THERMODYNAMIC COEFFICIENTS FOR STELLAR ATMOSPHERES AND PLASMA SPECTROSCOPY

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1 he Astrophysical Journal, 69:8 864, 2009 April 20 C he American Astronomical Society. All rights reserved. rinted in the U.S.A. doi: / x/69/2/8 HERMODYNAMIC COEFFICIENS FOR SELLAR AMOSHERES AND LASMA SECROSCOY O. Cardona 1, E. Simonneau 2, L. Crivellari 3 1 Instituto Nacional de Astrofísica, Óptica y Electrónica, L. E. Erro 1, M72840 onantzintla, uebla, Mexico; ocardona@inaoep.mx 2 Institut d Astrophysique de aris, CNRS, 98 bis Boulevard Arago, F7014 aris, France 3 Instituto de Astrofísica de Canarias, Vía Lactea s/n, E38200 La Laguna,enerife, Spain Dipartimento di Astronomia, Università degli Studi di rieste, Via G. B. iepolo, 11, I rieste, Italy; luc@iac.es Received 2008 September 11; accepted 2009 January 14; published 2009 April 3 ABSRAC A new method to obtain the thermodynamic coefficients in an analytic exact form for applications in radiative convective transport in thermodynamic equilibrium for stellar atmosphere plasma spectroscopy is developed. he resulting exact expressions are formed by sums of the degrees of ionization of the components of the system. herefore, they are easy to calculate are numerically stable. he method is developed initially for two elements: hydrogen helium, with constant partition functions, to show the simplicity of the procedure. he method is very easy to generalize to any number of elements for partition functions dependent on temperature pressure as well as for the first negative ions. he thermodynamic coefficients derived are the adiabatic gradient, the specific heats for constant volume constant pressure, the dilatation coefficient, the velocity of sound in the given medium. he derivation is based on the perfect gas Saha ionization equations. herefore, the results are valid for the regime where these equations are valid. hese results will be of value to astrophysics in stellar structure atmospheres, in geophysics, in planetary atmospheres, in plasma physics in spectroscopic analysis diagnostics. Key words: equation of state stars: atmospheres 1. INRODUCION In the radiative convective energy transport for stellar atmospheres plasma spectroscopy, it is necessary to obtain the principal coefficients that enter into the equations of radiative convective transport such as the opacities emissivities, the specific heats, the adiabatic gradient, the thermal dilatation coefficient, the speed of sound, the populations of the excited ionic states in the system, others. For these applications, it is necessary to have these coefficients for a great variety of physical, numerical, computational situations in an accessible simple form, to improve the stability speed of the numerical computations of the methods used to calculate the energy transport equations for any application. he thermodynamic coefficients expressed as analytical, exact equations are very useful for this purpose, especially if they are expressed in terms of the main variables of the system. For a gaseous medium, these variables are the temperature, the pressure, the total degree of ionization, particularly for partially ionized perfect gases in thermodynamic equilibrium at low densities moderate temperatures where the Boltzmann the Saha equations are valid Cox 1968; Eliezer et al. 2002; Griem 1964, 1997; Smirnov here has already been much wor on the subject of the evaluation of the so-called equation of state calculations, especially in astrophysical problems, such as stellar structure Cox 1968 stellar atmosphere model calculations Unsöld 19; Mihalas 1978, in planetary atmospheres laboratory plasma analysis Griem 1964, Most of these calculations are carried out entirely numerically Rouse 1961, 1962a, 1962b, including the derivatives. Since the conception of the equation of ionization Saha 1920, it has been applied together with the Boltzmann equation for the calculation of the thermodynamic parameters for stellar atmospheres. Unsöld 19 developed a procedure for approximately deriving the specific heats the adiabatic gradient for stellar atmospheres by calculating the derivatives numerically. Our method is a generalization based on that idea. A series of papers Hummer et al. 1988; Mihalas et al. 1988; Dappen et al. 1988; Mihalas et al developed a method completely different to ours for obtaining the thermodynamic coefficients for more general physical conditions using an approach based on statistical mechanics, the free-energy minimization method in the chemical picture. Roger 1986 has introduced the physical picture to minimize the free-energy. hose methods produce numerical results similar to those obtained with the simple method for the conditions where the Saha equation the ideal gas equation of state are valid far from the center of the stars where the radiation pressure can be neglected Cox he methods in these papers are based on sophisticated approximate partition functions their derivatives. Our method starts with an exact equation, the Saha ionization equation, thereby results in exact equations. he calculation of the partition functions their derivatives with respect to temperature pressure can be evaluated with any of the desired approximations Hummer et al. 1988; Cardona et al It should be emphasized that all the thermodynamic parameters are obtained fairly easily in a simple, exact, analytic form as a function of the main physical parameters of the system. We do not now any other mathematical procedures for deriving these coefficients in such a simple, analytic form from the Saha equation. All these thermodynamic parameters are necessary for calculating the coefficients of radiative convective transport for the spectroscopic diagnostics of different plasma types. he equation of state of a system in thermodynamic equilibrium is the function that interrelates the main variables that describe the physical state of the system. In a perfect gas, this function relates the temperature, the pressure the volume, or the total number of particles to each other of the thermodynamic system under study. his seemingly simple equation represents all the physical information of the state that has all the particles of the system. It is necessary to now the distributions of the particles in their different excited ionized states for obtaining the populations of these states the thermodynamic coefficients that are useful for many applications. We begin 8

2 86 CARDONA, SIMONNEAU, & CRIVELLARI Vol. 69 our procedure with the Saha equation, which is a closed exact expression. he method is directly applicable to the local thermodynamic equilibrium LE problems. For non-le NLE problems, the equations that relate the populations to the radiative collisional rates the statistical equilibrium equations should be solved through an iterative procedure where the starting values are the values in LE Mihalas 1978; Crivellari et al Also, the underlying material has to be considered in LE because there is no explicit equation of state for NLE. herefore, the method developed here is useful for many applications. In what follows we describe the basic thermodynamic equations definitions in Section 2. In Section 3, the method to obtain the fundamental results is developed. he numerical procedure that we follow to calculate the thermodynamic coefficient some numerical results are presented together with some graphs in Section 4. Finally, Section has the conclusions. 2. BASIC DEFINIIONS AND EQUAIONS his section describes the fundamental concepts, the most important definitions that will be used for the mathematical development of our procedure are presented. he equation of state for a perfect gas is N, 1 where is the pressure, is the temperature, N is the number density of gas particles of the thermodynamic system, B is the Boltzmann constant. In terms of the gas mass density ρ,this is ρ, 2 μm H where μ is the mean mass of the material or mean molecular weight, m H is the mass of the hydrogen atom. It is actually more appropriate to use the unit of atomic mass but, for simplicity, the mass of the hydrogen atom is used to obtain round numbers in some of the expressions. Considering that the total number of atoms is the sum of all the atomic species in different ionization states, we have N N N j, 3 j0 by definition the density of the material is given by ρ m N, 4 where N is the number of atoms per unit of volume of the chemical species with mass m, N j is the number of atoms per unit of volume in the ionization state j, where j 0, 1, 2,..., for the neutral atoms, ionized once, twice, so on; it can also tae negative values for the negative ions. he Saha equation between the two consecutive states of ionization j j 1is 3 N j1 n e 2πmB 2 2Uj1 e χ j N j h 2 U j where the U j U j1 are the partition functions for the two contiguous ionization states, χ j is the ionization potential of the j ionized state of the element. Defining the degree of ionization of the element by X j N j N, 6 we can write Equation as 3 X j1 n e 2πmB 2 U j1 2 e χ j X j h 2. 7 U j Considering that the total number of particle in the gas is composed of free electrons particles with nuclei, regardless of their state of ionization, we write N N N n e, 8 with N N as the number of particles with nuclei per unit volume n e as the number of free electrons per unit volume, we can write Equation 1 as 1 X n N V B, 9 where we have defined the total degree of ionization of the gas as X n e, 10 N N the total numbers of particles with nuclei in the volume V by n N. We assume that the gaseous system is neutral, that is, the free electrons are produced only by the ionization of atoms ions. herefore, n e N N jx j, 11 α where we have defined the abundance of the element by α N, 12 N N From Equations 8, 10, 11, we can obtain the relationship between the number of free electrons the total number of particles by the following expression: n e X N. 13 1X With expression 13 Equation 1, Equation 7 becomes X j1 X j X 1X 2πmB h U j1 U j 2 e χ j. 14 his is the fundamental equation for developing our method to find all the important thermodynamic parameters for stellar atmospheres other applications. However, we define the internal energy of a partially ionized gas as Unsöld 19 U X B α X j χ l n N. 1

3 No. 2, 2009 HERMODYNAMIC COEFFICIENS 87 One can obtain from the thermodynamic definition of the enthalpy that the enthalpy of a partially ionized gas can be expressed by H 2 1 X B α X j χ l n N. 16 For Equation 2 we need the mean molecular weight for the partially ionized mixture is defined as μ α A 1X, 17 where A m /m H is the atomic weight. he differential of the total ionization degrees as obtained from Equations is α j j, 18 the differential of the internal energy, Equation 1, is du 1 X 3 2 Bd 3 2 B α j χ l ]n N, 19 of the enthalpy, Equation 16, is dh 1 X 2 Bd 2 B α j χ l ]n N. 20 In the following section, we present the procedure for obtaining the differentials of the main variables in terms of d, d, as these are necessary for our method. 3. MAHEMAICAL MEHOD his section presents the procedure for obtaining the exact analytic expressions for the necessary thermodynamic parameters to describe a gas in local thermodynamic equilibrium. Starting from the equation of the conservation of the number of particles considering that there are no nuclear reactions in the system, for each chemical species we have N N j. 21 j0 We develop a method for the case of two elements, hydrogen helium. We do this for simplicity. Because it shows all the peculiarities of the procedure, we generalize our results afterward for all the elements that might be necessary in the application in which one might be interested. For hydrogen, from Equation 21, we have N 1 N 01 N Dividing Equation 22 byn 1 using Equation 6 gives X 01 X 11 1, 23 which when differentiated produces the following condition: Differentiating Equation 14, we obtain j1 X j X j X1 X d d 2 χ j X j1. 2 Equation 2 for the case of hydrogen becomes 01 d 11 X1 X 2 χ 01 d X 11, 26 that together with Equations becomes 11 1 X 11 X1 X d 2 χ ] 01 d X 11, 27 that is, the form required for our procedure. For helium, from Equation 21, we obtain N 2 N 02 N 12 N then X 02 X 12 X 22 1, As for hydrogen, now for helium we have from Equation 2, X 02 X1 X d 2 χ 02 d X 12, X 12 X1 X d 2 χ 12 d X Substituting Equation 31 into Equation 32 produces X 02 X1 X 2d 2 χ ] 12 d 2 χ 02 X 22, 33 substituting Equations into Equation 30 using Equation 29, we have 02 X 12 2X 22 X1 X d ] 2 χ 02 X 12 X 22 2 χ ] 12 d X 22 X

4 88 CARDONA, SIMONNEAU, & CRIVELLARI Vol. 69 his expression, in turn, is substituted into Equations to yield 12 1 X 12 X 22 X1 X d ] 2 χ 02 1 X 12 X 22 2 χ ] 12 d X 22 X 12, X 12 X 22 X1 X d ] 2 χ 02 1 X 12 X 22 2 χ ] 12 d 1 X 22 X hese again are in the desired form. he development for helium clearly shows the general procedure that one should follow to obtain the differentials of the degrees of ionization for other elements in terms of only d, d,. hese differentials of the ionization degree Equations 26, 27, 34, 3, 36 can be used in Equations 18, 19, 20 for the differentials of the total degree of ionization, internal energy, enthalpy. Substituting the equations of the differentials of the degrees of ionization of the different ions into Equation 18 produces where in this case S X X1 X S d S d, 37 S X S α 1 1 X 11 X 11 α 2 1 X 12 2X 22 X 12 α 2 22 X 12 2X 22 X 22, 38 S α 1 2 χ 02 α 2 2 χ 02 α 2 2 χ 02 1 X 11 X 11 1 X 12 2X 22 X 12 2 χ 12 ] 2 X 12 2X 22 X Solving for the differential of the total degree of ionization from Equation 37, we have S d S d 1 S X X1X. 40 his is the basic expression of our procedure can be used to obtain the different thermodynamic variables for the given applications physical processes. his completes the first part of our method. he second part is accomplished using some thermodynamic concepts the expression derived above. Substituting the differentials of the degrees of ionization of the elements into the differential of the internal energy, Equation 19, yields ] 3 d du Σ 2 Σ X X1 X ] d XΣ n N, 41 into the differential of the enthalpy, Equation 20, becomes ] dh 2 Σ X d Σ X1 X ] d 2 1 XΣ n N 42 with χ 01 Σ X Σ α 1 1 X 11 X 11 χ 02 α 2 1 X 12 2X 22 X 12 χ02 χ 12 α 2 2 X 12 2X 22 X 22, 43 Σ α 1 2 χ 01 χ 01 1 X 11 X 11 α 2 2 χ 02 1 X 12 2 χ 02 2 χ 12 α 2 2 χ 12 X 12 2 χ ] 12 1 X 22 ] χ 02 X 22 X 12 2 χ 02 χ 02 χ 12 X We can generalize Equations 38, 39, 43, 44 for any number of elements by following the same procedure: S X S α jx j j jx j, 4 S α Σ X Σ Σ α X j j α X j j X j 2 χ l jx j 2 χ l, B 46 jx j χ l, 47 1 jx j χ l. 48

5 No. 2, 2009 HERMODYNAMIC COEFFICIENS 89 Figure 1. Adiabatic temperature gradient d ln d ln for the mixture given in the text. ad When the pressure is constant, Equation 40 is transformed into X1 XS. 2 d X1 XS X Substituting the above into Equation 42 using the definition for the specific heat at constant pressure gives us C ] ] 2 1 XΣ 2 Σ X X1 X n N. d 3 For processes at constant volume, Equation 0 becomes d 1X d. 4 Figure 2. Same as Figure 1, in a different representation to show the convective ionization zones. Now it is necessary to include the most important physical processes. For adiabatic processes using the first law of thermodynamics, one has from Equation 9 d du V dv V, 49 1X dv V d. 0 Using Equations 9, 49, 0, we arrive at the expression d 1X du 1 Xn N d. 1 Substituting Equation into Equation 40, we obtain d V S S 1 S X X1X S. 1X Now using the definition for the specific heat at constant volume substituting Equation 6 into Equation 41, we obtain C V XΣ Σ d V ] 3 2 Σ X X1 X Σ ] 1X B n N. 6 o express the specific heats in grams, using the equation of state Equation 1 Equation 9, we replace n N in C C V by R/ B μ 0, where R is the gas constant μ 0 is the mean molecular weight of the unionized mixture, i.e., X 0, in Equation 17.

6 860 CARDONA, SIMONNEAU, & CRIVELLARI Vol. 69 Figure 3. Specific heat at constant pressure per gram c p /R of the material. For adiabatic processes, substituting Equation 41 into Equation 0 using Equation 40 gives d ad Figure 4. Specific heat at constant volume per gram c v /R of the material. 1 X Σ S 2 1 XΣ ] S 1 X Σ 1 S ] X X1X 2 Σ ], X S X1X 7 therefore we have from Equations 41 1 the adiabatic gradient d ln d ln ad 2 1 XΣ 1X Σ 2 Σ X X1X ]. d ad 8 From Equation 17, one derives dμ μ 1X, 9 using Equation 2, we find that d ln μ d ln X1 XS X1 XS X. 60 hen from the coefficient of thermal dilatation d ln ρ d ln μ 1 61 d ln d ln

7 No. 2, 2009 HERMODYNAMIC COEFFICIENS 861 Figure. Mean molecular weight μ of the material. Figure 6. Sound speed C s d dρ 1 2 ad in m s 1. one obtains d ln ρ d ln X1 X1 S S X X1 XS X. 62 Finally, from Equations 60, 41, 1, we have d ln μ d ln ad 2 Σ ] d ln 1X d ln ad 1 Σ ] 1X 2 Σ. 63 X] From this last equation together with Equation 9, the speed of sound in the medium can be obtained ρ d ln d ln μ C s 1 d ln d ln ad ad ] his mostly completes the whole list of necessary thermodynamic parameters for any calculation of the state equation. hese expressions are functions of the ionization degrees that can be easily obtained using the Saha equation. he derivatives are expressed analytically only with those parameters. For simplicity, we have considered the partition functions as constants; this is not true, because they depend on the

8 862 CARDONA, SIMONNEAU, & CRIVELLARI Vol. 69 Figure 7. Negative of the dilatation coefficient d ln ρ d ln. temperature the pressure due to the interaction of the particles with the medium that surrounds them. his interaction reduces the number of levels in the atoms, maing the partition function dependent on the gas pressure Inglis & eller 1939; Hummer et al. 1988; Cardona et al For this purpose, it is necessary to introduce into our equations the derivatives of the partition functions with respect to temperature pressure. In the Appendix, we consider the derivatives of the partition functions the contribution of the first negative ions of the elements to complete the procedure for obtaining the thermodynamic parameter. 4. NUMERICAL ROCEDURE AND RESULS When the number density of electrons n e the total number density of particles N as well as the temperature are given, the solution is simple. From Equation for each element ionization state the populations are obtained then the ionization degrees that enter into the previously obtained equations. When only the number of electron is given one can calculate the total number of particles using Equations, 6, 10, 13, therefore the thermodynamic coefficient derived above. When only the total number density of particles N is given the number of electrons must be found through an iterative procedure. Starting from Equation 11, which relates the number of electrons with the total number of particles through the total degree of ionization, considering that the material is, say, half-ionized, one obtains the initial number of electrons begins the iterative procedure. For each pressure temperature one obtains, from Equation 1, the total number of particles. he numbers of atoms of the different chemical species are obtained using the given abundances Equation 12. he partition functions the derivatives of the different ionic species are evaluated using the analytic expressions given by Hummer et al Cardona et al he populations of the different states of ionization of the elements of the mixture under study are calculated using the Saha equation. he number of electrons is calculated from the equation of charge conservation 11, the result is compared with the current value of the iteration to decide if the iterations are stopped for the desired precision. Once the populations the number density of electrons are obtained, the different ionization degrees as well as the total degree of ionization are calculated which will be used in the calculations of the desired thermodynamic coefficients using the formulas given previously. When the density ρ is given, one can obtain the total number density of atoms N N from Equations 6,, 10, 13, the number of atoms of the different elements are obtained from Equation 12 using the given abundances. he initial number of electron can be taen equal to half the total number of hydrogen atoms derived in the previous step. hen the above iterative procedure can be followed. Also, one can calculate the populations of the excited states of the different ionization states of the chemical elements in the given mixture using the Boltzmann equation. We calculate some thermodynamic coefficients from the analytic results derived above for a mixture of H, H,H,He, He,He,He, two mean atoms Z 1 Z 2 with four ionization states each. hese mean atoms are formed by the average with respect to the abundances of the most abundant elements with high ionization potentials, such as for C, N, O with moderate abundances low ionization potentials such as for Na, Mg, Ca, Al, Si for Z 1 Z 2.he abundances are obtained from Allen s Astrophysical Quantities Cox he calculations are carried out for temperatures that go from to K for pressures from 10 to 10 1 dyn cm 2. he behavior of the adiabatic gradient is shown in Figures 1 2, where one can see the different ionization zones. his gradient is important in the convective energy transport in gaseous media determines the location of the ionization zones when the gradient is less than 0.4 for material in stellar atmospheres. he variation of the specific heat at constant pressure with respect to temperature pressure is

9 No. 2, 2009 HERMODYNAMIC COEFFICIENS 863 shown in Figure 3. he changes of the specific heat at constant volume are given in Figure 4. Figure shows the changes of the mean molecular weight of the mixture. he velocity of sound is shown in Figure 6. Lastly, Figure 7 shows the coefficient of thermal dilatation. Also, one can obtain graphs for the number density of electrons the populations of all the ions of the different elements in the mixture.. CONCLUSIONS We have developed a method for obtaining the thermodynamic coefficients for a perfect gas in thermodynamic equilibrium in an exact analytic form. We start with the equation of state for perfect gases from the Saha equation, solving for the differential of the total ionization degree this produces a general expression that can be used for any type of physical process; those together with some results of thermodynamics are used to derive the thermodynamic coefficients, such the adiabatic gradient, the thermal dilatation coefficient, the velocity of sound. From the internal energy the enthalpy, the specific heats for processes at constant volume pressure are derived. All the derivatives are expressed in analytic forms that permit them to be evaluated without numerical difficulty. he expressions are given in terms of sums of the ionization degrees of the components of the system, herefore, they are very stable numerically, one can select certain components of the system delete them analytically to follow the behavior of some variables in the calculation. he results are comparable with other derivations of the thermodynamic coefficients, except that in some of them, they are calculated numerically; this is especially true for the derivatives. he procedure is very easy to generalize for molecules any other type of negative ionic species. AENDIX EMERAURE AND RESSURE-DEENDEN ARIION FUNCIONS AND NEGAIVE IONS For obtaining the thermodynamic parameters when the partition functions depend on temperature pressure, we differentiate Equation 14 to obtain instead of Equation 2 j1 j X j X1 X 1 duj1 U j1 d U j 2 χ j duj d U j1 ] d U j duj1 d X j1. duj d ] d A1 Following the procedure developed above, Equation 37 becomes where S S S X X1 X S α d S d, A2 S X S X, A3 X j j jx j U 0,U j ], A4 S S α X j j jx j Equation 41 is transformed into 3 du 2 Σ ] X Σ X1 X ] 3 d 2 1 XΣ n N, similarly Equation 42 dh 2 Σ ] X Σ X1 X ] d 2 1 XΣ n N, with U j,u 0 ]. d d A A6 A7 Σ X Σ X, A8 Σ Σ α X j U 0,U j ] X j U 0,U j ] χ l Σ Σ α X j U j,u 0 ] X j U j,u 0 ] χ l, where we have defined the square bracets as U m,u l ] 1 dum 1 U m d U l d U m,u l ] 1 U m dum d 1 U l dul dul d A9 A10, A11 A12 for any two indexes m l. When one taes into account the negative ions of hydrogen helium H He, the first negative ions of the other elements in the ionization equilibrium, then following the method described above we add to each of Equations A3 A, A8 A10 respectively, the following expressions S S X α X 1 jx j, α X 1 1 U 1,U 0 ] jx j, A13 A14

10 864 CARDONA, SIMONNEAU, & CRIVELLARI Vol. 69 S α X 1 2 χ All these expressions represent the main results of our 1 U 1,U 0 ] procedure for any mixture of chemical species ionization degrees. jx j, A1 REFERENCES Σ Σ Σ X α X 1 α X 1 1 χ l, X j χ l, X j X j U 1,U 0 ] α X 1 2 χ 1 U 1,U 0 ] χ l. X j A16 A17 A18 Cardona, O., Crivellari, L., & Simonneau, E. 200, Rev. Mex. Fis., 1, 476 Crivellari, L., Cardona, O., & Simonneau, E. 2002, Astrophysics, 4, 480 Cox, A. N. 2000, Allen s Atrophysical Quantities 4th ed.; New Yor: Springer Cox, J , rinciples of Stellar Structure New Yor: Gordon Breach Dappen, W., Mihalas, D., & Hummer, D. G. 1988, ApJ, 332, 361 Eliezer, S., Ghata, A., & Hora, H. 2002, Fundamentals of Equation of State Hacensac, NJ: World Scientific Griem, H. R. 1964, lasma Spectroscopy New Yor: McGraw-Hill Griem, H. R. 1997, rinciples of lasma Spectroscopy Cambridge: Cambridge Univ. ress Hummer, D. G., & Mihalas, D. 1988, ApJ, 331, 794 Inglis, D., & eller, E. 1939, ApJ, 90, 439 Mihalas, D. 1978, Stellar Atmospheres San Francisco, CA: Freeman Mihalas, D., Dappen, W., & Hummer, D. G. 1988, ApJ, 331, 81 Mihalas, D., Hummer, D. G., Mihalas, B. W., & Dappen, W. 1990, ApJ, 30, 300 Roger, F. J. 1986, ApJ, 310, 723 Rouse, C. A. 1961, ApJ, 134, 43 Rouse, C. A. 1962a, ApJ, 13, 99 Rouse, C. A. 1962b, ApJ, 13, 636 Saha, M. N. 1920, hil. Mag., 40, 47 Smirnov, B. M. 2001, hysics of Ionized Gases New Yor: Wiley Unsöld, A. 19, hysi der Sternatmosphären Berlin: Springer