# Prospects for the Observation of Continuous Gravitational Waves from
Deformed Fast-spinning White Dwarfs

Manoel F. Sousa,1,2 Edson Otoniel,3 Jaziel G. Coelho2,4 and José C. N. de
Araujo2
1Programa de Pós-Graduação em Física e Astronomia, Universidade Tecnológica
Federal do Paraná, Avenida sete de setembro 3165, 80230-901, Curitiba, PR,
Brazil
2 Divisão de Astrofísica, Instituto Nacional de Pesquisas Espaciais, Avenida
dos Astronautas 1758, São José dos Campos, SP 12227-010, Brazil
3Instituto de Formação de Educadores, Universidade Federal do Cariri, R.
Olegário Emidio de Araujo, s/n – Aldeota, 63260-000 Brejo Santo, CE, Brazil
4Núcleo de Astrofísica e Cosmologia (Cosmo-Ufes) & Departamento de Física,
Universidade Federal do Espírito Santo, 29075–910, Vitória, ES, Brazil E-mail:
<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

There has been a growing interest within the astrophysics community in highly
magnetized and fast-spinning white dwarfs (WDs), commonly referred to as
HMWDs. WDs with these characteristics are quite uncommon and possess magnetic
fields $\geqslant 10^{6}$ G, along with short rotation periods ranging from
seconds to just a few minutes. Based on our previous work, we analyze the
emission of Gravitational Waves (GWs) in HMWDs through two mechanisms: matter
accretion and magnetic deformation, which arise due to the asymmetry
surrounding the star’s rotational axis. Here, we perform a thorough self-
consistent analysis, accounting for rotation and employing a realistic
equation of state to investigate the stability of stars. Our investigation
focuses on the emission of gravitational radiation from six rapidly spinning
WDs: five of them are situated within binary systems, while one is an AXP,
proposed as a magnetic accreting WD. Furthermore, we apply the matter
accretion mechanism alongside the magnetic deformation mechanism to assess the
influence of one process on the other. Our discoveries indicate that these WDs
could potentially act as GW sources for BBO and DECIGO, depending on specific
parameters, such as their mass, the angle ($\alpha$) between the magnetic and
rotational axes, and the accumulated mass ($\delta m$) at their magnetic
poles, which is influenced by the effect of matter accretion. However,
detecting this particular class of stars using the LISA and TianQin space
detectors seems unlikely due to the challenging combination of parameters such
as a large $\delta m$, a large $\alpha$ angle and a small WD mass value.

###### keywords:

gravitational waves – (stars:) white dwarfs – stars: magnetic field

††pubyear: 2023††pagerange: Prospects for the Observation of Continuous
Gravitational Waves from Deformed Fast-spinning White Dwarfs–Prospects for the
Observation of Continuous Gravitational Waves from Deformed Fast-spinning
White Dwarfs

##  1 Introduction

White dwarfs (WDs) represent the ultimate stage of stellar evolution for stars
with masses ranging from $\sim 0.4~{}\textrm{M}{\odot}$ to
$8.0~{}\textrm{M}{\odot}$. Some of these remnants rotate with periods ranging
from seconds to years. In general, isolated WDs have rotation periods of days
or even years. However, isolated WDs rotating with periods of a few seconds to
minutes have been observed recently; as examples, we can mention SDSS
J221141.80+113604.4 (SDSS J2211+1136) with a rotation period of $P=70.32$ s
(Kilic et al., 2021) and ZTF J190132.9+145808.7 (ZTF J1901+1458) with
$P=416.20$ s (Caiazzo et al., 2021). WDs in binary systems, on the other hand,
spin with periods ranging from seconds to hours. Recently, Pelisoli et al.
(2022) reported on a WD exhibiting the most fast rotation ever observed. In
their study, the authors present observations conducted with the Gran
Telescopio Canarias (GTC), unveiling that LAMOST J0240.51+1952 is a
cataclysmic variable hosting a WD with an extremely fast rotation of $24.93$
s. Another WD with a very similar rotation period was detected in the binary
system CTCV J2056-3014 from the XMM-Newton observations that point out that
the WD has a rotation period of $29.6$ s (Lopes de Oliveira et al., 2020).

From a theoretical perspective, WDs can have an extremely short rotation
period and still maintain rotational stability. For instance, Boshkayev et al.
(2013b) investigated the stability of WDs for different nuclear compositions
using General Relativity to describe uniformly rotating WDs within Hartle’s
formalism (Hartle, 1967; Hartle & Thorne, 1968). In this scenario, they found
that the minimum rotation periods for a WD composed of 4He, 12C, 16O and 56Fe
be stable are $P_{min}\sim$ ($0.3$; $0.5$; $0.7$; $2.2$) s with a maximum mass
of $M\sim$ ($1.500$; $1.474$; $1.467$; $1.202$) $\textrm{M}_{\odot}$,
respectively.

WDs can also have magnetic fields with intensities ranging from $10^{3}$ G to
$10^{9}$ G (Ferrario et al., 2015). In recent years, there has been an
increasing interest within the astrophysics community in highly magnetized
white dwarfs (HMWDs). These HMWDs, with surface magnetic fields ranging from
$10^{6}$ G to $10^{9}$ G, have already been confirmed in several works (see
e.g., Külebi et al., 2009; Kuelebi et al., 2010; Kepler et al., 2013, 2015;
Amorim et al., 2023). In addition to their high magnetic fields, most of these
WDs are massive; for instance: REJ 0317–853 has $M\approx
1.35~{}\textrm{M}_{\odot}$ and $B\approx(1.7$–$6.6)\times 10^{8}$ G (Barstow
et al., 1995; Külebi et al., 2010); ZTF J1901+1458 has $M\approx
1.31~{}\textrm{M}_{\odot}$ and $B\approx(6.0$–$9.0)\times 10^{8}$ G (Caiazzo
et al., 2021); and PG 1031+234 has the highest magnetic field $B\approx
10^{9}$ G (Schmidt et al., 1986; Külebi et al., 2009). The existence of ultra-
massive WDs has been revealed in several studies (see e.g., Althaus et al.,
2005, 2007; Hermes et al., 2013; Curd et al., 2017; Camisassa et al., 2019;
Gentile Fusillo et al., 2018; Jiménez-Esteban et al., 2018).

Some rapidly rotating magnetic WDs have been observed to emit pulsed
electromagnetic radiation and have been proposed as WD pulsars. In particular,
a WD pulsar was recently discovered in a binary system known as AR Scorpii (AR
Sco). This WD has a rotation period of $1.97$ min and emits pulses over a wide
range of frequencies (Marsh et al., 2016). The spindown luminosity of this WD
is an order of magnitude greater than the observed X-ray luminosity, which,
with the absence of accretion signals, suggests that AR Sco is probably a
rotation-powered pulsar (Buckley et al., 2017). More recently, another WD
pulsar, considered to be a sibling of AR Sco, has been discovered in the
system J191213.72$-$441045.1 (Pelisoli et al., 2023). This pulsar rotates with
a period of $5.30$ min and exhibits pulsed emission ranging from radio to
X-ray wavelengths. The spindown value for this source has not yet been
measured. With that, constraints on the strength of the magnetic field and the
mechanism that powers the pulsar have not been estimated (Pelisoli et al.,
2023).

Other sources have also been proposed as WD pulsar candidates. A specific
example is AE Aquarii (AE Aqr), identified as an X-ray pulsar with a short
rotation period of $P=33.08$ s (Patterson, 1979; Terada et al., 2008). Another
example is the HD 49798/RX J0648.0-4418 binary system. Mereghetti et al.
(2009) showed that the pulsed X-ray emission from this system came from a
massive WD with mass $M=1.28M_{\odot}$ and a very short rotation period of
$P=13.2$ s. However, the nature of this compact object is still unclear (see
Mereghetti et al., 2016; Popov et al., 2018).

Some isolated, very massive, magnetic, and fast WDs seem to have been
originated by a merger of two WDs. Numerical simulations show that these WDs
might indeed originate in double WD mergers (see e.g., Lorén-Aguilar et al.,
2009; Dan et al., 2014). The general configuration after the merger is a
central WD formed by the mass of the primary WD, which is rigidly rotating.
This central object is surrounded by a hot corona with differential rotation
and a Keplerian disc of rapidly rotating mass. The mass of the interrupted
secondary star is distributed between the envelope and the disk. The hot,
convective corona works as an efficient dynamo that can lead to magnetic
fields of $\lesssim 10^{10}$ G (García-Berro et al., 2012). From this post-
merger configuration, Sousa et al. (2022) recently investigated the
possibility that the isolated WDs, SDSS J2211+1136 and ZTF J1901+1458, are
products of the coalescence of double WDs. For this, the authors calculated
the post-merger rotational evolution of the central WD and inferred the system
parameters for which the rotational age coincides with the cooling age of the
WD. They concluded that the observed parameters of SDSS J2211+1136 and ZTF
J1901+1458 are consistent with an origin in the double WD mergers. More
recently, Sousa et al. (2023) characterized the optical signatures of these
mergers. They found that the merger ejecta’s expansion and cooling produce a
peak optical emission $1$–$10$ days post-merger, with luminosities of
$10^{40}$–$10^{41}$ erg s-1 and that LSST at the Vera C. Rubin Observatory
will likely detect a few hundred of these events annually.

Several theoretical works regarding very magnetic, massive, and fast WDs have
been published in the last few years. One of these studies, in particular,
involves WD pulsars in an alternative description for the Soft Gamma Repeater
(SGRs) and Anomalous X-Ray Pulsars (AXPs) (Malheiro et al., 2012; Coelho &
Malheiro, 2014; Mukhopadhyay & Rao, 2016). From this perspective, a canonical
model of a pulsar powered by rotation can explain the process of energy
emission in a WD (Usov, 1988; Coelho et al., 2017). In another proposed
scenario, Rueda et al. (2013) suggest that AXP 4U 0142+61 is a WD pulsar
resulting from the merger of two common WDs, which is surrounded by an
accretion disk produced during the merger. The optical and infrared
observational data from this source would thus be explained by the WD
photosphere and the disk, while the X-ray emission would be caused by a
pulsar-like emission. However, Borges et al. (2020) argue that, within this
perspective, quiescent spectral energy distribution from mid-infrared down to
hard X-rays can be described by a magnetic and isolated WD that accretes
matter from a debris disk that surrounds it.

From a gravitational perspective, direct observations of gravitational waves
(GWs) have recently been made by the LIGO-Virgo-KAGRA collaboration. The first
event was detected in 2015 by the LIGO detector (Abbott et al., 2016). Since
then, a few dozen detections of GWs have been made that come from the
coalescence of compact binaries, namely black hole binaries, neutron star
binaries, and binaries formed by a black hole and a neutron star (Abbott et
al., 2019, 2021; The LIGO Scientific Collaboration et al., 2021). All these
GWs detections are in a frequency band ranging from $10$ Hz to $1000$ Hz,
which is the operational band of LIGO, Virgo, and KAGRA. However, there are
several proposals for space missions for detection at lower frequencies, such
as LISA and TianQin, whose frequency band is of $(10^{-4}-10^{-1})$ Hz, and
BBO and DECIGO over a frequency band ranging from $0.01$ Hz to $10$ Hz (see
e.g., Amaro-Seoane et al., 2017; Luo et al., 2016; Harry et al., 2006; Yagi &
Seto, 2017; Kawamura et al., 2006).

Different possibilities of generation of continuous GWs have already been
proposed (see e.g., Bonazzola & Gourgoulhon, 1996; de Araujo et al., 2016;
Mukhopadhyay et al., 2017; de Araujo et al., 2017; Gao et al., 2017; Pereira
et al., 2018; Kalita et al., 2020, and references therein). Franzon & Schramm
(2017) investigated the possibility that fast-spinning and magnetic WDs are
sources of detectable GW emission. They computed the quadrupole moment of the
mass distribution and performed an estimate of the GW of such WDs from the
construction of numerical stellar models at different baryon masses. They
found that magnetic and fast WDs might generate GW radiation that lies in the
bandwidth of BBO and DECIGO. Kalita & Mukhopadhyay (2019) also showed that
fast and magnetic WDs can be prominent sources for space detectors such as
DECIGO, BBO, and LISA. The authors explored very massive WDs (close to the
Chandrasekhar limit) and super-Chandrasekhar WDs and analyzed the GW amplitude
for different WD parameters. They found that several super-Chandrasekhars,
rotating in a frequency range of $0.01-1$ Hz and with a distance of $100$ pc,
emit gravitational radiation detectable by BBO, DECIGO and LISA, considering a
1-year observation (see also Kalita et al., 2020; Mukhopadhyay & Bhattacharya,
2022).

The main goal of this paper is to extend our previous study (Sousa et al.,
2020) in which we investigated the continuous gravitational radiation from
three fast-spinning magnetized WDs, which rotate with periods from seconds to
minutes and have magnetic fields around $10^{6}$ G to $\sim 10^{9}$ G. We
considered two mechanisms of emission of GWs: matter accretion and magnetic
deformation. In both cases, WDs generate continuous GWs due to the asymmetry
around their axis of rotation formed as a result of the accumulated mass at
the magnetic poles of a WD accreting mass, so that the accretion flux follows
the field lines, or due to the intense magnetic field, which can break the
star’s spherical symmetry, making it oblate or prolate depending on the field
configuration. Here, we increase the number of fast-spinning and magnetic WDs
and investigate the combined impact of accretion and magnetic deformations on
the emission of GWs. Specifically, we examine how the asymmetry induced in the
WD by accretion and the magnetic field synergistically influence the emission
of gravitational radiation. Additionally, we calculate the GW amplitude for a
mass range derived from theoretical limits (minimum and maximum) obtained
through numerical and self-consistent determination of stability regions for
WDs. This determination incorporates constraints imposed by the Kepler
frequency (mass-shedding) and pycnonuclear fusion reactions.

This paper is organized as follows. In Sec. 2, we present the basic model
adopted, brief descriptions of the Hartle’s stellar rotation formalism and the
pycnuclear fusion reactions. The gravitational emission mechanism is
considered in Sec. 3. In Sec. 4 the results and discussions are considered.
Finally, in Sec. 5 the main conclusions are presented.

##  2 The model

Recent advancements in observation and detection techniques for WDs have
sparked significant interest in theoretical studies of their structure and
evolution. This is accompanied by sophisticated calculations of fermionic
matter properties under extreme conditions (Eisenstein et al., 2006; Kepler et
al., 2013, 2015; Córsico et al., 2019). In this context, the equation of state
(EoS) for relativistic and degenerate WD matter plays a crucial role. The EoS
accounts for the excess free energy of an ion-based one-component plasma (OCP)
and is computed using the Helmholtz free energy (Boshkayev et al., 2013b;
Coelho et al., 2014; Chamel et al., 2014; Becerra et al., 2018; Otoniel et
al., 2019, 2022).

The Helmholtz free energy, denoted by $F$, is given by:

$F=F_{\mathrm{id}}^{\mathrm{ion}}+F_{\mathrm{id}}^{(\rm e)}+F_{\rm ee}+F_{\rm
ii}+F_{\rm ie}\,.$ (1)

The free energy of a nonrelativistic classical gas,
$F_{\mathrm{id}}^{\mathrm{ion}}$, is described by:

$F_{\mathrm{id}}^{\mathrm{ion}}=N_{\rm j}k_{\rm B}T\left[\ln\left(n_{\rm
j}\lambda_{\rm j}^{3}/g_{\rm j}\right)-1\right].$ (2)

Herein, $N_{\rm j}=n_{\rm j}V$, $k_{\rm B}$ symbolizes the fundamental
Boltzmann constant, $T$ corresponds to the temperature of the gas, $n_{\rm j}$
encapsulates the comprehensive number density of ions of an ensemble of
particles in volume V and $\lambda_{\rm j}^{3}$ epitomizes the thermal de
Broglie wavelength, whereas $\lambda_{\rm j}=\left(2\pi\hbar^{2}/m_{\rm
j}k_{\rm B}T\right)^{1/2}$ encapsulates the profound essence of the ion’s mass
$m_{\rm j}$ and the multiplicity of its spin, ingeniously denoted by $g_{\rm
j}$. A meticulous examination of the seminal work by Chabrier & Potekhin
(1998) allows us to glean a profound understanding of its intricate fabric.

Similarly, the free energy of electrons, $F_{\mathrm{id}}^{(\rm e)}$, can be
expressed as:

$F_{\mathrm{id}}^{(\rm e)}=\mu_{e}N_{e}-P_{\mathrm{id}}^{(\rm e)}V.$ (3)

The pressures and electron number density are given by: $P_{\text{id}}^{(\rm
e)}$ and $n_{\rm e}=N_{\rm e}/V$, where $N_{e}$ represents the number of
electrons and $\mu_{e}$ denotes the electron chemical potential excluding the
rest energy $m_{e}c^{2}$. This expression can be further elaborated in terms
of Fermi-Dirac integrals, $I_{\rm\nu}\left(\chi_{\rm e},\tau\right)$, where
$\chi_{\rm e}=\mu_{\rm e}/k_{\rm B}T$ and $\nu=1/2,3/2,$ and $5/2.$ The
chemical potential can be obtained by inverting the function $n_{\rm
e}\left(\chi_{\rm e},T\right)$ numerically. The remaining terms in equation
(1) represent free energies arising from electron–electron, ion–ion, and
ion–electron interactions (Chabrier & Potekhin, 1998; Potekhin & Chabrier,
2000). The plasma’s pressure ($P$) and entropy ($S$) with a fixed number of
particles in a given volume ($V$) are obtained from the thermodynamic
relations $P=-(\partial F/\partial V)_{T,N}$ and $S=-(\partial F/\partial
T)_{V,N}$, where the internal energy is defined as $U=F+TS$ (Chabrier &
Potekhin, 1998; Potekhin & Chabrier, 2000).

###  2.1 Hartle’s Stellar Rotation Formalism

The structural characteristics of WDs are governed by a state of hydrostatic
equilibrium, wherein the gravitational forces are precisely counterbalanced by
the outward pressure exerted by a relativistic electron gas. Recent studies
have underscored the importance of accounting for the contributions of general
relativity in accurately modeling the structural properties of WDs (see
Boshkayev et al., 2013a). The Tolman-Oppenheimer-Volkoff (TOV) equation serves
as the cornerstone for describing the configuration of non-rotating stellar
objects, providing crucial insights into their radii and masses. However, to
fully incorporate the effects of rotation, it is imperative to solve
Einstein’s field equations using a metric that accommodates rotational
deformations and the accompanying dragging of local inertial frames (see
Hartle, 1967; Hartle & Thorne, 1968; Friedman et al., 1986).

In this study, we adopt Hartle’s formalism to investigate the equilibrium
configurations of rotating stellar bodies within the framework of general
relativity. Hartle’s formalism is founded upon two critical assumptions:
uniform rotation and slow rotation, with a quadratic dependence on the angular
velocity. These assumptions provide the necessary mathematical framework to
model the behavior of rotating stars, enabling a systematic analysis of their
structure and dynamics. By incorporating these assumptions, a deeper
understanding of the physical processes that govern the behavior of rotating
stellar objects is achieved. In this context, the metric formulation employed
can be expressed as follows:

$ds^{2}=-e^{2\nu}dt^{2}+e^{2\psi}(d\phi-\omega
dt)^{2}+e^{2\mu}d\theta^{2}+e^{2\lambda}dr^{2},$ (4)

where the metric functions $\nu$, $\psi$, $\mu$, and $\lambda$, along with the
frame dragging frequency $\omega$, are reliant upon the radial coordinate $r$,
the polar angle $\theta$, and implicitly on the rotational frequency $\Omega$
of the star (see Friedman et al., 1986; Weber, 1999, for further details).
Accordingly, we conducted a comprehensive two-dimensional analysis that
incorporates the rotational deformations described by the polar angle
$\theta$, while assuming axial symmetry in compact object configurations. The
range of $\Omega$ considered was confined within $0\leq\Omega\leq\Omega_{K}$,
where $\Omega_{K}$ $(=2\pi/P_{K})$ denotes the Kepler frequency, signifying
the threshold for stable rotation termination due to mass-shedding.
Remarkably, $\Omega_{K}$ serves as an absolute upper limit for rapid rotation.

In our study of relativistic stars, we applied Hartle’s perturbative
formalism, which approximates the behavior of rotating stars up to the
Keplerian frequency. While Hartle’s work defines the slow rotation limit using
$\sqrt{M/R^{3}}$ as a reference, it does not specify the upper limit of the
rotation rate for perturbative calculations (Hartle, 1967; Hartle & Thorne,
1968). Nevertheless, Weber & Glendenning (1992) have demonstrated through
numerical analysis that Hartle’s perturbative approach remains valid within a
certain margin of deviation from exact results at the Keplerian frequency.
Therefore, despite the Keplerian frequency being a characteristic frequency of
the system, our justification for extending Hartle’s formalism to encompass
this frequency range lies in Weber $\&$ Glendenning’s validation of its
applicability beyond the traditional slow rotation limit.

The self-consistent computation of the Kepler frequency was performed
concurrently with Einstein’s field equations pertaining to the metric
functions. This procedure entailed utilizing the following equation:

$\Omega_{\rm
K}=\omega+\frac{\omega^{\prime}}{2\psi^{\prime}}+e^{\nu-\psi}\sqrt{\frac{\nu^{\prime}}{\psi^{\prime}}+\left(\frac{\omega^{\prime}}{2\psi^{\prime}}e^{\psi-\nu}\right)^{2}},$
(5)

where, the primes indicate partial derivatives of the metric functions with
respect to the radial coordinate (Weber & Glendenning, 1992; Glendenning &
Weber, 1994). Perturbative treatments, such as the one developed by Hartle,
have been found to yield results that demonstrate remarkable agreement with
those obtained through numerically exact approaches to Einstein’s field
equations. This notable congruity is particularly evident when examining the
increase in mass attributable to rapid rotation at the Kepler frequency (Weber
& Glendenning, 1992).

To summarize, a comprehensive understanding of WD structure necessitates the
consideration of general relativity effects and the inclusion of rotational
phenomena within appropriate metric formulations. These advancements provide
invaluable insights into the interplay between gravitational forces, pressure,
and rotation, thereby enriching our knowledge of these intriguing remnants of
stellar evolution (Hartle & Thorne, 1968).

###  2.2 Pycnonuclear fusion reactions

Pycnonuclear fusion reactions are nuclear fusion reactions between heavy
atomic nuclei that occur in regions of high density and are schematically
expressed as
${}_{A}^{Z}\textrm{X}+\,_{A}^{Z}\textrm{X}\to\,_{2A}^{2Z}\textrm{Y}$. Examples
of this type of reaction are the fusion of carbon with carbon
(${}^{12}\textrm{C}\,+\,^{12}\textrm{C}$) and oxygen with oxygen
(${}^{16}\textrm{O}\,+\,^{16}\textrm{O}$). These fusion reactions are
significantly density sensitive but nearly temperature independent (Gasques et
al., 2005). At these high densities, the probability of nuclei undergoing the
tunneling process becomes considerable, and therefore fusion reactions are
likely to occur often enough to begin to affect the star’s stability.

It is well known that pycnonuclear fusion reactions make WDs unstable.
However, the nuclear fusion rates at which these reactions occur are highly
uncertain because of some poorly constrained parameters (see, e.g., Gasques et
al., 2005; Yakovlev et al., 2006). However, the occurrence of these reactions
has already been theoretically analyzed for a considerable density range that
includes values of densities that are found inside WDs (Chamel et al., 2013;
Chatterjee et al., 2017). Gasques et al. (2005), for example, developed a set
of analytical equations to calculate the rates of pycnonuclear reactions
between identical atomic nuclei and applied them to carbon fusion inside WDs.
The authors found that carbon burning becomes important for a temperature
$T\sim(4-15)\times 10^{8}$ K assuming a density $\rho\lesssim 3\times 10^{9}$
g cm-3, and for $\rho\sim(3-50)\times 10^{9}$ g cm-3 considering $T\lesssim
10^{8}$ K.

Several studies have been carried out with the aim of performing a stability
analysis of the matter in the cores of WDs against pycnonuclear fusion
reactions and electron capture reactions (see, e.g., Boshkayev et al., 2013b;
Otoniel et al., 2019; Nunes et al., 2021). In a recent work, Otoniel et al.
(2021) investigated the stability of the rapidly rotating WD of the CTCV
J2056–3014 system, where they determined mass density limits for which
pycnonuclear fusion reactions start, thus inferring a maximum value for the
mass of this WD. The authors also studied the impact of rapid rotation on the
structure and stability of the WD to calculate a minimum mass value for this
remnant. The procedure performed in this work to find the mass limits of the
WDs follows the methodology presented in that study.

Therefore, to obtain the upper mass limit, we assume that WD has a uniform
chemical composition throughout the star, and we investigate the stability of
the WD matter to pycnonuclear fusion reactions using up-to-date theoretical
models (see, e.g., Gasques et al., 2005; Golf et al., 2009). These models
analyze nuclear fusion rates in dense matter, covering various burning
regimes. By addressing the Coulomb barrier problem and proposing a versatile
formula for reaction rates, they offer a comprehensive solution adaptable to
current theoretical uncertainties. The model’s efficacy is demonstrated
through the examination of the ${}^{12}\text{C}+\,^{12}\text{C}$ fusion
reaction, crucial for understanding stellar phenomena like nuclear burning in
evolved stars, type Ia supernovas, and accreting neutron stars. Leveraging a
parameter-free model for the energy-dependent S factor calculation, accounting
for Pauli nonlocality effects, enhances accuracy in estimating fusion
efficiencies, particularly emphasizing carbon ignition at densities exceeding
$10^{9}$ g cm-3 (Gasques et al., 2005; Golf et al., 2009).

##  3 Gravitational emission mechanism

We investigated the gravitational radiation emitted by fast-spinning magnetic
WDs with structural deformation caused by matter accretion. Thus, within this
GW generation mechanism, we consider a WD accreting matter through magnetic
poles that do not coincide with the spin axis of the star. In this scenario,
the WD receives matter via an accretion column, which may come from a remnant
debris disk generated after the merger of two WDs (see e.g., Dan et al., 2014;
García-Berro et al., 2012) or from a secondary star that transfers part of its
mass through a disk truncated by field lines, or even without forming a disk
(see e.g., Welsh et al., 1998; Warner, 2003). In this way, the WD accumulates
an amount of matter at its magnetic poles causing irregularities in the shape
of the star.

Below, we describe the mechanism of matter accretion and deduce the GW
amplitude emitted in this process. Moreover, we found an expression for
amplitude that takes into account the effect of magnetic deformation on GW
emission.

###  3.1 Accretion of matter: basic equations

We consider a rotating rigid object whose axes of symmetry ($x_{1}$, $x_{2}$,
$x_{3}$) rotate with the object, and the corresponding principal moments of
inertia are $I_{1}$ , $I_{2}$ and $I_{3}$, respectively. This solid rotates
with angular velocity $\Omega$ relative to the axis $x^{\prime}_{3}$ of an
inertial frame whose axes are ($x^{\prime}_{1}$, $x^{\prime}_{2}$,
$x^{\prime}_{3}$). We also consider that the axis of the magnetic dipole is
given by the axis $x_{3}$ and that this axis forms an angle $\alpha$ with the
axis $x^{\prime}_{3}$. In other words, the angle of inclination of the
magnetic axis relative to the axis of rotation is given by $\alpha$. Thus,
making $I_{1}=I_{2}$ and $d$ being the distance to the emitting source, the GW
amplitude, $h_{ac}$, can be represented by (Shapiro & Teukolsky, 1983;
Maggiore, 2008)

$h_{ac}=\frac{4G}{c^{4}}\frac{(I_{1}-I_{3})\Omega^{2}}{d}\sin^{2}\alpha.$ (6)

It is worth mentioning that the amplitude equation above is deduced
considering that the angle between the axis of rotation ($x^{\prime}_{3}$) and
the line of sight is zero. With this assumption, we maximize the value of GW
amplitude when we consider only this parameter (see Maggiore, 2008, for
details on this dependency).

Now, we need to determine the moments of inertia $I_{1}$ and $I_{3}$. Assuming
a deformation created only by accretion of mass, we can consider that the star
has deformities or mountains of matter on the $x_{3}$ axis, so that these
deformities can be approximated to small spheres of masses close to the star’s
magnetic poles. Thus, with this configuration, the term $I_{1}-I_{3}$ and the
GW amplitude can be expressed as (Sousa et al., 2020)

$I_{1}-I_{3}=2\delta m~{}R^{2},$ (7)

$h_{ac}=\frac{8G}{c^{4}}\frac{\delta m~{}R^{2}\Omega^{2}}{d}\sin^{2}\alpha,$
(8)

where $R$ is the radius of the star and $\delta m$ is the amount of mass
accumulated on one magnetic pole.

However, together with the deformations caused by $\delta m$, we can also
include the effect of the magnetic field on the spherical shape of the star.
In other words, we can find an equation that expresses the irregularities in
the star’s structure due to accretion and the intense magnetic field. For this
task, we must assume that the star, previously considered as a sphere in the
above assumption, contracts along the magnetic field, becoming an oblate
spheroidal shape. Thus, in addition to the small spheres we assume in the
above configuration, we must consider the star as an oblate object.

Assuming that $R$ is the polar radius of the oblate spheroid ($x_{3}$ axis)
and $R_{\rm eq}=R+b$ is the radius at the star’s equator ($x_{1}$ and $x_{2}$
axes), we can write the new moments of inertia $I_{1}$ and $I_{3}$ as

$I_{1}=\frac{1}{5}M(R_{\rm eq}^{2}+R^{2})+2\,\delta m\,R^{2},$ (9)

$I_{3}=\frac{2}{5}MR_{\rm eq}^{2}+2\frac{2}{5}\delta
m\,a^{2}\approx\frac{2}{5}MR_{\rm eq}^{2},$ (10)

where $M$ is the mass of the star and $a$ is the radius of the amount of mass
accumulated on one magnetic pole. In the last term of equation (10), we assume
$R_{\rm eq}\gg a$. The first term on the right side of equations (9) and (10)
corresponds to the moment of inertia of an oblate spheroid, whereas the second
term represents the moment of inertia of the two mountains of matter on the
$x_{3}$ axis.

Now, considering that $R_{\rm eq}\gg b$, the term $I_{1}-I_{3}$ can be
expressed as follows

$I_{1}-I_{3}=2\,\delta m\,R^{2}-\frac{2}{5}M\,R^{2}\frac{b}{R}.$ (11)

By recalling that the moment of inertia for a sphere of radius $R$ is given by
$I=(2/5)MR^{2}$ and that the definition of the eccentricity
$\varepsilon=\sqrt{R_{\rm eq}^{2}-R^{2}}/R$ returns $\varepsilon=b/R$, one
obtains

$I_{1}-I_{3}=2\,\delta m\,R^{2}-I\varepsilon.$ (12)

It is worth noting that the first term on the right side of the equation (12)
indicates the deformation produced by the accretion of matter mechanism, while
the second term on the right side represents the deformability due to the
intense magnetic field. Furthermore, the minus sign between these two terms
should be noted, suggesting that the two GW emission effects act to cancel
each other out. That is, the eccentricity created by the magnetic field does
not contribute to the accretion process to produce GWs with larger amplitudes
but rather weakens the wave amplitude. This occurs because the deformations
produced by the two processes appear on different semi-axes, reducing the
irregularities in the star shape.

Substituting equation (12) into equation (6), it follows that

$h_{\rm acmg}=\frac{4G}{c^{4}}\frac{\left|2\,\delta
m\,R^{2}-I\varepsilon\right|\Omega^{2}}{d}\sin^{2}\alpha,$ (13)

where the modulus function appears because $h_{\rm acmg}$ is defined as the
strain and we introduce the $acmg$ index to represent the amplitude that takes
into account the effect of the magnetic field. Moreover, the eccentricity
caused by the magnetic field can also be represented as (see, e.g., Bonazzola
& Gourgoulhon, 1996; Konno et al., 2000; Regimbau & de Freitas Pacheco, 2006)

$\varepsilon=\kappa\frac{B^{2}R^{4}}{GM^{2}},$ (14)

where, as before, $B$ is the magnetic field strength on the star’s surface and
$\kappa$ is the distortion parameter, which depends on the magnetic field
configuration and equation of state (EoS) of the star.

Thereby, we find an equation that represents the GW emission for the process
of accretion of matter together with the mechanism of magnetic deformation
that depend on the amount of mass accumulated on the WD’s magnetic poles and
also on the eccentricity caused by the magnetic field.

##  4 Results and Discussions

Here we extend our previous study (Sousa et al., 2020) by expanding the number
of fast-spinning magnetic WDs and establishing constraints on the mass and
radius for these WDs from the construction of stability regions. These limits
are determined by the Kepler frequency and by the pycnonuclear fusion
reactions. We also approach the effect of the deformation of the WD produced
by the intense magnetic field on the GW amplitude emitted by the accretion
mechanism.

Table 1 shows the six fast-spinning magnetic WDs that we analyzed in this work
along with their main parameters. These sources are five binary systems:
LAMOST J0240+1952, CTCV J2056-3014, AE Aqr, V1460 Her, RX J06480-4418; and an
AXP: 4U 0142+61 for which we assume the model described by Borges et al.
(2020) in which this object is a massive remnant WD accreting from a debris
disk formed during the merger of two other WDs. The magnetic field strength of
this AXP is in an interval inferred from the rotation variation due to the
coupling between the magnetic field and the disk to reproduce the spindown
value of this object (Borges et al., 2020).

Table 1: Six fast-spinning magnetic WDs along with their main parameters:
Period (P), magnetic field (B), Temperature (T) and distance to source (d).

SYSTEMS |  | P
---
(s)
| B
---
(G)
| T
---
(K)
| d
---
(pc)
AXP 4U 0142+61 a,b | 8.69 |  $5.0\times 10^{7}$ b | $10^{5}$ | 3780 a
RX J06480-4418 c | 13.18 | $10^{8}$ | $10^{5}$ | 520.94 h
LAMOST J0240+1952 d | 24.93 | $10^{7}$ | $\leqslant 2.5\times 10^{4}$ | 592.98 h
CTCV J2056-3014 e | 29.61 | $10^{7}$ | $10^{4}$ | 256.29 h
AE Aquarii f | 33.08 | $5.0\times 10^{7}$ | $10^{7}$ | 91.87 h
V1460 Herculis g | 38.87 | $10^{7}$ | $10^{3}$ | 266.03 h

a Olausen & Kaspi (2014); b Borges et al. (2020); c Israel et al. (1997);
Mereghetti et al. (2009, 2021, 2011); d Pelisoli et al. (2022); e Lopes de
Oliveira et al. (2020); Otoniel et al. (2021); f Terada et al. (2008);
Kitaguchi et al. (2014); g Ashley et al. (2020); Kjurkchieva et al. (2017); h
Gaia Collaboration et al. (2022)

Furthermore, to analyze the effect of the magnetic field on the GW amplitude
emitted by the accretion mechanism, we established a value of $B=10^{7}$ G for
the WDs that do not have known magnetic field values, such as the LAMOST
J0240+1952 and V1460 Her. This is a moderate field value for magnetic
cataclysmic variables (see e.g., Figure 30 by Ferrario et al., 2020) and one
that is also not so intense that the magnetic interaction with the secondary
star locks the WD into a rotation synchronized with the orbital period. For
CTCV J2056-3014, although magnetic field estimates show that this WD has
possibly a low field value ($\sim 10^{4}$ G; Lopes de Oliveira et al. (2020)),
we also assume here a magnetic field of $B=10^{7}$ G, in order to investigate
the influence of an intense magnetic field on the emission of gravitational
radiation from this system.

Using the approach described in Sec. 2, we numerically calculate the minimum
mass (maximum radius) and maximum mass (minimum radius) from the WD stability
conditions. This approach is presented in Otoniel et al. (2021), where the
authors model a WD rotating uniformly within Hartle’s formalism for stellar
rotation, considering an EoS that takes into account not only the contribution
of a Fermi gas but also the contributions of electron-ion, electron-electron
and ion-ion interactions, and explore instabilities due to the effects of
rapid rotation and microscopic stability on the structure of WDs.

Table 2 presents the mass-radius limits found for each WD of the systems
presented in Table 1. The minimum mass values are obtained assuming the WD is
rotating near its mass-shedding limit, while the maximum mass values are
calculated from the occurrence of pycnonuclear reactions. The WD composition
adopted in each simulation is also presented in this table. It is worth
mentioning that the composition of WDs is not well established. Therefore, we
simulate canonical rotating WDs composed purely of C, except for those that
show some indication of a different chemical composition or having already
been modeled as WDs composed of more than one chemical element. We simulate,
for example, the systems RX J06480-4418 and 4U 0142+61 twice to cover two
different compositions. We did this to investigate whether the mass-radius
limits can vary significantly with this parameter.

Table 2: The minimum (maximum radius) and maximum (minimum radius) mass values
of some fast-spinning magnetic WDs inferred from the stability conditions. The
minimum mass value is obtained assuming that the WD rotation must be close to
the mass-shedding limit, while the maximum mass value is calculated from the
occurrence of pycnonuclear reactions. The WD composition in each case is also
shown.

SYSTEMS |  |  Mmin
---
($M_{\odot}$)
|  Rmax
---
($10^{8}$ cm)
|  Mmax
---
($M_{\odot}$)
|  Rmin
---
($10^{8}$ cm)
Composition
LAMOST J0240+1952 | 0.653 | 10.067 | 1.380 | 1.307 | C
CTCV J2056-3014 | 0.554 | 10.867 | 1.380 | 1.309 | C
AE Aquarii | 0.545 | 11.214 | 1.380 | 1.307 | C a
V1460 Herculis | 0.433 | 11.775 | 1.379 | 1.318 | C
AXP 4U 0142+61 | 1.136 | 6.325 | 1.380 | 1.312 | C
AXP 4U 0142+61 | 1.132 | 6.310 | 1.377 | 1.312 | C 50%/O 50% b
RX J06480-4418 | 0.938 | 7.703 | 1.376 | 1.311 | C 50%/O 50% c
RX J06480-4418 | 0.932 | 7.686 | 1.369 | 1.308 | O 50%/Ne 50% c

a Kitaguchi et al. (2014); b Borges et al. (2020); c Mereghetti et al. (2021,
2011)

It can be seen from Table 2 that the minimum and maximum mass values, as well
as the maximum and minimum radius values, of AXP 4U 0142+61 and RX J06480-4418
change very little for the different compositions of the WD. These values
differ only to the second or third decimal place. Thus, when we consider the
GW amplitude, this difference in mass and radius is not relevant, showing that
the amplitude value is practically the same for both cases. Thus, we chose
only one of the values for these sources in the analysis of gravitational
radiation. We then select AXP 4U 0142+61 with a composition of C and RX
J06480-4418 with a composition of C(50$\%$)/O(50$\%$).

Now, we can proceed with the estimate of the GW emission considering the
parameters of the systems presented in Tables 1 and 2. We use equation (13) to
calculate the GW amplitude from the effects of accretion and magnetic field
strength on the star’s structure. This equation shows that the GW amplitude
depends on the amount of accumulated mass; however, it is not easy to predict
the amount of matter that may have been accreted by WD and how much of it was
dispersed on its surface. In non-magnetic systems, the accreted material on
the surface of the WD can reach
$10^{-4}\,\rm{M}_{\odot}-10^{-6}\,\rm{M}_{\odot}$; nonetheless, the amount of
accumulated mass can be reduced if accretion onto the WD is confined to a
smaller fraction of its surface area (see, e.g., Mitrofanov, 1980; Shara,
1982; José et al., 2020; Scaringi et al., 2022b). Here we attribute four
values to the amount of matter deposited at the magnetic poles: $\delta m=$
($10^{-3}\,\rm{M}_{\odot}$, $10^{-4}\,\rm{M}_{\odot}$,
$10^{-5}\,\rm{M}_{\odot}$, $10^{-6}\,\rm{M}_{\odot}$). These values are
assessed to determine the necessary accreted mass for generating gravitational
radiation detectable by space-based GW instruments. It is observed that lesser
accumulations of matter may result in GW amplitudes that could potentially
escape detection, as we will see below. Furthermore, we consider a range for
the angle of inclination $\alpha$ from $10^{\circ}$ to $90^{\circ}$.

However, before evaluating the GW amplitude values, we must understand the
effect of the deformation created by the magnetic field on the amplitude
generated by the mountain of matter due to accretion. In Sec. 3.1, we note
that the irregularity formed by the magnetic field acts by decreasing the wave
amplitude produced by accretion (see equations (12) and (13)). Therefore, if
the deformation effects created by the two processes become similar, they may
cancel each other and emit no GWs.

Thus, to determine the values of the parameters for which the equation (12) is
null, an expression that represents the dependence of $\delta m$ in relation
to $B$, $M$ and $R$ can be found, namely

$\displaystyle 2\,\delta m\,$ $\displaystyle R^{2}-I\varepsilon=0,$
$\displaystyle\delta m$
$\displaystyle=\frac{1}{5}\,\kappa\,\frac{B^{2}R^{4}}{GM}.$ (15)

Here, we assign the value of $B$ from Table 1 and we adopt $\kappa\simeq 10$,
which is a conservative value that holds for an incompressible fluid star with
a dipole magnetic field (see, e.g. Ferraro, 1954). Similar values are obtained
when relativistic models based on a polytropic EoS with dipole magnetic field
are considered (see, e.g. Konno et al., 2000). Thereby, the equation for
$\delta m$ is only a function of the mass/radius and we can calculate the
value of $\delta m$ for which $h_{\rm acmg}=0$ using the established mass
interval by the $M_{\rm min}$ and $M_{\rm max}$ of each source. The result of
this analysis is shown in Fig. 1 where the dotted lines form the interval of
$\delta m$ that contains the four values of $\delta m$ that we considered
above.

Figure 1: Relation between $\delta m$ and $M$, given by equation (15), for
which $h_{\rm acmg}=0$, that is, when the deformation in the star structure
created by accretion and magnetic field are equal. The mass range presented by
the solid curves is established by the $M_{\rm min}$ and $M_{\rm max}$ of each
system. The dotted lines form a $\delta m$ range that contains the four
considered $\delta m$ values.

It can be observed from this figure that all the analyzed systems are below
the interval of $10^{-6}M_{\odot}\leqslant\delta m\leqslant 10^{-3}M_{\odot}$
and, thus, the GW amplitude is not zero for any value in the mass range of
these WDs when we take these $\delta m$ values into account. Therefore, the
effect of accretion on the WD shape, and consequently on the GW amplitude, is
dominant when compared to magnetic effects on the structure of the star. For
$h_{\rm acmg}=0$ at some point in the mass range, these WDs must have $\delta
m\lesssim 10^{-7}M_{\odot}$ for RX J06480-4418 and AE Aqr, and $\delta
m\lesssim 10^{-9}M_{\odot}$ for LAMOST J0240+1952, CTCV J2056-3014, V1460 Her
and 4U 0142+61. With this, we realize that the GW amplitude of these sources
is not significantly affected by the deformation produced by the magnetic
field. It is worth mentioning that the mass accumulation at the magnetic poles
is significantly influenced by the interplay between the mass-accretion rate
and other factors, such as the magnetic field (see, e.g., Scaringi et al.,
2022a). This interaction can lead to smaller values of $\delta m$ than those
mentioned previously, sometimes resulting in the cancellation of the effects
of deformations caused by accretion and the magnetic field, or with the
magnetic field exerting a greater effect. Nevertheless, the resulting GW
amplitudes in both cases do not generate observable signals for spatial
detectors due to the small $\delta m$ values and the insufficient ellipticity
generated by the analyzed sources’ magnetic field intensity (see Sousa et al.,
2020).

With the issue discussed above, we can now proceed to calculate the GW
amplitude values for the six fast-spinning magnetic WDs considered in this
work. Using equation (13) and assuming the mass range established by $M_{\rm
min}$ and $M_{\rm max}$ from Table 2, the GW amplitude is presented in Fig. 2
for each system, where we set two values for the inclination angle,
$\alpha=90^{\circ}$ and $\alpha=10^{\circ}$. We chose these two values of
$\alpha$ in order to analyze the amplitude values for a small inclination
($10^{\circ}$), close to the alignment of the axes of rotation and magnetic
field, and for an angle ($90^{\circ}$) at which the GW amplitude value is
greatest. Figure 2 displays four panels where the GW amplitude is presented as
a function of the frequency of the WDs, so that the vertical bars represent
$M_{\rm min}\leq M\leq M_{\rm max}$ , from top to bottom. In each panel, a
different value of $\delta m$ is considered. We also show the sensitivity
curves of the space-based detectors LISA, TianQin, BBO and DECIGO (Robson et
al., 2019; Lu et al., 2019; Mei et al., 2021; Yagi & Seto, 2017). It is worth
mentioning that to plot the sensitivity curves, we use the minimum amplitude,
$h_{min}$, that can be measured by the detector, for a periodic signal, for a
given signal-to-noise ratio (SNR) and observation time $\tau_{\rm obs}$ (see
section 7.6 and equation (7.134) of Maggiore, 2008). Thereby, the sensitivity
curves are set to SNR $=8$ and $\tau_{\rm obs}=1$ year.

Figure 2: GW amplitude for different mass values taking into account
$\alpha=90^{\circ}$ (red color) and $\alpha=10^{\circ}$ (blue color). The
vertical bars stands for M${}_{\rm min}\leq$ M${}_{\rm WD}\leq$ Mmax, from top
to bottom. The different panels represent a quantity of accreted mass at the
WD’s magnetic poles, $\delta m=$ ($10^{-3},10^{-4},10^{-5},10^{-6}$)
$M_{\odot}$. The sensitivity curves of the space-based detectors LISA,
TianQin, BBO and DECIGO are shown considering an SNR $=8$ and $\tau_{\rm
obs}=1$ yr.

Observing Fig. 2, we notice that LISA possibly does not have enough
sensitivity to observe the fast-spinning magnetic WDs analyzed in this study.
AE Aqr, for example, which is a standout font for this detector, would need to
have $\delta m\approx 10^{-3}\,\rm{M}_{\odot}$, an angle of inclination $\sim
90^{\circ}$ and mass values close to $M_{\rm min}$ to be detected by this
instrument. A difficult combination of parameters since $\delta m\approx
10^{-3}\,\rm{M}_{\odot}$ is already big enough for a mountain of matter
accreted in a WD. TianQin, on the other hand, would observe all fonts if they
had $\delta m\approx 10^{-3}\,\rm{M}_{\odot}$, $\alpha\sim 90^{\circ}$ and
were not so massive; or it would only detect AE Aqr and RX J06480-4418 systems
with $\delta m\approx 10^{-4}\,\rm{M}_{\odot}$, $\alpha\sim 90^{\circ}$ and
with mass close to $M_{\rm min}$.

For BBO and DECIGO, the results are more optimistic. Assuming $\alpha\sim
90^{\circ}$, for example, these instruments would observe WDs over the entire
mass range as long as $\delta m\geq 10^{-4}\,\rm{M}_{\odot}$. For a $\delta
m\approx 10^{-5}\,\rm{M}_{\odot}$, only the RX J06480-4418 and 4U 0142+61
systems are detectable for the entire mass range. For a $\delta m\approx
10^{-6}\,\rm{M}_{\odot}$, systems must have small mass values to be observed,
except for RX J06480-4418 and 4U 0142+61 which can have larger mass values.
When we assume $\alpha\sim 10^{\circ}$, the WDs will hardly be detected by BBO
and DECIGO if they have $\delta m\leq 10^{-5}\,\rm{M}_{\odot}$, except RX
J06480-441 and 4U 0142+61, which are measurable for some mass values $M$
considering $\delta m\approx 10^{-5}\,\rm{M}_{\odot}$.

In general, we noticed that regarding the emission of GWs from matter
accretion mechanism, the six fast-spinning and magnetic WDs are unlikely to be
observed by the LISA and TianQin detectors and, thus, are not candidate
sources for these instruments. However, these systems are good candidates for
BBO and DECIGO and may be detected if they have a $\delta m\geq
10^{-5}~{}\rm{M}_{\odot}$. For an amount of $\delta m$ smaller than this
value, only the 4U 0142+61 and RX J06480-441 systems stand out, since for
$\delta m=10^{-6}~{}\rm{M}_{\odot}$, these systems are still detected if we
assume an inclination angle $\alpha$ close to the maximum value.

The results obtained here for the AE Aqr and RX J06480-4418 systems are in
agreement with the results presented in Sousa et al. (2020). However, in this
study, it was possible to obtain an amplitude interval for a value of $\delta
m$, in which we can observe the dependence of this parameter with the
mass/radius of the WD, with the inclination of the magnetic axis and with the
strength of the magnetic field.

##  5 Conclusions

In this article, we have investigated the emission of GWs from a class of
stellar remnants that has attracted great attention from the astrophysics
community in recent years: the fast, massive, and magnetic WDs. These stars,
which can be found both isolated and in binary systems, have a high magnetic
field ranging from $10^{6}$ G to $10^{9}$ G and a short rotation period of
around seconds to a few minutes. Some of these WDs are also classified as WD
pulsars because of their pulsed electromagnetic emission, similar to that
found in NS pulsars. Therefore, we analyze the continuous gravitational
radiation from these sources using two mechanisms: matter accretion and
magnetic deformation. In both mechanisms, the GW emission is generated by the
asymmetry around the axis of rotation of the star due to the accumulated mass
at the magnetic poles and the intense magnetic field, respectively.

Our efforts in this scope started with Sousa et al. (2020), where we
investigated the continuous gravitational radiation from three fast-spinning
magnetized WDs; however, here we extend our analysis by applying it to a
larger number of sources, establishing constraints on the WD mass and radius
from the construction of stability regions, and modifying some system
parameters such as the inclination of the magnetic field in relation to the
axis of rotation. For this, we investigated the GW emission in six fast and
magnetic WDs: five in binary systems (LAMOST J0240+1952, CTCV J2056-3014, AE
Aqr, V1460 Her and RX J06480-4418) and one AXP (4U 0142+61) described as a WD
pulsar. We applied the matter accretion mechanism along with the magnetic
deformation mechanism to evaluate the influence that the asymmetry caused by
one process has on the deformability caused by the other process. We further
calculated the GW amplitude for a mass range established by the inferred
minimum and maximum mass self-consistently with rotation and use a realistic
equation of state to explore the star’s stability. The angle of inclination of
the magnetic field in relation to the axis of rotation and the amount of mass
accumulated in the magnetic poles are also parameters considered in the
calculation of the amplitude.

We have shown that the effect of magnetic field deformation is negligible when
we take into account the GW amplitude generated by the accretion mechanism for
all sources. However, it is important to mention that if the WDs have stronger
magnetic fields (AXP 4U 0142+61 described as a pulsar WD, for example, can
have $B$ of up to $\sim 10^{10}$ G; see e.g., Malheiro et al. (2012); Coelho &
Malheiro (2014)), the effect produced by the field can become relevant by
creating regions where the magnetic deformation mechanism is dominant over the
accretion process. These regions are defined by the mass accumulation value
$\delta m$ and the WD mass.

We have also inferred the mass and radius limits from the stability conditions
of the WDs, where the minimum mass is obtained assuming that the observed
rotation period corresponds to the Kepler period, that is, the WD rotates
close to the mass-shedding limit. The maximum mass, in turn, is calculated
from the occurrence of pycnonuclear reactions. Thus, we found that the
analyzed WDs are possible GW sources for BBO and DECIGO given a certain
combination of parameters, such as mass, inclination between the magnetic and
rotational axes, and the accumulation of mass at the magnetic poles. However,
this class of stars is unlikely to be detected by the LISA and TianQin space
detectors due to a difficult combination of parameters such as a large
accumulated mass at their magnetic poles, a large angle between the magnetic
and rotational axes and a small WD mass value close to $M_{\rm min}$.

In addition, it is worth stressing the advantages of the detailed WD modeling
that takes rotation and a realistic EoS into account, as it made it possible
to obtain a range of mass/radius values that are possible for the analyzed
sources. With the application of this model, it was possible to estimate
constraints on the GW amplitude to verify the detectability of these WDs for
different values of mass/radius, since these parameters are not yet well
known.

## Acknowledgements

We thank the anonymous referees for the relevant suggestions which helped
improve this work. M.F.S. and J.C.N.A. thank the Brazilian Ministry of
Science, Technology and Innovation (MCTI) and the Brazilian Space Agency (AEB)
who supported the present work under the PO 20VB.0009. M.F.S. thanks CAPES and
Fundação Araucária for the financial support. J.G.C. is grateful for the
support of FAPES (1020/2022, 1081/2022, 976/2022, 332/2023), CNPq
(311758/2021-5), and FAPESP (2021/01089-1). M.F.S. and J.G.C. gratefully
acknowledge the financial support of the “Fenômenos Extremos do Universo" of
the Fundação Araucária. E. Otoniel thanks for the support received from the
productivity scholarship program, encouragement of internalization and
technological innovation - BPI, notice (BP5-0197-00153.01.00/22). J.C.N.A.
thanks CNPq (307803/2022-8) for partial financial support.

## Data Availability

The data underlying this article will be shared on reasonable request to the
corresponding author.

## References

  * Abbott et al. (2016) Abbott B. P., et al., 2016, Phys. Rev. Lett., 116, 061102
  * Abbott et al. (2019) Abbott B. P., et al., 2019, Physical Review X, 9, 031040
  * Abbott et al. (2021) Abbott R., et al., 2021, Physical Review X, 11, 021053
  * Althaus et al. (2005) Althaus L. G., García-Berro E., Isern J., Córsico A. H., 2005, A&A, 441, 689
  * Althaus et al. (2007) Althaus L. G., García-Berro E., Isern J., Córsico A. H., Rohrmann R. D., 2007, A&A, 465, 249
  * Amaro-Seoane et al. (2017) Amaro-Seoane P., et al., 2017, preprint (arXiv:1702.00786)
  * Amorim et al. (2023) Amorim L. L., Kepler S. O., Külebi B., Jordan S., Romero A. D., 2023, ApJ, 944, 56
  * Ashley et al. (2020) Ashley R. P., et al., 2020, MNRAS, 499, 149
  * Barstow et al. (1995) Barstow M. A., Jordan S., O’Donoghue D., Burleigh M. R., Napiwotzki R., Harrop-Allin M. K., 1995, MNRAS, 277, 971
  * Becerra et al. (2018) Becerra L., Rueda J. A., Lorén-Aguilar P., García-Berro E., 2018, ApJ, 857, 134
  * Bonazzola & Gourgoulhon (1996) Bonazzola S., Gourgoulhon E., 1996, A&A, 312, 675
  * Borges et al. (2020) Borges S. V., Rodrigues C. V., Coelho J. G., Malheiro M., Castro M., 2020, ApJ, 895, 26
  * Boshkayev et al. (2013a) Boshkayev K., Rueda J., Ruffini R., 2013a, International Journal of Modern Physics: Conference Series, 23, 193
  * Boshkayev et al. (2013b) Boshkayev K., Rueda J. A., Ruffini R., Siutsou I., 2013b, ApJ, 762, 117
  * Buckley et al. (2017) Buckley D. A. H., Meintjes P. J., Potter S. B., Marsh T. R., Gänsicke B. T., 2017, Nature Astronomy, 1, 0029
  * Caiazzo et al. (2021) Caiazzo I., et al., 2021, Nature, 595, 39
  * Camisassa et al. (2019) Camisassa M. E., et al., 2019, A&A, 625, A87
  * Chabrier & Potekhin (1998) Chabrier G., Potekhin A. Y., 1998, Phys. Rev. E, 58, 4941
  * Chamel et al. (2013) Chamel N., Fantina A. F., Davis P. J., 2013, Phys. Rev. D, 88, 081301
  * Chamel et al. (2014) Chamel N., Molter E., Fantina A., Arteaga D. P., 2014, Phys. Rev. D, 90, 043002
  * Chatterjee et al. (2017) Chatterjee D., Fantina A. F., Chamel N., Novak J., Oertel M., 2017, MNRAS, 469, 95
  * Coelho & Malheiro (2014) Coelho J. G., Malheiro M., 2014, PASJ, 66, 14
  * Coelho et al. (2014) Coelho J. G., Marinho R. M., Malheiro M., Negreiros R., Cáceres D. L., Rueda J. A., Ruffini R., 2014, ApJ, 794, 86
  * Coelho et al. (2017) Coelho J. G., Cáceres D. L., de Lima R. C. R., Malheiro M., Rueda J. A., Ruffini R., 2017, A&A, 599, A87
  * Córsico et al. (2019) Córsico A. H., Althaus L. G., Miller Bertolami M. M., Kepler S. O., 2019, A&ARv, 27, 7
  * Curd et al. (2017) Curd B., Gianninas A., Bell K. J., Kilic M., Romero A. D., Allende Prieto C., Winget D. E., Winget K. I., 2017, MNRAS, 468, 239
  * Dan et al. (2014) Dan M., Rosswog S., Brüggen M., Podsiadlowski P., 2014, MNRAS, 438, 14
  * Eisenstein et al. (2006) Eisenstein D. J., et al., 2006, The Astrophysical Journal Supplement Series, 167, 40
  * Ferrario et al. (2015) Ferrario L., de Martino D., Gänsicke B. T., 2015, Space Sci. Rev., 191, 111
  * Ferrario et al. (2020) Ferrario L., Wickramasinghe D., Kawka A., 2020, Advances in Space Research, 66, 1025
  * Ferraro (1954) Ferraro V. C. A., 1954, ApJ, 119, 407
  * Franzon & Schramm (2017) Franzon B., Schramm S., 2017, MNRAS, 467, 4484
  * Friedman et al. (1986) Friedman J. L., Ipser J. R., Parker L., 1986, ApJ, 304, 115
  * Gaia Collaboration et al. (2022) Gaia Collaboration Vallenari A., Brown A., Prusti T., de Bruijne J., Arenou F., 2022, Astronomy & Astrophysics
  * Gao et al. (2017) Gao H., Cao Z., Zhang B., 2017, ApJ, 844, 112
  * García-Berro et al. (2012) García-Berro E., et al., 2012, ApJ, 749, 25
  * Gasques et al. (2005) Gasques L. R., Afanasjev A. V., Aguilera E. F., Beard M., Chamon L. C., Ring P., Wiescher M., Yakovlev D. G., 2005, Phys. Rev. C, 72, 025806
  * Gentile Fusillo et al. (2018) Gentile Fusillo N. P., et al., 2018, Monthly Notices of the Royal Astronomical Society, 482, 4570
  * Glendenning & Weber (1994) Glendenning N. K., Weber F., 1994, Phys. Rev. D, 50, 3836
  * Golf et al. (2009) Golf B., Hellmers J., Weber F., 2009, Phys. Rev. C, 80, 015804
  * Harry et al. (2006) Harry G. M., Fritschel P., Shaddock D. A., Folkner W., Phinney E. S., 2006, Classical and Quantum Gravity, 23, 4887
  * Hartle (1967) Hartle J. B., 1967, ApJ, 150, 1005
  * Hartle & Thorne (1968) Hartle J. B., Thorne K. S., 1968, ApJ, 153, 807
  * Hermes et al. (2013) Hermes J. J., Kepler S. O., Castanheira B. G., Gianninas A., Winget D. E., Montgomery M. H., Brown W. R., Harrold S. T., 2013, ApJ, 771, L2
  * Israel et al. (1997) Israel G. L., Stella L., Angelini L., White N. E., Kallman T. R., Giommi P., Treves A., 1997, ApJ, 474, L53
  * Jiménez-Esteban et al. (2018) Jiménez-Esteban F. M., Torres S., Rebassa-Mansergas A., Skorobogatov G., Solano E., Cantero C., Rodrigo C., 2018, MNRAS, 480, 4505
  * José et al. (2020) José J., Shore S. N., Casanova J., 2020, A&A, 634, A5
  * Kalita & Mukhopadhyay (2019) Kalita S., Mukhopadhyay B., 2019, MNRAS, 490, 2692
  * Kalita et al. (2020) Kalita S., Mukhopadhyay B., Mondal T., Bulik T., 2020, ApJ, 896, 69
  * Kawamura et al. (2006) Kawamura S., et al., 2006, Classical and Quantum Gravity, 23, S125
  * Kepler et al. (2013) Kepler S. O., et al., 2013, Monthly Notices of the Royal Astronomical Society, 429, 2934
  * Kepler et al. (2015) Kepler S. O., et al., 2015, MNRAS, 446, 4078
  * Kilic et al. (2021) Kilic M., Kosakowski A., Moss A. G., Bergeron P., Conly A. A., 2021, ApJ, 923, L6
  * Kitaguchi et al. (2014) Kitaguchi T., et al., 2014, ApJ, 782, 3
  * Kjurkchieva et al. (2017) Kjurkchieva D. P., Popov V. A., Vasileva D. L., Petrov N. I., 2017, New Astronomy, 52, 8
  * Konno et al. (2000) Konno K., Obata T., Kojima Y., 2000, A&A, 356, 234
  * Kuelebi et al. (2010) Kuelebi B., Jordan S., Euchner F., Gaensicke B. T., Hirsch H., 2010, VizieR Online Data Catalog, pp J/A+A/506/1341
  * Külebi et al. (2009) Külebi B., Jordan S., Euchner F., Gänsicke B. T., Hirsch H., 2009, A&A, 506, 1341
  * Külebi et al. (2010) Külebi B., Jordan S., Nelan E., Bastian U., Altmann M., 2010, A&A, 524, A36
  * Lopes de Oliveira et al. (2020) Lopes de Oliveira R., Bruch A., Rodrigues C. V., Oliveira A. S., Mukai K., 2020, ApJ, 898, L40
  * Lorén-Aguilar et al. (2009) Lorén-Aguilar P., Isern J., García-Berro E., 2009, A&A, 500, 1193
  * Lu et al. (2019) Lu X.-Y., Tan Y.-J., Shao C.-G., 2019, Phys. Rev. D, 100, 044042
  * Luo et al. (2016) Luo J., et al., 2016, Classical and Quantum Gravity, 33, 035010
  * Maggiore (2008) Maggiore M., 2008, Gravitational waves: volume 1: theory and experiments. Gravitational Waves, OUP Oxford
  * Malheiro et al. (2012) Malheiro M., Rueda J. A., Ruffini R., 2012, PASJ, 64, 56
  * Marsh et al. (2016) Marsh T. R., et al., 2016, Nature, 537, 374
  * Mei et al. (2021) Mei J., et al., 2021, Progress of Theoretical and Experimental Physics, 2021, 05A107
  * Mereghetti et al. (2009) Mereghetti S., Tiengo A., Esposito P., La Palombara N., Israel G. L., Stella L., 2009, Science, 325, 1222
  * Mereghetti et al. (2011) Mereghetti S., La Palombara N., Tiengo A., Pizzolato F., Esposito P., Woudt P. A., Israel G. L., Stella L., 2011, ApJ, 737, 51
  * Mereghetti et al. (2016) Mereghetti S., Pintore F., Esposito P., La Palombara N., Tiengo A., Israel G. L., Stella L., 2016, MNRAS, 458, 3523
  * Mereghetti et al. (2021) Mereghetti S., et al., 2021, MNRAS, 504, 920
  * Mitrofanov (1980) Mitrofanov I. G., 1980, in Close Binary Stars: Observations and Interpretation. pp 431–436
  * Mukhopadhyay & Bhattacharya (2022) Mukhopadhyay B., Bhattacharya M., 2022, Particles, 5, 493
  * Mukhopadhyay & Rao (2016) Mukhopadhyay B., Rao A. R., 2016, J. Cosmology Astropart. Phys., 2016, 007
  * Mukhopadhyay et al. (2017) Mukhopadhyay B., Rao A. R., Bhatia T. S., 2017, MNRAS, 472, 3564
  * Nunes et al. (2021) Nunes S. P., Arbañil J. D. V., Malheiro M., 2021, ApJ, 921, 138
  * Olausen & Kaspi (2014) Olausen S. A., Kaspi V. M., 2014, ApJS, 212, 6
  * Otoniel et al. (2019) Otoniel E., Franzon B., Carvalho G. A., Malheiro M., Schramm S., Weber F., 2019, The Astrophysical Journal, 879, 46
  * Otoniel et al. (2021) Otoniel E., Coelho J. G., Nunes S. P., Malheiro M., Weber F., 2021, A&A, 656, A77
  * Otoniel et al. (2022) Otoniel E., Coelho J. G., Malheiro M., Weber F., 2022, in , Astrophysics in the XXI Century with Compact Stars. Edited by C.A.Z. Vasconcellos. eISBN 978-981-12-2094-4. Singapore: World Scientific. World Scientific, pp 121–152, doi:10.1142/9789811220944_0004
  * Patterson (1979) Patterson J., 1979, ApJ, 234, 978
  * Pelisoli et al. (2022) Pelisoli I., et al., 2022, MNRAS, 509, L31
  * Pelisoli et al. (2023) Pelisoli I., et al., 2023, Nature Astronomy,
  * Pereira et al. (2018) Pereira J. P., Coelho J. G., de Lima R. C. R., 2018, European Physical Journal C, 78, 361
  * Popov et al. (2018) Popov S. B., Mereghetti S., Blinnikov S. I., Kuranov A. G., Yungelson L. R., 2018, MNRAS, 474, 2750
  * Potekhin & Chabrier (2000) Potekhin A. Y., Chabrier G., 2000, Physical Review E, 62, 8554
  * Regimbau & de Freitas Pacheco (2006) Regimbau T., de Freitas Pacheco J. A., 2006, A&A, 447, 1
  * Robson et al. (2019) Robson T., Cornish N. J., Liu C., 2019, Classical and Quantum Gravity, 36, 105011
  * Rueda et al. (2013) Rueda J. A., Boshkayev K., Izzo L., Ruffini R., Lorén-Aguilar P., Külebi B., Aznar-Siguán G., García-Berro E., 2013, ApJ, 772, L24
  * Scaringi et al. (2022a) Scaringi S., Groot P. J., Knigge C., Lasota J. P., de Martino D., Cavecchi Y., Buckley D. A. H., Camisassa M. E., 2022a, MNRAS, 514, L11
  * Scaringi et al. (2022b) Scaringi S., et al., 2022b, Nature, 604, 447
  * Schmidt et al. (1986) Schmidt G. D., West S. C., Liebert J., Green R. F., Stockman H. S., 1986, ApJ, 309, 218
  * Shapiro & Teukolsky (1983) Shapiro S. L., Teukolsky S. A., 1983, Black holes, white dwarfs and neutron stars: the physics of compact objects. John Wiley & Sons
  * Shara (1982) Shara M. M., 1982, ApJ, 261, 649
  * Sousa et al. (2020) Sousa M. F., Coelho J. G., de Araujo J. C. N., 2020, MNRAS, 492, 5949
  * Sousa et al. (2022) Sousa M. F., Coelho J. G., de Araujo J. C. N., Kepler S. O., Rueda J. A., 2022, ApJ, 941, 28
  * Sousa et al. (2023) Sousa M. F., Coelho J. G., de Araujo J. C. N., Guidorzi C., Rueda J. A., 2023, ApJ, 958, 134
  * Terada et al. (2008) Terada Y., et al., 2008, PASJ, 60, 387
  * The LIGO Scientific Collaboration et al. (2021) The LIGO Scientific Collaboration the Virgo Collaboration the KAGRA Collaboration Abbott R., Abbott T. D., et al., 2021, arXiv e-prints, p. arXiv:2111.03606
  * Usov (1988) Usov V. V., 1988, Pisma v Astronomicheskii Zhurnal, 14, 606
  * Warner (2003) Warner B., 2003, Cataclysmic Variable Stars. Cambridge Astrophysics, Cambridge University Press, doi:10.1017/CBO9780511586491
  * Weber (1999) Weber F., 1999, Pulsars as Astrophysical Laboratories for Nuclear and Particle Physics, 1st edition edn. CRC Press, Bristol ; Philadelphia
  * Weber & Glendenning (1992) Weber F., Glendenning N. K., 1992, ApJ, 390, 541
  * Welsh et al. (1998) Welsh W. F., Horne K., Gomer R., 1998, MNRAS, 298, 285
  * Yagi & Seto (2017) Yagi K., Seto N., 2017, Phys. Rev. D, 95, 109901
  * Yakovlev et al. (2006) Yakovlev D. G., Gasques L. R., Afanasjev A. V., Beard M., Wiescher M., 2006, Phys. Rev. C, 74, 035803
  * de Araujo et al. (2016) de Araujo J. C. N., Coelho J. G., Costa C. A., 2016, ApJ, 831, 35
  * de Araujo et al. (2017) de Araujo J. C. N., Coelho J. G., Costa C. A., 2017, European Physical Journal C, 77, 350