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Radiative properties of stellar envelopes: Comparison of asteroseismic results to opacity calculations and measurements for iron and nickel
High Energy Density Physics 9 (2013) 473e479
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High Energy Density Physics journal homepage: www.elsevier.com/locate/hedp
Radiative properties of stellar envelopes: Comparison of asteroseismic results to opacity calculations and measurements for iron and nickel S. Turck-Chièze a, *, D. Gilles a, M. Le Pennec a, T. Blenski b, F. Thais b, S. Bastiani-Ceccotti c, C. Blancard d, M. Busquet e, T. Caillaud d, J. Colgan f, P. Cossé d, F. Delahaye h, J.E. Ducreta a,1, G. Faussurier d, C.J. Fontes g, F. Gilleron d, J. Guzik f, J.W. Harris i, D.P. Kilcrease f, G. Loisel b, 2, N.H. Magee f, J.C. Pain d, C. Reverdin d, V. Silvert d, B. Villette d, C.J. Zeippen h a
CEA/DSM/IRFU/SAp, UMR 7158, CE Saclay, 91190 Gif sur Yvette, France CEA/DSM/IRAMIS/SPAM, CE Saclay, 91190 Gif sur Yvette, France c LULI, Ecole Polytechnique, CNRS, CEA, UPMC, 91128 Palaiseau Cedex, France d CEA DAM/DIF, F-91297 Arpajon, France e ARTEP, Ellicott City, MD 21042, USA f Theoretical Division, LANL, Los Alamos NM 87545, USA g Computational Physics Div., LANL, Los Alamos NM 87545, USA h LERMA Observatoire de Paris, 5 Place Jules Janssen, 92195 Meudon, France i AWE, Reading, Berkshire RG7 4PR, UK b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 April 2013 Accepted 3 April 2013 Available online 16 April 2013
The international OPAC consortium consists of astrophysicists, plasma physicists and experimentalists who examine opacity calculations used in stellar physics that appear questionable and perform new calculations and laser experiments to understand the differences and improve the calculations. We report on iron and nickel opacities for envelopes of stars from 2 to 14 M1 and deliver our first conclusions concerning the reliability of the used calculations by illustrating the importance of the configuration interaction and of the completeness of the calculations for temperatures around 15e27 eV. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Stellar plasma Stellar opacity calculations Opacity measurements
1. Unresolved problems in asteroseismic observations Along the last decade, the description of the energy transfer in stars linked to plasma opacities, has been identified as a potential problem in stellar evolution. A summary of the questions was provided in Ref. [1]. In this paper, we give some first answers for the description of the radiative envelope of stars more massive than the Sun. We concentrate on the opacity peak found in stellar envelopes of 2e14 M1 around log T ¼ 5.25 (T ¼ 15.3 eV). This peak is dominated by the transition Dn ¼ 0, n ¼ 3 to n ¼ 3 of the iron and nickel constituents.
* Corresponding author. E-mail address: [email protected] (S. Turck-Chièze). 1 Present address: CELIA, Univ. Bordeaux 1, 351 Cours de la Libération, 33405 Talence, France. 2 Present address: Sandia National Laboratories, Albuquerque, NM 87185-1196, USA. 1574-1818/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.hedp.2013.04.004
That case corresponds to a problem, first expressed by A. Cox in the eighties [2,3] when there were difficulties interpreting the observations of Cepheid variables. The Livermore group [4,5] found an increase up to a factor of 4 in the photon cross section when compared to the extant Los Alamos library (LAOL), which improved the opacity calculation of iron. These theoretical results were then confirmed by related experiments [6,7] and by the Opacity project group using independent approaches and methods [14]. The attention of the stellar community is today mainly focused on the internal dynamics of Sun and stars. Seismic data of three space missions: SoHO for the Sun [8,9], COROT and KEPLER for other stars [10,11], will contribute to better introduce the different dynamical processes (rotation, magnetic field, waves). Slow Pulsating B and b Cephei stars are interesting pulsators that might put important constraints on rotation and magnetic fields, two key processes for massive supernovae type II stars, but they show much more complex patterns than in the case of the Sun. Presently, the 40 observed stars in Kepler field exhibit acoustic and gravity modes probably up to [ ¼ 20 due to the deformation of these stars, but
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each mode is difficult to label without ambiguity due to the forest of modes [12]. However, the primary function of these missions is to check their internal structure which is directly related to the microscopic physics. The accuracy of the seismic information clearly reveals some misunderstanding of the opacity tables as we illustrate in this section. Effectively the acoustic time s, the inverse of the acoustic mode period is directly related to the sound speed c(r) between two turning points rmin, rmax:
s¼
Zrmax rmin
dr Pðr; tÞ Tðr; tÞ with cðr; tÞ ¼ g1 ðr; tÞ f rðr; tÞ mðr; tÞ cðrÞ
(1)
where P(r,t), r(r,t), and m(r,t) are the pressure, temperature and mean molecular weight at a position r and time t, respectively. This information gives direct access to the thermodynamic properties of the plasma. The first difficulty comes from the iron group opacity peak that drives their pulsations. One observed stable modes which were considered like unstable modes in the theoretical predictions using the OPAL or OP tables [13,14]. Depending on the mass of the star, some of them are better predicted in using OP or OPAL tables. This fact suggests that some of these opacities could be badly determined in both tables [15]. Among different studies, Salmon and collaborators [16] have examined some B stars pulsators observed in both Magellanic clouds. If one believes the theoretical predictions, some observed modes might not be visible due to the reduction of the opacity bump at low Z where Z is the mass fraction of the elements above helium called the heavy element contribution. They suggest that an underestimate of the nickel opacity contribution in OP tables could explain that these modes are not predicted to be excited, they also note that the opacity nickel contribution is slightly displaced to higher temperature (log T ¼ 5.46) in comparison with the iron peak (log T ¼ 5.3). The analysis of the OPAC consortium confirms real problems with OPAL and OP tables. We compare OP calculations to new opacity calculations in Section 2 [17e19]. In Section 3 we give some interpretation to the differences between calculations [33]. Sections 4 and 5 are dedicated to comparisons with previous iron and recent nickel experiments to validate our analysis. Then we conclude and describe the next step. 2. Comparison of dedicated calculations of the consortium The iron opacity bump has received a lot of attention in the nineties to solve the Cepheids problem [4,5]. Nevertheless, large differences are still present between the tables delivered to astrophysicists and provided respectively by Ref. [13] for OPAL and Ref. [14] for OP, as mentioned by the academic OP consortium [20] and recalled in Ref. [21]. The OP tables include also the photon spectra to estimate the radiative acceleration and have been built for the description of the envelopes of stars. The differences between OP and OPAL have stimulated our interest because the asteroseismic community has shown that these differences complicate their understanding of SPB and b Cephei stars. Thus, our consortium OPAC has been formed to perform new calculations and measurements and to help to understand the existing differences and the directions of progress to deliver improved tables. A stellar plasma is mainly composed by hydrogen and helium. But, the Sun is the unique case where one measures at the photosphere by spectroscopy, the detailed relative contribution of species from carbon to uranium relative to hydrogen. After more than 20
years of study, Asplund and collaborators [22] have converged on the contributions of the main species as a fraction relative to the number of hydrogen atoms, C: 3.31 104, N: 8.32 105, O: 6.76 104, Fe: 3.2 105 and Ni: 1.78 106. From solar model constraints on age and luminosity, one deduces also the helium contribution to be 8.5 102. The solar internal relative contribution of these elements is generally only slightly changed by gravitational settling or radiative acceleration leading to changes of the order of 10%. Even if the heavy element contribution is very small, it is now well established that these elements, in particular iron and oxygen but not only, play a crucial role in the mean Rosseland opacity in different regions of stars. This is mainly due to their bound free and bound bound contributions, as they are generally partly ionized contrary to the lightest elements, H þ He, which are generally completely ionized, except very near the surface. So opacity tables for astrophysics are always calculated for some specific mixture and contain calculations for at least 21 elements from hydrogen to nickel. So, one considers always a mixture of species in stellar calculations generally based on the relative composition of the Sun. When one would like to discuss the validity of a calculation, one concentrates on a specific element. Iron and nickel dominate the total opacity in the range of temperature discussed in Section 1, so one needs to calculate some equivalent plasma conditions in term of distribution of ionization state for these specific elements. We follow the assumptions of Ref. [23], also used to pass from one mixture to another which is consistent with keeping the free electron number Ne constant:
Ne ¼ r
X Qi c
i
i
Ai
¼
r0 QFe cFe AFe
(2)
In Eq. (2) r is the stellar density and r0 is the corresponding density for the pure iron case, while Qi, ci and Ai are the ionization charge, the relative contribution in mass and the atomic weight of species i in the stellar plasma, respectively. This relationship is justified here as density stays always small. We first compared new opacity calculations to the OP calculations for conditions that are not far from equivalent conditions of the stellar cases and which are relevant for opacity measurements at the available laser facilities. Fig. 1 shows, the opacity iron spectrum and the distribution of ion charge at T ¼ 27 eV, r ¼ 3.4 103 g/cm3 compared to a typical astrophysical case corresponding to stellar envelopes. Here the electron number Ne is 3 1020 instead of the astrophysical case shown in Fig. 1 where it is located around 6 1018 (there exists some large variation of this number because it depends on the mass of the stars). Our first papers [17,18] give a brief description of the codes used in the OPAC consortium and shows clear differences between the spectrally-resolved opacity for mentioned conditions of Table 1. Then a more detailed study has been performed that also shows the ionic distribution and some details of the opacity spectra [19]. Table 1 and Fig. 2 summarize the comparisons of the consortium. The differences of the Rosseland mean values increase at low temperature. Further, they are larger for Rosseland than Planck mean values. The differences correspond to about 30% for iron and go up to a factor near 4 for nickel at 27 eV. Fig. 2 reveals the regions where the differences are the largest and illustrate two tendencies. STA [24], CASSANDRA [25] are pure statistical calculations, OPAS [26], SCO-RCG [27], LEDCOP [28] include lines computation, they all lead to similar results. The detailed OP calculations are strongly different from the others both for iron and nickel. So the next step consists in understanding the origins of such differences and determining the most appropriate codes for astrophysical calculations.
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Fig. 1. Comparison of the OP iron opacity spectrum (left) and the OP distribution of ionization (right) in stellar conditions: T ¼ 15 eV (178 000 K) and r ¼ 3.4 106 g/cm3and in reliable conditions for laboratory experiments T ¼ 27 eV (319 000 K) and r ¼ 3.4 103 g/cm3 (a grid point of the OP table). Superimposed is the weighting of the mean Rosseland value for T ¼ 15 eV, the maximum is around 4hn/kT.
3. The interaction of configuration in the iron and nickel spectra The code HULLAC [29,30] has been used to estimate the origin of the differences. This code is well adapted to check the role of the full Configuration Interaction (CI) which is included in the OP calculations, but is not present in any of the other codes. Its effect on spectral opacities is well known and several approaches have been developed to take this effect into account [29,31]. Full CI inclusion assumes that the diagonalization of the Hamiltonian is performed taking into account all atomic levels. The effects of CI on the spectral opacity are level energy shifts toward high energy and possible large change in the intensity distribution. When the number of levels is so large that full CI treatment becomes impossible, due to computational difficulties appearing for such complex system, it is possible to diagonalize the Hamiltonian only for block of levels (sub-CI modes). Among them, the most used submodes are CI in a non-relativistic configuration, CIinNRC submode and the CIinLAYZER submode. In the HULLAC-v9 code [32] used in the present analysis, it is possible to select the CI mode (no CI, CI in NRC, CI in LAYZER, full CI) and these possibilities help us to compare HULLAC results to other opacity code results [33e35]. The transitions Dn ¼ 0, n ¼ 3, and Dn ¼ 1, n ¼ 3, are the dominant contributions of the iron and nickel spectra around the maximum Rosseland mean value (typically 4hn/kT ¼ 108 eV in the present study). For these transitions CI effects are important with larger effect occurring for iron. In Fig. 3 left, we present a comparison for the iron case at a temperature of 27.3 eV and a density of 3.4 mg/cm3, hZi 8.5 for Table 1 Comparison of the Rosseland mean values for different codes studied by our consortium at a temperature of 27 eV and a density of 3.4 103 g/cm3. Codes
KR Fe 15.3 eV
KR Fe 27.3 eV
STA OP Cassandra OPAS SCO-RCG LEDCOP Codes
19 330 8160 27 650 18 906 18 209 18 578 KR Ni 15.3 eV
20 500 14 642 20 250 23 323 19 335 18 790 KR Ni 27.3 eV
STA OP Cassandra SCO-RCG
11 910 5910 19 370 11 695
20 180 5990 19 110 19 737
OPAS, SCO-RCG, OP and HULLAC-v9 (CIinNRC and full CI modes) codes. The agreement is very good between OP and HULLAC-v9 full CI. These two spectra significantly differ from the SCO-RCG, LEDCOP, OPAS and HULLAC-V9 CI in NRC results. When full CI is taken into account, lines are shifted and the shape of the spectrum changes noticeably. This effect plays a large role in the differences noted for the Rosseland mean values shown in Table 1, see also Table 2. Such important consequences can be explained by the fact that the shift is produced where the opacity cross section dominates near 70e80 eV. The two families of results correspond to full CI treatment (HULLAC-v9 full CI and OP) and CI in NRC treatment for all the other codes. In the nickel case, on the right side of Fig. 3, HULLAC-v9 full CI and OP opacities do not show the same agreement. Only the Dn ¼ 0, n ¼ 3 are included in HULLAC calculations. The CI interaction plays also a noticeable role but at lower energy between 50 and 60 eV. At higher photon energy, HULLAC spectrum follows the other codes and disagrees with OP spectrum. Consequently the impact of CI on the mean Rosseland values is not so large compared to the iron case. OP opacities are smaller than all the other calculations, so one can suspect an insufficient number of excited levels or some inappropriate extrapolation from the iron case since OP opacities for nickel are not calculated but extrapolated from iron ones. Full OP calculations for nickel are under way. The atomic description i.e. the number of configurations taken into account in the CI groups is of course very important. This point has already been noted in Ref. [32]. For the iron case, Fe VIII to Fe X gives the most important contributions to the boundebound opacity term. For each ion, e.g. Fe VIII, all levels of two states, the one under study and the next ionization stage, i.e. Fe VIII and Fe IX, must be included in the computation. The number of configurations (ground state and excited levels) is respectively 307 216, 210 631 and 117 427 and the number of configuration interaction coefficients varies from 3 065 545 067 to 493 527 664. This situation represents a real challenge (large matrices and huge number of CI coefficients) but precise results cannot be obtained otherwise. In the nickel case at 27 eV, ions Ni VIII to Ni XI are involved in the calculation. So the difficulty is even increased because a larger number of configurations must be included for these ions. In the OPAC collaboration, three opacity codes are now available to include full CI treatment: OP [14,20], HULLAC [32] and ATOMIC, this last code has been developed in Los Alamos [36]. Other codes (SCO-RCG, LEDCOP, OPAS) include only partial CI (CI in NRC). The
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Fig. 2. Comparison between opacity spectra for Fe and Ni obtained by different codes for conditions near from the conditions of laboratory experiments.
lack of a full CI calculation seems the main explanation for the Rosseland differences in the iron case, but not for the nickel one. Table 2 shows the results obtained for the mean Rosseland values for the two new detailed codes ATOMIC and HULLAC when they are available. One sees a rather close agreement with OP in the case of iron while the disagreement with OP for nickel is clear also for ATOMIC. In that case the calculation is really complex, and justifies more effort and more comparison.
5. Comparison of calculations to recent experiment on nickel
4. Comparison of calculations to past iron experiment In laboratory, one measures the transmission spectrum T(n) of the photons through a foil of thickness r and density r. This spectrum is directly linked to the opacity k(n). It is obtained in determining the ratio between the spectrum with and without the foil of the species we analyze:
TðnÞ ¼ expkðnÞrr ¼
In In;ref
On Fig. 4 right, one sees different options of the code HULLAC and their impact on the theoretical spectrum, see Refs. [33,34] for more details. These results confirm the previous conclusion concerning the influence of CI in the calculation of opacity. As the density is higher, the Fe ion stages involved have slightly less number of configurations so most of the n ¼ 3, Dn ¼ 1 transitions have been included in the opacity calculation.
(3)
Fig. 4 left [37], shows a comparison of the transmission spectrum of iron between OPAL and OP calculations and an experiment performed twenty years ago [7]. One sees clearly the effect of CI between OPAL and OP, that is, a displacement of the spectrum toward high energy. It is visible that OP provides a better fit to the experiment than OPAL, even though the calculation is not in total agreement with the experiment.
In parallel with the theoretical efforts, the experimental team of the OPAC consortium performed XUV domain experiments at LULI 2000 installation at the Ecole Polytechnique for the following elements: chromium, iron, nickel and copper at temperature near 27 eV and some 103 g/cm3. The principle of the experiment has been previously described in Ref. [17], see also Fig. 1 of [38] for a schematic of the experiment. Ref. [39] will provide more detailed information on it. The foil to study (iron, nickel, chromium) is heated by a nanosecond laser. In the two campaigns, we limit the temperature gradient inside the foil to be less than 10% by splitting the incident beam in two parts. These two beams heat two cavities and the foil is placed between them to receive a rather similar flux on both sides. The rapid expansion of the foil during the heating is also limited by placing the element to study between two thin samples of a low-Z material (here carbon), playing the role of inertial confinement to
Fig. 3. Importance of the role of the full interaction of configuration. Comparison between calculated opacity spectra for Fe (left) and Ni (right). HULLAC-v9 (different options) is compared to OP for iron and to other calculations. See Refs. [33e35].
S. Turck-Chièze et al. / High Energy Density Physics 9 (2013) 473e479 Table 2 Rosseland mean values for ATOMIC and HULLAC at a temperature of 27 eV and density of 3.4 103 g/cm3. These values are directly comparable to Table 1. ATOMIC calculations: intermediate coupling corresponds to complete configuration set of about 20,000 configurations per ion stage but no CI, in the full CI calculation, the configuration set is largely reduced. Codes
KR Fe 27.3 eV
Ni 27.3 eV
ATOMIC full CI (intermediate coupling) HULLAC full CI for n ¼ 3 to n ¼ 3
13 759 (17 777) 13 500
9 360 (16 938)
prevent a dramatically decrease of the density (see Ref. [40]). The success of these measurements depends on our ability to form a plasma in LTE with an ionization distribution equivalent to stellar conditions. This goal depends upon a good simulation of the entire experiment, which will provide the best time (typical 1.5 or 2 ns later) to probe the foil with a second picosecond laser, to get the appropriate sample temperature and density while minimizing the X-ray background due to the emission of the gold cavity. The photon spectrum was obtained with an XUV-ray spectrometer [41] specifically designed for this purpose, having a spectral range from 60 to 180 eV, the energy resolution varies from 1.5 to 3 eV. We measure also the radiative temperature of the X-rays in the cavities with a multi channel X-ray spectrometer [42]. A streak camera provides the time dependence of the different phases of the experiment and one discriminates between backlight signal corresponding to the picosecond laser and self-emission of the cavity. In adding some shots without foil, we estimate the ratio between the absorbed spectrum and the reference one. The new calculations show the high sensitivity of the calculations to the temperature and density of the foil so the exact conditions of the experiment need to be determined carefully. The results of the two campaigns performed around 15 eV and 25 eV are still in discussion, we present here a first comparison on nickel because up to now no spectrum of nickel has been obtained in the XUV domain, region which appears central today in the astrophysical context. The thermodynamic conditions are estimated to be T ¼ 24 3 eV and r ¼ 2 1 mg/cm3. Fig. 5 compares the obtained spectrum to OP, LEDCOP and SCO-RCG calculations. This spectrum has been normalized by the emission of the cavity between nickel absorption spectrum and reference spectrum, see Eq. (3). The figure confirms the theoretical conclusion of Section 3: the agreement is reasonable for LEDCOP
477
and SCO-RCG, showing no evident shift in the range of interest, i.e., 60e120 eV, that is around the maximum of the mean Rosseland value. One notices a better agreement with LEDCOP which has been calculated for conditions nearer to that of the experiment, the other ones are performed as in the previous sections at 27 eV and 3.4 mg/cm2. But the measured spectrum disagrees totally with OP confirming the problems with OP extrapolation for temperatures around 25 eV. A dedicated study of the configuration interaction using HULLAC shows that the change in the spectrum appears at lower energy (around 40e60 eV) for nickel (unfortunately not measured in the present experiments), so it has less effect on the mean Rosseland value. Fig. 6 left focuses on the region where the bound bound component is maximum, it shows that there is a reasonable agreement between HULLAC (n ¼ 3 to n ¼ 3) configuration interaction calculation and the experimental absorption feature observed in the spectrum around 80 eV, the hole is not displaced contrary to the OP case. The width of the hole (which corresponds to the bump in opacity) has not exactly the same value, it can come partly from the choice of the thermodynamic conditions which are not exactly the good ones and also from the contributions n ¼ 3, Dn ¼ 1, 2 not yet introduced in the HULLAC calculation. Fig. 6 right presenting the comparison with the ATOMIC calculation [36] for the same thermodynamical conditions seems to confirm this result. In the figure the horizontal error on the experimental spectrum corresponds to the resolution of the spectrometer and the vertical errors are not final because if the statistical errors are small, another error will come from granulation effect due to the luminance amplification and to the gold emission. A detailed comparison is under way where the real thermodynamical conditions will be calculated together with some integration on the temperature and density domains of the experiment. Moreover the resolution of the spectrometer will be included in the calculations for a better comparison.
6. Initial conclusions on the reliability of the opacity calculations for astrophysics Asteroseismology places strong constraints on the required accuracy of the predicted stellar models. So improved benchmark calculations and experiments are necessary to resolve known discrepancies that are currently creating ambiguities when the
Fig. 4. Left: Comparison between the iron experimental opacity spectrum from Ref. [7] and OP and OPAL spectra, extracted from Ref. [37]. Right: Comparison showing different options of the HULLAC calculations [34].
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Fig. 5. Comparison between nickel opacity spectrum taken at LULI 2000 and respectively OP, LEDCOP (26 eV, 2 mg/cm3) and SCO-RCG. The OP and SCO-RCG calculations are given at 27 eV, 3.4 mg/cm3.
Fig. 6. Zoom on the nickel spectrum performed at LULI 2000 at 24 3 eV and compared to the full CI HULLAC calculation of only the n ¼ 3 to n ¼ 3 transition (left) and to the full CI ATOMIC calculation (right). The vertical error bars are statistical ones, the horizontal ones show the spectral energy resolution. The calculations have been performed at 27 eV and 3.4 mg/cm3.
relevant opacity tables are employed to study the stellar interior of low mass stars or the envelopes of more massive stars. The present report is restricted to thermodynamic conditions found in stellar radiative envelopes. Our consortium OPAC provides new theoretical calculations and laser-based experiments to validate the microscopic physics used in stellar evolution and to improve the absorption spectra of stellar plasma. We show that, in the case of iron for temperature around 24e27 eV and density of several mg/cm3, the OP calculations appear clearly superior to the calculations neglecting part of the interaction of configurations like LEDCOP, OPAS, SCO-RCG. The OPAL results are in broad agreement with these codes, because it includes spin-orbit coupling in atomic data calculations [43]. This fact has been shown in using the full relativistic HULLAC code and confirmed by the semi relativistic ATOMIC calculations. In the present study, we show that calculations that include configuration interaction are required in order to obtain accurate mean Rosseland values. This conclusion is also in agreement with the first experiment on iron realized 20 years ago. On the other hand, in the case of nickel for temperature around 24e27 eV, the configuration interaction has an effect at rather low
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energy. Moreover we notice that OP extrapolated calculations for nickel do not agree with HULLAC calculations and other calculations which all agree in the spectral region near the peak of the Rosseland mean weighting function. This fact is also confirmed by the recent measurement that we have done, which better agree with LEDCOP, SCO-RCG, OPAS, HULLAC and ATOMIC than with OP calculations. So a significant increase compared to OP predictions is awaited in nickel opacity mean Rosseland values. The difficulty in that case is to perform a sufficiently complete calculation which is a real challenge for new computers. These results could explain why for some envelopes of stars, the OPAL tables were preferred and why the OP tables for other cases. One better understands also why the difficulties of interpretation appear on the high temperature side of the “iron” opacity bump. The present conclusions concern a limited set of plasma conditions and must be confirmed by the final detailed analysis of the two campaigns for nickel, chromium, iron and copper realized at LULI 2000 and for calculations performed precisely for the experimental conditions. Then, further calculations will focus on the real astrophysical conditions to see the impact of this new computational effort for stellar physics. It is obvious that statistical calculations are less time consuming than detailed calculations so it is important to establish the thermodynamical conditions for which detailed calculations are absolutely necessary. This general study will lead to updated tables for stellar physics where it is necessary. References [1] [2] [3] [4] [5] [6] [7] [8]
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Radiative properties of stellar envelopes: Comparison of asteroseismic results to opacity calculations and measurements for iron and nickel S. Turck-Chièze a, *, D. Gilles a, M. Le Pennec a, T. Blenski b, F. Thais b, S. Bastiani-Ceccotti c, C. Blancard d, M. Busquet e, T. Caillaud d, J. Colgan f, P. Cossé d, F. Delahaye h, J.E. Ducreta a,1, G. Faussurier d, C.J. Fontes g, F. Gilleron d, J. Guzik f, J.W. Harris i, D.P. Kilcrease f, G. Loisel b, 2, N.H. Magee f, J.C. Pain d, C. Reverdin d, V. Silvert d, B. Villette d, C.J. Zeippen h a
CEA/DSM/IRFU/SAp, UMR 7158, CE Saclay, 91190 Gif sur Yvette, France CEA/DSM/IRAMIS/SPAM, CE Saclay, 91190 Gif sur Yvette, France c LULI, Ecole Polytechnique, CNRS, CEA, UPMC, 91128 Palaiseau Cedex, France d CEA DAM/DIF, F-91297 Arpajon, France e ARTEP, Ellicott City, MD 21042, USA f Theoretical Division, LANL, Los Alamos NM 87545, USA g Computational Physics Div., LANL, Los Alamos NM 87545, USA h LERMA Observatoire de Paris, 5 Place Jules Janssen, 92195 Meudon, France i AWE, Reading, Berkshire RG7 4PR, UK b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 3 April 2013 Accepted 3 April 2013 Available online 16 April 2013
The international OPAC consortium consists of astrophysicists, plasma physicists and experimentalists who examine opacity calculations used in stellar physics that appear questionable and perform new calculations and laser experiments to understand the differences and improve the calculations. We report on iron and nickel opacities for envelopes of stars from 2 to 14 M1 and deliver our first conclusions concerning the reliability of the used calculations by illustrating the importance of the configuration interaction and of the completeness of the calculations for temperatures around 15e27 eV. Ó 2013 Elsevier B.V. All rights reserved.
Keywords: Stellar plasma Stellar opacity calculations Opacity measurements
1. Unresolved problems in asteroseismic observations Along the last decade, the description of the energy transfer in stars linked to plasma opacities, has been identified as a potential problem in stellar evolution. A summary of the questions was provided in Ref. [1]. In this paper, we give some first answers for the description of the radiative envelope of stars more massive than the Sun. We concentrate on the opacity peak found in stellar envelopes of 2e14 M1 around log T ¼ 5.25 (T ¼ 15.3 eV). This peak is dominated by the transition Dn ¼ 0, n ¼ 3 to n ¼ 3 of the iron and nickel constituents.
* Corresponding author. E-mail address: [email protected] (S. Turck-Chièze). 1 Present address: CELIA, Univ. Bordeaux 1, 351 Cours de la Libération, 33405 Talence, France. 2 Present address: Sandia National Laboratories, Albuquerque, NM 87185-1196, USA. 1574-1818/$ e see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.hedp.2013.04.004
That case corresponds to a problem, first expressed by A. Cox in the eighties [2,3] when there were difficulties interpreting the observations of Cepheid variables. The Livermore group [4,5] found an increase up to a factor of 4 in the photon cross section when compared to the extant Los Alamos library (LAOL), which improved the opacity calculation of iron. These theoretical results were then confirmed by related experiments [6,7] and by the Opacity project group using independent approaches and methods [14]. The attention of the stellar community is today mainly focused on the internal dynamics of Sun and stars. Seismic data of three space missions: SoHO for the Sun [8,9], COROT and KEPLER for other stars [10,11], will contribute to better introduce the different dynamical processes (rotation, magnetic field, waves). Slow Pulsating B and b Cephei stars are interesting pulsators that might put important constraints on rotation and magnetic fields, two key processes for massive supernovae type II stars, but they show much more complex patterns than in the case of the Sun. Presently, the 40 observed stars in Kepler field exhibit acoustic and gravity modes probably up to [ ¼ 20 due to the deformation of these stars, but
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each mode is difficult to label without ambiguity due to the forest of modes [12]. However, the primary function of these missions is to check their internal structure which is directly related to the microscopic physics. The accuracy of the seismic information clearly reveals some misunderstanding of the opacity tables as we illustrate in this section. Effectively the acoustic time s, the inverse of the acoustic mode period is directly related to the sound speed c(r) between two turning points rmin, rmax:
s¼
Zrmax rmin
dr Pðr; tÞ Tðr; tÞ with cðr; tÞ ¼ g1 ðr; tÞ f rðr; tÞ mðr; tÞ cðrÞ
(1)
where P(r,t), r(r,t), and m(r,t) are the pressure, temperature and mean molecular weight at a position r and time t, respectively. This information gives direct access to the thermodynamic properties of the plasma. The first difficulty comes from the iron group opacity peak that drives their pulsations. One observed stable modes which were considered like unstable modes in the theoretical predictions using the OPAL or OP tables [13,14]. Depending on the mass of the star, some of them are better predicted in using OP or OPAL tables. This fact suggests that some of these opacities could be badly determined in both tables [15]. Among different studies, Salmon and collaborators [16] have examined some B stars pulsators observed in both Magellanic clouds. If one believes the theoretical predictions, some observed modes might not be visible due to the reduction of the opacity bump at low Z where Z is the mass fraction of the elements above helium called the heavy element contribution. They suggest that an underestimate of the nickel opacity contribution in OP tables could explain that these modes are not predicted to be excited, they also note that the opacity nickel contribution is slightly displaced to higher temperature (log T ¼ 5.46) in comparison with the iron peak (log T ¼ 5.3). The analysis of the OPAC consortium confirms real problems with OPAL and OP tables. We compare OP calculations to new opacity calculations in Section 2 [17e19]. In Section 3 we give some interpretation to the differences between calculations [33]. Sections 4 and 5 are dedicated to comparisons with previous iron and recent nickel experiments to validate our analysis. Then we conclude and describe the next step. 2. Comparison of dedicated calculations of the consortium The iron opacity bump has received a lot of attention in the nineties to solve the Cepheids problem [4,5]. Nevertheless, large differences are still present between the tables delivered to astrophysicists and provided respectively by Ref. [13] for OPAL and Ref. [14] for OP, as mentioned by the academic OP consortium [20] and recalled in Ref. [21]. The OP tables include also the photon spectra to estimate the radiative acceleration and have been built for the description of the envelopes of stars. The differences between OP and OPAL have stimulated our interest because the asteroseismic community has shown that these differences complicate their understanding of SPB and b Cephei stars. Thus, our consortium OPAC has been formed to perform new calculations and measurements and to help to understand the existing differences and the directions of progress to deliver improved tables. A stellar plasma is mainly composed by hydrogen and helium. But, the Sun is the unique case where one measures at the photosphere by spectroscopy, the detailed relative contribution of species from carbon to uranium relative to hydrogen. After more than 20
years of study, Asplund and collaborators [22] have converged on the contributions of the main species as a fraction relative to the number of hydrogen atoms, C: 3.31 104, N: 8.32 105, O: 6.76 104, Fe: 3.2 105 and Ni: 1.78 106. From solar model constraints on age and luminosity, one deduces also the helium contribution to be 8.5 102. The solar internal relative contribution of these elements is generally only slightly changed by gravitational settling or radiative acceleration leading to changes of the order of 10%. Even if the heavy element contribution is very small, it is now well established that these elements, in particular iron and oxygen but not only, play a crucial role in the mean Rosseland opacity in different regions of stars. This is mainly due to their bound free and bound bound contributions, as they are generally partly ionized contrary to the lightest elements, H þ He, which are generally completely ionized, except very near the surface. So opacity tables for astrophysics are always calculated for some specific mixture and contain calculations for at least 21 elements from hydrogen to nickel. So, one considers always a mixture of species in stellar calculations generally based on the relative composition of the Sun. When one would like to discuss the validity of a calculation, one concentrates on a specific element. Iron and nickel dominate the total opacity in the range of temperature discussed in Section 1, so one needs to calculate some equivalent plasma conditions in term of distribution of ionization state for these specific elements. We follow the assumptions of Ref. [23], also used to pass from one mixture to another which is consistent with keeping the free electron number Ne constant:
Ne ¼ r
X Qi c
i
i
Ai
¼
r0 QFe cFe AFe
(2)
In Eq. (2) r is the stellar density and r0 is the corresponding density for the pure iron case, while Qi, ci and Ai are the ionization charge, the relative contribution in mass and the atomic weight of species i in the stellar plasma, respectively. This relationship is justified here as density stays always small. We first compared new opacity calculations to the OP calculations for conditions that are not far from equivalent conditions of the stellar cases and which are relevant for opacity measurements at the available laser facilities. Fig. 1 shows, the opacity iron spectrum and the distribution of ion charge at T ¼ 27 eV, r ¼ 3.4 103 g/cm3 compared to a typical astrophysical case corresponding to stellar envelopes. Here the electron number Ne is 3 1020 instead of the astrophysical case shown in Fig. 1 where it is located around 6 1018 (there exists some large variation of this number because it depends on the mass of the stars). Our first papers [17,18] give a brief description of the codes used in the OPAC consortium and shows clear differences between the spectrally-resolved opacity for mentioned conditions of Table 1. Then a more detailed study has been performed that also shows the ionic distribution and some details of the opacity spectra [19]. Table 1 and Fig. 2 summarize the comparisons of the consortium. The differences of the Rosseland mean values increase at low temperature. Further, they are larger for Rosseland than Planck mean values. The differences correspond to about 30% for iron and go up to a factor near 4 for nickel at 27 eV. Fig. 2 reveals the regions where the differences are the largest and illustrate two tendencies. STA [24], CASSANDRA [25] are pure statistical calculations, OPAS [26], SCO-RCG [27], LEDCOP [28] include lines computation, they all lead to similar results. The detailed OP calculations are strongly different from the others both for iron and nickel. So the next step consists in understanding the origins of such differences and determining the most appropriate codes for astrophysical calculations.
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Fig. 1. Comparison of the OP iron opacity spectrum (left) and the OP distribution of ionization (right) in stellar conditions: T ¼ 15 eV (178 000 K) and r ¼ 3.4 106 g/cm3and in reliable conditions for laboratory experiments T ¼ 27 eV (319 000 K) and r ¼ 3.4 103 g/cm3 (a grid point of the OP table). Superimposed is the weighting of the mean Rosseland value for T ¼ 15 eV, the maximum is around 4hn/kT.
3. The interaction of configuration in the iron and nickel spectra The code HULLAC [29,30] has been used to estimate the origin of the differences. This code is well adapted to check the role of the full Configuration Interaction (CI) which is included in the OP calculations, but is not present in any of the other codes. Its effect on spectral opacities is well known and several approaches have been developed to take this effect into account [29,31]. Full CI inclusion assumes that the diagonalization of the Hamiltonian is performed taking into account all atomic levels. The effects of CI on the spectral opacity are level energy shifts toward high energy and possible large change in the intensity distribution. When the number of levels is so large that full CI treatment becomes impossible, due to computational difficulties appearing for such complex system, it is possible to diagonalize the Hamiltonian only for block of levels (sub-CI modes). Among them, the most used submodes are CI in a non-relativistic configuration, CIinNRC submode and the CIinLAYZER submode. In the HULLAC-v9 code [32] used in the present analysis, it is possible to select the CI mode (no CI, CI in NRC, CI in LAYZER, full CI) and these possibilities help us to compare HULLAC results to other opacity code results [33e35]. The transitions Dn ¼ 0, n ¼ 3, and Dn ¼ 1, n ¼ 3, are the dominant contributions of the iron and nickel spectra around the maximum Rosseland mean value (typically 4hn/kT ¼ 108 eV in the present study). For these transitions CI effects are important with larger effect occurring for iron. In Fig. 3 left, we present a comparison for the iron case at a temperature of 27.3 eV and a density of 3.4 mg/cm3, hZi 8.5 for Table 1 Comparison of the Rosseland mean values for different codes studied by our consortium at a temperature of 27 eV and a density of 3.4 103 g/cm3. Codes
KR Fe 15.3 eV
KR Fe 27.3 eV
STA OP Cassandra OPAS SCO-RCG LEDCOP Codes
19 330 8160 27 650 18 906 18 209 18 578 KR Ni 15.3 eV
20 500 14 642 20 250 23 323 19 335 18 790 KR Ni 27.3 eV
STA OP Cassandra SCO-RCG
11 910 5910 19 370 11 695
20 180 5990 19 110 19 737
OPAS, SCO-RCG, OP and HULLAC-v9 (CIinNRC and full CI modes) codes. The agreement is very good between OP and HULLAC-v9 full CI. These two spectra significantly differ from the SCO-RCG, LEDCOP, OPAS and HULLAC-V9 CI in NRC results. When full CI is taken into account, lines are shifted and the shape of the spectrum changes noticeably. This effect plays a large role in the differences noted for the Rosseland mean values shown in Table 1, see also Table 2. Such important consequences can be explained by the fact that the shift is produced where the opacity cross section dominates near 70e80 eV. The two families of results correspond to full CI treatment (HULLAC-v9 full CI and OP) and CI in NRC treatment for all the other codes. In the nickel case, on the right side of Fig. 3, HULLAC-v9 full CI and OP opacities do not show the same agreement. Only the Dn ¼ 0, n ¼ 3 are included in HULLAC calculations. The CI interaction plays also a noticeable role but at lower energy between 50 and 60 eV. At higher photon energy, HULLAC spectrum follows the other codes and disagrees with OP spectrum. Consequently the impact of CI on the mean Rosseland values is not so large compared to the iron case. OP opacities are smaller than all the other calculations, so one can suspect an insufficient number of excited levels or some inappropriate extrapolation from the iron case since OP opacities for nickel are not calculated but extrapolated from iron ones. Full OP calculations for nickel are under way. The atomic description i.e. the number of configurations taken into account in the CI groups is of course very important. This point has already been noted in Ref. [32]. For the iron case, Fe VIII to Fe X gives the most important contributions to the boundebound opacity term. For each ion, e.g. Fe VIII, all levels of two states, the one under study and the next ionization stage, i.e. Fe VIII and Fe IX, must be included in the computation. The number of configurations (ground state and excited levels) is respectively 307 216, 210 631 and 117 427 and the number of configuration interaction coefficients varies from 3 065 545 067 to 493 527 664. This situation represents a real challenge (large matrices and huge number of CI coefficients) but precise results cannot be obtained otherwise. In the nickel case at 27 eV, ions Ni VIII to Ni XI are involved in the calculation. So the difficulty is even increased because a larger number of configurations must be included for these ions. In the OPAC collaboration, three opacity codes are now available to include full CI treatment: OP [14,20], HULLAC [32] and ATOMIC, this last code has been developed in Los Alamos [36]. Other codes (SCO-RCG, LEDCOP, OPAS) include only partial CI (CI in NRC). The
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Fig. 2. Comparison between opacity spectra for Fe and Ni obtained by different codes for conditions near from the conditions of laboratory experiments.
lack of a full CI calculation seems the main explanation for the Rosseland differences in the iron case, but not for the nickel one. Table 2 shows the results obtained for the mean Rosseland values for the two new detailed codes ATOMIC and HULLAC when they are available. One sees a rather close agreement with OP in the case of iron while the disagreement with OP for nickel is clear also for ATOMIC. In that case the calculation is really complex, and justifies more effort and more comparison.
5. Comparison of calculations to recent experiment on nickel
4. Comparison of calculations to past iron experiment In laboratory, one measures the transmission spectrum T(n) of the photons through a foil of thickness r and density r. This spectrum is directly linked to the opacity k(n). It is obtained in determining the ratio between the spectrum with and without the foil of the species we analyze:
TðnÞ ¼ expkðnÞrr ¼
In In;ref
On Fig. 4 right, one sees different options of the code HULLAC and their impact on the theoretical spectrum, see Refs. [33,34] for more details. These results confirm the previous conclusion concerning the influence of CI in the calculation of opacity. As the density is higher, the Fe ion stages involved have slightly less number of configurations so most of the n ¼ 3, Dn ¼ 1 transitions have been included in the opacity calculation.
(3)
Fig. 4 left [37], shows a comparison of the transmission spectrum of iron between OPAL and OP calculations and an experiment performed twenty years ago [7]. One sees clearly the effect of CI between OPAL and OP, that is, a displacement of the spectrum toward high energy. It is visible that OP provides a better fit to the experiment than OPAL, even though the calculation is not in total agreement with the experiment.
In parallel with the theoretical efforts, the experimental team of the OPAC consortium performed XUV domain experiments at LULI 2000 installation at the Ecole Polytechnique for the following elements: chromium, iron, nickel and copper at temperature near 27 eV and some 103 g/cm3. The principle of the experiment has been previously described in Ref. [17], see also Fig. 1 of [38] for a schematic of the experiment. Ref. [39] will provide more detailed information on it. The foil to study (iron, nickel, chromium) is heated by a nanosecond laser. In the two campaigns, we limit the temperature gradient inside the foil to be less than 10% by splitting the incident beam in two parts. These two beams heat two cavities and the foil is placed between them to receive a rather similar flux on both sides. The rapid expansion of the foil during the heating is also limited by placing the element to study between two thin samples of a low-Z material (here carbon), playing the role of inertial confinement to
Fig. 3. Importance of the role of the full interaction of configuration. Comparison between calculated opacity spectra for Fe (left) and Ni (right). HULLAC-v9 (different options) is compared to OP for iron and to other calculations. See Refs. [33e35].
S. Turck-Chièze et al. / High Energy Density Physics 9 (2013) 473e479 Table 2 Rosseland mean values for ATOMIC and HULLAC at a temperature of 27 eV and density of 3.4 103 g/cm3. These values are directly comparable to Table 1. ATOMIC calculations: intermediate coupling corresponds to complete configuration set of about 20,000 configurations per ion stage but no CI, in the full CI calculation, the configuration set is largely reduced. Codes
KR Fe 27.3 eV
Ni 27.3 eV
ATOMIC full CI (intermediate coupling) HULLAC full CI for n ¼ 3 to n ¼ 3
13 759 (17 777) 13 500
9 360 (16 938)
prevent a dramatically decrease of the density (see Ref. [40]). The success of these measurements depends on our ability to form a plasma in LTE with an ionization distribution equivalent to stellar conditions. This goal depends upon a good simulation of the entire experiment, which will provide the best time (typical 1.5 or 2 ns later) to probe the foil with a second picosecond laser, to get the appropriate sample temperature and density while minimizing the X-ray background due to the emission of the gold cavity. The photon spectrum was obtained with an XUV-ray spectrometer [41] specifically designed for this purpose, having a spectral range from 60 to 180 eV, the energy resolution varies from 1.5 to 3 eV. We measure also the radiative temperature of the X-rays in the cavities with a multi channel X-ray spectrometer [42]. A streak camera provides the time dependence of the different phases of the experiment and one discriminates between backlight signal corresponding to the picosecond laser and self-emission of the cavity. In adding some shots without foil, we estimate the ratio between the absorbed spectrum and the reference one. The new calculations show the high sensitivity of the calculations to the temperature and density of the foil so the exact conditions of the experiment need to be determined carefully. The results of the two campaigns performed around 15 eV and 25 eV are still in discussion, we present here a first comparison on nickel because up to now no spectrum of nickel has been obtained in the XUV domain, region which appears central today in the astrophysical context. The thermodynamic conditions are estimated to be T ¼ 24 3 eV and r ¼ 2 1 mg/cm3. Fig. 5 compares the obtained spectrum to OP, LEDCOP and SCO-RCG calculations. This spectrum has been normalized by the emission of the cavity between nickel absorption spectrum and reference spectrum, see Eq. (3). The figure confirms the theoretical conclusion of Section 3: the agreement is reasonable for LEDCOP
477
and SCO-RCG, showing no evident shift in the range of interest, i.e., 60e120 eV, that is around the maximum of the mean Rosseland value. One notices a better agreement with LEDCOP which has been calculated for conditions nearer to that of the experiment, the other ones are performed as in the previous sections at 27 eV and 3.4 mg/cm2. But the measured spectrum disagrees totally with OP confirming the problems with OP extrapolation for temperatures around 25 eV. A dedicated study of the configuration interaction using HULLAC shows that the change in the spectrum appears at lower energy (around 40e60 eV) for nickel (unfortunately not measured in the present experiments), so it has less effect on the mean Rosseland value. Fig. 6 left focuses on the region where the bound bound component is maximum, it shows that there is a reasonable agreement between HULLAC (n ¼ 3 to n ¼ 3) configuration interaction calculation and the experimental absorption feature observed in the spectrum around 80 eV, the hole is not displaced contrary to the OP case. The width of the hole (which corresponds to the bump in opacity) has not exactly the same value, it can come partly from the choice of the thermodynamic conditions which are not exactly the good ones and also from the contributions n ¼ 3, Dn ¼ 1, 2 not yet introduced in the HULLAC calculation. Fig. 6 right presenting the comparison with the ATOMIC calculation [36] for the same thermodynamical conditions seems to confirm this result. In the figure the horizontal error on the experimental spectrum corresponds to the resolution of the spectrometer and the vertical errors are not final because if the statistical errors are small, another error will come from granulation effect due to the luminance amplification and to the gold emission. A detailed comparison is under way where the real thermodynamical conditions will be calculated together with some integration on the temperature and density domains of the experiment. Moreover the resolution of the spectrometer will be included in the calculations for a better comparison.
6. Initial conclusions on the reliability of the opacity calculations for astrophysics Asteroseismology places strong constraints on the required accuracy of the predicted stellar models. So improved benchmark calculations and experiments are necessary to resolve known discrepancies that are currently creating ambiguities when the
Fig. 4. Left: Comparison between the iron experimental opacity spectrum from Ref. [7] and OP and OPAL spectra, extracted from Ref. [37]. Right: Comparison showing different options of the HULLAC calculations [34].
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Fig. 5. Comparison between nickel opacity spectrum taken at LULI 2000 and respectively OP, LEDCOP (26 eV, 2 mg/cm3) and SCO-RCG. The OP and SCO-RCG calculations are given at 27 eV, 3.4 mg/cm3.
Fig. 6. Zoom on the nickel spectrum performed at LULI 2000 at 24 3 eV and compared to the full CI HULLAC calculation of only the n ¼ 3 to n ¼ 3 transition (left) and to the full CI ATOMIC calculation (right). The vertical error bars are statistical ones, the horizontal ones show the spectral energy resolution. The calculations have been performed at 27 eV and 3.4 mg/cm3.
relevant opacity tables are employed to study the stellar interior of low mass stars or the envelopes of more massive stars. The present report is restricted to thermodynamic conditions found in stellar radiative envelopes. Our consortium OPAC provides new theoretical calculations and laser-based experiments to validate the microscopic physics used in stellar evolution and to improve the absorption spectra of stellar plasma. We show that, in the case of iron for temperature around 24e27 eV and density of several mg/cm3, the OP calculations appear clearly superior to the calculations neglecting part of the interaction of configurations like LEDCOP, OPAS, SCO-RCG. The OPAL results are in broad agreement with these codes, because it includes spin-orbit coupling in atomic data calculations [43]. This fact has been shown in using the full relativistic HULLAC code and confirmed by the semi relativistic ATOMIC calculations. In the present study, we show that calculations that include configuration interaction are required in order to obtain accurate mean Rosseland values. This conclusion is also in agreement with the first experiment on iron realized 20 years ago. On the other hand, in the case of nickel for temperature around 24e27 eV, the configuration interaction has an effect at rather low
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energy. Moreover we notice that OP extrapolated calculations for nickel do not agree with HULLAC calculations and other calculations which all agree in the spectral region near the peak of the Rosseland mean weighting function. This fact is also confirmed by the recent measurement that we have done, which better agree with LEDCOP, SCO-RCG, OPAS, HULLAC and ATOMIC than with OP calculations. So a significant increase compared to OP predictions is awaited in nickel opacity mean Rosseland values. The difficulty in that case is to perform a sufficiently complete calculation which is a real challenge for new computers. These results could explain why for some envelopes of stars, the OPAL tables were preferred and why the OP tables for other cases. One better understands also why the difficulties of interpretation appear on the high temperature side of the “iron” opacity bump. The present conclusions concern a limited set of plasma conditions and must be confirmed by the final detailed analysis of the two campaigns for nickel, chromium, iron and copper realized at LULI 2000 and for calculations performed precisely for the experimental conditions. Then, further calculations will focus on the real astrophysical conditions to see the impact of this new computational effort for stellar physics. It is obvious that statistical calculations are less time consuming than detailed calculations so it is important to establish the thermodynamical conditions for which detailed calculations are absolutely necessary. This general study will lead to updated tables for stellar physics where it is necessary. References [1] [2] [3] [4] [5] [6] [7] [8]
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