On slowly rotating magnetized white dwarfs Diana Alvear Terrero1,∗, Daryel Manreza Paret2,3,†, 7 Aurora Perez Martinez1,3,‡ 1 0 2 1Departamento de F´ısica Te´orica, Instituto de Cibern´etica Matema´tica y F´ısica n Calle E esq 15 No. 309, Vedado, La Habana 10400, Cuba a J 2Facultad de F´ısica, Universidad de la Habana 0 San La´zaro y L, Vedado, La Habana 10400, Cuba 3 3Instituto de Ciencias Nucleares, Universidad Nacional Auto´noma de M´exico ] A. P.70-543, 04510 C. M´exico, M´exico E H ∗[email protected], †dmanreza@fisica.uh.cu, ‡[email protected] . h p February 1, 2017 - o r t Abstract s a [ Rotating magnetized white dwarfs are studied within the framework of general relativity usingHartle’s formalism. Matter inside magnetized white dwarfs isdescribed byan equation 1 ofstateofparticlesundertheactionofaconstantmagneticfieldwhichintroducesanisotropic v pressures. Our study is done for values of magnetic field below 1013 G -a threshold of the 2 maximum magnetic field obtained by the cylindrical metric solution- and typical densities of 9 WDs. The effects of the rotation and magnetic field combined are discussed, we compute 7 8 relevant magnitudes such as the moment of inertia, quadrupole moment and eccentricity. 0 1. Keywords: white dwarfs; magnetic fields; general relativity; slow rotation. 0 7 1 Introduction 1 : v Whitedwarfsareverywell-knowncompactobjectswithtypicalvaluesofmassaroundasolarmass i X and the size of the Earth. Composed mainly by carbon, they counteract the gravitationalpull by r means of the pressure of the degenerate electron gas while the carbon nuclei are the principal a contribution to the mass. Observations estimate magnetized white dwarfs (MWDs) surface magnetic fields in the range of 106G to 109G [1,2] whereas internal magnetic fields are determined indirectly using theoret- ical models based on macroscopic and microscopic analyses. Moreover, there are observations of superluminous thermonuclear supernovae, whose progenitor could be super-Chandrasekhar WDs (MWD >1.44M⊙) [3]. Consequently, in Refs. [4–6] it was proposed to justify their existence with the presence of strong magnetic fields above 1013G. A magnetic field acting on a fermions system breaks the SO(3) symmetry, giving rise to an anisotropy in the equations of state (EoS) [7]. Furthermore, the anisotropy of the energy momentum tensor caused by the magnetic field can be included considering an axi-symmetric and poloidal strong magnetic field [8], which allows 1 to model rotating magnetized white dwarfs in a self-consistent way by solving Einstein-Maxwell equations [9,10]. Thepresenceoftheanisotropicpressuressuggeststhatintroducinganaxiallysymmetricmetric to solve Einstein equation is crucial. Previously we have used a cylindrical metric and obtained 13 a maximum bound of 1.5×10 G for the magnetic field of stable MWDs [11], ruling out the possibility of super-ChandrasekharWDs with strong magnetic fields. Rotation is another plausible cause for the increment in mass of white dwarfs. In order to investigatethis issue,we solvethe rotatingstructureequationsemergingfromsphericalsymmetry by Hartle’s method –despite spherical metric is no longer adequate when considering anisotropic EoS–. This allows us to determine if the deformation of the rotating stars accounts for stable RMWDs with mass above 1.44M⊙. With that aim, we first describe the equilibrium of RMWDs by solving Einstein equations in section2whileconsideringbothpressures-oneparallelandtheotherperpendiculartothemagnetic field- independently. Then, in section 3 we present numerical results, and finally in section 4, our conclusions. 2 Slowly rotating structure equations for RMWDs When discussing the structure of compacts objects, it must be analyzed both local and global properties of the involved matter. The first ones are described by an equation of state (EoS), while the latter ones comprises the dynamical response of matter at large scales to, for instance, gravity and rotation. In this paper, we consider the magnetized equations of state obtained in Refs. [12,13] for carbon/oxygen WDs whose matter is composed by particles under the action of a constant magnetic field, which leads to a splitting of the pressure into a component parallel to the magnetic field and a perpendicular one. The values of the magnetic field are chosenbelow the 13 10 G threshold mentioned before. Regarding the structure equations for a slowly rotating compact object, we take into account the angular velocity (Ω) of the star uniform and sufficiently slow so that R3Ω2 ≪ M, where M and R are the mass and the radius of the non-rotating WDs respectively. Then, the angular velocity provokes small changes in the pressure P, energy density E =ρc2 and gravitational field with respect to the corresponding quantities of the static configuration. These changes can be considered as perturbations of the non-rotating solution. So, to consider that a star is rigidly and slowly rotating implies calculating its equilibrium properties reckoning small perturbations on static configuration. Introducing new coordinates (r,θ,φ), where r(R,θ) = R+ξ(R,θ) takes into account deviations from spherical symmetry, the metric of the rotating configuration becomes [14,15] −1 ds2 = −eν{1+2[h0+h2P2(cosθ)]}dt2+ 1+2[m0+m2P2(cosθ)][r−2M] dr2 1−2M/r 2 2 2 2 3 +r [1+2(v2−h2)P2(cosθ)] dθ +sin θ(dφ−ωdt) +O(Ω ). (1) h i Here P2(cosθ) is the Legendre polynomial of second order, eν and eλ =[1−2M(r)/r]−1 are the static metric functions, andω(r)=ω¯(r)+Ω is the angularvelocityof the localinertialframe, where ω¯(r) is the fluid’s angular velocity relative to the local inertial frame. The functions h0= h0(r), h2=h2(r), v2=v2(r), and mass perturbation factors m0=m0(r) and m2=m2(r) are all proportional to Ω2. Besides, we must define the pressure perturbation factors p∗ and p∗ on the 0 2 order of Ω2, which modify the energy-momentum tensor [15]. 2 Once computed Einstein equations considering perturbations up to O(Ω2) with the metric (1), the structure of the perturbed rotating stars is described by the static equations of Tolman- Oppenheimer-Volkoff for the pressure P, the mass and ν in addition to the equations for ω¯, m0, p∗0, h2, v2, m2 and p∗2. The system to integrate outward is dP (E+P)(M +4πr3P) = − , (2) dr r(r−2M) dM 2 = 4πr E, (3) dr dν 2 dP = − , (4) dr E+P dr dω¯ = κ, (5) dr dκ 4πr(E+P)(rκ+4ω¯) κ = −4 , (6) dr r−2M r dm0 = 4πr2(E+P)dEp∗+ r3e−ν (r−2M) κ2 + 8πr(E+P)ω¯2 , (7) 0 dr dP 3 4 r−2M (cid:20) (cid:21) dp∗0 = −m0(1+8πr2P)−4πr2(E+P)p∗+r3e−ν κ2 +ω¯2 2 − dν +2κω¯ , (8) dr (r−2M)2 r−2M 0 3 4 r r dr r (cid:20) (cid:18) (cid:19) (cid:21) dv2 dν r3e−ν 2 dν κ2 4πr(E+P)ω¯2 = −h2 + (r−2M) + + , (9) dr dr 3 r dr 4 r−2M (cid:20) (cid:21)(cid:20) (cid:21) dh2 dν 8πr3(E+P)−4M dν −1 4v2 dν −1 dr = h2 −dr+ r2(r−2M) dr − r(r−2M) dr " (cid:18) (cid:19) # (cid:18) (cid:19) r3e−ν dν −1 κ2 dν 2 2 + (r−2M) − 3 dr 4 dr r (cid:18) (cid:19) ( " (cid:18) (cid:19) # 4πr(E+P)ω¯2 dν 2 2 + (r−2M) + , (10) r−2M dr r " (cid:18) (cid:19) #) alongside with expressions 1 m2 = (r−2M) r3e−ν (r−2M)κ2+16πr(E+P)ω¯2 −h2 , (11) 6 (cid:26) (cid:27) 1 (cid:2) (cid:3) p∗2 = − r2e−ν −h2. (12) 3 These equations must be solved with the proper boundary conditions. This means to contem- platevaluesofthecentralenergydensitywithintypicalvaluesforWDs. Thepressureismaximum in the center of the star and must go to zero at the surface. Hence, the integration is carried out until P vanishes. The value for ω¯ atthe center is arbitraryand the restofthe variables areset up to zero initially. The total angular momentum is J = R4κ(R)/6, the angular velocity of the rotating WD is 3 Ω=ω¯(R)+2J/R , the moment of inertia is I =J/Ω, and the total mass is J2 MT =M(R)+m0(R)+ R3 . (13) Also, we compute the quadrupolar momentum [14,15] 3 Q= 8M3 h2+v2− MJR23 + J2, (14) 5 2M Q 1 R 1 +Q 2 R 1 M √R(R−2M) 2 M − 2 M − (cid:0) (cid:1) (cid:0) (cid:1) where Q n(x) is the associated Legendre function of second kind. The rotational deformation of m the WD can be depicted through the displacement of the surface of constant density at radius r in the static configuration to r(R,θ)=R+ξ (R)+ξ (R)P (cosθ), (15) 0 2 2 ∗ dP −1 ∗ dP −1 ξ = p (E+P) , ξ = p (E+P) (16) 0 − 0 dr 2 − 2 dr (cid:18) (cid:19) (cid:18) (cid:19) when rotating. The eccentricity is R 2 ε= 1 p , (17) s −(cid:18)Req(cid:19) with Rp = r(R,0)=R+ξ0(R)+ξ2(R), (18) Req = r(R, π2)=R+ξ0(R)− ξ2(2R). (19) 3 Results and discussion Fig. 1 shows the behavior of the mass for the static and the rotating configurations. In the left panel, we superpose the curves corresponding to M versus R, M versus R and M versus R . T p T eq Intherightpanel,bothmassesareshownasafunctionofthecentraldensity. Allplotsincludethe non-magnetic configuration as well as the solutions for the parallel and perpendicular pressures 12 corresponding to B =10 G. 1.44 1.44 B=0G 12 P B=10 G 1.38 1.38 P B=1012 G MT A A M1.32 M1.32 M/ M/ MT vs Req 1.26 1.26 MT vs Rp M B=0 G 1.20 1.20 12 P B=10 G M vs R 12 1.14 1.14 P B=10 G 0.2 0.4 0.6 0.8 1 1.2 108 109 1010 3 100R / RA c [g/cm] Figure 1: Left panel: Mass versus radius for static configuration and total mass as a function of equatorial and polar radii. Rigth panel: Static and total masses as a function of central density. As described in the previews section, considering slow rotation increasesthe mass of the stars. However,thisincrementdiminishesasthedensityincreases,sothattheoutcomeforthetotalmass is lower than the Chandrasekhar mass even for higher densities solutions, at least for the values of the magnetic fields below the Schwinger critical magnetic field for which our EoS are valid. Furthermore, the precision of our stable solutions increases with the central density of the star. 4 In left panel of Fig. 2 we present the moment of inertia I and the quadrupolar momentum Q as a function of density for B =0, andB =1012 G, for both parallelandperpendicular pressures. The right panel shows the eccentricity also as a function of ρ . The change of the magnetized c solutions respect to the non-magnetic ones are not substantial. Contraryto the behavior of I and Q, the eccentricity decreases with the increment of energy density. 1.0 1.0 B=0 G B=0 G 12 0.8 I Q P B=10 G 0.8 P B=1012 G 12 P B=10 G 12 ** Q, I/I 0.6 0.6 P B=10 G Q/ 0.4 0.4 * 49 2 0.2 Q=10 g cm 0.2 * 50 2 I=10 g cm 0.0 108 109 1010 0.0 8 9 10 10 10 10 3 3 c [g/cm] c [g/cm] Figure 2: Left panel: Moment of inertia and quadrupolar momentum as a function of central density. Right panel: Eccentricity versus central density. Additionally, the facts that 0<ε<1, R <R and Q>0 implies that the solutions correspond p eq to oblate WDs configurations. This can be easily pictured from Fig. 3, where we have plotted the polarradiusversusthe equatorialradiusonleft panel,and,in the rightpanelwe haveconstructed a parametrical surface of the non-magnetized solution at ρ = 2.49547×108 g cm−3 using the c corresponding values of R and R . eq p B=0G 12 0.4 P B=10 G 12 P B=10 G A R / Rp 0.3 0 0 1 0.2 0.1 0.2 0.4 0.6 0.8 1.0 1.2 100Req / RA Figure 3: Left panel: Polar radius versus equatorial radius, with density increasing towards left. The orange point corresponds to the B = 0 solution at ρ = 2.49547×108 g cm−3. Right panel: c representationofthe oblatespheroidshape ofthe WD associatedto the orangepoint inthe graph of the left panel. 5 4 Conclusions We haveimplemented analgorithmto study slowlyrotatingMWDs using the formalismproposed byHartleformagnetizedequationsofstate. Numericalsolutionshavebeencomputedforthetotal mass of the rotating star as well as its equatorial and polar radii and the static couterpart. The moment of inertia, the quadrupolar momentum and the eccentricity were analyzed, confirming thatrotationinthis waydeformsthe star,thatarenowoblatespheroids. Inallcases,resultswere obtained for the non-magnetic configuration and for a fixed value of 1012 G, lower than 1013 G, a critical field beyond which solutions are unstable. Our results for non-magnetized slowly rotating WDs are in agreement with Refs. [16,17]. Also, the stable slowly rotating solutions obtained are bounded by the condition of applicability of Hartle’s method, which is satisfied more accurately for WDs of higher densities. Taking into account the splitting of the pressure, it would be interesting to investigate the possibility of an alternative method to the one discussed here in order to include both, parallel and perpendicular pressures at the same time. This could give an insight into a more precise descriptionofsuchanisotropicWDs,andwouldallowto comparethedeformationofthe starsdue to the effect of the magnetic field with the rotational deformation. Acknowledgements D.A.T, D.M.P and A.P.M have been supported by the grant CB0407 and the ICTP Office of External Activities through NET-35. A.P.M thanks Consejo Nacional de Ciencia y tecnologia (CONACYT) for the support with the sabbatical Grant 264150 at ICN-UNAM, M´exico, where this work was developed. D.M.P has been also supported by a DGAPA-UNAM fellowship. References [1] 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. 60, p. 387 (2008). [2] G. D. Schmidt and P. S. Smith, Astrophys. J. 448, p. 305 (1995). [3] D. A. Howell et al., Nature 443, p. 308 (2006). [4] U. Das and B. Mukhopadhyay, Phys. Rev. D86, p. 042001 (2012). [5] U. Das and B. Mukhopadhyay, Phys. Rev. Lett. 110, p. 071102(2013). [6] U. Das and B. Mukhopadhyay, JCAP 1406, p. 050 (2014). [7] M. Chaichian, S. S. Masood, C. Montonen, A. P´erez Mart´ınez and H. P´erez Rojas, Physical Review Letters 84, 5261 (June 2000). [8] B. Franzon and S. Schramm, Phys. Rev. D 92, p. 083006(2015). [9] S. Bonazzola, E. Gourgoulhon, M. Salgado and J. A. Marck, Astron. Astrophys. 278, 421 (1993). [10] M. Bocquet, S. Bonazzola, E. Gourgoulhon and J. Novak, Astron. Astrophys. 301, p. 757 (1995). [11] D. M. Paret,J.E. HorvathandA. P.Mart´ınez, Research in Astronomy and Astrophysics 15, p. 1735 (2015). 6 [12] D.AlvearTerrero,M.CastilloGarc´ıa,D.ManrezaParet,J.E.HorvathandA.P´erezMart´ınez, Astronomische Nachrichten 336, 851 (2015). [13] D. Alvear Terrero,D. Manreza Paretand A. P´erezMart´ınez, ArXiv e-prints (January 2016). [14] J. B. Hartle, ApJ 150, p. 1005 (December 1967). [15] J. B. Hartle and K. S. Thorne, ApJ 153, p. 807 (September 1968). [16] K. Boshkayev, J. A. Rueda, R. Ruffini and I. Siutsou, Astrophys. J. 762, p. 117 (2013). [17] K.Boshkayev,H. QuevedoandB.Zhami, Monthly Notices of the Royal Astronomical Society 464, p. 4349 (2016). 7