Compact star deformation and universal relationship for magnetized white dwarfs
aa r X i v : . [ nu c l - t h ] J u l Compact star deformation and universal relationship for magnetized white dwarfs
Sujan Kumar Roy ∗§ , Somnath Mukhopadhyay §† and D. N. Basu ∗§ ∗ Variable Energy Cyclotron Centre, 1/AF Bidhan Nagar, Kolkata 700 064, India § Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India and † National Institute of Technology, Tiruchirapalli - 620015, Tamil Nadu, India ∗ (Dated: July 13, 2020)Recently super-Chandrasekhar mass limit has been derived theoretically in presence of strongmagnetic field to complement experimental observations. In the framework of Newtonian physics,we have studied the equilibrium configurations of such magnetized white dwarfs by using the rela-tivistic Thomas-Fermi equation of state for magnetized white-dwarfs. Hartle formalism, for slowlyrotating stars, has been employed to obtain the equations of equilibrium. Various physical quanti-ties of uniformly rotating and non-rotating white dwarfs have been calculated within this formal-ism. Consequently, the universality relationship between the moment of inertia(I), rotational lovenumber( λ ) and spin induced quadrupole moment(Q), namely the I-Love-Q relationship, has beeninvestigated for such magnetized white dwarfs. The relationship between I, eccentricity and Q i.e.I-eccentricity-Q relationship has also been derived. Further, we have found that, the I-eccentricity-Qrelationship is more universal in comparison to I-Love-Q relationship. PACS numbers:
I. INTRODUCTION
Black holes are said as bald because their multipolemoments can be expressed in terms of their mass, chargeand spin angular momentum. There are numbers of stud-ies carried out recently on slowly rotating neutron stars,quark stars and dark stars, which show the equation ofstate independent nature in the relations between certainmultipole moments of these stars [1–10]. Binary neutronstars are one of the most promising gravitational wave(GW) sources [11, 12] for Advanced LIGO, AdvancedVIRGO, KAGRA, Einstein telescope. Neutron star bi-naries can be used to extract information about the equa-tion of state (EoS) by detecting the GWs emitted in thelate inspiral during which neutron stars are tidally de-formed. Therefore, GWs emitted by neutron star bina-ries in the late inspiral must incorporate corrections in-duced by the neutron star internal structure, thereby pro-viding information about the EoS [13–15]. Unlike blackholes, exterior gravitational fields of neutron stars arenot only determined by their mass, radius, spin angularmomentum but also by their higher multipole moments.The extraction of the higher multipole moments from ob-servations might be erroneous if EoS dependent descrip-tion is neglected. But, degeneracies between the neutronstar quadrupole moment and the spin prevent detectionsfrom separately measuring these quantities. Again, de-generacies between the effect of the neutron star EoS andcorrections due to modified gravity on observables pre-vent serious tests of general relativity which are internalstructure independent.The I-Q relationship breaks down as the rotation in- ∗ E-mail 1: [email protected]; E-mail 2: [email protected];E-mail 3: [email protected] creases to the mass-shedding limit and the deviation maybe as high as 40% for neutron stars and 75% for strangestars [16]. Considering the effect of strong magnetic fieldon slowly rotating neutron stars it has been found that,for the case of a realistic magnetic field configuration, therelation again depends significantly on the EoS, losing itsuniversality [17]. Therefore, the use of the universalityrelationships in the subject of GW physics is limited bydifferent conditions like rotation, magnetic field etc.White dwarfs are another important type of compactstars. As the most common end product of evolution sce-nario, white dwarfs account for almost 97% of all evolvedstars. Therefore, the properties and distribution of whitedwarfs contain plenty of information. Consequently, thedetections of white dwarfs and study of their propertiesare of paramount importance. From GW physics pointof view, white dwarfs in binaries are potential source ofthe powerful gravitational waves for LIGO/VIRGO typeinterferometers [18]. Therefore, in the similar fashion ofneutron stars the study of relationship between differentparameters of white dwarfs under different conditions hasattracted certain attention in the literature [19, 20]. Ithas already been shown in Ref.-[19] that, the universalityrelationships hold good between the moment of inertia,tidal love number and spin induced quadrupole momentfor the case of cold white dwarfs as well. But as the effectof temperature is incorporated into the EoS it turns outthat, the universality breaks down as the temperatureapproaches millions of Kelvin [20]. As, whether theseuniversality relationships hold in presence of magneticfield is yet to be investigated, in this work we have stud-ied the effects of magnetic field on these relationships.The effects of the magnetic field have been incorporatedto obtain the EoS for the Helium, Carbon, Oxygen, Ironwhite dwarfs and consequently, in the framework of New-tonian physics equilibrium configurations of such magne-tized white dwarfs have been obtained. Finally, all thephysically relevant parameters and multipole momentshave been calculated to check out the validity of the uni-versal relationships between different multipole moments.
II. FORMALISM AND EQUATIONS OFEQUILIBRIUM
The formalism we have used in this work, is that de-vised by Hartle [21] for uniformly rotating compact starsadopted in the framework of Newtonian gravity. There-fore, the basic formalism to determine the equilibriumconfigurations would assume the slow rotation approxi-mation within the framework of Newtonian physics andthus the rotation in such slowly, uniformly rotating starswill be treated as perturbation to the non-rotating spher-ical stars. As white dwarfs can have low compactnesscompared to neutron stars, they can be well treatedwithin the framework of Newtonian physics as the gen-eral relativistic corrections for various physical quantitiesturn out to be very small [19, 22].It is worth mentioning the fact that, this particularapproach becomes advantageous in the sense that, fromphysical and geometric point of view it is very much intu-itive because of the use of coordinates make direct corre-spondence to the configuration and dynamics of rotatingbodies in equilibrium. The relativistic Hartle formalism,when applied in the context of Newtonian gravity, theequilibrium configurations can be described through or-dinary differential equations. Moreover, these equationsi.e. equations of equilibrium, can be solved with requiredphysical conditions to extract all the relevant physicalquantities such as the mass M , the equatorial radius r e ,the polar radius r p , the moment of inertia I , angular mo-mentum J , ellipticity ǫ and the tidal love number λ ofthe star. A. Slowly and uniformly rotating stars inNewtonian gravity
When a star is said to be rotating slowly and uniformly,it means that all the particles within the star must movewith a speed much lower than the speed of light. Thiscan be simply represented by,Ω << GMR , (1)where Ω is the angular velocity of the star, G is the uni-versal gravitational constant, M is the mass and R isthe radius of the star. Therefore, Eq.(1) is the conditionunder which we can treat the rotation as a small per-turbation to the already known non-rotating sphericalconfiguration having axial and reflection symmetry.For a rotating configuration, each of the particle of thestar in spherical polar coordinate ( r , θ , φ ) will acquire a centrifugal force given by, (cid:18) d ~rdt (cid:19) C = − ˆ rr Ω sin θ − ˆ θr Ω sin θ cos θ, (2)which incurs no φ dependent force, thus retaining theaxial symmetry. Moreover, as the force terms dependonly on Ω , the configuration will be symmetric under thereversal of the rotation axis as well. Hence, the rotatingconfiguration will have reflection symmetry as well.Hence, the equation of gravitational potential Φ( r, θ )becomes, ∇ Φ( r, θ ) = 4 πGρ ( r, θ ) , (3)where ρ ( r, θ ) is the mass density of the star. The θ depen-dence in the equation is the consequence of the rotationof the star.The EoS for the star is assumed to be of the form p = p ( ρ ), i.e. for the cold matter. Then the equation ofthe hydrostatic equilibrium turns out to be, − ∇ p ( r, θ ) ρ ( r, θ ) − ∇ Φ( r, θ ) = (cid:18) d ~rdt (cid:19) C ⇒ Z dp ( r, θ ) ρ ( r, θ ) + Φ( r, θ ) = 12 Ω r sin θ + Constant. (4)The rotating configurations of stars in equilibrium canbe found by solving Eqs.-(3) & (4) by assuming that, allthe points on a spherical surface of a particular densityin non-rotating configuration now lie on a deformed sur-face of the same density in the rotating configuration.Assuming slow rotation, the introduction of the Hartlecoordinate for the rotating deformed star gives, r = R + ξ ; Θ = θ, (5)where R , Θ represent a point in the non-rotating con-figuration and the same point is represented by r , θ inthe rotating configuration. As the rotating configura-tion of the star retains axial and reflection symmetriesthere would be no φ dependence in the equations andfor the same reason the perturbation term ξ can containonly Ω terms. Hence, the perturbations in radius, mass,potential, moment of inertia due to rotation can be rep-resented in terms of Ω powers. Throughout this workperturbation terms only up to first power in Ω will beconsidered for calculation purpose. Hence, Eq.(5) can berepresented as below, r ( R, Θ) = R + ξ ( R, Θ) + O (Ω ); Θ = θ, (6)where the term ξ ( R, Θ) ∼ Ω and for the slow rotationapproximation to be valid the condition need to be ful-filled is, ξ ( R, Θ) R << , ∀ R . (7)Corresponding to all these, the density profile and con-sequently the EoS for the rotating configuration of thestar can be represented as follows, ρ ( r, θ ) = ρ ( R, Θ) = ρ ( R ) = ρ (0) ( R ) ,p ( ρ ) = p ( R, Θ) = p ( R ) = p (0) ( R ) , (8)where ρ (0) ( R ) and p (0) ( R ) are the corresponding quanti-ties in the non-rotating configuration. B. Equations of Equilibrium
Now, the equations of equilibrium for the rotating con-figurations can be found by transforming the Eqs.-(3) &(4) in R, Θ coordinate and decomposing deformation ξ ,potential Φ as, ξ ( R, Θ) = X l ξ l ( R ) P l (cos Θ) , Φ( R, Θ) = Φ (0) ( R ) + Φ (2) ( R, Θ) + O (Ω ) , Φ (2) ( R, Θ) = X l Φ (2) l ( R ) P l (cos Θ) . (9) P l (cos Θ) is the Legendre polynomial of the order l andΦ (0) ( R ) is the potential in the non-rotating configura-tion, whereas, the Φ (2) ( R, Θ) term in rotating configura-tion represents the perturbation ∼ Ω . Ultimately, theseyields ∇ Φ (0) ( R ) = 4 πGρ (0) ( R ) , ( Ω ) , (10) ξ ( R ) ddR ∇ Φ (0) ( R ) + ∇ Φ (2)0 ( R ) = 0 , (Ω & l = 0) , (11) ξ ( R ) ddR ∇ Φ (0) + ∇ Φ (2)2 ( R ) − R Φ (2)2 ( R ) = 0 , (Ω & l = 2) , (12)from Eq.-(3) and from Eq.-(4) Z p dp (0) ( R ) dR + Φ (0) ( R ) = Constant, ( Ω ) , (13) ξ ( R ) d Φ (0) ( R ) dR + Φ (2)0 ( R ) −
13 Ω R = 0 , (Ω & l = 0) , (14) ξ ( R ) d Φ (0) ( R ) dR + Φ (2)2 ( R ) + 13 Ω R = 0 , (Ω & l = 2) . (15)As the Eq.-(4) contains only sin Θ term in the right handside we get equations for l = 0 & 2 only. The Eqs.(10-15) altogether describe the equilibrium configuration ofthe rotating star with ξ l and Φ (2) l ( l = 0 ,
2) being theunknowns and ξ l = 0 , Φ (2) l = 0 , f or l > C. Equations of Spherical Background
The equations corresponding to Ω i.e. Eqs.-10 & 13together represent the background spherical configura-tion or the non-rotating configuration on which the ro-tation is assumed to be a perturbation. Simplification ofthese equations gives, dp (0) ( R ) dR = − ρ (0) ( R ) GM (0) ( R ) R ,dM (0) ( R ) dR = 4 πR ρ (0) ( R ) , (17)which is the non-relativistic limit ( c →∞ ) of the Tolman-Oppenheimer-Volkov [23, 24] equations. M (0) ( R ) inthese equations, is simply the mass within radius R ofthe spherical configuration and corresponding momentof inertia I (0) ( R ) can be found from the equation, I (0) ( R ) = 8 π Z R ρ (0) ( R ) R dR (18)Boundary conditions used are, ρ (0) ( R ) → ρ c & M → R → p (0) → R → a , with a beingthe radius of the unperturbed star and ρ c is the centraldensity of the configuration. It is worth mentioning thefact that, the Hartle formalism allows one to determinethe physical quantities like mass, radius, moment of in-ertia etc, for the rotating configuration with the samecentral density ρ c as the non-rotating configuration. Italso turns out that, the gravitational potential inside thestar is connected to the mass in the following way, d Φ (0) ( R ) dR = GM (0) ( R ) R (19) D. Equations of l=0 Deformation
The total mass of the star M tot ( R ) of the rotating starcan be defined as follows, M tot ( R ) = Z V ρ ( r, θ ) r sin θdrdθdφ = Z V ρ (0) ( R )( R + ξ ) sin Θ( dR + dξ ) d Θ dφ = Z V ρ (0) ( R ) R sin Θ dRd Θ dφ + Z V ρ ( R ) R (cid:18) ξ ( R, Θ) R + dξ ( R, Θ) dR (cid:19) sin Θ dRd Θ dφ. (20)The first integral in the right hand side of this equationis clearly providing the definition of M (0) ( R ) as given inEq.-(17). The second integral is defined as M (2) ( R ) andcan be further simplified with the help of the decompo-sition of ξ ( R, Θ) (Eq.-(9)), as follows, M (2) ( R ) = 4 π Z R ρ ( R ) R (cid:18) ξ ( R ) R + dξ ( R ) dR (cid:19) dR = 4 π Z R (cid:18) − ξ ( R ) dρ (0) ( R ) dR (cid:19) R dR. (21)Eqs.-11 & 14 provide the perturbation terms ∼ Ω corre-sponding to l = 0, represent the spherical deformation ofthe star due to the rotation. Simplifying these two equa-tions and incorporating the definition of M (2) ( R ) we get, − dp ∗ ( R ) dR + 23 Ω R = GM (2) ( R ) R ,dM (2) ( R ) dR = 4 πR ρ (0) ( R ) dρ (0) dp (0) p ∗ ( R ) , (22)where M (2) ( R ), the extra mass of the rotating star overits spherical background, can be supported with the cen-trifugal force while the rotating star is having the samecentral density as its non-rotating background. The newvariable p ∗ ( R ) is defined by, p ∗ ( R ) = ξ ( R ) d Φ (0) ( R ) dR . (23)The equation set Eq.-22 is solved with the boundary con-ditions M (2) ( R ) → R → p ∗ ( R ) → Ω R as R →
0. The perturbation in the potential Φ (2)0 ( R ) isthen given by, d Φ (2)0 ( R ) dR = GM (2) ( R ) R (24) E. Equations of l=2 Deformation
Eqs.-12 & 15, the equations provide the perturba-tion terms ∼ Ω corresponding to l = 2, represent thequadrupolar deformation of the star due to the rotation.Simplification of these equations and introduction of thenew variable χ provide, dχ ( R ) dR = − GM (0) ( R ) R Φ (2)2 ( R ) + 8 π G Ω ρ (0) ( R ) R ,d Φ (2)2 ( R ) dR = (cid:18) πR ρ (0) ( R ) M (0) ( R ) − R (cid:19) Φ (2)2 ( R )+ 4 π M (0) ( R ) Ω ρ (0) ( R ) R − χ ( R ) GM (0) ( R ) . (25)This set of equations are integrated numerically outwardswith necessary conditions which turn out to be, as R → (2)2 ( R ) → AR & χ ( R ) → BR , (26)where A and B are arbitrary constants related through,2 πG Aρ c + B = 2 πG ρ c . (27)The other condition applies is, Φ (2)2 ( R ) → R → ∞ .This large R behavior of Eqs.-(25) shows, as R → ∞ , χ ( R ) → K GM (0) ( a )2 R & Φ (2)2 ( R ) → K R , (28) where the M (0) ( a ) is the total mass of the unperturbedstar and K is the arbitrary constant need to be de-termined from the condition of continuity of χ ( R ) andΦ (2)2 ( R ) at the surface of the star. III. CALCULATION OF PHYSICALQUANTITIES
Having determined all the equations of structure, inthis section all the required physical quantities for thepresent work will be summarized.
A. Mass and Radius
Following the coordinate transformation in Eq.-(5) theradius of the star at surface can be expressed as, r ( a, Θ) = a + ξ ( a ) P (cos Θ) + ξ ( a ) P (cos Θ) , (29)where Θ = 0 or π represent the polar points and theequator is represented by Θ = π/
2. Hence, the polarradius r p and the equatorial radius r e are given by, r p = r ( a,
0) = r ( a, π ) = a + ξ ( a ) + ξ ( a ) ,r e = r ( a, π/
2) = a + ξ ( a ) − ξ ( a ) / , (30)where ξ ( a ) is found from Eq.-(23). As the solution ofΦ (2)2 ( R ) is found by solving Eq.-(25), ξ ( a ) can easily bedetermined using Eq.-(15), i.e. ξ ( a ) = − Φ (2)2 ( R ) + Ω R d Φ (0) ( R ) dR ! R = a (31)The eccentricity of the rotating spheroid is given through, e = s − (cid:18) r p r e (cid:19) . (32)The definition of the total mass is given in Eq.-(20).Hence, we can write the total mass of the rotating con-figuration M tot as, M tot = M ( a ) + M ( a ) (33)= Z a πR ρ (0) ( R ) dR + Z a πR ρ (0) ( R ) dρ (0) dp (0) p ( ∗ )0 ( R ) dR. B. Total Moment of Inertia
In the similar fashion of Eq.-(20), we can write theexpression for the total moment of inertia I tot as follows, I tot ( R ) = Z V ρ ( r, θ )( r sin θ ) r sin θdrdθdφ = Z V ρ (0) ( R )( R + ξ ) sin Θ( dR + dξ ) d Θ dφ = Z V ρ (0) ( R ) R sin Θ dRd Θ dφ + Z V ρ (0) ( R ) R sin Θ (cid:18) ξ ( R, Θ) R + dξ ( R, Θ) dR (cid:19) dRd Θ dφ. (34)The first integral can easily be recognized as the I (0) ( R )of Eq.-(18). Hence, the total moment of inertia of therotating star is, I tot ( a ) = I (0) ( a ) + I (2) ( a ) (35)where I (2) ( a ) is the correction in the moment of inertia ∼ Ω and can be expressed as, I (2) ( a ) = a Z π Z π Z ρ (0) ( R ) R (cid:20)(cid:18) ξ ( R ) R + 4 ξ ( R ) R P (cos Θ) (cid:19) + (cid:18) dξ ( R ) dR + dξ ( R ) dR P (cos Θ) (cid:19)(cid:21) sin Θ dRd Θ dφ = 8 π Z a ρ (0) ( R ) R (cid:20) R (cid:18) ξ ( r ) − ξ ( R ) (cid:19) + (cid:18) dξ ( R ) dR − dξ ( R ) dR (cid:19)(cid:21) dR = 8 π Z a (cid:18) ξ ( R ) − ξ ( R ) (cid:19) dρ (0) ( R ) dR R dR. (36) C. Quadrupole Moment
As the potential of the rotating spheroid, Φ( R, Θ), isgiven in Eq.-(9), one can write the potential for
R > a as follows,Φ( R, Θ) = Φ (0) ( R ) + Φ (2)0 ( R ) + Φ (2)2 ( R ) P (cos Θ) . (37)The behavior of the terms Φ (0) ( R ), Φ (2)0 ( R ) for R > a can be found from Eqs.-(19) & (24), respectively. Onthe other hand, for the external behavior of Φ (2)2 ( R ) oneneeds to look into the Eq.-(28). Therefore,Φ( R, Θ) = − GM (0) ( a ) R − GM (2) ( a ) R + K R P (cos Θ)= − G M tot R + K R P (cos Θ) . (38)In the coefficient of the P (cos Θ), the denominator de-pends on R and therefore, this term represents the con-tribution of the quadrupole moment in the potential.Hence, the quadrupole moment Q of the rotating starcan be defined as, Q = K G , (39)where K has already been fixed from the condition ofcontinuity of χ ( R ) and Φ (2)2 ( R ) at the star’s surface inEq.-(28). D. Rotational Love Number
A constant density spherical surface of radius R be-comes oblate shaped spheroid under the action of rota-tion and therefore, its equatorial radius r e ( R ) becomesdifferent from its polar radius r p ( R ). This deformationis quantified through ellipticity ǫ ( R ) and is defined as[25], ǫ ( R ) = r e ( R ) − r p ( R ) R ⇒ ǫ ( R ) = − ξ ( R )2 R (40)This quantity satisfies the following equation [25, 26], M ( R ) R d ǫ ( R ) dR + 2 R dM ( R ) dR dǫ ( R ) dR +2 dM ( R ) dR ǫ ( R ) R − M ( R ) ǫ ( R ) R = 0 . (41)Introducing the average mass density variable, ρ m ( R ) =3 M (0) ( R ) / πR , for the non-rotating star, it can beshown that, the Eq.-(41) takes the form [27], R d ǫ ( R ) dR + 6 ρ ( R ) ρ m ( R ) (cid:20) R dǫ ( R ) dR + ǫ ( R ) (cid:21) − ǫ ( R ) = 0 . (42)Then Eq.-(42) can be shown to take the form, R dη ( R ) dR +6 D ( R )[ η ( R )+1]+ η ( R )[ η ( R ) −
1] = 6 , (43)where D ( R ) = ρ ( R ) /ρ m ( R ) and η ( R ) is the new variablewhich is given by, η ( R ) = Rǫ ( R ) dǫ ( R ) dR . (44)Now, the Eq.-(43) is integrated outwards from the centerwith the obvious conditions, D ( R ) → η ( R ) → → . (45)It is worth mentioning the fact that, the η ( R ) turns outto be Ω independent and hence the rotational apsidalconstant k is defined as, k = [3 − η ( a )]2[2 + η ( a )] . (46)Finally, the tidal love number λ is defined by the relation, λ = 23 G a k (47) E. The Angular Velocity Ω The slow rotation approximation here, is assumed tobe valid up to angular velocity Ω K , the Keplerian angular M ( M S un ) ρ c (gm/cc) Static He(0B c )Static He(5B c )Static He(10B c )Rotating He(0B c )Rotating He(5B c )Rotating He(10B c ) FIG. 1: Plots of the mass of the Helium white dwarfs withrespect to the central density for different magnetic fieldstrengths. M ( M S un ) ρ c (gm/cc) Static C(0B c )Static C(5B c )Static C(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c ) FIG. 2: Plots of the mass of the Carbon white dwarfs withrespect to the central density for different magnetic fieldstrengths. velocity. It is determined from,Ω K = s GM tot r e . (48)The initial value for Ω K to start the computation is cho-sen to be the one corresponding to the Keplerian angularvelocity of the non-rotating spherical configuration i.e. p GM (0) ( a ) /a . After each step of computation Ω K iscomputed from Eq.-(48) and using this value of Ω K thewhole computation is carried out again. Until, a certainlevel of accuracy is achieved this process goes on and thevalues of the physical quantities, provided in the finalstep, are considered for further investigation. M ( M S un ) ρ c (gm/cc) Static O(0B c )Static O(5B c )Static O(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c ) FIG. 3: Plots of the mass of the Oxygen white dwarfs withrespect to the central density for different magnetic fieldstrengths. M ( M S un ) ρ c (gm/cc) Static Fe(0B c )Static Fe(5B c )Static Fe(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 4: Plots of the mass of the Iron white dwarfs with respectto the central density for different magnetic field strengths. M ( M S un ) r e (10 Km)
Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Static He(0B c )Static He(5B c )Static He(10B c ) FIG. 5: Mass-Radius relationship for Helium white dwarfs fordifferent magnitudes of magnetic field. M ( M S un ) r e (10 Km)
Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Static C(0B c )Static C(5B c )Static C(10B c ) FIG. 6: Mass-Radius relationship for Carbon white dwarfs fordifferent magnitudes of magnetic field. M ( M S un ) r e (10 Km)
Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Static O(0B c )Static O(5B c )Static O(10B c ) FIG. 7: Mass-Radius relationship for Oxygen white dwarfsfor different magnitudes of magnetic field. M ( M S un ) r e (10 Km)
Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c )Static Fe(0B c )Static Fe(5B c )Static Fe(10B c ) FIG. 8: Mass-Radius relationship for Iron white dwarfs fordifferent magnitudes of magnetic field. I( g m . c m ) ρ c (gm/cc) Static He(0B c )Static He(5B c )Static He(10B c )Static C(0B c )Static C(5B c )Static C(10B c )Static O(0B c )Static O(5B c )Static O(10B c )Static Fe(0B c )Static Fe(5B c )Static Fe(10B c )Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 9: Plots of moment of inertia for Helium, Carbon, Oxy-gen, Iron white dwarfs with respect to varying central densityunder different field strengths. I / ( M R ) ρ c (gm/cc) Static He(0B c )Static He(5B c )Static He(10B c )Static C(0B c )Static C(5B c )Static C(10B c )Static O(0B c )Static O(5B c )Static O(10B c )Static Fe(0B c )Static Fe(5B c )Static Fe(10B c )Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 10: Plots of normalized moment of inertia for Helium,Carbon, Oxygen, Iron white dwarfs with respect to varyingcentral density under different field strengths. I- ρ c (gm/cc) Static He(0B c )Static He(5B c )Static He(10B c )Static C(0B c )Static C(5B c )Static C(10B c )Static O(0B c )Static O(5B c )Static O(10B c )Static Fe(0B c )Static Fe(5B c )Static Fe(10B c )Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 11: Plots of dimensionless moment of inertia for Helium,Carbon, Oxygen, Iron white dwarfs with respect to varyingcentral density under different field strengths. λ ( g m . c m . s ) ρ c (gm/cc) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 12: Plots of love number for Helium, Carbon, Oxygen,Iron white dwarfs with respect to varying central density un-der different field strengths. λ - ρ c (gm/cc) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 13: Plots of dimensionless love number for Helium, Car-bon, Oxygen, Iron white dwarfs with respect to varying cen-tral density under different field strengths. ecce n t r i c it y ρ c (gm/cc) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 14: Plots of eccentricity for Helium, Carbon, Oxygen,Iron white dwarfs with respect to varying central density un-der different field strengths. Q ( g m . c m ) ρ c (gm/cc) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 15: Plots of spin quadrupole moment for Helium, Car-bon, Oxygen, Iron white dwarfs with respect to varying cen-tral density under different field strengths. Q / ( M R ) ρ c (gm/cc) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 16: Plots of normalized spin quadrupole moment forHelium, Carbon, Oxygen, Iron white dwarfs with respect tovarying central density under different field strengths. Q - ρ c (gm/cc) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 17: Plots of dimensionless spin quadrupole moment forHelium, Carbon, Oxygen, Iron white dwarfs with respect tovarying central density under different field strengths. l n (I-) ln( λ -) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 18: Plots of logarithmic dimensionless moment of iner-tia with respect to logarithmic dimensionless love number fordifferent white dwarfs and different field strengths. l n (I-) ln(Q- ) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 19: Plots of logarithmic dimensionless moment of inertiawith respect to logarithmic dimensionless quadrupole momentfor different white dwarfs and different field strengths. l n ( Q -) ln( λ -) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 20: Plots of logarithmic dimensionless quadrupole mo-ment with respect to logarithmic dimensionless love numberfor different white dwarfs and different field strengths. I / ( M R ) eccentricity Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 21: Plots of normalized moment of inertia with respectto eccentricity for different white dwarfs and different fieldstrengths. I / ( M R ) Q/(MR ) Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 22: Plots of normalized moment of inertia with respectto normalized quadrupole moment for different white dwarfsand different field strengths. Q / ( M R ) eccentricity Rotating He(0B c )Rotating He(5B c )Rotating He(10B c )Rotating C(0B c )Rotating C(5B c )Rotating C(10B c )Rotating O(0B c )Rotating O(5B c )Rotating O(10B c )Rotating Fe(0B c )Rotating Fe(5B c )Rotating Fe(10B c ) FIG. 23: Plots of normalized quadrupole moment with respectto eccentricity for different white dwarfs and different fieldstrengths. IV. THE EQUATION OF STATE
The equation of state that we adopted here is the onedeveloped in Ref.-[28] i.e. the extension of the Feynman-Metropolis-Teller(FMT) treatment to treat magnetizedsuper-Chandrasekhar white dwarfs. For this each of theatomic configuration with atomic number Z and massnumber A has been considered as Wigner-Seitz cell andthe electrons within the cell occupy Landau quantizedstates when subjected to magnetic field B and the maxi-mum number of particles per Landau level per unit areais eB (2 s +1) hc . When the magnetic field B is in z-directionthe Fermi energy E F of the electron at ν th Landau levelis given by, E F = (cid:2) p F ( ν ) c + m e c (1 + 2 νB D ) (cid:3) − m e c − eV ( r )(49)where e is the electronic charge, m e is the electronic restmass, p F is the Fermi momentum of the electron, c isthe speed of light and V ( r ) is the Coulomb potential atradius r . The term B D = B/B c , with B c being themagnetic field corresponding to which the Landau quan-tization energy becomes equal to that of the electron restmass energy.The number density of electrons under the influence ofCoulomb screening is given by, n e ( r ) = 2 B D (2 π ) λ e ν m X ν =0 g ν x F ( ν ) (50)where g ν is the degeneracy of the ν th Landau level and x F ( ν ) = p F ( ν ) cm e c . ν m is the highest occupied landau leveldetermined from positive semi-definiteness of p F ( ν ). Theoverall Coulomb potential V ( r ) can be obtained by solv-ing the Poisson equation ∇ V ( r ) = − πe [ n p ( r ) − n e ( r )] ⇒ ∇ b V ( r ) = − πe [ n p ( r ) − n e ( r )] (51)where n p ( r ) = 3 Z/ πR c is the constant proton densitywithin the nuclear radius R c and ˆ V ( r ) = eV ( r ) + E F .Introducing the dimensionless quantities x = rλ π and y ( x ) = r b V ( r )¯ hc = x b V ( x ) m π c , the Eq.(51) can be rewritten as,1 x d y ( x ) dx = − αθ ( x c − x )∆ + 2 e B D π ¯ hc (cid:18) m π m e (cid:19) (cid:18) λ π λ e (cid:19) × ν m X ν =0 g ν (cid:20)(cid:18) − νν m (cid:19) (cid:26)(cid:16) yx (cid:17) + 2 (cid:18) m e m π (cid:19) (cid:16) yx (cid:17)(cid:27)(cid:21) , (52)where ¯ h is the Planck’s constant, α = e / ¯ hc is the finestructure constant, ∆ = R c /λ π Z , θ ( x c − x ) is theHeavyside step function with x c = R c /λ π , m π is the pion rest mass, λ π = ¯ hc/m π c is the Compton wave-length for pion and the Compton wavelength for electronis λ e = ¯ hc/m e c . The Eq.(52) is integrated numericallyto find out the electronic distribution within the cell andother subsequent quantities where the constraints andboundary conditions arise out of the physical require-ments of the system itself. More details on the numericalintegration and electron distribution within the Wigner-Seitz cell can be found on [28, 29] and references therein.The kinetic energy density including the electronic restmass within the Wigner-Seitz cell then turns out to be, ε k ( x ) = B D m e c π λ e ν m X ν =0 g ν (1 + 2 νB D ) ψ (cid:18) x F ( ν )(1 + 2 νB D ) / (cid:19) (53)where it should be noted that, ψ ( z ) = Z z (1 + y ) / dy = 12 [ z p z + ln( z + p z )] . (54)The total kinetic energy E k of the cell excluding the elec-tronic rest mass can be calculated as, E k = 4 πλ π Z x WS x c x [ ε k ( x ) − n e ( x ) m e c ] dx (55)where R W S is the radius of the Wigner-Seitz cell and x W S = R W S /λ π .The total potential energy E c of the cell can be evalu-ated using, E c = 4 πλ π Z x WS x [ n p ( x ) − n e ( x )] eV ( x ) dx ⇒ E c = − πλ π Z x WS x c x n e ( x ) eV ( x ) dx (56) ⇒ E c = − πλ π m π c Z x WS x c xn e ( x ) y ( x ) dx + E F Z. The energy density ε can now be given by ε = E k + E c + M ( A, Z ) c π R W S + B π (57)where M ( A, Z ) is atomic mass of the uncompressed atomand the last term accounts for the magnetic contributionto the energy density. The pressure P is simply given by, P = ( B D m e c π λ e ν m X ν =0 g ν (1 + 2 νB D ) η (cid:18) x F ( ν )(1 + 2 νB D ) / (cid:19) + B π (cid:27) x WS (58)1where the function η is defined as, η ( z ) = z p z − ψ ( z )= 12 [ z p z − ln( z + p z )] . (59) A. Onset of β -instability High central density may lead to the onset of electroncapturing by the nucleus and hence, the inverse β -decaysets in. In this work the highest value for the central den-sity ρ c has been chosen to be the critical limit where theinverse β -decay just starts. The limits for He, C, Oand Fe have been tabulated in Table-I, where ǫ β ( Z )is the experimental inverse β -decay energy [28, 30–33], ρ β,unifcrit is the critical density calculated assuming uni-form electron distribution inside Wigner-Seitz cell and ρ β,relF MTcrit is the one calculated under magnetic field as-suming non-uniform electron distribution. TABLE I: The critical limits for the onset of β -instability.Decay channel ǫ β ( Z ) ρ β,relF MTcrit (5B c ) ρ β,unifcrit MeV g cm − g cm − He → H+n →
4n 20.596 1.368 × × C → B → Be 13.370 3.812 × × O → N → C 10.419 1.785 × × Fe → Mn → Cr 3.695 1.139 × × B. Magnetic Field
On the basis of the discussion provided in Ref.-[28]the actual calculations have been performed with vary-ing magnetic field including the effects of energy densityand pressure arising due to magnetic field. The densitydependent magnetic field [34] inside white dwarf is takento be of the form, B D = B s + B [1 − exp {− β ( ρ/ρ ) γ } ] (60)where ρ is taken as ρ β,relF MTcrit /10 and β , γ are constants.We choose constants β = 0 . γ = 0 . B D at center of thewhite dwarf has been kept up to 10 B c which is 4 . × gauss [35, 36]. The parameter value B is fixed bysetting a particular value for the magnetic field B D at thecenter with the maximum central density i.e. ρ β,relF MTcrit .The surface magnetic field B s ∼ gauss estimated byobservations [37–39]. V. RESULTS, DISCUSSION AND CONCLUSION
In equilibrium configuration the gravitational pull isbalanced by the pressure and centrifugal force due torotation. The Figs.-1-4 are showing mass variations ofthe equilibrium configurations with the central densityfor Helium, Carbon, Oxygen, Iron white dwarfs, respec-tively. Here the rotational frequency of the star has beentaken to be the one determined from Eq.-(48), i.e. theKeplerian angular velocity. Magnetic field profile underwhich the calculations of equilibrium configurations havebeen carried out is given by Eq.-(60). The central mag-netic field for each of the configuration is the one foundby setting the ρ in Eq.-(60) equal to the central densityfor the configuration and other parameters of the equa-tion are determined following the description in Section-IV B. It can be seen from these figures that, the change inthe critical mass due to the presence of magnetic field isleast for He white dwarfs and highest for the Fe whitedwarfs. The rotation has its common effect in increasingthe critical mass. The decrement in the central densityshows continuous decrease of the mass of the equilibriumconfigurations irrespective of any applied magnetic fieldstrength. Unlike the general relativistic case He and Cwhite dwarfs show no peak in the mass-central densityplots. Therefore, having no secular instabilities for restof the calculations the ρ β,relF MTcrit has been taken as thehighest limit for the central density for all the elements.Figs.-5-8 are depicting the corresponding mass-radiusrelationships for He, C, O and Fe white dwarfs, re-spectively. Applied magnetic field and other parametersare the same as those used for Figs.-1-4. These figures areagain showing the minimum effect of the magnetic fieldon the He white dwarf mass-radius relationship and themaximum effect on the Fe white dwarf mass-radius re-lationship. According to the Eq.-(60) and the descriptionprovided in the Section-IV B, the central magnetic fieldis lower for lower central density and consequently it hasbeen found that the mass-radius relationship is more orless unaltered in the lower central density regime irrespec-tive of the field strength. Only the Fe white dwarf inFig.-8 retains some difference in the mass-radius relation-ship in the lower density region. This in turn proving thevalidity of the statement proposed in section-I that, themagnetic field pressure is small compared to the matterpressure and hence the pressure splitting due to magneticfield as well.Figs.-9-11 shows the moment of inertia counterpart forall the elements, where along y axes moment of inertiain unit of 10 gm.cm , normalized moment of inertia,dimensionless moment of inertia have been plotted, forboth the cases of rotating and non-rotating stars. InFig.-9 all the curves assume more or less the same fea-ture of initially increasing with increasing density andthen falling, whereas, in Figs.-10 & 11 all the curves fallsmoothly when central density is increased. The normal-ization of moment of inertia has been done by dividingthe moment of inertia by MR , where M and R are the2mass and equatorial radius of the non-rotating configura-tion, respectively. In Fig.-11 the dimensionless momentof inertia is given by, ¯ I = c G IM , (61)where I is I (0) for non-rotating configuration and I tot forthe rotating configuration with c and G being the speedof light and gravitational constant, respectively. In allthese figures rotation has its obvious effect in increasingmoment of inertia.From physics point of view it is intuitive that, more isthe compactness of the star less will be its deformabilityunder external field and rotation. For this work, as al-ready described in preceding paragraphs, secular instabil-ity does not occur. Hence, Fig.-12 & Fig.-13, the plots ofrotational tidal love number and dimensionless rotationaltidal love number with respect to the central density fordifferent elements under different field strengths, showthe nature of continuously increasing tidal deformabilitywith decreasing central density. On the other hand, Fig.-14, the plot of eccentricity, depicts the increasing natureof eccentricity while density is decreasing. Figs.-15-17 arethe plots showing the variations of quadrupole moment inunit of 10 gm.cm , normalized quadrupole moment anddimensionless quadrupole moment with respect to thecentral density for different elements under different fieldstrengths. The normalization of quadrupole moment isdone by dividing the spin quadrupolar moment by masstimes the equatorial radius square of the non-rotatingconfiguration. It is interesting to notice that, the tidaldeformability, eccentricity and quadrupole moments, allthese quantities follow almost an universal trend for allkind of white dwarfs except Iron white dwarfs under dif-ferent field strengths. The Iron white dwarfs only showsome deviations in all these cases. Due to presence ofmagnetic field, only the Iron white dwarfs show signif-icant changes in the mass-radius relationship. In pres-ence of magnetic field Iron white dwarfs consist of hugecores compared to other white dwarfs as the magneticfield energy and pressure contributions have been takeninto the equation of state explicitly [28]. Then, it turnsout that, the eccentricity in the higher central density re-gion increases for Iron white dwarfs [Fig.-14]. As, lower isthe eccentricity higher is the universality in the emergentproperties of the compact stars, the Iron white dwarfsdeviate most [1, 7, 8]. It is also worth mentioning that,higher the central density more is the universality in thetrends.Fig.-18 is the plot between the first pair of the pa-rameters (I-Love-Q) which follow the universality undercertain circumstances i.e. the dimensionless moment ofinertia is plotted against the dimensionless love number.The definition of dimensionless moment of inertia is given in Eq.-(61) and the dimensionless love number can be de-fined as follows, ¯ λ = c G λM . (62)All the calculated data points in this figure can well befitted to a straight line given by, ln( ¯ I ) = − . . λ ) and hence is obeying an universal rela-tionship. It is the case of the Iron again for which, therelative error is highest ∼ I ) = − . . Q ), it turnsout that, under high magnetic field and with higher cen-tral densities Iron white dwarfs tend to show some devia-tion from this trend. 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