study the thermodynamic properties of the Sun. In principle, the equation (4.30) can be continued to include all the higher order terms by following the same methodology discussed in the previous chapter. This exact virial expansion, even though is rigorous in each order of terms, are derived under a perturba- tive way. This approach restricts the domain of its validity where the related perturba- tive parameters need to be small enough. Otherwise, the higher order correction would become more important than the lower order terms, and thus the perturbative expansion will fail. The truncated expansion becomes unreliable. In practice, the infinity series of the perturbative expansion cannot be obtained since the terms at each order do not have simple relations, and moreover the complex for calculating higher order terms grows quickly; the number of diagrams included will grow exponentially. Besides, there is a more fundamental problem about the perturbation method. The infinite perturbation expansion cannot be a convergent series by its nature, but only asymptotically conver- gent. However, both flaws will not stop us from making some applications on it. It turns out that they are avoided by the same reason. This exact expansion has only been calculated explicitly up to the order 5=2 , with nobody so far investing the effort to go to the higher terms. The most important question is if the values calculated by using this ”truncated” expansion is good enough for our purpose or not. Of course, in general, 56 one would expect that including more higher-order terms would improve the quality of the truncated expansion, even though this is not always guaranteed, but the decision of where to stop needs to be made smartly given the considerable effort for higher terms. We will continue this discussion later, but here we will stay on applying the formula (4.30) what we do have now. 5.1 DomainofApplicabilityofVirialEquationofState The first thing before starting to apply the FK virial EOS is to confirm if it is valid to be used in our case. This can be qualitatively estimated by doing the scaling analyses. To facilitate this approach, we indicate the relevant scales of the system we are considering. Basically, those scales determines its fundamental properties, and the phase of the a Coulomb system can be thus estimated by its density and temperature. For our purpose, the relevant characteristic scales for the Coulomb system can be elegantly reduced to these characteristic length scales: d = 3 4n 1=3 mean interparticle distance, (5.1) = ~ 2 m 1=2 de Broglie wavelength, (5.2) l =e e Landau length, (5.3) a 0 = ~ 2 me 2 Bohr radius. (5.4) However, we should still keep it in mind that the scaling analyses cannot be exact due to its simple estimation and some ambiguity. For example, any factor like 2 or can be randomly added to it without changing its meaning. Beside, most criteria for the test also include some simplification without further serious check. If the result is at extreme 57 condition like>> 1 or<< 1, there will be no doubt, otherwise we cannot use it as the confirmed proof to simply conclude. In that situation, deeper study will be required. 5.1.1 CoulombCouplingParameter Figure 5.1: Coupling parameters throughout the Sun for different charged particles. The coupling parameter can be defined as = l d : (5.5) This ratio represents the qualitative estimate of the relative strength for the Coulomb interaction in the system with temperatureT and densityn. Since the virial expansion (4.30) is a perturbation expansion with weak coupling, this means should satisfy << 1. Throughout the whole region of the Sun, this coupling parameter is tested for pairs of the same charged particles and also pairs of different charged particles. The results are concluded in Fig. 5.1. It clearly shows that all the coupling parameters are all at least smaller than the critical value 1, except around 5 10 5 K for the electron and helium nuclear pair and the proton and helium nuclear pair. Overall, the solar plasma is only weakly coupled in the region with higher temperature (in the deeper region of the Sun). 58 5.1.2 DegeneracyParameter Figure 5.2: The electron degeneracy parameter is below the critical value 1 throughout the Sun. The quantum and interaction effects are entangled in the manner of perturbation in the virial expansion. Among them, the exchange effects in the FK virial expansion are treated as perturbative corrections to the reference system: Maxwell-Boltzmann parti- cles. The degeneracy parameter is a criterion to estimate the importance of the quantum effects, like the exchange effects, and therefore can qualitatively test the validity of that approach. The degeneracy parameter can be defined as =n 3 ; (5.6) and the condition for weak degeneracy is n 3 << 1. The result from Fig. 5.2 displays that the electrons are weakly degenerate throughout the whole region of the Sun. Nevertheless, the electron degeneracy increases significantly within the solar core, which hints the quantum effects might become important for electrons there. For the other heavier elements, like proton and helium, their degeneracy parameter are even 59 Reactant X z geg z+1 gz H 13.54eV 1 He 24.48eV 4 He + 54.17eV 1 Table 5.1: Ionization energy and statistical ratios. smaller due to their bigger mass. Thus, the approach of weak degeneracy expansion should be valid for the solar plasma. 5.1.3 IonizationDegree Figure 5.3: The ionization degrees of different states of H and He are estimated by using the Saha equation. As discussed in the section 3:2:3, the virial expansion as a truncated density expan- sion has intrinsic problem of dealing the process of recombination and ionization. This restricts its applicability to almost fully ionized plasm. In order to confirm if the equa- tion (4.30) is valid in applications to the solar modeling, we need to check the ionization degrees of different elements in the Sun. This can be easily estimated by using the Saha 60 equation [53]. For an ionization-recombination reactionZ!Z + 1, their densities can be described in first approximation by using the mass-action law at thermal equilibrium N z+1 N e N z = g e g z+1 g z V 3 e exp X z k B T : (5.7) N is the particle number of the specie with ion charge,g is its statistical weight, and is the ionization potential to ionize one electron from it. The ionization degrees can be defined as the ratio x z = N z+1 N 0 z ; (5.8) whereN 0 z =N z +N z+1 is the total particle number before the reaction happens. After substituting Eq. (5.8) into Eq. (5.7) and using the condition of neutral chargeN z+1 = N e , the ionization degreex z can be solved as x z = n + p n 2 + 4n 2 ; (5.9) n = 1 N 0 z g e g z+1 g z V 3 e exp X z k B T : (5.10) The estimated ionization degrees ofH andHe in whole region of the Sun are shown in Fig. 5.3 by using the data in the Tab. 5.1. If we choose the criterion ofx z for nearly full ionization as 90%, it is clear that a large part of the solar interior will be satisfied except the region around the center for hydrogen plasma. The Saha equation does not provide correct ionization degree in that situation (high pressure and high temperature) because of its lack of the mechanism of pressure ionization. If we ignore that incorrect feature, according to Fig. 5.4, the valid area is within at least 90% of the solar radius (for H&He). This confirms that the FK virial EOS might be a useful EOS for solar models. 61 5.1.4 Conclusion The simple tests performed above indicate that the applicability of the FK virial EOS for the Sun is mainly limited by the condition of full ionization. Despite of this limitation, the FK virial EOS can still be applied to a large region of the solar interior. Those results suggest that the FK virial EOS could be potentially the most exact EOS for the inner region of the Sun. In addition, from the experimental point of view, the local power of helioseismology will be also able to test the FK virial EOS locally. However, in order to have a complete EOS for the Sun, we will need another suitable EOS for the exterior region of the Sun where the FK virial EOS cannot be applied. This can be done by using some other proper EOS such as OPAL, MHD, or Saha EOS for the outer part of the Sun. Ideally, this joint EOS with two suitable EOSs at two different regions might lead to a more accurate solar model. 5.2 PreliminaryDiscussions Before we start to apply the FK virial EOS to the Sun, we want to discuss the relevant solar data and the parameters we are going to use in our study. Besides, we have to solve several issues about realizing the numerical calculations of the FK virial EOS. This will be done here. 5.2.1 SolarDataandParameters The constituents of the Sun are electrons, hydrogen (about 73% of the Sun’s mass), helium (about 25% of the Sun’s mass), and the other heavy elements (mainly represented by oxygen). We adopt standard solar model provided by Christensen-Daalsgaard [15] as the input for the temperatureT , the total mass density, the hydrogen abundanceX, and the heavy element abundanceZ (represented mainly by oxygen) at different depth 62 Figure 5.4: The temperature-radius map expresses the relation between the temperature and the relative radius throughout the Sun. r R = 1 is at the solar surface. Figure 5.5: The abundances of different elements involved in the solar data. throughout the Sun. The helium abundanceY at each depth can be computed by using the normalization conditionX +Y +Z = 1. The depth is usually denoted as the relative radius r=R where the R is the solar radius. In the Sun, the temperature decreases monotonically when r=R increases as shown in Fig. 5.4. This diagram shows how to convert the temperature in the Sun to its depth, and is useful when we only use the temperature as the horizontal axis but want to know the corresponding position in the Sun. The relations betweenT and the mass abundanceX, Y , Z are indicated in Fig. 63 Electron Protron He nucleus O nucleus m [kg] 9:10953 10 31 1:67263 10 27 6:64466 10 27 2:65626 10 26 e [C] 1:60219 10 19 1:60219 10 19 3:20438 10 19 1:28175 10 18 1 2 1 2 0 0 Physical constant k B ~ C 1 C 2 C E Value [SI units] 1:38065 10 23 1:05457 10 34 15:205 14:733 0:577216 Table 5.2: The physical constants used in the computation. 5.5. In the formula of the FK virial EOS, the number densities =n are used instead of the mass density, but they can be obtained from the total mass density by using the relation = A m ; (5.11) whereA is the mass abundance, andm is the atom’s mass for the element. Besides, the number density of electrons can be easily calculated from the neutral charge condi- tion. For example, if we consider the Sun is constituted by H, He, and O. The total electron number density can be obtained from e = H + 2 He + 8 O : (5.12) The values of the physical constants and some quantities used in all the numerical cal- culations are summarized in Tab. 5.2. The discussions in the section 5:1 already indicate that the FK virial EOS is not appli- cable closer to the solar surface, yet we still can perform the computation throughout the Sun. This can not only help us to find out where is beyond the region of applicability more accurately but also to understand why the problem arises. Of course we should not trust the consequent results near the solar surface, and they will need to be examined more carefully. 64 5.2.2 IssuesofComputingQandEfunctions It turns out that the most complex to apply the FK virial EOS is about to calculate the Q and E functions efficiently. Even though both Q and E have very elegant forms with convergent infinite series, they are not very useful in practice. Both the infinite sums need to be either truncated or approximated in order to obtain the accurate values within reasonable time. In order to find out a better strategy to calculate Q and E, first we need to know the range of the Born parameterx for our study. Fig. 5.6 represents the values ofx for different charged particles throughout the Sun. Figure 5.6: The ranges ofx for different particles in the Sun. For the pairs of different species of charge particles, the values ofx can be easily inferred from Fig. 5.6 with their charge and mass. Overall, their magnitude are all within the curves ofee andO 8+ O 8+ , but might have extra negative signs depend on their charge polarities. From that result, we know that we have to calculate theQ and E functions for the values ofx up to hundreds in order to at least involveHe 2+ . Second, we also need to understand the behaviors ofQ andE within that relevant region ofx . The results are presented in Fig. 5.7 and Fig. 5.8. It is clear that theE function approaches zero quickly asx << 0 (the argumentx contained in the E can only be zero or negative). On the other hand, the Q function 65 Figure 5.7: The behaviors of the Q and E functions for smallerx values. Figure 5.8: The behaviors of the Q function for largeerjxj values. grows dramatically asjxj increases. Nevertheless, theQ has a useful asymptotic form for largejxj Q(x ) = 2 p H(x )[ 1 X s=1 s 2 [exp( x 2 4s 2 ) 1 x 2 4s 2 ] x 2 8 ] x 3 6 [lnjxj + 2C E + ln 3 11 6 ] x 12 1 60x +O( 1 x 3 ); (5.13) where H is the Heaviside function. Within this formula, the dominate part is the so- called PLPF 1 X s=1 s 2 [exp( x 2 4s 2 ) 1 x 2 4s 2 ]; (5.14) 66 which is the same as Eq. (3.55). Thus, we can further estimate the asymptotic behavior ofQ by using the identity 1 X s=1 s 2 [exp( x 2 4s 2 ) 1 x 2 4s 2 ] = 1 X n=2 (2n 2) n! ( x 2 4 ) n (5.15) to find the lower and upper bonds of the PLPF. Because of the property of the zeta function: 1<(n)< 2 forn 2, we can obtain the inequalities exp( x 2 4 ) 1 x 2 4 < 1 X n=2 (2n 2) n! ( x 2 4 ) n < 2 exp( x 2 4 ) 1 x 2 4 : (5.16) This result shows the Q function grows asymptotically like exp( x 2 4 ) for x >> 1. On other hand,Q only grows as x 3 6 lnjxj forx<<1 which is much slower than in the positive region ofx. The numerical calculations ofQ andE can be performed in different ways: Truncate the infinite series in the equation (4.33) at some order with small enough producing error. Use the Pad´ e approximant to approximateQ. ApproximateQ with a simple asymptotic infinite series that converges faster and can choose how accurate as we want. The first one is the most straightforward but cannot be efficient since the original series converges slowly asx increases. The second method is widely adopted and its advantage is to have an analytical form can substitute the originalQ function. The Pad´ e approx- imant uses a fractional polynomial function to reasonably simulate the Q function by smoothly connecting two asymptotic functions for opposite extremex values. However, the errors produced by this method might not be small enough for us. For example, a 67 constructed Pad´ e approximant from [45] is compared with the originalQ, and the result is plotted in Fig. 5.9. Another example is to compare with the Pad´ e approximant from [58], and the comparison with the exactQ is shown in Fig. 5.10. In conclusion, even the Pad´ e approximant can offer a very easy way to estimateQ for either some part of the range or the whole region, its accuracy is not enough for our demand. Figure 5.9: Difference between the Pad´ e approximant in [45] and the exact function. Figure 5.10: Compare the Pad´ e approximant in [58], the asymptotic functions, and the exact function. The third method is based on the behavior of the Riemann zeta function as described in Fig. 5.11. To begin with, theQ function can be modified as 68 Figure 5.11: The behavior of the zeta function(x). Q(x ) = x 6 p 8 x 2 1 6 ( 1 2 C E + ln 3 1 2 )x 3 + [ 1 X n=0 p (1 + n 2 ) ( x 2 ) n 3 X n=0 p (1 + n 2 ) ( x 2 ) n ] + 1 X n=4 p ((n 2) 1) (1 + n 2 ) ( x 2 ) n (5.17) Q(x ) = x 6 p 8 x 2 1 6 ( 1 2 C E + ln 3 1 2 )x 3 + [[ p exp( x 2 4 ) p 4x exp( x 2 4 )(1 +erfc( p x 2 2 )) 2 p x 2 ] 3 X n=0 p (1 + n 2 ) ( x 2 ) n ]+ 1 X n=4 p ((n 2) 1) (1 + n 2 ) ( x 2 ) n (5.18) with the complementary error function erfc(x), which can be computed quickly. In addition, this method also introduces much less error to truncate the infinite series 1 X n=4 p ((n 2) 1) (1 + n 2 ) ( x 2 ) n (5.19) at the same order. For this reason, we uses this method to calculateQ, and have verified that the accuracy can be controlled to be good enough. Another important issue about theQ function is some authors claimed that the lin- ear term x 6 in the Q function should not exist [33, 35]. However, the disputed term 69 does not only appear in the analysis of Alastuey and Perez [9], but also in Brown&Yaffe [12]. Recently, Alastuey responded that question and claimed that the hight temperature expansion of the Q function does contain the linear term x 6 as the first leading term in their derivation by using the FK path integral [4]. While no unambiguous refuta- tion exists, this linear term still remains controversial. If Kraeft and Kremp, Schlanges & Kraeft are right, The equation (4.33) would have to be modified. The comparison betweenQ with the linear term and without linear term is plotted in Fig. 5.12. The dif- ference is only manifest around the origin since Q grows rapidly withx. The singularity appeared in the right graph is artificial and due to the zero value of Q around x = 2. Thus, the linear term is only important in Q for small values of x. In practical, there are two Q functions appear in the FK virial EOS (Eq. (4.31)). The first one is in the terms of the order 2 and the second is in the order 5=2 . Nevertheless, that linear term does not have any effect in the first place for the plasma with neutral charge. This can be easily explained by simplifying the expression of the term contains the linear therm ofQ in the order 2 as p 2 X ; 3 ( x 6 ) X ; e e m = 2( X e a m )( X e ): (5.20) As long as the neutral charge condition is satisfied (means P e = 0), this term has no contributions. On the other hand, the second term contains similar linear term in the orderrho 5=2 is not zero in general because 3 2 p 2 X ; e e D 3 ( x 6 ) X ; e 2 e 2 m = 2( X e 2 m )( X e 2 ) (5.21) will not be zero in any case. Therefore, even the linea term ofQ might play some impor- tant effects, but would be in the order 5=2 . We mention this controversy in the hope that 70 the calculations are being checked and perhaps re-done. Should the controversy not be solved by theoretical arguments, then perhaps our planned solar application can shed light on the issue. This point will be discussed with more results later. Figure 5.12: The left graph shows the behaviors ofQ with the linear term and without the linear term. The right graph expresses the relative difference between them. 5.3 ResultsandDiscussions In order to be able to understand the results from the complex solar plasma clearly, we will simplify the model of the Sun to make the analysis simpler at beginning. The simplest case is to assume the Sun is composed of electrons and homogeneous positive charge background, and the whole system stays charge neutral. However, we jump over this naive one-component system and begin with a more real situation that the solar plasma only contains the electrons and the protons: the hydrogen plasma. This should play as a good first approximation since the 92% number of particles in the Sun are hydrogen. Later, we will consider the helium nuclei (about 7:8% by particle) and even to further include the oxygen nuclei (less than 0:2% by particle) to make our system closer to the real Sun. Besides, in order to be consistent with Alastuey’s and Perez’s theory, we use the Q function with the linear term in our computation. However, In 71 order to understand wether the linear term is negligible or not for our applications, we will also discuss its effects to some relevant thermodynamic properties of the Sun. 5.3.1 PhysicsofPressure For the first application of the FK virial EOS, we can straightforward calculate the pres- sure from the density, the abundances, and the temperature at every depth of the Sun. To start with, we can rearrange Eq. (4.31) as P P id 1 = 1 P f 3 D 24 + 6 (ln 2 1) X ; 3 e 3 e 3 p 2 X ; 3 Q(x ) 3 3 X ; e 3 e 3 ln ( D ) + p 2 X (1) 2+1 (2 + 1) 3 2 E(x ) 3 2 p 2 X ; e e D 3 Q(x ) 2 4 X ; e 4 e 4 D ln ( D ) + 3 2 p 2 X (1) 2+1 (2 + 1) 3 2 e 2 D E(x ) + 1 16 X 2 ~ 2 e 2 m 3 D +( 1 3 3 4 ln 2 + 1 2 ln 3) X ; 4 e 4 e 4 D +C 1 X ;; 5 e 3 e 4 e 3 1 D +C 2 X ;; ; 6 e 3 e 3 e 3 e 3 3 D +O( 3 ln)g; (5.22) whereP id = 1 P is the corresponding ideal gas pressure calculated from the ideal gas law with the same conditions, i.e. the same density, abundances, and temperature. Thus, the total pressure or the partial pressure for each term on the right hand side of Eq. (5.22) are normalized with the ideal gas pressure and become dimensionless. The ratio 1 represents that the pressure is the same as its ideal gas pressure. In this way, we can focus on the details about the effects from the non-ideal contributions and ignore the uninteresting ideal gas pressure, which covers very broad range of value. 72 H system Figure 5.13: Compare the normalized total pressure and some normalized partial pres- sure for theH system throughout the Sun. P id andP DH express the ideal gas pressure and Debye-H¨ uckle pressure respectively. The normalized total pressure and the classical pressureP id 1 3 D 24 throughout the Sun are plotted in Fig. 5.13. One main signature of the total pressure is that it decreases as the temperature decreases (or closer to the solar surface). However, the total pressure will finally decrease to become negative at about 1:5 10 4 K, which cannot be correct. This feature of the virial expansion of pressure has been already pointed out in the section 3:2:3. Here, it is clearly observed that this subtle defect of the the virial EOS can be demonstrated by the exact, truncated density expansion of the pressure for a quantum plasma. In this case, the incorrect result is due to the formation of hydrogen atoms. When the temperature is close to the ionization energy of the hydrogen atom, the electrons and protons will bound together to form hydrogen atoms significantly. The deeper reason can be clarified by looking at the behaviors of its partial pressure later. Another information from Fig. 5.13 is that the idea gas pressure can already approximate very well within the Sun. To roughly estimate its thermodynamic quantities, the simple ideal gas law is already good enough, and has been also extensively applied in astronomy under high temperature and low density conditions. In the Sun, the discrepancy between 73 the idea gas pressure and the FK virial pressure can be within 1% as the temperature is above 1 10 6 K (or within 0:85 solar radius). This simple analysis also shows the non- ideal effects to the total pressure are roughly about few percent of its ideal gas pressure for the most region of the solar interior. Secondly, by including the first order correction which is from the mean field of the interaction (also called Debye-H¨ uckle term here) to the ideal gas pressure can improve the accuracy eminently. The curve ofP id +P DH is already very close to the ”exact” total pressure between 6 10 4 K and 2 10 6 K. When the temperature is below about 6 10 4 K, the curve ofP total starts to deviate from the curve ofP id +P DH manifestly. Put the other way around, the beginning of the significant discrepancy at lower temperature indicates the temperature where the great quantity of recombination starts to occur. Of course, we should not also trust thatP id +P DH can be still a good approximation below that temperature since according to the simple analysis from Saha equation (see Fig. 5.3), many hydrogen atoms have already been generated. In contrast to the curve of P total , the curve of P id +P DH starts to curve upward and becomes closer to 1 when the temperature is even lower. In this situation, the Debye- H¨ uckle screening interaction is no longer a correct mean field for theHH interaction, but the Van der Waals force is a more accurate effective interaction betweenH atoms instead. This reminds us that if we want to develop an extended exact EOS to be also suitable for lower temperature, we need to be able to recover the Van der Waals force under the low temperature limit in the theory. This topic will be discussed with more details in the next chapter. Another manifest feature of theP total curve in Fig. 5.13 is that it exceeds the ideal pressure above the temperature is about 8 10 6 K (or within about 0:23 solar radius). This region happens to be also about the solar core where the nuclear fusion occurs. In addition, the curve ofP id +P DH decrease and begins to deviate from the curve ofP total again. It always stays belowP id even at solar center. As we have already discussed, scienceP HD describes the major classical interaction effect, 74 this result shows the importance of quantum effects in the deep region of the Sun even though the correction is only within about 1%. Besides, this is also consistent with the estimate of electron degeneracy in Fig.5.2. Figure 5.14: The behaviors of different normalized partial pressure defined in Eq. (5.22) of theH system. The curve number represents the order of the term calculated on the right hand side of Eq. (5.22). In order to have detailed understanding about the behavior of the total pressure, we can study the contributions from different terms in Eq. (5.22) toP total . This is one of the advantages of the viral expansion formula. It helps us to understand the importance of each effects throughout the Sun. All those partial pressure except the ideal pressure are plotted together in Fig. 5.14. Because their order of magnitude covers very wide range and some are negative, they are plotted for positive and negative values of pressure separately in two log-log graphes. The curves are labeled with the number which is the same as the order of the term on the right had side of Eq. (5.22). Thus, the curve 1 75 represents the first normalized partial pressure 1 P ( 3 D 24 ) and so on and so forth. The same rule also applies for the other similar graphs hereafter. For the discussions, first of all, the first term which is the Debye-H¨ uckle term must be negative and can be confirmed with the curve 1. The physical reason is because the screening Coulomb interaction produces the attractive forces in the hydrogen plasma, and reduces the total pressure from the ideal gas pressure. Even though its relative contribution is just a few percent, this can already improve the naive ideal gas law sig- nificantly as already indicated in Fig. 5.13. Second, the reason why the total pressure drops dramatically and becomes negative eventually is because of the third term (the curve 3). It contributes negative partial pressure unless Q(x) is negative at very high temperature (when 0 < x . 2), and the Q function becomes great number at lower temperature whenx 1. In order to compare the contributions of positive and negative partial pressure toP total conveniently, their absolute values are plotted together in Fig. 5.15, and the temperature is divided into three different regions. It becomes clear that in regionIII (above 8 10 6 K), the partial pressure curve 5 is dominant and positive. The fifth term in Eq. (5.22) becomes more relevant than the Debye-H¨ uckle term (the first term) and controls the trend of the total pressure. ThusP total turns to be a little bit above the ideal pressure. The deeper reason is because the fifth term mainly contains theE function which describes the exchange effect of identical particles and is purely a quantum effect. This is consistent with what we have argued previously:the degeneracy parameter for electrons is close to 1 near the solar core. When the temperature decreases and reaches regionII, the curve 1 becomes prevailing and its contribution is negative. The net effect is the total pressure deviates and becomes lower than the ideal pressure. Therefore the total pressure keeps decreasing as the temperature decreases in this region. This explains why P id +P DH can approximate P total very well in this region and the other corrections are not so important. In this classical mean filed region, the quantum 76 effects and interaction fluctuation become unimportant. As the temperature is below about 6 10 4 K and within regionIII, the curve 3 overcomes curve 1 and the others, and its contribution is negative. Since now the curve 3 controls the trend of the total pressure,P total turns out to deviate fromP id +P DH significantly as described before. In this region, the behavior of pressure results from interplay between complicated interac- tion and the other quantum effects except the exchange effect. Meanwhile, even though the contribution from curve 6 is positive and also increases dramatically as tempera- ture decreases, because curve 6 increases slower than curve 3, the total pressure is still dominated by curve 3 and decreases as temperature decreases. From the formulism, by comparing the sixth term with the third term of partial pressure, the ratioj P 6 P 3 j is about the ratio between the Landau length and the Debye length (e e ). This turns out to be about 3=2 where is the coupling parameter. The small coupling parameter for electron and proton pair (see Fig. 5.1) guarantees the curve 3 overwhelms the curve 6 at lower temperature. At the end, the total pressure becomes negative and decreases dramatically due to the behavior of theQ function. This explains the reason why the recombination of the H plasma causes P total to decrease and becomes negative finally from the first principle. The Q function contains the Hamiltonian of two charge particles. For two particles with opposite charge, the PLPF for H atom appears inQ. As the temperature is about its ionization energy,the PLPF increases dramatically and causes the value ofQ to increase significantly as well. Besides, the explicit decrease of the total pressure happens at the temperature roughly the same as the ionization energy ofH atom. This turns out to be true in general. The temperature of prominent change of the total pressure is controlled by the highest ionization energy of the elements in the system. This offers more quantitative way to discuss how the recombination between the electron and the proton can drag the total pressure lower than its ideal pressure by considering the ratioj P 3 P id j at lower temperature. 77 II I III Figure 5.15: Absolute values of the different normalized partial pressure of theH sys- tem. Because Q increases much at low temperature (when x 1), all the other contribu- tions to the ideal gas pressure become irrelevant; Meanwhile, we knowQ exp( x 2 4 ) as x>> 1. Under this condition, this ratio can be approximated as j P 3 P id jn ep 3 ep Q(x ep )n ep 3 ep exp( x 2 ep 4 ) = (n ep 3 ep ) exp(E H ); (5.23) whereE H is the ionization energy of the hydrogen atom (about 13:6eV 1:5810 5 K). From the previous discussion about the degenerate parameter,n 3 is about 1% around the temperature we are interested here. The condition for the ratio to be bigger than 1 is exp(E H ) & 10 2 . Thus we can conclude thatj P 3 P id j becomes greater than 1 for the temperature T . 3:5 10 4 . This confirms why the deep drop of the total pressure appears to happen roughly at the temperature of the ionization energy. Because the temperature is contained in the exponential function, the ratioj P 3 P id j changes significantly in the range within an order of magnitude of the ionization energy. 78 H&Hesystem Figure 5.16: Compare the normalized total pressure and some normalized partial pres- sure for theH&He system throughout the Sun. In Fig. 5.16, the total pressure for H system from the previous discussion is also added (theP H total curve) in order to see the distinct effects by includingHe. TheP H total curve has been normalized by the ideal gas pressure of the H system but not H&He system. Its value should not be compared with the other curves directly because the H&He system has higher total density than theH system, and thus has higher idea gas pressure. Nevertheless, we can still compare their relative deviations from their won ideal gas pressure qualitatively. Overall, except the strange peak around 4 10 4 K, the the total pressure of H&He system has similar behavior as P H total . As can be seen on the right diagram,P total also deviates fromP id +P DH but begins at higher temperature about 1 10 5 K. This shows the valid region of temperature for applying the mean-field approximation becomes smaller. This is due to the higher ionization energy of He + compare to the ionization energy ofH. Even though this reason shrinks the applicable region for the truncated virial equations, we need to stress that the valid region still cov- ers over 90% of the solar interior. Above about 8 10 6 K, it is similar to theH system. P total is higher than bothP id andP id +P DH . In addition, it is also above theP H total curve 79 in this region. According to the previous discussion, this indicates the quantum correc- tions which are mainly the exchange effects become greater asHe is added. The reason for this phenomenon will become apparent when we look at the behaviors of partial pressure later. On the other hand,P total becomes lower thanP H total below the same tem- perature. This means the other non-ideal effects for the pressure also become stronger in this region by includingHe. This can be quickly understood from theP id +P DH curve. Due to stronger collective interaction of the Debye-H¨ uckle term and it contributes neg- ative partial pressure, theP total curve becomes lower. This interaction effect dominates the total pressure in this region until the temperature decreases to about 110 5 K. From the formulism of the Debye-H¨ uckle partial pressure, we know its relative contribution is proportional to the square root of density and charge. Since theH&He system has more particles andHe carries more charge, this effect becomes stronger as well. Besides, a more subtle problem arises whileHe particles are added. Even it cannot be shown on the graphes, we can still know it form the fundamental theory. The up-to- date FK virial EOS we have on hand is only up to the order 5=2 . Even though it already contains various quantum and interaction effects, the three body effect is still absent. It is only contained at least in third power of density (or three loops in filed theory terminology). The lack of the three body effect makes the calculation do not involve any states of a He atom. This neglection could cause significant error when the He abundance is crucial. Similar issue will also appear as we want to consider other heavier elements. Nevertheless, the energy corresponding to those excited states are lower than the second ionization energy of He. As long as we restrict our computation above the temperature higher than that, the amount ofHe atom formations can be neglected. Therefore the FK virial EOS up to the order 5 2 should be still a good approximation as long as the three body effects is not relevant at the temperature of calculation. 80 Figure 5.17: The behaviors of different normalized partial pressure defined in Eq. (5.22) of theH&He system. The situation becomes more complicated as includingHe into the system. There are six possible particle pairs instead of three for theH system. The competition between different pairs cause the subtle differences and makes the total pressure behave little dissimilar. By looking at the behaviors of partial pressure in Fig. 5.17, in the region of higher temperature (T >> 8 10 6 K), It is similar to Fig. 5.14 forH system. The total pressure is also dominated by curve 5 (contains the exchange integral inE) but the curve 5 does not bend downward as in Fig. 5.14. Thus the total pressure in Fig. 5.16 follows similar trend in this region. However this phenomenon does not originate mainly from the He nuclei. The adding He provides extra electrons to the system. This creates more electron-electron pairs and thus results in more prominent contribution from the exchange effect. On the other hand, as the temperature decreases, the curve 5 goes 81 down and cannot compensate the negative partial pressure of curve 1. Similar to the H system, the behavior of the total pressure is mainly controlled by the Debye-H¨ uckle term. However curve 3 decreases as negative pressure faster than in theH system.P total begins to deviate fromP id +P DH and decreases at higher temperature (about 2 10 5 K) than in theH system. Unlike theH system, a strange wide peak ofP total appears below the temperature about 110 5 K. This turns out to be caused by the competition between curve 3 and curve 6. Curve 6 contributes large positive pressure. As we discussed before, the ratioj P 6 P 3 j is about 3=2 , and the coupling parameter foreHe 2+ pair is different fromeH + pair. It increases to be about 1 aroundT t 4 10 4 K (see Fig. 5.1). WhenP 6 wins, the total pressure will increase. Until at lower temperature (about 2 10 4 K), the curve 3 starts to dominate again and makes the total pressure decrease rapidly, and finally becomes negative pressure. Thus, this different phenomenon is due to the stronger coupling effect for theHe nuclei. Besides, theQ function only depends on the Born parameterx = p 2 l . For theHe 2+ e pair, thex value is four times bigger than the H + e pair at the same temperature. Thus He makes the Q function increase much faster as the temperature decreases. This can also explain why the significant deviation happens at higher temperature. H&He&O system Finally, we can further include theO nuclei (about 0:1% of total particle number) into the calculations in order to make our calculation closer to the real solar model. First of all, we can expect that the recombination of the electron and theO nuclei will cause the erroneous pressure at even higher temperature due to the higher ionization energy of theO 7+ ion. This makes the FK virial EOS only valid in even smaller region of the solar interior. From the result of total pressure in the Fig. 5.18, the valid region of the temperature shrinks to above about 1:6 10 6 K; nevertheless, this still covers the area 82 Figure 5.18: Compare the normalized total pressure and some normalized partial pres- sure for theH&He&O system throughout the Sun. within 0:8 solar radius and thus is still useful for solar applications. In Fig. 5.18, the total pressure of theH system and theH&He system are also plotted in order to see the difference between different constituents. Overall the systems including The heavier mass and more charge of theO nucleus makes its Born parameter even bigger. Although the particle density of O is many order of magnitude less than the particle densities ofH andHe, because theQ function turns out to be an asymptotical exponential function of the square of the Born parameter, the contributions from the O nuclei increases fast as the temperature is low enough. The complex competition between different elements at lower temperature induces the unpredicted behavior of the total pressure as in the left graph of Fig. 5.18. Unlike in the other two systems, the total pressure turns upward as the temperature is below about 7 10 5 K. Other than that, its behavior is not that different from the total pressure of H&He system, except it has stronger screening effect, and thus the relative pressure is overall lower. In Fig. 5.19, there is one feature different from the previous systems. The third curve which represents the normalized partial pressure P 3 P id becomes totally negative and more significant within solar core. However, the dominate effect here is still controlled by the exchange integral term ( P 5 P id ), and therefore the net behavior turns to be similar to the 83 Figure 5.19: The behaviors of different normalized partial pressure defined in Eq. (5.22) of theH&He&O system. H&He system. Finally, by comparing Fig. 5.19 with Fig. 5.14 and Fig. 5.17, we can also understand the reason why the total pressure of theH&He&O system increase to large positive pressure, instead of negative pressure like the other two. The sixth partial pressure (it is also the second term containing theQ function) turns to grow faster and overcome the third partial pressure at lower temperature. Thus, unlike the other two systems, its total pressure is dominated by the sixth partial pressure and become large positive value. At the middle, the competition betweenP 3 andP 6 causes the big dip of the total pressure before it turns to large value. In conclusion, the simple analysis of the FK virial pressure can offers us the more accurate criterions about the valid region of the Sun to apply the FK virial EOS although the pressure is not a directly measurable quantity from the helioseismology experiments. 84 The FK virial EOS provides an exact and analytical expression that shows the non-ideal contributions to the pressure beyond the Debye-H¨ uckle approximation clearly. By per- forming the numerical comparisons with the solar data, we can already obtain useful information and understand the details about how the non-ideal effects (both interac- tion and quantum effects) influence the pressure in the solar model. From this point of view, we understand the FK virial EOS plays a useful EOS. The complicated behavior of the total pressure results form the complex competition between various non-ideal effects. Inside the solar core, the exchange effect becomes more important. This is mainly contributed form the electrons and is described in theE function. In more detail, the exchange integral contains entangled exchange and interaction effect. If the inter- action can be treated pertubatively, the leading contribution to the E function would be the pure exchange effect and the other higher order contributions should consists of exchange interaction effect. On the other hand, in the outer region, the correction due to the phenomenon of recombination becomes relevant. This is mainly described by theQ function for electron-ion pairs. Higher ionization energy results in higher temperature of the drop of total pressure. Nevertheless, the FK virial EOS should be more accurate and is still useful, as long as we confine its applications to the valid region of the Sun. Last but not least, by aid of this EOS, we can analytically calculate all the other thermody- namic quantities which are more important to understand the thermodynamic properties of the Sun. This is one of the advantages that the FK virial EOS surpasses the exten- sively used OPAL EOS. In principle, all the thermodynamic quantities can be derived from the FK virial EOS as analytically closed forms. Those derivations are exact and will not introduce any extra errors. Besides, The resulting formalism can be computed directly to obtain its numerical values without using complicated numerical technics. 85 5.3.2 PhysicsofThermodynamicQuantities After simple and direct calculations of the pressure, we can start to compute the other thermodynamic quantities. They are useful and important because they can be used to probe the thermodynamic properties of the Sun and some can also be measured by the experiments of helioseismology. From the FK virial EOS, we can already derive some useful thermodynamical quantities by using the thermodynamic relations. To obtain all the other quantities, we can begin with the Helmholtz free energy, and all the thermody- namic quantities can be obtained with its derivatives. The Helmholtz free energy density f can be derived (up to the same order 5=2 ) as f = F V = X ln (2 2 ) 3=2 1 3 D 12 + 6 ln 2 X ; 3 e 3 e 3 p 2 X ; 3 Q(x ) 3 3 X ; e 3 e 3 ln ( D ) + p 2 X (1) 2+1 (2 + 1) 3 2 E(x ) p 2 X ; e e D 3 Q(x ) 3 4 X ; e 4 e 4 D ln ( D ) + p 2 X (1) 2+1 (2 + 1) 3 2 e 2 D E(x ) + 1 24 X 2 ~ 2 e 2 m 3 D +( 1 3 1 2 ln 2 + 1 3 ln 3) X ; 4 e 4 e 4 D + 2 3 C 1 X ;; 5 e 3 e 4 e 3 1 D + 2 3 C 2 X ;; ; 6 e 3 e 3 e 3 e 3 3 D : (5.24) The pressure is the first derivative of the free energy and satisfies the relation P = X n @(f) @n f : (5.25) Many other useful thermodynamic quantities contain the the second order derivatives of the free energy. Because there are only two independent thermodynamic variables, 86 we can choose any three independent thermodynamic second derivatives as the basis. A convenient choice would be the specific heat at constant volumec v , and two power-law exponents T and . They can be computed by using the relations derived from their definitions: c v = T ( @ 2 f @T 2 ) ; (5.26) T = @ lnP @ lnT = T P @P @T ; (5.27) = @ lnP @ ln T = P @P @ T : (5.28) is the particle number density, thus the specific heatc v is the heat capacity per particle, not per mass here. Base on this, all the other thermodynamic second derivatives can be calculated as the combinations with this basis and some other lower order deriva- tives. For example, the dimensionless adiabatic exponents measure the thermodynamic response of a system to adiabatic changes. They are used extensively in stellar modeling and are defined by the following expressions: 1 = @ lnP @ ln ad ; (5.29) 2 2 1 = @ lnP @ lnT ; ad = 1 r ad (5.30) 3 1 = @ lnT @ ln ad ; (5.31) wherer ad is the adiabatic gradient and can be expressed as r ad = P Tc p T : (5.32) 87 Therefore, these three adiabatic exponents can be expressed as 1 = T P T T c v + ; (5.33) 2 = 1 1r ad ; (5.34) 3 = 1 + ( 2 1) 1 2 : (5.35) Furthermore, the sound speed can also be derived by using the adiabatic approximation and is related to 1 as v s = s 1 P : (5.36) Besides, the specific heat at constant pressure can be computed from the relation c P =c v + P T 2 T : (5.37) When some thermodynamic derivatives are computed by using the FK virial EOS, the Figure 5.20: The power exponents and T are calculated by using the FK-virial EOS for theH system, and both contain the singularities. singularities might appear at some temperature. For examples, in Fig. 5.20, both T 88 and of the H system have similar singularities at about 1:4 10 4 K. Those non- physical behaviors can be understood with their mathematical expressions. Because their denominators contain the pressure, T and will become infinity as the pressure is zero. Owing to the same reason, some the other thermodynamic second derivatives (or even higher order derivatives) may also contain similar singularities. Those unreal defects originate from the same fail of truncated virial expansion: the pressure decreases rapidly and becomes negative beyond the valid region of the FK virial EOS. All those abnormal behaviors happen when the recombination become significant. However, once we avoid to to approach or cross the boundary of applicable region and stay at higher temperature, we should not concern about this problem anymore. From the previous dis- cussions, we confirmed that the FK virial EOS is only applicable above some tempera- ture depends on the ionization energy of the elements in the system. In order to focus on the phenomena of real physics, we will restrict the computation of the thermodynamic quantities within their valid ranges in this section. They are again sorted according to their constituents and discussed separately. Although we only discuss some important thermodynamic second derivatives, all the other related quantities can be easily deduced by using the simple thermodynamic relations. H system First, we begin with the simple situation by only focusing theH atoms. The behaviors of , T ,c v , 1 , and sound speedv s are plotted separately in Fig.5.21 to Fig.5.25. For a monoatomic ideal gas, we know = T = 1,c v = 1:5, and 1 = 2 = 3 = 5=3. The non-ideal effects can be also clearly observed as the deviations from those ideal gas values for each thermodynamic quantities. At higher temperature (also higher density for the Sun), we might expect the system would behave more like ideal gas; nevertheless, as we already know that the interaction coupling is not very weak and the quantum 89 effects are also relevant in the region of solar core. Similar signatures should also appear in the behaviors of some thermodynamic quantities. Figure 5.21: The behavior of the the power exponent of theH system in the solar interior. Figure 5.22: The behavior of the power exponent T of theH system in the solar inte- rior. Quick survey of those figures shows the power exponents and T do approach the value 1 as the temperature increases, but in different directions. Moreover, both quan- tities finally cross 1 around the temperature about 8 10 6 K. This behavior is similar to the total pressure in Fig. 5.13 in which the total pressure approaches and cross the 90 ideal pressure around 8 10 6 K. However, the crossover temperatures are not exactly the same because any thermodynamic quantity is equal to its ideal value does not imply that it behaves equivalently as the ideal gas. From the discussion in the previous sec- tion, we understand that the competition and interplay within the contributions from different effects result in the complicated behaviors of the thermodynamic quantities: the screened classical interaction and bounding effect dominate at lower temperature end, but the exchange interaction effect starts to overwhelm at the other end of the tem- perature. Thus, it would not be surprising that both the specific heat c v (Fig. 5.23) Figure 5.23: The behavior of the specific heatc v of theH system in the solar interior. and the adiabatic exponent 1 (Fig. 5.24) approaches but do not cross their ideal values, at least under the conditions of temperature and density in the Sun. The specific heat c v increases as the temperature decreases because moreH atoms formed at lower tem- perature and make more contribution to the specific heat and increases rapidly at lower temperature end in Fig. 5.23. However, the FK virial EOS cannot be applied to the low temperature region where the plasma becomes partially ionized and thus cannot be used to obtained the correct values ofc v below that temperature. In addition, similar behavior also appears in the figure of 1 . Last but not least, bothc v and 1 turn out to be not as sensitive as T and to detect the details about changes caused by non-ideal effects. 91 Figure 5.24: The behavior of the adiabatic exponent 1 of the H system in the solar interior. Figure 5.25: The behavior of the sound speedv s of theH system in the solar interior. H&HesystemandH&He&O system By adding helium or oxygen into the calculation, the valid domain becomes more restric- tive. The thermodynamic quantities of different systems with different constituents are plotted together in the valid domain of theH&He&O system in order to compare all different systems at once. The same thermodynamic quantity of different systems are graphed in Fig. 5.26 to Fig. 5.30 respectively for different thermodynamic quantity. For the convenience, the previous results of the H system are also added in order to 92 compare the differences caused by adding different particles. This causes two different effects: the particle number density and the composition both change. In principle, the ideal values of those thermodynamic quantities do not depend on the particle number density except the pressure; however, the interaction and quantum effects do depend on that. Overall, we can observe that the major effect by addingHe orHe&O is to increase the particle number of electrons and nuclei. Therefore, the behaviors of theH&He and H&He&O systems are similar because addingO does not increase the particle number significantly as addingHe. This increase does not only enhance the screened interac- tion behavior but also the effect of the quantum exchange interaction. As a result, their competition enhances the slops of each thermodynamic quantities. Figure 5.26: Compare the behaviors of the the power exponent for the systems with different constituents. 5.3.3 Physicsoftheheavyelements In this section, we want to simulate the influence of changing the composition of the elements heavier thanHe which is about 2% of the solar mass. Our working assump- tion was that all the elements heavier than H and He can be represented by O. This hypothesis must be of course confirmed more quantitatively. Besides, this test could also help us to quantitatively understand how the detail about that 2% mass matters. 93 Figure 5.27: Compare the behaviors of the the power exponent T for the systems with different constituents. Figure 5.28: Compare the behaviors of the the power exponentc v for the systems with different constituents. Figure 5.29: Compare the behaviors of the the adiabatic exponent 1 for the systems with different constituents. 94 Figure 5.30: Compare the behaviors of the the sound speed v s for the systems with different constituents. First, we suppose all the heavier elements are represented byHe. This is equivalent to replace theO atoms byHe but keep the same mass density used in the previous com- putation. By comparing this sample with the originalH&He&O system in Fig. 5.31 for different thermodynamic quantities, it might indicate how the heavier elementO in the system influence its thermodynamic quantities (within the formulism of FK virial expansion). As a result, the induced differences for those thermodynamic quantities are Figure 5.31: TheO atoms in the system are substituted by theHe atoms. The left graph represents the absolute values of relative discrepancies of the thermodynamic quantities between the substituted system and the original system. The right graph expresses the relative discrepancy of the squared sound speed between the substituted system and the original system. 95 about 10 3 s 10 4 for most region of solar interior. In addition, the induced discrep- ancy of the squared sound speed is also graphed in Fig. 5.31 and has similar value. This result is important and will be discussed more in the next chapter. If this induced difference is smaller than the precision of helioseismology (see Fig. 4.1), this might indicate that how the accurate component of the heavier elements is not relevant in prac- tice, at least cannot be resolved by the experiments now. This indicates that although theO andHe are very different in mass and charge, they have fine distinction for the thermodynamic quantities of the Sun. Among them, the adiabatic exponent 1 turns out to be the most insensitive. On the other hand, the pressure is the most sensitive. This can be understood and estimated by comparing the charge-mass ratios of theHe andO atoms. Even thoughO is about 16 times heavier thanHe, it only contributes 8 electrons. This is the reason to introduce significant difference between their amount of particles and thus causes different pressure. For the second case, theO atoms are substituted by C. This example could show how the detail about the component of heavier elements changes the thermodynamic quantities since the second abundant heavier element isC. Because their mass and charge are not that different, we could expect the differences between their thermodynamic quantities should be smaller than in the first case. Indeed, Fig. 5.32 shows less relative discrepancies between each thermodynamic quantities. Furthermore, the difference of the squared sound speed is only about 2 10 4 . To con- clude, it is crucial to understand the accurate composition of heavier elements in order to match the precise experiments with helioseismology which can measure some physical quantities with the precision as high as 10 5 . 5.3.4 EffectsoftheLinearTermintheQFunction To see the effects of the linear term ofQ for our applications, the thermodynamic quan- tities of theH system are calculated by using theQ functions with the linear term and 96 Figure 5.32: The O atoms in the system are substituted by C atoms. The left graph represents the absolute values of relative discrepancies of the thermodynamic quantities between the substituted system and the original system. The right graph expresses the relative discrepancy of the squared sound speed between the substituted system and the original system. without the linear term separately. As has been discussed previously, the FK virial EOS contains two linear terms and the first one does not have any contributions. This state- ment is also true for the formula of the free energy in Eq. (5.24) since they have similar structure. Therefore, the first linear term should not contribute to any thermodynamic quantities, but the second one in the order 5=2 does. In order to tell the difference, the results with and without the linear term inQ are compared. As shown in in Fig. 5.33, the relative discrepancies caused by the linear term of Q for some important thermo- dynamic quantities are quite small in the solar interior (includes H,He, andO). Both quantities have the relative uncertainty far below the precision of the helioseismology today (up to 10 5 ). This shows the linear term in theQ function is not very relevant in our study, at least in the solar interior. To conclude, we should be free from the debate of existence of the linear term x 6 in theQ function for our practical applications. 97 Figure 5.33: The influences of the linear term for the thermodynamic quantities of the H&He&O system. 5.3.5 Conclusion From the calculation of the pressure, we can observe that the FK virial EOS fails at lower temperature when the recombination of theH atom takes place greatly. By ana- lyzing the behaviors of partial pressure within the FK virial EOS, we can understand the fundamental reason to cause it. This does not only make the pressure drop dramat- ically and becomes negative, but also make nonphysical results of the thermodynamic quantities. Furthermore, they will happen at higher temperature if heavier elements are involved in the system. Those irregular behaviors can be understood from the first place by analyzing the formulism of the free energy and pressure. Although the FK virial EOS fails as the temperature is below the ionization energy of the constituting elements, it is still useful for the solar applications. The valid range for applying it has covered most part of the interior of the Sun. Only the outer region of the Sun would request another approach. In addition, the composition of the elements heavier than helium can be already well approximated by only consideringC,N,O, andNe. The induced errors for the thermodynamic quantities under the calculation with FK virial EOS should be around 10 3 s 10 4 , and thus, whether the question about the composition of heavier elements in the Sun is relevant or not might depend on which thermodynamic quantities 98 we want to study and how precise the measuring tool is. Of course, this statement is based on the formula of the FK virial EOS. In the next chapter we will discuss another EOS but developed by using similar cluster expansion approach to handle the outer region of the Sun. 99 Chapter6 Low-TemperatureExtensionsofthe VirialEquationofStateforSolar Modeling From the discussions in the previous chapter, we understand the FK virial EOS can only be applied to a nearly full ionized plasma. In practice this can already cover more than 90% of the solar radius inside the Sun (if considerH&He). To further have a complete EOS suitable for the entire Sun, another EOS suits the solar exterior where the plasma is only partially ionized is requested. This can be fulfilled in different ways. However, instead of using precalculated tabulating EOS or phenomenologically optimized EOS like OPAL or MHD, we want to pursue an analytical and exact EOS as the FK virial EOS. For this demand, the Saha EOS (from the Saha equation), even though is also derived within the chemical picture, can play as the first order correction to the ideal gas pressure to take into account the formation of atoms. This simple and analytical formulism can be proved to be asymptotically exact under scaled low temperature and low density limit [40]. Beyond that, it cannot be accurate due to the absence of the interaction and the internal structure of the atoms. Thus a more delicate EOS would be needed. In this chapter, we use the simple Saha EOS as the first trial. We compare its results of thermodynamic quantities for theH system with the results from the FK virial EOS. This contains twofold intentions. It can show how the Saha EOS can qualitatively fix the problem that the FK virial EOS fails within the partially ionized regime. Second, 100 we can see what the Saha theory is deficient by comparing with the FK virial EOS at nearly full ionized regime. In its following, the theory of scaled low temperature (SLT) extension is introduced [6]. In this framework, the corrections to the Saha EOS due to the interaction and quantum effects are derived within the physical picture. Thus, it shares the same virtue with the FK virial EOS: the corrections are exact, analytical and systematical. Under the calculation of the thermodynamic quantities for the solar exterior, it should show that this SLT EOS can paly a potentially accurate EOS to com- pensate the FK virial EOS at the other end of the Sun. About the middle region, more subtle discussions will be needed in order to deal their cross-over. This work can show the combination of two exact EOS (SLT EOS and FK virial EOS) might be potentially the most accurate EOS throughout the entire Sun. 6.1 SahaEquationofState 6.1.1 IntroductiontoTheory As the simplest EOS from the chemical picture, the Saha equation successfully describes the feature of ionization/recombination in the plasma. Although it might not be accurate enough to satisfy our request, as a prime and simple theory, it is still worth to study. To be more specific here, we will consider the ionization/recombination process for theH atom: p + +e $H: (6.1) 101 By using the Saha equation, we can simply approximate the ionization rate of the H atoms. Follow the discussion in the section 5:1:3, the Saha EOS can be expressed as the ideal gas pressure pressure combined with the ionization degreex, P Saha = (1 +x) : (6.2) By substituting the equation (5.9) intox, the Saha EOS for the hydrogen plasma can be expressed as P Saha = + ( r 1 + 2 1) (6.3) with the temperature-dependent density = exp(E H ) 2 3 ; (6.4) where is the thermal De Broglie Wavelength and E H is the ground state energy of hydrogen. It turns out that this characteristic density controls the cross-over between the full ionization and full recombination as P = 2, for ; (6.5) P =, for : (6.6) This simple theory shows, even though it lacks of the many non-ideal effects, it can still involves the correct features of partially ionized plasma: ionization and recombination. In addition to the total pressure, we can further derive the internal energy density from the Saha theory as u Saha =(1 +x) 3 2 +xE H : (6.7) 102 With the aid of this formula and the Saha EOS, all the relevant thermodynamic quantities can be derived by using the same thermodynamic relations discussed before. As in the section 5:3:2, T and can be directly obtained by taking the derivative of the Saha pressure. The specific heat c v (per volume) can be derived from the internal energy density and the result is c v = ( @u @T ) =( 3 2 (1 +x) + ( 3 2 +E H ) 2 x(1x) (2x) )k B ; (6.8) where k B is the Boltzmann constant. Thereafter, 1 and sound speed can be easily obtained as well. 6.1.2 ResultsandDiscussions Figure 6.1: Compare the normalized total pressure calculated from different EOS for theH system (the dot line represents P P id = 1). In Fig. 6.1, the computed Saha pressure is normalized by the corresponding ideal gas pressure but under full ionization condition. Therefore, if the plasma is fully ionized, the curve of Saha pressure is 1. On the other hand, if the plasma becomes fully recombined, the normalized Saha pressure should become 0:5 since the particle number reduces to 103 half. As we expect, the Saha pressure shows the correct trend as the temperature and density decrease in the Saha regime where the plasma is partially ionized. Unlike the FK virail pressure, it decreases monotonically with the temperature from about 1 to 0:5 as 4 10 4 K.T . 4 10 6 K. In the regionT . 10 6 K, the normalized Saha pressure is close to 1 and thus shows the plasma is nearly fully ionized. Meanwhile, it is also above the FK virial pressure. This is due to the absence of Coulomb interaction between the charge particles in the Saha theory, and the corresponding Debye-H¨ uckel correction introduces negative partial pressure. Therefore, although the simple Saha pressure can- not be accurate, at least is qualitatively correct in these regions. Furthermore, we might expect it becomes quantitatively correct as the temperature and density are low enough. However, at even higher temperature (T & 4 10 6 K), the Saha pressure decreases sig- nificantly as the temperature increases. Though this specious result seems to indicate the plasma is also only partially ionized at solar core, it contradicts the result from FK viral pressure. The FK pressure keeps increasing and becomes greater than the corre- sponding ideal gas pressure as has been discussed in the section 5:3:1 (see Fig. 5.13). In addition, we do not expect hydrogen atoms should exist at such high temperature in the solar core. Therefore the Saha EOS cannot be correct in this region. The Saha theory should not be valid under such high temperature and density. In the other words, this is beyond the Saha regime. On the other hand, the FK virial includes interaction and quantum effects and can thus describe the real pressure in that region correctly. Actually, the qualitative behavior of the Saha pressure can be easily captured by looking at the behavior of the ratio throughout the Sun in Fig. 6.2 and using the properties of the Saha pressure in equations (6.5) and (6.6). This is also similar to its ionization degree in Fig. 5.3. It turns out that 1 is within the same range of the temperature 4 10 5 K . T . 4 10 6 K as the Saha pressure is close to the ideal gas pressure of fully ionized plasma. Below that temperature range, increases rapidly 104 Figure 6.2: The behavior of ratio throughout the Sun. and becomes much greater than 1 as the temperature decreases. This consistently shows the H plasma transits into H atomic gas. On the other hand, increases with the temperature above that temperature range. Even though it does not become bigger than 1, this already causes its normalized Saha pressure to decrease significantly as shown in Fig. 6.1. Nevertheless, as we have mentioned before, the simple assumption of single atom in the Saha theory is no longer satisfied under such high density condition. The interaction effect becomes important and should not be neglected. Thus, the mechanism of pressure ionization which is not considered in the Saha theory turns the H plasma into fully ionized. From the simple analysis of Saha pressure, we know that the Sun covers the cross- over between full ionization and full recombination. This is indeed consistent with the result for H in Fig. 5.3. Following this conclusion, we can expect the related thermodynamic quantities from the Saha EOS should also change between that two extreme conditions except in the solar core where the Saha EOS is not valid. Under the full ionization or the full recombination, the Saha theory predicts the same values as 105 the ideal thermodynamic quantities since the ideal gas law is actually their asymptotic results, i.e. T = 1, = 1, c v = 1:5, and = 3 5 . Therefore, the thermodynamic quantities from the Saha EOS should vary from about those idea values in the region 4 10 5 K.T . 4 10 6 K and return to about the same values at very low temperature (T . 8 10 3 K). The computed results of some relevant thermodynamic quantities are summarized in Fig. 6.3 to Fig. 6.7. Figure 6.3: Compare calculated by using the Saha EOS with the FK virial EOS. Figure 6.4: Compare T calculated by using the Saha EOS with the FK virial EOS. 106 Figure 6.5: Comparec v calculated by using the Saha EOS with the FK virial EOS. Figure 6.6: Compare 1 calculated by using the Saha EOS with the FK virial EOS. In conclusion, the Saha theory can qualitatively describe the physics of the Sun within the temperature rangeT . 4 10 6 K. In contrast to the thermodynamic quanti- ties from the FK virial EOS, the results from the Saha EOS do not show any singularity or irregularity throughout the Sun. Furthermore, the Saha EOS can already capture the correct trends of those thermodynamic quantities in the partial ionization region. This also demonstrates the power of the chemical picture approach: it easily describes the phenomenon of recombination, yet the values might not be very accurate. The compar- isons with the results from the FK virial EOS show that the Saha EOS is not accurate 107 Figure 6.7: Compare the sound speedsv s calculated by using the Saha EOS with the FK virial EOS. enough in the solar interior. For our purpose, even though we should use the exact FK virial EOS for the interior region (T & 6 10 4 K), the simple Saha EOS turns out to be not sufficiently accurate for the rest of the Sun, especially around the junction, we would expect its results might become asymptotically exact in the low temperature (T . 10 4 K) and low density region. This statement might be able to be confirmed by using the more accurate SLT EOS and will be discussed in next section. Nevertheless, we might already have two suitable EOSs on each sides of the Sun: the FK virial EOS in the deeper region and the Saha EOS in the outmost region. As usual, the middle region turns to be the most difficult and this is the challenge needs to be overcomed in the next section. 6.2 ScaledLowTemperatureExpansions 6.2.1 IntroductiontoTheory Like the ideal gas EOS, the simple Saha theory also does not consider the interaction at first place. Even though the recombination of the electron and the proton is involved, this 108 is done in a phenomenological way by introducing the potential energy (the ground state energy ofH) of forming theH atom as usually did in the chemical picture. Even more, the details about the internal structure of theH atom are usually ignored for simplifica- tion. However, under weak coupling, low density, and low temperature conditions, we should expect the Saha equation to represent as an effectively asymptotic form. On the other hand, despite the exact FK virial EOS takes into account the contribution of the two-particle atom in its quantum second virial function, its perturbation method does not capture the full mechanism of recombination. Moreover, the truncated FK virial EOS causes significant deviation as the recombination becomes important. Base on this, a more fragile theory similar to the exact quantum virial equation but go beyond the Saha theory was developed. In addition to that, in order to be consistent, the Saha equation should be reobtained under certain limits in the same theory. Follow this idea, Alastuey and the cooperators have extended their work of the FK virial EOS to develop an exact asymptotic expansion in the Saha regime by using the similar method: FK path integral combined with the Mayer diagrammatical method [5]. Within this framework, the recombination and screening can be treated simultaneously and systematically. The resulting grand canonical partition function in screened cluster representation is free from divergences which originate from the long range Coulomb interaction. In con- trast to the chemical picture approach, this does not introduce any uncontrolled internal partition function like PLPF. In this approach, the involved cluster partition functions represent the familiar chemical species, and include their possible bound and diffusive states. This so-called scaled low temperature (SLT) expansion solves the problem of dealing the bound states of ions, atoms, and molecules within the physical picture, and provides the rigourous extension of the FK virial EOS into the Saha regime [6, 4]. Their result shows the first five leading corrections to the ideal Saha pressure to exactly include the effects of interaction and the possible internal structure of bound states up to the 109 order exp(E H ). The even higher order corrections decay exponentially faster than exp(E H ) and can be neglected. After decomposition and rearrangement, each correc- tion can be expressed as a function of times a temperature-dependent function. Thus, the according SLT expansion of the pressure can be written in the form of P = P Saha + 1 X k=1 b k ( ) k (): (6.9) As the k function decays exponentially fast at low temperature limit, i.e. k s exp( k ), this expansion is arranged in the order of its exponentially decay rates k . Their physical content and decay rate values are summarized in Tab. 6.1. As the result, Correction (k) Physical content δ k (in eV) 1 plasma polarization around ionized charges |E H |/2 6.8 2 formation of molecules, atom-atom interactions |3E H −E H2 | 9.1 3 atomic excitations, charge-charge interactions 3|E H |/4 10.2 4 formation of ions, atom-charge interactions |2E H −E H + 2 | 11.0 5 fluctuations of plasma polarization |E H | 13.6 Table 6.1: The physical content and decay rate values for the first five corrections in the SLT expansion (cited from [3]). the first five coefficientsb k are obtained as following b 1 ( ) = 3=2 s ( s 2) 3( s + 1) ; (6.10) b 2 ( ) = 4 s ( s + 3) 2( s + 1) ; (6.11) b 3 ( ) = 2 s + 1 ; (6.12) b 4 ( ) = 3 s ( s + 4) 3( s + 1) ; (6.13) b 5 ( ) = 2 s (2 2 s ( s + 1) 3 ; (6.14) 110 with s = q 1 + 2 1 defined from the Saha equation. They are functions of both temperature and density. Their asymptotic behaviors for relatively high density and low density at fixed temperature are summarized in Tab. 6.2 Thus the first, the third, k ρ ρ ∗ ρ ρ ∗ 1 ρ 3/2 ρ 3/4 2 ρ 4 ρ 2 3 ρ 2 ρ 1/2 4 ρ 3 ρ 3/2 5 ρ 2 ρ 1/2 Table 6.2: Asymptotic behaviors of each corrections (cited from [3]). and the fifth corrections should recover the virial expansion at least up the same order of the density 2 at low density as the temperature is also low enough. All the other corrections, include k 6, contribute even higher power of density. This has been checked and confirmed, thus this SLT expansion is consistent with the virial expansion under that conditions. In addition, the function k only depends on the temperature. They are related to theh k function as 1 () =h 1 () 2 () =h 2 () 3 () =h 3 () 4 () =h 4 () 5 () =h 1 () 2 ; (6.15) 111 and the expressions ofh k are h 1 () = (E H ) 3=4 1=4 exp(E H =2); (6.16) h 2 () = ( 1 64 ( 2m M ) 3=2 Z(2; 2) +W (1; 1j1; 1)) exp(3E H ); (6.17) h 3 () = 1 2 + [1 + 1 12 ln( 4m M )] (jE H j) 3=2 1=2 exp(E H ) + 1 8 1=2 f2Q(x pe ) + ( 2m m p ) 3=2 [Q(x pp ) 1 2 E(x pp )] + ( 2m m e ) 3=2 [Q(x ee ) 1 2 E(x ee )]g exp(E H ); (6.18) h 4 () = ( 3 64 (( m e (M +m p ) M 2 ) 3=2 Z(2; 1) + ( m p (M +m e ) M 2 ) 3=2 Z(1; 2)) +S 3 (1; 1) + 3 2 (W (1; 1j1; 0) +W (1; 1j0; 1))) exp(2E H ): (6.19) Within these formulas, the cluster partition functionZ(N p ;N e ) withN p protons andN e electrons is related to the truncated trace of Gibbs operator exp(H Np;Ne ) as Z(N p ;N e ) = 2 2 Np;Ne lim !1 1 Tr[exp(H Np;Ne )] T ; (6.20) where Np;Ne = (~ 2 =(N p m p + N e m e )) 1=2 is the thermal de Broglie wave length of the cluster. This partition function does not only include the contribution from cluster excited states but also the related diffusive states describing its dissociation. W (N p ;N e jN 0 p ;N 0 e ) is also similarly defined for the pair of two clusters contain (N p ;N e ) and (N 0 p ;N 0 e ) protons and electrons separately. Whileh 1 andh 3 can be written in closed forms with analytic functions,h 2 andh 4 do not have similar expressions. They contain the internal partition functions ofH 2 , H , andH + 2 , but the spectra of three-body and four-body Hamiltonian do not have the exact formulas similar to the H atom. There- fore, in order to evaluate theh 2 andh 4 functions, some phenomenological models with approximations for low temperature will be used. 112 Among the phenomenological approximations of theh 2 and theh 4 functions in [3], Z(2; 2) is approximated as factorized partition function of moleculeH 2 Z(2; 2)Z H 2 = exp(E H 2 )Z (rot) H 2 Z (vib) H 2 : (6.21) The electronic partition function only counts the electronic ground state ofH 2 . The rota- tional dynamics can be modeled as a rigid rotator. Due to different spin configurations, its rotational partition function can be expressed as Z (rot) H 2 = [ 1 X l=0 (4l+1) exp(2l(2l+1) (rot) H 2 )+3 1 X l=0 (4l+3) exp((2l+1)(2l+2) (rot) H 2 )]: (6.22) with the rotational energy quantum energy (rot) H 2 . The first sum counts the rotational states of para-hydrogen and the second sum is for ortho-hydrogen. The vibrational dynamics can be approximated with a simple harmonic oscillator, and results in the vibrational partition function: Z (vib) H 2 = 1 1 exp( (vib) H 2 ) : (6.23) with vibrational quantum energy (vib) H 2 . In addition, the function h 4 can be also con- structed similarly. It contains the partition functions of the H + 2 ion expressed as the product of its corresponding internal partition functions as Z(2; 1)Z H + 2 = 2 exp(E H + 2 )Z (rot) H + 2 Z (vib) H + 2 (6.24) 113 with Z (rot) H + 2 = [ 1 X l=0 (4l+1) exp(2l(2l+1) (rot) H + 2 )+3 1 X l=0 (4l+3) exp((2l+1)(2l+2) (rot) H + 2 )]; (6.25) and Z (vib) H + 2 = 1 1 exp( (vib) H + 2 ) ; (6.26) where (rot) H + 2 and (vib) H + 2 are the corresponding rotational and vibrational quantum energy. For the partition function of theH ion, due to its simple ionic structure, its partition function can be simpler approximated as Z(1; 2)Z H = 2 exp(E H ): (6.27) For the contributions of atom-atom interactions, they can be approximated by using the effective potentials as W HH = m 3=2 8(2~ 2 ) 3=2 [ 4 3 3 HH + Z R> HH dR(exp(U HH (R)) 1)]; (6.28) where HH is the classical cross section of theH atom andU HH (R) =A HH =R 6 is its attractive van der Waals potential (quantum fluctuation induced dipole-dipole inter- action) between two neutral atoms. For the contributions of interactions between an atom and a charged particle, they are expressed respectively as W Hp = 3m 3=2 4(2~ 2 ) 3=2 [ 4 3 3 Hp + Z R> Hp dR(exp(U Hp (R)) 1)]; (6.29) 114 and W He = 3m 3=2 4(2~ 2 ) 3=2 [ 4 3 3 He + Z R> He dR(exp(U He (R)) 1)]; (6.30) whereU Hp (R) =A Hp =R 4 andU He (R) =A He =R 4 are the effective potential generated by quantum fluctuations of dipole-charge interaction. From the discussion in [4], the low-temperature behaviors ofh 2 andh 4 can be rea- sonably estimated as h 2 ()w 1 64 ( 2m M ) 3=2 Z(2; 2) exp(3E H ); (6.31) and h 4 ()w 3 64 ( m e (M +m p ) M 2 ) 3=2 Z(2; 1) exp(2E H ) + 3 64 ( m p (M +m e ) M 2 ) 3=2 Z(1; 2) exp(2E H ) + ( c at 8 3=2 (jE H j) 1=2 ) exp(2E H ): (6.32) Furthermore, their asymptotic behaviors at low temperature can be simplified by only counting the dominate exponentially decay rates in their internal partition functions [6]. From the simple comparison between various decay rates,h 2 andh 4 can be expressed separately as h 2 () 1 64 ( 2m M ) 3=2 exp((3E H E H 2 )); (6.33) h 4 () 3 64 ( m e (M +m p ) M 2 ) 3=2 exp((2E H E H + 2 )): (6.34) Even though the interplay between theb k andh k functions generates the complicated behaviors of pressure, some simple conclusions can still be shortly made with the aid of the asymptotic behaviors ofh k andb k (see Tab. 6.2). As marked in [6], the SLT EOS 115 (6.2.1) displays consistent behaviors with respect to different range of ratio as fixed is sufficiently larger: 1 The series of equation (6.2.1) can be expanded in powers of and leads to the virial expansion in power of. The contributions from the termsk 5 provides the complete virial expansion up to the order 2 : P = 2 2 3=2 (2) 3=4 3 ( ep ) 3=2 exp(E H =2)h 1 () 3=2 (2) 3=2 ( pe ) 3 exp(E H )[2h 3 () + 1 2(h 1 ()) 2 ] 2 +O( 5=2 ) = 2 (8e 2 ) 3=2 24 p 2 f2 3 pe Q(x pe ) + 3 pe [Q(x pp ) 1 2 E(x pp )] + 3 ee [Q(x ee ) 1 2 E(x ee )]g 2 6 ln( 4m M ) 3 e 6 2 +O( 5=2 ): (6.35) This coincides with the expression of the FK virial EOS up to the same order 2 . It is remarkable that both uncertainh 2 andh 4 functions are not involved there. The contribution forh 2 is at least of order 4 , and forh 4 is at least of order 3 . This indicates that we might not need to worry about the higher temperature expres- sions ofh 2 andh 4 , and their low-temperature models may be already enough for our application: the solar modeling. 1 The leading termP Saha combined with higher order correctionsP k fork 5 expressed the accurate expansion of pressure. Each correction has its physical meaning as in Tab. 6.1, and correctly count the effects of screening and recombi- nation. The corrections of even higher order fork 6 should be unimportant due to that they decay much faster with temperature. 116 1 Under this condition, the first termP Saha behaves as P Saha ; (6.36) which means the atoms are almost fully formed. Furthermore, its leading correc- tion comes fromP 2 P 2 2h 2 () ( ) 2 : (6.37) This shows the molecular recombination prevails all the other contributions to the Saha pressure. However, if the ratio is such large that the corrections become larger than the Saha pressure, the SLT expansion (6.2.1) is no longer valid. Thus the result cannot be easily extrapolated. In addition to the pressure, the SLT expansion of the internal energy density can be also obtained similarly as u =u Saha + 1 X k=1 u k : (6.38) Within this expression, the first five leading correction have been derived as [4] u 1 = 1=2 s (1 + s ) 1 [(1 + s 1 )k B T + 2(1 s 1 )E H ]h 1 () (6.39) u 2 = 4 s 1 h 0 2 ()=2 + 3 s (1 + s ) 1 (1 + s 1 =2)(E H 3 2 k B T )h 2 () (6.40) u 3 = 2 s 1 h 0 3 () + 3 s 1 (1 + s ) 1 (E H 3 2 k B T )h 3 () (6.41) u 4 =2 3 s 1 h 0 4 ()=3 + 2(1 + s 1 ) 2 s (1 + s ) 1 (E H 3 2 k B T )h 4 ()=3 (6.42) u 5 = (1 + s ) 3 (3k B T + 2 s (3 s + 4)(2 + s ) 1 E H )h 1 () 2 (6.43) Hereafter, all the relevant thermodynamic quantities can be easily computed by using the same thermodynamic relations as discussed previously. 117 6.2.2 ResultsandDiscussions 01 5 10 15 20 25 30 35 40 T Rydberg /T 10 -9 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 1 density (g/cm 3 ) 157801 K 15780 K 7890 K 5260 K 3945 K photosphere Sun adiabat c (T) *(T) Plasma Atomic fluid Molecular fluid Figure 6.8: Qualitative phase diagram of hydrogen gas (cited from [4]). First of all, a qualitatively description of the Sun as hydrogen plasm can be obtained from the simple analysis in Fig. 6.8. It represents a simple phase diagram of hydrogen gas within broad ranges of temperature and density. The line of crossover density for hydrogen and the line of criterion density defined as c (T ) = (T ) 20jh 2 (T )j ; (6.44) are also added as simple criteria to distinguish the properties of hydrogen gas. They divide the phase diagram into thee different phases: Plasma, atomic fluid, and molecular fluid. In addition, the line of constant coupling = 0:5 (the dashed line) is also included as the criterion to check the validity of perturbative approach with small coupling. As the state of the entire sun is drew as a dot line denoted as ”Sun adiabate”, this figure 118 shows that the hydrogen gas in the Sun covers two different phases: the solar interior is in plasma phase and the solar surface is in atomic fluid phase. Meanwhile, both two parts of the sun adiabat stay far below the criterion density c (T ) indicates the SLT expansion should be effective for the Sun except the condition with very high temperature and density in the solar core. In the plasma region, the SLT expansion might approach the same result from the FK virial expansion. Moreover, as has already been discussed in section 5:1:1, since the solar tract also remains small coupling < 0:5 in this region, the virial expansion should also become valid here. From the qualitative discussion above, we can already make a simple conclusion that the SLT EOS is suitable to the solar exterior and the FK virial EOS (contains complete expansion up to 3=2 ) should paly better for the solar interior. Figure 6.9: The behavior of s throughout the Sun. To be more accurate, we need to step into the quantitative analysis of the formulas of SLT expansion with the solar conditions. According to the discussion in the previous section, although the five leading terms in the SLT expansion are ordered according to the exponentially decay behaviors of k at low temperature limit, their respective contributions might not follow the same order for the case of the Sun since it covers 119 wide range of temperature and density. Furthermore, the coefficient functions b k ( s ) also covers wide range of value and varies rapidly as can be seen in Fig. 6.10. Since all the five b k functions have closed analytical forms, their behaviors throughout the Sun can be certainly computed. The result in Fig. 6.10 shows the five b k ( s ) have large values at solar surface but decay quickly as going toward the solar interior. Inside the Sun, they remain with smaller values and do not vary much. Besides, b 2 and b 4 dominate asT . 6 10 4 K but become smaller than the others above that temperature range. Interestingly, b 1 , b 2 , b 3 , and b 4 cross within very small range of temperature around 6 10 4 K. On the contrary, the h 2 () and h 4 () functions only depend on Figure 6.10: The behaviors of theb k functions throughout the Sun. the temperature but need to be approximated with some phenomenological approaches. In order to have certain understanding about the behaviors of each phenomenological approaches and compare their values for the temperature range of the Sun, we plot all of them together in Fig. 6.11. Within it, the h 3 curve and the h 4 curve represent the equations (6.31) and (6.32). Theh 2 curve and theh 4 curve show the equations (6.33) and (6.34). Overall,h 1 has significant contribution asT . 10 5 K. Beyond that region h 3 becomes important. Unless the internal approximations are included (theh 2 andh 4 curves),h 2 andh 4 functions are relatively unimportant. 120 Figure 6.11: The behaviors of theh k functions throughout the Sun. The various behaviors of theb k andh k functions result in the complicated behavior of the total pressure of hydrogen gas. Their net contributions can be calculated as each partial pressureP k = b k ( ) k () and the results are all shown in the Fig. 6.12 and Fig. 6.12. The k = 2 and k = 4 partial pressure are approximated with equations (6.31) and (6.32). Equations (6.33) and (6.34) are used for the k = 2 and k = 4 curves. Besides, the absolute values of the partial pressure are also plotted together as two different graphes for the convenience. In Fig. 6.13, the equations (6.31) and (6.32) are used for computing P 2 and P 4 . In Fig. 6.14, the equations (6.33) and (6.34)are applied for calculatingP 2 andP 4 . Fig. 6.13 shows the consistent result as we can expect Figure 6.12: The corrections to the Saha pressureP Saha throughout the Sun. from the precious discussion about the Eq. 6.35. It happens that the ratio is less than 121 Figure 6.13: The absolute values of the corrections to the Saha pressureP Saha through- out the Sun. one within the solar interior except around the solar core (Fig. 6.2). Within this plasma region, the fist, third, and fifth corrections dominate the corrections to the Saha pressure. Meanwhile the weak coupling condition in this region guarantees the higher order terms contribute less. Thus the net result asymptotically approaches the FK virial EOS, at least up to the order 2 as can be expected from Eq. 6.35. However, once we include the internal partition functions into the second and the forth correction terms, they intervene the result of pressure at larger temperature and do not match the FK virial pressure (Fig. 6.14). The incomplete phenological models ofh 2 andh 4 cause significant errors. They contribute the corrections for higher order of density ( 3 ), but do not involve all the corrections. This causes bizarre behavior of the total pressure and should not be treated as a very accurate result. Fortunate, we already have a more accurate EOS from FK virial expansion, at least is exact up to the order 3=2 . We should transform the SLT EOS to the FK virial EOS in the solar interior where the plasm is in well defined plasma region. From those results, we might expect the second and forth corrections are not important to 122 Figure 6.14: The absolute values of the corrections to the Saha pressureP Saha through- out the Sun. the application in the Solar interior. As in the Fig. 6.15, both the pressure computed by using the SLT EOS with different approximations approach the Saha pressure under low temperature region (T . 10 4 K) of the Sun. On the hand, they begin to deviate from each other asT s 2 10 5 K. The one contains the internal partition functions of molecule decreases rapidly at about 10 6 K and even becomes negative pressure. This situation is quite similar to what happens to the FK virial pressure as in the Fig. 5.13. Nevertheless, the SLT pressure with the simplest approximation (SLT 1 curve) can match the FK virial pressure well within 8 10 4 K.T . 3 10 6 K. The reason has been discussed previously and it is because P 2 and P 4 do not contribute to the virial expansion up to the power 5=2 . Their simplest approximations do not contain any irregular behavior and can be neglected by comparing with the FK-virila pressure. On the other hand, if the approximations involve the internal partition functions, ie. Eq. (6.33) and Eq. (6.34), they increases exponentially fast, even though do not contribute significantly at lower temperature, finally intervene the pressure at higher temperature (T & 10 5 K). 123 Figure 6.15: Compare the normalized total pressure calculated by using different EOS. At the end, they contribute large negative partial pressure, especially the k = 4 term, and thus the total pressure turns downward and the total pressure drops rapidly around T s 10 6 K. As we should focus on the low temperature region of the Sun for applying the SLT expansion (T 10 6 K), compare to all the other partial pressure, the Saha pressure always dominates. First, this result confirms the validity of SLT expansion in this region since all the other corrections are smaller. Second, all the corrections are not important untilT & 2 10 4 K. However, the contributions fromP 2 andP 4 are still very minor. Different approach of the approximation does not intervene the result as the temperature T . 10 5 K. Thus the choice about the approximation for theh 2 andh 4 functions are not relevant for our purpose. On the other hand, although different approximations might intervene the pressure asT & 10 5 K, as we have discussed, the FK virial should start to become a more accurate EOS in this region. This can be clarified with the net result of the total pressure in Fig. 6.15. It is consistent that both theSLT 1 (use the equations (6.31) and (6.32)) and SLT 2 (use equations (6.33) and (6.34))curves match the Saha pressure whenT . 10 4 and match the FK virial pressure asT s 10 5 K. In addition, 124 theSLT 2 curve begins to deviate form the FK virial pressure asT & 2 10 5 K. This concludes that we might not need to worry about how accurate the approximations are for theh 2 andh 4 functions. We can avoid it for the application of the solar modeling since bothh 2 andh 4 are not relevant in the solar exterior where their contributions are very minor ,and in the solar interior we have a more accurate FK virial EOS can be used. To completely confirm this, we also compute the relevant thermodynamic quantities by using different EOS. The comparisons in Fig. 6.16 show the consistent result as Figure 6.16: Compare the thermodynamic quantities calculated by using different EOS. the comparisons for the total pressure in Fig. 6.15. The thermodynamic quantities calculated from the SLT EOS under the solar conditions can match the Saha EOS on one side and the FK virial EOS on the other side during some range of temperature (10 5 K.T . 410 6 K). Beyond that, the SLT EOS starts to deviate from the FK virial 125 EOS for those thermodynamic quantities. In the solar core, the FK virial represents the more accurate EOS and the SLT expansion turns out to be unable to approach it there. 6.3 Conlcusion Even though the discussions above only focuses on the hydrogen only system under the solar conditions, as the most abundant element, the results are still valuable and can instruct us in improving the application of the EOS for the solar modeling. To be a use- ful EOS for the solar applications, the other elements must be also included. This will rely on the extension of the SLT expansion to the multi-component plasma. Neverthe- less, from our initial study, the SLT expansion turns to be the most accurate EOS can compensate the FK virial expansion within partially ionized condition. To summarize, we have confirmed that the SLT expansion can be consistent with the Saha equation at low temperature end of the Sun. Furthermore, the SLT expansion does not only go beyond the Saha theory but can also approach the FK virial expansion (at least up to the power 2 ) somewhere at middle. The result shows we can connect these two exact expansion EOSs: the SLT EOS and the FK virial EOS, somewhere within the range that they can match well to reduce the error to be minimum. As long as the induced discrep- ancy at the joint is small enough for our practical applications with the Sun, this ultimate unified expansion formulas will offer the most exact and also might be potentially the most accurate EOS for the solar applications. In order to understand the accuracy of this unified EOS, we compare our results above with the results computed from OPAL, and CEFF. Since they have been extensively tested with the inversion results from the helio- seismology experiment, they can offer as some kind of standard to verify the accuracy of our unified EOS. This will be discussed in the next chapter. 126 Chapter7 ComparisonswiththeOPALand CEFFEquationsofStates In order to determine whether the SLT EOS and the FK virial EOS are accurate, we should compare their results with the ones computed from the well studied EOS, like OPAL, MHD, or CEFF. Among the possible choices, although OPAL is accepted to be the most accurate EOS for solar and stellar modeling, we also compare our results with CEFF, a simple but popular equation of state for stellar modeling. CEFF is based on the original EFF EOS proposed by P. Eggleton, J. Faulkner and B. Flannery in 1973 [23]. Their basic idea was to extend the Saha EOS to involve the effect of pressure ionization. Later it was modified to include the Coulomb interaction effect transferred from the MHD package and thus is called CEFF [16]. CEFF has become the most prevailing practical equation of state among solar evolutionary model and helioseismic analysis, its compact form and fast calculation speed yet still with good performance are the main reasons [39]. Our reasons to compare our new equation of state with both OPAL and CEFF are twofold. First, the two EOS are fundamentally different: OPAL is derived in the physical picture, while CEFF is based on the chemical picture. Second, because of the proprietary nature of the OPAL code, OPAL results are only available in the form of pre-computed tables released from Livermore. In particular, the relative abundances of the heavy ele- ments (heavier than He) cannot be specified. New releases with custom composition have not been announced. In order to compare our results of theH&He&O system, we 127 need to use another EOS that not only can accept the same input as we used but is still accurate. It turns out that a recent modification of CEFF that allows it to successfully emulate OPAL [39] is the perfectly appropriate tool for our task. Since the SLT EOS is only designed forH gas, but the FK virial EOS is suited to multi-component plasma, we divide the discussions into two parts. In the first parts, we compare the results computed from the SLT and FK virial EOSs with OPAL’s and CEFF’s forH only system. In the second part, we only compare the results calculated by using the FK virial EOS with OPAL’s and CEFF’s. However, for theH&He&O sys- tem, only the CEFF’s results are available due to the limitation of OPAL just remarked. Besides, the comparisons are presented in two different types. The similar thermody- namic quantities are graphed together to tell their distinct behaviors. Second, the abso- lute values of their relative discrepancies are also presented in order to quantitatively analyze their difference. These comparisons can show how ”accurate” the SLT and FK virial EOSs are. In addition, they might also help to indicate where is suitable to connect two different EOSs: the SLT EOS and the FK virial EOS, to obtain the best unified EOS. 7.1 CompareSLT&FKVirialwithOPALandCEFFfor H System Due to the SLT expansion is only available to the hydrogen gas, we first compare the results from both FK virial and SLT EOSs with OPAL’s and CEFF’s for hydrogen system under the same solar conditions. The results for the comparisons with the OPAL’s are graphed in Fig. 7.1 and Fig. 7.2 separately with the different approaches of the SLT EOS, i.e. SLT 1 and SLT 2. To be more precise, the absolute values of their relative discrepancies are also plotted in Fig. 7.3. From the trends of each results computed by using different EOS, we see that the OPAL’s results can basically match the SLT’s 128 Figure 7.1: Compare the thermodynamic quantities computed by using the SLT EOS, the FK virial EOS, and OPAL (the approximations Eq. (6.31) and Eq. (6.32) are used for the curveSLT 1). and FK virial’s at low and high temperature regions respectively. Compare to the SLT1 curve at higher temperature region (T & 410 6 K), the OPAL’s results fit the FK virial’s results better. This confirms again that the SLT EOS is not accurate for the deeper solar interior, but both FK virial and OPAL EOSs can give consistent results in this region. In addition, the differences between SLT1 and SLT2 are not apparent below the valid temperature of SLT2 (T & 10 6 K). The more precise comments can be made by using Fig. 7.3. First, by comparing theSLT 1OPAL andSLT 2OPAL curves, it can be determined which one is better below the temperature 2 10 5 K. It turns out that SLT 1 has less discrepancy thenSLT 2 at some temperature but has bigger discrepancy at the others. However, SLT 1 is suitable for broader range of temperature due to its lack of irregular behavior as SLT 2’s. The discrepancy for SLT 2 increases rapidly 129 Figure 7.2: Compare the thermodynamic quantities computed by using the SLT EOS, the FK virial EOS, and OPAL (the approximations Eq. (6.33) and Eq. (6.34) are used for the curveSLT 2). above the temperature 2 10 5 K. Secondly, the major discrepancy betweenSLT 1 (or SLT 2) and OPAL occurs at around 510 4 K within the valid region of the temperature. This also happens to be similar to the temperature that the recombination ofH plasma becomes significant (see Fig. 5.3). The discrepancy between FK virial’s and OPAL’s is even bigger here since the plasma has started to transit into partially ionized regime. Therefore, the SLT expansion still offers a better EOS than the FK virial expansion here, but the relative discrepancy is about 10 3 . On the other hand, the discrepancies between SLT’s and OPAL’s results are overall smaller as the temperature decreases below that temperature if we ignore the dips caused by the fluctuations of the data. This shows the SLT EOS becomes more accurate as close to the solar surface (also close to the Saha limit), and the discrepancies can be as low as to be about 10 5 . Besides, it is 130 Figure 7.3: The absolute values of the relative discrepancy of the thermodynamic quan- tities between different EOS and OPAL. also unexpected that the ”more accurate”SLT 2 turns to be broadly rather worse than SLT 1 by comparing with OPAL around the solar surface. The conclusion for comparing with the CEFF’s results is similar to what we discussed above, thus we only show their absolute values of relative discrepancies in Fig. 7.4. However, it shows that OPAL offers more consistent results than CEFF by comparing with the FK virial EOS asT & 5 10 6 K although the improvement is only about 10 3 . The reason is already known for that CEFF does not contain the exchange interaction contributions; it only contains the exchange effect in its ideal free energy. However, the induced error is only about 10 3 , at least forH system. On the other hand, OPAL contains the contribution from exchange-interaction effect and thus can match the FK virial EOS better. Nevertheless, the discrepancies between the OPAL’s and FK virial’s results are still about 10 3 to 131 Figure 7.4: The absolute values of the relative discrepancy of the thermodynamic quan- tities between different EOS and CEFF. 10 4 . Besides, both Fig. 7.3 and Fig. 7.4 indicate that the discrepancies forSLT 1’s are roughly rather better thanSLT 2 except for a few data at the lowest temperature (at solar surface). At there, the comparisons with CEFF and OPAL do not give the consistent results. Fig. 7.4 shows the discrepancies forSLT 2 increase apparently below 10 4 K, but Fig. 7.3 indicates the discrepancies forSLT 1 increase manifestly there. In practice, this might not be an important problem because this area is on the top of solar photosphere and thus cannot be detected accurately by helioseismology. 132 7.2 Compare FK Virial with OPAL and CEFF for H&HeSystem To considerHe atoms in the Sun, we can only compare the results from the FK virial EOS with OPAL and CEFF in the solar interior where the FK virial EOS is valid. The comparisons of different thermodynamic quantities are in Fig. 7.5, and their absolute values of relative discrepancies are graphed in Fig. 7.6. Even though the results of the solar exterior (T . 2 10 5 K) are not available from the FK virial EOS, we need to emphasize again that this range already covers about 90% of the solar interior. Besides, the H&He system also covers almost 99:8% (by particle) of all solar particles. This should still provide valuable results for practical applications in solar modeling. From Fig. 7.5, the OPAL’s results are overall closer to the FK virial’s results again. Above about 2 10 6 K, the CEFF’s pressure appears to be overestimated by comparing with FK virial pressure. On the other hand, the OPAL pressure is underestimated and the CEFF pressure also becomes underestimated below 2 10 6 K. Those behaviors lead to the suspicious results in Fig. 7.6, in which the relative discrepancies between the CEFF’s and FK virial’s results are smaller around 10 6 K. This should not be misunder- stood as that CEFF might be better than OPAL in this area. It is because the lack of exchange interaction contributions causes CEFF pressure change smoothly from below the FK virial’pressure to its top. The more precise statement can be made with the aid of the discussion in section 5:3:1. The main exchange interaction effect (contained in the E function) contributes negative partial pressure. Thus, its absence would cause overestimated pressure. Since this effect is more important in the deeper solar core, it causes the crossover between the CEFF pressure and the FK virial pressure. Within that range containing the crossover, the discrepancies would be around zero and show the dips for P , T , and in Fig. 7.5. On the contrary, 1 turns to be not so sensitive. 133 If we ignore that dips in the diagram, obviously, OPAL’s results are closer to the FK virial’s results. Within the solar core, the discrepancies are even about one order of magnitude smaller than the CEFF’s. Furthermore, by comparing with the results in Fig. 7.3 with the solar core, addingHe element to the system does not introduce significant changes about the relative discrepancies between OPAL’s and FK virial’s results. How- ever, the CEFF does not behave likewise. This might show that OPAL and FK virial EOSs are more fundamentally similar. In addition, we also compute the sound speed for different EOS and plot the result in Fig. 7.5. The sound speed can be calculated straight- forward from 1 by using Eq. 5.36. This quantity is important due to be measurable by using helioseismology. The absolute values of the relative discrepancies of squared sound speed between different EOSs are also graphed in Fig. 7.6. The similar quantity can be directly obtained by using numerical inversion with the measured frequencies from helioseismology experiments [11], and the relative difference of the squared sound speed between the experiment and solar model with different EOS is as in Fig. 4.1. Even though our results in Fig. 7.6 are obtained by only consideringH&He plasma in the Sun, those comparisons can still hint us, compare to OPAL, whether the accuracy of the FK virial EOS is good enough for the solar modeling. If possible, this might be able to be detected by helioseismology with up to 10 5 precision. This statement might become more solid as we extend the system to includeO, or even more details about the heavier elements in the Sun. Nevertheless, if theH&He system can already simulate the situation of the Sun well, it might make some sense to infer here by comparing Fig. 7.7 with Fig. 4.1. 134 7.3 Compare FK Virial with CEFF forH&He&O Sys- tem Finally, we compare our results from the FK virial EOS for theH&He&O system with the corresponding results from CEFF. As we can find from the previous discussion, CEFF does not provide very accurate results in the region near the solar core. However, we can still look at the influence caused by adding theO atoms into the system. This will complete the comparisons and might make some valuable conclusions as well. Both similar comparisons are shown in Fig. 7.8 and Fig.7.9. In the lower temperature regions, similar dips of the relative discrepancies also appear. If we only focus in the deeper region of the solar core, the CEFF’s results only match the FK virial’s results with about 10 3 . Besides, we can observe that addingO into the system does not introduce apparently different discrepancy between the CEFF’s results and FK virial’s. Because the increase of the number of the electron by adding O into H&He system is not so significant as we addHe into theH system, this confirms again the main discrepancy between the CEFF and FK virial EOSs is contributed by the exchange interaction effect for electrons. 7.4 Conclusion Those comparisons indicate the OPAL’s results are closer to the results from the unified EOS of the SLT EOS and FK virial EOSs. The OPAL can approach the FK virial EOS better than the CEFF at the solar core. This tiny deficiency of the CEFF is due to its lack of the interplay between the exchange effect and the interaction effect mainly for electrons. However, CEFF does contain the major part of exchange effect in its ideal free energy. Therefore it can still close to the FK virial at solar core even though not as 135 good as the OPAL. In contrast, the discrepancy at solar surface turns out to be not very significant, and both OPAL and CEFF can match SLT EOS well, at least compare to the results in solar core. For a multi-component system, the SLT EOS becomes unavailable and only the FK virial expansion can be used. Similar comparisons concludes similarly. Overall, OPAL can fit the FK virial EOS better than CEFF if we ignore those suspicious dips of the relative discrepancies with CEFF’s results. To conclude, the SLT EOS can already provide very close results as OPAL’s and CEFF’s, especially near the solar sur- face. In addition, FK virial EOS gives consistent results as OPAL’s, and thus should be more accurate than CEFF although it is only restricted in the solar interior. 136 Figure 7.5: Compare the thermodynamic quantities computed by using the FK virial EOS with CEFF and OPAL. 137 Figure 7.6: The absolute values of the relative discrepancy between the thermodynamic quantities of theH&He System from the FK virial EOS and OPAL/CEFF. Figure 7.7: The discrepancy between the squared sound speed of the H&He System from the FK virial EOS and OPAL/CEFF. 138 Figure 7.8: Compare the thermodynamic quantities of theH&He&O System computed by using different EOS. 139 Figure 7.9: The absolute values of the relative discrepancy between the thermodynamic quantities of theH&He&O System from the FK virial EOS and CEFF. Figure 7.10: The discrepancy between the squared sound speed of theH&He&O Sys- tem from the FK virial EOS and CEFF. 140 Chapter8 Conclusion In this thesis, we have performed the necessary tasks for practical solar applications of the combination of the FK virial EOS with the SLT EOS. Given their rigorous nature, both have the potential to become the best diagnostic tools to probe the inside the Sun. Using these two exact, analytical equations, we have been able to investigate details of thermodynamic quantities in the Sun. A full implementation will facilitate to construc- tion of a better solar model. In the course of our study, we found that although the FK virial EOS can be only applied to the inside of the Sun, where the plasma is almost fully ionized, its applicable region still covers most part of the solar interior. To be more precise, the valid region will depend on the highest ionization energy of the elements involved in the model. We have not only used simple tests to find the possible domain of applicability of the FK virial EOS, but we also performed a detailed computation based on the FK virial EOS to locate the region of its breakdown more exactly. In practice, this can already cover over 90% (forH&He) of the solar interior, and the exact boundary of that valid domain will depend on the heaviest element used in the models. For our case, in which we have represented all particles heavier than He by O, the boundary of its domain of validity is located where temperature is about 1:6 10 6 K. Since the full mixture of all heavy elements will not affect this result much, our result should be essentially the same in the real Sun. On the other hand, the SLT EOS shares the advantages of the FK virial EOS (analytic and exact), but opposite to the latter, it can be used only below that boundary where the plasma becomes fully ionized. Thus, extending to the SLT EOS can overcome the 141 limitation of the FK virial EOS. However, as of now, the SLT EOS is restrict to a one- element plasma, limiting us toH in our practical computations. However, as the most abundant element,H does play the most important role in the Sun (and the other stars as well). In our preliminary calculations with the SLT EOS to a pure H plasma, and comparing them with results obtained from OPAL and CEFF, we have confirmed its accuracy in the outer region of the Sun. Not only is the SLT EOS equivalent to the Saha equation in the outermost region of the Sun, giving more accurate results than the Saha equation, it also provides as consistent results as the FK virial EOS in the solar interior except in the deep interior of the Sun. We therefore conclude that combining SLT with the FK virial EOS will create a “global” analytic and exact equation of state that is applicable to the entire Sun. Nevertheless, before it can be realized to perform practical applications to the Sun, the theory of the SLT expansion method will be required to be extended to multi-species plasmas. We expect that significant improvements can already be made with simple extensions to multi-species plasmas, which still neglect theh 2 and h 4 functions with their complicated many-body internal partition functions. From the results of comparing the FK virial EOS with OPAL and CEFF, the relative discrepancies of different thermodynamic quantities are roughly about 10 3 s 10 4 . Among those discrepancies, the comparisons of the relative differences of squared sound speed indicate that they are possible to be resolved with the aid of helioseismology com- bined with the numerical inversion methodology. Observations will ultimately decide if either OPAL or the FK virial EOS (up to the order 5=2 ) is a more accurate description of the physics of the solar interior. 142 Bibliography [1] R. Abe. 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Abstract (if available)
Abstract
This study deals with solar-physics applications of a recent equation-of-state formalism based on the formulation of the so-called ""Feynman-Kac (FK) representation"". This formalism leads to an exact virial expansion of the thermodynamic functions in powers of the particle densities of a Coulomb plasma (""exact"" here refers to the accuracy of the low-order virial coefficients of the expansion). By taking advantage of the exact and analytic form of this virial expansion, we can probe the thermodynamic properties of the solar interior, both in detail and to an accuracy that has so far not been achieved with currently available equation-of-state formalisms. For reacting plasmas, virial-expansion equation of state have an intrinsic problem when the plasma is less than fully ionized. Fortunately, in most parts of the Sun's interior, the plasma is almost completely ionized. Therefore, the FK virial equation of state can be applicable, but only in the deeper (and hotter) solar interior. The precise boundary of the domain of validity of the virial expansion depends on the elements included in the formalism. Since the computational effort increases tremendously with an increasing number of chemical elements, here, we choose to represent the heavier elements by a single one, oxygen. This approximation is reasonable for the following reason. In the Sun, the major heavy elements (that is, the elements other than H and He which together comprise more than 98% of the mass fraction) are C, N and O. They all have a similar effect in the equation of state. Then, the other elements, such as Fe and Ne, are so little abundant that they have altogether a very small effect in the equation of state (although they are very important in spectroscopy). Assuming the relevant constituents of the Sun to be H, He, and O, we have examined the effects on thermodynamic quantities from each of these components. With the aid of the FK formalism, we have studied the influences of the contributions to the respective different partial pressure. This not only helps us to understand the intrinsic challenge of applying the virial expansion, but also to analyze the various effects quantitatively. We have also tackled the aforementioned main limitation of the virial expansion, that is, its breakdown for relatively low temperatures, where the plasma is clearly less than fully ionized. For this, we have replaced the virial equation for these outer regions for the Sun by another, similar equation of state, the so-called ""scaled low temperature (SLT)"" expansion. Current versions of SLT are so far only for one chemical element, therefore restricted to the hydrogen part of the solar plasma. However, since hydrogen is the most abundant element in the Sun (more than 90% by number), SLT in its current form is already useful, if one adds the contributions of the remaining elements (He and heavy elements) using conventional equation-of-state formalisms. In such a procedure, an improved H part and a conventional treatment of the other elements will none the less lead to a net improvement of the overall equation of state compared to entirely conventional formalisms. Our results show (i) that the SLT EOS is not only consistent with the conventional reacting ideal-gas equation of state (the so-called ""Saha equation"") at low temperature and low densities, but (ii) that there is also a smooth matching with the FK virial expansion EOS up to the order of density ρ². Comparing our resulting combined equation of state (FK plus SLT) with currently popular equations of state, such as OPAL (developed at Livermore), the relative discrepancies of the relevant thermodynamic quantities are about 10⁻³ to 10⁻⁴. Since these differences are of the same order as the accuracy of helioseimic inversions for thermodynamic quantities, we have demonstrated that the FK equation of state extended by SLT will be a serious player in solar modeling. Further technical steps still have to be taken before the FK equation of state can be used in a fully-fledged helioseismic study. Our result demonstrates the feasibility of such an application, which will then tell whether the FK virial EOS is more accurate than the so-far best equation of state OPAL. With the exact nature of FK (devoid of several approximations used in OPAL), there is a promise of a successful outcome. However, at the moment, a final comparison is not possible because of the current state of the SLT EOS. Once it will be available for a multi-element plasmas, our product consisting of the SLT EOS for the solar exterior, and the FK virial EOS for the solar interior, will likely become the most accurate EOS for solar and stellar modeling.
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Creator
Liang, Yung-Ching
(author)
Core Title
On the Feynman path into the sun
School
College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
Degree Program
Physics
Publication Date
07/24/2013
Defense Date
05/31/2013
Publisher
University of Southern California
(original),
University of Southern California. Libraries
(digital)
Tag
CEFF,EOS,equation of state,Feynman-Kac path integral,MHD,OAI-PMH Harvest,OPAL,quantum Coulomb plasma,scaled low temperature expansion,Sun,virial expansion
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application/pdf
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English
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Electronically uploaded by the author
(provenance)
Advisor
Haas, Stephan (
committee chair
), Bickers, Nelson Eugene (
committee member
), Daeppen, Werner (
committee member
), Dappen, Werner (
committee member
), Nakano, Aiichiro (
committee member
), Wang, Joseph (
committee member
)
Creator Email
yclemail@gmail.com,yungchin@usc.edu
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-299212
Unique identifier
UC11293509
Identifier
etd-LiangYungC-1835.pdf (filename),usctheses-c3-299212 (legacy record id)
Legacy Identifier
etd-LiangYungC-1835.pdf
Dmrecord
299212
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Liang, Yung-Ching
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
CEFF
EOS
equation of state
Feynman-Kac path integral
MHD
OPAL
quantum Coulomb plasma
scaled low temperature expansion
virial expansion