Braking index of isolated uniformly rotating magnetized pulsars
Oliver Hamil, Jirina Stone, Martin Urbanec, Gabriela Urbancova
aa r X i v : . [ a s t r o - ph . H E ] M a r Compact Stars in the QCD Phase Diagram IV (CSQCD IV)September 26-30, 2014, Prerow, Germany
Braking index of isolated uniformly rotatingmagnetized pulsars
Oliver Hamil Jirina Stone , Martin Urbanec and Gabriela Urbancova , Department of Physics and Astronomy, University of Tenessee, Knoxville, Tenessee37996, USA Department of Physics, Oxford University, Oxford, United Kingdom Institute of Physics, Faculty of Philosophy and Sciences, Silesian University inOpava, CZ 74601 Opava, Czech Republic Speaker who presented the work described herein
The slowing down of rotating neutron stars has been observed and modeled fordecades. The simplest models relate the loss the kinetic rotational energy of thestar to the emission of magnetic radiation from a rotating dipolar magnetic field(MDR), attached to the star [1, 2, 3, 4, 5]. The calculated energy loss by a rotatingpulsar is assumed proportional to a model dependent power of Ω. This relation leadsto the power law ˙Ω = -K Ω n where n is called the braking index. The value of n canbe, in principle, determined from observation of higher-order frequency derivativesrelated to n by [6] n = Ω ¨Ω˙Ω (1)(2)When the star is taken as a magnetized sphere, rotating in vacuum , with a constantmoment of inerta (MoI) and a constant magnetic dipole moment, missaligned at afixed angle to its axis of rotation, n is equal to 3 (for derivation see Section 2).Extraction of the rotational frequency and its time derivatives from observationinvolves a detailed analysis of the time evolution of the pulses, and of the spectraand luminosity of radiation from the related nebulae in a wide range of wavelengths.Although data on many pulsars are available in the literature, there are only eightpulsars generally accepted to yield reliable data on the pulsar’s spin-down (see Ta-ble. 1, recent compilation [7] and Refs. therein). The third derivative is known onlyfor the Crab pulsar [8], and PSR B1509-58 [9].1SR Frequency n Ref.(Hz)B1509 −
58 6.633598804 2.839 ± − ± − ± ± ± −
69 19.8344965 2.140 ± − ± −
45 (Vela) 11.2 1.4 ± − ± n = 3 does not agree with observation. Therehave been many attempts to extend/modify the basics of the MDR model. Theseinclude consideration of magnetic field activity (e.g. [12, 18, 19, 20, 21, 22, 23]), su-perfluidity and superconductivity of the matter within pulsars (e.g. [24, 25, 26]), andmodifications of the power law and related quantities (e.g. [27, 7]). Time dependenceof the constants in the MDR model has also been considered [18, 28, 29, 30]. Inparticular, time evolution of the inclination angle between spin and magnetic dipoleaxes as been recently addressed [6, 31]. However, there is no model currently avail-able which would yield, consistently, the typical spread of values of n as illustratedin Table 1.In this work we focus on determination of the maximum deviation of the brakingindex from the value n = 3 by introducing two modifications of the simple MDRmodel: frequency dependence of MoI, related to the change of shape of a deformablestar due to rotation, and the macroscopic effect of superfluidity of the pulsar core.The correction to the expression the braking index arising from these modificationsfollowing Glendenning [32] is derived, and included in the calculation, using fourrealistic Equations of State (EoS) over range of baryonic mass (M B ) . We studythe relation between the softness of the EoS and the rate of change of the brakingindex as a function of frequency and the M B ). The four EoS’s were also used to obtainmass density profiles of the pulsars needed to determine the transition region betweenthe crust and core. These results were utilized in simulation of an effect of superfluidconditions which eliminate the angular momentum exchange at the threshold betweenthe crust and core. The calculation is performed over a full range of frequencies ofthe pulsar from zero to the Kepler frequency and a range of M B from 1.0 to 2.2 M ⊙ ,representing the gravitational mass range from about 0.8 to 2.0 M ⊙ .2 Simple MDR Model
The total energy loss by a rotating magnetized sphere can be expressed in terms ofthe time derivative of the radiated energy as [2, 32, 6] dEdt = − µ Ω sin α, (3)where µ is the magnetic dipole moment of the pulsar, µ = B R . R is the radialcoordinate of a surface point with the surface magnetic field strength B , Ω is therotational frequency, and α is the angle of inclination between the dipole momentand the axis of rotation [32].Substituting the kinetic energy of a rotating body, dependent on the MoI I , E = 12 I Ω , (4)into (3) yields ddt (cid:18) I Ω (cid:19) = − µ Ω sin α. (5)Assuming constant MoI, dI/dt = 0, we get˙Ω = − µ I Ω sin α. (6)Setting K = µ I sin α in (6) and taking µ and α constant leads to the commonlyused braking power law describing the pulsar spin-down due to dipole radiation:˙Ω = − K Ω . (7)Differentiating (7) with respect to time¨Ω = − K Ω ˙Ω , (8)and combining (7) and (8) to eliminate K we get the value of the braking index nn = Ω ¨Ω˙Ω = 3 . (9) The simple MDR value n = 3 (9) is derived taking the I , µ and α as independentof frequency and constant in time. However, in reality the MoI of rotating pulsarschanges with frequency and, consequently with time. [33, 32]. The equlibrium state3f a rotating pulsar includes the effect of centrifugal forces, acting against gravity.The shape of the pulsar is ellipsoidal with decrease (increase) in radius along theequatorial (polar) direction with respect to the rotation axis as pulsar spins down.Thus the MoI, and, consequently, the braking index, are both frequency dependent.It is convenient to re-write (5) as ddt (cid:18) I Ω (cid:19) = − C Ω , (10)where C = µ sin α . Assuming this time that dI/dt is non-zero, differentiation of(10) with respect to time gives2 I ˙Ω + Ω ˙ I = − C Ω . (11)Differentiating once more gives2 I ¨Ω + 2 ˙Ω ˙ I + ˙Ω ˙ I + Ω ¨ I = − C Ω ˙Ω . (12)Using the chain rule we can write ˙ I in terms of ˙Ω dIdt = d Ω dt dId Ω , (13)and obtain ˙ I = I ′ ˙Ω (14)¨ I = ˙Ω I ′′ + I ′ ¨Ω , (15)where the primed notation represents the derivative with respect to Ω.Substituting the identities shown above into (11) and (12), we get the followingrelations for ˙Ω and ¨Ω, ˙Ω = − C Ω (2 I + Ω I ′ ) (16)¨Ω = − C ˙ΩΩ − ˙Ω (3 I ′ + Ω I ′′ )(2 I + Ω I ′ ) . (17)After some algebra it is easy to show that the expression of the braking index as afunction of angular velocity reads n (Ω) = Ω ¨Ω˙Ω = 3 − (3Ω I ′ + Ω I ′′ )(2 I + Ω I ′ ) . (18)We note that the magnetic dipole moment of the non-spherical pulsar may in prin-ciple also change with frequency. Estimation of this effect would require knowledgeof the origin and distribution of the dipole moment, which is lacking. We thereforeignore such change here and restrict ourselves to analysis of the two effects described.4 Calculation Method
Previous modeling of the braking index using the simple MDR model with constantMoI assumed a pulsar with 1 . M ⊙ gravitational mass and a radius ∼
10 km. Inthis work, which includes frequency dependent MoI and varying M B , we solve theequations of motion of rotating stars with realistic EoS using two different numericalmethods. The PRNS9 code, developed by Weber [35, 34], is based on a perturbative approachto the equations of motion of slowly rotating near-spherical objects [36, 37]. To en-sure the reliability of the PRNS9 code results, we also used the RPN code. Thiscode by Rodrigo Negreiros [38] is based on a publically available algorithm, RNS,developed by Stergioulas and Friedman [39]. The equations of motion are deriveddirectly from Einstein’s equations, following the Cook, Shapiro and Teukolsky ap-proach [40], described in detail in [41]. Both codes are applicable to rotating starswith all frequencies up to the Kepler limit.A comparison of the results of the two codes is demonstrated in Figure 1 whichshows MoI as a function of frequency for a pulsar with the QMC700 EoS and M B = 2.0 M ⊙ (see Section 3.2). They differ most, but by less than 10%, as the Keplerfrequency is approached. The small difference at near zero frequency (about 1.25%),due to the difference in behavior of the two low density EoS’s (see Section 3.2 ), isnegligible in the context of calculating neutron star macro-properties. An essential input to the calculation of macroscopic properties of rotating neutronstars is the EoS. The EoS is constructed for two physically different regimes, the highdensity core and the relatively low density crust.The microscopic composition of high density matter in the cores of neutron starsis not well understood. We have chosen two EoS’s, which assume that the core ismade only of nucleons, KDE0v1 [42] and NRAPR [43]. These EoS were selected byDutra et al. [44] as being among the very few which satisfied an extensive set ofexperimental and observational constraints on properties of high density matter. Inaddition, we use two more realistic EoS which include in the core the heavy strangebaryons (hyperons) as well as nucleons. The QMC700 EoS has been derived in theframework of the Quark-Meson-Coupling (QMC) model [45, 46] and the Hartree V(HV) EoS [47] is based on a relativistic mean-field theory of nuclear forces. Themaximum mass of a static star, calculated using the Tolman-Oppenheimer-Volkoff(TOV) equation, is 1.96, 1.93, 1.98 and 1.98 M ⊙ for KDE0v1, NRAPR, QMC7005
100 200 300 400 500 600 700 800 900 1000Frequency (Hz)160180200220240260 M o m e n t o f I n e r ti a ( k m ) PRNS9RPN
Figure 1: MoI as a function of frequency for a pulsar with M B = 2.0 M ⊙ as calculatedwith both, RPN and PRNS9 numerical codes.and HV, respectively, which is close to the gravitational mass of the heaviest knownneutron stars [48, 49]. The EoS’s are illustrated in Figure 2 which shows pressureas a function of energy density ǫ in units of nuclear saturation energy density ǫ =140 MeV/fm . We observe that the pressure increases as a function of energy densityalmost monotonically for KDE0v1, NRAPR and HV, whereas QMC700 EoS predictsa change in the rate of increase at about 4 ǫ . This change, and the subsequentsoftening of the EoS, happens at the transition energy density marking the thresholdfor appearance of hyperons in the matter. Such a change is not observed in theHV EoS. The main reason for the difference between the two hyperonic models isthat the QMC700 distinguishes between the nucleon-nucleon, and nucleon-hyperoninteractions (neglecting the poorly known hyperon-hyperon interaction), whereas theHV model uses a universal set of parameters for all hadrons. Inclusion of both theQMC700 and HV EoS in this work reflects the uncertainty in the theory of densematter in the cores of neutron stars. As detailed in the previous section, the calculation of the frequency dependence ofthe braking index has been done for a multiple combination of codes, EoS’s and M B of the rotating star. We show only typical examples of the results, usually for theQMC700 EoS, unless stated otherwise. 6 ε / ε P ( M e V / f m ) KDE0v1NRAPRQMC700HV
Figure 2: Pressure vs. energy density ǫ (in units of the energy density of symmetricnuclear matter at saturation ǫ ) as predicted by the four EoS’s used in this work. As a general feature, we find that any appreciable deviation of the braking indexfrom the generic value n = 3 is observed only at rotational frequencies higher thanabout 250 Hz. The sensitivity of this deviation to the EoS and M B is demonstratedin Figure 3. As can be seen in Figure 3, the biggest change in the braking index of a2.0 M ⊙ star pulsar is predicted by the HV EoS, followed by the QMC700, reaching ∼ values 1.75 and 2.15 at 750 Hz, respectively. The two nucleon-only EoS’s, KDE0v1and NRAPR behave in a very similar way and predict a larger value of n = 2.5 at thisfrequency. These trends can be directly related to the properties of the EoS’s. Figure3 shows the sensitivity to M B for the QMC700 EoS. The effect clearly increases withdecreasing M B . The effects demonstrated in Figure 3 were calculated assuming that the whole body ofa pulsar contributes to the total (core+crust) MoI. However, some theories suggest theconditions inside a pulsar are consistent with the presence of superfluid/superconductingmatter, both in the crust and in the core [24, 25, 26, 50]. Superfluid material wouldnot contribute to the rotation thus reducing the MoI.In this work we considered an extreme case in which the whole contribution ofthe core to the total MoI is removed. This scenario could be realized, for example, ifeither the whole core is superfluid or there is a layer of superfluid material between7
100 200 300 400 500 600 700
Frequency (Hz) B r a k i ng I nd e x KDE0v1NRAPRQMC700HV
Frequency (Hz) B r a k i ng I nd e x sun sun sun sun Figure 3: Braking index as a function of frequency. Left Panel: Braking index cal-culated for a pulsar with M B = 2.0 M ⊙ with all EoS’s adopted in this work. RightPanel: Braking index calculated of pulsars with M B = 1.0 - 2.2 M ⊙ .the core and the inner crust of the star, preventing an angular momentum transferbetween the core and the crust. Either scenario simply results in removal of thecontribution of the core to the MoI.Elimination of the core contribution can lead to a dramatic lowering of the totalMoI by more than a factor of three, as shown, as an example, in Figure 4 for the1.0 M ⊙ baryon mass and the QMC700 EoS. The difference between the total andcrust-only MoI shows a weak frequency dependence with a slight increase above about600 Hz. In turn, the reduction of the MoI by removal of the core contribution leads toadditional changes in the braking index, on top of the changes due to the frequencydependent MoI (see Figure 3) as shown in Figure 4. This change is, as expected,larger for higher mass stars which contain a more significant proportion of dense corematerial then for lower mass stars being more crust-like throughout. The variation of braking index of isolated rotating neutron stars with M B = 1.0 M ⊙ ,1.5 M ⊙ , 2.0 M ⊙ and 2.2 M ⊙ with rotational frequency from zero to the Kepler limitwithin the MDR model with frequency dependent MoI has been investigated. Themicrophysics of the star was included through utilizing realistic EoS’s of the pulsarmatter. An illustration of the possible effect of superfluidity in the star core has beenincluded in the study.Compiling results of all models used in this work, including the superfluidity effect,8
100 200 300 400 500 600 700
Frequency (Hz) M o m e n t o f I n e r ti a ( k m ) Total MoICrust-only MoI
Frequency (Hz) ∆ n sun sun Figure 4: Left panel: Total (crust and core) and the crust-only MoI as a function offrequency, calculated for a pulsar with M B = 1.0 M ⊙ . Right panel: ∆ n represent thedifference in braking index as a function of frequency between stars with and withoutcore contribution to the MoI. Each curve is displayed up to the Kepler frequency ofthe star.we deduce a definitive upper and lower limit on the braking index as a function offrequency, shown in Figure 5. The maximum change in the braking index is obtainedwith the QMC700 EoS and 1.0 M ⊙ , the least effect is found for KDE0v1 and the2.2 M ⊙ star. Reduction of the braking index from the simple MDR model value n = 3 happens only at frequencies that are some significant fraction of the Keplerfrequency. The calculation predicts that isolated pulsars with the braking index mostdeviating from n = 3 have low M B . For the frequencies of known isolated pulsars withaccurately measured braking indices (see Table 1), the reduction away from n = 3found in this model, is negligible.Finally, we have shown that the simple exclusion of the core due the superfluidity,or some superfluid barrier between the crust and core, does not have a strong effecton braking in the frequency range of observed isolated pulsars. Further developmentof the idea of a macroscopic description of superfluidity would be interesting. Changeof the magnetic field due to superfluidity and possible magnetic field expulsion, anda consequential increase in surface magnetic field strength B could also be usefullyexplored. 9
200 400 600 800 1000 1200
Frequency (Hz) B r a k i ng I nd e x KDE0v1 2.2M sun
QMC700 1.0M sun
Figure 5: Lower and upper limits on values of the braking index as a function of fre-quency, including results from both numerical codes, all EoS’s, M B , and the superfluidcondition. The (yellow) shaded area between the two lines defines the location of allresults within the limits. The pulsar with baryonic M B = 2.2 M ⊙ and the KDE0v1EoS has the highest Kepler frequency and defines the frequency limit in this work. Acknowledgement
We express our thanks to the organizers of the CSQCD IV conference for providingan excellent atmosphere which was the basis for inspiring discussions with all partic-ipants. We would like to acknowledge that this work was supported in part by CzechGrant No. GACR 209/12/P740 and Grant No. CZ.1.07/2.3.00/20.0071 ”Synergy,”aimed to foster international collaboration of the Institute of Physics of the SilesianUniversity, Opava. The research was also supported by the Department of Physicsand Astronomy, University of Tennessee.
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