Heavy Magnetic Neutron Stars

Neutron stars (NSs) are considered to be the densest objects that are formed as a result of the gravitational collapse of supernovae. Their interior covers a wide range of densities (reaching $\sim$ 10 times the normal nuclear density) and, hence, invites the possibility of exotic degrees of freedom like quarks, hyperons, and kaons. They also provide a distinctive environment to investigate many unknowns in physics and astrophysics. The properties of the interior dense matter affect NS macroscopic properties, such as mass, radius, and the amount by which it can be deformed.

The most precise measurement of a NS mass is 1.4$M_{\sun}$, that of the Hulse-Taylor pulsar (Hulse & Taylor, 1975). Recently, higher masses of binary neutron-star (BNS) systems like millisecond pulsar (MSP) PSR J1614-2230 (M= 1.97 $\pm$ 0.04 $M_{\sun}$) (Demorest et al., 2010), PSR J0348+0432 (M= 2.01 $\pm$ 0.04 $M_{\sun}$) (Antoniadis & Freire et al., 2013), and PSR J0740+6620 (M= 2.04${}^{+0.10}_{-0.09}$ $M_{\sun}$) (Cromartie & Fonseca et al., 2019) have been measured with high precision. The gravitational wave detection by LIGO and Virgo Collaborations (LVC) of a BNS merger GW170817 event (Abbott et al., 2017, 2018) further provided a precise measurement of the NS mass, 1.16-1.60 $M_{\sun}$ for low spin priors and a maximum mass approaching 1.9 $M_{\sun}$ for high spin priors (Abbott et al., 2019), which allowed one to study dense matter properties at extreme conditions due to the measurement of a new property of NSs, the tidal deformability. The latter provided a new constraint on the NS EoS and its variation with the radius approximated by the relation $\Lambda\propto R^{5}$ (Hinderer et al., 2010; Kumar et al., 2017) resulting in a strong constraint on the nuclear EoS at intermediate density.

Very recently, a new gravitational wave event reported by LVC as GW190814 (Abbott et al., 2020a) was observed with a 22.2-24.3 $M_{\sun}$ black hole and 2.50-2.67 $M_{\sun}$ secondary component. The secondary component of GW190814 attracted a lot of attention as it contained no measurable tidal deformability signatures. As the mass of the secondary component of GW190814 lies in the lower region of the mass-gap (2.5$M_{\sun}$ $<$ M $<$ 5$M_{\sun}$), it raised the question whether it is a light black hole or a very massive NS. To understand this object, many interesting theories have been proposed recently regarding its nature as the most massive NS, lightest black hole, or fastest pulsar (Godzieba et al., 2021; Most et al., 2020; Zhang & Li, 2020; Tsokaros et al., 2020; Fattoyev et al., 2020; Lim et al., 2020a; Tews et al., 2021; Rather et al., 2021c; Sedrakian et al., 2020a; Beniamini et al., 2020; Biswas et al., 2020; Roupas et al., 2020; Zhou et al., 2021; Wu et al., 2021; Khadkikar et al., 2021; Bombaci et al., 2020; Goncalves & Lazzari, 2020; Christian & Schaffner-Bielich, 2021a; Demircik et al., 2021a; Li et al., 2020; Riahi & Kalantari, 2021; Dhiman et al., 2007; Shahrbaf et al., 2020; Thapa & Sinha, 2020; Huang et al., 2020a; Lim et al., 2020b; Tan et al., 2020; Gupta et al., 2020; Dexheimer et al., 2021).

Different values of NS radii have been obtained from the the analysis of X-ray spectra emitted by NSs (Özel et al., 2010; Lattimer & Steiner, 2014; Lattimer & Prakash, 2016; Özel & Freire, 2016). Lattimer & Prakash determined the range for radius at the NS canonical mass 10.7 $\leq R_{1.4}$ $\leq$ 13.1 km from the properties of symmetric matter and nuclear experiments (Lattimer & Prakash, 2016). The recent measurement of the radius for the NS canonical mass inferred from PSR J0030+0451 by the Neutron Star Interior Composition Explorer (NICER) (Gendreau et al., 2016) is R=13.02${}_{-1.06}^{+1.24}$ km for M=1.44${}_{-0.14}^{+0.15}$ $M_{\sun}$ (Miller et al., 2019) or R=12.71${}_{-1.19}^{+1.14}$ km for M=1.34${}_{-0.16}^{+0.15}$ $M_{\sun}$ (Riley et al., 2019), obtained using a Bayesian inference approach. In Ref. (Fattoyev et al., 2018), an upper limit on the NS radius for 1.4 $M_{\sun}$, $R_{1.4}$ $\leq$ 13.76 km was deduced. The measurements from NICER provide an estimate of the canonical mass NS radius within a precision of 1 km. For a canonical mass NS with small radius, a softening of the pressure of neutron matter at intermediate density is required, which results in a small value of the nuclear symmetry energy around saturation density $\rho_{0}$ (Lattimer & Prakash, 2007; Özel & Freire, 2016). However, the requirement of the 2$M_{\sun}$ NS doesn’t allow the pressure to be reduced too much. The small radius and large mass constraints are satisfied simultaneously by very few models that provide an additional accurate description of finite nuclei properties (Chen & Piekarewicz, 2015; Jiang et al., 2015; Guichon et al., 2018).

At densities of a few times normal nuclear density, the composition of matter inside NS is not known. With increasing density, the appearance of exotic degrees of freedom inside NSs is possible and has been studied over the past decade. In particular, the appearance of hyperonic matter under NS inner core conditions is energetically favored (Ambartsumyan & Saakyan, 1960; Glendenning, 1985). The onset of hyperons reduces the pressure, leading to a softer EoS as they open new channels filling their Fermi sea, which decreases the maximum mass of NSs by about 0.5 $M_{\sun}$ (Glendenning, 1982; Chatterjee & Vidaña, 2016). Several works has been performed recently considering hyperonic matter in NSs (Li & Sedrakian, 2019a; Li et al., 2018b; Li & Sedrakian, 2019b; Sedrakian et al., 2020a; Fortin et al., 2017; Li et al., 2018a; Baldo et al., 2000; Tolos et al., 2017; Colucci & Sedrakian, 2014; Gusakov et al., 2014; Bhowmick et al., 2014; Oertel et al., 2015; Chatterjee & Vidaña, 2016; Lopes & Menezes, 2018; Isaka et al., 2017; Spinella & Weber, 2019; Guichon et al., 2018; Vidaña, 2018; Fortin et al., 2020; Tolos & Fabbietti, 2020; Gomes et al., 2015; Dexheimer & Schramm, 2008; Fortin et al., 2021).

Heavy NSs are expected to contain exotic matter in their interior, even if they are rotating fast (Avancini et al., 2018; Dexheimer et al., 2021). Very massive and/or fast rotating stars could be the result of accretion, or even a previous stellar merger, both of which have been shown to enhance stellar magnetic fields (Pons & Viganò, 2019; Bransgrove et al., 2017; Yakhshiev et al., 2019; Aguirre, 2020; Zhong & Dai, 2020; Most et al., 2019; Ciolfi et al., 2019; Giacomazzo et al., 2015; Xue et al., 2019; Raynaud et al., 2020; Sur & Haskell, 2021). With exceptionally high density, a magnetic field reaching $\approx$ 10⁹ to 10¹⁸ G (Bocquet et al., 1995; Cardall et al., 2001; Chatterjee et al., 2015) is attainable in massive NS centers. Among various classes of compact stars available, Anomalous X-ray pulsars (AXPs) and the Soft gamma repeaters (SGRs), usually called magnetars (Mereghetti et al., 2015; Kaspi & Beloborodov, 2017), are considered to have surface magnetic field within the order 10¹⁴-10¹⁵ G (Harding & Lai, 2006; Turolla et al., 2015). Fast radio bursts (FRBs) have also shown evidence of magnetars (Margalit et al., 2020; Beniamini et al., 2020; Beloborodov, 2020; Levin et al., 2020; Zanazzi & Lai, 2020).

In the interior of the magnetars, the magnetic field cannot be measured directly, and hence only estimated using theoretical models. The absolute largest value of magnetic field a star can possess in the interior (usually taken as an upper bound for the magnetic field) can be estimated using the relativistic version of the virial theorem Bonazzola et al. (1993), for which the negative total energy implies that the magnitude of the gravitational potential energy must be greater than the magnetic field energy, which results in magnetic field $\approx$ 10¹⁸ G. As discussed in Refs. Bocquet et al. (1995); Cardall et al. (2001), assuming a poloidal configuration, the maximum allowed central magnetic field that still fulfills Einstein’s and Maxwell’s equations is around a few times 10¹⁸ G, the exact value being dependent on the equation of state. Similar limits exist in purely toroidal configurations Pili et al. (2017). As discussed in Refs. Dexheimer et al. (2017b); Pili et al. (2017), the magnitude of magnetic field does not increase much beyond one order of magnitude within a star, regardless of the equation of state or magnetic field configuration. This implies that, regardless of its feasibility, the star would have to present $\geq 10\times^{16}$ G on the surface in order to reach a magnetic field $\geq 10\times^{17}$ G or higher in the center. Most of the magnetar observations inferred magnetic fields of $\sim 10\times^{15}$ G or below, the only exception being data from the source 4U 0142+61 for slow phase modulations in hard x-ray pulsations (interpreted as free precession) that suggests magnetic fields of the order of $10\times^{16}$ G Makishima et al. (2014). Ref. Dall’Osso et al. (2018) provides $10\times^{16}$ G as a low bound for the same source. To be strongly structurally changed by magnetic fields, either more sources have not yet been discovered because their magnetic fields do not fit in the magnetar model (used to infer magnetic fields through period-period derivative diagrams), their magnetic fields decay quickly due to misalignment of rotation and magnetic axes Lander & Jones (2018); Kuzur & Mallick (2019), or these stars are not stable, being formed for example in previous mergers Price & Rosswog (2006); Giacomazzo et al. (2015); Most et al. (2019). It should be mentioned that magnetic fields $\geq 10\times^{15}$ G could be produced by three-dimensional simulations of supernova explosions with main requirement to produce a strong field dynamo being sufficient angular momentum in the progenitor star Raynaud et al. (2020). Previous works have studied the effect of the magnetic field on the NS Equation of state (EoS) and on the stellar properties like mass and radius (Chakrabarty et al., 1997; Broderick et al., 2000; Rabhi et al., 2008b; Mallick & Schramm, 2014; Bandyopadhyay et al., 1997; Aguirre, 2019; Gomes et al., 2017; Dexheimer et al., 2012; Pili et al., 2017; Felipe et al., 2008; Paulucci et al., 2013; Casali et al., 2014; Rabhi et al., 2008a). The presence of strong magnetic fields deviates neutron-star structures from spherical symmetry very strongly and, hence, the spherically symmetric Tolman–Oppenheimer–Volkoff (TOV) equations can no longer be applied for studying their macroscopic structure (Chatterjee et al., 2015; Gomes et al., 2019).

In the present work, we employ recently proposed density-dependent Relativistic Mean-Field (DD-RMF) parameter sets to study the properties of NSs. By reproducing different hyperon-hyperon optical potentials, the values of the hyperon couplings are obtained, using several different coupling schemes, which are then used to model hyperons in our calculations. The presence of hyperons lowers NS maximum masses (to a value still larger than 2 $M_{\sun}$) by softening the EoS, although the radius at the NS canonical mass remains insensitive to them. We further analyze the effect of magnetic fields on nucleonic and hyperonic matter by employing a realistic chemical potential-dependent magnetic field (Dexheimer et al., 2017a). The macroscopic stellar matter properties for the magnetic EoSs are obtained using the publicly available Language Objet pour la RElativité NumériquE (LORENE) library (LORENE, -; Bocquet et al., 1995; Bonazzola et al., 1993), which solves the coupled Einstein-Maxwell field equations in order to determine stable magnetic star configurations. In this case, NSs become more massive, especially the ones containing hyperons in their interior.

The main motivation behind our work is to demonstrate that a possible measurement of a neutron star mass $\sim 2.5$ M_⊙ does not necessarily rule out exotic degrees of freedom in its interior, as several works in the literature claim. In this regard, despite the fact that the GW190814 data has been available for about a year Abbott et al. (2020b), our conclusions differ significantly from all other works on the subject that have been published. While some argue that if the secondary object in GW190814 is a neutron star, it either has no exotic degrees of freedom Li et al. (2020); Sedrakian et al. (2020b) or, if it does, it rotates extremely fast Dexheimer et al. (2021); Rather et al. (2021a) or uses an unusual approach to model deconfined quark matter Christian & Schaffner-Bielich (2021b); Roupas et al. (2021); Gonçalves & Lazzari (2020); Bombaci et al. (2021); Demircik et al. (2021b); Cao et al. (2020), We investigate the possibility of a star with a strong magnetic field inside. We investigate how different particle populations and nuclear interactions affect microscopic and macroscopic stellar properties, and we illustrate how, if the secondary object in GW190814 is a massive neutron star, we can learn about the dense matter inside them.

The paper is organized as follows: in section 2, the density-dependent RMF model is presented and the inclusion of the magnetic field for the beta-equilibrium EoS is discussed. The various DD-RMF parameter sets, nuclear matter properties, and hyperon couplings are discussed in section 3. The nucleonic and hyperonic EoSs are described in subsection 4.1, along with spherical solutions for star matter properties. Subsection 4.2 deals with the neutron and hyperon EoS with effects of magnetic fields, together with the discussion of their stellar properties. Subsection 4.3 presents additional results produced assuming different hyperon couplings. We summarize our results and outline an outlook for the future in sec. 5. Finally, the equations of motion and the expressions for the energy and baryon density in the presence of magnetic fields are shown in Appendix A.

The Lagrangian density is usually the starting point in RMF theory: the baryons interact through meson exchanges, including the scalar-isoscalar sigma $\sigma$, the vector-isoscalar $\omega$, and the vector-isovector $\rho$. The Lagrangian density contains contributions from the baryonic octet, leptons and the mesons, followed by the interactions between them. Within the DD-RMF model, the coupling constants are density-dependent (Brockmann & Toki, 1992), as shown in the following.

The DD-RMF Lagrangian density is written as (Huang et al., 2020b)

where $B$ sums over the baryon octet ($n,p,\Lambda,\Sigma^{+},\Sigma^{0},\Sigma^{-},\Xi^{0},\Xi^{-}$) and $l$ over $e^{-}$ and $\mu^{-}$. $\psi_{B}$ and $\psi_{l}$ represent the baryonic and leptonic Dirac fields, respectively. The mesonic tensor fields and covariant derivative are defined as

The Lagrangian density for the pure electromagnetic part is written as

where, $F_{\mu\nu}$ is the electromagnetic field tensor, $F_{\mu\nu}$=$\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$. Hence, the total Lagrangian density in the presence of a magnetic field is

The density dependent coupling constants in the DD-RMF model are parametrized by the relation

for $i=\sigma,\omega$. The function $f_{i}(x)$, where $x=\rho_{B}/\rho_{0}$ with $\rho_{0}$ as the nuclear matter saturation density, is defined as

For the $\rho$ meson, the density-dependent coupling constant is given by an exponential relation as

This parametrization has eight parameters, which are reduced to three using the constraint conditions from refs (Typel & Wolter, 1999; Rather et al., 2021c) to reproduce symmetric and asymmetric nuclear matter properties.

For the present case that includes all baryons from the octet, the NS chemical equilibrium condition between different particles are

The charge neutrality condition follows as

The expressions for the energy density and pressure in presence of a magnetic field can be obtained by solving the energy-momentum tensor relation

where (Huang et al., 2010; Khalilov, 2002)

Here, $\mathcal{M}$ is the magnetization per unit volume and $B^{\mu}B_{\mu}=-B^{2}$. For the nuclear matter in presence of a magnetic field, the single particle energy of all charged baryons and leptons is quantized in the direction perpendicular to the magnetic field.

For a uniform magnetic field locally pointing in the $z$-direction, $B=B\hat{z}$, the total energy of a charged particle becomes

The quantity $\nu=\Big{(}n+\frac{1}{2}-\frac{1}{2}\frac{q}{|q|}\sigma_{z}\Big{)}=0,1,2,...$ indicates the Landau levels of fermions with electric charge $q$. $n$ is the orbital angular momentum quantum number and $\sigma_{z}$ is the Pauli matrix. The Fermi momentum of all baryons charged $k_{F,\nu}^{cb}$ and leptons $k_{F,\nu}^{l}$ with Fermi energies $E_{F}^{cb}$ and $E_{F}^{l}$, respectively, are given as (Broderick et al., 2000)

The highest value of $\nu$ is obtained with sum over Landau levels under the condition that the Fermi momentum of each particle is positive

for charged baryons $cb$ and leptons, respectively.

The expressions for the matter energy density and baryon density obtained in presence of magnetic field are shown in Appendix A. From Eq. 2, the total energy density is

The total pressure in the perpendicular and the parallel directions to the local magnetic field are

where the magnetization is calculated as

For the present study, we employ two recently proposed density dependent DD-MEX (Taninah et al., 2020), and DD-LZ1 (Wei et al., 2020) parameter sets. These include the necessary tensor couplings of the vector mesons to nucleons and were obtained by fitting the ground state properties of finite nuclei, which considered parametric corrections and shell evaluations of mesons. Additionally, widely used parameter sets DD-ME1 (Nikšić et al., 2002) and DD-ME2 (Lalazissis et al., 2005) are also used.

Table 1 displays the nucleon and meson masses and the coupling constants between nucleon and mesons for the given parameter sets. The parameters $a,b,c,d$ for $\sigma$, $\omega$, and $\rho$ mesons are also shown.

Some important nuclear matter properties for the DD-LZ1, DD-ME1, DD-ME2, and DD-MEX parameter sets are also shown in table (1). The symmetry energy parameter $J$ for the listed parameter sets is compatible with the values $J=31.6\pm 2.66$ MeV (Li & Han, 2013). The symmetry energy slope parameter $L$ also satisfies standard constraints $L=59.57\pm 10.06$MeV (Zhang et al., 2020; Danielewicz & Lee, 2014). A very recent publication by the PREX collaboration reported a value for the neutron skin thickness of 208 Pb of $R_{skin}$ = (0.283 $\pm$ 0.071) fm (Adhikari et al., 2021). Such a value is significantly larger than the $R_{skin}$ = 0.20 fm predicted by DD-ME1. Given the strong correlation between $R_{skin}$ and $L$, symmetry energy slope parameter values as large as $L$ $\approx$ 100 MeV have been proposed (Reed et al., 2021). Although our parametrizations reproduce a lower symmetry energy and slope, they are still in excellent agreement with a large amount of constraints, as shown in Fig. 1 from Ref. (Li et al., 2021). The currently accepted value of incompressibility determined from the isoscalar giant monopole resonance (ISGMR) lies in the range $K_{0}=240\pm 20$MeV. The $K_{0}$ value for all the given parameter sets is within this range, except for the DD-MEX, which predicts a value little higher.

The density-dependent coupling constants of the hyperons to the vector mesons are determined from the SU(6) symmetry as (Banik & Bandyopadhyay, 2001; Schaffner & Mishustin, 1996; Banik et al., 2014; Tolos et al., 2016)

These couplings are calculated by fitting the hyperon optical potential obtained from the experimental data. See Refs. (Inoue, 2019a, b) for current lattice calculations and experimental data for these potentials.

The hyperon coupling to the $\sigma$ field is obtained, so as to reproduce the hyperon potential in symmetric nuclear matter (SNM) derived from the hypernuclear observables (Banik et al., 2014; Klevansky, 1992)

For the present study, we reproduce the following optical potentials for the hyperons (Thapa et al., 2020)

These potentials correspond to the value of the density-dependent scalar couplings $g_{\sigma\Lambda}/g_{\sigma N}=0.6105$, $g_{\sigma\Xi}/g_{\sigma N}=0.3024$, and $g_{\sigma\Sigma}/g_{\sigma N}=0.4426$.

We modeled massive nucleonic and hyperonic stars that fulfill the constraints set by the observation of the possibly most massive neutron star (NS) ever detected (in the secondary object of the gravitational wave event GW190814) using a density-dependent relativistic mean field model (DD-RMF). This simple, yet powerful formalism provides the freedom necessary to fulfill simultaneously nuclear and astrophysical constraints. The results obtained from the TOV equations show that the hyperons soften the EoS, lowering the maximum mass of NS to around 2.2$M_{\sun}$, thereby satisfying more conservative massive constraints from the astrophysical observations. The radius of the NS canonical mass is seen to be insensitive to the appearance of hyperons. Both the nucleonic and the hyperonic stars satisfy the constraints from mass-radius limits inferred from NICER observations and tidal deformability constraints from the LIGO and VIRGO collaborations.

We also studied the effects of strong magnetic fields on DD-RMF nucleonic and hyperonic EoSs. We investigated the EoS and particle populations using a realistic chemical potential-dependent magnetic field that was developed by solving Einstein-Maxwell equations. For very strong magnetic fields, the spherical symmetric solutions obtained by solving the TOV equations lead to an overestimation of the mass and the radius and, hence, cannot be used for determining stellar properties. For this reason, we used the LORENE library to determine stellar properties of magnetic NSs. For low values of the magnetic dipole moment, implying lower strengths of magnetic fields, the EoS resembles the non-magnetic one. For higher magnetic dipole moments, the EoS stiffens at higher energy density. The amount of stiffness is larger in case of hyperonic EoSs than for the case of pure nucleonic ones. As the magnetic field strength is increased, the particle fractions of leptons ($e^{-}$ and $\mu^{-}$) increases at higher densities and the appearance of charged hyperons ($\Xi^{-}$) is delayed.

The stiffening in the EoS caused by the changes in population described above increases the maximum mass of magnetic stars. For a small dipole moment of 5$\times$ 10³⁰ Am², the nucleonic and hyperonic mass-radius profiles resemble the non-magnetic case due to the lower magnetic field produced. For higher magnetic dipole moments, although the variation in the maximum mass is small, a variation of about 1 km is seen in the radius of the NS canonical mass. For hyperonic stars, the maximum mass increases by $\approx$ 0.3 $M_{\sun}$, allowing them to satisfy the GW190814 possible mass constraint for a dipole moment $\mu\geq 10^{32}$ Am² when different coupling schemes are considered. For strong magnetic fields, the different coupling schemes for hyperons predict increases in radius of $\approx 0.2-1$ km. Thus, we see that different hyperon couplings and different hyperon potentials populate stellar matter differently and, hence, can change stellar properties significantly.

The change in the dimensionless tidal deformability for NS was studied. It was found that, for higher value of the magnetic dipole moment, the tidal deformability of the canonical mass surpasses the upper limit of 800 set by the measurement from GW1701817 at 90% confidence, which is consistent with the acknowledgement of such objects possessing weak magnetic fields. No such measurement of tidal deformability is available for GW190814, as it has no tidal signatures. But if in the future, measurements of tidal deformability of massive NSs such as the one measured for the secondary object in GW190814 are $\leq$ 800, this would imply that object could not be a hyperonic magnetar.

In the future, we will investigate the effect of considering more exotic particles in our modelling of magnetic NSs. These include, kaons, $\Delta$-resonances and deconfined quark matter (with and without mixtures of phases). We also intend to analyze combined effects of magnetic fields and fast rotation on stars containing exotic matter, as both of these aspects could be important in young stars (Lasky et al., 2019; Yu et al., 2019; Lasky et al., 2017).