The two component model of neutron star dynamics describing the behaviour of the observed crust coupled to the superfluid interior has so far been applied to radio pulsars for which the external torques are constant on dynamical timescales. We recently solved this problem under arbitrary time dependent external torques. Our solutions pertain to internal torques that are linear in the rotation rates, as well as to the extremely non-linear internal torques of the vortex creep model. Two-component models with linear or nonlinear internal

torques can now be applied to magnetars and to neutron stars in binary systems, with strong variability and timing noise. Time dependent external torques can be obtained from the observed spin-down (or spin-up)time series

Alpar, M. Ali

Gugercinoglu, Erbil

Gügercinoğlu, Erbil

Journal of Physics : Conference Series

arXiv

The two-component model describes neutron star dynamics incorporating the response of the superfluid interior. Conventional solutions and applications involve constant external torques, as appropriate for radio pulsars on dynamical time-scales. We present the general solution of two-component dynamics under arbitrary time-dependent external torques, with internal torques that are linear in the rotation rates, or with the extremely non-linear internal torques due to vortex creep. The two-component model incorporating the response of linear or nonlinear internal torques can now be applied not only to radio pulsars but also to magnetars and to neutron stars in binary systems, with strong observed variability and noise in the spin-down or spin-up rates. Our results allow the extraction of the time-dependent external torques from the observed spin-down (or spin-up) time series. Applications are discussed

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Neutron star dynamics under time-dependent external torques

Journal of Physics: Conference SeriesPAPER • OPEN ACCESSNeutron star dynamics under time dependentexternal torquesTo cite this article: M A Alpar and E Gügercinolu 2017 J. Phys.: Conf. Ser. 932 012036 View the article online for updates and enhancements.Related contentVORTEX CREEP AGAINST TOROIDALFLUX LINES, CRUSTAL ENTRAINMENT,AND PULSAR GLITCHESErbil Gügercinolu and M. Ali Alpar-Spin Evolution of PulsarsArmen Sedrakian and James M. Cordes-Torque measurement issuesJ Goszczak-This content was downloaded from IP address 212.252.164.95 on 07/01/2018 at 10:441Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distributionof this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.Published under licence by IOP Publishing Ltd1234567890International Conference Physics of Neutron Stars - 2017. 50 years after. IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 932 (2017) 012036  doi :10.1088/1742-6596/932/1/012036Neutron star dynamics under time dependentexternal torquesM A Alpar and E Gu¨gercinog˘luFaculty of Engineering and Natural Sciences, Sabancı University, Orhanlı -Tuzla, I˙stanbul34956, TurkeyE-mail: alpar@sabanciuniv.eduAbstract. The two component model of neutron star dynamics describing the behaviour ofthe observed crust coupled to the superﬂuid interior has so far been applied to radio pulsarsfor which the external torques are constant on dynamical timescales. We recently solved thisproblem under arbitrary time dependent external torques. Our solutions pertain to internaltorques that are linear in the rotation rates, as well as to the extremely non-linear internaltorques of the vortex creep model. Two-component models with linear or nonlinear internaltorques can now be applied to magnetars and to neutron stars in binary systems, with strongvariability and timing noise. Time dependent external torques can be obtained from the observedspin-down (or spin-up) time series, Ω˙(t).1. The two component model with constant external torquesThe two component model for neutron star dynamics was proposed immediately after theobservation of the ﬁrst pulsar glitches [1]. This model and its extension to nonlinear vortexcreep coupling between the crust and the superﬂuid [2] have been applied with remarkablesuccess to the postglitch relaxation data from radio pulsars, for which the external torques areconstant on dynamical timescales. This talk presents, in summary, our recent solution of thetwo component models for linear and nonlinear coupling for the general case of arbitrary timedependent external torques [3]. These general solutions are applicable to the timing behaviourof neutron stars in X-ray binaries and magnetars, as well as timing noise from radio pulsars.Postglitch exponential relaxation on timescales of days to weeks supported the theoreticalexpectations of superﬂuidity in the neutron star interior. As exponential relaxation is producedin linear systems, the original two component model of Baym et al [1] posits a linear couplingbetween the crust and the superﬂuid interior. The equations of the linear two component modelareIcΩ˙c + IsΩ˙s = Next, (1)Ω˙s = −Ωs − Ωcτ0= − ωτ0. (2)Ic and Is denote the moments of inertia and Ωc and Ωs are the rotation rates of the crust andthe superﬂuid, respectively. The rotational velocity lag between the two components is denotedby ω ≡ Ωs − Ωc. The internal torque that the superﬂuid exerts on the crust isNint ≡ −IsΩ˙s, (3)21234567890International Conference Physics of Neutron Stars - 2017. 50 years after. IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 932 (2017) 012036  doi :10.1088/1742-6596/932/1/012036while Next denotes the external torque on the neutron star. The crust (normal matter)rotation rate Ωc obeys the Navier Stokes Equation, but is already in rigid body rotation onsuperﬂuid timescales. The superﬂuid rotation rate Ωs does not obey the Navier Stokes Equation.Physically, the superﬂuid spins down by a ﬂow of quantized vortices away from the rotation axis.Under many kinds of physical interactions between the vortices and normal matter, the vortexcurrent, and the superﬂuid rate are linear in the angular velocity lag ω.Under a constant external torque Next, the system has a steady state,ω˙ = 0,Ω˙s = Ω˙c =NextI. (4)The steady state value ω∞ of the lag is determined byω∞τ0= −NextI. (5)The linear response to any glitch induced oﬀset δω(0) from steady state is exponential relaxationΩ˙c = Ω˙c(0)− IsIδω(0)τe−t/τ . (6)Several components i of the exponential relaxation are indeed observed following pulsar glitches.An analysis of a postglitch relaxation yields Is,i/I ∼ 10−3 − 10−2 for each component ofthe exponential relaxation. This points at the crust superﬂuid. With the eﬀective masses(entrainment) [4] taken into account, Is,i/I  10−1, pointing to the crust + outer core [5]. Thecore superﬂuid is already coupled tightly to the crust; the core is eﬀectively a part of the crust,dynamically, on glitch and postglitch relaxation timescales [6]. An analysis in terms of thetwo component model is enough: Since Is,i/I  1, the diﬀerent superﬂuid components i withmoments of inertia Is,i can be handled with the crust in separate two component models andthen the response of the crust to each can be superposed.2. Vortex creep and the nonlinear two component modelA superﬂuid component with vortex pinning can spin down by the thermally activated ﬂow(creep) of vortices against pinning potentials. The vortex creep model gives the spindown rateΩ˙s = −4Ωsv0rexp(−EpkT)sinh( ω)≡ −f(ω), (7)where ≡ kTEpωcr, (8)to replace (2). Here, Ep is the pinning energy and T is the temperature. The distance r ofthe vortex lines from the rotation axis is approximately equal to the neutron star radius R inthe crust superﬂuid where creep takes place against the pinning sites in the crust lattice. Themicroscopic vortex velocity v0 around pinning centres is ≈ 107 cm/s [7]. The two componentmodel is now nonlinear in the lag ω.The values of the parameters, in particular of Ep/kT , can be such that (7) requiressinh (ω/)  1, so that sinh (ω/)  (1/2) exp (ω/). Equations (1) and (7) then yieldΩ˙c =NextIc+IsIc2τlexp(ω/), (9)31234567890International Conference Physics of Neutron Stars - 2017. 50 years after. IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 932 (2017) 012036  doi :10.1088/1742-6596/932/1/012036whereτl ≡ kTEpRωcr4Ωsv0exp(EpkT). (10)For nonlinear coupling, the steady state value of the lag, ω∞, is given by (4) and (7).The solution for the crust spindown rate under a constant external torque Next = IΩ˙∞ isΩ˙c(t) =IIcΩ˙∞ − IsIcΩ˙∞[1 +[exp(t0τnl)− 1]exp(− tτnlIIc)]−1, (11)with a nonlinear creep relaxation timeτnl ≡ kTEpIωcrNext=kTEpωcr|Ω˙|∞, (12)and recoupling (waiting) timescalet0 ≡ IδωNext=δω|Ω˙|∞. (13)Nonlinear creep regions are responsible for glitches through vortex unpinning avalanches leadingto a decrease δΩs(r) in those superﬂuid regions through which the unpinned vortices moverapidly during the glitch. Angular momentum conservation leads to δΩs = (Ic/Is)ΔΩc  ΔΩc,and δω ∼= δΩs. Because of the very sensitive dependence of the creep rate on δω this glitchinduced oﬀset from the steady state lag ω∞ will stop creep altogether in the superﬂuid regionseﬀected. As the same external torque is now acting on less moment of inertia, the observedspindown rate of the crust will suﬀer a glitch associated jumpΔΩ˙cΩ˙c∼= IsIc. (14)Creep restarts after a waiting time t0 ∼= δΩs/|Ω˙|∞. This “Fermi function” response was predictedas a standard signature of the vortex creep response to a glitch involving vortex unpinning (seeﬁgure 4 in [2]) and later observed in the Vela pulsar [8]. The common, almost ubiquitous form ofnonlinear response is the constant second derivative Ω¨c interglitch timing behaviour observed inthe Vela and other pulsars [9,10]. This is the response to a uniform density of vortices unpinnedat the glitch leading to a range of waiting times throughout the superﬂuid regions. Such powerlaw behaviour is characteristic of nonlinear dynamics.Two component models with linear or nonlinear internal torques have so far been applied topost-glitch or inter-glitch timing behaviour of radio pulsars [9,11–14], where the external torque,with a secular (characteristic) timescale τc ≡ Ωc/2|Ω˙c| ∼ 103 − 106 yr is constant for timescalesof observed postglitch relaxation.While constant secular external torques are characteristic for radio pulsars, neutron stars inX-ray binaries, magnetars and transients exhibit strong variations in the observed spin-down orspin-up rates. This indicates variable external torques, including strong torque noise.3. Linear two-component model with a time dependent external torqueThe two component model postglitch response with a linear internal torque and a time varyingexternal torque is described byω˙ = −ωτ− Next(t)Ic. (15)41234567890International Conference Physics of Neutron Stars - 2017. 50 years after. IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 932 (2017) 012036  doi :10.1088/1742-6596/932/1/012036The relaxation time τ is determined by the physics of the internal torque. The solution isω(t) = e−t/τ[ω(0)− 1Ic∫ t0et′/τNext(t′) dt′],Ω˙c(t) =Next(t)Ic+IsIe(−t/τ)τ[ω(0)− 1Ic∫ t0e(t′/τ)Next(t′) dt′]. (16)4. Vortex Creep Model with a Time Dependent External TorqueFor vortex creep under a time dependent external torque (1) and (7) lead toω˙ = − I2Icτleω/ − Next(t)Ic. (17)In terms of y ≡ exp(−ω/), equation (17) becomesdydt− Next(t)Icy − I2Icτl= 0. (18)This has an integration factor exp(−X(t)Ic)whereX(t) =∫ t0Next(t′) dt′. (19)The solution for the angular velocity lag ω exhibits an exponential dependence on the postglitchlag ω(0) and on X(t), the cumulative angular momentum transfer by the external torque:e−(ω/) = e−ω(0)/ exp(X(t)Ic)+ exp(X(t)Ic)∫ t0I2Icτlexp(−X(t′)Ic)dt′. (20)From equations (1), (17), and (20) the response of nonlinear creep to a glitch in the presence ofa time dependent external torque is obtained,Ω˙c(t) =Next(t)Ic+IsIc2τl[exp(X(t)Ic− ω(0))+ exp(X(t)Ic)∫ t0I2Icτlexp(−X(t′)Ic)dt′]−1. (21)The nonlinear creep timescale and the waiting time are now time dependent, involving therunning time average 〈Next(t)〉 ≡ X(t)/t of the external torque:τnl(t) ≡ kTEpIωcr〈Next(t)〉 , (22)t0(t) ≡ Iδω(0)〈Next(t)〉 . (23)51234567890International Conference Physics of Neutron Stars - 2017. 50 years after. IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 932 (2017) 012036  doi :10.1088/1742-6596/932/1/0120365. ApplicationsEquations (16) and (21) show that the observed Ω˙c(t) gives Next(t)/Ic to lowest order inIs/Ic  0.1. The residuals after a ﬁrst ﬁt to the Ω˙c(t) time series will be Is/Ic times a certainconvolution of the external torque with the internal torque, as deﬁned in (16) or (21), allowingfor a consistency check of the model.Let us consider three kinds of time dependent external torques; (i) an exponentially decayingexternal torque, (ii) power law time dependence and (iii) timing noise.(i) Taking an exponentially decaying torque with time scale τd added to the preglitch externaltorque N0, we haveNext(t) = N0 + δNe−t/τd = IΩ˙∞ + δNe−t/τd . (24)For linear internal torques, (16) givesΩ˙c(t) = Ω˙∞ +δNIce−t/τd[1− IsIτdτd − τ]+IsIe−t/τ(ω(0)τ+IIcΩ˙∞ +δNIcτdτd − τ). (25)For vortex creep, the solution is obtained by substituting in (21) the integrated angularmomentum transferX(t) = N0t+ δNτd[1− e−t/τd ]. (26)(ii) For a power law torque with the index α added to the preglitch torque we haveNext(t) = N0 +δNtα0(t+ t0)α= IΩ˙∞ +δNtα0(t+ t0)α. (27)Then (16) givesΩ˙c(t) = Ω˙∞ +δNIctα0(t+ t0)α[1− IsI(t+ t0)(1− α)τ]+IsIe−t/τ(ω(0)τ+IIcΩ˙∞ +δNIct0(1− α)τ +IsIδNIctα0(1− α)τ2∫ t0et′/τ dt′(t+ t0)α−1), (28)for linear internal torques.The integrated angular momentum transfer X(t) isX(t) = N0t+δNt0α− 1[1− tα−10(t+ t0)α−1], (29)deﬁning the response with nonlinear internal torques through Equation (21).(iii) Take white torque noise:Next =∑iαiδ(t− ti), (30)where αi are amplitudes of torque variations. For linear coupling equations (16) and (30) leadto the power spectrumP (f) =1√2π[2〈α2〉I2c+(Is/I1 + (2πτf)2)(〈α〉〈ω〉Ic− 〈α2〉I2c)+((Is/I)21 + (2πτf)2)(2〈α2〉I2c+ 2〈ω2〉 − 〈α〉〈ω〉Ic)], (31)61234567890International Conference Physics of Neutron Stars - 2017. 50 years after. IOP PublishingIOP Conf. Series: Journal of Physics: Conf. Series 932 (2017) 012036  doi :10.1088/1742-6596/932/1/012036where 〈α〉 is the mean external torque variation amplitude and 〈ω〉 denotes the mean value ofthe angular velocity lag. This result was obtained earlier for the linear two component model[15–17]. To lowest order we ﬁnd the model power spectrum, a constant P (f) for white noise.To order Is/I we ﬁnd the power spectrum of the integrated process, which is ﬂat at low f anda random walk spectrum, P (f) ∝ f−2, at high frequencies f  τ−1. The ﬁrst ﬁt will givethe strength of the noise process, the term proportional to 〈α2〉. The residuals to order Is/Iwill constrain 〈α〉〈ω〉 and 〈ω2〉. The term proportional to (Is/I)2 can be neglected to a goodapproximation. Noise processes other than white noise can be handled similarly for the lineartwo component model, by inserting the noise model in (16) and calculating the power spectrum.For the nonlinear two component model, equation (21) does not lead to explicit expressions forany time dependent external torque, but a well deﬁned algorithm can be constructed to obtainthe power spectrum, as for the times series in cases (i) and (ii).AcknowledgmentsThis work was supported by the Scientiﬁc and Technological Research Council of Turkey(TU¨BI˙TAK) under the grant 113F354. M.A.A. is a member of the Science Academy (BilimAkademisi), Turkey.References[1] Baym G, Pethick C J, Pines D and Ruderman M 1969 Nature 224 872[2] Alpar M A, Anderson P W, Pines D and Shaham J 1984 ApJ 276 325[3] Gu¨gercinog˘lu E and Alpar, M A 2017 MNRAS 471 4827[4] Chamel N 2012 Phys. Rev. C 85 035801[5] Gu¨gercinog˘lu E and Alpar, M A 2014 ApJ 788 L11[6] Alpar M A, Langer S A and Sauls J A 1984b ApJ 282 533[7] Gu¨gercinog˘lu E and Alpar M A 2016 MNRAS 462 1453[8] Buchner S and Flanagan C 2008 American Institute of Physics Conference Series vol 983, ed C Bassa, ZWang, A Cumming and V Kaspi p 145[9] Akbal O, Alpar MA, Buchner S and Pines D 2017 MNRAS 469 4183[10] Yu M et al 2013, MNRAS 429 688[11] Alpar M A, Chau H F, Cheng K S and Pines D 1993 ApJ 409 345[12] Alpar M A, Chau H F, Cheng K S and Pines D 1996 ApJ 459 706[13] Akbal O, Gu¨gercinog˘lu E, S¸as¸maz Mus¸ S and Alpar M A 2015 MNRAS 449 933[14] Gu¨gercinog˘lu E 2017, MNRAS 469 2313[15] Lamb F K, Pines D and Shaham J 1978a ApJ 224 969[16] Lamb F K, Pines D and Shaham J 1978b ApJ 225 582[17] Baykal A, Alpar M A and Kızılog˘lu U¨ 1991, A&A 252 664

Neutron star dynamics under time dependent external torques

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Neutron star dynamics under time dependent external torques

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