11institutetext: INAF-Osservatorio Astronomico d’Abruzzo, via M. Maggini, sn.
64100, Teramo, Italy<EMAIL_ADDRESS>22institutetext: INFN - Sezione
di Pisa, Largo Pontecorvo 3, 56127 Pisa, Italy 33institutetext: Ioffe
Institute, Politekhnicheskaya 26, Saint Petersburg 194021, Russia
44institutetext: Astrophysics Research Institute, Liverpool John Moores
University, IC2, Liverpool Science Park, 146 Brownlow Hill, Liverpool, L3 5RF,
UK

# Electron conduction opacities at the transition between moderate and strong
degeneracy: Uncertainties and impact on stellar models

Santi Cassisi 1122 Alexander Y. Potekhin 33 Maurizio Salaris 44 Adriano
Pietrinferni 11

(Received ; accepted )

Electron conduction opacities are one of the main physics inputs for the
calculation of low- and intermediate-mass stellar models, and a critical
question is how to bridge calculations for moderate and strong degeneracy,
which are necessarily performed adopting different methods. The density-
temperature 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 hydrogen/helium envelopes of white dwarfs. Prompted by recently published
new, improved calculations of electron thermal conductivities and opacities
for moderate degeneracy, we study different, physically motivated
prescriptions to bridge these new computations with well established results
in the regime of strong degeneracy. We find that these different prescriptions
have a sizable impact on the predicted He-core masses at the He-flash (up to
0.01$M_{\odot}$ for initial total masses far from the transition to non-
degenerate He-cores, and up to $\sim 0.04M_{\odot}$ for masses around the
transition), the tip of the red giant branch (up to $\sim$0.1 mag) and the
zero age horizontal branch luminosities (up to 0.03 dex for masses far from
the transition, and up to $\sim$0.2 dex around the transition), and white
dwarf cooling times (up to 40-45% at high luminosities, and up to $\sim$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 allow yet
to definitely exclude any of these alternative options for the conductive
opacities. Tests against observations of slowly-cooling faint WDs in old
stellar populations will need to be performed to see whether they can set some
more stringent constraints on how to bridge calculations of conductive
opacities for moderate and strong degeneracy.

###### Key Words.:

Opacity – Stars: interiors – Stars: late-type – Stars: low mass – white dwarfs

##  1 Introduction

Calculations of the thermal conductivity of degenerate electrons and the
corresponding electron conduction opacities are a crucial input for the
calculation of 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 low-mass stars (masses below $\sim$2.0–2.3 $M_{\odot}$)
during their red giant branch (RGB) evolution (see, e.g., Salaris et al.,
2002, for a review of RGB models), the carbon-oxygen cores of low- and
intermediate mass stars (masses below 6-7$M_{\odot}$) during the asymptotic
giant branch (AGB) phase (see, e.g., Cassisi & Castellani, 1993), the oxygen-
neon cores of super-AGB stars with masses between $\sim$6–7 and $\sim
10\leavevmode\nobreak\ M_{\odot}$ (see, e.g., Garcia-Berro & Iben, 1994;
Siess, 2007), the cores and parts of the H-He 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 decades-long history, starting with the works by
Marshak (1941), Lee (1950), Mestel (1950), elaborated 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
coworkers (Flowers & Itoh, 1976, 1979, 1981; Itoh et al., 1984; Mitake et al.,
1984; Itoh & Kohyama, 1993), and Yakovlev and coworkers (Yakovlev & Urpin,
1980; Urpin & Yakovlev, 1980; Raikh & Yakovlev, 1982; Yakovlev, 1987; Baiko &
Yakovlev, 1995; Baiko et al., 1998), which were summarized, refined, and used
in extensive calculations by Potekhin et al. (1999, hereafter P99).

Each of these sources of opacities had its own limitations and shortcomings.
For instance, Spitzer & Härm (1953) considered non-degenerate electrons, while
Hubbard & Lampe (1969) used different methods of calculations in the cases of
weak and strong electron degeneracy, i.e. when $T\gg T_{\mathrm{F}}$ and $T\ll
T_{\mathrm{F}}$, where $T$ is the temperature and $T_{\mathrm{F}}$ is the
Fermi temperature (see Sect. 2.1), leaving some gaps in the intermediate range
of partially degenerate electrons, where $T\sim T_{\mathrm{F}}$. Besides,
Hubbard & Lampe tabulations covered a very limited set of chemical mixtures,
and neither Spitzer & Härm nor Hubbard & Lampe took into account 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 electron-ion plasmas,
as well as the electron-phonon scattering which replaces the electron-ion
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, i.e. a regime where $T\ll T_{\mathrm{F}}$,
which is not really fulfilled in the He-cores 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 electron-ion (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 finite-
temperature effects on the Shubnikov–De Haas oscillations of the electron
transport coefficients of degenerate electron-ion plasmas in quantizing
magnetic fields.111Note that here we consider non-magnetized 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. Unfortunately, this method is not
applicable to the electron-electron (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, suggested at the time by Shternin & Yakovlev (2006).

The electron conduction theory has undergone substantial progress in the last
decade, enabling yet 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 non-trivial way at $T\sim T_{\mathrm{F}}$, 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 compared to C07 near the boundary of the
temperature-density 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 $\sim$2 Gyr, compared to
calculations with C07 opacities.

The important point to notice is that, as also stated by Shaffer & Starrett
(2020) and B20, the traditional (e.g., C07) results are superior at strong
degeneracy, because they, unlike B20, ensure the known asymptotic limits at
$T/T_{\mathrm{F}}\ll 1$. Besides, the theory underlying the B20 results is
non-relativistic and therefore it is restricted to mass densities
$\rho\lesssim 10^{6}$ g cm-3. Therefore, we need to bridge B20 results for
mildly degenerate, non-relativistic 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, cover a range
of the degeneracy parameter $\theta\equiv T/T_{\mathrm{F}}$ 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 $\theta\sim 1$ with the opacities at $\theta\ll 1$, studying
their impact on the cooling times of WD models, on the mass of the electron
degenerate helium cores of low-mass stellar models at the He-flash, and the
resulting effect on the RGB lifetime, the luminosities of the tip of the RGB
and of the start of quiescent core He-burning 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 summarize 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. A
summary and conclusions follow in Sect. 4.

##  2 Conductive opacities

In stationary and non-convective layers of a star, the heat transport is
governed by the Fourier law $\bm{F}=-\lambda\,\nabla T,$ where $\bm{F}$ is the
heat flux, $T$ is the temperature, and $\lambda$ the thermal conductivity. The
last quantity is related to the opacity $\kappa$ by the equation (see, e.g.,
Kippenhahn et al. 2012)

$\kappa=\frac{16\sigma T^{3}}{3\rho\lambda},$ (1)

where $\sigma$ is the Stefan-Boltzmann constant and $\rho$ is the mass
density.

In general, radiative and conductive energy transports work in parallel, hence
the total thermal conductivity is the sum
$\lambda=\lambda_{\mathrm{r}}+\lambda_{\mathrm{e}},$ where
$\lambda_{\mathrm{r}}$ and $\lambda_{\mathrm{e}}$ denote the radiative (r) and
electron (e) components of the thermal conductivity $\lambda$. Accordingly,
$\kappa^{-1}=\kappa_{\mathrm{r}}^{-1}+\kappa_{\mathrm{c}}^{-1}$, where the
radiative (r) and conductive (c) opacities are related, respectively, to
$\lambda_{\mathrm{r}}$ and $\lambda_{\mathrm{e}}$ by Eq. (1).

The transport coefficients of the electron-ion plasmas in the case of non-
degenerate and non-relativistic electrons ($T\gg T_{\mathrm{F}}$,
$x_{\mathrm{r}}\ll 1$) and weakly coupled ions ($\Gamma_{\mathrm{i}}\ll 1$,
where $\Gamma_{\mathrm{i}}=({4\pi
n_{\mathrm{ion}}/3})^{1/3}{(Ze)^{2}/k_{\mathrm{B}}T}\approx(2.275\times
10^{7}\mbox{\leavevmode\nobreak\ K}/T)\,Z^{5/3}x_{\mathrm{r}}$ is the Coulomb
coupling parameter) have been 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 non-relativistic stellar interior has been
reviewed by Hubbard & Lampe (1969). An extension to the degenerate electron
gas with allowance for the special relativity effects has been 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 $\nu$ does not depend on the electron energy,
$\lambda_{\mathrm{e}}$ can be written as (Ziman, 1960)

$\lambda_{\mathrm{e}}=\frac{3}{2}\,\frac{n_{\mathrm{e}}k_{\mathrm{B}}^{2}T}{m_{\mathrm{e}}^{*}\nu}\mbox{\leavevmode\nobreak\
\leavevmode\nobreak\ at\leavevmode\nobreak\ }T\gg
T_{\mathrm{F}},\quad\lambda_{\mathrm{e}}=\frac{\pi^{2}}{3}\,\frac{n_{\mathrm{e}}k_{\mathrm{B}}^{2}T}{m_{\mathrm{e}}^{*}\nu}\mbox{\leavevmode\nobreak\
\leavevmode\nobreak\ at\leavevmode\nobreak\ }T\ll T_{\mathrm{F}},$ (2)

where $n_{\mathrm{e}}$ is the electron number density, $m_{\mathrm{e}}^{*}$ is
the effective dynamical electron mass, $k_{\mathrm{B}}$ is the Boltzmann
constant, $c$ is the speed of light, and $T_{\mathrm{F}}$ is the electron
Fermi temperature. At $T\ll m_{\mathrm{e}}c^{2}/k_{\mathrm{B}}=5.93\times
10^{9}$ K, the effective electron mass is given by
$m_{\mathrm{e}}^{*}=m_{\mathrm{e}}\sqrt{1+x_{\mathrm{r}}^{2}}$, where
$m_{\mathrm{e}}$ is the true electron mass, and $x_{\mathrm{r}}$ is the
relativity parameter

$x_{\mathrm{r}}\equiv\frac{p_{\mathrm{F}}}{m_{\mathrm{e}}c}\approx\left(\rho_{6}\,\frac{Z}{A}\right)^{1/3}.$
(3)

Here, $\rho_{6}\equiv\rho/10^{6}\,\mbox{g\leavevmode\nobreak\ cm${}^{-3}$}$,
$p_{\mathrm{F}}=\hbar(3\pi^{2}n_{\mathrm{e}})^{1/3}$ 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 $x_{j}=n_{j}/n_{\mathrm{ion}}$ of the components as
weights, viz. $\langle Z\rangle\equiv\sum_{j}x_{j}Z_{j}$ (where $n_{j}$ is the
number of ions of species $j$ and $n_{\mathrm{ion}}=n_{\mathrm{e}}\langle
Z\rangle^{-1}$ the total number of ions, per unit volume); the same holds for
$A_{j}$. The Fermi temperature

$T_{\mathrm{F}}=\frac{m_{\mathrm{e}}c^{2}}{k_{\mathrm{B}}}\left(\sqrt{1+x_{\mathrm{r}}^{2}}-1\right)$
(4)

determines whether the electrons are non-degenerate ($T_{\mathrm{F}}\ll T$),
strongly degenerate ($T_{\mathrm{F}}\gg T$), or mildly degenerate
($T_{\mathrm{F}}\sim T$). In the non-relativistic theory, which is valid at
$x_{\mathrm{r}}\ll 1$, we have $m_{\mathrm{e}}^{*}=m_{\mathrm{e}}$ and
$T_{\mathrm{F}}\approx 3\times 10^{9}\,(\rho_{6}\,Z/A)^{2/3}$ K.

Beyond the elementary transport theory, it is still convenient to use Eq. (2),
but in this case $\nu$ 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

$\nu=\nu_{\mathrm{ei}}+\nu_{\mathrm{ee}},$ (5)

where $\nu_{\mathrm{ei}}$ and $\nu_{\mathrm{ee}}$ 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)

$\nu_{\mathrm{ei}}=4\pi
Z^{2}e^{4}n_{\mathrm{ion}}\,m_{\mathrm{e}}^{*}\,p_{\mathrm{F}}^{-3}\Lambda(p_{\mathrm{F}}),$
(6)

where $\Lambda(p_{\mathrm{F}})$ 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
energy-dependent effective collision frequency, described by Eq. (6) at every
isoenergetic surface, corresponding to a given $p_{\mathrm{F}}$ (e.g., C07;
Potekhin & Yakovlev, 1996; Ventura & Potekhin, 2001).

Although the ei scattering is usually most important for degenerate plasmas,
the ee scattering is non-negligible for $Z\lesssim 10$, being especially
important for H and He. Lampe (1968a) treated the ee scattering using the
Chapman-Enskog solution of the quantum Lenard-Balescu kinetic equation for the
system of degenerate electrons and pointlike non-degenerate, weakly coupled
classical ions. The dynamical screening of the electrons was treated in the
random-phase approximation, applicable at $T\ll T_{\mathrm{F}}$. The author
showed that the character of the scattering is different at temperatures $T\ll
T_{\mathrm{pl}}$ and $T_{\mathrm{pl}}\ll T\ll T_{\mathrm{F}}$, where

$T_{\mathrm{pl}}=\frac{\hbar}{k_{\mathrm{B}}}\,\left(\frac{4\pi
n_{\mathrm{e}}e^{2}}{m_{\mathrm{e}}^{*}}\right)^{1/2}$ (7)

is the electron plasma temperature. In a subsequent paper, Lampe (1968b)
applied the Chapman-Enskog solution of the quantum Lenard-Balescu kinetic
equation to the non-degenerate and weakly degenerate electrons.

Hubbard & Lampe (1969) combined these calculations with earlier results of
Hubbard (1966), who considered the ei opacities $\kappa_{\mathrm{ei}}$ in a
non-relativistic degenerate electron gas, taking into account ion-ion
correlations. Hubbard and Lampe provided conductive opacities in tabular form
for various chemical compositions. Due to the use of different approximations
for non-degenerate 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 non-relativistic theory. The expression of
$\nu_{\mathrm{ee}}$ for the relativistic degenerate electrons was obtained by
Flowers & Itoh (1976) at $T\ll T_{\mathrm{pl}}$, and extended by Urpin &
Yakovlev (1980) to higher temperatures, where $T_{\mathrm{pl}}\lesssim T\ll
T_{\mathrm{F}}$. 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 $\nu_{\mathrm{ee}}$ in
the domain of $x_{\mathrm{r}}\gtrsim 1$, $\theta\ll 1$, and $T\ll
T_{\mathrm{pl}}$. Their fit to to $\nu_{\mathrm{ee}}$ 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

The Matthiessen’s rule (see Eq. 5) results in the additivity of the ei and ee
opacities

$\kappa_{\mathrm{c}}=\kappa_{\mathrm{ei}}+\kappa_{\mathrm{ee}}.$ (8)

and can be derived in the lowest (one-polynomial) approximation of the
Chapman-Enskog method (Chapman, 1954; Hubbard & Lampe, 1969). Although, as
stated by Hubbard & Lampe (1969), at least the two-polynomial 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
$\nu_{\mathrm{ei}}+\nu_{\mathrm{ee}}\leq\nu\leq\nu_{\mathrm{ei}}+\nu_{\mathrm{ee}}+\delta\nu$,
where $\delta\nu\ll\min(\nu_{\mathrm{ei}},\nu_{\mathrm{ee}})$ (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

$\kappa_{\mathrm{c}}=S_{\kappa}\kappa_{\mathrm{ei}}+\kappa_{\mathrm{ee}},$ (9)

where $S_{\kappa}$ 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 Kohn-Sham density-functional theory together with a Kubo-
Greenwood response framework, and compared the results with the quantum
Lenard-Balescu solution in the regime of weak ion coupling
($\Gamma_{\mathrm{i}}\ll 1$) and moderate degeneracy ($T\sim T_{\mathrm{F}}$),
where both methods are applicable. They found that the reshaping factor can be
as low as $S_{\kappa}\sim 0.6$.

Daligault (2016, 2017, 2018) extended the formulas for the transport
coefficients of classical plasmas inside the dense plasma regime by applying
the Chapman-Enskog method to solve the quantum Landau-Fokker-Planck (qLFP)
kinetic equation. The qLFP equation extends the classical LFP equation by
accounting for the Pauli principle while retaining the small-angle collision
approximation. This extension has become possible due to modifications to the
classical Chapman-Enskog 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 $T\gg
T_{\mathrm{F}}$, to lower temperatures. For example (see Fig. 1 in Daligault
2018, where $r_{s}=0.014/x_{\mathrm{r}}$), the 10% accuracy is ensured by the
classical LFP approximation at $T>4T_{\mathrm{F}}$, while the qLFP
approximation provides the same accuracy at $T>1.7T_{\mathrm{F}}$ and
$x_{\mathrm{r}}>0.005$, at $T>T_{\mathrm{F}}$ and $x_{\mathrm{r}}>0.16$, and
also at $T>0.3T_{\mathrm{F}}$ and $x_{\mathrm{r}}>0.25$.

In both LFP and qLFP cases the plasma was assumed weakly coupled and non-
relativistic, and these approximations impose supplementary restrictions on
the validity domain, which can be roughly put as $\Gamma_{\mathrm{i}}\ll 1$
and $x_{\mathrm{r}}\lesssim 1$. 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 mean-force scattering, where the scattering cross-sections 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 $\Gamma_{\mathrm{i}}\ll 1$. They found
a significant suppression of the effective ee scattering rate in a finite
temperature interval, caused by non-monotonicities in the ee mean-force
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, $T/T_{\mathrm{F}}\to 0$, 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 small-angle approximation
and traced a connection to the argument by Lampe (1968a), who noted that
large-angle ee collisions are more strongly Pauli blocked than ei ones,
whereas small-angle 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 above-mentioned
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 long-range order effects (Baiko et
al., 1998; Wetta & Pain, 2020).

###  2.3 Bridging the opacities of mildly and strongly degenerate H and He
plasmas

B20 have 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 electron-electron interactions are less important, so that this
method is expected to produce results more similar to the conductive opacities
$\kappa_{\mathrm{c}}$ obtained using the Matthiessen’s rule, that is, by
assuming $S_{\kappa}=1$. Hereafter, following B20, we will denote the latter
opacities $\kappa_{\mathrm{c}}^{\mathrm{Ioffe}}$. 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 $\kappa_{\mathrm{c}}^{\mathrm{Ioffe}}$. Their Tables 1
and 2 provide $\kappa_{\mathrm{c}}$ for pure H and pure He compositions, which
at fixed temperature $T$ reach densities corresponding to $\theta\equiv
T/T_{\mathrm{F}}$ generally between 0.2 and 0.1. The difference with
$\kappa_{\mathrm{c}}^{\mathrm{Ioffe}}$ exceeds a factor of 2 on the verge of
this density-temperature domain.

To facilitate the implementation of their new opacity calculations in stellar
evolution codes, B20 have 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 read

$\kappa_{\mathrm{c}}=\kappa_{\mathrm{c}}^{\mathrm{Ioffe}}/F,\qquad\lambda_{\mathrm{e}}=F\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}.$
(10)

The correction factor is written as

$F=1+g(x,y)H(g(x,y)),$ (11)

where $x=\log(\rho/\rho_{0})$, $y=\log(T/T_{0})$, $\rho_{0}=10^{5.45}$ g cm-3
and $T_{0}=10^{8.40}$ K for hydrogen, $\rho_{0}=10^{6.50}$ 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 $\theta>0.1$ and ensures that the correction vanishes when
$\theta\to 0$. Hereafter $\kappa_{\mathrm{c}}^{\mathrm{B20}}$ and
$\lambda_{\mathrm{e}}^{\mathrm{B20}}$ denote, respectively, the opacities and
thermal conductivities given by Eqs. (10) and (11).

Figure 1: Thermal conductivity for hydrogen as function of mass density for
several constant temperatures $T$. Dashed lines show the traditional
conductivities $\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$, and dotted lines
display the conductivities
$\lambda_{\mathrm{e}}^{\mathrm{B20}}=F\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$
enhanced by the $F$ factor in Eq. (11). The solid and dot-dashed lines show
the conductivities with the weakly and strongly damped corrections given by
Eq. (12), $\lambda_{\mathrm{e}}^{\mathrm{B20wd}}$ and
$\lambda_{\mathrm{e}}^{\mathrm{B20sd}}$ respectively. They provide two
different transitions from the new calculations for partially degenerate
electrons $\lambda_{\mathrm{e}}^{\mathrm{B20}}$ to the traditional results
$\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ in the strong degeneracy regime (see
text). Numbers near the curves mark $\log\,T$ (K) values.  Figure 2: Same as
in Fig. 1 but for helium.

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 $\rho$ ranges. The convergence of
$\lambda_{\mathrm{e}}^{\mathrm{B20}}$ to the traditional estimate
$\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ is rather slow at high densities, if
the temperature is also high. In this case the B20 correction does not vanish
until $T\ll 0.1T_{\mathrm{F}}$, 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.

This is shown even more clearly in Figs. 3 and 4, which display the ratio
$\lambda_{\mathrm{e}}^{\mathrm{B20}}/\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ as
a function of $\rho$ for several temperatures $T$; densities corresponding to
$\theta=0.1$ and 1.0 are also marked. For example, at $T>10^{7}$ K convergence
is reached only at densities corresponding to $T_{\mathrm{F}}$ values well
above $10\,T$, that is at $\theta\ll 0.1$. Besides, large differences between
the dashed and dotted lines are observed in the case of helium at
$\rho>10^{6}$ g cm-3, where the electrons are relativistic.

To achieve a faster convergence to the degenerate asymptote in the regime of
strong degeneracy, we have introduced a damping factor
$D(\theta)=(1+a\theta^{-b})^{-1}$ ($\theta\equiv T/T_{\mathrm{F}}$). The
damped enhancement factor $F$ for the electron thermal conductivity (a
reduction factor for the conductive opacities) then reads

$F=1+\frac{g(x,y)H(g(x,y))}{1+a\,(T_{\mathrm{F}}/T)^{b}}.$ (12)

Given that the qLFP equation is non-relativistic, we use the non-relativistic
approximation for $T_{\mathrm{F}}$ 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 $D(\theta)$ does not change $F$ by more than 1% at
$T>T_{\mathrm{F}}$ and that it does not exceed 1% (ensuring that $F\approx 1$)
at $T<0.01T_{\mathrm{F}}$. 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 $\lambda_{\mathrm{e}}^{\mathrm{B20wd}}$
are shown in Figs. 1 and 2 (the corresponding opacities will be denoted by
$\kappa_{\mathrm{c}}^{\mathrm{B20wd}}$), while the ratio of
$\lambda_{\mathrm{e}}^{\mathrm{B20wd}}$ to
$\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ as a function of $\rho$ is shown in
Figs. 3 and 4.

We can see that $\lambda_{\mathrm{e}}^{\mathrm{B20}}$ and
$\lambda_{\mathrm{e}}^{\mathrm{B20wd}}$ almost coincide at $T>T_{\mathrm{F}}$
(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 $0.5\,T_{\mathrm{F}}$, while at the
same time it almost fully converges to the traditional results at $T\lesssim
0.03\,T_{\mathrm{F}}$. The ratio
$\lambda_{\mathrm{e}}^{\mathrm{B20wd}}/\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$
is however sometimes still quite substantial (up to a factor of $\sim 1.5$) at
$T\sim 0.1T_{\mathrm{F}}$, where $\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ may
be already preferable, as discussed also by B20. Indeed, as we have seen in
Sect. 2.2, results and discussions by Daligault (2017, 2018) and Shaffer &
Starrett (2020) prompt that the approximations inherent to the qLFP method (in
particular, the small-angle scattering approximation) may lead to an
uncertainty of $\sim 10\%$ at $T=T_{\mathrm{F}}$ and to implausible results at
$T\ll T_{\mathrm{F}}$.

We cannot therefore exclude that in reality the conductivity should converge
to $\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ more rapidly in the transitional
range $0.1T_{\mathrm{F}}\lesssim T\lesssim T_{\mathrm{F}}$. To this purpose,
we have considered a much more extreme, but probably still realistic ‘strong
damping’ choice, defined by the requirements that $D(\theta)$ does not affect
$F$ by more than 10% at $T>T_{\mathrm{F}}$ and that $D(\theta)$ does not
exceed 1% at $T<0.1T_{\mathrm{F}}$. 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
$\lambda_{\mathrm{e}}^{\mathrm{B20sd}}$ and
$\kappa_{\mathrm{c}}^{\mathrm{B20sd}}$, respectively.

Figure 3: Ratio of the electron thermal conductivity for hydrogen enhanced
according to B20 ($\lambda_{\mathrm{e}}^{\mathrm{B20}}$) to
$\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ results, as a function of $\log\,\rho$
(dotted lines), for the labelled values of $\log\,T$, with $T$ given in
Kelvin. Solid lines display the analogous ratio for the thermal conductivities
$\lambda_{\mathrm{e}}^{\mathrm{B20wd}}$ obtained using the weakly damped
enhancement factor $F$. Dot-dashed lines show the corresponding ratio for the
strong damping, $\lambda_{\mathrm{e}}^{\mathrm{B20sd}}$. The vertical lines
mark the boundaries of the range of densities corresponding to $\theta=1$ and
$\theta=0.1$ (higher $\rho$ implies lower $\theta$, at constant $T$).  Figure
4: Same as Fig. 3 but for helium.

The ratio of $\lambda_{\mathrm{e}}^{\mathrm{B20sd}}$ to
$\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ as a function of $\rho$ is also
displayed in Figs. 3 and 4, which show how
$\lambda_{\mathrm{e}}^{\mathrm{B20sd}}$ converges to
$\lambda_{\mathrm{e}}^{\mathrm{Ioffe}}$ at $T\sim 0.1\,T_{\mathrm{F}}$, whilst
it is almost equal to $\lambda_{\mathrm{e}}^{\mathrm{B20}}$ at
$T>T_{\mathrm{F}}$. The values of $\lambda_{\mathrm{e}}^{\mathrm{B20sd}}$ may
noticeably (up to $\sim 30$%) differ from the B20 calculations already at
$T\sim 0.5\,T_{\mathrm{F}}$, 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 $T\sim T_{\mathrm{F}}$, hence
it is not unrealistic to assume still larger inaccuracies at $T\sim
0.5\,T_{\mathrm{F}}$.

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 $\theta\sim 0.1$ (within a factor of 3), which is a $\theta$
range encountered in RGB He-cores and WD envelopes, as shown in Figs. 5 and 6.

Figure 5 displays the run of $\theta$ across the structure of the He-core at
three selected luminosities during the RGB evolution of a typical low-mass
(0.8$M_{\odot}$), metal poor stellar model (from Pietrinferni et al., 2021).
In all three cases, $T/T_{\mathrm{F}}$ ranges between $\sim$0.05 at the centre
of the He-core, and $\sim$1 at its outer edge.

Figure 5: Run of the ratio $\theta\equiv T/T_{\mathrm{F}}$ as a function of
the mass (in solar units) enclosed within a distance $r$ from the centre, for
the He-core 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 $\alpha$-enhanced, with [$\alpha$/Fe]=0.4,
typical of Galactic halo stars) corresponding to [Fe/H]=$-$1.5. The
discontinuity at $M_{r}\approx 0.2M_{\odot}$ in the structure corresponding to
the TRGB stellar model is due to off-center He-burning ignition that starts
removing the electron degeneracy.

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$M_{\odot}$,
bracketing the typical mass range of carbon-oxygen WDs. For both masses, the
layers where $T/T_{\mathrm{F}}$ 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).

Figure 6: Evolution with the surface luminosity $L$ of selected physical and
chemical quantities across WD models with masses equal to 0.55 and
1$M_{\odot}$. 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,
normalized to the total WD mass $M_{\mathrm{WD}}$. Dashed lines mark the
boundaries of the carbon-oxygen core, 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 $\theta$=1 and 0.1.

##  3 Effects on stellar models

In this section we quantify the effect of using alternatively
$\kappa_{\mathrm{c}}^{\mathrm{Ioffe}}$ (C07),
$\kappa_{\mathrm{c}}^{\mathrm{B20}}$ (B20),
$\kappa_{\mathrm{c}}^{\mathrm{B20sd}}$ (B20sd), and
$\kappa_{\mathrm{c}}^{\mathrm{B20wd}}$ (B20wd) opacities on RGB (and the
following horizontal branch stage) and WD models. For the RGB computations we
rely 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 employ the code and physics inputs described by Salaris
et al. (2010).

###  3.1 Red giant branch and horizontal branch models

We have computed models for initial masses in the range 0.8–$2.4M_{\odot}$ and
various chemical compositions, from the pre-main sequence stage until the He-
burning 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).

Let us start by analysing the results for the lower-end of the explored mass
regime, i.e. for the $0.8M_{\odot}$ models, which are characterized 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 He-core mass at helium ignition (${\rm
M_{cHe}}$). Going from C07 to B20 opacities increases ${\rm M_{cHe}}$ by $\sim
0.01M_{\odot}$, independent of $Z$. If the opacities with weak damping B20wd
are used instead, ${\rm M_{cHe}}$ increases by $\sim$0.007$M_{\odot}$ compared
to calculations with C07. Finally, the opacities with strong damping B20sd
increase ${\rm M_{cHe}}$ by just $\sim$0.001$M_{\odot}$.

Figure 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 $\omega$ Centauri.

Given that the TRGB brightness depends on the He-core 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 He-core masses), an increase which is reduced to about 0.07 mag
when calculations with $\kappa_{\mathrm{c}}^{\mathrm{B20wd}}$ are considered
instead. The use of $\kappa_{\mathrm{c}}^{\mathrm{B20sd}}$ instead of the C07
opacities has a negligible impact of the TRGB brightness of the models.

For the sake of comparison, we also show in Fig. 7 the absolute magnitudes of
the TRGB determined for the Galactic globular clusters $\omega$ 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 $I$-band
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 have 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 He-core. The upper panel of Fig. 8 shows ${\rm M_{cHe}}$ at
the ignition of core He-burning for models with $Z=0.001$ and $Y=0.248$, and
masses between 1.4 and 2.4 $M_{\odot}$. For masses larger than $\sim
1.4M_{\odot}$, the effect of choosing a different set of opacities increases,
reaching a maximum between 2.1 and $2.2M_{\odot}$, to then vanish for larger
masses, that do not develop electron degeneracy after the main sequence. For
an initial mass of 2.1–$2.2M_{\odot}$, the B20wd and B20 opacities increase
${\rm M_{cHe}}$ by $\sim 0.035M_{\odot}$ and $\sim 0.043M_{\odot}$,
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.

Figure 8: Upper panel: Trend of the He-core mass at He-ignition as a function
of the initial total mass for models with $Z=0.001$ and $Y=0.248$, and various
assumptions about the conductive opacities. Lower panel: As the upper panel
but for the bolometric luminosity at the beginning of the quiescent core He-
burning, after the electron degeneracy has been lifted.

These variations of the degenerate He-core masses at helium ignition affect
the properties of the following core He-burning 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 $M_{\odot}$ and the labelled initial
composition (we denote this stage by zero age horizontal branch –ZAHB– like
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 ${\rm M_{cHe}}$, as
expected. The effect is maximized for masses around $2.2M_{\odot}$, where
$\Delta\log(L/L_{\odot})\sim 0.2$ dex when passing from C07 to B20 opacities
(the corresponding $\Delta\log(L/L_{\odot})$ for the 1.4$M_{\odot}$ models is
equal to $\sim 0.03$ dex), and $\sim 0.1$ when B20wd opacities are used
instead of C07.

The change of luminosity impacts directly on the core He-burning lifetime: For
models with $1.4\,M_{\odot}$ 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 $2.2\,M_{\odot}$.

Figure 9: The 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).

Figure 9 displays the theoretical ZAHB luminosity of lower mass models, at the
typical temperature of the RR Lyrae instability strip
($\log{T_{\mathrm{eff}}}=3.85$, with $T_{\mathrm{eff}}$ 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 ${\rm
M_{cHe}}$ at He-ignition. Models calculated with the C07 opacities are the
faintest ones, whilst those calculated with B20 opacities are about
$\Delta\log(L/L_{\odot})=0.03$ more luminous. The calculations with the B20sd
opacities are basically identical to the C07 ones, while the B20wd opacities
provide ZAHB models $\Delta\log(L/L_{\odot})\sim 0.02$ brighter than the C07
ones. The corresponding core He-burning lifetimes are affected at the level of
at most 6-8% when adopting these different opacities.

The same figure displays also 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 $\kappa_{\mathrm{c}}^{\mathrm{Ioffe}}$ opacities in the calculations
with the smaller $\kappa_{\mathrm{c}}^{\mathrm{B20}}$ values. Here we make
similar comparisons, but including also the cases of B20wd and B20sd
opacities. We have considered a WD model with mass
$M_{\mathrm{WD}}$=1.0$M_{\odot}$, made of a carbon-oxygen core with chemical
stratification taken from the solar progenitors’ models by Salaris et al.
(2010), surrounded by a helium envelope with mass equal to
$10^{-2}M_{\mathrm{WD}}$, and a more external hydrogen envelope with mass
equal to $10^{-4}M_{\mathrm{WD}}$. 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: Relative differences of the cooling times as a function of the
surface luminosity, among different evolutionary models of a 1.0$M_{\odot}$ 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
dot-dashed line the difference between B20sd and C07 calculations, and the
solid line the difference between B20wd and C07 models.

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
$\log(L/L_{\odot})\sim-1.5$, 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 $\log(L/L_{\odot})=-1.5$.

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 $\log(L/L_{\odot})\sim-2.6$, 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 carbon-oxygen phase separation (see, e.g., Salaris et al., 2010, and
references therein). At the typical luminosity of the faintest observed WDs
($\log(L/L_{\odot})\sim-4.5$) the model calculated with B20 opacities has a
cooling age of $\sim$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 5-10% compared to the previous case of using
B20 opacities. Like for RGB models and their core He-burning progeny,
calculations with the B20sd opacities provide results almost identical to C07
models, with differences of the cooling times within $\pm$5%.

##  4 Summary and conclusions

Electron conduction opacities are a key ingredient in the calculation of
stellar models for low- and intermediate-mass stars, and a critical issue is
how to bridge computations of conductive opacities in the regimes of moderate
($\theta\sim 1$) and strong ($\theta\lesssim 0.1$) degeneracy, which are
necessarily calculated adopting different methods. In fact, the density-
temperature 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 have discussed in detail 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 considered first
B20 own analytical approximation, which however converges to C07 results only
at $\theta\ll$0.1, well into the regime of strong degeneracy. We have then
modified B20 formula by introducing a physically motivated damping factor,
which depends on the ratio $\theta=T/T_{\mathrm{F}}$, tuned in two alternative
ways (weak and strong damping) to converge faster than B20 fit to C07 results
in the regime of strong degeneracy. Both damping prescriptions keep almost
intact the B20 fit at $\theta>1$. The weak damping option provides opacities
still different from C07 at $\theta=0.1$, whilst the more extreme strong
damping converges to C07 opacities at $\theta=0.1$, but changes B20
calculations already by 30% at $\theta\sim 0.5$, in the moderate degeneracy
regime. As a consequence, these three sets of conductive opacities have large
differences (up to a factor $\sim$2) in the critical region around $\theta\sim
0.1$, which in turn have a major impact on the predicted RGB He-core masses
(up to 0.01$M_{\odot}$ for low-mass models far from the transition regime to
non-degenerate He-cores, and up to $\sim 0.04M_{\odot}$ for masses around the
transition), TRGB (up to $\sim$0.1 mag) and ZAHB luminosities (up to 0.03 dex
for masses far from the transition, and up to $\sim$0.2 dex around the
transition), and WD cooling times (up to 40–45% at high luminosities, and up
to $\sim$25% at low luminosities).

Current observational constraints on TRGB and ZAHB absolute magnitudes do not
allow 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 have updated the table of non-magnetic electron conductivities available at
the Ioffe Institute website222http://www.ioffe.ru/astro/conduct/index.html 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 extreme possibility. We have not implemented this correction
directly in the computer code presented a that website, but provided the
corresponding subroutine and envisioned a possibility of its use (in the
absence of a strong magnetic field) to correct the result of the main
computation.

###### Acknowledgements.

We thank our referees for constructive comments that have improved the
presentation of our results. SC acknowledges support from Premiale INAF MITiC,
from INFN (Iniziativa specifica TAsP), and from PLATO ASI-INAF agreement
n.2015-019-R.1-2018. The work of AYP was supported by the Russian Science
Foundation (grant 19-12-00133).

## References

  * Baiko et al. (1998) Baiko, D. A., Kaminker, A. D., Potekhin, A. Y., & Yakovlev, D. G. 1998, Phys. Rev. Lett., 81, 5556
  * Baiko & Yakovlev (1995) Baiko, D. A. & Yakovlev, D. G. 1995, Astronomy Letters, 21, 702
  * Baumgardt & Vasiliev (2021) Baumgardt, H. & Vasiliev, E. 2021, MNRAS, 505, 5957
  * Bellazzini et al. (2004) Bellazzini, M., Ferraro, F. R., Sollima, A., Pancino, E., & Origlia, L. 2004, A&A, 424, 199
  * Beznogov et al. (2021) Beznogov, M. V., Potekhin, A. Y., & Yakovlev, D. G. 2021, Phys. Rep, in press [arXiv:2103.12422]
  * Blouin et al. (2020) Blouin, S., Shaffer, N. R., Saumon, D., & Starrett, C. E. 2020, ApJ, 899, 46
  * Braginskii (1958) Braginskii, S. I. 1958, Sov. Phys. – JETP, 6, 358
  * Cassisi & Castellani (1993) Cassisi, S. & Castellani, V. 1993, ApJS, 88, 509
  * Cassisi et al. (2007) Cassisi, S., Potekhin, A. Y., Pietrinferni, A., Catelan, M., & Salaris, M. 2007, ApJ, 661, 1094
  * Cassisi & Salaris (2013) Cassisi, S. & Salaris, M. 2013, Old Stellar Populations: How to Study the Fossil Record of Galaxy Formation (Wiley-VCH)
  * Catelan (2007) Catelan, M. 2007, in American Institute of Physics Conference Series, Vol. 930, Graduate School in Astronomy: XI Special Courses at the National Observatory of Rio de Janeiro (XI CCE), ed. F. Roig & D. Lopes, 39–90
  * Chabrier & Baraffe (2000) Chabrier, G. & Baraffe, I. 2000, ARA&A, 38, 337
  * Chapman (1954) Chapman, S. 1954, ApJ, 120, 151
  * Chapman & Cowling (1970) Chapman, S. & Cowling, T. G. 1970, The Mathematical Theory of Non-Uniform Gases, 3rd edn. (Cambridge: Cambridge University Press)
  * Daligault (2016) Daligault, J. 2016, Physics of Plasmas, 23, 032706
  * Daligault (2017) Daligault, J. 2017, Phys. Rev. Lett., 119, 045002
  * Daligault (2018) Daligault, J. 2018, Physics of Plasmas, 25, 082703
  * de Santis & Cassisi (1999) de Santis, R. & Cassisi, S. 1999, MNRAS, 308, 97
  * Desjarlais et al. (2017) Desjarlais, M. P., Scullard, C. R., Benedict, L. X., Whitley, H. D., & Redmer, R. 2017, Phys. Rev. E, 95, 033203
  * Flowers & Itoh (1976) Flowers, E. & Itoh, N. 1976, ApJ, 206, 218
  * Flowers & Itoh (1979) Flowers, E. & Itoh, N. 1979, ApJ, 230, 847
  * Flowers & Itoh (1981) Flowers, E. & Itoh, N. 1981, ApJ, 250, 750
  * Fontaine et al. (2001) Fontaine, G., Brassard, P., & Bergeron, P. 2001, PASP, 113, 409
  * Garcia-Berro & Iben (1994) Garcia-Berro, E. & Iben, I. 1994, ApJ, 434, 306
  * Hubbard (1966) Hubbard, W. B. 1966, ApJ, 146, 858
  * Hubbard & Lampe (1969) Hubbard, W. B. & Lampe, M. 1969, ApJS, 18, 297
  * Itoh & Kohyama (1993) Itoh, N. & Kohyama, Y. 1993, ApJ, 404, 268
  * Itoh et al. (1984) Itoh, N., Kohyama, Y., Matsumoto, N., & Seki, M. 1984, ApJ, 285, 758
  * Kippenhahn et al. (2012) Kippenhahn, R., Weigert, A., & Weiss, A. 2012, Stellar Structure and Evolution (Berlin–Heidelberg: Springer)
  * Lampe (1968a) Lampe, M. 1968a, Physical Review, 170, 306
  * Lampe (1968b) Lampe, M. 1968b, Phys. Rev., 174, 276
  * Lee (1950) Lee, T. D. 1950, ApJ, 111, 625
  * Marshak (1941) Marshak, R. E. 1941, Annals of the New York Academy of Sciences, 41, 49
  * Mestel (1950) Mestel, L. 1950, Mathematical Proceedings of the Cambridge Philosophical Society, 46, 331–338
  * Mitake et al. (1984) Mitake, S., Ichimaru, S., & Itoh, N. 1984, ApJ, 277, 375
  * Pietrinferni et al. (2021) Pietrinferni, A., Hidalgo, S., Cassisi, S., et al. 2021, ApJ, 908, 102
  * Potekhin et al. (1999) Potekhin, A. Y., Baiko, D. A., Haensel, P., & Yakovlev, D. G. 1999, A&A, 346, 345
  * Potekhin et al. (2015) Potekhin, A. Y., Pons, J. A., & Page, D. 2015, Space Sci. Rev., 191, 239
  * Potekhin & Yakovlev (1996) Potekhin, A. Y. & Yakovlev, D. G. 1996, A&A, 314, 341
  * Raikh & Yakovlev (1982) Raikh, M. E. & Yakovlev, D. G. 1982, Ap&SS, 87, 193
  * Salaris et al. (2013) Salaris, M., Althaus, L. G., & García-Berro, E. 2013, A&A, 555, A96
  * Salaris et al. (2010) Salaris, M., Cassisi, S., Pietrinferni, A., Kowalski, P. M., & Isern, J. 2010, ApJ, 716, 1241
  * Salaris et al. (2002) Salaris, M., Cassisi, S., & Weiss, A. 2002, PASP, 114, 375
  * Shaffer & Starrett (2020) Shaffer, N. R. & Starrett, C. E. 2020, Phys. Rev. E, 101, 053204
  * Shternin & Yakovlev (2006) Shternin, P. S. & Yakovlev, D. G. 2006, Phys. Rev. D, 74, 043004
  * Siess (2007) Siess, L. 2007, A&A, 476, 893
  * Spitzer & Härm (1953) Spitzer, L. & Härm, R. 1953, Phys. Rev., 89, 977
  * Urpin & Yakovlev (1980) Urpin, V. A. & Yakovlev, D. G. 1980, Sov. Astron., 24, 126
  * Ventura & Potekhin (2001) Ventura, J. & Potekhin, A. 2001, in NATO science series C: Mathematical and physical sciences, Vol. 567, The Neutron Star - Black Hole Connection, ed. C. Kouveliotou, J. Ventura, & E. van den Heuvel, 393–414
  * Wetta & Pain (2020) Wetta, N. & Pain, J.-C. 2020, Phys. Rev. E, 102, 053209
  * Yakovlev (1987) Yakovlev, D. G. 1987, Sov. Ast., 31, 347
  * Yakovlev & Urpin (1980) Yakovlev, D. G. & Urpin, V. A. 1980, Sov. Ast., 24, 303
  * Ziman (1960) Ziman, J. M. 1960, Electrons and Phonons (Oxford: Oxford University Press)


#
Study of speaker localization under dynamic and reverberant environments

Daniel A<EMAIL_ADDRESS>and Boaz<EMAIL_ADDRESS>
### ABSTRACT

Speaker localization in a reverberant environment is a fundamental problem in
audio signal processing. Many solutions have been developed to tackle this
problem. However, previous algorithms typically assume a stationary
environment in which both the microphone array and the sound sources are not
moving. With the emergence of wearable microphone arrays, acoustic scenes have
become dynamic with moving sources and arrays. This calls for algorithms that
perform well in dynamic environments. In this article, we study the
performance of a speaker localization algorithm in such an environment. The
study is based on the recently published EasyCom speech dataset recorded in
reverberant and noisy environments using a wearable array on glasses. Although
the localization algorithm performs well in static environments, its
performance degraded substantially when used on the EasyCom dataset. The paper
presents performance analysis and proposes methods for improvement.
Keywords: Direction of Arrival (DOA) estimation, Direct Path Dominance (DPD)
test, wearable microphone arrays, EasyCom dataset

## 1 INTRODUCTION

Localizing multiple sound sources recorded with a microphone array in an
enclosure is an important task used in a wide range of applications such as
speech enhancement, source separation and video conferencing [2]. Therefore,
many direction-of-arrival (DOA) estimation methods have been developed for
this task. These include methods based on beam-forming [11], subspace methods
such as multiple signal classification (MUSIC) [9], and time-delay of arrival
estimation methods [3]. Many of the algorithms based on these methods were
designed assuming a free-field environment, but when these algorithms are used
in a more common reverberant environment their DOA performance degrades. The
reason for this is that in a reverberant environment, room reflections mask
the direct sound which carries the DOA information. Recently, however, several
methods have been developed for DOA estimation of multiple speakers that are
robust to reverberation. One such method processes the microphone signals in
the time-frequency domain, and employs a direct-path dominance (DPD) test [7]
to identify time-frequency bins that are dominated by the direct sound from
the source. Algorithms which use this method have been widely studied assuming
a static environment, where both the sound sources and the microphone array
are stationary. On the other hand, these algorithms have been less intensely
studied in a more realistic, dynamic environment, where the sound sources
and/or the microphone array are moving.

In dynamic environments, the motion of the sound sources and/or microphone
array may lead to the rapid change of DOAs in time. Thus, to accurately trace
the DOA of speakers requires a short interval between successive DOA
estimates. Additionally, the DOA estimates may be smoothed in time using a
tracking algorithm. Although DOA estimation and tracking algorithms in dynamic
environments have been the subject of several recent studies [5, 12],
including the Acoustic Source Localization and Tracking (LOCATA) [6]
challenge, none of these have included experiments with wearable microphone
arrays - which may bring new challenges. Such scenes are becoming increasingly
popular due to the increased interest in applications involving augmented
reality.

In this article we address the problem of DOA estimation in a noisy dynamic
environment involving a wearable microphone array. The experiments were
performed using the Easy Communication (EasyCom) dataset [4], which was
explicitly designed to represent a realistic cocktail-party environment. The
DOA estimates were computed using a computationally efficient algorithm, which
had been shown previously to have good source localization performance in a
static reverberant environment [10]. The algorithm incorporates a DPD test and
operates in the time-frequency domain. We study the performance and
limitations of this algorithm on the EasyCom dataset under different operating
parameters. We also introduce two modifications of the algorithm and study the
consequent improvement in performance.

## 2 MATHEMATICAL MODEL

In this section we first briefly present the model assumed for the recorded
signal as captured by the microphone array at each time-frequency bin $(t,f)$.
Then, we describe the local space domain distance (LSDD) algorithm [10] for
the DOA estimation at each bin $(t,f)$.

### 2.1 Signal model

Assume a microphone array with $M$ microphones arranged according to a pre-
defined geometry. Next, consider a sound field comprised of $K$ far-field
sources, arriving from directions $\Psi_{k},k\in\\{1,2,\ldots,K\\}$. These
sources represent the direct sound from the speakers in the scene, as well as
reflections (reverberations) due to objects and room boundaries.

In the next step, the recorded microphone signals are transformed into the
joint time-frequency domain by applying the short-time Fourier transform
(STFT). This is done by first separating the speech signal into short time
intervals of length $\delta t$. A fast Fourier transform (FFT) is then applied
to each time segment. Following these pre-processing steps, the signal
received by the microphone array can then be described in the STFT domain as

$\textbf{x}(t,f)=\sum_{k=1}^{K}s_{k}(t,f)\textbf{v}(f,\Psi_{k})+\textbf{n}(t,f)\;,$
(1)

where $\textbf{x}(t,f)=[x_{1}(t,f),x_{2}(t,f),\ldots,x_{M}(t,f)]^{T}$ is an
$M\times 1$ complex vector denoting the signal as measured by the microphones
composing the array;
$\textbf{v}(f,\Psi_{k})=[v_{1}(f,\Psi_{k}),v_{2}(f,\Psi_{k}),\ldots,v_{M}(f,\Psi_{k})]^{T}$
is an $M\times 1$ vector denoting the response of the microphone array to a
unit-amplitude plane wave at frequency $f$ arriving from the $k$th source in
direction $\Psi_{k}$; $s_{k}(t,f)$ is a scalar which represents the amplitude
of the $k$th sound source signal; and
$\textbf{n}(t,f)=[n_{1}(t,f),n_{2}(t,f),\ldots,n_{M}(t,f)]^{T}$ is an $M\times
1$ vector denoting the noise in the signal $\textbf{x}(t,f)$.

### 2.2 Local Space Domain Distance (LSDD) algorithm

The LSDD algorithm is a recently developed DOA estimation algorithm
characterized by DOA performance that is robust to reverberation. The
algorithm was first proposed in [10] and works as follows. The directional
spectrum $\textbf{S}(t,f)=[S_{1}(t,f),S_{2}(t,f),\ldots,S_{L}(t,f)]^{T}$
computed within this algorithm is an $L\times 1$ vector defined over a grid of
DOAs $\Theta_{l},l\in\\{1,2,\ldots,L\\}$. The $l$th component is defined as

$S_{l}(t,f)=d\left(\textbf{x}\left(t,f\right),\textbf{v}\left(f,\Theta_{l}\right)\right)=d\left([x_{1}(t,f),\ldots,x_{M}(t,f)]^{T},[v_{1}(f,\Theta_{l}),\ldots,v_{M}(f,\Theta_{l})]^{T}\right)\;,$
(2)

where $d(\textbf{a},\textbf{b})$ is a function which measures the similarity
between two vectors a and b. In [10], $d(\textbf{a},\textbf{b})$ was defined
as

$d(\textbf{a},\textbf{b})=\frac{1}{{\underset{\beta}{\min}}\big{(}\frac{\|\textbf{a}-\beta\textbf{b}\|}{\|\textbf{a}\|}\big{)}}\;,$
(3)

where $\|.\|$ is the 2-norm. However, in this article we shall use the more
conventional cosine similarity measure:

$d(\textbf{a},\textbf{b})=\frac{|<\textbf{a},\textbf{b}>|}{\|\textbf{a}\|\|\textbf{b}\|}\;,$
(4)

where $<\textbf{a},\textbf{b}>$ denotes the inner product between a and b.

Given the spectrum vector $\textbf{S}(t,f)$, the estimated DOA for bin $(t,f)$
is computed by

$\hat{\theta}(t,f)=\underset{l}{\arg\max}\\{S_{l}(t,f)\\}\;.$ (5)

However, some of the bins $(t,f)$ do not contain a valid $\hat{\theta}(t,f)$
value. These are bins in which the direct signal from the speaker is masked by
noise and reverberations. We eliminate these bins by calculating a DPD measure
value ${\chi}(t,f)$ for each bin which we then test against a threshold
$\lambda$. Although there are different methods available for calculating
${\chi}(t,f)$, we shall, for simplicity, use the following:

${\chi}(t,f)=\underset{l}{\max}\left\\{S_{l}(t,f)\right\\}\;.$ (6)

Together, Eqns. (5) and (6) define a (joint) LSDD DOA/DPD algorithm.

### 2.3 Energy Weighted Local Space Domain Distance (LSDDe) algorithm

We describe an energy weighted modification for calculating the DPD test value
${\chi}(t,f)$. In this equation, we weight the DPD test value in Eqn. (6) with
its corresponding signal energy. The energy weighted DPD test value is then:

${\chi}(t,f)=\underset{l}{\max}\left\\{S_{l}(t,f)\cdot
MED[|x_{1}(t,f)|^{2},|x_{2}(t,f)|^{2},\ldots,|x_{M}(t,f)|^{2}]\right\\}\;,$
(7)

where $MED$ is the median operator. Together, Eqns. (5) and (7) define a
(joint) LSDDe DOA/DPD algorithm.

## 3 PROPOSED DOA ESTIMATION ALGORITHM

We propose a new DOA estimation algorithm which was the outcome of
investigating the performance of the LSDD algorithm with the EasyCom dataset.
It is clear from Eqns. (2), (4) and (5) that the LSDD algorithm does not use
any information about the behavior of $\textbf{S}(t,f)$ with respect to
$\Theta_{l}$. As such information may be useful, it is proposed to incorporate
this information using a correlation process, as follows. For each frequency
$f$ we define an ïdeal two-dimensional spectrum represented by matrix W, whose
elements, $W_{lh}\equiv W(\Theta_{l},\Theta_{h})$, represent the similarity
between the $l$th steering vector $\textbf{v}(f,\Theta_{l})$ and the $h$th
steering vector $\textbf{v}(f,\Theta_{h}$) defined as

$\begin{split}W(\Theta_{l},\Theta_{h})&=d(\textbf{v}(f,\Theta_{l}),\textbf{v}(f,\Theta_{h}))\\\
&=d([v_{1}(f,\Theta_{l}),\ldots,v_{M}(f,\Theta_{l})]^{T},[v_{1}(f,\Theta_{h}),\ldots,v_{M}(f,\Theta_{h})]^{T})\;,\forall
l,h\in\\{1,2,\ldots,L\\}\;,\end{split}$ (8)

where the measure $d$ is defined in (4). Now, the similarity between each
column vector in W and the spectrum $\textbf{S}(t,f)$ is computed, which
provides the following indication. Suppose that $\Theta_{h}$ is a DOA of an
actual source. Then, we expect the $h$th column of W to be similar to
$\textbf{S}(t,f)$. This, in effect, defines the new directivity based Space
Domain Distance (dSDD) DOA estimation algorithm. The corresponding DOA
estimate for bin $(t,f)$ is now computed using

$\hat{\theta}(t,f)=\underset{h}{\arg\max}\left\\{d(\textbf{S}(t,f),\textbf{W}_{h})\right\\}\;,$
(9)

where $\textbf{W}_{h}$ denotes the $h$th column in W. We follow Eqn. (6) and
define a corresponding DPD test measure as

${\chi}(t,f)=\underset{h}{\max}\left\\{d(\textbf{S}(t,f),\textbf{W}_{h})\right\\}\;.$
(10)

Together, Eqns. (9) and (10) define a (joint) dSDD DOA/DPD algorithm. It
should be noted that under ideal conditions where signal x in Eqn. (1) is
composed of a single plane wave, the two algorithms, LSDD and dSDD, should
provide the same estimate as they both rely on the same set of steering
vectors. However, the motivation for proposing dSDD is the expected robustness
against potential noise and reverberation due to the comparison of entire
functions, or vectors. This is in contrast to the LSDD where DOA estimates are
based on looking for a peak in a function.

As in the case of the LSDD algorithm, we describe an energy weighted dSDD
algorithm (dSDDe), in which, for each bin $(t,f)$, we weight the dSDD DPD test
measure with the corresponding signal energy. The energy weighted DPD test
value is therefore given by

${\chi}(t,f)=\underset{h}{\max}\left\\{d(\textbf{S}(t,f),\textbf{W}_{h})\right\\}\cdot
MED\left[|x_{1}(t,f)|^{2},|x_{2}(t,f)|^{2},\ldots,|x_{M}(t,f)|^{2}\right]\;.$
(11)

## 4 EXPERIMENTS

This section presents an experimental study that aimed to investigate the
performance of the LSDD and the dSDD algorithms with the EasyComm dataset.
First, the experimental setup and methodology are presented. This is then
followed by the evaluation of the results.

### 4.1 Set-up

The experiments described in this article were performed on the EasyCom
dataset [4]. This dataset was designed with the aim of analyzing the cocktail
party effect with audio signals captured by augmented reality (AR) glasses
equipped with an egocentric six-channel microphone array. Figure 1 shows a
schematic drawing of the glasses with locations of the microphones [4].

The dataset contains recordings of natural conversations in a noisy restaurant
environment. Participants were equipped with close-talk microphones, a camera
and tracking markers. They were asked to engage in conversations during
several tasks, including introductions, ordering food, solving puzzles,
playing games and reading sentences. The recordings also contain an egocentric
video viewpoint of the participants. The pose (position and rotation) of every
participant was also recorded. The dataset was additionally labelled with
annotators of voice activity.

Figure 1: Illustration of the AR glasses with locations of microphones [4].
Four of the microphones are fixed rigidly to the glasses and two of the
microphones are placed in the user’s ears.

The signals recorded by the microphones were sampled at a rate of $48\textrm{
kHz}$. The recorded data was transformed into the STFT domain using a 1024
samples ($\simeq 20\textrm{ msec}$) Hann window with an overlap of 512
samples. The microphone signals in the STFT domain were employed as an input
to the algorithms under study.

### 4.2 Methodology

Evaluation of the DOA/DPD algorithms incorporated a direction search with a
resolution of $6^{\circ}$, which was limited to the horizontal plane. The
ground truth azimuthal DOA ($\Psi_{k}$) was obtained from the EasyCom dataset
as a function of time. Altogether, a series of three experiments was performed
with the EasyCom dataset. The first experiment measured the effective
frequency range of the array $[f_{low},f_{high}]$, and the second and third
experiments investigated the effect of frequency smoothing and the length of
the time interval $\Delta T$ on the performance.

DOA estimation performance was evaluated as follows. For each $(t,f)$ bin, the
absolute error,

$\varepsilon(t,f)=|\Psi(t)-\hat{\theta}(t,f)|\;,$ (12)

between the true DOA $\Psi(t)$ and the estimated DOA $\hat{\theta}(t,f)$ was
computed. Note that Eqn. (12) assumes that both $\Psi(t)$ and
$\hat{\theta}(t,f)$ are measured with respect to the same axis. In practice,
in this dataset, $\Psi(t)$ is measured with respect to an axis defined
relative to the room, while $\hat{\theta}(t,f)$ is measured with respect to
the orientation of the glasses. Thus, before calculating $\varepsilon(t,f)$,
$\hat{\theta}(t,f)$ is transformed to the fixed axis of the room by
incorporating head tracking information. In addition, each bin was labeled a
"hit" if $\varepsilon(t,f)\leq 10^{\circ}$; otherwise it was labeled a "miss".

The bins ($t,f$) were divided into time blocks of length $\Delta T$. For each
block, only bins which satisfied the following four conditions were regarded
as valid and were used for DOA estimation:

  1. 1.

The frequency $f$ lies within the effective operating frequency range:
$f\in[f_{low},f_{high}]$.

  2. 2.

The time $t$ lies within the selected block. Denoting $T$ as the middle time
of the block, then $t\in[T-\Delta T/2,T+\Delta T/2]$ belongs to block $T$.

  3. 3.

Voice activity was detected at time $t$.

  4. 4.

The DPD test value $\chi(t,f)$ exceeds a threshold $\lambda$. For the purpose
of this study, the threshold was determined from the percentage $p$ of bins
which satisfy the first three conditions with the highest value $\chi$. Note
that $\lambda$ is computed independently for each block $T$.

The mean value of $\varepsilon(t,f)$ was computed over valid $(t,f)$ bins,
i.e. bins which satisfy the above four conditions. By definition, this is the
mean absolute DOA estimation error $E(p,T,\Delta T)$ for the block $T$.
Similarly, the mean hit ratio $H(p,T,\Delta T)$ was computed on valid $(t,f)$
bins by dividing the number of valid bins labelled "hit" by the total number
of valid bins. Finally, the mean absolute error, $\bar{E}(p,\Delta T)$, and
the mean hit ratio, $\bar{H}(p,\Delta T)$, for the entire experiment was
computed by averaging $E(p,T,\Delta T)$ and $H(p,T,\Delta T)$ over all blocks
$T$.

### 4.3 Effective operating frequency band

The EasyCom dataset involves speech sound which naturally limits the frequency
range of interest [8]. This frequency band is reduced in practice by aliasing
effects which arise from the microphone array. For a specific steering vector
$\textbf{v}(f,\Theta_{h})$ (corresponding to a frequency $f$ and direction
$\Theta_{h}$), the similarity between $\textbf{v}(f,\Theta_{h})$ and the set
of steering vectors $\textbf{v}(f,\Theta_{l}),l\in\\{1,2,...,L\\}$ was
computed using Eqn. (4). This was repeated for all frequencies $f$ leading to
the following measure:

${\Lambda}(f,\Theta_{l})=d(\textbf{v}(f,\Theta_{h}),\textbf{v}(f,\Theta_{l}))\;.$
(13)

Figure 2 shows $\Lambda$ for $\Theta_{h}=0^{\circ}$. Visual inspection shows
that the preferred frequency band is about $1100-2000$ Hz, where at lower
frequencies the directivity may be too wide, while at higher frequencies
significant side lobes may degrade spatial processing. While this is a
relatively narrow band of frequencies, in this work it led to the best
performance. Extending the range of operation for both lower and higher
frequencies is proposed for future work.

Figure 2: The similarity matrix $\Lambda$ for $\Theta_{h}=0^{\circ}$.

### 4.4 DOA error and frequency smoothing

The spectrum $\textbf{S}(t,f)$ plays a key role in the DOA/DPD algorithms. In
particular, the authors in [1] have shown that it is beneficial to smooth
$\textbf{S}(t,f)$ in frequency. We investigated the effect of smoothing
$\textbf{S}(t,f)$ over frequency using a moving average filter of length
$(2R+1)$. Let
$\bar{\textbf{S}}(t,f)=[\bar{S}_{1}(t,f),\bar{S}_{2}(t,f),\ldots,\bar{S}_{L}(t,f)]^{T}$
denote the smoothed spectrum, which is computed by

$\bar{S}_{l}(t,f)=\sum_{r=-R}^{R}S_{l}(t,f+r\Delta f)/(2R+1)\;,$ (14)

where $\Delta f$ is the STFT frequency bin width and $\textbf{S}_{l}(t,f)$ is
defined in Eqn. (2).

DOA estimation experiments were carried out on several 1-minute segments
extracted from the EasyCom dataset. Altogether, $\textbf{S}(t,f)$ was
frequency smoothed using (i) a 9-element filter, (ii) a 3-element filter and
(iii) no smoothing. In this experiment, $\Delta T$ was fixed to $200$ msec.
The corresponding results are shown in Fig. 3.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3: The mean absolute error $\bar{E}(p,\Delta T)$ and the mean hit ratio
$\bar{H}(p,\Delta T)$ as a function of $p$ when averaged over ten 1-minute
data segments. The time interval was $\Delta T=200$ msec. (a)-(c) Show,
respectively, $\bar{E}(p,\Delta T)$ with a $9$-element smoothing filter, a
$3$-element smoothing filter and no smoothing. (d)-(f) show the corresponding
$\bar{H}(p,\Delta T)$ curves.

Figure 3 shows that best overall performance, in terms of both
$\bar{E}(p,\Delta T)$ and $\bar{H}(p,\Delta T)$, for the original LSDD
algorithm as described in Sec. 2.2 is obtained with the $9$-element filter. On
the other hand, for both LSDDe as in Sect. 2.3 and dSDDe as in Sect. 3, the
best overall performance is obtained when no smoothing is used. Moreover, for
both algorithms the change in performance with respect to smoothing is
relatively small.

Finally, we compare the results obtained with the variation of the LSDD
algorithm (i.e with a $9$-element smoothing filter) and the best variations of
the LSDDe and dSDDe algorithms (i.e. no smoothing). At low percentages $p$ the
mean absolute error $\bar{E}(p,\Delta T)$ obtained with the dSDDe algorithm is
approximately $9^{\circ}$ lower than that obtained with the LSDD algorithm.
Similarly, the mean hit ratio obtained with the dSDDe algorithm is
approximately $5\%$ higher than that obtained with the LSDD algorithm.

### 4.5 DOA error and time interval $\Delta T$

The choice of time interval $\Delta T$ may be directly related to the dynamic
nature of the dataset. In general, we would like to use a value of $\Delta T$
which is small enough such that the environment can be considered spatially
stationary within the interval. However, if the chosen value of $\Delta T$ is
too small, DOA performance may degrade. In this experiment, three different
values of $\Delta T$ were investigated: (a) $\Delta T=200$ msec; (b) $\Delta
T=300$ msec; and (c) $\Delta T=500$ msec. The DOA/DPD algorithms used the best
frequency smoothing as presented in the previous section. Figure 4 shows
$\bar{E}(p,{\Delta T})$ averaged over ten 1-minute segments for three $\Delta
T$ values. The figure shows a consistent and significant improvement in
performance as $\Delta T$ increases. At $\Delta T=500$ msec and $p=1\%$, the
dSDDe algorithm gave a mean absolute DOA error $\bar{E}\approx 20^{\circ}$.
Similar to Fig. 3, the dSDDe algorithm achieved the best performance, and the
LSDD algorithm the worst.

(a)

(b)

(c)

Figure 4:  $\bar{E}(p,{\Delta T})$ as a function of percentage $p$ for the
LSDD, LSDDe and dSDDe algorithms for three different ${\Delta T}$ values,
averaged over ten 1-minute segments. (a) ${\Delta T}=200$ msec; (b) ${\Delta
T}=300$ msec; (c) ${\Delta T}=500$ msec. Best frequency smoothing was applied
as in Sec. 4.4.

### 4.6 Summary

Figure 5 illustrates the performance of the dSDDe algorithm together with a
timeline of several experimental data values, for an example taken from the
EasyCom dataset. In this example there are two active speakers. The time
interval $\Delta T$ used in the figure is $200$ msec.

Figure 5: Ground-truth and estimated DOAs for a 1-minute long segment taken
from the EasyCom dataset, including two speakers. Ground-truth directions are
presented using dashed thick lines. The orientation of the microphone array is
marked using a thick red line. The DOA estimates $\hat{\theta}(t,f)$ provided
by the dSDDe algorithm with $p=1\%$ and $\Delta T=200$ msec are shown as
colored circles.

The figure illustrates the dynamic nature of the environment. While many of
the DOA estimates fall close to active speakers, some fall very far away from
the true DOA. There is also a clear bias in the estimation toward a preferred
direction. These findings raise the need for further research to better
understand the dataset and the algorithms, and propose improved solutions.

## 5 CONCLUSIONS

This work presented three experiments for DOA estimation based on the EasyCom
dataset. These preliminary experiments showed that:

  1. 1.

The baseline performance using the original LSDD algorithm showed limited
performance.

  2. 2.

The dSDD algorithm improved performance by incorporating more detailed spatial
information.

  3. 3.

Energy weighting DPD test values was found to be useful.

  4. 4.

Both the LSDDe and the dSDDe DOA algorithms do not require frequency
smoothing.

## ACKNOWLEDGEMENTS

This research was supported by Reality Labs at Meta.

## REFERENCES

##

  * [1] H. Beit-On, V. Tourbabin, and B. Rafaely.  The importance of time-frequency averaging for binaural speaker localization in reverberant environments.  In INTERSPEECH, pages 5071–5075, 2020.
  * [2] M. Brandstein and D. Ward.  Microphone arrays: signal processing techniques and applications.  Springer Science & Business Media, 2001.
  * [3] M. S. Brandstein and H. F. Silverman.  A robust method for speech signal time-delay estimation in reverberant rooms.  In 1997 IEEE International Conference on Acoustics, Speech, and Signal Processing, volume 1, pages 375–378. IEEE, 1997.
  * [4] J. Donley, V. Tourbabin, J.-S. Lee, M. Broyles, H. Jiang, J. Shen, M. Pantic, V. K. Ithapu, and R. Mehra.  Easycom: An augmented reality dataset to support algorithms for easy communication in noisy environments.  2021\.
  * [5] C. Evers and P. A. Naylor.  Acoustic slam.  IEEE/ACM Transactions on Audio, Speech, and Language Processing, 26(9):1484–1498, 2018.
  * [6] H. W. Löllmann, C. Evers, A. Schmidt, H. Mellmann, H. Barfuss, P. A. Naylor, and W. Kellermann.  The locata challenge data corpus for acoustic source localization and tracking.  In 2018 IEEE 10th Sensor Array and Multichannel Signal Processing Workshop (SAM), pages 410–414. IEEE, 2018.
  * [7] L. Madmoni and B. Rafaely.  Improved direct-path dominance test for speaker localization in reverberant environments.  In 2018 26th European Signal Processing Conference (EUSIPCO), pages 2424–2428. IEEE, 2018.
  * [8] B. B. Monson, E. J. Hunter, A. J. Lotto, and B. H. Story.  The perceptual significance of high-frequency energy in the human voice.  Frontiers in Psychology, 5:587, 2014.
  * [9] R. Schmidt.  Multiple emitter location and signal parameter estimation.  IEEE Transactions on Antennas and Propagation, 34(3):276–280, 1986.
  * [10] V. Tourbabin, D. L. Alon, and R. Mehra.  Space domain-based selection of direct-sound bins in the context of improved robustness to reverberation in direction of arrival estimation.  In Proc. 11th European Congress and Exposition on Noise Control Engineering (EURONOISE18), pages 2589–2596, 2018.
  * [11] B. D. Van Veen and K. M. Buckley.  Beamforming: A versatile approach to spatial filtering.  IEEE ASSP Magazine, 5(2):4–24, 1988.
  * [12] W. Wang and R. Wu.  High-dynamic doa estimation based on weighted l 1 minimization.  Progress In Electromagnetics Research C, 42:253–265, 2013.