Very Magnetized White Dwarfs with Axisymmetric Magnetic Field and the Importance of the Electron Capture and Pycnonuclear Fusion Reactions for their Stability
In this work, we study the properties of magnetized white dwarfs taking into account possible instabilities 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 poloidal magnetic field in a fully general relativistic approach. The stellar interior is composed of a regular crystal lattice made of carbon ions immersed in a degenerate relativistic electron gas. The onsets of electron capture reactions and pycnonuclear reactions are determined with and without magnetic fields. We find that magnetized white dwarfs violate the standard Chandrasekhar mass limit significantly, even when electron capture and pycnonuclear fusion reactions are present in the stellar interior. We obtain a maximum white dwarf mass of around for a central magnetic field of G, which indicates that magnetized white dwarfs may play a role for the interpretation of superluminous type Ia supernovae. Furthermore, we show that the critical density for pycnonuclear fusion reactions limits the central white dwarf density to g/cm. As a result, equatorial radii of white dwarfs cannot be smaller than km. Another interesting feature concerns the relationship between the central stellar density and the strength of the magnetic field at the core of a magnetized white dwarf. For high magnetic fields, we find that the central density increases (stellar radius decrease) with magnetic field strength, which makes ultramagnetized white dwarfs more compact. The opposite is the case, however, if the central magnetic field is less than G. In the latter case, the central density decreases (stellar radius increases) with central magnetic field strengths.
It is generally accepted that stars born with masses below around 10 solar masses end up their evolutions as white dwarfs (WDs) Weber (1999); Shapiro and Teukolsky (2008); Glendenning (2012). With a typical composition mostly made of carbon, oxygen, or helium, white dwarfs possess central densities up to . They can be very hot (Althaus et al., 2009), fast rotating Arutyunyan et al. (1971); Boshkayev et al. (2013); Hartle (1967) and strongly magnetized Coelho et al. (2014); Lobato et al. (2016); Mukhopadhyay and Rao (2016). The observed surface magnetic fields range from G to G Terada et al. (2008); Reimers et al. (1996); Schmidt and Smith (1995); Kemp et al. (1970); Putney (1995); Angel (1978). The internal magnetic fields of white dwarfs are not known, but they are expected to be larger than their surface magnetic fields. This is due to the fact that in ideal magneto hydrodynamics (MHD), the magnetic field, , is ‘frozen-in’ with the fluid and , with being the local mass density (see, e.g., Refs. Mestel (2012); Landau et al. (1958)). A simple estimate of the internal magnetic field strength follows from the virial theorem by equating the magnetic field energy with the gravitational binding energy, which leads to an upper limit 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 fields as large as G (see, e.g., Refs. Angel (1978); Shapiro and Teukolsky (2008); Das and Mukhopadhyay (2014a); Bera and Bhattacharya (2014a, 2016a); Franzon and Schramm (2015, 2016); Das and Mukhopadhyay (2015)).
The relationship between the gravitational stellar mass, , and the radius, , of non-magnetized white dwarfs was first determined by Chandrasekhar Chandrasekhar (1939). Recently, mass-radius relationships of magnetic white dwarfs have been discussed in the literature (see, e.g., Refs. Suh and Mathews (2000); Bera and Bhattacharya (2014a); Franzon and Schramm (2015)). These studies show that the masses of white dwarfs increase in the presence of strong magnetic fields. This is due to the Lorentz force, which acts against gravity, therefore supporting stars with higher masses.
Based on recent observations of several superluminous type Ia supernovae (SN 2006gz, SN 2007if, SN 2009dc, SN 2003fg) Silverman et al. (2011); Scalzo et al. (2010); Howell et al. (2006); Hicken et al. (2007); Yamanaka et al. (2009); Taubenberger et al. (2011); Kepler et al. (2007), it has been suggested that the progenitor masses of such supernovae significantly exceed the Chandrasekhar mass limit of (Ilkov and Soker, 2012). Super-heavy progenitors were studied as a result of mergers of two massive white dwarfs Moll et al. (2014); Ji et al. (2013); van Rossum et al. (2016). Alternatively, the authors of Ref. Liu et al. (2014) obtained super-Chandrasekhar white dwarfs for magnetically charged stars. In addition, super-Chandrasekhar white dwarfs were investigated in the presence of strong magnetic fields in Refs. Das and Mukhopadhyay (2014b). In Refs. Adam (1986); Ostriker and Hartwick (1968), WDs models with magnetic fields were calculated in the framework of Newtonian physics. A recent study of differential rotating, magnetized white dwarfs has shown that differential rotation might increase the mass of magnetized white dwarfs up to 3.1 Subramanian and Mukhopadhyay (2015). Also, as shown in Ref. Bera and Bhattacharya (2016b), purely toroidal magnetic field components can increase the masses of white dwarfs up to .
According to Refs. Das and Mukhopadhyay (2012), effects of an extremely large and uniform magnetic field on the equation of state (EOS) of a white dwarf could increase its critical mass up to . This mass limit is reached for extremely large magnetic fields of G. Nevertheless, as already discussed in Refs. Coelho et al. (2014); Chamel et al. (2013), the breaking of spherical symmetry due to magnetic fields and micro-physical effects, such as electron capture reactions and pycnonuclear reactions, can severely limit the magnetic field inside white dwarfs.
In Ref. Franzon and Schramm (2015), mass-radius relationships of highly magnetized white dwarfs were computed using a pure degenerate electron Fermi gas. However, according to Ref. Salpeter (1961), many-body corrections modify the EOS and, therefore, the mass-radius relationship of white dwarfs. The purpose of our paper is two-fold. Firstly, we model white dwarfs using a model for the equation of state which takes into account not only the electron Fermi gas contribution, but also the contribution from electron-ion interactions Chamel and Fantina (2015). Secondly, we perform a stability analysis of the matter in the cores of white dwarfs against electron capture and pycnonuclear fusion reactions. The Landau energy levels of electrons are modified by relativistic effects if the magnetic field strength is higher than the critical QED magnetic field strength of G. However, as already shown in Ref. Bera and Bhattacharya (2014b), the global properties of white dwarfs, such as masses and the radii, are nearly independent of Landau quantization. For this reason, we do not take into account magnetic fields 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 the equation of state used in our study to model white dwarfs. This is followed, in Sec. III, by a brief discussion of the equations 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 of General 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. Salpeter (1961); Hamada and Salpeter (1961). Typically, a white dwarf is composed of atomic nuclei immersed in a fully ionized electron gas. In this work, we make use of the latest experimental atomic mass data Wang et al. (2012); Audi et al. (2012) used to determine the equation of state. Modifications of the equation of state due to the interactions between electrons and atomic nuclei are taken into account too. The model adopted to describe the nuclear lattice was derived for the outer crust of a neutron star in Refs. Pearson et al. (2011); Chamel et al. (2013) and later applied to WDs in Ref. Chamel et al. (2014). According to Ref. Chamel et al. (2014), the cores of white dwarfs are subjected to the degenerate electron and ionic lattice pressures. The total pressure is then given by
where denotes the electron pressure, determined in Salpeter (1961), and is the lattice pressure for two different type of ions. The lattice pressure is given by the energy density of the ionic lattice (see Ref. Pearson et al. (2011)),
with and being the proton number of two different ions. In our case, the white dwarf is composed of carbon ions, i.e., ==12. Following the Bohr-van Leeuwen theorem Pearson et al. (2011), the lattice pressure of ions arranged in a regular body-centered-cubic (bcc) crystal does not depend on the magnetic field, apart from a small contribution due to the quantum zero-point motion of ions. In this case, the lattice energy density reads Chamel et al. (2013)
with given by
The quantities , , are lattice constants and is the ratio of ions and in the lattice Chamel et al. (2014) (see also Table 1). If only a single ion is present in the lattice, Eq. 4 does not depend on and so that Eq. (3) becomes
The energy density is given in terms of the degenerate electron energy, the energy density of the ions, and the energy density of the ionic lattice,
where and are the number densities of atomic nuclei with masses and , respectively. As already mentioned above, here we adopt the most recent experimental values for (see Refs. Wang et al. (2012); Audi et al. (2012)).
Figure 1 shows the impact of lattice contributions on the white dwarf equation of state studied in this paper. The black lines (no lattice contribution) and red dashed line (with lattice contribution) are for white dwarf matter with zero magnetic field (). One sees that adding the lattice contribution to the equation of state lowers the pressure somewhat, which in turn makes white dwarfs less massive. It also follows from this figure that the presence of lattice contributions reduces the radii of white dwarfs (renders them more compact) with comparable central pressures.
Iii Instabilities in strongly magnetized white dwarfs
iii.1 Inverse decay
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) , one obtains the Gibbs free energy, , per nucleon as
with being the neutron mass and denoting the excess mass of nuclei, which, for magnetic field strengths G, is independent of the magnetic field, see, e.g., Ref Chamel et al. (2013). For we have and , with being the electron chemical potential. Inverse -decay reactions are believed to occur in the cores of white dwarfs if the condition Chamel et al. (2014)
is fulfilled, where and follow from Eq. (7) and the possible choices for and are , , and .
From the inequality (8), we obtain the following relation
with the electron number density and mass density of a magnetized electron gas given respectively by
with and given by
Another important quantity is , which describes the difference of , defined in Eq. (4), before and after an inverse decay reaction. It is given by
For an electron gas consisting of only one type of ion, we have
In the limit where only the ground state () is fully occupied by electrons, one has
where with G being the critical magnetic field (see Ref. Haensel et al. (2007) for more details about ). The chemical potential of the electrons in this case is given by
where denotes the Compton wavelength of electrons. In Ref. Chamel et al. (2014) it was estimated that the maximum magnetic field inside of white dwarfs, before the onset of -inverse reactions, is given by
with the fine structure constant. We note that because of the second term on the right-hand-side of Eq. (18), which originates from lattice contributions, the maximum value of increases if lattice contributions are taking into account.
In Fig. 2 we show the numerical solution of Eq. (9) for white dwarf matter made of only carbon ions immersed in a magnetized electron gas. The oscillatory behavior is caused by the Landau level contributions to the number density, given by Eq. (11). For high values of with only the ground state occupied, the dependence of density on becomes linear, as can seen in Fig. 2.
iii.2 Pycnonuclear reaction
In this section, we will focus on nuclear fusion reactions (pycnonuclear fusion reactions) among heavy atomic nuclei ,, schematically expressed as . An example of such a reaction is carbon on carbon, . Pycnonuclear reactions have been found to occur over a significant range of stellar densities (see, for instance, Ref. Gasques et al. (2005)), including the density range found in the interiors of white dwarfs Chamel et al. (2013, 2014). The nuclear fusion rates at which pycnonuclear reactions proceed, however, are highly uncertain because of some poorly constrained parameters (see Ref. Gasques et al. (2005); Yakovlev et al. (2006)). The reaction rates have been calculated for different models. In Ref. Gasques et al. (2005), the pycnonuclear reaction rates are defined as
where is the astrophysical S-factor used in Ref. Gasques et al. (2005) for the NL2 nuclear model parametrization. According to Ref. Gasques et al. (2005), an analytic equation for the S-factor is given by
where is in units of MeV barn. The factors and in Eq. (19) are given by
with , and are dimensionless parameters for a regular bcc-type crystal lattice (see at zero temperature). Their values are listed in Table 2.
|bcc; static lattice||2.638||3.90||1.25|
For number densities less than neutron drip density one has Gasques et al. (2005) and the pycnonuclear reaction rates are given by
with given in units of cm s. The zero-point oscillation energy of C nuclei at g/cm is given by Ref. Shapiro and Teukolsky (2008)
As already mentioned above, the reaction rates are rather uncertain, and the analytic astrophysical S-factor has an uncertainly of 3.5, which considerably affects the density thresholds of pycnonuclear reaction and their reaction times. Finally, in Fig. 3 we show pycnonuclear fusion reaction rates and pycnonuclear reaction time scales for carbon burning at zero temperature as functions of mass density. The bcc crystal lattice for nuclear model NL2 was employed to produce Fig. 3.
Iv White dwarfs with axisymmetric magnetic fields
The numerical technique used in this work to study axisymmetric magnetic fields was first applied to neutron stars in Refs. Bonazzola et al. (1993); Bocquet et al. (1995), and more recently in Ref. Franzon et al. (2016a, b, c). The same formalism was used to study rotating and magnetized white dwarfs in Ref. Franzon and Schramm (2015). Here we build stellar equilibrium configurations by solving the Einstein-Maxwell field equations in a fully general relativistic approach. For more details about the theoretical formalism and numerical procedure, see, for instance, Ref. Gourgoulhon (2012). Below we show the basic electromagnetic equations which, combined with the gravitational equations, are solved numerically by means of a spectral method. In this context, the stress-energy tensor is composed of the matter and the electromagnetic source terms,
Here is the antisymmetric Faraday tensor defined as , with denoting the electromagnetic four-potential . The total energy density of the system is , the pressure is denoted by , is the fluid 4-velocity, and the metric tensor is . 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 field contribution.
The metric tensor in axisymmetric spherical-like coordinates can be read of from the line element
where is the heat function defined in terms of the baryon number density ,
The quantity in Eq. (29) is defined as , and the magnetic potential is given by
where denotes the current function. Magnetic stellar models are obtained by assuming a constant value, , for the latter Franzon et al. (2016b). According to Ref. Bocquet et al. (1995), other choices for are possible, but the general conclusions as presented in this work remain the same. The constant current function is a standard way to generate self-consistently a dipolar magnetic field throughout the star.
In this section, we discuss the effects of strong magnetic fields on the global properties of stationary white dwarfs taking into account instabilities due to inverse -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 computed for the latest experimental atomic mass data. The instabilities related to the microphysics are fundamental since they put constraints on the equilibrium configurations and also limit the maximum magnetic fields which these stars can have Chamel et al. (2013). In addition to the magnetic profiles, which have already been computed in Ref. Franzon and Schramm (2015), we also compute stellar models at constant magnetic dipole moments . In Ref. Franzon and Schramm (2015), a simple Fermi gas model was used to model white dwarfs, and the microphysical issues were not addressed. In our study, the maximum white dwarf mass for non-magnetized stars is smaller than the one considered in Ref. Franzon and Schramm (2015), 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, , and current functions, . The magnetic dipole moment is defined as (see Ref. Bonazzola et al. (1993))
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 leads first to an increase in the white dwarf maximum mass. However, if we increase further, the maximum mass begins to drop. This is due to the fact that the stellar radius becomes larger (see also Fig. 5), which reduces the magnetic field (see Eq. 32). As a consequence, the Lorentz force becomes 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. The behavior is very different if the value of the current function is kept constant, in which case non-monotonic (in some cases even multivalued) mass-density relationships are obtained. The cross-hatched area in Fig. 4 shows the density regime where pycnonuclear fusion reactions become possible. The position of the white dwarf with just the right threshold density ( g/cm) for this reaction to occur is marked with a solid black square in Fig. 4. The pycnonuclear reaction time at that density is 10 Gyrs. For a central white dwarf density of g/cm the fusion reaction time decreases to 0.1 Myrs (see Fig. 3). White dwarfs subject to inverse -decay reactions in their cores are located in the yellow area (marked “-inverse”) of Fig. 4. The most massive stable white dwarf which is not subject to microscopic instability reactions in its core (end point of the curve with Am), has a mass of and a radius (see Fig. 5) of km. Finally, we note that the condition for stability against radial oscillations is fulfilled for all white dwarf sequences for which the magnetic dipole moment is kept fixed, as can be seen in Fig. 4. An overview of the density thresholds discussed just above is provided in Table 3 for white dwarfs with different magnetic field values and magnetic dipole moments.
The mass-radius relationship of magnetized white dwarfs for different magnetic dipole moments is shown in Fig. 5. One sees that increasing values of 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 of surface () and central () magnetic fields, the circumferential equatorial radius (), and the baryon number density (), for two sample magnetic dipole moments of Am (red line) and Am (orange line).
In Fig. 6 (top panels), one sees that the curves with and cross each other. This is due to the fact that the magnetic field scales as , with being the stellar radius (see Eq. (32)). The locations of stars with fixed baryon masses of and are shown in Fig. 6 by dashed horizontal lines. According to Eq. (32), the magnetic field is determined by the size of the star along the curves with . However, along the lines with fixed baryon masses, the strength of the magnetic field is a combination of the magnetic dipole moment and the stellar radius .
Next, we discuss the behavior of the magnetic dipole moments of white dwarfs whose magnetic fields are weakening. From the versus and versus relationships shown in Fig. 6 (top panels), one sees that two different scenarios are possible, depending on the star mass and on the magnetic field strengths of white dwarfs. If located above the crossing point of the (red line) and (orange line) curve, white dwarfs with weakening magnetic fields would be evolving from right to left in the two upper panels of Fig. 6, as shown (back arrow) for a white dwarf with a constant baryon number of . The magnetic dipole moment of such white dwarfs would increase, from Am to for the sample star shown in Fig. 6. This is accompanied by an increase in the stellar radius (see versus diagram) and a decrease in the central baryon density (see versus diagram). The situation is reversed for white dwarfs located below the the crossing. For such 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 (black arrows). In this case, white dwarfs become smaller and therefore more dense at the center (see versus and versus diagrams shown in Fig. 6).
As discussed just above (Fig. 6), the equatorial radii of white dwarfs located above the crossing point increase as their 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 . This means that for a star with a mass of , the increase in the magnetic dipole moment, , is canceled by the increase in the radius, reducing the magnetic field. This is the opposite of what is expected for stars with lower masses. For example, a star with decreases its magnetic dipole moment and its radius. However, in this case, the decrease in the radius is not enough to cancel the reduction in . The net result is a decrease of the magnetic field. This can be understood by looking at the variation in the circular equatorial radius of the stars with and . For the latter, the change in radius is much smaller than the radial change for the star, for a change in the magnetic dipole moment of Am.
In Fig. 7, we show the global properties of two white dwarfs with fixed baryon masses of and . The top panels show the central baryon density as a function of the central magnetic field (top-left left panel) and the circular equatorial radius (top-right panel) for a white dwarf with . For such stars, as the magnetic field decreases, the central baryon density becomes smaller due to the fact that the radius is increasing. On the other hand, for lighter white dwarfs, with a mass of , the central baryon number density increases as the magnetic field decreases, since the stellar radius is getting smaller.
In this work, we presented axisymmetric and stationary models of magnetized white dwarfs obtained by solving the Einstein-Maxwell equations self-consistently and taking into stability considerations related to neutronization due to electron 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 the explanation of observed ultra-luminous explosions of supernovae Type Ia. The Lorentz force induced by strong magnetic fields breaks the spherical symmetry of stars and increases their masses, since the force acts in the radial outward 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 of state becomes softer if nuclear lattice contributions are included in addition to the electron pressure. This is due to the fact that the repulsive force between electrons is smaller in the presence of an ionic lattice, causing a softening of the equation of state (see Fig. 1). We note that the density thresholds for pycnonuclear fusion reactions and inverse -reactions are reduced when magnetic fields are present in the stellar interior, as can be seen in Table 3.
We have shown that the masses of white dwarfs increase up to (with a corresponding magnetic dipole moment of Am (see, e.g., Fig. 4) if microphysical instabilities are considered. This star has an equatorial radius of km with magnetic fields of G and G at the center and at the stellar surface, respectively. For this white dwarf, the ratio between the magnetic pressures and the matter pressure at the center is 0.789. Although the surface magnetic fields obtained here are higher than the observed ones for white dwarfs, these figures provide an idea of the maximum possible magnetic field strength that can be reached inside of these objects, and may also be used to assess the effects of strong magnetic fields on both the microphysics and 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. Franzon and Schramm (2015). This is because we modeled the stellar interior with a more realistic equation of state than just a simple electron gas, and we considered the density threshold for pycnonuclear fusion reactions for a 10 Gyrs fusion reaction time scale, which restricts the central density of white dwarfs to g/cm (see Table 3), limiting the stellar masses and, therefore, their radii, which for very massive and magnetized white dwarfs cannot be smaller than R km. However, it is important to mention that the pycnonuclear reaction time scales are somewhat uncertain. In our case, for example, we have a factor of uncertainty of approximately 3.5 in the calculation of the astrophysical S-factor (see Refs. Gasques et al. (2005); Yakovlev et al. (2006)).
Our results show that the surface magnetic field, , is about one order of magnitude smaller than the magnetic field reached at the stellar center, . If the magnetic field weakens for massive white dwarfs, we found that the magnetic 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. The situation is reversed for less massive white dwarfs, for which smaller the magnetic fields imply smaller stellar magnetic dipole moments. The radii of massive (light) white dwarfs are found to increase (decrease) for decreasing central magnetic fields (Fig. 7). This opens up the possibility that massive white dwarfs, with central magnetic fields greater than G, increase their magnetic fields through continued compression. This phenomenology differs from previous studies carried out for magnetic fields less than G Suh and Mathews (2000); Ostriker and Hartwick (1968), where an increase of the central magnetic field 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., Armaza et al. (2015); Mitchell et al. (2015); Braithwaite (2006)). Moreover, according to Ref. Goldreich and Reisenegger (1992), many different mechanisms can affect the magnetic fields and their distributions inside of white dwarfs. In this work, in the framework of a fully general relativistic treatment, we model the properties of magnetized white dwarfs with purely poloidal magnetic field components. Although this is not the most general magnetic field profile, and a dynamical stability of these stars still needs to be addressed, magnetic fields considerably increase the masses of white dwarfs, even when microphysical instabilities are considered. As a consequence, 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 an Alfven velocity of cm/s, which, for a white dwarf with a typical radius of km, leads to an Alfven crossing time of s Durisen (1973); Yakovlev and Urpin (1980); Cumming (2002). This is close to the hydrostatic equilibration time of white dwarfs. As a consequence, although magnetized white dwarfs seem to be short-lived stars, they might still be supported by magnetic fields. Our results represent magnetostatic equilibrium conditions. The stability analysis of such systems is beyond the scope of this study, which constitutes a first step toward a more complete discussion of the possible existence of super-Chandrasekhar white dwarfs. Studies which address issues such as the role of different (poloidal and toroidal) magnetic field configurations, stellar rotation, and different compositions of the stellar cores will be presented in a series of forthcoming papers.
We acknowledge financial support from the Brazilian agencies CAPES, CNPq, and we would like to thank FAPESP for 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.
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