# General relativistic pulsations of ultra-massive ZZ Ceti stars

Alejandro H. Córsico1,2, S. Reece Boston3,4, Leandro G. Althaus1,2, Mukremin
Kilic5, S. O. Kepler6, María E. Camisassa7 and Santiago Torres7,8,
1 Facultad de Ciencias Astronómicas y Geofísicas, Universidad Nacional de La
Plata, Paseo del Bosque s/n, 1900 La Plata, Argentina
2 Instituto de Astrofísica de La Plata, UNLP-CONICET, Paseo del Bosque s/n,
1900 La Plata, Argentina
3 University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA
4 Computer Science and Mathematics Division, Oak Ridge National Laboratory,
Oak Ridge, TN 37830, USA
5 Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma,
440 W. Brooks St., Norman, OK 73019, USA
6 Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-900
Porto-Alegre, RS, Brazil
7 Departament de Física, Universitat Politècnica de Catalunya, c/Esteve
Terrades 5, 08860 Castelldefels, Spain
8 Institute for Space Studies of Catalonia, c/Gran Capita 2–4, Edif. Nexus
104, 08034 Barcelona, Spain E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Ultra-massive white dwarf stars are currently being discovered at a
considerable rate, thanks to surveys such as the Gaia space mission. These
dense and compact stellar remnants likely play a major role in type Ia
supernova explosions. It is possible to probe the interiors of ultra-massive
white dwarfs through asteroseismology. In the case of the most massive white
dwarfs, General Relativity could affect their structure and pulsations
substantially. In this work, we present results of relativistic pulsation
calculations employing relativistic ultra-massive ONe-core white dwarf models
with hydrogen-rich atmospheres and masses ranging from $1.29$ to
$1.369M_{\sun}$ with the aim of assessing the impact of General Relativity on
the adiabatic gravity ($g$)-mode period spectrum of very-high mass ZZ Ceti
stars. Employing the relativistic Cowling approximation for the pulsation
analysis, we find that the critical buoyancy (Brunt-Väisälä) and acoustic
(Lamb) frequencies are larger for the relativistic case, compared to the
Newtonian case, due to the relativistic white dwarf models having smaller
radii and higher gravities for a fixed stellar mass. In addition, the $g$-mode
periods are shorter in the relativistic case than in the Newtonian
computations, with relative differences of up to $\sim 50$ % for the highest-
mass models ($1.369M_{\sun}$) and for effective temperatures typical of the ZZ
Ceti instability strip. Hence, the effects of General Relativity on the
structure, evolution, and pulsations of white dwarfs with masses larger than
$\sim 1.29M_{\odot}$ cannot be ignored in the asteroseismological analysis of
ultra-massive ZZ Ceti stars.

###### keywords:

stars: evolution — stars: interiors — stars: white dwarfs — stars: pulsations
— asteroseismology — relativistic processes

††pubyear: 2023††pagerange: General relativistic pulsations of ultra-massive
ZZ Ceti stars–2

##  1 Introduction

ZZ Ceti variables are pulsating DA (H-rich atmosphere) white dwarf (WD) stars
with effective temperatures in the range $10\,500\lesssim T_{\rm eff}\lesssim
13\,500$ K and surface gravities in the interval $7.5\lesssim\log g\lesssim
9.35$. They exhibit periods from $\sim 100$ s to $\sim 1400$ s due to
nonradial gravity($g$) modes with harmonic degree $\ell=1$ and $\ell=2$
(Winget & Kepler, 2008; Fontaine & Brassard, 2008; Althaus et al., 2010). The
interiors of these compact stars, which constitute the evolutionary end of
most stars in the Universe, can be investigated through the powerful tool of
asteroseismology by comparing the observed periods with theoretical periods
computed using large grids of WD stellar models (e.g. Córsico et al., 2019).

Although most ZZ Ceti stars have masses between $\sim 0.5$ and $\sim
0.8~{}M_{\odot}$, at least seven ultra-massive ($M_{\star}\gtrsim
1.05M_{\odot}$) ZZ Ceti stars have been discovered so far: BPM 37093
($M_{\star}=1.13M_{\odot}$; Kanaan et al., 1992; Bédard et al., 2017), GD 518
($M_{\star}=1.24~{}M_{\odot}$; Hermes et al., 2013), SDSS J084021.23+522217.4
($M_{\star}=1.16~{}M_{\odot}$; Curd et al., 2017), WD J212402.03$-$600100.05
($M_{\star}=1.16~{}M_{\odot}$; Rowan et al., 2019), J0204+8713 and J0551+4135
($M_{\star}=1.05~{}M_{\odot}$ and $M_{\star}=1.13~{}M_{\odot}$, respectively;
Vincent et al., 2020), and WD J004917.14$-$252556.81 ($M_{\star}\sim
1.30~{}M_{\odot}$; Kilic et al., 2023b). With such a high stellar mass, the
latter is the most massive pulsating WD currently known. The discovery and
characterisation of pulsating ultra-massive WDs through asteroseismology is
important for understanding the supernovae type Ia explosions. We know that
accreting CO-core WDs are the progenitors of these explosions (e.g., Nugent et
al., 2011; Maoz et al., 2014), but we have not been able to probe the interior
structure of such WDs near the Chandrasekhar limit.

Modern photometric data of pulsating WDs collected by spacecrafts such as the
ongoing Transiting Exoplanet Survey Satellite mission (TESS; Ricker et al.,
2014) and the already finished Kepler/K2 space mission (Borucki et al., 2010;
Howell et al., 2014), brought along revolutionary improvements to the field of
WD asteroseismology in at least two aspects (Córsico, 2020, 2022). First, the
space missions provide pulsation periods with an unprecedented precision.
Indeed, the observational precision limit of TESS for the pulsation periods is
of order $\sim 10^{-4}$ s or even smaller (Giammichele et al., 2022). Second,
these space missions also enable the discovery of large numbers of new
pulsating WDs. For example, Romero et al. (2022) used the TESS data from the
first three years of the mission, for Sectors 1 through 39, to identify 74 new
ZZ Ceti stars, which increased the number of already known ZZ Cetis by $\sim
20$ per cent. It is likely that many more pulsating WDs, not only average-mass
($M_{\star}\sim 0.60~{}M_{\sun}$) objects, but also ultra-massive WDs, will be
identified by TESS and other future space telescopes such as the Ultraviolet
Transient Astronomy Satellite (ULTRASAT, Ben-Ami et al., 2022) in the coming
years, though TESS’s relatively small aperture limits its ability to observe
intrinsically fainter massive WDs. In addition, large-scale wide-field ground-
based photometric surveys like the Vera C. Rubin Observatory’s Legacy Survey
of Space and Time and the BlackGEM (Groot et al., 2022) will significantly
increase the population of WD pulsators, including massive WDs.

The use of space telescopes for WD asteroseismology has opened up a new window
into the interiors of these stars and led to some new and interesting
questions. For example, the availability of pulsation periods with high
precision supplied by modern space-based photometric observations has, for the
first time, raised the question of whether it is possible to detect very
subtle effects in the observed period patterns, such as the signatures of the
current experimental 12C$(\alpha,\gamma)^{16}$O reaction rate probability
distribution function (Chidester et al., 2022), or the possible impact of
General Relativity (GR) on the pulsation periods of ZZ Ceti stars (Boston et
al., 2023). In particular, the possibility that relativistic effects can be
larger than the uncertainties in the observed periods when measured with space
missions has led Boston et al. (2023) to conclude that, for average mass WDs,
the relative differences between periods in the Newtonian and relativistic
calculations can be larger than the observational precision with which the
periods are measured. Hence, to fully exploit the unprecedented quality of the
observational data from TESS and similar space missions, it is necessary to
take into account the GR effects on the structure and pulsations of WDs.

The impact of GR is stronger as we consider more massive WD configurations. In
particular, WDs with masses close to the Chandrasekhar mass ($M_{\rm Ch}\sim
1.4M_{\sun}$). Carvalho et al. (2018); Nunes et al. (2021) and Althaus et al.
(2022) used static WD models and evolutionary ONe-core WD configurations,
respectively, to explore the effects of GR on the structure of ultra-massive
WDs. These investigations found that GR strongly impacts the radius and
surface gravity of ultra-massive WDs. In addition, Althaus et al. (2022) found
that GR leads to important changes in cooling ages and in mass-radius
relationships when compared with Newtonian computations. Furthermore, Althaus
et al. (2023) have extended the relativistic computations to CO-core ultra-
massive WD models.

In the present work, we aim to assess the impact of GR on the $g$-mode period
spectra of ultra-massive ZZ Ceti stars with masses $\gtrsim 1.29M_{\sun}$.
This is the lower limit for the WD mass from which the effects of GR begin to
be relevant (Althaus et al., 2022). Our analysis is complementary to that of
Boston et al. (2023), which is focused on average-mass pulsating DA WDs ($\sim
0.60M_{\sun}$; the bulk of pulsating WD population). For these average-mass DA
WDs, the difference of Newtonian physics and GR was shown to be on the order
of the surface gravitational redshift $z\sim 10^{-4}$, though for stars with
very high central concentration of mass this difference could be an order of
magnitude larger. Since the ultra-massive WDs are highly centrally condensed,
GR might be even more important for these objects. The study of ultra-massive
WDs is of particular interest at present, given the increasing rate of
discovery of these objects (Gagné et al., 2018; Kilic et al., 2020, 2021;
Hollands et al., 2020; Caiazzo et al., 2021; Torres et al., 2022; Kilic et
al., 2023a) and the prospect of finding pulsating ultra-massive WDs more
massive than WD J004917.14$-$252556.81 (Kilic et al., 2023b). This last point
is particularly relevant in view of the capabilities of the current (e.g.
TESS) and upcoming (e.g., ULTRASAT, LSST, BlackGEM) surveys.

The formalism of stellar pulsations in GR began with Thorne & Campolattaro
(1967), using the Regge-Wheeler gauge to treat the pulsations as linear
perturbations on top of a static, spherically symmetric background (Regge &
Wheeler, 1957). The result was a reduction in the Einstein Field Equations
(EFE) that describe spacetime curvature in GR to only five complex-valued
equations for the perturbation amplitudes. Further theoretical work showed
this system was only 4th-complex-order, with two degrees of freedom describing
the fluid perturbations and two describing the gravitational perturbations
(Ipser & Thorne, 1973). Later, Detweiler & Lindblom (1985) (see also Lindblom
& Detweiler, 1983) reduced the perturbed EFE to the explicit form of four 1st-
order complex-valued equations describing the normal mode perturbations. For
quadrupole modes or higher ($\ell\geq 2$), the two gravitational degrees of
freedom at the surface produce outgoing gravitational radiation (i.e.
gravitational waves) which will gradually damp any excitations, so that
stellar perturbations in GR can be at best quasinormal.

In asteroseismology, the outgoing gravitational radiation is largely an
undesired complication, requiring specialized methods to avoid carrying the
boundary condition out to spatial infinity (Chandrasekhar & Detweiler, 1975;
Fackerell, 1971; Lindblom et al., 1997; Andersson et al., 1995). The outgoing
gravitational waves can be easily removed using a form of the Cowling
approximation within GR, first developed by McDermott et al. (1983) and
further studied by Lindblom & Splinter (1990), McDermott et al. (1985), and
Yoshida & Lee (2002). In this relativistic Cowling approximation, the
gravitational degrees of freedom are set to zero, retaining only the fluid
perturbations. Further, there is no intrinsic damping, so that the problem
becomes real-valued and the modes are stationary. This treatment is widely
used to study the pulsation and stability of compact stellar objects in
situations where knowledge of the outgoing gravitational waves is irrelevant,
and especially in stars with surface crystallization (Flores et al., 2017;
Yoshida & Lee, 2002). Another approach to include the relativistic effects in
stellar pulsations is to use the Post-Newtonian approximation (Cutler, 1991;
Poisson & Will, 2014; Boston et al., 2023). This approach is able to include
gravitational perturbations in the form of two scalar potentials and a vector
potential, without also producing gravitational radiation (Boston, 2022).

Most interest in pulsations of relativistic stars has focused on neutron stars
(e.g. McDermott et al., 1988; Cutler & Lindblom, 1992; Lindblom & Splinter,
1989a). The earliest calculations of pulsations in WDs involving GR tried to
address the origin of radio sources discovered by Hewish et al. (1968), as an
alternative to a neutron star origin (Thorne & Ipser, 1968). These studies,
which date back to the late 1960s, were devoted to computing the fundamental
radial pulsation mode of Hamada-Salpeter WD models (Hamada & Salpeter, 1961)
including GR effects (Faulkner & Gribbin, 1968; Skilling, 1968; Cohen et al.,
1969). Boston et al. (2023) have recently renewed interest in this topic by
focusing on relativistic pulsations of ZZ Ceti stars and other pulsating WDs,
concentrating on average-mass WDs. In the present paper, we study the impact
of GR on realistic evolutionary stellar models of ultra-massive DA WDs
computed by Althaus et al. (2022), which are representative of very high-mass
ONe-core ZZ Ceti stars. As a first step, in this work we adopt the
relativistic Cowling approximation described above to incorporate relativistic
effects in the pulsation calculations, following the treatment provided in
Yoshida & Lee (2002). In future papers we plan to examine the Post-Newtonian
and full 4th-order GR equations, applied to ultra-massive ONe-core WDs and to
ultra-massive CO-core WDs (Córsico et al. 2023b, in prep.).

The paper is organised as follows. In Sect. 2 we briefly describe the
relativistic WD models computed by Althaus et al. (2022), emphasising the
impact of GR on the stellar structure. We devote Sect. 3 to describe our
approach for the relativistic nonradial stellar pulsations, particularly the
formalism of the relativistic Cowling approximation (Sect. 3.1, 3.2, 3.3 and
3.4). The pulsation results for our ultra-massive WD models are described in
Sect. 4. Finally, in Sect. 5 we summarise our findings. We present in Appendix
A a derivation of the relativistic version of the "modified Ledoux" treatment
of the Brunt-Väisälä frequency, and in Appendix B the results of a validation
of the main results of the paper using a toy model based on Chandrasekhar’s
models.

##  2 Relativistic ultra-massive WD models

To determine whether to employ GR or Newtonian gravity in a system like a
star, a qualitative general criterion commonly used is to assess the magnitude
of the "relativistic correction factor", $\varepsilon$, defined as
$\varepsilon=GM_{\star}/c^{2}R_{\star}$, where $G$ is the Newtonian
gravitational constant, $c$ is the speed of light, and $M_{\star}$ and
$R_{\star}$ are the stellar mass and radius, respectively (Poisson & Will,
2014)111The parameter $\varepsilon$ is nothing more than the surface
gravitational redshift in the Newtonian limit, $z$.. The larger $\varepsilon$,
the worse the approximation of Newtonian gravity. For instance, for a neutron
star, $\varepsilon$ is of order $\sim 0.1$, while for a black hole,
$\varepsilon\sim 1$. For average mass ($\sim 0.6M_{\sun}$) WDs, $\varepsilon$
is $\sim 10^{-4}$, and that is why until recently the relativistic effects
have been neglected in the calculation of their structures. If we instead
consider an ultra-massive WD star with $M\sim 1.3M_{\sun}$ and
$\varepsilon\sim 0.001$, at first glance, it is not clear if the relativistic
effects should be included or not. However, Carvalho et al. (2018) showed that
for the most massive WDs, the importance of GR for their structure and
evolution cannot be ignored. In fact, numerous works based on static WD
structures have shown that GR effects are relevant for the determination of
the radius of massive WDs (Rotondo et al., 2011; Mathew & Nandy, 2017;
Carvalho et al., 2018; Nunes et al., 2021). In particular, these studies have
demonstrated that for fixed values of mass, deviations of up to $50\%$ in the
Newtonian WD radius are expected compared to the GR WD radius. Recently,
Althaus et al. (2022) have presented the first set of constant rest-mass ONe-
core ultra-massive WD evolutionary models with masses greater than $\sim
1.30M_{\sun}$ (and up to $1.369M_{\sun}$) that fully take into account the
effects of GR. This study demonstrates that the GR effects must be considered
to assess the structural and evolutionary properties of the most massive WDs.
This analysis has been extended recently by Althaus et al. (2023) to ultra-
massive WDs with CO cores that result from the complete evolution of single
progenitor stars that avoid C-ignition (Althaus et al., 2021; Camisassa et
al., 2022).

Althaus et al. (2022) employed the LPCODE stellar evolution code,
appropriately modified to take into account relativistic effects. They
considered initial chemical profiles as predicted by the progenitor
evolutionary history (Siess, 2007, 2010; Camisassa et al., 2019), and computed
model sequences of $1.29,1.31,1.33,1.35$, and $1.369M_{\odot}$ WDs. The
standard equations of stellar structure and evolution were generalised to
include the effects of GR following Thorne (1977). In particular, the modified
version of LPCODE computes the dimensionless GR correction factors
$\mathscr{H,G,V,}$ and $\mathscr{R}$ which turn to unity in the Newtonian
limit. These factors correspond, respectively, to the enthalpy, gravitational
acceleration, volume, and redshift correction parameters. For comparison
purposes, Althaus et al. (2022) have also computed the same WD sequences for
the Newtonian gravity case. All these sequences included the energy released
during the crystallisation process, both due to latent heat and the induced
chemical redistribution, as in Camisassa et al. (2019).

Figure 1: The stellar radius (upper panels) and gravity (bottom panels) in
terms of the outer mass fraction coordinate corresponding to ultra-massive DA
WD models with $M_{\star}=1.29M_{\odot}$ (left) and $M_{\star}=1.35M_{\odot}$
(right), the GR case (black curves) and for the N case (red curves) ($T_{\rm
eff}\sim 12\,000$ K).

We briefly describe below some of the properties of the representative models
of ultra-massive ONe-core WD stars, emphasising the impact of GR on their
structure. We refer the reader to the paper by Althaus et al. (2022) for a
detailed description of the effects of GR on the structural properties of
these models. Here, we choose two template WD models characterised by stellar
masses $M_{\star}=1.29M_{\sun}$ and $M_{\star}=1.35M_{\sun}$, H envelope
thickness of $\log(M_{\rm H}/M_{\star})\sim-6$, and an effective temperature
of $T_{\rm eff}\sim 12\,000$ K, typical of the ZZ Ceti instability strip. We
distinguish two cases: one in which we consider Newtonian WD models (N case),
and another one in which the WD structure is relativistic (GR case). In Fig. 1
we plot the run of the stellar radius and gravity in terms of the outer mass
fraction coordinate, corresponding to WD models with $1.29M_{\sun}$ (left
panels) and $1.35M_{\sun}$ (right panels), for the GR case (black curves) and
the N case (red curves). Clearly, GR induces smaller radii and larger
gravities, and this effect is much more pronounced for larger stellar masses.

Figure 2: Left: the absolute relative difference between relativistic and
Newtonian stellar radius versus stellar mass. Right: the relative difference
between the relativistic and Newtonian surface gravity in terms of the stellar
mass.

In Table 1, which is a shortened version of Table 1 of Althaus et al. (2022),
we include the values of the stellar radius and the surface gravity for models
with $T_{\rm eff}=10\,000$ K and masses between $1.29M_{\sun}$ and
$1.369M_{\sun}$ in the GR and N cases. As can be seen, the impact of GR on the
radius and gravity of the models is noticeable. In Fig. 2 we plot the relative
differences $\Delta R_{\star}=|R_{\star}^{\rm GR}-R_{\star}^{\rm
N}|/R_{\star}^{\rm GR}$ (left panel) and $\Delta g=(g^{\rm GR}-g^{\rm
N})/g^{\rm GR}$ (right panel) in terms of the stellar mass. The stellar radius
is lower by $\sim 3\%~{}(1.29M_{\sun})$ to $\sim 34\%~{}(1.369M_{\sun})$, and
the surface gravity is higher by $\sim 6\%~{}(1.29M_{\sun})$ to $\sim
44\%~{}(1.369M_{\sun})$ compared to the case where GR is neglected. The
typical observational uncertainties in the radii and surface gravities of the
most massive WDs in the Montreal White Dwarf Database 100 pc sample (Kilic et
al., 2021) are 3% and 6%, respectively. Hence, the differences between the GR
and N cases can be detected observationally for WDs with masses above $\sim
1.3~{}M_{\odot}$. These discrepancies must have important consequences for the
pulsational properties of ultra-massive WDs, as we will see in Sect. 4.2.

Table 1: Stellar masses, radius and surface gravities of the ultra-massive ONe-core WD models at $T_{\rm eff}=10\,000$ K in the relativistic and Newtonian cases. $M_{\star}/M_{\odot}$ | $R_{\star}^{\rm GR}$ | $R_{\star}^{\rm N}$ | $\log g^{\rm GR}$ | $\log g^{\rm N}$
---|---|---|---|---
| [$\times 10^{8}$ cm] | [$\times 10^{8}$ cm] | [cm/s2] | [cm/s2]
1.29 | 2.609 | 2.685 | 9.401 | 9.375
1.31 | 2.326 | 2.426 | 9.507 | 9.470
1.33 | 2.005 | 2.157 | 9.643 | 9.579
1.35 | 1.543 | 1.829 | 9.878 | 9.728
1.369 | 1.051 | 1.409 | 10.217 | 9.961

##  3 Relativistic nonradial stellar pulsations in WDs

In order to incorporate the relativistic effects in the pulsations of WDs, we
adopt the relativistic Cowling approximation in the form developed by Yoshida
& Lee (2002), and follow the GR formalism provided in Boston (2022).

###  3.1 The relativistic Cowling approximation

The Cowling approximation of Newtonian nonradial pulsations (named after T. G.
Cowling’s pioneer paper; Cowling, 1941) is based on neglecting the
gravitational potential perturbations during the fluid oscillations. This
approximation has been widely used in Newtonian nonradial pulsation
computations in the past, because it constitutes a 2nd-order differential
eigenvalue problem, thus simplifying the complete 4th-order problem (Unno et
al., 1989). It is also a very good approximation to periods of $g$-modes in
WDs, which are primarily envelope modes (Montgomery et al., 1999). The Cowling
approximation has been frequently used in asymptotic treatment of stellar
pulsations (see, for instance, Tassoul, 1980), and also in numerical
treatments of $g$-mode pulsations in rapidly rotating WDs (e.g., Saio, 2019;
Kumar & Townsley, 2023), although it has fallen out of use in the context of
present-day numerical calculations of Newtonian nonradial stellar pulsations
and asteroseismology. The relativistic Cowling approximation (McDermott et
al., 1983), on the other hand, is generally employed in the field of
pulsations of relativistic objects such as neutron stars (Lindblom & Splinter,
1989b; Yoshida & Lee, 2002; Sotani & Takiwaki, 2020) and hybrid (hadron plus
quark matter phases) neutron stars (Tonetto & Lugones, 2020; Zheng et al.,
2023).

In the next sections, we first describe the relativistic correction factors
involved in the pulsation problem. Then, we provide relativistic expressions
to calculate the critical frequencies (Brunt-Väisälä and Lamb frequencies),
after which we assess the coefficients of the pulsation differential equations
in the relativistic Cowling form. Finally, we provide the two first order
differential equations to be solved, along with the boundary conditions of the
eigenvalue problem.

###  3.2 Relativistic correction factors $\mathscr{R}$, $\mathscr{V}$, and
potentials $\nu$, $\lambda$

We start by considering the Schwarzschild metric of GR for spacetime inside
and around a star (Thorne, 1977):

$ds^{2}=-e^{2\Phi/c^{2}}c^{2}dt^{2}+\left(1-\frac{2Gm}{c^{2}r}\right)^{-1}dr^{2}+r^{2}d\Omega^{2},$
(1)

where $m$ is the "total mass inside radius $r$", which includes the rest mass,
nuclear binding energy, internal energy, and gravity. $\Phi$ is a
gravitational potential, which in the Newtonian limit $c\rightarrow\infty$
corresponds to the scalar Newtonian potential.

Following Thorne (1977) in his treatment of relativistic stellar interiors, it
is convenient to write the metric in the form

$ds^{2}=-\mathscr{R}^{2}c^{2}dt^{2}+\mathscr{V}^{2}dr^{2}+r^{2}d\Omega^{2},$
(2)

where the redshift correction factor $\mathscr{R}$, and the volume correction
factor $\mathscr{V}$, are defined as (Thorne, 1977):

$\mathscr{R}=e^{\Phi/c^{2}},\ \ \
\mathscr{V}=\left(1-\frac{2Gm}{c^{2}r}\right)^{-1/2}.$ (3)

The metric is usually written also as a function of two relativistic
gravitational potentials $\nu$ and $\lambda$ (Tolman, 1939; Oppenheimer &
Volkoff, 1939), so that:

$ds^{2}=-e^{\nu}c^{2}dt^{2}+e^{\lambda}dr^{2}+r^{2}d\Omega^{2}.$ (4)

Equating Eqs. (1) and (4), we have

$\nu=\frac{2\Phi}{c^{2}},\ \ \ {\rm and}\ \ \
\lambda=-\ln\left(1-\frac{2Gm}{c^{2}r}\right).$ (5)

We obtain $\nu$ and $\lambda$ in terms of the variables $\mathscr{R}$ and
$\mathscr{V}$, that are the output of the relativistic LPCODE version (Althaus
et al., 2022) by equating Eqs. (2) and (4):

$\mathscr{R}^{2}=e^{\nu},\ \ \ \mathscr{V}^{2}=e^{\lambda},$ (6)

so that,

$\nu=2\ln\mathscr{R},\ \ \ \lambda=2\ln\mathscr{V}.$ (7)

In the Newtonian limit, we have $\mathscr{R}=e^{\nu/2}\rightarrow 1$ and
$\mathscr{V}=e^{\lambda/2}\rightarrow 1$, so that $\nu,\lambda\rightarrow 0$.

We compute the derivatives of $\nu$ and $\lambda$ by calculating the numerical
derivatives of $\mathscr{R}$ and $\mathscr{V}$ as:

$\nu^{\prime}\equiv\frac{d\nu}{dr}=\frac{2}{\mathscr{R}}\left(\frac{d\mathscr{R}}{dr}\right),\
\ \
\lambda^{\prime}\equiv\frac{d\lambda}{dr}=\frac{2}{\mathscr{V}}\left(\frac{d\mathscr{V}}{dr}\right).$
(8)

Figure 3: The relativistic potentials $\nu$ (black) and $\lambda$ (blue), in
terms of the outer mass fraction coordinate, corresponding to two template
ONe-core WD models with masses $M_{\star}=1.29M_{\sun}$ and
$M_{\star}=1.35M_{\sun}$, and effective temperature $T_{\rm eff}\sim 12\,000$
K. Note the high peaks in the stellar center, indicating the increased
gravitational strength of the core, and that the more massive star shows more
extreme values. At the surface, $\nu\rightarrow-\lambda$. Figure 4: The
derivatives of the relativistic potentials $\nu^{\prime}$ (black) and
$\lambda^{\prime}$ (blue), in terms of the outer mass fraction coordinate,
corresponding to the same template models shown in Fig. 3. Note again the
sharp peaks in the core, and more extreme values in the more masive star.
Figure 5: The second derivative of $\nu$, that is, $\nu^{\prime\prime}$, in
terms of the outer mass fraction coordinate, corresponding to the same
template models shown in Fig. 3. Figure 6: Upper panels: abundances by mass of
the different chemical species as a function of the fractional mass,
corresponding to the template WD models with masses $M_{\star}=1.29M_{\sun}$
(left) and $M_{\star}=1.35M_{\sun}$ (right), and effective temperature $T_{\rm
eff}\sim 12\,000$ K. Lower panels: logarithm of the squared Brunt-Väisälä and
Lamb ($\ell=1$) frequencies for the GR case (solid lines) and the N case
(dashed lines). The grey areas correspond to the crystallised core regions, in
which $g$ modes cannot propagate.

The numerical derivatives of $\mathscr{R}$ and $\mathscr{V}$, as well as
$\nu^{\prime}$ and $\lambda^{\prime}$, are usually noisy when computed
following Eqs. (8). To avoid this, we compute $\nu^{\prime}$ and
$\lambda^{\prime}$ by employing solutions to the Einstein field equation for
the static, spherically symmetric distribution of matter, given by Tolman
(1939) and Oppenheimer & Volkoff (1939) (see also Tooper, 1964):

$e^{-\lambda}\left(\frac{1}{r^{2}}-\frac{\lambda^{\prime}}{r}\right)-\frac{1}{r^{2}}=-\frac{8\pi
G}{c^{4}}\rho c^{2},$ (9)

$e^{-\lambda}\left(\frac{1}{r^{2}}+\frac{\nu^{\prime}}{r}\right)-\frac{1}{r^{2}}=\frac{8\pi
G}{c^{4}}P,$ (10)

$\frac{1}{2}e^{-\lambda}\left[\nu^{\prime\prime}+\frac{1}{2}\left(\nu^{\prime}-\lambda^{\prime}\right)\left(\nu^{\prime}+\frac{2}{r}\right)\right]=\frac{8\pi
G}{c^{4}}P,$ (11)

where $P$ is the pressure and $\rho$ is the mass-energy density (not just the
mass density). With some rearranging, we can write:

$\lambda^{\prime}=\frac{1}{r}+\left(\frac{8\pi G}{c^{2}}\rho
r-\frac{1}{r}\right)e^{\lambda}$ (12)

$\nu^{\prime}=-\frac{1}{r}+\left(\frac{8\pi
G}{c^{4}}Pr+\frac{1}{r}\right)e^{\lambda}$ (13)

$\nu^{\prime\prime}=\frac{16\pi
G}{c^{4}}Pe^{\lambda}-\frac{1}{2}\left(\nu^{\prime}-\lambda^{\prime}\right)\left(\nu^{\prime}+\frac{2}{r}\right)\equiv\frac{d^{2}\nu}{dr^{2}}$
(14)

To summarise, in our numerical treatment we employ Eqs. (12) and (13) to
compute $\lambda^{\prime}$ and $\nu^{\prime}$ using the value of $\lambda$
calculated with Eq. (7). We employ Eq. (14) to assess $\nu^{\prime\prime}$
using $\lambda$, $\lambda^{\prime}$, and $\nu^{\prime}$ derived above. The
quantity $\nu^{\prime\prime}$ is required to compute one of the coefficients
of the pulsation differential equations (Sect. 3.4).

###  3.3 Relativistic adiabatic exponent, sound speed, Lamb and Brunt-Väisälä
frequencies

The relativistic adiabatic exponent, defined as
$\Gamma_{1}=\left(\frac{\partial\log P}{\partial\log n}\right)_{\rm ad}$,
where $n$ is the baryon number density, can be expressed as (Thorne, 1967;
Meltzer & Thorne, 1966):

$\Gamma_{1}=\frac{\rho+(P/c^{2})}{P}\left(\frac{\partial
P}{\partial\rho}\right)_{\rm
ad}=\frac{\rho+(P/c^{2})}{\rho}\left(\frac{\partial\log
P}{\partial\log\rho}\right)_{\rm ad},$ (15)

This should be compared with the Newtonian case, where
$\Gamma_{1}=\left(\frac{\partial\log P}{\partial\log\rho}\right)_{\rm ad}$.
The relativistic sound speed, $v_{s}$, is given by (Curtis, 1950):

$v_{s}^{2}=\frac{\Gamma_{1}P}{\rho+(P/c^{2})},$ (16)

whereas in the Newtonian case, $v_{s}^{2}=\left(\frac{\partial
P}{\partial\rho}\right)_{\rm ad}=\Gamma_{1}\frac{P}{\rho}$.

The squared Lamb and Brunt-Väisälä critical frequencies of the nonradial
stellar pulsations, $L_{\ell}^{2}$ and $N^{2}$, can be written as (Boston,
2022):

$L_{\ell}^{2}=\ell(\ell+1)\frac{v_{s}^{2}}{r^{2}},$ (17)

$N^{2}=\frac{c^{2}}{2r}\nu^{\prime}e^{-\lambda}\left[\frac{1}{\Gamma_{1}}\left(\frac{d\log
P}{d\log r}\right)-\frac{\rho}{\rho+(P/c^{2})}\left(\frac{d\log\rho}{d\log
r}\right)\right].$ (18)

This expression for $N^{2}$ is analogous to the Newtonian version of $N^{2}$
for $g\rightarrow\frac{c^{2}}{2}\nu^{\prime}$ and an additional relativistic
correction factor $e^{-\lambda}$.

The relativistic prescription given by Eq. (18) for the assessment of the
Brunt-Väisälä frequency is not well-defined numerically, due to the high
degree of electronic degeneracy prevailing in the core of the ultra-massive
WDs, similar to the case of Newtonian pulsations (Brassard et al., 1991). In
particular, the use of Eq. (18) leads to unacceptable numerical noise of $N$,
which can lead to miscalculations of the adiabatic $g$-mode periods. To avoid
this problem, we employ a numerically convenient relativistic expression,
analogous to the Newtonian recipe known as the modified Ledoux prescription
(Tassoul et al., 1990). The appropriate relativistic expression for $N^{2}$,
which is derived in Appendix A, is:

$N^{2}=e^{-\lambda}\left(\frac{c^{2}}{2}\nu^{\prime}\right)^{2}\frac{\rho+(P/c^{2})}{P}\frac{\chi_{\rm
T}}{\chi_{n}}\left[\nabla_{\rm ad}-\nabla+B\right],$ (19)

where $B$ is the Ledoux term, defined as:

$B=-\frac{1}{\chi_{\rm T}}\sum_{i=1}^{M-1}\chi_{X_{i}}\frac{d\ln X_{i}}{d\ln
P},$ (20)

$M$ being the number of different atomic species with fractional abundances
$X_{i}$ that satisfy the constraint $\sum_{i=1}^{M-1}X_{i}+X_{M}=1$. The
compressibilities $\chi_{\rm T}$, $\chi_{n}$, and $\chi_{X_{i}}$ are defined
as, similar to the Newtonian problem:

$\chi_{\rm T}=\left(\frac{\partial\ln P}{\partial\ln
T}\right)_{n,\\{X_{i}\\}},\ \ \ \chi_{n}=\left(\frac{\partial\ln
P}{\partial\ln n}\right)_{T,\\{X_{i}\\}},\ \ \ \chi_{\rm
X_{i}}=\left(\frac{\partial\ln P}{\partial\ln X_{i}}\right)_{T,n}.$ (21)

Using $(d\ln\rho/d\ln n)=(\rho+P/c^{2})/\rho$ (Eq. 5.90 of Boston, 2022, see
also Thorne 1967), the compressibility $\chi_{n}$ can be computed as:

$\chi_{n}=\frac{\rho+(P/c^{2})}{\rho}\chi_{\rho},$ (22)

where $\chi_{\rho}=\left(\frac{\partial\ln
P}{\partial\ln\rho}\right)_{T,\\{X_{i}\\}}$. Here, $\nabla_{\rm ad}$ and
$\nabla$ are the adiabatic and actual temperature gradients, respectively,
defined as:

$\nabla_{\rm ad}=\left(\frac{\partial\ln T}{\partial\ln P}\right)_{{\rm
ad},\\{X_{i}\\}},\ \ \ \nabla=\frac{d\ln T}{d\ln P}~{}.$ (23)

Eq. (19) is completely analogous to the Newtonian expression for the squared
Brunt-Väisälä frequency, $N^{2}=g^{2}(\rho/P)(\chi_{\rm
T}/\chi_{\rho})\left[\nabla_{\rm ad}-\nabla+B\right]$. In the relativistic
formula, $g$ has been replaced by $c^{2}\nu^{\prime}/2$, and the ratio
$\rho/P$ becomes $(\rho+P/c^{2})/P$. There is an additional relativistic
factor, $e^{-\lambda}$, and the compressibility $\chi_{\rho}$ is replaced by
$\chi_{n}$, where $n$ is the baryonic number density.

###  3.4 Differential equations of the relativistic Cowling approximation

Here, we formulate the system of differential equations of the nonradial
pulsations in the relativistic Cowling approximation form, that results when
we ignore Eulerian metric perturbations in the pulsation equations (McDermott
et al., 1983). This reduces the 4th-complex-order problem of nonradial
pulsations in GR to a 2nd-real-order problem, which can be written as two
real, 1st-order differential equations. Following Yoshida & Lee (2002), we
define the dimensionless variables $\omega$, $y_{1}$, and $y_{2}$, analogous
to Dziembowski’s variables in Newtonian pulsations (Dziembowski, 1971)222At
variance with Boston (2022), we use $\sigma$ for the physically meaningful
oscillation frequency, and $\omega$ for the dimensionless frequency, following
Unno et al. (1989).:

$\omega^{2}=\frac{R_{\star}^{3}}{GM_{\star}}\sigma^{2},\ \ \
y_{1}=\frac{\xi_{r}}{r}e^{-i\sigma t},\ \ \
y_{2}=\xi_{h}C_{1}\omega^{2}e^{-i\sigma t},$ (24)

where $\xi_{r}$ and $\xi_{h}$ correspond to the Lagrangian radial and
horizontal displacements, respectively. We also define the following
dimensionless functions, analogous to Dziembowski’s coefficients (Dziembowski,
1971), calculated with respect to the stellar equilibrium model (Boston,
2022):

$V_{g}(r)=-\frac{1}{\Gamma_{1}}\left(\frac{d\log P}{d\log
r}\right)=\frac{\rho+(P/c^{2})}{\Gamma_{1}P}\frac{rc^{2}}{2}\nu^{\prime},$
(25)

$U_{1}(r)=2+r\frac{\nu^{\prime\prime}}{\nu^{\prime}},$ (26)

$U_{2}(r)=r\lambda^{\prime},$ (27)

$C_{1}(r)=\frac{GM_{\star}}{c^{2}R_{\star}^{3}}\frac{2r}{\nu^{\prime}}e^{-\nu},$
(28)

and (Thorne, 1966)

$A^{\ast}(r)=\frac{1}{\Gamma_{1}}\left(\frac{d\log P}{d\log
r}\right)-\frac{\rho}{\rho+(P/c^{2})}\left(\frac{d\log\rho}{d\log
r}\right)=\frac{rN^{2}}{c^{2}\nu^{\prime}/2}e^{\lambda}.$ (29)

In the Newtonian limit, $A^{\ast}$, $V_{g}$, and $C_{1}$ will limit to their
conventional expressions (Unno et al., 1989). On the other hand, in the
Newtonian limit we have that $U_{1}$ tends to $U$, which is defined in Unno et
al. (1989), and $U_{2}\rightarrow 0$. Using these definitions, and defining
$x=r/R_{\star}$, the resulting differential equations for the relativistic
Cowling approximation (McDermott et al., 1983; Lindblom & Splinter, 1989b;
Yoshida & Lee, 2002; Boston, 2022) are:

$x\frac{dy_{1}}{dx}=\left(V_{g}-3+U_{2}\right)y_{1}+\left(\frac{\ell(\ell+1)}{C_{1}\omega^{2}}-V_{g}\right)y_{2},$
(30)

$x\frac{dy_{2}}{dx}=\left(e^{\lambda}C_{1}\omega^{2}-A^{\ast}\right)y_{1}+\left(1+A^{\ast}-U_{1}\right)y_{2}.$
(31)

In the Newtonian limit, we have $e^{\lambda}\rightarrow 1$ and
$U_{2}\rightarrow 0$, and the equations adopt exactly the form of the
Newtonian Cowling approximation (Cowling, 1941; Unno et al., 1989). The
boundary conditions for this system of differential equations are, at the
stellar (fluid) center $(x=0)$:

$y_{1}C_{1}\omega^{2}-\ell y_{2}=0,$ (32)

and at the stellar surface ($x=1$):

$y_{1}-y_{2}=0,\ \ \ {\rm and}\ \ \ y_{1}=1\ \ ({\rm normalisation\
condition}).$ (33)

These are the same boundary conditions as for the Newtonian Cowling
approximation.

For the ulta-massive WD models considered in this work, the stellar core is
crystallised, so that the so called “hard-sphere” boundary conditions
(Montgomery et al., 1999) may be adopted, which exclude the $g$-mode
oscillations from the solid core regions. In that case, Eq. (32) is replaced
by the condition:

$y_{1}=0\ \ \ {\rm and}\ \ \ y_{2}=\ {\rm arbitrary}$ (34)

at the radial shell $x=x_{\rm crys}$ associated with the outward-moving
crystallisation front, instead of the center of the star ($x=0$). To maintain
consistency between Newtonian and GR calculations and for a clean comparison,
we assume the same internal boundary condition for the GR case as for the N
case, that the eigenfunctions are approximately zero in the solid core, and
can be treated with a hard-sphere boundary condition.

In this work, to take into account the relativistic effects on $g$-mode
pulsations of crystallised ultra-massive WD models, the LP-PUL pulsation code
(Córsico & Althaus, 2006) has been appropriately modified to solve the problem
of relativistic pulsations in the Cowling approximation as given by Eqs. (30)
and (31), with boundary conditions given by Eqs. (33) and (34).

Figure 7: Logarithm of the quantity $V_{g}$ for the GR case (black solid
lines), defined by Eq. (25), as a function of the fractional mass,
corresponding to the template WD models with masses $M_{\star}=1.29M_{\sun}$
(left) and $M_{\star}=1.35M_{\sun}$ (right), and effective temperature $T_{\rm
eff}\sim 12\,000$ K. For comparison, we include the function $V_{g}$ computed
for the N case (red dashed lines). The vertical dashed blue line indicates the
location of the boundary of the crystallised core region (grey zone). Figure
8: Same as Fig. 7, but for the quantity $U_{1}$ (Eq. 26). Figure 9: Same as
Fig. 7, but for the quantity $U_{2}$ (Eq. 27). Since $U_{2}$ has no
counterpart in the Newtonian pulsation equations, only the GR case is plotted
(black curves). Figure 10: Same as Fig. 7, but for the quantity $C_{1}$ (Eq.
28). Figure 11: Same as Fig. 7, but for the logarithm of the quantity $A^{*}$
(Eq. 29).

##  4 Pulsation results

###  4.1 Properties of template models

It is illustrative to examine the metric parameters $\nu$, $\lambda$,
$\nu^{\prime}$, $\lambda^{\prime}$ and $\nu^{\prime\prime}$. In Figs. 3, 4,
and 5 we show the $\nu$ and $\lambda$ and their derivatives
$\nu^{\prime},\lambda^{\prime}$ and $\nu^{\prime\prime}$, in terms of the
outer mass fraction coordinate, corresponding to the two template WD models
with masses $M_{\star}=1.29M_{\sun}$ (left) and $M_{\star}=1.35M_{\sun}$
(right), and effective temperature $T_{\rm eff}\sim 12\,000$ K. As can be
seen, $\nu,\lambda$ quantities are very small throughout the star, being of a
similar order with $\varepsilon\sim 0.001.$ However, in the center the
gravitational values are more extreme than near the surface, pointing to the
high central concentration of the mass of these stars.

The chemical profiles (abundances by mass, $X_{i}$) of the different nuclear
species corresponding to the template models are plotted in the upper panels
of Fig. 6 as a function of the fractional outer mass. In the lower panels, we
depict the logarithm of the squared Brunt-Väisälä (black lines) and dipole
($\ell=1$) Lamb (red lines) frequencies for the GR case (solid lines) and the
N case (dashed lines). We have emphasised the crystallised regions of the core
with grey. The chemical interface of 12C,16O, and 20Ne, which is located at
$-\log(1-M_{r}/M_{\star})\sim 1.5$, is embedded in the crystalline part of the
core for both template models. Since we assume that $g$-mode eigenfunctions
cannot penetrate the solid regions (due to the hard-sphere boundary condition,
Eq. 34), this chemical interface is not relevant for the mode-trapping
properties of the models. The chemical transition region between 12C, 16O, and
4He [$-\log(1-M_{r}/M_{\star})\sim 4.5$], which is located in the fluid region
in both models, also does not have a significant impact on the mode-trapping
properties. Thus, mode-trapping properties are almost entirely determined by
the presence of the 4He/1H transition, which is located in the fluid external
regions, at $-\log(1-M_{r}/M_{\star})\sim 6$.

By closely inspecting Fig. 6, we conclude that the Brunt-Väisälä and Lamb
frequencies for the N and GR cases are similar for the model with
$1.29M_{\sun}$, although they are significantly different for the
$M_{\star}=1.35M_{\sun}$ model, with both critical frequencies being higher
for the GR case than for the N case. Because of this, it is expected that
$g$-mode frequencies shift to larger values so that all periods experience a
global offset towards shorter values in the relativistic case, compared to the
Newtonian case. This will be verified with the calculations of the $g$-mode
period spectra in both situations (Sect. 4.2).

We close this section by comparing the coefficients of the differential
equations of the relativistic Cowling approximation with their Newtonian
counterparts. In Figs. 7 to 11 we depict with black curves the dimensionless
functions $V_{g},U_{1},U_{2},C_{1}$ and $A^{*}$ in the GR case, as defined by
Eqs. (25) to (29), along with the same quantities corresponding to the N case
(red curves), computed according to their definition (see, e.g., Unno et al.,
1989). We include the cases of the two template WD models with
$M_{\star}=1.29M_{\odot}$ (left panel) and $M_{\star}=1.35M_{\odot}$ (right
panel). We marked the crystallised region in each model with a grey area with
a dashed blue boundary. These figures demonstrate that the dimensionless
quantities in the GR case are very similar to the ones for the N case, and
this is true for both of the representative models. This is not surprising,
since the relativistic correction factors $\nu$ and $\lambda$ and their
derivatives $\nu^{\prime},\lambda^{\prime}$ and $\nu^{\prime\prime}$, that are
included in the calculation of the dimensionless coefficients, are small. For
the specific case of $A^{*}$, some numerical noise is observed in the core
regions. This is irrelevant for the purposes of this investigation, since
those regions are contained in the crystallised core and do not affect the $g$
modes, which are prevented from propagating in the solid phase.

###  4.2 Newtonian and relativistic $g$-mode period spectra

We computed N and GR nonradial $g$-mode $\ell=1$ adiabatic pulsation periods
in the range $50\lesssim\Pi\lesssim 2000$ s using an updated version of the
LP-PUL pulsation code that includes the capability to solve the pulsations
equations in the relativistic Cowling approximation described in Sect. 3.1.
The N-case pulsation periods were calculated by solving the differential
problem of the Newtonian nonradial stellar pulsations (Unno et al., 1989). We
emphasise that in the GR case we are using evolutionary WD models calculated
in GR with relativistic, 2nd-order Cowling mode pulsations that ignore
gravitational (i.e. spacetime) perturbations, while in the N case we are using
evolutionary WD models calculated with Newtonian gravity and Newtonian, 4th-
order mode pulsations that include gravitational perturbations333This is at
variance with the preliminary results presented in Córsico et al. (2023), in
which Newtonian equations were used for the $g$-modes, with a fully
relativistic WD as the background.. We have also computed Newtonian periods by
solving the 2nd-order Newtonian Cowling approximation (Unno et al., 1989). For
$g$-modes, these 2nd-order periods are sufficiently similar to the 4th-order
periods used in the N case, that the results are not impacted.

In the analysis below, to study the dependence of the relativistic effects on
the stellar mass, we compare the $g$-mode period spectra calculated according
to the N and GR cases for ultra-massive WD models of different stellar masses
at effective temperatures typical of the ZZ Ceti instability strip.

Figure 12: Dipole $\ell=1$ asymptotic period spacing for ultra-massive WD
sequences with different stellar masses for the relativistic (GR) and
Newtonian (N) cases in terms of $T_{\rm eff}$ through the whole ZZ Ceti
instability strip (light grey area).

Before analysing the behaviour of the periods, we first examine the impact of
GR on the period spacing of $g$ modes. According to the asymptotic theory of
stellar pulsations, and in the absence of chemical gradients, the pulsation
periods of the $g$ modes with high radial order $k$ (long periods) are
expected to be uniformly spaced with a constant period separation given by
(Tassoul, 1980; Tassoul et al., 1990):

$\Delta\Pi_{\ell}^{\rm a}=\Pi_{0}/\sqrt{\ell(\ell+1)},$ (35)

where

$\Pi_{0}=2\pi^{2}\left[\int_{\rm fluid}\frac{N}{r}dr\right]^{-1},$ (36)

with the integral in Eq. (36) calculated only in the fluid part of the star.
Fig. 12 depicts the asymptotic period spacing for the sequences of
$1.29,1.31,1.33,1.35$ and $1.369~{}M_{\odot}$ WD models in terms of the
effective temperature along the ZZ Ceti instability strip. We find that
$\Delta\Pi_{\ell}^{\rm a}$, the asymptotic period spacing, is smaller for the
relativistic WD sequences compared to the Newtonian sequences. This is
expected, since the asymptotic period spacing is inversely proportional to the
integral of the Brunt-Väisälä frequency divided by the radius. Since the
Brunt-Väisälä frequency is larger for the relativistic case (see Fig. 6), the
integral is larger and its inverse is smaller than in the Newtonian case. The
differences of $\Delta\Pi_{\ell}^{\rm a}$ between the GR and the N cases are
larger for higher stellar masses, reaching a minimum difference of $\sim 0.6$
s (which represents a relative variation in period spacing of $\sim 3\%$) for
$1.29~{}M_{\sun}$, and a maximum difference of $\sim 3$ s (that constitutes a
relative variation of $\sim 48\%$) for $1.369~{}M_{\sun}$ for effective
temperatures within the ZZ Ceti instability strip.

Figure 13: Upper panel: $\ell=1$ $g$-mode periods in terms of the radial order
($k$) for a WD model with $M_{\star}=1.29M_{\sun}$ and $T_{\rm eff}\sim
12\,000$ K for the relativistic (GR, black dots) and Newtonian (N, red dots)
cases. Lower panel: the absolute value of the relative difference between the
periods of the GR and the N cases, $\delta=|\Pi_{\rm GR}-\Pi_{\rm N}|/\Pi_{\rm
GR}$ (blue). Figure 14: Same as Fig. 13, but for the case of a WD model with
$M_{\star}=1.369M_{\sun}$.

Since there are substantial differences in the separation of $g$-mode periods
in the GR and N cases, it is natural to expect significant differences in the
individual pulsation periods ($\Pi$). In the upper panels of Figs. 13 and 14,
we compare the periods of the GR and N cases for the less massive
($1.29M_{\sun}$) and the most massive ($1.369M_{\sun}$) WD model considered in
this work ($T_{\rm eff}\sim 12\,000$ K). $M_{\star}=1.369M_{\sun}$ corresponds
to the maximum possible value in the calculations of Althaus et al. (2022),
above which the models become unstable with respect to GR effects. It is clear
from these figures that the periods in the relativistic case are shorter than
those in the Newtonian case, with the absolute differences becoming larger
with increasing $k$. This is mainly due to the structural differences of the
equilibrium models in the GR case in relation to the N case (smaller radii and
larger gravities characterising the relativistic WD models, see Fig. 1), and
to a much lesser extent, due to the differences in the relativistic treatment
of the pulsations in comparison with the Newtonian one.

To quantify the impact of GR on the period spectrum, we have plotted in the
lower panel of each figure the absolute value of the relative differences
between the GR periods and the N periods, $\delta=|\Pi_{\rm GR}-\Pi_{\rm
N}|/\Pi_{\rm GR}$, versus the radial order. These differences are smaller than
$\sim 0.035$ for the less massive model ($1.29~{}M_{\sun}$, Fig. 13), but they
become as large as $\sim 0.5$ for the most massive models ($1.369~{}M_{\sun}$,
Fig. 14). We conclude that, for ultra-massive WDs with masses in the range
$1.29\leq M_{\star}/M_{\sun}\leq 1.369$, the impact of GR on the pulsations is
important, resulting in changes from $\sim 4\%$ to $\sim 50\%$ in the values
of $g$-mode periods.

Another way to visualise the impact of GR on the pulsation periods is to plot
the periods for the GR and N cases in terms of stellar mass. We display in the
upper panel of Fig. 15 the periods of selected $g$ modes (with radial order
$k=5,10,20,40$ and 70) in terms of the stellar mass for the GR and the N
cases. In the lower panel, we show the absolute value of the relative
difference $\delta$ (percentage %) between the relativistic and Newtonian
periods, as a function of the stellar mass. The relative differences in the
periods exhibit an exponential growth with stellar mass, without appreciable
dependence on the radial order (see also Figs. 13 and 14). The behaviour of
$\delta$ with the stellar mass visibly mirrors the exponential increase in the
relative differences between the relativistic and Newtonian stellar radii and
surface gravities, as seen in Fig. 2.

At first glance, the relative differences $\delta$ might seem larger than
expected, given recent work on periods in average-mass WDs by Boston et al.
(2023). For the simple models considered there, it was shown that $\delta\sim
z\sim 10^{-4}$ for a WD with $M_{\star}\approx 0.6M_{\odot}$. However,
considering their Figure 4, it is possible for stars with high central
concentrations, such as ultra-massive WDs, that $\delta$ can be larger than
$z$, consistent with our present findings. To confirm this, we also carried
out pulsational calculations on a simplified stratified Chandrasekhar-type
equilibrium model that mimics a $\sim 1.3~{}M_{\sun}$ ultra-massive WD, in the
case of Newtonian gravity and in the Post-Newtonian approximation, following
the process in Boston et al. (2023). These calculations and their results are
presented in the Appendix B. The comparison of the periods in both cases
indicates a relative difference of the order of $10^{-2}$, in complete
agreement with the results obtained here for our WDs models of $1.29M_{\sun}$
and $1.31M_{\sun}$ (see Figs. 13 and 15).

Figure 15: Upper panel: the periods of selected $\ell=1$ $g$ modes
($k=5,10,20,40$ and 70) in terms of the stellar mass, for the GR case (solid
lines with filled dots) and the N case (dashed lines with hollow dots). Lower
panel: the absolute relative difference $\delta$ (percentage %) between the
relativistic and Newtonian periods, as a function of the stellar mass.

It is interesting to examine how the period spacings versus periods change
depending on whether we consider the GR case or the N case. We define the
forward period spacing as $\Delta\Pi=\Pi_{k+1}-\Pi_{k}$. The dipole ($\ell=1$)
forward period spacing in terms of the periods is plotted in Figs. 16 to 20
for WD models with stellar masses between 1.29 and $1.369~{}M_{\odot}$ and
$T_{\rm eff}=12\,000$ K. We have adopted the same range in the $y-$axis in
order to make the comparison of the results between the different stellar
masses clearer. These figures show that, in general, the period spacing is
larger in the N case than in the GR case, and that this difference becomes
larger as the stellar mass increases. This is expected based on the behaviour
of the asymptotic period spacing (see Fig.12), which is indicated with
horizontal dashed lines.

Figure 16: Dipole ($\ell=1$) forward period spacing
($\Delta\Pi=\Pi_{k+1}-\Pi_{k}$) versus periods, corresponding to a WD model
with $M_{\star}=1.29M_{\sun}$ and $T_{\rm eff}\sim 12\,000$ K for the
relativistic case (red), and the Newtonian case (black). The horizontal dashed
lines correspond to the asymptotic period spacings for both cases. Figure 17:
Same as Fig. 16, but for a WD model with $M_{\star}=1.31M_{\sun}$. Figure 18:
Same as Fig. 16, but for a WD model with $M_{\star}=1.33M_{\sun}$. Figure 19:
Same as Fig. 16, but for a WD model with $M_{\star}=1.35M_{\sun}$. Figure 20:
Same as Fig. 16, but for a WD model with $M_{\star}=1.369M_{\sun}$.

###  4.3 The case of the ultra-massive ZZ Ceti star WD J0049$-$2525

The ultra-massive DA WD star WD J004917.14$-$252556.81 ($T_{\rm
eff}=13\,020\pm 460$ K, $\log g=9.341\pm 0.036$) is the most massive pulsating
WD known to date (Kilic et al., 2023b). It shows only two periods, at $\sim
209$ s and $\sim 221$ s, which are insufficient to find a single seismological
model that would give us details of its internal structure. Extensive follow-
up time-series photometry could allow discoveries of a significant number of
additional pulsation periods that would help to probe its interior.
Considering the ONe-core WD evolutionary models of Althaus et al. (2022), WD
J0049$-$2525 has $M_{\star}=1.283\pm 0.008~{}M_{\sun}$ in the Newtonian
gravity, or $M_{\star}=1.279\pm 0.007~{}M_{\sun}$ if we adopt the GR
treatment. This heavyweight ZZ Ceti, in principle, could be considered as an
ideal target to explore the relativistic effects on ultra-massive WD
pulsations. However, the difference between the relativistic and Newtonian
mass of this target is tiny. A difference of only $0.004M_{\sun}$ is even
smaller than the uncertainties in the mass estimates. This small difference is
due to the star being just slightly below the lower mass limit for the
relativistic effects to be important444That is, $M_{\star}\sim 1.3M_{\sun}$,
the lower limit of the mass regime of the so-called ”relativistic ultra-
massive WDs (Althaus et al., 2023)..

Fig. 15 (see also Fig. 16) demonstrates that the effects of the GR on the
$g$-mode periods of WD J0049$-$2525 are less than $\sim 1\%$. Although
extremely important for being the most massive pulsating WD star known, WD
J0049$-$2525 is not massive enough for the exploration of the GR effects on WD
pulsations. We conclude that, to be able to study the effects of GR on WD
pulsations, we have to wait for the discovery and monitoring of even more
massive pulsating WDs, especially the ones with $M_{\star}\gtrsim
1.33~{}M_{\sun}$.

###  4.4 Prospects for finding pulsating WDs where GR effects are significant

Figure 21: Masses and effective temperatures for high probability WD
candidates with $M_{\star}\geq 1.3~{}M_{\odot}$ in the Gaia EDR3 WD sample
from Gentile Fusillo et al. (2021) assuming CO cores. For ONe cores, the
masses would be lower on average by $0.04-0.05~{}M_{\odot}$. The blue and red
lines show the empirical boundaries of the ZZ Ceti instability strip from
Vincent et al. (2020) extrapolated to higher masses. Blue and red dots show
the spectroscopically confirmed DA and DC/magnetic WDs, respectively.

Figure 21 shows the masses and effective temperatures for high probability
($P_{\rm WD}\geq 0.9$) WD candidates with $M_{\star}\geq 1.3~{}M_{\odot}$ in
the Gaia EDR3 WD sample from Gentile Fusillo et al. (2021) assuming CO cores.
Here we limit the sample to the temperature range near the ZZ Ceti instability
strip. The blue and red lines show the boundaries of the instability strip
from Vincent et al. (2020) extrapolated to higher masses. There are 78 objects
in this sample, including 7 spectroscopically confirmed DA WDs (labelled in
the figure) and 6 magnetic or DC WDs. Kilic et al. (2023a) found that only 48%
of the $M_{\star}\approx 1.3\,{\rm M}_{\odot}$ WDs within 100 pc are DA WDs,
with the rest being strongly magnetic (40% of the sample) or WDs with unusual
atmospheric compositions (hot DQ, DBA, DC etc). Hence, follow-up spectroscopy
is required to identify the DA WDs in this sample.

Kilic et al. (2023a) presented time-series photometry for the five DA WDs
cooler than $13\,000$ K in Fig. 21. They did not detect any significant
variations in four of the targets, and their observations were inconclusive
for J0959$-$1828\. Nevertheless, there are a number of relativistic ultra-
massive WD candidates that may fall within the ZZ Ceti instability strip, and
therefore may exhibit pulsations. The masses shown here are based on the CO-
core evolutionary models, for ONe cores the masses would be lower on average
by $0.04-0.05~{}M_{\odot}$. Even then, there are 9 candidates with
$M_{\star}>1.35~{}M_{\odot}$ and up to $1.39~{}M_{\odot}$ (assuming a CO core)
near the instability strip. If confirmed, such targets would be prime examples
of objects where GR effects would have a significant impact on their pulsation
properties.

Unfortunately, the observational errors in temperatures and masses of these
targets based on Gaia photometry and parallaxes (Gentile Fusillo et al., 2021)
are too large to effectively identify the best targets for follow-up. For
example, 64 of the 78 objects shown here have temperature errors larger than
2000 K, roughly the width of the instability strip, and 62 have errors in mass
that are larger than $0.1~{}M_{\odot}$. Hence, further progress on
understanding the GR effects on WD pulsation will require spectroscopic and
time-series observations of a relatively large sample of candidates to
identify genuine pulsating ultra-massive WDs with $M_{\star}\gtrsim
1.33~{}M_{\sun}$. In addition, the median $G-$band magnitude for these 78
objects is 20.25 mag. Hence, 4-8m class telescopes would be needed to confirm
pulsating DA WDs in this sample.

##  5 Summary and conclusions

In this paper, we have assessed for the first time the impact of GR on the
$g$-mode period spectra of ultra-massive ZZ Ceti stars. To this end, we
pulsationally analysed fully evolutionary ONe-core ultra-massive WD models
with masses from $1.29$ to $1.369M_{\sun}$ computed in the frame of GR
(Althaus et al., 2022). We employed the LPCODE and LP-PUL evolutionary and
pulsation codes, respectively, adapted for relativistic calculations. In
particular, for the pulsation analysis, we considered the relativistic Cowling
approximation. Our study is consistent with Boston et al. (2023), considering
the high central compactness of the stars studied here. The study of pulsating
ultra-massive WDs in the context of GR is timely considering the increasing
rate of discovery of very high-mass objects (e.g., Kilic et al., 2020, 2021;
Hollands et al., 2020; Caiazzo et al., 2021; Torres et al., 2022; Kilic et
al., 2023a), the discovery of the ZZ Ceti WD J004917.14$-$252556.81 (the most
massive pulsating WD currently known, Kilic et al., 2023b), and the
possibility of finding even more massive pulsating objects in the near future.
This is particularly relevant in view of the space-based surveys like TESS and
ULTRASAT and wide-field ground-based surveys like the LSST and BlackGEM.

We find that the Brunt-Väisälä and Lamb frequencies are larger for the
Relativistic case compared to the Newtonian case, as a result of relativistic
models having smaller radii and higher gravities. This has the important
consequence that the typical separation between consecutive $g$-mode periods
is smaller in the relativistic case than in the Newtonian computations, with
percentage differences of up to 48% in the case of the most massive model
($1.369~{}M_{\sun}$). We assessed the dipole period spectrum of $g$ modes of
our ultra-massive WD models for the Newtonian and the relativistic cases, and
found that the periods in the GR case are shorter than in the Newtonian
computations. In particular, for the less massive model ($1.29~{}M_{\sun}$),
these relative differences are smaller than $\sim 0.035$, but the variations
reach values as large as $\sim 0.5$ for the most massive model
($1.369~{}M_{\sun}$).

We conclude that, for ultra-massive DA WDs models with masses in the range
that we have considered in this paper ($1.29\leq M_{\star}/M_{\sun}\leq
1.369$) and effective temperatures typical of the ZZ Ceti instability strip,
GR does matter in computing the adiabatic $g$-mode pulsations, resulting in
periods that are between $\sim 4$ and $\sim 50\%$ shorter, depending on the
stellar mass, when a relativistic treatment is adopted instead of a Newtonian
one. This suggests that the effects of GR on the structure and pulsations of
WDs with masses $\gtrsim 1.29M_{\sun}$ cannot be ignored in
asteroseismological analysis of ultra-massive ZZ Ceti stars and likely other
classes of pulsating WDs.

## Acknowledgements

We wish to thank the suggestions and comments of an anonymous referee that
improved the original version of this work. Part of this work was supported by
AGENCIA through the Programa de Modernización Tecnológica BID 1728/OC-AR, by
the PIP 112-200801-00940 grant from CONICET, by the National Science
Foundation under grants AST-2205736, PHY-2110335, the National Aeronautics and
Space Administration under grant 80NSSC22K0479, and the US DOE under contract
DE-AC05-00OR22725. ST acknowledges support from MINECO under the
PID2020-117252GB-I00 grant and by the AGAUR/Generalitat de Catalunya grant
SGR-386/2021. MC acknowledges grant RYC2021-032721-I, funded by
MCIN/AEI/10.13039/ 501100011033 and by the European Union
NextGenerationEU/PRTR. This research has made use of NASA Astrophysics Data
System.

## Data Availability Statement

The data underlying this article are available upon request.

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##  Appendix A Relativistic expression for the Brunt-Väisälä frequency in the
“modified Ledoux” prescription

We start from the relativistic expression for the Brunt-Väisälä frequency,
obtained according to its definition (Eq. 18). This expression can be derived
by considering slight buoyant perturbations of a fluid packet within the
stellar medium, as detailed in Boston (2022). In the GR case, $N^{2}$ is given
by:

$N^{2}=-\frac{c^{2}}{2}\nu^{\prime}e^{-\lambda}\left[\frac{1}{\rho+P/c^{2}}\left(\frac{d\rho}{dr}\right)-\frac{1}{\Gamma_{1}P}\left(\frac{dP}{dr}\right)\right],$
(37)

which reduces to the Newtonian result in the limit $c\rightarrow\infty$. In
what follows, we will derive expressions for the first and second members
inside the brackets of Eq. (37). In the Newtonian case, if stellar plasma is
composed by $M$ atomic species with fractional abundances $X_{i}$, the
equation of state can be written as:

$P=P(\rho,T,\\{X_{i}\\}),$ (38)

where $i=1,\cdots,M-1$, and $\sum_{i=1}^{M-1}X_{i}+X_{M}=1$. In the
relativistic case, we have, instead:

$P=P(n,T,\\{X_{i}\\}),$ (39)

where $n$ is the baryonic number density. Following Brassard et al. (1991), we
can differentiate $P$ from Eq. (39) and write:

$d\ln P=\chi_{n}d\ln n+\chi_{T}d\ln T+\sum_{i=1}^{M-1}\chi_{X_{i}}d\ln X_{i},$
(40)

where $\chi_{\rm T}$, $\chi_{n}$, and $\chi_{X_{i}}$ are given by Eqs. (21).
Now, from Eq. (40) we have:

$\frac{d\ln n}{d\ln
P}=\frac{1}{\chi_{n}}-\frac{\chi_{T}}{\chi_{n}}\nabla-\frac{1}{\chi_{n}}\sum_{i=1}^{M-1}\chi_{X_{i}}\frac{d\ln
X_{i}}{d\ln P},$ (41)

The relativistic adiabatic exponent, $\Gamma_{1}$, is defined as:

$\Gamma_{1}=\left(\frac{\partial\ln P}{\partial\ln n}\right)_{\rm ad}$ (42)

Following Kippenhahn & Weigert (1990, Eqs. 6.6 and 13.24), we can write

$\Gamma_{1}=\frac{1}{\alpha-\delta\nabla},$ (43)

where:

$\alpha=\left(\frac{d\ln\rho}{d\ln P}\right)_{T},\ \
\delta=-\left(\frac{d\ln\rho}{d\ln T}\right)_{P}.$ (44)

From the definition of $\chi_{T}$ and $\chi_{n}$ (Eqs. 21), and using the
property of the partial derivatives $(\partial f/\partial y)_{x}=-(\partial
f/\partial x)_{y}\cdot(\partial x/\partial y)_{f}$, we have:

$\alpha=\frac{1}{\chi_{n}},\ \ \ \delta=\frac{\chi_{T}}{\chi_{n}}.$ (45)

so that $\Gamma_{1}$ can be written as:

$\frac{1}{\Gamma_{1}}=\frac{\left(1-\chi_{T}\nabla_{\rm
ad}\right)}{\chi_{n}}.$ (46)

The first law of thermodynamics in GR can be written (Eq. 2.12 of Thorne,
1967, converted to standard, non-geometrized units) as

$d\rho=\left[\frac{\rho+P/c^{2}}{n}\right]dn+\frac{T}{c^{2}}n\
ds+\sum_{k}\mu_{k}n\ dX_{k},$ (47)

where $T$, $s$, and $\mu_{k}$ are the temperature, the entropy per baryon, and
the nuclear chemical potential of the species $k$, respectively.

If we now assume isentropic changes ($ds=0$) and suppose that the abundances
of the nuclear species do not change ($dX_{k}=0$) (Eq. 2.14 of Thorne, 1967),
then by differentiating with respect to $r$, we finally have:

$\frac{d\rho}{dr}=\left[\frac{\rho+P/c^{2}}{n}\right]\frac{dn}{dr}.$ (48)

Eq. (48) can be written as:

$\frac{1}{\rho+P/c^{2}}\left(\frac{d\rho}{dr}\right)=\frac{1}{n}\frac{dn}{dr}$
(49)

Substituting Eq. (49) in Eq. (37), using the static TOV equation of GR
(Tolman, 1939; Oppenheimer & Volkoff, 1939),

$\frac{dP}{dr}=-\frac{1}{2}\left(\rho+\frac{P}{c^{2}}\right)c^{2}\nu^{\prime},$
(50)

and employing Eqs. (41) and (46), we obtain:

$N^{2}=\left(\frac{1}{2}c^{2}\nu^{\prime}\right)^{2}e^{-\lambda}\left(\frac{\rho+P/c^{2}}{P}\right)\frac{\chi_{T}}{\chi_{n}}\left[\nabla_{\rm
ad}-\nabla-\frac{1}{\chi_{T}}\sum_{i=1}^{M-1}\chi_{X_{i}}\frac{d\ln Xi}{d\ln
P}\right],$ (51)

where the last term inside the brackets is the Ledoux term $B$ (Eq. 20). Thus,
we finally obtain:

$N^{2}=\left(\frac{1}{2}c^{2}\nu^{\prime}\right)^{2}e^{-\lambda}\left(\frac{\rho+P/c^{2}}{P}\right)\frac{\chi_{T}}{\chi_{n}}\left[\nabla_{\rm
ad}-\nabla+B\right],$ (52)

##  Appendix B Validation with a toy model based on Chandrasekhar’s models

As validation of the results presented in this paper, in particular the size
of the relative difference in the periods, we have carried out pulsation
calculations on a toy model based on Chandrasekhar’s models, with a stellar
mass $M_{\star}\sim 1.3M_{\odot}$. This model has a cold degenerate-electron
equation of state featuring a near-surface chemical transition from
$\mu_{e}=2$ to $\mu_{e}=1$, simulating a surface H layer. Thus, this simple
model mimics the structure of a stratified realistic ultra-massive WD model.
Following the Post-Newtonian method described in Boston et al. (2023), we have
compared the fourth-order nonradial Newtonian pulsations to the nonradial GR
pulsations for this toy model for several $g$, $f$, and $p$ modes with low
radial orders $k$ for harmonic degree $\ell=1,2$ and 3. We show the results in
Table 2. The relative differences we obtain for $g$ modes are, on average,
$\sim 2.65\times 10^{-2}$ (column 4), which is consistent with the results
shown in Figs. 13 and 15 for the cases of ultra-massive WD models with
$M_{\star}=1.29M_{\sun}$ and $M_{\star}=1.31M_{\sun}$, respectively.

Table 2: Periods for several low-order $p$, $f$, and $g$ modes corresponding to the Newtonian gravity computations (column 2) and the GR computations (column 3) for a stratified degenerate electron gas model, both with $M_{\star}=1.29966M_{\odot}$. The Newtonian model uses $y_{0}=6.385$, with $z=6.6012\times 10^{-4}$, while the GR model uses $y_{0}=6.779$, with $z=6.3047\times 10^{-4}$. Column 3 gives the relative difference, which is commensurate with those in Figure 13.  | Newtonian | Post-Newtonian |
---|---|---|---
mode ℓ,k |  $\Pi$ (s) |  $\Pi$ (s) | rel. diff.
$p_{1,1}$ | 1.0495784 | 0.9782581 | $1.52\times 10^{-3}$
$g_{1,1}$ | 16.2675050 | 14.7865834 | $2.66\times 10^{-2}$
$g_{1,2}$ | 35.7515552 | 32.4932485 | $2.67\times 10^{-2}$
$g_{1,3}$ | 54.4264093 | 49.4686549 | $2.67\times 10^{-2}$
$g_{1,4}$ | 72.9174383 | 66.2778687 | $2.66\times 10^{-2}$
$g_{1,5}$ | 91.3355203 | 83.0211550 | $2.66\times 10^{-2}$
$g_{1,6}$ | 109.7169819 | 99.7313589 | $2.66\times 10^{-2}$
$g_{1,7}$ | 128.0773819 | 116.4225439 | $2.66\times 10^{-2}$
$g_{1,8}$ | 146.4245279 | 133.1017651 | $2.65\times 10^{-2}$
$f_{2}$ | 1.2909440 | 1.1915768 | $1.12\times 10^{-2}$
$g_{2,1}$ | 9.4164937 | 8.5607991 | $2.64\times 10^{-2}$
$g_{2,2}$ | 20.6825679 | 18.8000960 | $2.66\times 10^{-2}$
$g_{2,3}$ | 31.4771109 | 28.6128454 | $2.66\times 10^{-2}$
$g_{2,4}$ | 42.1642251 | 38.3283552 | $2.65\times 10^{-2}$
$g_{2,5}$ | 52.8083359 | 48.0049281 | $2.65\times 10^{-2}$
$g_{2,6}$ | 63.4306072 | 57.6617194 | $2.65\times 10^{-2}$
$g_{2,7}$ | 74.0401429 | 67.3069695 | $2.65\times 10^{-2}$
$g_{2,8}$ | 84.6415378 | 76.9448380 | $2.65\times 10^{-2}$
$f_{3}$ | 1.0796099 | 1.0003529 | $7.39\times 10^{-3}$
$g_{3,1}$ | 6.6828617 | 6.0771126 | $2.62\times 10^{-2}$
$g_{3,2}$ | 14.6647687 | 13.3323883 | $2.64\times 10^{-2}$
$g_{3,3}$ | 22.3087624 | 20.2815701 | $2.64\times 10^{-2}$
$g_{3,4}$ | 29.8754567 | 27.1606785 | $2.64\times 10^{-2}$
$g_{3,5}$ | 37.4108454 | 34.0113796 | $2.64\times 10^{-2}$
$g_{3,6}$ | 44.9300978 | 40.8474210 | $2.64\times 10^{-2}$
$g_{3,7}$ | 52.4397798 | 47.6747566 | $2.64\times 10^{-2}$
$g_{3,8}$ | 59.9432319 | 54.4964175 | $2.64\times 10^{-2}$