Very Magnetized White Dwarfs with Axisymmetric Magnetic Field and the Importance of the Electron Capture and Pycnonuclear Fusion Reactions for their Stability
Edson Otoniel, Bruno Franzon, Manuel Malheiro, Stefan Schramm, Fridolin Weber
aa r X i v : . [ a s t r o - ph . S R ] M a r Very Magnetized White Dwarfs with Axisymmetric Magnetic Field and theImportance of the Electron Capture and Pycnonuclear Fusion Reactions for theirStability
Edson Otoniel , , ∗ Bruno Franzon , † Manuel Malheiro , ‡ Stefan Schramm , § and Fridolin Weber ¶ Departamento de F´ısica, Instituto Tecnol´ogico de Aeron´autica, Pra¸ca Marechal Eduardo Gomes,50 - Vila das Acacias, 12228–900 S˜ao Jos´e dos Campos, SP, Brazil Frankfurt Institute for Advanced Studies, Ruth-Moufang-1 60438 Frankfurt am Main, Germany and Department of Physics, San Diego State University,5500 Campanile Drive, San Diego, California 92182, USA andCenter for Astrophysics and Space Sciences, University of California at San Diego, La Jolla, California 92093, USA (Dated: July 11, 2018)In this work, we study the properties of magnetized white dwarfs taking into account possibleinstabilities due to electron capture and pycnonuclear fusion reactions in the cores of such objects.The structure of white dwarfs is obtained by solving the Einstein-Maxwell equations with a poloidalmagnetic field in a fully general relativistic approach. The stellar interior is composed of a regularcrystal lattice made of carbon ions immersed in a degenerate relativistic electron gas. The onsets ofelectron capture reactions and pycnonuclear reactions are determined with and without magneticfields. We find that magnetized white dwarfs violate the standard Chandrasekhar mass limit sig-nificantly, even when electron capture and pycnonuclear fusion reactions are present in the stellarinterior. We obtain a maximum white dwarf mass of around 2 . M ⊙ for a central magnetic fieldof ∼ . × G, which indicates that magnetized white dwarfs may play a role for the inter-pretation of superluminous type Ia supernovae. Furthermore, we show that the critical density forpycnonuclear fusion reactions limits the central white dwarf density to 9 . × g/cm . As a re-sult, equatorial radii of white dwarfs cannot be smaller than ∼ ∼ G. In the latter case, the central density decreases (stellar radius increases) with centralmagnetic field strengths. ∗ [email protected] † franzon@fias.unifrankfurt.de ‡ [email protected] § schramm@fias.unifrankfurt.de ¶ [email protected] I. INTRODUCTION
It is generally accepted that stars born with masses below around 10 solar masses end up their evolutions as whitedwarfs (WDs) [1–3]. With a typical composition mostly made of carbon, oxygen, or helium, white dwarfs possesscentral densities up to ∼ g / cm . They can be very hot [4], fast rotating [5–7] and strongly magnetized [8–10].The observed surface magnetic fields range from 10 G to 10 G [11–16]. The internal magnetic fields of white dwarfsare not known, but they are expected to be larger than their surface magnetic fields. This is due to the fact that inideal magneto hydrodynamics (MHD), the magnetic field, B , is ‘frozen-in’ with the fluid and B ∝ ρ , with ρ being thelocal mass density (see, e.g., Refs. [17, 18]). A simple estimate of the internal magnetic field strength follows from thevirial theorem by equating the magnetic field energy with the gravitational binding energy, which leads to an upperlimit for the magnetic fields inside WDs of about ∼ G. On the other hand, analytic and numeric calculations,both in Newtonian theory as well as in General Relativity theory, show that WDs may have internal magnetic fieldsas large as 10 − G (see, e.g., Refs. [2, 16, 19–24]).The relationship between the gravitational stellar mass, M , and the radius, R , of non-magnetized white dwarfswas first determined by Chandrasekhar [25]. Recently, mass-radius relationships of magnetic white dwarfs have beendiscussed in the literature (see, e.g., Refs. [20, 22, 26]). These studies show that the masses of white dwarfs increasein the presence of strong magnetic fields. This is due to the Lorentz force, which acts against gravity, thereforesupporting stars with higher masses.Based on recent observations of several superluminous type Ia supernovae (SN 2006gz, SN 2007if, SN 2009dc,SN 2003fg) [27–33], it has been suggested that the progenitor masses of such supernovae significantly exceed theChandrasekhar mass limit of M Ch ∼ . M ⊙ [34]. Super-heavy progenitors were studied as a result of mergers of twomassive white dwarfs [35–37]. Alternatively, the authors of Ref. [38] obtained super-Chandrasekhar white dwarfs formagnetically charged stars. In addition, super-Chandrasekhar white dwarfs were investigated in the presence of strongmagnetic fields in Refs. [39]. In Refs. [40, 41], WDs models with magnetic fields were calculated in the frameworkof Newtonian physics. A recent study of differential rotating, magnetized white dwarfs has shown that differentialrotation might increase the mass of magnetized white dwarfs up to 3.1 M ⊙ [42]. Also, as shown in Ref. [43], purelytoroidal magnetic field components can increase the masses of white dwarfs up to 5 M ⊙ .According to Refs. [44], effects of an extremely large and uniform magnetic field on the equation of state (EOS) ofa white dwarf could increase its critical mass up to 2 . M ⊙ . This mass limit is reached for extremely large magneticfields of ∼ G. Nevertheless, as already discussed in Refs. [45, 46], the breaking of spherical symmetry due tomagnetic fields and micro-physical effects, such as electron capture reactions and pycnonuclear reactions, can severelylimit the magnetic field inside white dwarfs.In Ref. [22], mass-radius relationships of highly magnetized white dwarfs were computed using a pure degenerateelectron Fermi gas. However, according to Ref. [47], many-body corrections modify the EOS and, therefore, themass-radius relationship of white dwarfs. The purpose of our paper is two-fold. Firstly, we model white dwarfs usinga model for the equation of state which takes into account not only the electron Fermi gas contribution, but also thecontribution from electron-ion interactions [48]. Secondly, we perform a stability analysis of the matter in the coresof white dwarfs against electron capture and pycnonuclear fusion reactions. The Landau energy levels of electrons aremodified by relativistic effects if the magnetic field strength is higher than the critical QED magnetic field strengthof B cr = 4 . × G. However, as already shown in Ref. [49], the global properties of white dwarfs, such as massesand the radii, are nearly independent of Landau quantization. For this reason, we do not take into account magneticfields effects in the equation of state to calculate the global properties of WD’s.Our paper is organized as follows. In Sec. II, we discuss the stellar interior of white dwarfs and details of theequation of state used in our study to model white dwarfs. This is followed, in Sec. III, by a brief discussion of theequations that are being solved numerically to obtain the structure of stationary magnetized white dwarfs. In Sec.IV, we briefly discuss the Einstein-Maxwell tensor and the metric tensor used to solve Einstein’s field equations ofGeneral Relativity. The results of our study are discussed in Sec. V and summarized in Sec. VI.
II. STELLAR INTERIOR
The properties of fermionic matter have been studied many decades ago in Refs. [47, 50]. Typically, a white dwarfis composed of atomic nuclei immersed in a fully ionized electron gas. In this work, we make use of the latestexperimental atomic mass data [51, 52] used to determine the equation of state. Modifications of the equation ofstate due to the interactions between electrons and atomic nuclei are taken into account too. The model adoptedto describe the nuclear lattice was derived for the outer crust of a neutron star in Refs. [46, 53] and later applied toWDs in Ref. [54]. According to Ref. [54], the cores of white dwarfs are subjected to the degenerate electron and ionic ρ [g/cm ] P [ dyn e s / c m ] B = 0B = 0 (bcc ; lattice)
FIG. 1. (Color online) Equation of state for B = 0 with (red curve) and without (black curve) lattice contributions. lattice pressures. The total pressure is then given by P = P e + P L ( Z, Z ′ ) , (1)where P e denotes the electron pressure, determined in [47], and P L ( Z, Z ′ ) is the lattice pressure for two different typeof ions. The lattice pressure is given by the energy density of the ionic lattice (see Ref. [53]), P L ( Z, Z ′ ) = 13 E L , (2)with Z and Z ′ being the proton number of two different ions. In our case, the white dwarf is composed of carbonions, i.e., Z ′ = Z =12. Following the Bohr-van Leeuwen theorem [53], the lattice pressure of ions arranged in a regularbody-centered-cubic (bcc) crystal does not depend on the magnetic field, apart from a small contribution due to thequantum zero-point motion of ions. In this case, the lattice energy density reads [46] E L = Ce n / e G ( Z, Z ′ ) , (3)with G ( Z, Z ′ ) given by G ( Z, Z ′ ) = αZ + γZ ′ + (1 − α − γ ) ZZ ′ ( ξZ + (1 − ξ ) Z ′ ) / . (4)The quantities C , α , γ are lattice constants and ξ is the ratio of ions AZ Y and A ′ Z ′ Y in the lattice [54] (see also TableI). If only a single ion is present in the lattice, Eq. 4 does not depend on α and γ so that Eq. (3) becomes E L = Ce n / e Z / , (5)The energy density is given in terms of the degenerate electron energy, the energy density of the ions, and theenergy density of the ionic lattice, E = n x M ( Z, A ) c + n x ′ M ( Z ′ , A ′ ) c + E e + E L − n e m e c , (6)where n x and n ′ x are the number densities of atomic nuclei with masses M ( Z, A ) and M ( Z ′ , A ′ ), respectively. Asalready mentioned above, here we adopt the most recent experimental values for M (see Refs. [51, 52]).Figure 1 shows the impact of lattice contributions on the white dwarf equation of state studied in this paper. Theblack lines (no lattice contribution) and red dashed line (with lattice contribution) are for white dwarf matter withzero magnetic field ( B = 0). One sees that adding the lattice contribution to the equation of state lowers the pressuresomewhat, which in turn makes white dwarfs less massive. It also follows from this figure that the presence of latticecontributions reduces the radii of white dwarfs (renders them more compact) with comparable central pressures. TABLE I. Lattice constants C , α , γ and parameters (1 − α − γ ) and ξ for a body-centered-cubic (bcc) structure, as obtainedby the method of Coldwell-Horsfall and Maradudin (see Ref. [54]) for more details.Lattice C α γ (1 − α − γ ) ξ bcc -1.444231 0.389821 0.389821 0.220358 0.5 III. INSTABILITIES IN STRONGLY MAGNETIZED WHITE DWARFSA. Inverse β decay As shown in Refs. [54, 55], the matter inside of white dwarfs is unstable due to inverse β -decay, A ( N, Z ) + e − → A ( N + 1 , Z −
1) + ν e . Because of this reaction, atomic nuclei become more neutron rich and the energy density of the matter is being reduced,at a given pressure, leading to a softer EoS. Using the thermodynamic relation (at zero temperature) E e + P e = n e µ e ,one obtains the Gibbs free energy, g , per nucleon as g ( Z, z ′ ) = mc + ξξA + (1 − ξ ) A ′ ∆( A, Z ) + (1 − ξ ) ξA + (1 − ξ ) A ′ ∆( A ′ , Z ′ ) + γ e (cid:20) µ e + m e c + 43 E L n e (cid:21) , (7)with m being the neutron mass and ∆( A, Z ) denoting the excess mass of nuclei, which, for magnetic field strengths < G, is independent of the magnetic field, see, e.g., Ref [46]. For γ e = ¯ Z/A we have ¯ Z = ξZ + (1 − ξ ) Z ′ and¯ A = ξA + (1 − ξ ) A ′ , with µ e being the electron chemical potential. Inverse β -decay reactions are believed to occur inthe cores of white dwarfs if the condition [54] g ( Z, Z ′ ) ≥ g ( Z − ∆ Z, Z ′ − ∆ Z ′ ) (8)is fulfilled, where g ( Z, Z ′ ) and g ( Z − ∆ Z, Z ′ − ∆ Z ′ ) follow from Eq. (7) and the possible choices for ∆ Z and ∆ Z ′ are∆ Z = 1 & ∆ Z ′ = 0, ∆ Z = 0 & ∆ Z ′ = 1, and ∆ Z ′ = 1 & ∆ Z ′ = 1.From the inequality (8), we obtain the following relation∆ ¯ Z (cid:20) µ e + 43 Ce n / e ∆( ¯ ZG ( Z, Z ′ )) (cid:21) ≥ ¯ µ βe (9)with the electron number density n e and mass density ρ of a magnetized electron gas given respectively by n e = 2 B ⋆ (2 π ) λ X ν g ν q x F − − νB ⋆ . (10) ρ = 1 γ e mn e . (11)where only the ground-state Landau level ν = 0 is occupied, ν max = 1. For two occupied levels, ν = 0 and ν = 1,one has ν max = 2, and similarly for the higher levels. The quantities x F in Eq. (11) and ¯ µ βe in Eq. (9) are defined as x F ≡ p F /m e c and ¯ µ βe = ξµ βe ( A, Z ) + (1 − ξ ) µ βe ( A ′ , Z ′ ) , (12)with µ βe ( A, Z ) and µ βe ( A ′ , Z ′ ) given by µ βe ( A, Z ) ≡ ∆( A, Z − ∆ Z ) − ∆( A, Z ) + m e c (13) µ βe ( A ′ , Z ′ ) ≡ ∆( A ′ , Z ′ − ∆ Z ′ ) − ∆( A ′ , Z ′ ) + m e c . (14)Another important quantity is ∆ ¯ ZG ( Z, Z ′ ), which describes the difference of G , defined in Eq. (4), before and afteran inverse β decay reaction. It is given by∆( ¯ ZG ( Z, Z ′ )) = G ( Z, Z ′ ) − G ( Z − ∆ , Z ′ − ∆) . (15)For an electron gas consisting of only one type of ion, we have∆( ¯ ZG ( Z, Z )) = Z / − ( Z − / − Z / . (16) B / B c ρ [ g / c m ] C Only the ground state occupied
FIG. 2. Mass density thresholds for the onset of electron capture as a function of magnetic field strength (in units of the criticalmagnetic field, B c ), computed from Eq. (9) for matter made of only carbon ions. In the limit where only the ground state ( ν = 0) is fully occupied by electrons, one has n e = n eB ∝ B / ⋆ , where B ⋆ = B/B c with B c = 4 . × G being the critical magnetic field (see Ref. [56] for more details about n eB ). The chemical potential of the electrons in this case is given by µ e ≈ π m e c λ e n eB B ⋆ , (17)where λ e = ~ /m e c denotes the Compton wavelength of electrons. In Ref. [54] it was estimated that the maximummagnetic field inside of white dwarfs, before the onset of β -inverse reactions, is given by B β⋆ ≈ (cid:18) ¯ µ βe ( A, Z ) m e c ∆ ¯ Z (cid:19) " (cid:18) π (cid:19) / Cα ZG ( Z, Z ′ )) − , (18)with α = e / ( ~ c ) the fine structure constant. We note that because of the second term on the right-hand-side ofEq. (18), which originates from lattice contributions, the maximum value of B β⋆ increases if lattice contributions aretaking into account.In Fig. 2 we show the numerical solution of Eq. (9) for white dwarf matter made of only carbon ions immersed in amagnetized electron gas. The oscillatory behavior is caused by the Landau level contributions to the number density,given by Eq. (11). For high values of B with only the ground state occupied, the dependence of density on B becomeslinear, as can seen in Fig. 2. B. Pycnonuclear reaction
In this section, we will focus on nuclear fusion reactions (pycnonuclear fusion reactions) among heavy atomicnuclei , AZ Y , schematically expressed as AZ Y + AZ Y → A Z Y . An example of such a reaction is carbon on carbon, C + C. Pycnonuclear reactions have been found to occur over a significant range of stellar densities (see, forinstance, Ref. [57]), including the density range found in the interiors of white dwarfs [46, 54]. The nuclear fusion ratesat which pycnonuclear reactions proceed, however, are highly uncertain because of some poorly constrained parameters(see Ref. [57, 58]). The reaction rates have been calculated for different models. In Ref. [57], the pycnonuclear reactionrates are defined as R pyc = n i S ( E pk ) ~ mZ e P pyc F pyc (19)where S ( E pk ) is the astrophysical S-factor used in Ref. [57] for the NL2 nuclear model parametrization. Accordingto Ref. [57], an analytic equation for the S-factor is given by S ( E pk ) = 5 . × exp " − . E pk − E . e . − E pk ) , (20)where S ( E pk ) is in units of MeV barn. The factors P pyc and F pyc in Eq. (19) are given by P pyc = exp (cid:16) − C exp / √ λ (cid:17) , (21) F pyc = 8 C pyc . /λ C pl , (22)with C exp , C pyc and C pl are dimensionless parameters for a regular bcc-type crystal lattice (see at zero temperature).Their values are listed in Table II. TABLE II. Coefficients C exp , C pyc , C pl related to pycnonuclear reaction rates at zero temperature, computed for nuclear modelNL2 (see Refs. [59, 60]). Model C exp C pyc C pl bcc; static lattice 2.638 3.90 1.25 The inverse-length parameter λ in Eq. (21) and Eq. (22) has the form Refs. [57, 58] λ = ~ mZ e (cid:16) n i (cid:17) / = 1 AZ (cid:18) A ρX i . × g cm − (cid:19) / . (23)For number densities ρ less than neutron drip density one has X i = 1 [57] and the pycnonuclear reaction rates aregiven by R pyc = ρX i AZ S ( E pk ) C pyc λ − C pl exp (cid:16) − C exp / √ λ (cid:17) , (24)with R pyc given in units of cm − s − . The zero-point oscillation energy E pk of C nuclei at ρ = 10 g/cm is givenby Ref. [2] E pk = ~ ω = ~ (cid:18) πe Z ρA M (cid:19) / . (25)The time it takes for the complete fusion of atomic nuclei of mass Am is obtained from [6, 57] τ pyc = n x R pyc = ρ Am R pyc . (26)As already mentioned above, the reaction rates are rather uncertain, and the analytic astrophysical S-factor has anuncertainly of ∼ IV. WHITE DWARFS WITH AXISYMMETRIC MAGNETIC FIELDS
The numerical technique used in this work to study axisymmetric magnetic fields was first applied to neutron starsin Refs. [61, 62], and more recently in Ref. [63–65]. The same formalism was used to study rotating and magnetized
Log ρ [g cm -3 ] -40-2002040 L og R py c [ c m - s - ] C + C = MgT = 0 t burn = 10 Gyrst burn = 0.1 Myrst burn = 0.1 s t burn = 10 yrs Log ρ [g cm -3 ] -10-5051015 L og ( τ py c / y ea r s ) C + C FIG. 3. Left: pycnonuclear fusion reaction rates for carbon burning at zero temperature as functions of mass density, fornuclear model NL2 and a bcc crystal lattice. Right: pycnonuclear reaction time scales at zero temperature for C+C fusion asa function of mass density. The S-factor is given by Eq. (20) and the zero-point oscillation energy is E pk ∼ .
034 MeV. white dwarfs in Ref. [22]. Here we build stellar equilibrium configurations by solving the Einstein-Maxwell fieldequations in a fully general relativistic approach. For more details about the theoretical formalism and numericalprocedure, see, for instance, Ref. [66]. Below we show the basic electromagnetic equations which, combined withthe gravitational equations, are solved numerically by means of a spectral method. In this context, the stress-energytensor T αβ is composed of the matter and the electromagnetic source terms, T αβ = ( e + p ) u α u β + pg αβ + 1 µ (cid:18) F αµ F µβ − F µν F µν g αβ (cid:19) . (27)Here F αµ is the antisymmetric Faraday tensor defined as F αµ = ∂ α A µ − ∂ µ A α , with A µ denoting the electromagneticfour-potential A µ = ( A t , , , A φ ). The total energy density of the system is e , the pressure is denoted by p , u α is the fluid 4-velocity, and the metric tensor is g αβ . The first term in Eq. (27) represents the isotropic (ideal)matter contribution to the energy momentum-tensor, while the second term is the anisotropic electromagnetic fieldcontribution.The metric tensor in axisymmetric spherical-like coordinates ( r, θ, φ ) can be read of from the line element ds = − N dt + Ψ r sin θ ( dφ − N φ dt ) + λ ( dr + r dθ ) , (28)where N , N φ , Ψ and λ are functions of the coordinates ( r, θ ) [61]. As in Ref. [61], the equation of motion for a starendowed with magnetic fields reads H ( r, θ ) + ν ( r, θ ) + M ( r, θ ) = const , (29)where H ( r, θ ) is the heat function defined in terms of the baryon number density n , H = Z n e ( n ) + p ( n ) dPdn ( n ) dn . (30)The quantity ν ( r, θ ) in Eq. (29) is defined as ν = ln N , and the magnetic potential M ( r, θ ) is given by M ( r, θ ) = M ( A φ ( r, θ )) ≡ − Z A φ ( r,θ ) f ( x ) dx , (31)where f ( x ) denotes the current function. Magnetic stellar models are obtained by assuming a constant value, f , forthe latter [64]. According to Ref. [62], other choices for f ( x ) are possible, but the general conclusions as presented inthis work remain the same. The constant current function is a standard way to generate self-consistently a dipolarmagnetic field throughout the star. ρ [g/cm ] M / M s un B = 0Pycnonuclear reaction β -Inversef = 100f = 200f = 500f = 700f = 1000f = 1500 µ = 5.0 x 10 Am µ = 1.0 x 10 Am µ = 2.0 x 10 Am µ = 3.0 x 10 Am µ = 4.0 x 10 Am P y c nonu c l ea r r eac ti on β -I nv e r s e FIG. 4. (Color online) Gravitational mass as a function of central mass density for magnetized white dwarfs, for different valuesfor the current function, f , and magnetic dipole moment, µ . Stars located in the colored areas are subject to pycnonuclearreactions and inverse β -decay. The threshold of these reactions are shown in Table III. The solid square and triangle mark thedensities at which pycnonuclear and inverse β -decay reactions set in, respectively. V. RESULTS
In this section, we discuss the effects of strong magnetic fields on the global properties of stationary white dwarfstaking into account instabilities due to inverse β -decay and pycnonuclear fusion reactions in their cores. In addition,we make use of an equation of state for white dwarf matter that accounts for electron-ion interactions and is computedfor the latest experimental atomic mass data. The instabilities related to the microphysics are fundamental since theyput constraints on the equilibrium configurations and also limit the maximum magnetic fields which these stars canhave [46]. In addition to the magnetic profiles, which have already been computed in Ref. [22], we also computestellar models at constant magnetic dipole moments µ . In Ref. [22], a simple Fermi gas model was used to modelwhite dwarfs, and the microphysical issues were not addressed. In our study, the maximum white dwarf mass fornon-magnetized stars is smaller than the one considered in Ref. [22], since the lattice contribution softens the EOS.In Fig. 4, we show the gravitational mass versus central density of white dwarf sequences computed for different(fixed) magnetic dipole moments, µ , and current functions, f . The magnetic dipole moment is defined as (seeRef. [61]) 2 µ cos θr = B ( r ) | r →∞ , (32)which is the radial (orthonormal) component of the magnetic field of a magnetic dipole seen by an observer at infinity.As can be seen from Fig. 4, a larger magnetic moment µ leads first to an increase in the white dwarf maximum mass.However, if we increase µ further, the maximum mass begins to drop. This is due to the fact that the stellar radiusbecomes larger (see also Fig. 5), which reduces the magnetic field (see Eq. 32). As a consequence, the Lorentz forcebecomes smaller, rendering the maximum mass configurations less massive.As can be seen in Fig. 4, the masses of magnetized white dwarfs increase monotonically with central density. Thebehavior is very different if the value of the current function is kept constant, in which case non-monotonic (in somecases even multivalued) mass-density relationships are obtained. The cross-hatched area in Fig. 4 shows the densityregime where pycnonuclear fusion reactions become possible. The position of the white dwarf with just the rightthreshold density (9 . × g/cm ) for this reaction to occur is marked with a solid black square in Fig. 4. The R [km] M / M s un B = 0B max = 4.27 x 10 G , µ = 5.0 x 10 Am B max = 1.54 x 10 G , µ = 1.0 x 10 Am B max = 3.85 x 10 G , µ = 2.0 x 10 Am B max = 1.74 x 10 G , µ = 3.0 x 10 Am B max = 8.83 x 10 G , µ = 4.0 x 10 Am Chandrasekhar limit
M = 2.14 M sun
Pycnonuclear and β -Inverse area FIG. 5. (Color online) Mass-radius relationship of magnetized white dwarfs for different magnetic dipole moments, µ . The blackline represents the mass-radius relationship of non-magnetic white dwarfs. The horizontal line represents the Chandrasekharmass limit for spherical stars. Also shown are the values of the magnetic field B max (together with the corresponding magneticdipole moment µ ) at the centers of the respective maximum mass stars (end points of each curve with fixed µ ). White dwarfslocated in the colored (upper left) corner are subject to pycnonuclear fusion ( τ pyc = 10 Gyrs) or inverse β -decay reactions. pycnonuclear reaction time at that density is 10 Gyrs. For a central white dwarf density of 1 . × g/cm thefusion reaction time decreases to 0.1 Myrs (see Fig. 3). White dwarfs subject to inverse β -decay reactions in theircores are located in the yellow area (marked “ β -inverse”) of Fig. 4. The most massive stable white dwarf which isnot subject to microscopic instability reactions in its core (end point of the curve with µ = 2 × Am ), has a massof ∼ . M ⊙ and a radius (see Fig. 5) of ∼ dM/dρ c > TABLE III. Thresholds of inverse β -reactions and pycnonuclear fusion reactions (pycnonuclear reaction time of 10 Gyrs) incarbon white dwarfs for different magnetic fields, B , and magnetic dipole moments, µ . C µ (Am ) B max (G) ρ pyc (g/cm ) ρ β (g/cm )5 . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × . × The mass-radius relationship of magnetized white dwarfs for different magnetic dipole moments µ is shown in Fig. 5.One sees that increasing values of µ lead to white dwarfs with larger radii, because of the added magnetic field energy.The strength of the magnetic field can be inferred from Fig. 6, which shows the gravitational mass as a function ofsurface ( B s ) and central ( B c ) magnetic fields, the circumferential equatorial radius ( R circ ), and the baryon numberdensity ( n b ), for two sample magnetic dipole moments of µ = 0 . × Am (red line) and µ = 4 . × Am (orange line).In Fig. 6 (top panels), one sees that the curves with µ = 0 . × Am and µ = 4 . × Am cross each other.This is due to the fact that the magnetic field scales as ∼ µ/r , with r being the stellar radius (see Eq. (32)). The0 s [ x 10 G] M / M s un c [ x 10 G] R [km] M / M s un n b [1 / cm ] M b = 1.8 M sun M b = 1.8 M sun M b = 1.0 M sun M b = 1.8 M sun M b = 1.8 M sun M b = 1.0 M sun M b = 1.0 M sun M b = 1.0 M sun FIG. 6. (Color online) Global properties of magnetized white dwarfs for two different (sample) magnetic dipole moments, µ = 0 . × Am (red line) and µ = 4 . × Am (orange line). M denotes the gravitational mass, B S the magnetic fieldat the surface, B c the magnetic field at the center, R the equatorial radius, and n b the baryon number density. The horizontallines represent white dwarfs with fixed baryon masses of M B = 1 . M ⊙ (bottom), and M B = 1 . M ⊙ (top). The arrowsindicate the paths of these white dwarfs in case of a magnetic field reduction (see text for details). locations of stars with fixed baryon masses of M B = 1 . M ⊙ and M B = 1 . M ⊙ are shown in Fig. 6 by dashedhorizontal lines. According to Eq. (32), the magnetic field is determined by the size of the star along the curves with µ = const. However, along the lines with fixed baryon masses, the strength of the magnetic field is a combination ofthe magnetic dipole moment µ and the stellar radius r .Next, we discuss the behavior of the magnetic dipole moments of white dwarfs whose magnetic fields are weakening.From the M versus B s and M versus B c relationships shown in Fig. 6 (top panels), one sees that two different scenariosare possible, depending on the star mass and on the magnetic field strengths of white dwarfs. If located above thecrossing point of the µ = 0 . × Am (red line) and µ = 4 . × Am (orange line) curve, white dwarfs withweakening magnetic fields would be evolving from right to left in the two upper panels of Fig. 6, as shown (backarrow) for a white dwarf with a constant baryon number of M B = 1 . M ⊙ . The magnetic dipole moment of suchwhite dwarfs would increase, from µ = 0 . × Am to µ = 4 . × Am for the sample star shown in Fig. 6. Thisis accompanied by an increase in the stellar radius (see M versus R diagram) and a decrease in the central baryondensity (see M versus n b diagram). The situation is reversed for white dwarfs located below the the crossing. Forsuch white dwarfs, a reduction of the magnetic field is accompanied by a decrease of the magnetic dipole moment,as shown in Fig. 6 for a sample white dwarf with a constant baryon mass of M B = 1 . M ⊙ (black arrows). In thiscase, white dwarfs become smaller and therefore more dense at the center (see M versus R and M versus n b diagramsshown in Fig. 6).As discussed just above (Fig. 6), the equatorial radii of white dwarfs located above the crossing point increase astheir magnetic fields are getting smaller. The increases in radius (at a fixed baryon mass) is due to the Lorentz force.However, the stellar magnetic field scales as µ/r . This means that for a star with a mass of M B = 1 . M ⊙ , theincrease in the magnetic dipole moment, µ , is canceled by the increase in the radius, reducing the magnetic field. Thisis the opposite of what is expected for stars with lower masses. For example, a star with M B = 1 . M ⊙ decreases itsmagnetic dipole moment and its radius. However, in this case, the decrease in the radius is not enough to cancel thereduction in µ . The net result is a decrease of the magnetic field. This can be understood by looking at the variationin the circular equatorial radius of the stars with M B = 1 . M ⊙ and M B = 1 . M ⊙ . For the latter, the change1
100 200 300 400B c [ x 10 G]05e+341e+351.5e+352e+35 n b [ / c m ] n b [ / c m ] c [ x 10 G]1e+311.5e+312e+312.5e+31 n b [ / c m ] n b [ / c m ] M b = 1.8 M sun M b = 1.8 M sun M b = 1.0 M sun M b = 1.0 M sun FIG. 7. Central baryon number density, n b , as a function of central magnetic field strength, B c , and equatorial radius, R , ofmagnetized white dwarfs with fixed baryon masses of M B = 1 . M ⊙ and M B = 1 . M ⊙ . The arrows refer to changes in n b and R for weakening magnetic fields. in radius is much smaller than the radial change for the M B = 1 . M ⊙ star, for a change in the magnetic dipolemoment of | ∆ µ | = 3 . × Am .In Fig. 7, we show the global properties of two white dwarfs with fixed baryon masses of M B = 1 . M ⊙ and M B = 1 . M ⊙ . The top panels show the central baryon density as a function of the central magnetic field (top-leftleft panel) and the circular equatorial radius (top-right panel) for a white dwarf with M B = 1 . M ⊙ . For suchstars, as the magnetic field decreases, the central baryon density becomes smaller due to the fact that the radius isincreasing. On the other hand, for lighter white dwarfs, with a mass of M B = 1 . M ⊙ , the central baryon numberdensity increases as the magnetic field decreases, since the stellar radius is getting smaller. VI. SUMMARY
In this work, we presented axisymmetric and stationary models of magnetized white dwarfs obtained by solving theEinstein-Maxwell equations self-consistently and taking into stability considerations related to neutronization due toelectron capture reactions as well as pycnonuclear fusion reactions among carbon nuclei in the cores of white dwarfs.We investigated also the influence of magnetic fields on the structure of white dwarfs. This is an important problem,since super-massive magnetized WD’s, whose existence is partially supported by magnetic forces, could simplify theexplanation of observed ultra-luminous explosions of supernovae Type Ia. The Lorentz force induced by strongmagnetic fields breaks the spherical symmetry of stars and increases their masses, since the force acts in the radialoutward direction against the inwardly directed gravitational pull.In this paper, we make use of an equation of state for a degenerate electron gas with electron-ion interactions (body-centered-cubic lattice structure) to describe the matter inside of white dwarfs. We have shown that the equation ofstate becomes softer if nuclear lattice contributions are included in addition to the electron pressure. This is due tothe fact that the repulsive force between electrons is smaller in the presence of an ionic lattice, causing a softening ofthe equation of state (see Fig. 1). We note that the density thresholds for pycnonuclear fusion reactions and inverse β -reactions are reduced when magnetic fields are present in the stellar interior, as can be seen in Table III.We have shown that the masses of white dwarfs increase up to M = 2 . M ⊙ (with a corresponding magneticdipole moment of µ = 2 . × Am (see, e.g., Fig. 4) if microphysical instabilities are considered. This star has anequatorial radius of ∼ B c = 3 . × G and B s = 7 . × G at the center andat the stellar surface, respectively. For this white dwarf, the ratio between the magnetic pressures and the matter2pressure at the center is 0.789. Although the surface magnetic fields obtained here are higher than the observed onesfor white dwarfs, these figures provide an idea of the maximum possible magnetic field strength that can be reachedinside of these objects, and may also be used to assess the effects of strong magnetic fields on both the microphysicsand the global structure of magnetized white stars.The maximum magnetic field found in this work is an order of magnitude smaller than that of Ref. [22]. This isbecause we modeled the stellar interior with a more realistic equation of state than just a simple electron gas, andwe considered the density threshold for pycnonuclear fusion reactions for a 10 Gyrs fusion reaction time scale, whichrestricts the central density of white dwarfs to ∼ . × g/cm (see Table III), limiting the stellar masses and,therefore, their radii, which for very massive and magnetized white dwarfs cannot be smaller than R ∼ B s , is about one order of magnitude smaller than the magneticfield reached at the stellar center, B c . If the magnetic field weakens for massive white dwarfs, we found that themagnetic dipole moments of such stars may increase (Fig. 6), which is due to the fact that, for a fixed baryon mass,the magnetic field is determined by the interplay between the magnetic dipole moment and the stellar radius. Thesituation is reversed for less massive white dwarfs, for which smaller the magnetic fields imply smaller stellar magneticdipole moments. The radii of massive (light) white dwarfs are found to increase (decrease) for decreasing centralmagnetic fields (Fig. 7). This opens up the possibility that massive white dwarfs, with central magnetic fields greaterthan B ∼ G, increase their magnetic fields through continued compression. This phenomenology differs fromprevious studies carried out for magnetic fields less than ∼ G [26, 41], where an increase of the central magneticfield was found to make stars less dense and therefore bigger in size.We note that stellar configurations which contain only poloidal magnetic fields (no toroidal component) are unstable(see, e.g., [67–69]). Moreover, according to Ref. [70], many different mechanisms can affect the magnetic fields andtheir distributions inside of white dwarfs. In this work, in the framework of a fully general relativistic treatment, wemodel the properties of magnetized white dwarfs with purely poloidal magnetic field components. Although this is notthe most general magnetic field profile, and a dynamical stability of these stars still needs to be addressed, magneticfields considerably increase the masses of white dwarfs, even when microphysical instabilities are considered. As aconsequence, such white dwarfs ought to be considered as possible candidates of super-Chandrasekhar white dwarfs,thereby contributing to our understanding of superluminous type-Ia supernovae.Lastly, we note that for a typical magnetic field value of ∼ G and a density of ∼ g/cm , we obtain anAlfven velocity of v = 10 cm/s, which, for a white dwarf with a typical radius of R = 1500 km, leads to an Alfvencrossing time of ∼ . VII. ACKNOWLEDGMENTS
We acknowledge financial support from the Brazilian agencies CAPES, CNPq, and we would like to thank FAPESPfor financial support under the thematic project 13/26258-4 B. Franzon acknowledges support from CNPq/Brazil,DAAD and HGS-HIRe for FAIR. S. Schramm acknowledges support from the HIC for FAIR LOEWE program. F.Weber is supported by the National Science Foundation (USA) under Grant PHY-1411708. [1] F. Weber,
Pulsars as astrophysical laboratories for nuclear and particle physics (CRC Press, 1999).[2] S. L. Shapiro and S. A. Teukolsky,
Black holes, white dwarfs and neutron stars: the physics of compact objects (2008).[3] N. K. Glendenning,
Compact stars: Nuclear physics, particle physics and general relativity (Springer Science & BusinessMedia, 2012).[4] L. G. Althaus, J. A. Panei, M. M. M. Bertolami, E. Garca-Berro, A. H. Crsico, A. D. Romero, S. Kepler, and R. D.Rohrmann, The Astrophysical Journal , 1605 (2009).[5] G. Arutyunyan, D. Sedrakyan, and ´E. Chubaryan, Soviet Astronomy , 390 (1971).[6] K. Boshkayev, J. A. Rueda, R. Ruffini, and I. Siutsou, The Astrophysical Journal , 117 (2013). [7] J. B. Hartle, The Astrophysical Journal , 1005 (1967).[8] J. G. Coelho, R. M. Marinho, M. Malheiro, R. Negreiros, D. L. C´aceres, J. A. Rueda, and R. Ruffini, apj , 86 (2014),arXiv:1306.4658 [astro-ph.SR].[9] R. V. Lobato, M. Malheiro, and J. G. Coelho, International Journal of Modern Physics D , 007 (2016).[11] Y. Terada, T. Hayashi, M. Ishida, K. Mukai, T. u. Dotani, S. Okada, R. Nakamura, S. Naik, A. Bamba, and K. Makishima,Publ. Astron. Soc. Jap. , 387 (2008), arXiv:0711.2716 [astro-ph].[12] D. Reimers, S. Jordan, D. Koester, N. Bade, T. Kohler, and L. Wisotzki, Astron. Astrophys. , 572 (1996),arXiv:astro-ph/9604104 [astro-ph].[13] G. D. Schmidt and P. S. Smith, Astrophys. J. , 305 (1995).[14] J. C. Kemp, J. B. Swedlund, J. D. Landstreet, and J. R. P. Angel, Astrophys. J. , L77 (1970).[15] A. Putney, The Astrophysical Journal Letters , L67 (1995).[16] J. Angel, Annual Review of Astronomy and Astrophysics , 487 (1978).[17] L. Mestel, Stellar magnetism , Vol. 154 (OUP Oxford, 2012).[18] L. D. Landau, E. M. Lifshitz, J. B. Sykes, J. S. Bell, and M. E. Rose, Physics Today , 56 (1958).[19] U. Das and B. Mukhopadhyay, Journal of Cosmology and Astroparticle Physics , 050 (2014).[20] P. Bera and D. Bhattacharya, Monthly Notices of the Royal Astronomical Society , 3951 (2014).[21] P. Bera and D. Bhattacharya, Monthly Notices of the Royal Astronomical Society , 3375 (2016).[22] B. Franzon and S. Schramm, Phys. Rev. D92 , 083006 (2015), arXiv:1507.05557 [astro-ph.SR].[23] B. Franzon and S. Schramm, (2016), 10.1093/mnras/stx397, [Mon. Not. Roy. Astron. Soc.467,4484(2017)],arXiv:1609.00493 [astro-ph.SR].[24] U. Das and B. Mukhopadhyay, Journal of Cosmology and Astroparticle Physics , 016 (2015).[25] S. Chandrasekhar,
An Introduction to the Study of Stellar Structure (Chicago : Univ. Chicago Press, 1939).[26] I.-S. Suh and G. Mathews, The Astrophysical Journal , 949 (2000).[27] J. M. Silverman, M. Ganeshalingam, W. Li, A. V. Filippenko, A. A. Miller, and D. Poznanski, Monthly Notices of theRoyal Astronomical Society , 585 (2011).[28] R. A. Scalzo et al. , Astrophys. J. , 1073 (2010), arXiv:1003.2217 [astro-ph.CO].[29] D. A. Howell et al. (SNLS), Nature , 308 (2006), arXiv:astro-ph/0609616 [astro-ph].[30] M. Hicken, P. M. Garnavich, J. L. Prieto, S. Blondin, D. L. DePoy, R. P. Kirshner, and J. Parrent,Astrophys. J. , L17 (2007), arXiv:0709.1501 [astro-ph].[31] M. Yamanaka et al. , Astrophys. J. , L118 (2009), arXiv:0908.2059 [astro-ph.HE].[32] S. Taubenberger, S. Benetti, M. Childress, R. Pakmor, S. Hachinger, P. Mazzali, V. Stanishev, N. Elias-Rosa, I. Agnoletto,F. Bufano, et al. , Monthly Notices of the Royal Astronomical Society , 2735 (2011).[33] S. O. Kepler, S. J. Kleinman, A. Nitta, D. Koester, B. G. Castanheira, O. Giovannini,A. F. M. Costa, and L. Althaus, Monthly Notices of the Royal Astronomical Society , 1315 (2007),http://mnras.oxfordjournals.org/content/375/4/1315.full.pdf+html.[34] M. Ilkov and N. Soker, Monthly Notices of the Royal Astronomical Society , 1695 (2012),http://mnras.oxfordjournals.org/content/419/2/1695.full.pdf+html.[35] R. Moll, C. Raskin, D. Kasen, and S. Woosley, Astrophys. J. , 105 (2014), arXiv:1311.5008 [astro-ph.HE].[36] S. Ji, R. T. Fisher, E. Garca-Berro, P. Tzeferacos, G. Jordan, D. Lee, P. Lorn-Aguilar, P. Cremer, and J. Behrends,The Astrophysical Journal , 136 (2013).[37] D. R. van Rossum, R. Kashyap, R. Fisher, R. T. Wollaeger, E. Garca-Berro, G. Aznar-Sigun, S. Ji, and P. Lorn-Aguilar,The Astrophysical Journal , 128 (2016).[38] H. Liu, X. Zhang, and D. Wen, Physical Review D , 104043 (2014).[39] U. Das and B. Mukhopadhyay, Modern Physics Letters A , 1450035 (2014).[40] D. Adam, Astronomy and Astrophysics , 95 (1986).[41] J. P. Ostriker and F. Hartwick, The Astrophysical Journal , 797 (1968).[42] S. Subramanian and B. Mukhopadhyay, Monthly Notices of the Royal Astronomical Society , 752 (2015),http://mnras.oxfordjournals.org/content/454/1/752.full.pdf+html.[43] P. Bera and D. Bhattacharya, Monthly Notices of the Royal Astronomical Society , 3375 (2016).[44] U. Das and B. Mukhopadhyay, Physical Review D (2012), 10.1103/PhysRevD.86.042001.[45] J. G. Coelho, R. M. Marinho, M. Malheiro, R. Negreiros, D. L. Cceres, J. A. Rueda, and R. Ruffini,The Astrophysical Journal , 86 (2014).[46] N. Chamel, A. F. Fantina, and P. J. Davis, Physical Review D (2013), 10.1103/PhysRevD.88.081301.[47] E. E. Salpeter, The Astrophysical Journal , 669 (1961).[48] N. Chamel and A. F. Fantina, Phys. Rev. D , 023008 (2015).[49] P. Bera and D. Bhattacharya, Mon. Not. Roy. Astron. Soc. , 3951 (2014), arXiv:1405.2282 [astro-ph.SR].[50] T. Hamada and E. E. Salpeter, The Astrophysical Journal , 683 (1961).[51] M. Wang, G. Audi, A. H. Wapstra, F. G. Kondev, M. MacCormick, X. Xu, and B. Pfeiffer,Chinese Physics C , 1603 (2012).[52] G. Audi, M. Wang, A. H. Wapstra, F. G. Kondev, M. MacCormick, X. Xu, and B. Pfeiffer,Chinese Physics C , 1287 (2012).[53] J. M. Pearson, S. Goriely, and N. Chamel, Physical Review C (2011), 10.1103/PhysRevC.83.065810. [54] N. Chamel, E. Molter, A. Fantina, and D. P. Arteaga, Physical Review D (2014), 10.1103/PhysRevD.90.043002.[55] G. Gamow, Physical Review , 718 (1939).[56] P. Haensel, A. Y. Potekhin, and D. G. Yakovlev, Neutron stars 1: Equation of state and structure , Vol. 326 (SpringerScience & Business Media, 2007).[57] L. R. Gasques, A. V. Afanasjev, E. F. Aguilera, M. Beard, L. C. Chamon, P. Ring, M. Wiescher, and D. G. Yakovlev,Physical Review C (2005), 10.1103/PhysRevC.72.025806.[58] D. G. Yakovlev, L. R. Gasques, A. V. Afanasjev, M. Beard, and M. Wiescher,Physical Review C (2006), 10.1103/PhysRevC.74.035803.[59] M. A. Cˆandido Ribeiro, L. C. Chamon, D. Pereira, M. S. Hussein, and D. Galetti, Phys. Rev. Lett. , 3270 (1997).[60] L. C. Chamon, D. Pereira, M. S. Hussein, M. A. Cˆandido Ribeiro, and D. Galetti, Phys. Rev. Lett. , 5218 (1997).[61] S. Bonazzola, E. Gourgoulhon, M. Salgado, and J. Marck, Astronomy and Astrophysics , 421 (1993).[62] M. Bocquet, S. Bonazzola, E. Gourgoulhon, and J. Novak, Astron. Astrophys. , 757 (1995), arXiv:gr-qc/9503044 [gr-qc].[63] B. Franzon, V. Dexheimer, and S. Schramm, Monthly Notices of the Royal Astronomical Society , 2937 (2016).[64] B. Franzon, V. Dexheimer, and S. Schramm, Phys. Rev. D94 , 044018 (2016), arXiv:1606.04843 [astro-ph.HE].[65] B. Franzon, R. O. Gomes, and S. Schramm, (2016), 10.1093/mnras/stw1967, arXiv:1608.02845 [astro-ph.HE].[66] E. Gourgoulhon,
3+ 1 formalism in general relativity: bases of numerical relativity , Vol. 846 (Springer Science & BusinessMedia, 2012).[67] C. Armaza, A. Reisenegger, and J. A. Valdivia, The Astrophysical Journal , 121 (2015).[68] J. Mitchell, J. Braithwaite, A. Reisenegger, H. Spruit, J. Valdivia, and N. Langer, Monthly Notices of the Royal Astro-nomical Society , 1213 (2015).[69] J. Braithwaite, Astronomy & Astrophysics , 687 (2006).[70] P. Goldreich and A. Reisenegger, The Astrophysical Journal , 250 (1992).[71] R. H. Durisen, The Astrophysical Journal , 215 (1973).[72] D. Yakovlev and V. Urpin, Soviet Astronomy , 303 (1980).[73] A. Cumming, Monthly Notices of the Royal Astronomical Society333