limiting magnetic field for minimal deformation of a

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9Astronomy & Astrophysics manuscript no. lowestBfield-rev-final c©ESO 2019July 9, 2019

Limiting magnetic field for minimal deformation of a magnetised

neutron star

R.O. Gomes1, H. Pais2, 3, V. Dexheimer4, C. Providência2, and S. Schramm1⋆

1 Frankfurt Institute for Advanced Studies, Frankfurt am Main, Germany.2 CFisUC, University of Coimbra, P-3004-516 Coimbra, Portugal.3 INESC TEC, Campus da FEUP, P-4200-465 Porto, Portugal.4 Department of Physics, Kent State University, Kent OH 44242 [email protected], [email protected], [email protected], [email protected]

Received xxx Accepted xxx

ABSTRACT

Aims. In this work, we study the structure of neutron stars under the effect of a poloidal magnetic field and determine the limitinglargest magnetic field strength that induces a deformation such that the ratio between the polar and equatorial radii does not exceed2%. We consider that, under these conditions, the description of magnetic neutron stars in the spherical symmetry regime is stillsatisfactory.Methods. We describe different compositions of stars (nucleonic, hyperonic, and hybrid), using three state-of-the-art relativisticmean field models (NL3ωρ, MBF, and CMF, respectively) for the microscopic description of matter, all in agreement with standardexperimental and observational data. The structure of stars is described by the general relativistic solution of both Einstein’s fieldequations assuming spherical symmetry and Einstein-Maxwell’s field equations assuming an axi-symmetric deformation.Results. We find a limiting magnetic moment of the order of 2 × 1031Am2, which corresponds to magnetic fields of the order of 1016

G at the surface and 1017 G at the centre of the star, above which the deformation due to the magnetic field is above 2%, therefore,not negligible. We show that the intensity of the magnetic field developed in the star depends on the EoS, and, for a given baryonicmass and fixed magnetic moment, larger fields are attained with softer EoS. We also show that the appearance of exotic degrees offreedom, such as hyperons or a quark core, is disfavored in the presence of a very strong magnetic field. As a consequence, a highlymagnetized nucleonic star may suffer an internal conversion due to the decay of the magnetic field, which could be accompanied bya sudden cooling of the star or a gamma ray burst.

Key words. equation of state – magnetic fields – stars: neutron

1. Introduction

Neutron stars are one of the possible remnants of supernova ex-plosions that are triggered by the gravitational collapse of inter-mediate mass stars. During the collapse, because of angular mo-mentum and magnetic flux conservation, the rotation frequenciesand magnetic fields of these stars are exceptionally amplified,reaching values of P ∼ 1 s and Bs ∼ 1012 G, respectively. Also,because of the extreme densities reached inside these objects,neutron stars provide a unique environment for investigating fun-damental questions in physics and astrophysics. In particular,a class of objects named magnetars possess surface magneticfields that are even more extreme than regular pulsars, of the or-der of Bs ∼ 1013−1015 G, appearing in nature in the form of Softgamma repeaters (SGRs) and Anomalous X-ray pulsars (AXPs).The nature of these objects was explained in the magnetar modelproposed by Thompson & Duncan (Thompson & Duncan 1995,1996), which interprets them as neutron stars that present highlyenergetic gamma-ray (SGRs) and X-ray (AXPs) activity, pow-ered by their strong magnetic fields decay. For a broad reviewon magnetars from theoretical and observational points of view,see Mereghetti et al. (2015), Kaspi & Beloborodov (2017), andreferences therein. Although the internal magnetic fields of neu-

Send offprint requests to: H. Pais⋆ Deceased

trons stars can not yet be accessed by observations, Virial the-orem arguments (Lai & Shapiro 1991) predict that the internalmagnetic fields can reach values up to Bc ∼ 1018 G in stellarcenters, which agree with realistic general relativity calculations(Bocquet et al. 1995; Cardall et al. 2001; Frieben & Rezzolla2012; Chatterjee et al. 2015)

In the past, the effects of strong magnetic fields werestudied, first, separately on the equation of state (EoS) ofneutron star matter (Broderick et al. 2000, 2002) and in afully general relativistic formalism on the structure of neu-tron stars (Bonazzola et al. 1993; Bocquet et al. 1995). Onlymuch later, the effects of strong magnetic fields were stud-ied self-consistently on the EoS and structure of neutron stars(Chatterjee et al. 2015; Franzon et al. 2015). The latter con-cluded that magnetic field effects on the EoS of baryons andquarks do not play a significant role for determining the macro-scopic properties of stars with central values Bc . 1018 G(as expected from simple order of magnitude estimates), al-though the same cannot be said about direct magnetic field ef-fects on the macroscopic stellar structure. In particular, it wasshown that neglecting deformation effects by solving generalrelativity spherically symmetric solutions (TOV) (Tolman 1939;Oppenheimer & Volkoff 1939), leads to an overestimation of themass and an underestimation of the equatorial radius of stars(Gomes et al. 2017). This happens because, in this case, the ex-

Article number, page 1 of 10

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A&A proofs: manuscript no. lowestBfield-rev-final

tra magnetic energy that would deform the star (when the properformalism is applied) is being added to the mass due to the im-posed spherical symmetry.

The particle population of the core of stars is also di-rectly affected by the presence of strong magnetic fields. Themain reasons for this being the shift of the particle onsetdensity to higher densities if magnetic field effects are takeninto account in the EoS (Chakrabarty 1996; Chakrabarty et al.1997; Broderick et al. 2000, 2002; Yuan & Zhang 1999;Suh & Mathews 2001; Lai & Shapiro 1991) and, also, the de-crease of stellar central densities (Franzon et al. 2015, 2016b;Gomes et al. 2017; Chatterjee et al. 2018). Depending on the in-tensity of the magnetic fields considered, exotic particles suchas hyperons, delta resonances, meson condensates and quarkscan vanish completely from their core, changing, for exam-ple, the neutrino emission of these objects (Rabhi & Providencia2010). In other words, magnetic field decay over time canlead to a repopulation of stars, similarly to rotational spindown (Negreiros et al. 2013; Bejger et al. 2017), playing animportant role on key questions such as the hyperon puzzle(Zdunik & Haensel 2013; Chatterjee & Vidaña 2016), the Deltapuzzle (Cai et al. 2015; Drago et al. 2016), hadron-quark phasetransitions (Avancini et al. 2012; Ferreira et al. 2014; Costa et al.2014; Roark & Dexheimer 2018; Lugones & Grunfeld 2018)and stellar cooling (Sinha & Sedrakian 2015; Raduta et al. 2017;Negreiros et al. 2018; Fortin et al. 2018; Patiño et al. 2018).

In the light of the aforementioned results from studies ofmagnetic neutron stars modeled in a self-consistent formalism, itis clear that a careful determination of the magnetic field thresh-old beyond which a spherical symmetry for stars is no longervalid is needed, together with the determination of the lowestmagnetic field that makes exotic particles vanish from the coreof stars. We address these two questions in this work, first, byidentifying the intensity of the magnetic fields relevant for mod-ifying the macroscopic properties of neutron stars. For doing so,we compare the structure of magnetic neutrons stars using a for-malism that allows stars to deform due the effect of a poloidalmagnetic field and test different values of the magnetic dipolemoment of the star, including the zero dipole case. Second, weinvestigate the conditions that give rise to the conversion of ahadronic star to a hyperonic or hybrid one due to the decay of themagnetic field. We use three different stellar compositions (nu-cleonic, hyperonic, and hybrid) and different relativistic meanfield models, in order to make our results more general.

In the present work, we consider a poloidal magnetic fieldconfiguration for non-rotating stars. The joint effect of a toroidalmagnetic field and rotation on the structure of neutron stars wascarried out in Frieben & Rezzolla (2012). It was shown that nei-ther purely poloidal nor purely toroidal magnetic field configura-tions are stable and that stability requires twisted-torus solutions(Ciolfi 2014). In Uryu et al. (2014), the authors have numericallyobtained stationary solutions of relativistic rotating stars consid-ering strong mixed poloidal and toroidal magnetic fields. In par-ticular, the presence of a dominant toroidal component is essen-tial, for instance, to describe an increase of the inclination angleof a NS (Lander & Jones 2018). Therefore, a more complete de-scription of magnetised neutron stars should be addressed in thefuture.

In order to consistently describe magnetars in a general rel-ativistic framework, we use the formalism implemented in thepublicly available Langage Objet pour la RElativité NumériquE(LORENE) library (LORENE 2019; Bonazzola et al. 1993;Bocquet et al. 1995; Bonazzola et al. 1998; Gourgoulhon 2012)in its present online version. It solves the coupled Einstein-

Maxwell field equations in order to determine stable and station-ary magnetised configurations of stars by assuming a poloidalmagnetic field distribution. The metric used for the polar-spherical symmetry is the Maximal-Slicing-Quasi-Isotropic(MSQI) (Gourgoulhon 2012; Franzon et al. 2015), which allowsthe stars to deform by making the metric potentials depend on theradial r and angular θ coordinates with respect to the magneticaxis. As this formalism does not account for the hydrodynami-cal generation of electromagnetic fields, these are introduced viamacroscopic currents, which are free parameters in the calcula-tion. Alternatively, we can determine the currents by fixing thestellar magnetic dipole moment, which is the conserved quantityin our system.

In Bocquet et al. (1995), the authors present the first nu-merical results of the coupled Einstein-Maxwell equations forhighly magnetised rotating neutron stars. They study the struc-ture of magnetized stars considering for the core several neutronmatter (Diaz Alonso 1985; Haensel et al. 1981; Pandharipande1971), one polytropic, and a hyperonic matter (Bethe & Johnson1974) EoS. Non unified crust-core EoS are used, with BPS forthe outer crust and BBP or NV (see Sec. 4.1 of Salgado et al.(1994)) for the inner crust. In our study, we analyse againthe effect of the magnetic field on magnetized non-rotatingstars using LORENE and considering three state-of-the-art EoS,NL3ωρ (hadronic), MBF (hyperonic), and CMF (hybrid). As inBocquet et al. (1995), for the outer crust, we take the BPS EoS,but the inner crust EoS is built from a Thomas-Fermi pasta calcu-lation using the NL3ωρ model, so that the inner crust-core EoSis unified for this EoS.

It was shown in Bocquet et al. (1995) that the deformationof a star is only significant for B > 1014 G, that the maximumstellar mass increases with the magnetic field, and that the max-imum allowed magnetic field is of the order of 1018 G. In thepresent work, we more thoroughly quantify the deformation cre-ated by the magnetic field and find that the limiting magneticfield that causes a relative deformation of 2% is of the order of1016 G at the surface and 1017 G at the center. We also showthat the largest effects created when increasing the magneticfield are seen on the stellar radius and not on the mass, which isin agreement with the results of Chatterjee et al. (2015). A dif-ferent conclusion was drawn in Bocquet et al. (1995) becausethe maximum magnetic dipole moment considered was above1032 Am2, generating strong effects also on the stellar maxi-mum mass. For magnetic dipole moments below 1032 Am2 (ourcase), the mass is not significantly affected when compared to thezero-field case. We do not include magnetic field effects in theEOS, as previous studies concluded that magnetic fields do notplay significant role on the core EoS, (Chatterjee et al. (2015)),though more recently, it has been shown that they do have anon-negligible role in the outer crust EoS (Kondratyev et al.2001; Potekhin & Chabrier 2013; Chamel et al. 2012, 2015;Stein et al. 2016; Potekhin & Chabrier 2018) and in the innercrust EoS (Fang et al. 2016, 2017; Fang et al. 2017). In what fol-lows, in section II, we present the EoS models used in the study.Our results for different families of stars and for a single star areshown in section III. Finally, in section IV, some conclusions aredrawn.

2. Equations of State

The full EoS used in this work are constructed with an outercrust, an inner crust, and a core. The outer crust is mergedwith the inner crust at the neutron drip line, and the in-ner crust is matched with the core EoS at the density for


Gomes et al.: Limiting B−field for minimal deformation

which the pasta geometries melt, i.e., the so-called crust-core transition. For the outer crust, we use the Baym-Pethick-Sutherland (BPS) EoS (Baym et al. 1971), and for the innercrust, we perform a Thomas-Fermi pasta calculation (Grill et al.2014; Pais & Providência 2016) using the relativistic mean fieldNL3ωρ model (Pais & Providência 2016).

We point out that more up-to-date outer crust EoS have beencalculated, however, it has been shown in Fortin et al. (2016) thatthe use of the BPS EoS for the outer crust or more recent EoS,such as the ones discussed in Ruester et al. (2006), will practi-cally not affect the mass and radius of neutron stars. Also theauthors of Sharma et al. (2015) have shown that the behaviorof BPS EoS is very similar to the one of BCPM (Sharma et al.2015), or the one of BSk21 (Pearson et al. 2012; Potekhin et al.2013). Since BPS is a well known and frequently used EoS, wechoose to consider it in the following calculations.

The EoS of the magnetized outer crust has beencalculated in Potekhin & Chabrier (2013), and applied inPotekhin & Chabrier (2018) to study the cooling of magnetizedneutron stars, or in Chamel et al. (2012, 2015), where the authorshave demonstrated that the magnetic field could affect the massof the outer crust. However, no EoS for the magnetized innercrust is currently publicly available. In a consistent calculation,we should have considered an unified EoS of magnetized nu-clear matter but we believe the error we introduce by not usingthe magnetized outer crust EoS is within the uncertainties of notusing a completely unified EoS.

In order to make our results as general as possible, we usethree different models for the core EoS, each with a different stel-lar composition: nucleonic, hyperonic, and hybrid. For the nucle-onic core EoS, we consider the NL3ωρmodel, composed of npehomogeneous matter (Pais & Providência 2016). This model ful-fills constraints coming from microscopic neutron matter cal-culations, such as chiral effective models (Hebeler et al. 2013),and also generates two-solar-mass neutron stars. For the hyper-onic core EoS, we use the many-body force (MBF) (Gomes et al.2015; Dexheimer et al. 2018) model with npeΛ matter; and, forthe hybrid one, we take the chiral mean field (CMF) modelwith npeµΛǫq matter, taking into account chiral symmetryrestoration and allowing for the existence of a mixed phase(Dexheimer & Schramm 2010). One should note that all of theEoS used in this work are calculated without taking into accountmicroscopic magnetic field effects, since it has been shown inprevious works (Chatterjee & Vidana 2016; Franzon et al. 2015;Dexheimer et al. 2017; Gomes et al. 2017) that the magneticfield does not significantly affect the EoS or stellar central mag-netic fields Bc . 1018 G. However, note that in some morerecent studies, it was shown that strong magnetic fields of theorder of magnitude observed in stars can have important ef-fects in the outer crust EoS and its properties (Kondratyev et al.2001; Potekhin & Chabrier 2013; Chamel et al. 2012, 2015;Stein et al. 2016; Potekhin & Chabrier 2018) or in the inner crustEoS (Fang et al. 2016, 2017; Fang et al. 2017). The nuclear mat-ter properties at saturation density of the models discussed in thiswork are displayed in Table 1. Their astrophysical properties areshown in the following section.

3. Results

In this section we begin by calculating sequences of magneticstars with different central densities and baryonic number, usingthe LORENE library for fixed values of the magnetic dipole mo-ment, µ. This quantity is related to the radial component of the

Table 1. Nuclear properties for symmetric matter at saturation densityfor the three EoS models used in this work. The columns are saturationdensity ρ0, binding energy per nucleon, B/A, effective mass of the nu-cleon, M∗, incompressibility modulus, K0, symmetry energy, Esym , andthe slope of the symmetry energy, L. All quantities are given in MeV,except for the saturation density, which is given in fm−3.

Model ρ0 B/A M∗ K0 Esym L

NL3ωρ 0.148 −16.24 559 270 31.7 55MBF-ω 0.149 −15.75 620 297 26.4 46

CMF 0.150 −16.00 629 297 29.6 88

magnetic field, Br, by:

2µ cos θ

r3= Br |r→∞ ,

which provides us with different magnetic field strength distri-butions in each stellar sequence. In a second moment, we fix thestellar baryonic mass and only change the value of the stellarmagnetic dipole moment.

Our main goal in this work is to determine the maximumvalue of the magnetic field intensity for which neutron stars canstill be described by spherical equilibrium solutions of Einstein’sequations, i.e the TOV equations, in a reasonable good approxi-mation. The stellar matter EoS is described within the three mod-els presented in the previous section and the effect of the mag-netic field on several properties of magnetised stars is discussed.

3.1. Families of stars

In this subsection, Figs. 1 to 8 show results for several fami-lies of stars, using three different EoS: NL3ωρ, MBF and CMF.According to Haensel et al. (2002), the minimum gravitationalmass of cold catalysed stars could be as low as M ∼ 0.1M⊙, whereas according to Goussard et al. (1998); Strobel et al.(1999), lepton-rich matter as found in proto-neutron stars, is un-bound for stars with a mass below ∼ 0.9 − 1.1 M⊙. For this rea-son, in the following, we only show configurations for stars withMg > 0.7 M⊙.

In Fig. 1, we show the gravitational mass of several fam-ilies of stars with different values of the magnetic dipole mo-ment, µ, as a function of the circumferential radius, which char-acterises the equator of stars coordinate-independently. The hor-izontal bands indicate the mass uncertainties associated withthe PSR J0348 +0432 (Antoniadis et al. 2013) (upper) and PSRJ1614−2230 (Fonseca et al. 2016) (lower) masses. The top panelshows results only for the NL3ωρmodel. There, one can see thatthe smaller the mass, the larger the effects of the magnetic fieldon the stellar structure, as one would expect. A noticeable dif-ference from the non-magnetic case (µ = 0) takes place for val-ues of magnetic dipole moment above µ = 2 × 1031 Am2 andonly for low-mass stars. This, therefore, implies that, for higherdipole magnetic moments, stars shall have magnetic fields strongenough to make deformation effects non negligible.

Having this in mind, results for µ = 0, µ = 5×1031 Am2, andµ = 1032 Am2 are displayed for all three models in the bottompanel. The behavior of the magnetised families of stars calcu-lated with the MBF and CMF models is similar to the one calcu-lated with the NL3ωρ model. Magnetic dipole moments of theorder of 5 × 1031 Am2 and above clearly do have strong effectson the radius of the stars, and this applies to all models.

In Fig. 2, we show the gravitational mass of several familiesof stars with different values of the magnetic dipole moment, µ,


A & A pr o ofs: m a n us cri pt n o. l o w est B fi el d-r e v- fi n al

1

1. 5

2

2. 5

3

1 2 1 3 1 4 1 5

N L 3 ω ρ

Mg (

M sun)

R cir c ( k m)

µ = 0 A m2

µ = 5 x 1 03 0

A m2

µ = 2 x 1 03 1

A m2

µ = 5 x 1 03 1

A m2

µ = 1 03 2

A m2

1

1. 5

2

2. 5

3

1 2 1 3 1 4 1 5

N L 3 ω ρ

M B F

C M F

Mg (

M sun)

R cir c ( k m)

µ = 0 A m2

µ = 5 x 1 03 1

A m2

µ = 1 03 2

A m2

Fi g. 1. T h e m ass-r a di us r el ati o n f or t h e N L 3 ω ρ m o d el f or s e v er al v al u esof t h e m a g n eti c di p ol e m o m e nt, µ , (t o p), a n d f or t h e t hr e e m o d el s c o n-si d er e d i n t hi s p a p er wit h t hr e e di ff er e nt v al u es of t h e m a g n eti c di p ol em o m e nt ( b ott o m).

as a f u n cti o n of t h e s urf a c e m a g n eti c fi el d at t h e p ol e (t o p p a n el),t h e c e ntr al m a g n eti c fi el d ( mi d dl e p a n el), a n d t h e r ati o b et w e e nt h e c e ntr al a n d s urf a c e m a g n eti c fi el d s, B c / B s , ( b ott o m p a n el),c o n si d eri n g t h e t hr e e E o S m o d els. L o o ki n g at t h e t o p p a n el, o n ec a n s e e t h at µ = 5 × 1 0 3 1 A m 2 , a v al u e t h at gi v es a si g ni fi c a ntdi ff er e n c e t o t h e µ = 0 c as e i n Fi g. 1, c orr es p o n d s t o a s urf a c em a g n eti c fi el d at t h e p ol e i n t h e r a n g e of ∼ 5 × 1 0 1 6 G t o 1 0 1 7

G f or t h e t hr e e m o d els a n d f a mili es of st ar s c o n si d er e d. T h es es urf a c e fi el d s c orr es p o n d t o a c e ntr al m a g n eti c fi el d i n t h e r a n g eof ∼ 2 × 1 0 1 7 G t o ∼ 4 × 1 0 1 7 G, as s e e n fr o m t h e mi d dl e p a n el.I n a d diti o n, t h e st ell ar m ass es ar e n ot si g ni fi c a ntl y aff e ct e d b yt h e m a g n eti c fi el d s, w h er e as t h e r a di u s i n cr e as es wit h t h e B −fi el d, as t h e st ar s d ef or m i nt o a n o bl at e s h a p e. T h e b ott o m p a n els h o ws t h at, f or st ar s wit h M g 1 M ⊙ , w e h a v e B c ∼ 3 − 5 B s .T h e di ff er e n c e is l ar g er f or l o w m ass st ar s a n d w e a k er fi el d s, i. e.µ < 2 × 1 0 3 1 A m 2 . T his s m o ot h c h a n g e of m a g n eti c fi el d s i n si d est ar s ill u str at es o n c e m or e t h at a d- h o c e x p o n e nti al p ar a m etri z a-ti o n s f or t h e m a g n eti c fi el d ( B a n d y o p a d h y a y et al. 1 9 9 7) ar e u n-r e alisti c ( as alr e a d y dis c u ss e d i n D e x h ei m er et al. ( 2 0 1 7)), ast h e y pr o d u c e a n i n cr e as e of s e v er al or d er s of m a g nit u d e f or t h em a g n eti c fi el d i n si d e t h e st ar s.

T h e gr a vit ati o n al m ass f or st ell ar s e q u e n c es is pl ott e d as af u n cti o n of t h e c e ntr al d e n sit y i n Fi g. 3. T h e st ell ar c e ntr al d e n-sit y is n ot si g ni fi c a ntl y a ff e ct e d b y t h e m a g n eti c fi el d f or t h eN L 3 ω ρ : f or a 1. 8 M⊙ st ar t h e d e n sit y d e cr e as es b y 0 .0 1 f m − 3 ,w hil e f or a 1 M ⊙ st ar, t his di ff er e n c e i n cr e as es t o 0 .0 2 f m − 3 .T h e C M F st ar s s u ff er t h e l ar g er c h a n g es: i n d e p e n d e ntl y of t h est ell ar m ass, as t h e m a g n eti c fi el d str e n gt h c h a n g es, t h e c e ntr al

1

1. 5

2

2. 5

3

1 x 1 01 5

1 x 1 01 6

1 x 1 01 7

Mg (

M sun)

B s ( G)

1

1. 5

2

2. 5

3

1 x 1 01 6

1 x 1 01 7

1 x 1 01 8

Mg (

M sun)

B c ( G)

1

1. 5

2

2. 5

3

2 3 4 5 6

N L 3 ω ρ

M B F

C M F

Mg (

M sun)

B c / Bs

µ = 1 03 0

A m2

µ = 5 x 1 03 0

A m2

µ = 1 03 1

A m2

µ = 2 x 1 03 1

A m2

µ = 5 x 1 03 1

A m2

µ = 1 03 2

A m2

Fi g. 2. T h e gr a vit ati o n al m ass of s e v er al f a mili es of st ars wit h di ff er e ntv al u es of t h e m a g n eti c di p ol e m o m e nt, µ , as a f u n cti o n of t h e s urf a c ep ol ar m a g n eti c fi el d at t h e p ol e B s (t o p), t h e c e ntr al m a g n eti c fi el d B c

( mi d dl e), a n d t h e r ati o b et w e e n t h e c e ntr al a n d s urf a c e m a g n eti c fi el ds,B c / B s , ( b ott o m), f or t h e N L 3ω ρ (s oli d), M B F ( d as h e d) a n d C M F ( d ot-d as h e d) m o d el s.

d e n sit y d e cr e as es b y 0 .0 4 f m − 3 w h e n µ i n cr e as es fr o m z er ot o µ = 1 0 3 2 A m 2 . A str o n g m a g n eti c fi el d p u s h es t h e o n s et ofΛ s a n d q u ar k s t o st ar s wit h m ass es of ∼ 0 .0 8 M ⊙ l ar g er t h a n i nn o n- m a g n etis e d o n es, a n e ff e ct alr e a d y dis c u ss e d i n R a b hi et al.( 2 0 1 1). As a c o n s e q u e n c e, a n u cl e o ni c st ar m a y s uff er a tr a n-siti o n t o a h y bri d st ar or a h y p er o ni c st ar w h e n t h e m a g n eti cfi el d s d e c a y s. T h e o n s et of h y p er o n s a n d q u ar k s o p e n s n e w c h a n-n els f or n e utri n o e missi o n a n d, t h er ef or e, t h e p o ssi bilit y of af ast er c o oli n g pr o c ess ( Ya k o vl e v et al. 2 0 0 4; R a d ut a et al. 2 0 1 8;N e gr eir o s et al. 2 0 1 8; Gri g ori a n et al. 2 0 1 8; Pr o vi d ê n ci a et al.2 0 1 8). Si m ult a n e o u sl y, it als o a ff e cts t h e o n s et of t h e n u cl e o ni c

Arti cl e n u m b er, p a g e 4 of 1 0

G o m es et al.: Li miti n g B − fi el d f or mi ni m al d ef or m ati o n

1

1. 5

2

2. 5

3

0. 2 0. 4 0. 6 0. 8

N L 3 ω ρ

M B F

C M F

Mg (

M sun)

ρ c (f m- 3

)

µ = 0 A m2

µ = 5 x 1 03 1

A m2

µ = 1 03 2

A m2

1

1. 5

2

2. 5

3

0. 2 0. 4 0. 6 0. 8

N L 3 ω ρ

M B F

C M F

Λ

q

Mg (

M sun)

ρ c (f m- 3

)

Fi g. 3. T h e gr a vit ati o n al m ass of s e v er al f a mili es of st ars wit h di ff er e ntv al u es of t h e m a g n eti c di p ol e m o m e nt, µ , as a f u n cti o n of t h e c e ntr ald e nsit y f or t h e t hr e e m o d el s c o nsi d er e d i n t hi s w or k. T h e o ns et d e nsiti esof Λ a n d q u ar ks i s al s o s h o w n f or t h e C M F m o d el.

1 x 1 01 5

1 x 1 01 6

1 x 1 01 7

1 x 1 01 6

1 x 1 01 7

1 x 1 01 8

N L 3 ω ρM B FC M F

Bs

(G)

B c ( G)

µ = 1 03 0

A m2

µ = 5 x 1 03 0

A m2

µ = 1 03 1

A m2

µ = 2 x 1 03 1

A m2

µ = 5 x 1 03 1

A m2

µ = 1 03 2

A m2

5 x 1 01 6

7 x 1 01 6

1 x 1 01 7

2 x 1 01 7

3 x 1 01 7

4 x 1 01 7

µ = 5 x 1 03 1

A m2

M g = 1 M S u n

M g = 1. 4 M S u n

M g = 2 M S u n

Bs

(G)

B c ( G)

Fi g. 4. ( T o p) T h e c orr es p o n d e n c e b et w e e n t h e c e ntr al a n d s urf a c e ( att he p ol e) m a g n eti c fi el ds, B c a n d B s , r es p e cti v el y, f or t h e t hr e e m o d el s.( B ott o m) T h e s a m e as t h e a b o v e p a n el b ut o nl y f or µ = 5 × 1 0 3 1 A m 2 .T h e cir cl es s h o w t hr e e v al u es of t h e gr a vit ati o n al m ass: M = 1 M ⊙

( bl a c k), M = 1 .4 M ⊙ ( gr e e n), a n d M = 2 M ⊙ (r e d). T h e r e d cir cl e f ort h e C M F m o d el r e pr es e nt s t h e st ar wit h t h e m a xi m u m m ass, M = 1 .9 2M ⊙ .

dir e ct Ur c a pr o c ess ( F orti n et al. 2 0 1 6; Pr o vi d ê n ci a et al. 2 0 1 8).O n t h e ot h er h a n d, t h e c o n v er si o n of a n u cl e o ni c or h y p er o ni cst ar t o a h y bri d st ar c o ul d b e a c c o m p a ni e d b y t h e e missi o n of ar e as o n a bl e a m o u nt of e n er g y, as l o n g as t h er e is a si z a bl e r a di u sc h a n g e i n t h e i n v ol v e d st ell ar c o n fi g ur ati o n s ( G o m es et al. 2 0 1 8;

0. 6

0. 7

0. 8

0. 9

1

1 1. 5 2 2. 5 3


Rpol

e/

R eq

M g ( Ms u n )

µ = 1 03 0

A m2

µ = 5 x 1 03 0

A m2

µ = 1 03 1

A m2

µ = 2 x 1 03 1

A m2

µ = 5 x 1 03 1

A m2

µ = 1 03 2

A m2

Fi g. 5. D e vi ati o n fr o m s p h eri c al s y m m etr y, gi v e n b y t h e r ati o b et w e e nt he st ell ar r a di us at t h e p ol e a n d at t h e e q u at or, as a f u n cti o n of t h egr a vit ati o n al m ass f or t h e t hr e e m o d el s.

- 0. 2 5

- 0. 2

- 0. 1 5

- 0. 1

- 0. 0 5

0

1 1. 5 2 2. 5 3


Q (10

38 k

g.m2

)

M g ( Ms u n )

µ = 1 03 1

A m2

µ = 2 x 1 03 1

A m2

µ = 5 x 1 03 1

A m2

µ = 1 03 2

A m2

Fi g. 6. Q u a dr u p ol e m o m e nt of st ars as a f u n cti o n of t h e gr a vit ati o n alm a ss f or t h e t hr e e m o d el s.

D e x h ei m er et al. 2 0 1 9). I n t his w a y, t h e d et e cti o n of f ast c o oli n gor a n e n er g eti c e v e nt c o mi n g fr o m a hi g hl y m a g n etis e d n e utr o nst ar c o ul d b e ass o ci at e d wit h o n e of t h es e i nt er n al d e gr e es offr e e d o m c o n v er si o n s ( B o m b a ci & D att a 2 0 0 0; B er e z hi a ni et al.2 0 0 3).

I n Fi g. 4, w e s h o w t h e c orr es p o n d e n c e b et w e e n t h e c e ntr alm a g n eti c fi el d, B c , a n d t h e s urf a c e m a g n eti c fi el d at t h e p ol e, B s ,f or e a c h v al u e of t h e m a g n eti c di p ol e m o m e nt c o n si d er e d. Weo b s er v e t h at t h e f a mil y of st ar s wit h t h e l o w est µ c o n si d er e d h asa m a xi m u m s urf a c e m a g n eti c fi el d of B s ∼ 2 × 1 0 1 5 G. T hisis als o s e e n o n t h e t o p p a n el of Fi g. 2. T h e s e q u e n c e of st ar swit h t h e hi g h est µ h as a c orr es p o n di n g B s ∼ 2 × 1 0 1 7 G a n da c e ntr al m a g n eti c fi el d of t h e or d er of 1 0 1 8 G. T h e b e h a vi o urof B s v er s u s B c is n ot m o n ot o ni c a n d, i n t h e b ott o m p a n el, t h er es ults f or µ = 5 × 1 0 3 1 A m 2 ar e s h o w n wit h m or e d et ail f ort h e t hr e e E o S c o n si d er e d i n t his w or k. T h e o v er all b e h a vi or ism o d el-i n d e p e n d e nt, alt h o u g h t h e q u a ntit ati v e b e h a vi o ur d o es d e-p e n d o n t h e E o S c o n si d er e d. I n t h e b ott o m p a n el, t h e st ar s wit hM = 1 .0 , 1 .4 a n d 2 M ⊙ h a v e b e e n i d e nti fi e d. T h e mi ni m u m B c

of t h e c ur v e o c c ur s f or a st ar wit h a m ass of t h e or d er of 1. 4 M ⊙

a n d c orr es p o n d s t o t h e st ar w h er e t h e m ass-r a di u s c ur v e c h a n g esc ur v at ur e. F or st ar s wit h a m ass gr e at er or s m all er t h a n ∼ 1 .4M ⊙ , t h e s urf a c e fi el d i n cr e as es m o n ot o ni c all y wit h t h e c e ntr alm a g n eti c fi el d.



0

0. 5

1

1. 5

5 x 1 0 1 6 1 0 1 7 1. 5 x 1 0 1 7 2 x 1 0 1 7

D = 1 k p c

Fr e q = 1 H z

N L 3 ω ρ

h (10

-24 )

B s ( G)

0

0. 5

1

1. 5

1 1. 5 2 2. 5 3

h (10

-24 )

M g ( Ms u n )

µ = 5 x 1 0 3 1 A m2 , a)µ = 5 x 1 0 3 1 A m2 , b)

µ = 1 0 3 2 A m2 , a)µ = 1 0 3 2 A m2 , b)

Fi g. 7. Gr a vit ati o n al w a v e a m plit u d e fr o m E q. ( 1) ( a) a n d fr o m E q. ( 5) ( b) as a f u n cti o n of t h e m a g n eti c fi el d at t h e s urf a c e ( p ol e), B s , (l eft), a n dt h e gr a vit ati o n al m ass (ri g ht) f or t h e N L 3ω ρ m o d el. We ass u m e d a f a mil y of st ars l o c at e d at a di st a n c e of 1 k p c a n d s pi n ni n g wit h a fr e q u e n c y of1 H z. T h e bl a c k d ot s c orr es p o n d t o t h e 2-s ol ar- m ass st ar c o n fi g ur ati o ns.

0

0. 5

1

1. 5

5 x 1 0 1 6 1 0 1 7 1. 5 x 1 0 1 7 2 x 1 0 1 7 0

D = 1 k p c

Fr e q = 1 H z

N L 3 ω ρ

M B F

C M F

h (10

-24 )

B s ( G)

0

0. 5

1

1. 5

1 1. 5 2 2. 5 3

h (10

-24 )

M g ( Ms u n )

µ = 1 0 3 1 A m2

µ = 2 x 1 0 3 1 A m2

µ = 5 x 1 0 3 1 A m2

µ = 1 0 3 2 A m2

Fi g. 8. Gr a vit ati o n al w a v e a m plit u d e fr o m E q. ( 1), as a f u n cti o n t h e m a g n eti c fi el d at t h e s urf a c e ( p ol e), B s , (l eft), a n d t h e gr a vit ati o n al m ass (ri g ht)f or t h e t hr e e m o d el s. We o n c e m or e ass u m e d a f a mil y of st ars l o c at e d at a di st a n c e of 1 k p c a n d s pi n ni n g wit h a fr e q u e n c y of 1 H z. T h e bl a c k d ot sc orr es p o n d t o ∼ 2-s ol ar- m ass st ar c o n fi g ur ati o ns, a n d t h e bl u e o n es c orr es p o n d t o 1. 4 M ⊙ st ars.

I n Fi g. 5, w e s h o w t h e st ar s’ d e vi ati o n fr o m s p h eri c al s y m-m etr y, gi v e n b y t h e r ati o b et w e e n t h e r a di u s at t h e m a g n eti c p ol ea n d t h e r a di u s at t h e e q u at or, as a f u n cti o n of t h e gr a vit ati o n alm ass. We s e e t h at f or t h e l o w est m a g n eti c di p ol e m o m e nts c o n-si d er e d t h e d ef or m ati o n is v er y s m all a n d it h a p p e n s o nl y f or t h el o w- m ass st ar s, i. e., wit h M < 1 M ⊙ . F or a 1 M⊙ st ar, t h e di ff er-e n c e b et w e e n t h e p ol ar a n d e q u at ori al r a dii is b el o w 1 %. D ef or-m ati o n a p p e ar s f or st ar s wit h hi g h er m a g n eti c di p ol e m o m e nts.T his all o ws u s t o c o n cl u d e t h at w e o nl y h a v e si g ni fi c a nt d ef or-m ati o n f or v al u es of t h e m a g n eti c di p ol e m o m e nt µ 2 × 1 0 3 1

A m 2 , w hi c h c orr es p o n d t o a s urf a c e m a g n eti c fi el d of B s 2 − 4( 1 01 6 G), a n d B c = 1 − 2 ( 1 0 1 7 G) ( s e e Fi g. 2).

We c a n als o a n al y s e t h e d ef or m ati o n of st ar s b y l o o ki n g att h eir q u a dr u p ol e m o m e nts. Fi g. 6 s h o ws t h e q u a dr u p ol e m o-m e nt of t h e f a mili es of st ar s c o n si d er e d i n t his st u d y as a f u n c-ti o n of t h e gr a vit ati o n al m ass. As e x p e ct e d, t h e r es ults f or t h eq u a dr u p ol e m o m e nt ar e i n a gr e e m e nt wit h t h e o n es s h o w n f ort h e d ef or m ati o n: it b e c o m es n o n- n e gli gi bl e o nl y f or µ 2 × 1 0 3 1

A m 2 ( s e e t h e pr e vi o u s fi g ur e).

It h as b e e n s h o w n ( B o n a z z ol a & G o ur g o ul h o n 1 9 9 6) t h at,f or a sli g htl y d ef or m e d st ar, t h e a m plit u d e of gr a vit ati o n al w a v es

e mitt e d is gi v e n b y

h 0 = −6 G

c 4Q

Ω 2

D, ( 1)

w h er e G is t h e gr a vit ati o n al c o n st a nt, c t h e s p e e d of li g ht, Q t h eq u a dr u p ol e m o m e nt, D t h e dist a n c e t o t h e st ar, a n d Ω t h e r ot a-ti o n al v el o cit y of t h e st ar. A n ot h er w a y t o esti m at e t h e gr a vit a-ti o n al w a v e a m plit u d e is fr o m t h e m a g n eti c fi el d i n d u c e d d ef or-m ati o n ( B o n a z z ol a & G o ur g o ul h o n 1 9 9 6), w h er e c o n si d eri n g a ni n c o m pr essi bl e m a g n etis e d fl ui d, t h e q u a dr u p ol e m o m e nt c a n b ewritt e n as

Q = −µ 0 µ

2

1 6 π 2 G ρ R 3, ( 2)

w h er e µ 0 is t h e m a g n eti c p er m e a bilit y of fr e e s p a c e, s o t h at t h eelli pti cit y, ǫ B , a n d t h e m o m e nt of i n erti a, I, of t h e st ar ar e gi v e nb y

ǫ B =1 5 π B 2

s R4cir c

1 2 µ 0 G M 2g

, ( 3)

I =2

5M g R

2ci r c , ( 4)


G o m es et al.: Li miti n g B − fi el d f or mi ni m al d ef or m ati o n

1 x 1 0 1 6

1 x 1 01 7

2 x 1 01 7

3 x 1 01 7

4 x 1 01 7

0 0. 2 0. 4 0. 6 0. 8 1

M b = 1. 6 0 M s u n


Br (

G)

r/ Rp

µ = 1 0 3 1 A m2

µ = 2 x 1 0 3 1 A m2

µ = 5 x 1 0 3 1 A m2

µ = 1 0 3 2 A m2

- 4 x 1 01 7

- 3 x 1 01 7

- 2 x 1 01 7

- 1 x 1 01 7

0

1 x 1 01 7

0 0. 2 0. 4 0. 6 0. 8 1

Bt (

G)

r/ Re q

Fi g. 9. T h e r a di al c o m p o n e nt of t h e m a g n eti c fi el d (l eft) a n d t h e t a n g e nti al c o m p o n e nt of t h e m a g n eti c fi el d (ri g ht) as a f u n cti o n of t h e n or m ali s e dr adi us f or t h e t hr e e m o d el s c o nsi d eri n g o nl y st ars wit h fi x e d b ar y o n m ass M b = 1 .6 M ⊙ .

c o n si d eri n g u nif or m d e n sit y t hr o u g h o ut t h e st ar, a n d a p pr o xi-m ati n g t h e m o m e nt of i n erti a t o t h e o n e of a s p h er e. T h e gr a vi-t ati o n al w a v e a m plit u d e c a n t h e n b e e x pr ess e d as

h 0 =1 6 π 2 G ǫ B I

c 4 D P 2, ( 5)

w h er e P is t h e p eri o d of r ot ati o n of t h e st ar, P = 2 π / Ω . A ne sti m ati o n of t h e a m plit u d e of gr a vit ati o n al w a v es e mitt e d b yt h es e st ar s is d o n e b y s etti n g t h e dist a n c e t o 1 k p c (t h e dist a n c est o t h e p uls ar s P S R J 1 6 1 4- 2 2 3 0, Vel a, a n d Cr a b ar e ∼ 1 .2 k p c, ∼ 0 .5 k p c, a n d ∼ 2 k p c, r es p e cti v el y) a n d t h e fr e q u e n c y f =1 / P = 1 H z. R es ults ar e s h o w n i n t h e n e xt t w o fi g ur es.

I n Fi g. 7, w e s h o w t h e gr a vit ati o n al w a v e ( G W) a m plit u d ec al c ul at e d u si n g t h e e x pr essi o n s i n E q s. ( 1) a n d ( 5), f or t h eN L 3 ω ρ m o d el. We als o m ar k t h e 2 M ⊙ st ar c o n fi g ur ati o n s. E v e nt h o u g h t h e t w o c ur v es b e h a v e si mil arl y, t h e y d o n ot gi v e t h es a m e r es ults a n d, i n f a ct, t h e di ff er e n c e is si g ni fi c a nt: f or t h e 2M ⊙ c as e, wit h µ = 5 × 1 0 3 1 A m 2 , w e o bt ai n h 0 = 0 .1 5 × 1 0 − 2 4 f orE q. ( 1) a n d h 0 = 0 .1 × 1 0 − 2 4 f or E q. ( 5). T h e diff er e n c e i n cr e as esw h e n w e c o n si d er t h e µ = 1 0 3 2 A m 2 c as e: 0 .6 5 × 1 0 − 2 4 f or E q. ( 1)a n d 0 .4 5 × 1 0 − 2 4 f or E q. ( 5). T h e esti m ati o n o bt ai n e d fr o m E q.( 5) is w or s e f or t h e l o w er m ass st ar s, w hi c h h a v e t h e l ar g er d ef or-m ati o n s. T his ill u str at es t h at u si n g a n a p pr o xi m at e e x pr essi o n, asi n E q. ( 5), t h at c o n si d er s st ar s wit h u nif or m d e n sit y a n d a m o-m e nt of i n erti a gi v e n b y t h e o n e of a s p h er e, pr o d u c es di ff er e ntr es ults a n d e m p h asi z es t h e f a ct t h at t h es e a p pr o a c h es s h o ul d b ec o n si d er e d wit h c ar e w h e n c al c ul ati n g s e n siti v e q u a ntiti es li k eG W a m plit u d es.

I n t h e f oll o wi n g, G W a m plit u d es ar e esti m at e d fr o m E q. ( 1)f or t h e t hr e e m o d els w e h a v e c o n si d er e d. I n Fi g. 8, t h e gr a vi-t ati o n al w a v e a m plit u d e is s h o w n as a f u n cti o n of t h e m a g n eti cfi el d at t h e s urf a c e, B s , a n d as a f u n cti o n of t h e gr a vit ati o n alm ass, f or a f a mil y of st ar s l o c at e d at a dist a n c e of 1 k p c a n ds pi n ni n g wit h a fr e q u e n c y of 1 H z. T h e bl a c k d ots c orr es p o n dt o 2 M⊙ c o n fi g ur ati o n s a n d t h e bl u e o n es c orr es p o n d t o 1. 4 M ⊙

st ar s.B esi d es c al c ul ati n g t h e G W a m plit u d e, w e c a n als o esti-

m at e t h e c h ar a ct eristi c str ai n, S , t h e a ct u al q u a ntit y m e as ur e db y t h e i nt erf er o m et er d et e ct or s. It is d e fi n e d as t h e pr o d u ct b e-t w e e n t h e G W a m plit u d e a n d t h e s q u ar e r o ot of t h e i nt e gr a-ti o n ti m e ( B o n a z z ol a & G o ur g o ul h o n 1 9 9 6; M o or e et al. 2 0 1 5):

S = h 0

√P . If w e ass u m e t h at d at a w as c oll e ct e d d uri n g a p eri o d

o f t hr e e y e ar s, i. e.,√

P ∼ 1 0 4 s, t h e c h ar a ct eristi c str ai n b e c o m esp r o p orti o n al t o G W a m plit u d e b y a f a ct or of 1 0 4 .

T his m e a n s t h at f or a st ar wit h B s = 2 × 1 0 1 6 G, M g = 2M ⊙ , at 1 k p c a w a y a n d s pi n ni n g at 1 H z, w e o bt ai n h 0 = 0 .1 5 ×1 0 − 2 4 a n d S = 1 .5 × 1 0 − 2 1 H z − 1 / 2 . T his v al u e c o ul d i n d e e d stillb e d et e ct e d b y d et e ct or s li k e B B O, D E CI G O, A LI A. D et e ct or sLI G O a n d Vir g o d et e ct a hi g h er fr e q u e n c y r a n g e, f > 3 0 H z( C hrist o p h er M o or e & B err y 2 0 1 9).

0

5 0

1 0 0

1 5 0

0 0. 2 0. 4 0. 6

P (

Me

V f

m-3 )

ρ (f m- 3

)


Fi g. 1 0. E q u ati o n of st at e f or t h e t hr e e m o d el s c o nsi d er e d i n t hi s w or k.

3. 2. Si n gl e- st ar c o n fi g ur ati o n s wit h M b = 1 .6 M ⊙

I n t his s u b s e cti o n, w e dis c u ss t h e pr o p erti es of m a g n etis e d st ar swit h a fi x e d b ar y o ni c m ass of 1. 6 M ⊙ , wit h diff er e nt m a g n eti cdi p ol e m o m e nts, µ . St ar s wit h t his b ar y o ni c m ass h a v e a gr a vit a-ti o n al m ass of t h e or d er of 1. 4 M⊙ . T h e r a di al c o m p o n e nt of t h em a g n eti c fi el d (l eft) a n d t h e t a n g e nti al c o m p o n e nt of t h e m a g-n eti c fi el d (ri g ht) ar e pl ott e d i n Fi g. 9 as a f u n cti o n of t h e n or-m alis e d r a di u s f or di ff er e nt v al u es of µ . T h e r a di al c o m p o n e ntof B , B r , is c al c ul at e d at (r, θ = 0 , φ = 0) ( p ol ar) a n d t h e t a n-g e nti al (t o t h e e q u at ori al s urf a c e) c o m p o n e nt, B t, is c al c ul at e dat ( r, θ = π / 2 , φ = 0). T h e c h a n g e of dir e cti o n of t h e t a n g e nti alfi el d o c c ur s f or all m a g n eti c m o m e nt a at ar o u n d R e q / 4 fr o m t h es urf a c e of t h e st ar.



0

0. 1

0. 2

0. 3

0. 4

0 2 4 6 8 1 0 1 2 1 4

M b = 1. 6 0 M s u nN L 3 ω ρ

C M F

ρ (

fm-

3 )

R e q ( k m)

µ = 1 0 3 1 A m2

µ = 2 x 1 0 3 1 A m2

µ = 5 x 1 0 3 1 A m2

µ = 1 0 3 2 A m2

0

1 0

2 0

3 0

4 0

5 0

0 2 4 6 8 1 0 1 2 1 4

P (

Me

V f

m-3 )

R e q ( k m)

Fi g. 1 1. Pr o fil es of t h e st ars: (l eft) b ar y o n n u m b er d e nsit y a n d (ri g ht) pr ess ur e c al c ul at e d i n t h e e q u at ori al pl a n e f or t h e t hr e e m o d el s c o nsi d er e d i nt hi s w or k.

As e x p e ct e d, t h e hi g h er t h e µ , t h e hi g h er t h e m a g n eti c fi el dm a g nit u d e. H o w e v er, it is i nt er esti n g t o s e e h o w t h e i nt e n sit y oft h e fi el d d e p e n d s o n t h e m o d el: N L 3ω ρ h as t h e s m all est r a di ala n d t a n g e nti al c o m p o n e nt a b s ol ut e v al u es, w hil e t h e ot h er t w om o d els s h o w si mil ar m a g nit u d es. T his b e h a vi or c o m es fr o m t h ef a ct t h at t h e C M F m o d el h as t h e s oft est E o S a n d gi v es ris e t ol ar g er m a g n eti c fi el d i nt e n siti es f or t h e s a m e m a g n eti c di p ol em o m e nt ( s e e Fi g. 1 0, w h er e t h e t hr e e E o S u s e d i n t his w or k ar er e pr es e nt e d). Fr o m Fi g. 3, o n e c a n s e e t h at st ar s wit h M ∼ 1 .4M ⊙ h a v e si mil ar c e ntr al d e n siti es f or m o d els C M F a n d M B F b e-l o w ρ = 0 .4 f m − 3 . C M F is t h e s oft est E o S a b o v e s at ur ati o n a n dd e v el o p s t h e str o n g est fi el d s i n t h e i nt eri or of t h e st ar. B el o w s at-ur ati o n d e n sit y, h o w e v er, C M F is cl o s er t o N L 3 ω ρ a n d it is t h eM B F m o d el, wit h t h e s oft est l o w d e n sit y E o S, t h at d e v el o p s t h estr o n g est s urf a c e fi el d s. I n t h e p ast, it w as s h o w n t h at t h e c o m-p a ct n ess of st ar s is dir e ctl y r el at e d t o t h e str e n gt h of c e ntr al m a g-n eti c fi el d s, t hr o u g h t h e s oft n ess of t h e E o S ( G o m es et al. 2 0 1 7).O ur r es ults i n di c at e t h at t h e r el ati o n b et w e e n t h e m a g n eti c fi el di n t h e c e nt er of t h e st ar a n d t h e o n e o n t h e s urf a c e, as dis c u ss e di n Fi g. 4, is d e fi n e d b y t h e pr o p erti es of t h e E o S b ot h at t h e cr u sta n d i n t h e c or e.

T h e e ff e ct of m a g n eti c fi el d s o n t h e str u ct ur e of st ar s is als ocl e arl y s h o w n i n Fi g. 1 1, w h er e t h e pr o fil es of st ar s wit h di ff er-e nt m a g n eti c fi el d c o n fi g ur ati o n s ar e pl ott e d f or m o d els N L 3 ω ρa n d C M F: t h e d e n sit y, a n d t h e pr ess ur e ar e c al c ul at e d at t h ee q u at ori al pl a n e. E ff e cts of t h e m a g n eti c fi el d ar e i) a d e cr e as e oft h e d e n sit y a n d pr ess ur e i n t h e c or e of t h e st ar a n d ii) a n i n cr e as eof t h e pr ess ur e at t h e s urf a c e. As a c o n s e q u e n c e, m att er is p u s h e do ut w ar d s a n d t h e r a di u s of t h e st ar i n cr e as es. F or µ = 5 × 1 0 3 1

A m 2 , t h e eff e ct is alr e a d y s e e n b ut it is still s m all. F or µ = 1 0 3 2

A m 2 , a d e cr e as e of t h e d e n sit y i n t h e c or e h as as a dir e ct c o n s e-q u e n c e: t h e s u p pr essi o n of t h e p o ssi bl e o n s et of n o n- n u cl e o ni cd e gr e es of fr e e d o m, h y p er o n s or a q u ar k p h as e, as alr e a d y dis-c u ss e d i n R ef s. ( Fr a n z o n et al. 2 0 1 5, 2 0 1 6 a; G o m es et al. 2 0 1 7).T his e x pl ai n s s o m e r es ults s h o w n i n Fi g. 1 f or t h e C M F m o d el:i n t h e pr es e n c e of a m a g n eti c fi el d, t h e o n s et of Λ s a n d q u ar km att er o c c ur i n st ar s wit h l ar g er m ass es.

L et u s n o w c o n si d er a st ar d es cri b e d b y t h e C M F m o d el wit hstr o n g m a g n eti c fi el d i n t h e i nt eri or t h at h as a gr a vit ati o n al ( orb ar y o ni c) m ass a b o v e t h e m a xi m u m m ass p o ssi bl e f or a n o n-m a g n etis e d or w e a kl y m a g n etis e d st ar. A dir e ct c o n s e q u e n c e oft h e i n st a bilit y o c c urri n g w h e n t h e m a g n eti c fi el d d e c a y s t o v al-u es t h at all o w f or hi g h er c e ntr al st ell ar d e n siti es, a n d a p o ssi bl e

o n s et of q u ar k m att er, is t h at t h e st ar b e c o m es u n st a bl e a n d d e-c a y s i nt o a l o w m ass bl a c k h ol e. T his tr a n siti o n c o ul d b e p o ssi bl yi d e nti fi e d b y t h e o b s er v ati o n of a g a m m a-r a y b ur st.

4. Fi n al R e m ar k s

I n t h e pr es e nt w or k, w e h a v e c al c ul at e d t h e m a g n eti c fi el dstr e n gt h a b o v e w hi c h t h e d ef or m ati o n of t h e n e utr o n st ar str u c-t ur e is at l e ast 2 %, w hi c h is alr e a d y n o n- n e gli gi bl e. We h a v ec o n si d er e d t hr e e r e pr es e nt ati v e r el ati visti c m e a n fi el d m o d els t od es cri b e t h e E o S of n e utr o n st ar s wit h di v er s e c o m p o siti o n s ( n u-cl e o ni c, h y p er o ni c a n d h y bri d). T h e c al c ul ati o n s f or t h e str u ct ur eof st ar s w er e p erf or m e d wit hi n t h e g e n er al r el ati visti c fr a m e-w or k i m pl e m e nt e d i n t h e p u bli cl y a v ail a bl e L a n g a g e O bj et p o url a R El ati vit é N u m éri q u E ( L O R E N E) li br ar y ( B o n a z z ol a et al.1 9 9 3; B o c q u et et al. 1 9 9 5).

O ur c al c ul ati o n w as u n d ert a k e n b y q u a ntif yi n g t h e d ef or m a-ti o n d u e t o t h e m a g n eti c fi el d, i n p arti c ul ar, t h e r el ati o n b et w e e nt h e p ol ar a n d e q u at ori al r a dii, a n d d e fi ni n g t h e m a g n eti c di p ol em o m e nt t h at c a u s es a di ff er e n c e b el o w ∼ 1 − 2 %, c orr es p o n d-i n g t o a m a g n eti c fi el d of t h e or d er of ∼ 1 0 1 7 G i n t h e c e nt era n d ∼ 5 × 1 0 1 6 G at t h e s urf a c e. It h as b e e n s h o w n t h at, wit hi nt h e a d o pt e d f or m alis m, t h e m a g n eti c fi el d d e v el o p e d i n si d e t h est ar, a n d at t h e s urf a c e, d e p e n d s o n t h e E o S. F or a fi x e d m a g-n eti c di p ol e m o m e nt, str o n g er m a g n eti c fi el d s ar e o bt ai n e d f or as oft er E o S. Q u a ntit ati v el y, w e h a v e f o u n d t h at, f or t h e m a g n eti cfi el d r a n g e st u di e d, t h e r el ati o n b et w e e n t h es e t w o q u a ntiti es t ob e B c ∼ 3 − 5 × B s, wit h t h e r el ati o n b ei n g d e p e n d e nt o n t h e E o Sa n d t h e st ell ar m ass.

T h e d et er mi n ati o n of t h e t hr es h ol d m a g n eti c fi el d f or d e-f or m e d n e utr o n st ar s w as u n d ert a k e n i n t w o e q ui v al e nt a p-pr o a c h es. I n t h e p ol ar v er s u s e q u at ori al r a dii m et h o d, w e h a v ei d e nti fi e d t h e m a g n eti c di p ol e m o m e nt t h at c a u s es a diff er e n c e∼ 1 − 2 % b et w e e n b ot h. I n t h e q u a dr u p ol e m o m e nt m et h o d, w eu s e t h e n e utr o n st ar q u a dr u p ol e m o m e nt t o q u a ntif y a d ef or m a-ti o n of a si mil ar m a g nit u d e. We fi n d t h at t h e li miti n g m a g n eti cfi el d f or w hi c h t h e i nt e gr ati o n of t h e T O V e q u ati o n s still gi v esr e alisti c r es ults f or t h e str u ct ur e of m a g n etis e d n e utr o n st ar s is,i n g e n er al, m o d el i n d e p e n d e nt.

We r e p ort t h at f or st ar s wit h m ass es b el o w 1 .5 M ⊙ , a di p ol em a g n eti c m o m e nt of ∼ 2 × 1 0 3 1 A m 2 , w hi c h c orr es p o n d s t o as urf a c e m a g n eti c fi el d of t h e or d er of ∼ 2 × 1 0 1 6 G a n d a c e ntr alm a g n eti c fi el d of ∼ 8 × 1 0 1 6 G, is e n o u g h t o c a u s e d ef or m ati o n


Gomes et al.: Limiting B−field for minimal deformation

above 1 − 2% on stars. The same effect is seen for all masses,when stars are described with a dipole magnetic moment of ∼5 × 1031 Am2, corresponding to a surface magnetic field of theorder of ∼ 5 × 1016 G and a central magnetic field of ∼ 2 − 4 ×1017 G. For the analysis carried out in this work, our results areindependent of the model and the composition of stars.

We have also discussed under which conditions the decay ofthe magnetic field could give rise to the collapse of the neutronstar or the onset of hyperons. Let us refer to the fact that the onsetof hyperons inside stars has important effects on the cooling ofthe star, not only by affecting the onset density of the nucleonicelectron direct Urca process, but also by opening new direct Urcachannels involving hyperons.

For a matter of completeness, we have investigated the mag-netic field distribution inside individual Mb = 1.6 M⊙ stars, bothin equatorial and radial directions, showing that for the case ofapproximately spherical stars, all the models and compositionspresent a quantitative and qualitative similar behavior. The mod-els discussed in this work present different baryon density distri-butions, which is a direct consequence of the degree of stiffnessof their EoS.

Finally, we have compared two methods for determining thegravitational wave amplitude for single stars, considering boththe case of approximately spherical stars (slightly deformed) andstars with induced deformation through magnetic effects. Weshow that, even for the lowest magnetic field that generates arelevant deformation, calculations for the gravitational wave am-plitude using these two methods never agree. The disagreementbetween the two methods increases for higher magnetic field am-plitudes, as the stars become more deformed. Therefore, we em-phasise that such calculations should be carried out cautiously,especially for highly magnetised neutron stars.

Acknowledgements

The authors acknowledge support from PHAROS COST Ac-tion CA16214 and by the FCT (Portugal) Projects No.UID/FIS/04564/2016 and POCI-01-0145-FEDER-029912. H.P.was supported by Fundação para a Ciência e Tecnologia (FCT-Portugal) under Project No. SFRH/BPD/95566/2013, and VD bythe National Science Foundation under grant PHY-1748621.

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