Issue 
A&A
Volume 654, October 2021



Article Number  A149  
Number of page(s)  9  
Section  Stellar structure and evolution  
DOI  https://doi.org/10.1051/00046361/202141425  
Published online  26 October 2021 
Electron conduction opacities at the transition between moderate and strong degeneracy: Uncertainties and impacts on stellar models
^{1}
INAFOsservatorio Astronomico d’Abruzzo, via M. Maggini, sn. 64100, Teramo, Italy
email: santi.cassisi@inaf.it
^{2}
INFN – Sezione di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy
^{3}
Ioffe Institute, Politekhnicheskaya 26, Saint Petersburg 194021, Russia
^{4}
Astrophysics Research Institute, Liverpool John Moores University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool L3 5RF, UK
Received:
29
May
2021
Accepted:
25
August
2021
Electron conduction opacities are one of the main physics inputs for the calculation of low and intermediatemass stellar models. A critical question considers how to devise a bridge when calculating both moderate and strong degeneracy, which are necessarily performed adopting different methods. The densitytemperature regime at the boundary between moderate and strong degeneracy is, in fact, crucial for modelling the helium cores of red giant branch stars and the hydrogenhelium envelopes of white dwarfs. Prompted by newly published, improved calculations of electron thermal conductivities and opacities for moderate degeneracy, we study different, physically motivated prescriptions to bridge these new computations with wellestablished results in the regime of strong degeneracy. We find that these varied prescriptions have a sizable impact on the predicted Hecore masses at the Heflash (up to 0.01 M_{⊙} for initial total masses far from the transition to nondegenerate Hecores and up to ∼0.04 M_{⊙} for masses around the transition), the tip of the red giant branch (up to ∼0.1 mag), and the zeroage horizontal branch luminosities (up to 0.03 dex for masses far from the transition and up to ∼0.2 dex around the transition), and white dwarf cooling times (up to 40–45% at high luminosities, and up to ∼25% at low luminosities). Current empirical constraints on the tip of the red giant branch and the zero age horizontal branch absolute magnitudes do not yet allow for the definitive exclusion of any of these alternative options for the conductive opacities. Tests against observations of slowlycooling faint WDs in old stellar populations will need to be performed to see whether they are capable of setting some more stringent constraints on bridging calculations of conductive opacities for moderate and strong degeneracy.
Key words: opacity / stars: interiors / stars: latetype / stars: lowmass / white dwarfs
© ESO 2021
1. Introduction
Calculations of the thermal conductivity for degenerate electrons and the corresponding electron conduction opacities are a crucial input in the calculations involved in stellar evolution models. When electron degeneracy sets in the stellar interiors, electron conduction becomes the dominant energy transport mechanism and the values of the electron conduction opacities are critical for the accurate calculation of the models’ thermal stratification (see, e.g., Cassisi & Salaris 2013, and references therein). This is true for the interiors of brown dwarfs (see, e.g., Chabrier & Baraffe 2000, for a review), the helium cores of lowmass stars (masses below ∼2.0–2.3 M_{⊙}) during their red giant branch (RGB) evolution (see, e.g., Salaris et al. 2002, for a review of RGB models), the carbonoxygen cores of low and intermediate mass stars (masses below 6–7 M_{⊙}) during the asymptotic giant branch (AGB) phase (see, e.g., Cassisi & Castellani 1993), the oxygenneon cores of superAGB stars with masses between ∼6–7 and ∼10 M_{⊙} (see, e.g., GarciaBerro & Iben 1994; Siess 2007), the cores and parts of the HHe envelopes of white dwarfs (WDs – see, e.g., Fontaine et al. 2001), as well as the envelopes of neutron stars (see, e.g., Beznogov et al. 2021, for a review).
The calculation of the electron conduction opacity in astrophysical plasmas is an ongoing enterprise, with a decadeslong history, starting with the works by Marshak (1941), Lee (1950), Mestel (1950), elaborated upon in the seminal works by Spitzer & Härm (1953), Chapman (1954), Braginskii (1958), and the widely employed calculations by Hubbard & Lampe (1969) further developed by Itoh and coauthors (Flowers & Itoh 1976, 1979, 1981; Itoh et al. 1984; Mitake et al. 1984; Itoh & Kohyama 1993), and Yakovlev and coauthors (Yakovlev & Urpin 1980; Urpin & Yakovlev 1980; Raikh & Yakovlev 1982; Yakovlev 1987; Baiko & Yakovlev 1995; Baiko et al. 1998), which were summarized, refined, and employed in extensive calculations by Potekhin et al. (1999, hereafter P99).
Each of these sources of opacities carried its own limitations and shortcomings. For instance, Spitzer & Härm (1953) considered nondegenerate electrons, while Hubbard & Lampe (1969) used different methods of calculations in cases of weak and strong electron degeneracy, for instance, when T ≫ T_{F} and T ≪ T_{F}, where T is the temperature and T_{F} is the Fermi temperature (see Sect. 2.1), leaving some gaps in the intermediate range of partially degenerate electrons, where T ∼ T_{F}. In addition, Hubbard & Lampe tabulations covered a very limited set of chemical mixtures, and neither Spitzer & Härm nor Hubbard & Lampe took into account the relativistic effects or the regime of dense matter where the stellar plasma solidifies. The work by Itoh’s and Yakovlev’s research groups made significant improvements over the previous results, taking into account the effects of special relativity and more accurate structure factors for the electronion plasmas, as well as the electronphonon scattering that replaces the electronion scattering in the solid phase. Their results could be employed also to compute opacities for arbitrary chemical mixtures, however they covered only the case of strong electron degeneracy, namely, a regime where T ≪ T_{F}, which is not really fulfilled in the Hecores of RGB stars or the envelopes of WDs (see e.g., Catelan 2007, and the next sections for a deeper analysis of this issue).
For the conductive opacities due to the electronion (ei) scattering, a consistent way of filling the gap between the domains of weakly and strongly degenerate electrons is provided by the thermal averaging procedure (see e.g., Cassisi et al. 2007, hereafter C07), patterned after the method previously employed by Potekhin & Yakovlev (1996) to compute finitetemperature effects on the Shubnikov–De Haas oscillations of the electron transport coefficients of degenerate electronion plasmas in quantizing magnetic fields^{1}. Unfortunately, this method is not applicable to the electronelectron (ee) scattering: to overcome this difficulty, an interpolation formula has been proposed by C07, who have also taken into account an improved treatment of the ee scattering at high densities, which was suggested at the time by Shternin & Yakovlev (2006).
The electron conduction theory has undergone substantial progress in the last decade, enabling further refined studies of the heat transport by partially degenerate electrons (e.g., Desjarlais et al. 2017; Daligault 2018; Shaffer & Starrett 2020, and references therein). In particular, Shaffer & Starrett (2020) demonstrated that the ee scattering affects the thermal conductivity in a nontrivial way at T ∼ T_{F}, resulting in lower conductive opacities compared to the traditional approach. This effect is especially pronounced for light chemical elements in the regime of moderate coupling and moderate degeneracy. Based on this theory, Blouin et al. (2020, hereafter B20) calculated the conductive opacities for H and He compositions, finding a difference by up to a factor 2.5–3 as compared to C07 near the boundary of the temperaturedensity domain, where the new theory may be applied. They have also shown that this decrease of the conductive opacities has a sizable impact on the cooling times of WD models with H and He envelopes, such that the age of the coolest models is reduced by as much as ∼2 Gyr, as compared to calculations with C07 opacities.
The important point to notice is, as also stated by Shaffer & Starrett (2020) and B20, that the traditional (e.g., C07) results are superior at a strong degeneracy because unlike B20, these ensure the known asymptotic limits at T/T_{F} ≪ 1. In addition, the theory underlying the B20 results is nonrelativistic and, therefore, it is restricted to mass densities ρ ≲ 10^{6} g cm^{−3}. Therefore, we need to bridge B20 results for mildly degenerate, nonrelativistic plasmas and the traditional opacities at higher densities. This introduces some uncertainty, which can affect the calculation of both WD and RGB models, for sizable portions of the helium cores of RGB models and of the H and He envelopes of WD models, covering a range of the degeneracy parameter θ ≡ T/T_{F} that extends from a few times 0.01 to a few times 0.10 and above.
The purpose of this paper is to investigate different possible ways to merge B20 results at θ ∼ 1 with the opacities at θ ≪ 1, studying their impact on the cooling times of WD models and on the mass of the electron degenerate helium cores of lowmass stellar models at the Heflash, as well as the resulting effect on the RGB lifetime, the luminosities of the tip of the RGB, and the start of quiescent core Heburning after the degeneracy has been lifted. These luminosities are traditionally used to constrain the distance of old stellar populations (ages above 1–2 Gyr).
The plan of the paper is as follows. In Sect. 2, we summarise the theoretical background to the calculations of conductive opacities, give an overview of the recent updates for the partially degenerate domain, and discuss possible ways to treat the transition to the strong degeneracy regime. Section 3 presents our stellar evolution calculations and discusses the impact of the new conductive opacities and the related uncertainties on the results. Our summary and conclusions follow in Sect. 4.
2. Conductive opacities
In stationary and nonconvective layers of a star, the heat transport is governed by the Fourier law F = −λ ∇T, where F is the heat flux, T is the temperature, and λ the thermal conductivity. The last quantity is related to the opacity κ by the equation (see e.g., Kippenhahn et al. 2012)
where σ is the StefanBoltzmann constant and ρ is the mass density.
In general, radiative and conductive energy transports work in parallel; hence, the total thermal conductivity is the sum λ = λ_{r} + λ_{e}, where λ_{r} and λ_{e} denote the radiative (r) and electron (e) components of the thermal conductivity λ. Accordingly, , where the radiative (r) and conductive (c) opacities are related, respectively, to λ_{r} and λ_{e} by Eq. (1).
The transport coefficients of the electronion plasmas in the case of nondegenerate and nonrelativistic electrons (T ≫ T_{F}, x_{r} ≪ 1), and weakly coupled ions (Γ_{i} ≪ 1, where Γ_{i} = (4πn_{ion}/3)^{1/3}(Ze)^{2}/k_{B}T ≈ (2.275 × 10^{7}K/T) Z^{5/3}x_{r} is the Coulomb coupling parameter) were calculated long ago (e.g., Spitzer & Härm 1953; Chapman 1954; Braginskii 1958) using the classical theory by Chapman & Cowling (1970). The theory of the thermal conduction by electrons of arbitrary degeneracy in the fully ionized nonrelativistic stellar interior was reviewed by Hubbard & Lampe (1969). An extension to the degenerate electron gas with allowance for the special relativity effects was described in detail by Flowers & Itoh (1976).
2.1. Theoretical background
When dealing with electron conduction, according to the elementary theory based on the kinetic method and on the assumption that the effective electron scattering rate ν does not depend on the electron energy, λ_{e} can be written as (Ziman 1960):
where n_{e} is the electron number density, is the effective dynamical electron mass, k_{B} is the Boltzmann constant, c is the speed of light, and T_{F} is the electron Fermi temperature. At T ≪ m_{e}c^{2}/k_{B} = 5.93 × 10^{9} K, the effective electron mass is given by , where m_{e} is the true electron mass, and x_{r} is the relativity parameter:
Here, ρ_{6} ≡ ρ/10^{6} g cm^{−3}, p_{F} = ℏ(3π^{2}n_{e})^{1/3} is the Fermi momentum, Z and A are, respectively, the ion charge and mass numbers. In mixtures of elements with different charge numbers Z_{j}, they should be averaged using the number fractions x_{j} = n_{j}/n_{ion} of the components as weights, viz. ⟨Z⟩≡∑_{j}x_{j}Z_{j} (where n_{j} is the number of ions of species j and n_{ion} = n_{e}⟨Z⟩^{−1} the total number of ions, per unit volume); the same holds for A_{j}. The Fermi temperature:
determines whether the electrons are nondegenerate (T_{F} ≪ T), strongly degenerate (T_{F} ≫ T), or mildly degenerate (T_{F} ∼ T). In the nonrelativistic theory, which is valid at x_{r} ≪ 1, we have and T_{F} ≈ 3 × 10^{9} (ρ_{6} Z/A)^{2/3} K.
Beyond the elementary transport theory, it is still convenient to use Eq. (2), but in this case ν is some ‘effective’ collision frequency. In the fully ionized gas or liquid, the electron heat conduction is limited by the ei and ee scattering; the assumption that the scattering rates of different kinds are mutually independent results in the so called Matthiessen’s rule, positing that the collision frequencies simply add up. Then, in the fully ionized liquid or gas,
where ν_{ei} and ν_{ee} are the frequencies of the electron scattering on the ions and on the electrons, respectively.
The ei collision frequency in a strongly degenerate Coulomb liquid can be written in the form (e.g., Yakovlev & Urpin 1980):
where Λ(p_{F}) is a dimensionless Coulomb logarithm. It is possible to compute conductivities determined by elastic ei scattering at arbitrary degeneracy, using a specific thermal averaging, which involves an energydependent effective collision frequency, described by Eq. (6) at every isoenergetic surface, corresponding to a given p_{F} (e.g., C07; Potekhin & Yakovlev 1996; Ventura & Potekhin 2001).
Although the ei scattering is usually most important for degenerate plasmas, the ee scattering is nonnegligible for Z ≲ 10, which is especially important for H and He. Lampe (1968a) treated the ee scattering using the ChapmanEnskog solution of the quantum LenardBalescu kinetic equation for the system of degenerate electrons and pointlike, nondegenerate, weakly coupled classical ions. The dynamical screening of the electrons was treated in the randomphase approximation, applicable at T ≪ T_{F}. The author showed that the character of the scattering is different at temperatures T ≪ T_{pl} and T_{pl} ≪ T ≪ T_{F}, where
is the electron plasma temperature. In a subsequent paper, Lampe (1968b) applied the ChapmanEnskog solution of the quantum LenardBalescu kinetic equation to the nondegenerate and weakly degenerate electrons.
Hubbard & Lampe (1969) combined these calculations with the earlier results of Hubbard (1966), who considered the ei opacities, κ_{ei}, in a nonrelativistic degenerate electron gas, taking into account the ionion correlations. Hubbard and Lampe provided conductive opacities in tabular form for various chemical compositions. Due to the use of different approximations for nondegenerate and degenerate electrons, their tables for these two cases do not match each other smoothly and thus contain gaps at sufficiently low temperatures (see more details in C07).
Hubbard and Lampe used the nonrelativistic theory. The expression of ν_{ee} for the relativistic degenerate electrons was obtained by Flowers & Itoh (1976) at T ≪ T_{pl}, and extended by Urpin & Yakovlev (1980) to higher temperatures, where T_{pl} ≲ T ≪ T_{F}. Shternin & Yakovlev (2006) reconsidered the problem including the Landau damping of transverse plasmons, neglected by the previous authors. They showed that the Landau damping strongly increases ν_{ee} in the domain of x_{r} ≳ 1, θ ≪ 1, and T ≪ T_{pl}. Their fit to to ν_{ee} is widely used in studies of degenerate stars and, in particular, it was employed by C07.
2.2. Suppression of opacities in partially degenerate plasmas
Matthiessen’s rule (see Eq. (5)) results in the additivity of the ei and ee opacities:
and can be derived in the lowest (onepolynomial) approximation of the ChapmanEnskog method (Chapman 1954; Hubbard & Lampe 1969). However, as stated by Hubbard & Lampe (1969), at least the twopolynomial approximation should be used to obtain accurate results, the accuracy provided by the Matthiessen’s rule was deemed to be sufficient for astrophysical applications because using the variational principle of the kinetic theory, it can be shown that ν_{ei} + ν_{ee} ≤ ν ≤ ν_{ei} + ν_{ee} + δν, where δν ≪ min(ν_{ei}, ν_{ee}) (see e.g., chapter 7 of Ziman 1960). However, this relation implies that the shape of the electron distribution function is the same with and without the ee collisions. In fact, the electron distribution function takes on a different shape depending on whether or not the ee collisions occur.
Desjarlais et al. (2017) posited a modified Matthiessen’s rule in the form:
where S_{κ} is a ‘reshaping correction’, representing the indirect modification of the ei scattering term due to the ee interaction. Desjarlais et al. (2017) computed hydrogen electrical and thermal conductivities by the QMD method using the KohnSham densityfunctional theory together with a KuboGreenwood response framework and compared the results with the quantum LenardBalescu solution in the regime of weak ion coupling (Γ_{i} ≪ 1) and moderate degeneracy (T ∼ T_{F}), where both methods are applicable. They found that the reshaping factor can be as low as S_{κ} ∼ 0.6.
Daligault (2016, 2017, 2018) extended the formulas for the transport coefficients of classical plasmas inside the dense plasma regime by applying the ChapmanEnskog method to solve the quantum LandauFokkerPlanck (qLFP) kinetic equation. The qLFP equation extends the classical LFP equation by accounting for the Pauli principle while retaining the smallangle collision approximation. This extension has become possible due to modifications to the classical ChapmanEnskog method. In particular, Daligault (2018) replaced the expansion over the classical Sonine polynomials by a set of orthogonal ‘quantum’ polynomials. Moreover, he derived practical formulas for the calculation of transport coefficients (electrical and thermal conductivities, viscosity, diffusion coefficients) based on this new polynomial expansion. He has demonstrated that with his method we can extend the range of validity of the classical LFP equation, determined by the strong inequality T ≫ T_{F}, to lower temperatures. For example (see Fig. 1 in Daligault 2018, where r_{s} = 0.014/x_{r}), the 10% accuracy is ensured by the classical LFP approximation at T > 4T_{F}, while the qLFP approximation provides the same accuracy at T > 1.7T_{F} and x_{r} > 0.005, at T > T_{F} and x_{r} > 0.16, and also at T > 0.3T_{F} and x_{r} > 0.25.
In both LFP and qLFP cases the plasma was assumed to be weakly coupled and nonrelativistic, and these approximations impose supplementary restrictions on the validity domain, which can be roughly put as Γ_{i} ≪ 1 and x_{r} ≲ 1. In addition, the effects of electron exchange are neglected. The exchange effects always reduce the electron scattering rate, but no more than by a factor of two (see Daligault 2017).
Very recently, Shaffer & Starrett (2020) have combined the qLFP theory with the concept of meanforce scattering, where the scattering crosssections are calculated using the potential of mean force as the interaction potential. This way they can account for strong coupling effects in a plasma kinetic framework and alleviate the constraint Γ_{i} ≪ 1. They found a significant suppression of the effective ee scattering rate in a finite temperature interval, caused by nonmonotonicities in the ee meanforce potential as an indirect effect of strong ion coupling. The inclusion or omission of ee collisions in qLFP is rather unimportant for the electrical conductivity at low temperatures, whereas the thermal conductivity still depends on ee collisions at any temperature. In the limit of a fully degenerate electron gas, T/T_{F} → 0, the thermal conductivity obtained with this method is identical to that of an electron gas, which is clearly unrealistic. Shaffer & Starrett (2020) concluded that this unphysical behavior at low temperatures is an artifact of the smallangle approximation and these authors traced a connection to the argument by Lampe (1968a), who noted that largeangle ee collisions are more strongly Pauli blocked than ei ones, whereas smallangle collisions are less so. Therefore, the qLFP method, while successful over a wide range of temperatures, still breaks down for sufficiently degenerate plasmas, in agreement with the abovementioned considerations by Daligault (2017, 2018). In addition, the method may fail in the case of very strongly coupled Coulomb plasmas, where an accurate ion structure factor is needed to grasp the longrange order effects (Baiko et al. 1998; Wetta & Pain 2020).
2.3. Bridging the opacities of mildly and strongly degenerate H and He plasmas
B20 applied the method of Shaffer & Starrett (2020) to the calculation of conductive opacities for pure H and He compositions. In case of heavier elements, the electronelectron interactions are less important, so that this method is expected to produce results more similar to the conductive opacities κ_{c} obtained using the Matthiessen’s rule, that is, by assuming S_{κ} = 1. Hereafter, following B20, we denote the latter opacities as . They are essentially the C07 opacities but improved as described in Potekhin et al. (2015); the differences with the original C07 opacities for liquid H and He plasmas are at most within 2%.
B20 found a substantial reduction of the conductive opacities (corresponding to an enhancement of the thermal conductivity) in the domain of partial degeneracy, compared to . Their Tables 1 and 2 provide κ_{c} for pure H and pure He compositions, which at fixed temperature, T, reach densities corresponding to θ ≡ T/T_{F} generally between 0.2 and 0.1. The difference with exceeds a factor of 2 on the verge of this densitytemperature domain.
To facilitate the implementation of their new opacity calculations in stellar evolution codes, B20 devised an analytic expression for the factor (denoted hereafter by F) to reduce the the traditional opacity to fit their numerical results. Accordingly, the reduced conductive opacity and enhanced thermal electron conductivity is given as:
The correction factor is written as:
where x = log(ρ/ρ_{0}), y = log(T/T_{0}), ρ_{0} = 10^{5.45} g cm^{−3} and T_{0} = 10^{8.40} K for hydrogen, ρ_{0} = 10^{6.50} g cm^{−3} and T_{0} = 10^{8.57} K for helium, function g(x, y) is a tilted scaled Gaussian, and H(g) is a correction to the Gaussian shape at large g (see the explicit formulas given in B20). This fit accurately reproduces the numerical B20 results at θ > 0.1 and ensures that the correction vanishes when θ → 0. Hereafter and denote, respectively, the opacities and thermal conductivities given by Eqs. (10) and (11).
The electron thermal conductivities calculated with and without the correction factor by B20 are shown in Fig. 1 for hydrogen, and in Fig. 2 for helium, in the relevant T and ρ ranges. The convergence of to the traditional estimate is rather slow at high densities, if the temperature is also high. In this case the B20 correction does not vanish until T ≪ 0.1T_{F}, which is certainly far beyond the range of validity of Shaffer & Starrett (2020) method and most likely overestimates the true enhancement of the conductivities in these cases.
Fig. 1. Thermal conductivity for hydrogen as function of mass density for several constant temperatures T. Dashed lines show the traditional conductivities , and dotted lines display the conductivities enhanced by the F factor in Eq. (11). The solid and dotdashed lines show the conductivities with the weakly and strongly damped corrections given by Eq. (12), and , respectively. They provide two different transitions from the new calculations for partially degenerate electrons to the traditional results in the strong degeneracy regime (see details in the text). Numbers near the curves mark log T(K) values. 
This is shown even more clearly in Figs. 3 and 4, which display the ratio as a function of ρ for several temperatures T; densities corresponding to θ = 0.1 and 1.0 are also marked. For example, at T > 10^{7} K convergence is reached only at densities corresponding to T_{F} values well above 10 T, that is, at θ ≪ 0.1. In addition, broad differences between the dashed and dotted lines are observed in the case of helium at ρ > 10^{6} g cm^{−3}, where the electrons are relativistic.
To achieve a faster convergence to the degenerate asymptote in the regime of strong degeneracy, we introduced a damping factor D(θ)=(1 + aθ^{−b})^{−1} (θ ≡ T/T_{F}). The damped enhancement factor F for the electron thermal conductivity (a reduction factor for the conductive opacities) then reads
Given that the qLFP equation is nonrelativistic, we use the nonrelativistic approximation for T_{F} in Eq. (12). We have made two choices of the parameters a and b. A conservative choice (that we denote as ‘weak damping’) is to ensure that D(θ) does not change F by more than 1% at T > T_{F} and that it does not exceed 1% (ensuring that F ≈ 1) at T < 0.01T_{F}. These conditions are fulfilled for a = 0.01 and b = 2. The electron conductivities obtained using this weakly damped enhancement factor, which we denote by are shown in Figs. 1 and 2 (the corresponding opacities will be denoted by ), while the ratio of to as a function of ρ is shown in Figs. 3 and 4.
Fig. 2. Thermal conductivity for helium as a function of mass density for several constant temperatures T. The meaning of the various lines is the same as in Fig. 1. 
Fig. 3. Ratio of the electron thermal conductivity for hydrogen enhanced according to B20 () to results, as a function of log ρ (dotted lines), for the labelled values of log T, with T given in Kelvin. Solid lines display the analogous ratio for the thermal conductivities obtained using the weakly damped enhancement factor F. Dotdashed lines show the corresponding ratio for the strong damping, . The vertical lines mark the boundaries of the range of densities corresponding to θ = 1 and θ = 0.1 (higher ρ implies lower θ, at constant T). 
Fig. 4. Ratio of the electron thermal conductivity for helium enhanced according to B20 to , as a function of log ρ, for the labelled values of log(T). The meaning of the various lines is the same as in Fig. 3. 
We can see that and almost coincide at T > T_{F} (to the left of the left vertical line in Figs. 3 and 4), ensuring that our damping does not distort the B20 results in the domain where the underlying approximations are reliable. The weak damping is seen to provide a good agreement with B20 results for T down to 0.5 T_{F}, while at the same time it almost fully converges to the traditional results at T ≲ 0.03 T_{F}. The ratio is however sometimes still quite substantial (up to a factor of ∼1.5) at T ∼ 0.1T_{F}, where may be already preferable, as discussed also by B20. Indeed, as we see in Sect. 2.2, the results and discussions from Daligault (2017, 2018) and Shaffer & Starrett (2020) prompt the assumption that these approximations, which are inherent to the qLFP method (in particular, the smallangle scattering approximation) may lead to an uncertainty of ∼10% at T = T_{F} and to implausible results at T ≪ T_{F}.
We cannot therefore exclude that in reality the conductivity should converge to more rapidly in the transitional range 0.1T_{F} ≲ T ≲ T_{F}. To this end, we considered a much more extreme, but probably still realistic ‘strong damping’ choice, defined by the requirements that D(θ) does not affect F by more than 10% at T > T_{F} and that D(θ) does not exceed 1% at T < 0.1T_{F}. In this case, a = 0.1 and b = 3 in Eq. (12). The conductivities (see Figs. 1 and 2) and opacities obtained with such strongly damped enhancement factor will be denoted by and , respectively.
The ratio of to as a function of ρ is also displayed in Figs. 3 and 4, which show how converges to at T ∼ 0.1 T_{F}, whilst it is almost equal to at T > T_{F}. The values of may noticeably (up to ∼30%) differ from the B20 calculations already at T ∼ 0.5 T_{F}; however, we believe that this strong damping option is a plausible extreme choice. As we have discussed in Sect. 2.2, the inaccuracy of the qLFP method may reach 10% at T ∼ T_{F}, hence, it is not unrealistic to assume still greater inaccuracies at T ∼ 0.5 T_{F}.
The differences between these three choices of the electron conductivity (B20, B20 with weak damping, and B20 with strong damping) are, by construction, maximal around θ ∼ 0.1 (within a factor of 3), which is a θ range encountered in RGB Hecores and WD envelopes, as shown in Figs. 5 and 6.
Fig. 5. Run of the ratio θ ≡ T/T_{F} as a function of the mass (in solar units) enclosed within a distance r from the centre, for the Hecore at three different stages of the RGB evolution of a model with the labelled total mass, initial helium mass fraction Y and a metal mass fraction Z (metal distribution αenhanced, with [α/Fe] = 0.4, typical of Galactic halo stars) corresponding to [Fe/H] = −1.5. The discontinuity at M_{r} ≈ 0.2 M_{⊙} in the structure corresponding to the TRGB stellar model is due to offcenter Heburning ignition that starts removing the electron degeneracy. 
Fig. 6. Evolution with the surface luminosity L of selected physical and chemical quantities across WD models with masses equal to 0.55 and 1 M_{⊙}. The vertical axis displays the logarithm of the mass m_{r} enclosed between the surface and a point at a distance r from the centre, normalised to the total WD mass M_{WD}. Dashed lines mark the boundaries of the carbonoxygen core as well as the helium and the hydrogen envelope. The lower boundary of surface convection is denoted by dotted lines. The solid lines show the position of the mass layers where θ = 1 and 0.1. 
Figure 5 displays the run of θ across the structure of the Hecore at three selected luminosities during the RGB evolution of a typical lowmass (0.8 M_{⊙}), metal poor stellar model (from Pietrinferni et al. 2021). In all three cases, T/T_{F} ranges between ∼0.05 at the centre of the Hecore and ∼1 at its outer edge.
A sketch of the internal structures of two WD models (for DA WDs with He and H envelopes, from Salaris et al. 2010) and their evolution with the surface luminosity is shown in Fig. 6, for masses equal to 0.55 and 1 M_{⊙}, bracketing the typical mass range of carbonoxygen WDs. For both masses, the layers where T/T_{F} is around 0.1 are located in the He or the H layers, depending on the model luminosity. The same is true in models with just He envelopes (for DB WDs).
3. Effects on stellar models
In this section, we quantify the effect of using alternatively (C07), (B20), (B20sd), and (B20wd) opacities on RGB (and the following horizontal branch stage) and WD models. For the RGB computations, we relied on the same stellar evolution code, physical assumptions (including atomic diffusion), and input physics adopted by Pietrinferni et al. (2021). The calculations by Pietrinferni et al. (2021) make use of C07 conductive opacities, hence, they are taken as a reference in the following discussion. For the WD models, we employed the code and physics inputs described by Salaris et al. (2010).
3.1. Red giant branch and horizontal branch models
We computed models for initial masses in the range 0.8–2.4 M_{⊙} and various chemical compositions, from the premain sequence stage until the Heburning ignition at the tip of the RGB (TRGB), employing the same combinations of metallicity Z and initial helium abundance Y as in Pietrinferni et al. (2021).
We begin by analysing the results for the lowerend of the explored mass regime, that is, for the 0.8 M_{⊙} models, which are characterised by a stronger electron degeneracy in their helium cores. The different choices of conductive opacities have a negligible impact on the RGB lifetime but, as expected, a sizable effect on the Hecore mass at helium ignition (M_{cHe}). Going from C07 to B20 opacities increases M_{cHe} by ∼0.01 M_{⊙}, independent of Z. If the opacities with weak damping B20wd are used instead, M_{cHe} increases by ∼0.007 M_{⊙} compared to calculations with C07. Finally, the opacities with strong damping B20sd increase M_{cHe} by just ∼0.001 M_{⊙}.
Given that the TRGB brightness depends on the Hecore mass at the He ignition, these differences translate to changes in the magnitude of the TRGB, an observable quantity, also used to determine distances to old stellar populations in galaxies. Figure 7 displays the IJHK TRGB absolute magnitudes obtained from our calculations, for models with an age of 12–13 Gyr at the TRGB, and a large range of initial metallicities. Moving from C07 to B20 opacities makes the TRGB brighter by about 0.1 mag in all filters (because of the larger Hecore masses), an increase which is reduced to about 0.07 mag when calculations with are considered instead. The use of instead of the C07 opacities has a negligible impact of the TRGB brightness of the models.
Fig. 7. Theoretical TRGB absolute magnitudes in the I (Cousins) and JHK (Bessel & Brett system) filters as a function of [Fe/H], for the four choices of conductive opacities discussed in this section (see labels and text for details). We also show (circles with error bars) the measured TRGB absolute magnitude in the Galactic globular clusters 47 Tuc and ω Centauri. 
For the sake of comparison, we also show in Fig. 7 the absolute magnitudes of the TRGB determined for the Galactic globular clusters ω Centauri and 47 Tuc. We display Bellazzini et al. (2004) results, with small adjustments to take into account the recent distance determinations by Baumgardt & Vasiliev (2021).
The calculations using B20 and B20wd opacities predict TRGBs marginally brighter than the observed TRGB magnitudes in the I band – which, incidentally, have smaller measurement errors compared to the infrared data – when taking into account observational errors, while in JHK bands all sets of models are compatible with observations within the error bars. On balance these TRGB observations cannot definitely exclude any of the displayed four choices of conductive opacities. The marginal discrepancy with the more precise Iband data could for example be ascribed to some small (on the order of 0.01 mag) systematic errors in the calculations of the bolometric corrections, which might affect less severely the infrared bands.
We also investigated the impact of these different sets of opacities on RGB models with initial masses around the transition for the onset of electron degeneracy in the Hecore. The upper panel of Fig. 8 shows M_{cHe} at the ignition of core Heburning for models with Z = 0.001 and Y = 0.248, and masses between 1.4 and 2.4 M_{⊙}. For masses larger than ∼1.4 M_{⊙}, the effect of choosing a different set of opacities increases, reaching a maximum between 2.1 and 2.2 M_{⊙}, to then vanish for larger masses, that do not develop electron degeneracy after the main sequence. For an initial mass of 2.1–2.2 M_{⊙}, the B20wd and B20 opacities increase M_{cHe} by ∼0.035 M_{⊙} and ∼0.043 M_{⊙}, respectively. These differences still hold at other metallicities, the only change being systematic shifts of the values of the initial masses of the models around the transition, due to the effect of the initial metallicity (and He abundance) on the mass threshold for the onset of electron degeneracy.
Fig. 8. The trend of selected structural and evolutionary properties for various assumptions about the conductive opacities. Upper panel: Hecore mass at Heignition as a function of the initial total mass for models with Z = 0.001 and Y = 0.248; lower panel: as the upper panel but for the bolometric luminosity at the beginning of the quiescent core Heburning, after the electron degeneracy has been lifted. 
These variations of the degenerate Hecore masses at helium ignition affect the properties of the following core Heburning phase, as shown in both Figs 8 and 9. The lower panel of Fig. 8 shows the luminosity at the beginning of quiescent He burning, after the degeneracy has been lifted, for the models with initial masses between 1.4 and 2.4 M_{⊙} and the labelled initial composition (we denote this stage by zeroage horizontal branch (ZAHB), as for the lower mass models, which are the theoretical counterpart of the stars that populate the horizontal branches of globular clusters) and the labelled initial composition. The variation of this luminosity with varying choices of the conductive opacities mirrors qualitatively that of M_{cHe}, as expected. The effect is maximised for masses around 2.2 M_{⊙}, where Δlog(L/L_{⊙})∼0.2 dex when passing from C07 to B20 opacities (the corresponding Δlog(L/L_{⊙}) for the 1.4 M_{⊙} models is equal to ∼0.03 dex) and ∼0.1 when B20wd opacities are used instead of C07.
Fig. 9. Trend of the ZAHB brightness at the typical temperature of the RR Lyrae instability strip, as a function of [Fe/H] for the labelled choices of the conductive opacities. Points with error bars display semiempirical data for a sample of Galactic globular clusters (see text for details). 
The change in luminosity directly impacts on the core Heburning lifetime: For models with 1.4 M_{⊙} the maximum effect amounts to a reduction on the order of 7% when B20 opacities are employed instead of C07, which increases to about 18% for models with mass around 2.2 M_{⊙}.
Figure 9 displays the theoretical ZAHB luminosity of lower mass models, at the typical temperature of the RR Lyrae instability strip (log T_{eff} = 3.85, with T_{eff} in K) as a function of the initial [Fe/H], obtained using alternatively the same four sets of conductive opacities. Again, the trends reflect the behaviour of M_{cHe} at Heignition. Models calculated with the C07 opacities are the faintest ones, whilst those calculated with B20 opacities are about Δlog(L/L_{⊙}) = 0.03 more luminous. The calculations with the B20sd opacities are basically identical to the C07 ones, while the B20wd opacities provide ZAHB models Δlog(L/L_{⊙})∼0.02 brighter than the C07 ones. The corresponding core Heburning lifetimes are affected at the level of at most 6–8% when adopting these different opacities.
The same figure also displays the semiempirical ZAHB luminosities for a sample of Galactic globular clusters, as determined by de Santis & Cassisi (1999), from the pulsational properties of their RR Lyrae stars. Due to the small sample size and the associated error bars, the comparison with the models does not set any definitive constraint on the appropriate way to bridge the regimes of moderate and strong degeneracy. Models based on C07 and the extreme case of the B20sd opacities, but also those based on B20 and B20wd opacities are to various degrees consistent with the data.
3.2. White dwarf models
B20 have already shown how WD cooling models are strongly affected by replacing opacities in the calculations with the smaller values. Here, we make similar comparisons, but including also the cases of B20wd and B20sd opacities. We have considered a WD model with mass M_{WD} = 1.0 M_{⊙}, made of a carbonoxygen core with chemical stratification taken from the solar progenitors’ models by Salaris et al. (2010), surrounded by a helium envelope with mass equal to 10^{−2}M_{WD}, and a more external hydrogen envelope with mass equal to 10^{−4}M_{WD}. The code and input physics (except for the conductive opacities) are described in Salaris et al. (2010) and references therein. Such a high mass WD model is expected to display the strongest sensitivity to changes of the conductive opacities, as shown by B20.
Figure 10 shows the relative differences of the cooling times as a function of the surface luminosity, among our calculations with different opacity choices. Models calculated with B20 opacities have longer cooling times than C07 calculations – by up to about 40% – at luminosities above log(L/L_{⊙})∼ − 1.5, where neutrino cooling is very efficient. As discussed by B20 (see also Salaris et al. 2013), the lower conductive opacities cause a faster cooling of the core, which reduces the efficiency of neutrino cooling and increases the cooling times at a given luminosity. In absolute terms, the cooling times in this phase are relatively short, on the order of at most 100 Myr when log(L/L_{⊙})= − 1.5.
Fig. 10. Relative differences of the cooling times as a function of the surface luminosity, among different evolutionary models of a 1.0 M_{⊙} DA WD. The dotted line displays the difference between calculations with C07 and B20 opacities (B20 cooling times minus C07 values at the same luminosity), the dotdashed line the difference between B20sd and C07 calculations, and the solid line the difference between B20wd and C07 models. 
With decreasing luminosities, the cooling times with B20 opacities become increasingly shorter than C07 calculations, because of the faster cooling of the structure. This trend is temporarily broken in a narrow range of luminosities centred around log(L/L_{⊙})∼ − 2.6, due to the earlier start of the crystallization in the models with B20 opacities, and the associated earlier onset of the release of latent heat and the extra energy due to carbonoxygen phase separation (see e.g., Salaris et al. 2010, and references therein). At the typical luminosity of the faintest observed WDs (log(L/L_{⊙})∼ − 4.5) the model calculated with B20 opacities has a cooling age of ∼9.5 Gyr, about 2.5 Gyr shorter than the corresponding calculations with C07 opacities. These differences are consistent with the results obtained by B20.
Calculations using the B20wd opacities display differences compared to C07 models, which are reduced by about 510% compared to the previous case of using B20 opacities. As for RGB models and their core Heburning progeny, calculations with the B20sd opacities provide results almost identical to C07 models, with differences of the cooling times within ±5%.
4. Summary and conclusions
Electron conduction opacities are a key ingredient in the calculation of stellar models for low and intermediatemass stars, and a critical issue is centred on ways to bridge computations of conductive opacities in the regimes of moderate (θ ∼ 1) and strong (θ ≲ 0.1) degeneracy, which are necessarily calculated adopting different methods. In fact, the densitytemperature regime at the transition between moderate and strong degeneracy is crucial for modelling the helium cores of RGB stars and the envelopes of WDs.
We discuss the case of bridging the new, improved conductive opacities calculated by B20 for the regime of moderate degeneracy and the calculations by C07 in the regime of strong degeneracy. We first considered B20 own analytical approximation, which, however, converges to C07 results only at θ≪0.1, well into the regime of strong degeneracy. We then modified the B20 formula by introducing a physically motivated damping factor, which depends on the ratio θ = T/T_{F}, tuned in two alternative ways (weak and strong damping) to converge faster than the B20 fit to C07 results in the regime of strong degeneracy. Both damping prescriptions keep the B20 fit almost intact at θ > 1. The weak damping option provides opacities still different from C07 at θ = 0.1, whilst the more extreme strong damping converges to C07 opacities at θ = 0.1, but it changes the B20 calculations already by 30% at θ ∼ 0.5, in the moderate degeneracy regime. As a consequence, these three sets of conductive opacities have large differences (up to a factor of ∼2) in the critical region around θ ∼ 0.1, which, in turn, have a major impact on the predicted RGB Hecore masses (up to 0.01 M_{⊙} for lowmass models far from the transition regime to nondegenerate Hecores, and up to ∼0.04 M_{⊙} for masses around the transition), TRGB (up to ∼0.1 mag) and ZAHB luminosities (up to 0.03 dex for masses far from the transition, and up to ∼0.2 dex around the transition), and WD cooling times (up to 40–45% at high luminosities, and up to ∼25% at low luminosities).
Current observational constraints on the TRGB and ZAHB absolute magnitudes do not allow us to categorically exclude any of these options for the conductive opacities, also taking into account that there might be other sources of uncertainties on the theoretical predictions for these quantities. The much shorter cooling times predicted for faint, slowly evolving WDs by calculations with both the B20 fit and the weak damping option (compared to models calculated with opacities including the strong damping) will need to be tested against observations of WDs in old stellar populations.
We updated the table of nonmagnetic electron conductivities available at the Ioffe Institute website^{2} by implementing the correction factor in Eq. (12). We use the weak damping as our fiducial choice by default, but we consider also the strong damping as a realistic, albeit extreme possibility. We did not implement this correction directly in the computer code presented by the institute on their website, but we provided the corresponding subroutine and envisioned a possibility for its use (in the absence of a strong magnetic field) to correct the result of the main computation.
We note that here we consider nonmagnetized plasmas and focus on their thermal conductivity in the liquid phase. For a more general overview of the recent progress in the theory of conductivities in the Coulomb plasmas, including the solid phase and the magnetized plasmas, see e.g., Potekhin et al. (2015) and references therein.
Acknowledgments
We thank our referees for constructive comments that have improved the presentation of our results. S. C. acknowledges support from Premiale INAF MITiC, from INFN (Iniziativa specifica TAsP), and from PLATO ASIINAF agreement n.2015019R.12018. The work of AYP was supported by the Russian Science Foundation (grant 191200133).
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All Figures
Fig. 1. Thermal conductivity for hydrogen as function of mass density for several constant temperatures T. Dashed lines show the traditional conductivities , and dotted lines display the conductivities enhanced by the F factor in Eq. (11). The solid and dotdashed lines show the conductivities with the weakly and strongly damped corrections given by Eq. (12), and , respectively. They provide two different transitions from the new calculations for partially degenerate electrons to the traditional results in the strong degeneracy regime (see details in the text). Numbers near the curves mark log T(K) values. 

In the text 
Fig. 2. Thermal conductivity for helium as a function of mass density for several constant temperatures T. The meaning of the various lines is the same as in Fig. 1. 

In the text 
Fig. 3. Ratio of the electron thermal conductivity for hydrogen enhanced according to B20 () to results, as a function of log ρ (dotted lines), for the labelled values of log T, with T given in Kelvin. Solid lines display the analogous ratio for the thermal conductivities obtained using the weakly damped enhancement factor F. Dotdashed lines show the corresponding ratio for the strong damping, . The vertical lines mark the boundaries of the range of densities corresponding to θ = 1 and θ = 0.1 (higher ρ implies lower θ, at constant T). 

In the text 
Fig. 4. Ratio of the electron thermal conductivity for helium enhanced according to B20 to , as a function of log ρ, for the labelled values of log(T). The meaning of the various lines is the same as in Fig. 3. 

In the text 
Fig. 5. Run of the ratio θ ≡ T/T_{F} as a function of the mass (in solar units) enclosed within a distance r from the centre, for the Hecore at three different stages of the RGB evolution of a model with the labelled total mass, initial helium mass fraction Y and a metal mass fraction Z (metal distribution αenhanced, with [α/Fe] = 0.4, typical of Galactic halo stars) corresponding to [Fe/H] = −1.5. The discontinuity at M_{r} ≈ 0.2 M_{⊙} in the structure corresponding to the TRGB stellar model is due to offcenter Heburning ignition that starts removing the electron degeneracy. 

In the text 
Fig. 6. Evolution with the surface luminosity L of selected physical and chemical quantities across WD models with masses equal to 0.55 and 1 M_{⊙}. The vertical axis displays the logarithm of the mass m_{r} enclosed between the surface and a point at a distance r from the centre, normalised to the total WD mass M_{WD}. Dashed lines mark the boundaries of the carbonoxygen core as well as the helium and the hydrogen envelope. The lower boundary of surface convection is denoted by dotted lines. The solid lines show the position of the mass layers where θ = 1 and 0.1. 

In the text 
Fig. 7. Theoretical TRGB absolute magnitudes in the I (Cousins) and JHK (Bessel & Brett system) filters as a function of [Fe/H], for the four choices of conductive opacities discussed in this section (see labels and text for details). We also show (circles with error bars) the measured TRGB absolute magnitude in the Galactic globular clusters 47 Tuc and ω Centauri. 

In the text 
Fig. 8. The trend of selected structural and evolutionary properties for various assumptions about the conductive opacities. Upper panel: Hecore mass at Heignition as a function of the initial total mass for models with Z = 0.001 and Y = 0.248; lower panel: as the upper panel but for the bolometric luminosity at the beginning of the quiescent core Heburning, after the electron degeneracy has been lifted. 

In the text 
Fig. 9. Trend of the ZAHB brightness at the typical temperature of the RR Lyrae instability strip, as a function of [Fe/H] for the labelled choices of the conductive opacities. Points with error bars display semiempirical data for a sample of Galactic globular clusters (see text for details). 

In the text 
Fig. 10. Relative differences of the cooling times as a function of the surface luminosity, among different evolutionary models of a 1.0 M_{⊙} DA WD. The dotted line displays the difference between calculations with C07 and B20 opacities (B20 cooling times minus C07 values at the same luminosity), the dotdashed line the difference between B20sd and C07 calculations, and the solid line the difference between B20wd and C07 models. 

In the text 
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