Magnetospheric "anti-glitches" in magnetars
MMagnetospheric “anti-glitches” in magnetars
Maxim LyutikovDepartment of Physics, Purdue University, 525 Northwestern Avenue, West Lafayette, INThe Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street Toronto, Ontario,M5S 3H8 Canada
ABSTRACT
We attribute the rapid spindown of magnetar 1E 2259+586 observed by Archibald et al. (2013), termedthe “anti-glitch”, to partial opening of the magnetosphere during the X -ray burst, followed by changesof the structure of the closed field line region. To account for the observed spin decrease during the X -ray flare all that is needed is the transient opening, for just one period, of a relatively small fraction ofthe magnetosphere, of the order of only few percent. More generally, we argue that in magnetars all timingirregularities have magnetospheric origin and are induced either by (i) the fluctuations in the current structureof the magnetosphere (similar to the long term torque variations in the rotationally powered pulsars); or,specifically to magnetars, by (ii) opening of a fraction of the magnetosphere during bursts and flares - thelatter events are always accompanied by rapid spindown, an “anti-glitch”. Slow rotational motion of theneutron star crust, driven by crustal magnetic fields, leads beyond some twist limit to explosive instabilityof the external magnetic fields and transient opening of a large magnetic flux in a CME-type event, thepost-flares increase of magnetospheric currents accompanied by enhanced X -luminosity and spindown rate,changing profiles, as well as spectral hardening, all in agreement with the magnetospheric model of torquefluctuations.
1. Magnetars’ bursts and flares
Magnetar emission (Woods & Thompson 2006) is powered by dissipation of a non-potential (current-carrying)magnetic field (Thompson et al. 2002). The magnetic field exerts Lorentz force on the crust, which is balanced byinduced elastic stress. For strong enough magnetic fields, Lorentz force may induce a stress that exceeds the criticalstress of the lattice, leading to crustal failure. This leads to lattice failure of the crust (Horowitz et al. 2011), whichcan be either in a form of fast propagating crack (Thompson & Duncan 1995) (and sudden untwisting of the internalmagnetic field) or a slowly (plastically) propagating fault (Jones 2003) that leads to slow evolution of the externalmagnetic fields (Lyutikov 2006). In the latter case twisting of the external magnetic fields leads to a sudden relaxationof the twist outside of the star in analogy with solar flares and Coronal Mass Ejections (CMEs) (Forbes et al. 2006;Lyutikov 2006; Parfrey et al. 2012). Perhaps the best argument in favor of magnetic storage of energy outside of the staris a very short rise time of the 2004 giant flares, ∼ .
25 msec (Palmer et al. 2005). This points to the magnetosphericorigin of giant flares (Lyutikov 2006).Thompson et al. (2002) developed a model of the emission and spin-down behavior of the Anomalous X-ray Pulsars(AXPs) and Soft Gamma-ray Repeaters (SGRs) in which the dissipation of magnetic fields powers its unusually brightX-ray luminosity. In this model the magnetic fields anchored in a well conducting stellar interior are dissipated inthe magnetosphere. Wolfson (1995); Thompson et al. (2002); Pavan et al. (2009) constructed a self-similar force-freecurrent-carrying magnetosphere in which the current twist the field lines and causes them to expand with respect toa potential dipole field. Expansion of field lines results in a changing torque on the neutron star which is observedas a variable spin-down rate. Since the X-ray luminosity is proportional to the strength of the currents flowing in themagnetosphere, one expects that spin-down rate is positively correlated with luminosity. a r X i v : . [ a s t r o - ph . H E ] J un not crustal)events similar to the Solar flares and CMEs. According to Lyutikov (2006) magnetars bursts and flares are drivenby unwinding of the internal non-potential magnetic field which leads to a slow build-up of magnetic energy outsideof the neutron star. For large magnetospheric currents, corresponding to a large twist of the external magnetic field,magnetosphere becomes dynamically unstable on the Alfv´en crossing times scale of the inner magnetosphere. Thereconnection processes, e.g., the tearing mode, in the strongly magnetized plasma of magnetar magnetospheres developssimilarly to the non- relativistic plasmas (Lyutikov 2003; Komissarov et al. 2007).In addition, current carrying charges resonantly scatter thermal photons from the surface producing through multipleresonant scattering (resonant Comptonization) a non-thermal power-law X-ray tail (Lyutikov & Gavriil 2006; Fern´andez& Thompson 2007; Rea et al. 2008; Beloborodov 2013). The hardness of the power-law depends on the typical velocityof the current carrying particles and their density. For larger densities (and larger currents) a given photon has a largerprobability to be scattered at the cyclotron resonance and thus will experience greater amount of scattering, gainingenergy stochastically in each scattering through Doppler effect. Thus, there is a natural prediction of the model byLyutikov & Gavriil (2006), that the harness of the power-law X-ray tail should correlate positively with the spin downrate and the X -ray luminosity.As we discuss below, the observations of the rapid spindown of the magnetar 1E 2259+586 (Archibald et al. 2013)are in general agreement with the magnetospheric model. Thus, they are qualitatively different form glitches in therotationally powered pulsar, where they are driven by the presence of a faster component inside the neutron star.
2. Observations of the “anti-glitch” in magnetar 1E 2259+586
Archibald et al. (2013) reported a fast change of the rotation frequency of the magnetar 1E 2259+586, whichoccurred within two weeks of a bright X -ray flare. The accumulated change in frequency was ∆ ν ∼ − Hz over twoweeks. For approximately a few months after the flare both the flux and the spindown rate were about two times largerthan normal. In addition, the flux increase was also accompanied by hardening of the spectrum and a change in thepulse profile. Importantly, there was a GBM burst, presumably associated with the glitch, with peak luminosity of theorder 10 erg s − and duration of ∼
100 milliseconds.By analogy with glitches in radio pulsar, except, importantly, for the different sing of the period change, thepreferred interpretation of Archibald et al. (2013) for the rapid spindown is a glitch internal to the neutron star. Thisrequires a neutron star component that rotates slower than the crust. The origin of such a component is not obvious(see, though, Thompson et al. 2000).In radio pulsars the notion of a glitch is usually referred to a sudden change of the spin frequency, on time scalecomparable to the pulsar period. In magnetars, the timing measurements are usually separated by about a week. Incase of 1E 2259+586, the anti-glitch could be a sudden event, on the time scale of rotation, as well due to enhancedspindown rate by an order of magnitude over approximately a week.Below we discuss an interpretation alternative to that of Archibald et al. (2013). We argue that all the spin-downactivity, as well as spectrum hardening, flux increase and profile changes during the activity period, can be explainedvia purely magnetospheric effects within the twisted magnetosphere model of Thompson et al. (2002) described above,plus an additional effect of opening the magnetosphere during the prompt burst (Lyutikov 2006). Thus, we envisionthat a large spindown occurred during an opening of a modest fraction of the magnetosphere, on a time scale as shortas one rotation, followed by an extended period of high spin-down rate, driven by the magnetospheric currents. 3 –
3. Twisted Force-Free Equilibria
In this Section we review the solution of Wolfson (1995); Thompson et al. (2002) (see also Gourgouliatos (2008);Pavan et al. (2009)), giving a few simple approximations that are later used for numerical estimates. In a static force-freeapproximation the electric current flows along magnetic field, ∇ × B = α ( P ) B . (1)The non-rotating magnetic equilibria are then described by the second-order elliptic differential equation, the Grad-Shafranov equation (Shafranov 1966): ∇ (cid:18) r sin θ ∇P (cid:19) + 1 r sin θ I dId P = 0 (2)where poloidal and toroidal fields are B P = ∇P × ˆ φr sin θ , B T = − Ir sin θ ˆ φ (3) I = I ( P ) is the total current enclosed by the magnetic flux surfaces P . Choosing separable solutions, and assumingthat the current function is a power-law, I ∼ P (1+ p ) /p , Eq. (2) becomes r R (cid:48)(cid:48) rr R + (1 − µ ) F (cid:48)(cid:48) µµ F + C r r ∗ ( R F ) /p = 0 (4)where C is some dimensionless constant related to the strength of the currents ( I ∝ (cid:112) Cp/ ( p + 1) P (1+ p ) /p ) and r ∗ is aradius of a neutron star. Separable solutions of Eq. (4) satisfy R ∼ ˜ r − p p ( p + 1) F + (1 − µ ) ∂ F∂µ = − CF ( p +2) /p (5)where ˜ r = r/r ∗ has been introduced.The current-free case comprises the monopole solution p = 0 (with an equatorial current sheet), and vacuummultipolar solutions p = 1 , , .... . For non-integer p system (5) gives current-carrying solutions of type considered byLynden-Bell & Boily (1994). Magnetic fields and current densities are then given by B r B ∗ = − F (cid:48) ( µ )2˜ r p +2 B θ B ∗ = pF (cid:112) − µ ˜ r p +2 B φ B θ = − (cid:20) Cp (1 + p ) (cid:21) / F /p j φ cr ∗ πB ∗ = − CF ( p +2) /p (cid:112) − µ r (3+ p ) j θ cr ∗ πB ∗ = − (cid:115) pCp + 1 pF ( p +1) /p r ( p +2) (cid:112) − µ j r cr ∗ πB ∗ = − (cid:115) C (1 + p ) p r ( p +2) F /p F (cid:48) (6) 4 –Comparing with equation (1) shows that α ( P ) is proportional to P /p , and on dimensional grounds one can write α ( P ) = C / r ∗ (cid:18) p + 1 p (cid:19) / (cid:18) PP (cid:19) /p (7)The poloidal components of Eq. (1) can be integrated to give B φ = (cid:82) α ( P ) d P r sin θ = pp + 1 P α ( P ) r sin θ . (8)Substituting Eqs. (3)-(8) into the φ -component of Eq. (1) then gives the non-linear equation (5) for the angular function F = F ( µ ).The solution of Eq. (5), including the dependence p ( C ), is uniquely defined by the parameter C and by the three boundary conditions. The first is the requirement of zero B φ at the polar axis (absence of line current): B φ ( µ = 1) ∝ F (1) = 0 (nonzero B φ at the polar axis implies a line current), another is overall normalization of the field which we taketo be F (cid:48) = const = − µ = 1 (corresponding to a fixed flux density B ∗ at the magnetic pole). The third boundarycondition specifies the multipole structure of the magnetic field at small currents. For dipole field B R ∝ F (cid:48) = 0 at µ = 0,for quadrupole field B θ ∝ F = 0 at µ = 0.In the absence of current, C = 0, Eq. (5) with a boundary condition F (1) = 0 has solutions of a vacuum multipolefield, e.g. dipole field for p = 1 and F ∝ − µ = sin θ , or quadrupole for p = 2 and F ∝ µ (1 − µ ). Interestingly, thereis also a monopole solution, p = 0, F ∝ (1 − µ ). For nonzero current Eq. (5) has to be solved numerically.Presence of a current leads to twisting of magnetic field lines. A magnetic field line anchored at polar angle θ willtwist through a net angle ∆ φ ( θ ) = (cid:90) B φ B θ dθ sin θ = − (cid:20) Cp (1 + p ) (cid:21) / (cid:90) F /p dµ − µ (9)(integration is between the anchor points of a given field line) before returning to the stellar surface. Twisting of fieldsimply a net helicity H B = (cid:90) A · B dV = 3 π B ∗ R ∗ ) (cid:115) Cp ( p + 1) (cid:90) dµ [ F ( µ )] /p − µ . (10)where A is vector potential.Equation for flux surfaces can be derived from the equation d ln ˜ r/dθ = B r /B θ which gives˜ r = (cid:18) FF ∗ (cid:19) /p (11)For nonzero current Eq. (5) has to be solved numerically for functions F ( µ ) and p ( C ), Figs. 1, 2, 3. For each valueof C ≤ .
873 there are two solutions for p . The upper branch connects continuously to the vacuum dipole C = 0, p = 1( F = 1 − µ , B r , B θ ∝ / ˜ r − ), Eq. (14); and the lower branch connects to a split monopole C = 0, p = 0 configuration( F = 2(1 − µ ), B r = B ∗ / ˜ r , Appendix A).As the current increases the distribution of the poloidal magnetic flux evolves towards more spherical configuration,becoming isotropic in the limit p = 0 (Fig. 4). At the same time the distribution of a radial current at the surfaceevolves from an approximately isotropic to strongly concentrated near equator for small p . The field lines become moretwisted, but for small p → F ( µ ) for different values of p . For p = 0, F = 2(1 − µ ) almost everywhere except close to µ = 0point, see Appendix A 6 – p Fig. 2.— Function p ( C ). The extremum is at C = 0 . p = 0 .
63. Dashed line is a small twist approximation (Eq.(14)) 7 –Fig. 3.— The net twist angle between two poles versus current parameter C . Dotted line is the small current limit(Eq. (14)) 8 –little twist and also are straitened out (as compared with dipole) in the radial direction. A given field line has largesttwist where it is furthest from the star - at the equator.The radial dependence of the magnetic field softens to B ∝ r − . when ∆ φ N − S = 1 radian. The net twistapproaches ∆ φ N − S = π (one-half turn) in the split monopole limit ( p = 0). For comparison, a twisted quadrupolemagnetic field expands to infinity after 1 / √ p = 0 the field structureapproaches split monopole: F = 2(1 − µ ), with field line straightened out in the radial direction, B r /B ∗ = sign[ µ ] / ˜ r , B θ = 0 plus a current sheet in the equatorial plane which curries a surface current g φ = 2 B ∗ c/ (2 π ˜ r ). The magneticflux leaves the northern hemisphere and after reaching infinity returns to the southern hemisphere.The total current flowing through one hemisphere at radius ˜ r is given by integral of j r , I = 8 π eB ∗ r ∗ c (cid:115) Cpp + 1 1˜ r p F (0) ( p +1) /p ) . (12)Thus, a current that reaches to radius ˜ r for a given twist decreases as 1 / ˜ r p . On the other hand, the current flowingthrough the star (at ˜ r = 1) increases continuously with decreasing p and reaches I max = 8 pi B ∗ r ∗ /c in the limit p → p = 1 , C = 0 and F = 1 − µ . We find then p = 1 − C/
35 and F = (1 − µ ) (cid:18) − µ )(17 − µ ) C (cid:19) (13)In this approximation flux function α , magnetic field components and twist angle are given by α = √ C (1 − µ )˜ rB r B ∗ = µ ˜ r − C/ (cid:18) − µ + 5 µ ) C (cid:19) B θ B ∗ = (cid:112) − µ r − C/ (cid:18) − (15 + 22 µ − µ ) C (cid:19) B φ B ∗ = √ C (1 − µ ) / √ r ∆ φ = √ Cµ (14)These relations are useful for the order of magnitude estimates. For example, for small twists the total poloidal current I ∼ sin θ/ ˜ r is concentrated near the equator.The simple twisted self-similar magnetosphere described above is not a temporal sequence, but, importantly, itprovides simple and clear order-of-magnitude estimates of the effects involved. Numerical solutions (Parfrey et al. 2012)are in a general agreement with analytical estimates.
4. Variations of the spin-down rate4.1. The CME model of magnetar bursts
The processes that cause magnetar X-ray flares (and possibly the persistent emission) may be similar to thoseoperating in the Solar corona. The electric currents within the start are slowly pushed into the magnetosphere, gated by 9 – p (cid:61) (cid:61) (cid:61) (cid:61) Fig. 4.— Form of the magnetic flux surfaces for different p ; p = 1 case corresponds to a dipole, p = 0 is a monopolewith radial poloidal fields. 10 –slow, plastic deformations of the neutron star crust. This leads to gradual twisting of the magnetospheric field lines, ontime scales much longer than flares or burst, and creates active magnetospheric regions similar to Sun’s spots. Solutionsdescribed in Section 3 have energy stored in the magnetosphere that exceeds the energy of the dipolar configuration, Fig.5, so that the dissipation of the supporting currents can power the magnetar high energy activity. Initially, when theelectric current (and possibly magnetic flux) is pushed from the star into magnetosphere, the latter slowly adjusts to thechanging boundary conditions. As more and more current is pushed into the magnetosphere it eventually reaches a pointof dynamical instability beyond which the stable equilibrium can no longer be maintained. The loss of stability leads toa rapid restructuring of magnetic configuration, on the Alfv´en crossing time scale, formation of narrow current sheets,and onset of magnetic dissipation. As a result, a large amount of magnetic energy is converted into the bulk motionkinetic energy and radiation. Moreover, the change of magnetic topology allows formation of expanding magnetic loopsthat eventually break away from the star (Lyutikov 2006; Gourgouliatos & Lynden-Bell 2008).Adapting the the generally accepted magnetic breakout model of Solar Coronal Mass Ejections (Lynch et al. 2008)to magnetar environment, the underlying cause of all the manifestations of Solar activity - CMEs, eruptive flares andfilament ejections - is the disruption of a force balance between the upward pressure of the strongly sheared field of afilament channel and the downward tension of a potential (non-current carrying) overlying field. Thus, an eruption isdriven solely by magnetic free energy stored in a closed, sheared magnetic fields that open toward infinity during aneruption. Reconnection is thought to plays a critical role in opening of magnetic field lines: it removes the unshearedfield above the low-lying, sheared core flux, thereby allowing this core flux to burst open. Next we estimate parameter of the rapid spindown during an X -ray flare. Using the Goldrecih-Julian model, thespindown luminosity is related to the open magnetic flux Φ: L sd ≈ Φ Ω /c . In particular, for dipole magnetosphere,Φ ≈ B NS R NS (Ω R NS /c ). In the twisted magnetosphere the open flux is larger (Thompson et al. 2002), so that thespin-down rate is L sd ∼ B ∗ R ∗ c (cid:18) Ω R ∗ c (cid:19) p +1) , < p < ∝ Ω p +1 , n = 2 p + 1 , < n < n is the spindown index.The changes of the magnetospheric structure can naturally induce large changes in the spindown rate. For example,for a given surface magnetic field B and pulsar spin frequency Ω, the maximal theoretical spindown rate (correspondingto the monopolar fields, p = 0 , n = 1) is ˙Ω max = η − d ˙Ω d = 10 ˙Ω d η d = Ω r ∗ c ˙Ω d ∼ B NS R NS cI NS Ω (cid:18) Ω R ∗ c (cid:19) (17)where ˙Ω d is a dipole spindown rate, R NS is neutron star radius, B NS is surface magnetic field, I NS is the moment ofinertia and the numerical estimate is given for the particular magnetar. Thus, in long period neutron stars a relativelysmall change in the magnetospheric structure can induce huge variations in the spindown rate. This is due to the fact 11 –Fig. 5.— Energy contained in magnetosphere, compared with pure dipole field 12 –that in a dipolar-type configuration only a tiny fraction of the field lines, of the order of η d ∼ × − contributes tothe spindown. A change in the fraction of the open flux then can induce large fluctuations of the spindown rate.Let us assume that the persistent spindown rate ˙ ν ≈ − − Hz/s (Archibald et al. 2013) is determined mostly bythe dipolar-like spindown. Let the open magnetic flux during the outburst be a fraction η of the total flux, so that thespindown rate is a faction η of the maximal value (17), ˙Ω = η ˙Ω max . Then to produce an “anti-glitch” of the order of5 × − Hz within one period it is required that the fraction of the open flux is η ≈ (cid:18) Ω R ∗ c (cid:19) (cid:114) ∆ ν ˙ ν P = 2 . × − (18)Thus, all that is required to produce an “anti-glitch” is to open only a few percent of the total magnetic flux throughhemisphere. Or, in other words, the enhanced torque should be of the order of 10 − − − of the maximal possibletorque acting during one rotational period. For longer active spindown periods, the requirement on η is even smaller.These are very reasonable estimates. The amount of open magnetic flux during the flare is much larger than for dipolefield, η (cid:29) η d , yet it is much smaller than the total magnetic flux through a hemisphere, η (cid:28)
1. Using numericalsimulations of the twisted magnetosphere (Parfrey et al. 2012) came to similar conclusions, that opening a small fractionof the magnetosphere can lead to rapid spindown. The total energy involved in the opening of the magnetosphere is afraction ∼ η of the total magnetic field energy. In case of 1E 2259+586 (magnetic field of 6 × G) this amountsto ∼ ergs. This is consistent with the energy released in the prompt GBM burst, ∼ erg, after accounting forthe radiative efficiency of the resulting CME-like event. We conclude that the “anti-glitch” behavior can be naturallyexplained as a transient high spindown rate induced by the magnetospheric changes. X activity Immediately following the X -ray burst, the magnetar 1E 2259+586 showed a period of enhanced activity duringwhich the spindown rate was approximately 2-3 times higher than the average value. Below we show this this is due to amild overall increase in the magnetospheric current. Let us use the analytical small current approximation (14). In theself-similar model, for not very large twists, we have p = 1 − / C, C = ∆ φ /
2. If magnetospheric structure changes(e.g., there is a jump in the current parameter ∆ C due to increased twist), this will induce a jump in the spin-downparameter ∆ p = − / C and a corresponding jump in spin-down rate.Let us assume that in a normal state a typical twist is ∆ φ . As discussed above, it is required that spindownincreased by a factor N ≈ φ = ∆ φ + 3516 ln N ln R LC /r ∗ (19)Assuming ∆ φ (cid:29) ∆ φ and using the parameters of the magnetar and the required spin-down increase, we find∆ φ ≈ .
37 (20)This is a reasonable estimate: during the activity time (a weeks after the X -ray flare in 1E 2259+586) the magnetosphereon average should be twisted by about twenty degrees. Note, that for large twists the spindown rate is a highly sensitivefunction of the twist: for a twenty degrees turn the spindown rate is 2 times the dipole, while for a full turn it is 10 times the dipole, Eq. (17). Using the estimate of the twist, and the small current solutions (14), we find for the currentparameter C = 0 .
06 and p = .
98, all within the assumed limit.The fact that the twist (20) is smaller than unity is important for the self-consistency of the model. It is expectedthat highly twisted configurations are unstable towards kinks or buckling instabilities. Understanding stability of theconfigurations (6) remains an open question. 13 –Note, that the spindown of a neutron star on long time scales (much longer than the period) are not uniquelyrelated to the X -ray activity. The luminosity presumable comes from a large fraction of the magnetosphere, mostlyfrom closed field lines. At the same time, the spindown is determined by currents flowing along a very small bundle offield lines. Thus we do not expect an exact one-to-one correspondence between the X -ray luminosity and the spindownrate, only a positive correlation.On the other hand, suppose that the crustal motion twists a patch of the surface by a given angle. Twisting theflux tube originating near the magnetic pole and extending nearly up to the light cylinder is more likely to lead to kinkinsatiability and opening of the magnetosphere, since the instability of twisted closed field lines is likely to be suppressedby the overlying dipolar fields. But twisting the fields near the magnetic pole strongly affects the current responsiblefor the spindown of the star. Thus, it is natural to expect that a burst or a flares lead to temporary opening of themagnetosphere and large increase in the torque.Lyutikov & Gavriil (2006) (see also Fern´andez & Thompson 2007; Rea et al. 2008; Beloborodov 2013) have con-structed a model of resonant cyclotron scattering in the magnetar magnetospheres that relates the spindown propertiesto the hardness of the soft non-thermal radiation. According to the model, during the enhanced activity the magne-tosphere typically supports larger currents. Larger currents and larger particle density in the magnetosphere increasesthe importance of resonant scattering, making the spectra harder. The optical depth due to resonant scattering τ canbe expressed in terms of the current parameter C . For not very large twists τ ∼ √ C ∝ ∆ φ (21)(assuming a constant velocity of scatters). The resulting spectrum is not a pure power law, but is in agreement withthe data. Qualitatively, the higher is the twist, the larger is the luminosity (see Eq. (34) of Thompson et al. 2002), theharder is the spectrum.The total dissipated energy is proportional to the current (and the twist), L X ∝ ∆ φ , so that τ ∝ L X . Thus weexpect that a substantial (order of unity) change in the X -ray luminosity is accompanied by a similar change in thespindown rate and a change in the spectral properties. All these relations are in agreement with the observations of 1E2259+586.
5. Discussion
In this paper we showed that the transient (lasting for just one period) opening of a relatively small fraction of themagnetosphere (a few percent) can explain the rapid spindown of the magnetar 1E 2259+586 observed by Archibaldet al. (2013). We point out that in magnetars it is very easy to have large variations of the spin-down rate - the theoreticalupper limit is an enhancement by a factor up to ∼ . What is more, in the ensuing period of the enhanced activitythe behavior of the X -ray luminosity, spindown rates and spectral behavior are all consistent with the magnetosphericorigin. We suggest, there may be no real second glitch, only erratic behavior while the magnetosphere relaxes to a longterm steady state.Magnetars are also known for their erratic timing behavior ( e.g., Woods & Thompson 2006). The spindown ratescan vary by a factor of few over timescales of weeks (eg Woods et al. 2002), while the giant flares are often accompaniedby a sudden increase in the spin period (Woods et al. 1999). Note that in case of magnetars the time “resolution” - atypical interval between the measurements of the spin, are about a week. This, combined with the fact that torque canfluctuate by as much as billion times, make it hard to define a glitch - as opposed to smooth torque variation.In this paper we argue that timing fluctuations in magnetars are purely magnetospheric. (In contrast, Thompsonet al. 2000, argued that large timing irregularities in magnetars may have an internal origin, similar to glitches in 14 –rotationally powered pulsars.). We suggest that there are two types of glitches in magnetars: (i) due to topologicalchanges of the magnetosphere - opening up of a considerable magnetic flux (rapid spin down on the time scale of onerotation, associated with some radiative activity; this is specific to magnetars); (ii) smooth changes in the magnetosphericstructure (radiatively silent, no large X -ray bursts, spindown changes can be positive or negative; this is similar to thetorque variations in the rotationally powered pulsars). In the second case there is a clear prediction: larger averaged L X , larger spindown, harder spectra (this is generally consistent with the behavior of 1E 2259+586, as well as post-giantflare evolution, see discussion in Lyutikov (2006)). The first case (opening up of the magnetosphere) leads to transientspindown events during bursts and flares and is specific to magnetars, while the second case (changes in the currentflow in the magnetosphere), produces fluctuations around some mean value and is analogues to torque fluctuations inrotationally-powered pulsar.In comparison, in the rotationally powered pulsars there are two types of timing irregularities: glitches, suddenchanges of the spin frequency occurring on the time scale of single rotation period ( e.g., Espinoza et al. 2011) andlong term spindown variations (Kramer et al. 2006; Hermsen et al. 2013). Glitches are related to the internal neutronstar dynamics, how the superfluid vortexes interact with the crustal lattice (Chamel & Haensel 2008). The long termtorque variations are assumed to be related to the changes in the magnetospheres current distribution, though a lackof general understanding of pulsar magnetospheres impede the development of models (see, though Timokhin & Arons2013). We suggest that the second type, magnetospheric long term variations, is also operational in magnetars, thoughgenerally with larger spin variations. Magnetars also have new specific torque variation mechanism due to opening ofthe magnetosphere.We hypothesize that in cases of so called spin-up glitches, the “prompt” short spike, like the one detected by theGBM in 1E 2259+586, was missed by the all sky monitors. Then the ensuing erratic behavior of the spin-down torquecan lead to overcompensation of the prompt spin-down glitch.As we discuss next, the overall timing behavior of magnetars is in a general agreement with the magnetosphericmodel. Dib & Kaspi (in preparation) point out that radiative changes are almost always accompanied by some form oftiming anomaly - in our model the magnetospheric opening is accompanied by a formation of a CME and associatedprompt spike. Also, large long term magnetospheric changes should have correlated spindown and persistent X -ray fluxincrease. On the other hand, only occasionally timing anomalies in AXPs are accompanied by any form of radiativeanomaly. Most of the torque variations in our model come from the magnetospheric changes on the closed field lines -such changes are not necessarily expected to be associated with large radiative effects. Importantly, the changes in themagnetospheric structure of closed field lines should show in the changing pulse profiles. This agrees with observations.Superficially, the timing glitches in normal pulsars and magnetars look similar: roughly same amplitude, similarrecovery and inter-glitch times (about a day or so). We think these similarities are misleading. For example, the rate ofaccumulation of any mismatch between different components within the neutron star, ∝ Ω , is different by an order ofa million between the glitching pulsars (which are typically fast) and the slow magnetars.There are other possible hints in favor of the magnetospheric origin of magnetar timing anomalies. The magnetar1E 1048.1-5937 has the highest rotational noise, has one of the largest flux variations and has the highest surfacetemperature (Dib & Kaspi, in preparation). This is also consistent with the magnetospheric model: larger variationsof the current are related to large flux and spindown variations. The high surface temperature is related to higheroverall magnetospheric currents. In addition, the timing noise is correlated with the strength of the magnetic field: 1E2259+586 and 4U 0142+61 have the least noise and smallest magnetic field, while 1RXS J1708-4009 and 1E 1841-045are the opposite.Finally, there are magnetar activity events that seem to be associated with spin-up glitches ( e.g., the 2007 event in1E 1048.1-5937, Dib et al. 2009). We note that at the time of the event there were very large variations in ˙Ω, that, we 15 –hypothesize, when averaged over the interval between the observations produced an effective spin-up event.I would like to thank Robert Archibald, Andrei Beloborodov, Konstantinos Gourgouliatos, Victoria Kaspi, Christo-pher Thompson and David Tsang for discussions and Department of Physics of McGill University for hospitality. Thisresearch was partially supported by NASA grant NNX12AF92G and by Simons Foundation. REFERENCES
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0. In this case function F is almost astraight line with a slope − µ = 0 with F (0) ∼
2. Near this point we can neglect µ .Integrating Eq. (5) once we find p ( p + 1) F + Cpp + 1 F p +1) /p + F (cid:48) = 4 (A1)where constant of integration has been chose so that at µ = 1, where F = 0, F (cid:48) = 4. Since at µ → F ∼ p ( p + 1) F as compared with CF ( p +2) /p : Cpp + 1 F p +1) /p + F (cid:48) = 4 (A2) This preprint was prepared with the AAS L A TEX macros v5.2.
17 –Making a substitution F → (cid:18) pC p + 1) (cid:19) − p p +1) (cid:18) − pp + 1 ln q (cid:19) ≡ p + 1) C p (cid:18) − pp + 1 ln q (cid:19) , (A3)the equation for the function q reads q (cid:48) (cid:112) q − C (A4)where C = (cid:18) p + 1) p (cid:19) / ( p +1) (cid:18) C ( p + 1) p (cid:19) p/ (2( p +1)) (A5)Boundary conditions for q are 1 − pp +1 ln q = 0 at µ = 1 q (cid:48) = 0 at µ = 0 (A6)(two boundary conditions for the first order ODE (A4) are needed to find the solution and dependence p ( C )). Eq. (A4)has a solution q = cosh( C µ + C ) (A7)Boundary conditions give C = 0 C = cosh − exp ( p +1) /p ∼ ( p + 1) /p + ln 2 (A8)which resolves implicitly p ( C ) C = 2 − /p ( p + 1) p (A9)and determines the flux function F = 2cosh − exp ( p +1) /p (cid:18) p + 1 p − ln cosh( µ cosh − exp ( p +1) /p ) (cid:19) ∼ p + 1)1 + p + p ln 2 (cid:18) − pp + 1 ln cosh µ (1 + 1 /p + ln 2) (cid:19) . (A10)(compare with Lynden-Bell & Boily 1994). Value of F at µ = 0 is then F (0) = 2( p + 1)1 + p + p ln 2 ∼ − pp