MMon. Not. R. Astron. Soc. , 1–13 (2013) Printed 14 November 2018 (MN L A TEX style file v2.2)
Constraining the Origin of Magnetar Flares
Bennett Link (cid:63)
Department of Physics, Montana State University, Bozeman, Montana, 59717, USADepartment of Physics, Monash University, Melbourne, Victoria 3800, Australia
14 November 2018
ABSTRACT
Sudden relaxation of the magnetic field in the core of a magnetar produces me-chanical energy primarily in the form of shear waves which propagate to the surfaceand enter the magnetosphere as relativistic Alfv´en waves. Due to a strong impedancemismatch, shear waves excited in the star suffer many reflections before exiting thestar. If mechanical energy is deposited in the core and is converted directly to radia-tion upon propagation to the surface, the rise time of the emission is at least secondsto minutes, and probably minutes to hours for a realistic magnetic field geometry, atodds with observed rise times of < ∼
10 ms for both and giant flares. Mechanisms forboth small and giant flares that rely on the sudden relaxation of the magnetic fieldof the core are rendered unviable by the impedance mismatch, requiring the energythat drives these events to be stored in the magnetosphere just before the flare. Acorollary to this conclusion is that if the quasi-periodic oscillations (QPOs) seen ingiant flares represent stellar oscillations, they must be excited by the magnetosphere ,not by mechanical energy released inside the star. Excitation of stellar oscillations byrelativistic Alfv´en waves in the magnetosphere could be quick enough to excite stellarmodes well before a giant flare ends, unless the waves are quickly damped.
Key words: stars: neutron
Soft-gamma repeaters (SGRs) are strongly-magnetized neutron stars with magnetic fields of B = 10 − G that producefrequent, short-duration bursts ( < ∼ < ∼ ergs in hard x-ray and soft gamma-rays, with the peak luminosity in theburst typically being reached in under 10 ms ( e.g. , Woods & Thompson 2006; Mereghetti 2008). SGRs occasionally producegiant flares that last ∼
100 s; the first giant flare to be detected occurred in SGR 0526-66 on 5 March, 1979 (Barat et al. 1979;Mazets et al. 1979; Cline et al. 1980), releasing > ∼ × erg (Evans et al. 1980), and rising to near its peak luminosity in < > ∼ × erg, with a rise time of < ∼ > ∼ × ergs and a rise time of < ∼ . (cid:39) ∼
100 Hz, and in thecase of SGR 1806, all of the Fourier power above ∼
100 Hz is in the QPOs. Measured spin down parameters imply surface (cid:63)
E-mail: [email protected] (cid:13) a r X i v : . [ a s t r o - ph . S R ] M a y B. Link dipole fields of 6 × G for SGR 0526-66 (Tiengo et al. 2009), 7 × G for SGR 1900+14 (Mereghetti et al. 2006), and2 × G for SGR 1806-20 (Nakagawa et al. 2008). These strong inferred fields, along with other properties such as quiescentbrightness (Thompson & Duncan 1995, 1996; Woods & Thompson 2006), establish these objects as magnetars.Anomalous x-ray pulsars (AXPs) are also magnetars that exhibit bursts that are in most respects like SGR flares, butwith a wider range of burst durations. These bursts are generally less energetic than SGR flares, with the most energeticbursts showing much harder spectra than are seen in SGR bursts (see, e.g. , Gavriil et al. 2004 and Kaspi 2007).In addition to these high-field objects, there are now three known “low-field” magnetars that produce small flares. SGR0418+5729 has an inferred dipole field of 6 × (Rea et al. 2013), typical of radio pulsars. Swift J1822.3-1606 has an inferreddipole field of ∼ × G (Rea et al. 2012; Scholz et al. 2012), while 3XMM J185246.6+003317 has an inferred field of < × G (Rea et al. 2013). These sources show that magnetar activity does not require high dipolar fields. Instead, theactivity could be driven by decay of multipolar components that could be an order of magnitude or more larger than thedipolar component (Braithwaite 2009). These bursts are rather different than SGR bursts. The total energy release is < ∼ erg, like small flares in SGRs, but with decay times of hundreds of days. The timing resolution is insufficient to ascertain howthe rise times compare to bursts in SGRs. The long relaxation time is consistent with the thermal relaxation time predictedfor the crust (Brown & Cumming 2009; Scholz et al. 2012).While energetics considerations strongly suggest that SGR flares are driven by the release of magnetic energy (see § ∼ G, the corewill not be superconducting, the electrical conductivity will be lower, and the field might evolve to the point that an MHDinstability occurs. A third possibility is that the energy is released not inside the star, but in the magnetosphere through anMHD instability (Lyutikov 2003, 2006; Komissarov et al. 2007; Gill & Heyl 2010), such as the “tearing mode”, producing amagnetospheric explosion akin to coronal mass ejections seen in the Sun.Duncan (1998) predicted that magnetic stresses that arise as the internal field of a magnetar evolves will eventually shearthe crust to failure, producing a flare and exciting torsional modes in the crust. The QPOs seen in the giant flares of SGRs1900 and 1806 were interpreted initially as crustal modes ( e.g. , Piro 2005; Samuelsson & Andersson 2006; Watts & Reddy2007; Lee 2007; Sotani et al. 2007; Steiner & Watts 2009). Subsequent work has accounted for magnetic coupling betweenthe crust and liquid core, and attributes QPOs to global magneto-elastic oscillations of the neutron star ( e.g. , Levin 2006;Glampedakis et al. 2006; Levin 2007; Sotani et al. 2008; Colaiuda et al. 2009; Cerd´a-Dur´an et al. 2009; Colaiuda & Kokkotas2011; van Hoven & Levin 2011; Gabler et al. 2011; van Hoven & Levin 2012; Gabler et al. 2012, 2013; Passamonti & Lander2013a,b). A crucial ingredient in the interpretation of QPOs as stellar oscillations is to understand how crust movementcan produce the large observed modulations of the x-ray emission by 10-20%. Timokhin et al. (2008) propose that twistingof the crust, associated with a stellar mode, modulates the charge density in the magnetosphere, creating variations in theoptical depth for resonant Compton scattering of the hard x-ray photons that accompany the flare. In this model, the shearamplitude at the stellar surface must be as large as 1% of the stellar radius, and it is unknown if the stellar crust can sustainthe associated strain without failing. D’Angelo & Watts (2012) have shown that beaming effects can increase the amplitudeof the QPO emission by a factor of typically several. As the theory of neutron star “seismology” is further developed, theexciting possibility of constraining the properties of dense matter and the magnetic field configuration of the core is becomingfeasible.This paper is concerned with the basic question of whether the energy that drives SGR flares is stored in the stellarinterior or in the magnetosphere just before the flare occurs. A key physical feature is the existence of a large impedancemismatch between the stellar interior and the magnetosphere for the propagation of shear waves, as originally pointed outby Blaes et al. (1989); the mismatch is due to the fact that the wave propagation speed in the core is ∼ − of that in themagnetosphere. With minimal assumptions, I show that shear waves produced in the core through sudden global relaxationof the magnetic field are prevented from quickly entering the magnetosphere by the impedance mismatch; rather, the outerregions of the star and the magnetosphere are highly reflective to shear waves, causing waves to remain trapped in the core forat least seconds to minutes, and perhaps as long as minutes to hours for a realistic magnetic field geometry. The trapping time In general excitation of many stellar modes, including p -modes, g -modes, and f modes should occur, but torsional modes have thelowest frequencies and would be the easiest to detect. c (cid:13) , 1–13 rigin of Magnetar Flares greatly exceeds typical rise times of <
10 ms, requiring that the energy that powers both small and giant flares is stored in themagnetosphere just before the flare. The energy could then be quickly released through a magnetic instability as proposed byLyutikov (2003); see, also, Lyutikov (2006) and Komissarov et al. (2007). The energy could be stored in the magnetosphereif, for example, the internal field gradually untwists, slowly twisting the magnetosphere until it becomes unstable (Lyutikov2003).Thompson & Duncan (1995) and Thompson & Duncan (2001) have proposed that flares arise from the deposition ofmagnetic energy inside the star. In the model of Thompson & Duncan (2001), both small and giant flares are driven by sudden relaxation of a globally-twisted internal magnetic field, with the energy release gated by crust rigidity. When the crustis stressed to failure in this model, the magnetic foot points are suddenly sheared, and energy flows from the stellar interiorinto the magnetosphere, producing a radiative event through the dissipation of Alfv´en waves and magnetic reconnection.Thompson & Duncan (2001) assume that the failure of the crust along the fault allows flow of energy from the core into themagnetosphere over a time-scale of order the Alfv´en crossing time of the star or shorter. As shown in this paper, though,the impedance mismatch between the core and the magnetosphere slows the flow of mechanical energy to at least 10 to 10 Alfv´en crossing times (seconds to minutes). Crust failure cannot remove this fundamental mismatch, and so the proposal ofThompson & Duncan (1995) and Thompson & Duncan (2001) that both large and small magnetar flares are driven by thesudden release of magnetic energy stored inside the star appears to be unviable.If flares indeed originate as magnetospheric explosions, energy will be trapped on closed magnetic field lines in the formof relativistic Alfv´en waves. The impedance mismatch between the magnetosphere and the stellar interior makes the starhighly reflective to these waves. I obtain a lower limit for the time-scale required for relativistic Alfv´en waves excited bya magnetospheric explosion to excite magneto-elastic modes in the star, and find that such modes could be excited wellbefore a giant flare ends. Hence, a viable explanation for the QPOs is that they represent stellar oscillations excited by themagnetosphere , not the stellar interior, provided the excitation can occur before the waves are damped.In §
2, I discuss general considerations of the release of energy in giant flares. In §
3, I formulate the problem of transmissionfrom the deep stellar interior to the surface in a planar geometry, and estimate the transmission coefficient at low frequency.In §
4, I calculate the transmission coefficient as a function of frequency, accounting for the material properties of the crustand the strong gradient in the wave propagation speed. In §
5, I give numerical results. In §
6, I discuss trapping of energyin the core. In §
7, I discuss how a realistic magnetic field geometry will greatly decrease the transmission efficiency. In § Suppose that the magnetic configuration within a volume l inside the star adjusts, lowering the magnetic energy, ultimatelydriving a flare of radiative energy E . By energy conservation, E is bounded by E < B π l , (1)where B is the average field strength. This is an upper limit since the field will not be reduced to zero, and the conversion ofmagnetic energy to radiation will not be perfectly efficient. The length scale l has a lower limit of l > ∼ (cid:18) E erg (cid:19) / B − / km , (2)comparable to the stellar radius for a giant flare. Here B ≡ B/ , and B is measured in Gauss. Readjustment of the magnetic field configuration occurs through the production of Alfv´en waves in the core and magneto-elastic shear waves in the crust; see §
3. I will refer to both kinds of waves as “shear waves”, since their properties are thesame. Independent of how the magnetic energy is gated or released, the volume l will fill with shear waves over a time-scale T = l/c s , where c s is the speed of shear waves, typically 3 × − c throughout most of the star; see eqs. [8] and [9] below.The power spectrum of the shear waves that are produced will have a cut-off at ν c ∼ /T : ν c ≡ T = c s l < ∼
200 Hz B / (cid:16) E erg (cid:17) − / , (3) c (cid:13)000
200 Hz B / (cid:16) E erg (cid:17) − / , (3) c (cid:13)000 , 1–13 B. Link assuming that the protons of the core form a type II superconductor, so that c s is given by eq. [8]. If the core protons arenormal, the estimate remains the same, though the scaling with B changes to B / . From the energy yields of the three giantflares to date, eq. [3] gives ν c ∼ . ν c ∼ c s /R ∼
100 Hz.This estimate does not depend on whether the magnetic energy is released through a global instability, or if it is gated bycrust rigidity.Based on these estimates, the subsequent analysis will be for the propagation of seismic energy at frequencies below 1kHz.
I now calculate the efficiency with which energy deposited in the stellar interior through global readjustment of the field istransmitted to the magnetosphere. For magnetic energy that is released in the stellar interior, the energy will be depositedas heat, sound waves, and shear waves. Thermal energy diffuses through the star relatively slowly - e.g. , over a time-scale ofmonths through the crust (Brown & Cumming 2009; Scholz et al. 2012); the energy propagates much more quickly to thesurface as mechanical waves. The core supports magnetic shear waves and sound waves. Shear waves propagate along fieldlines, and the fluid is essentially incompressible (Levin 2006).The crust supports shear waves, modified by the magnetic field, and sound waves. If a medium that supports shear isdriven to failure, most of the wave energy will be in the form of shear waves if the shear wave speed is less than the soundspeed (Blaes et al. 1989). This is the case throughout the core and near the base of the inner crust.For energy to leave the core, it must propagate along field lines that pass from the core, through the crust, and into themagnetosphere. The plasma density is low in the magnetosphere, so energy propagates there as relativistic Alfv´en waves andas magnetosonic waves. Hence, the most efficient way for energy to leave the core is to propagate along field lines that pointnearly radially outward. Outgoing shear waves will couple most directly to Alfv´en waves in the magnetosphere, and I ignorethe weaker coupling to magnetosonic waves. I also note that each reflection of a shear wave in the star will convert some ofthe energy to compressional waves. I ignore this effect as well, and consider the problem of the propagation of shear wavesfrom the core and crust, and their emission from the star as Alfv´en waves.Consider a simple planar geometry, with the crust-core boundary in the x − y plane at z = 0, and the stellar surface at z = z s . The magnetic field is constant and directed along the z axis. A linearly-polarized shear perturbation of displacement u ( z, t ) = u ( z )e − iωt obeys (Blaes et al. 1989) ddz (cid:16) ˜ µ dudz (cid:17) + ˜ ρ ω u = 0 , (4)where˜ µ ≡ µ + B π . (5)Here µ is the material shear modulus, which is non-zero only in the crust, and˜ ρ ≡ ρ d + B πc , (6)where ρ d is the dynamical mass density, that is, the mass density associated with matter that moves in response to a passingshear wave. In the inner crust, Bragg scattering of free neutrons with the nuclear lattice gives ρ d < ρ (Chamel 2005, 2012,2013; see eq. 19 below). The second term in eq. [6] is the contribution of the magnetic energy to the effective mass of thematter, and is important only near the stellar surface, for ρ < ∼ B g cm − . The speed of shear waves is c s = (cid:112) ˜ µ/ ˜ ρ . Fromeq. [4], continuity in u requires that the traction ˜ µdu/dz be everywhere continuous.The protons of the outer core are expected to form a type-II superconductor (SC) if the field is below the upper criticalfield H c ∼ G. Type-II superconductivity modifies the magnetic stress. Repeating the derivation of Blaes et al. (1989)using the magnetic stress tensor of Easson & Pethick (1977) gives˜ µ = H c B π (SC core protons) , (7)where H c (cid:39) G is the lower critical field. In the core, the protons and neutrons are expected to form distinct superfluids,with negligible nuclear entrainment of the neutron mass current by the proton mass current (Chamel & Haensel 2006; Link For material failure through local stresses, the lowest-order emission process of waves is quadrupolar. The energy density in a wave ofpropagation speed v p scales as v − p . c (cid:13) , 1–13 rigin of Magnetar Flares ρ d is nearly equal to the proton mass density. Ifthe core is a type II superconductor as predicted, magnetic disturbances propagate as vortex-cyclotron waves ( Mendell 2002)at speed c vc = (cid:114) BH c πρx p = 3 × − H / c , B / (cid:16) ρ (cid:17) − / (cid:16) x p . (cid:17) − / c (SC core protons), (8)where ρ ≡ ρ , ρ is in g cm − , and x p is the proton mass fraction. Here fiducial values typical of the outer core havebeen chosen.If the core protons are instead normal, waves propagate as Alfv´en waves at speed c A = B √ πρx p = 3 × − B (cid:16) ρ (cid:17) − / (cid:16) x p . (cid:17) − / c (normal core protons) (9)In the magnetosphere, ˜ µ = B / π and ˜ ρ (cid:39) B / πc , and the Alfv´en waves are relativistic: d udz + ω c u = 0 . (10)where the wavenumber in the magnetosphere is k = ω/c .The wavenumber in the core is k c = ω s /c s , where c s = c vc for SC core protons and c s = c A for normal core protons. Forsufficiently low frequencies that k c ∆ R << ν < ∼
100 Hz, the wave is insensitive to the gradientsin ˜ µ in the crust, and the crust can be treated as a thin discontinuity; crust structure is unimportant in this limit. In thiscase, the energy transmission coefficient takes the familiar form T = 4(˜ µ c k c )(˜ µ m k )(˜ µ m k + ˜ µ c k c ) , (11)where ˜ µ c = BH c / π for a superconducting core, B / π for a core of normal protons and superfluid neutrons, and ˜ µ m = B / π for the magnetosphere. (Recall that B is constant in the assumed planar geometry). Typically ˜ µ c k c >> ˜ µ m k , giving, for asuperconducting core, T (cid:39) (cid:16) BH c (cid:17) c vc c (cid:39) − (cid:16) BH c (cid:17) / (cid:16) x p . (cid:17) − / (cid:16) ρ (cid:17) − / (SC core protons) . (12)while for a core of normal protons T (cid:39) c A c (cid:39) − B (cid:16) x p . (cid:17) − / (cid:16) ρ (cid:17) − / (normal core protons) . (13)Because ˜ µ c k c >> ˜ µ m k , there is a strong impedance mismatch between the core and the magnetosphere, giving T << ν < ∼
100 Hz. We will see that T is further reduced by the structure of the crust for ν > ∼
100 Hz.Energy that is released primarily in the core cannot propagate directly to the surface, but becomes trapped in the core.For energy to propagate into the magnetosphere, it must then propagate from the core and through the crust, sufferingmultiple reflections before escaping to the magnetosphere.
I now turn to an exact calculation of the energy transmission coefficient T for ν > ∼
100 Hz, when k c ∆ R << µ = 0 . . / Γ) n i ( Ze ) a , (14)where n i is the number density of ions of charge Ze , a is the Wigner-Seitz cell radius given by n i πa / ≡ ( Ze ) / ( akT ) where k is Boltzmann’s constant. Typically in the crust, Γ >>
173 and the second term in the denominator isnegligible. For the composition of the crust, I use the results of Haensel & Pichon (1994) for the outer crust, and the results ofDouchin & Haensel (2001) for the inner crust, conveniently expressed analytically by Haensel & Potekhin (2004), who treatdensities from 10 g cm − to above nuclear density. The treatment by Douchin & Haensel (2001) of the inner crust givessomewhat higher values of the shear modulus at the base of the crust than do other studies. The equation of state of Akmalet al. (1998), for example, gives a shear speed at the base of the crust that is about 0.6 the shear speed of Douchin & Haensel(2001), and a corresponding shear modulus that is smaller by a factor of about 2.8. A higher shear speed in the crust decreasesthe impedance mismatch with respect to magnetosphere, giving somewhat higher values for the transmission coefficient thatmost other choices of the shear modulus would give. c (cid:13)000
173 and the second term in the denominator isnegligible. For the composition of the crust, I use the results of Haensel & Pichon (1994) for the outer crust, and the results ofDouchin & Haensel (2001) for the inner crust, conveniently expressed analytically by Haensel & Potekhin (2004), who treatdensities from 10 g cm − to above nuclear density. The treatment by Douchin & Haensel (2001) of the inner crust givessomewhat higher values of the shear modulus at the base of the crust than do other studies. The equation of state of Akmalet al. (1998), for example, gives a shear speed at the base of the crust that is about 0.6 the shear speed of Douchin & Haensel(2001), and a corresponding shear modulus that is smaller by a factor of about 2.8. A higher shear speed in the crust decreasesthe impedance mismatch with respect to magnetosphere, giving somewhat higher values for the transmission coefficient thatmost other choices of the shear modulus would give. c (cid:13)000 , 1–13 B. Link -500 0 500 1000 z (m) den s i t y ( g c m - ) core crust (cid:108) d (cid:108) Figure 1.
The density profile in the inner crust and outer core. The dashed curve gives the baryonic mass density profile from thesolution to eq. [15] with the equation of state of Haensel & Potekhin (2004). The solid curve shows the dynamical mass density of eq.[19]. In the denser regions of the inner crust, ρ d < ρ from the effects of nuclear entrainment (Chamel 2005, 2012, 2013); see eq. [19]. Theinset shows detail of the region where nuclear entrainment effects are most important; the squares are from the calculations of Chamel(2012). In the core, ρ d is equal to the proton mass density, here fixed to be the density x p ρ c = 6 . × g cm − at the crust-coreinterface. For a barytropic equation of state p ( ρ ), the density profile in the crust, neglecting the effects of General Relativity, followsfrom the equation of hydrostatic equilibrium dρdr = − gρ ( r ) (cid:18) dpdρ (cid:19) − , (15)where g is the gravitational acceleration. Henceforth I fix g = 2 × cm s − , appropriate to a neutron star of 1.4 solarmasses and a radius of 10 km. I take the crust to dissolve into the core at ρ c = 1 . × g cm − , about half of nuclearsaturation density. Under these assumptions, the crust thickness ∆ R is almost exactly 1 km. The density profile in the crustis given by the dashed line in Fig. 1.I take the core to be of constant density and infinite extent for z <
0, with a wave incident on the crust-core interface,and a reflected wave: u ( z ) = A e ik c z + B e − ik c z , (16)where A and B are constants. Requiring continuity in u and ˜ µ du/dz at the crust-core interface ( z = 0), gives the transmissioncoefficient T = 1 − (cid:12)(cid:12)(cid:12) BA (cid:12)(cid:12)(cid:12) = 1 − (cid:12)(cid:12)(cid:12)(cid:12) ˜ µ (0 − ) u (0+) k c + i ˜ µ (0+) u (cid:48) (0+)˜ µ (0 − ) u (0+) k c − i ˜ µ (0+) u (cid:48) (0+) (cid:12)(cid:12)(cid:12)(cid:12) , (17)where a prime denotes a derivative in z .At the surface z = z s = ∆ R , there is only an outgoing relativistic Alfv´en wave e ik ( z − z s ) with k = ω/c . Since A and B are unspecified, u is conveniently fixed to unity at the surface. Continuity in ˜ µ du/dz gives the surface boundary condition˜ µ ( z − z s = 0 − ) u (cid:48) ( z − z s = 0 − ) = ik ˜ µ ( z − z s = 0+) with u ( z − z s = 0 − ) = 1 . (18)The amplitude u becomes complex for z < z s . Calculation of the quantities u (0+) and u (cid:48) (0+) by numerical integration fromthe surface to z = 0 gives the transmission coefficient from eq. [17].In the inner crust, most of the baryonic mass is in the form of superfluid neutrons. As a shear wave passes, a fraction ofthe superfluid neutrons is non-dissipatively entrained by the nuclear clusters through Bragg scattering (Chamel 2005, 2012);the remaining “conduction” neutrons, that is, the neutrons that are not entrained, must be subtracted from the baryonicdensity to give the appropriate dynamical density (Pethick et al. 2010): ρ d = ρ (1 − n cn / ¯ n ) ≡ fρ, (19) c (cid:13) , 1–13 rigin of Magnetar Flares -500 0 500 1000 1500 z (m) -4 -3 -2 -1 w a v e s peed c s / c G10 G10 G core m agne t o s phe r e G Figure 2.
The shear speed c s = (cid:112) ˜ µ/ ˜ ρ as a function of position. As the magnetic field is increased, the effective shear modulus ˜ µ increases in the crust; see eq. [5]. For B = 10 G (cid:39) H c , the core is taken to be normal. The core neutrons are assumed to be superfluid.Outside the star, the shear speed is c , the speed of a relativistic Alfv´en wave. There is a small jump in µ at z (cid:39)
600 m, correspondingto the neutron drip density that is washed out by magnetic stresses for
B > ∼ × G. where n cn is the number density of conduction neutrons and ¯ n is the average baryon density in a unit cell. The speed of shearwaves in the crust (for B = 0) is c s = (cid:112) µ/ρ d ; the existence of conduction neutrons increases the shear speed. To account forthis effect, a fitting formula for f that gives a good approximation to the results of Chamel (2012) is f = 1 − . n (¯ n − . n < .
08 fm − . (20) f stays near unity up to ¯ n (cid:39) .
04 fm − , before falling to ∼ .
35 at ¯ n = 0 .
08 fm − , at which point the inner crust ends. Theinset of Fig. 1 shows the assumed dynamical density based on the results of Chamel (2012). The shear speed in the star isshown in Fig. 2.The inner core could reach a density of 5 − ρ c , but the gradients in ˜ ρ and ˜ µ are always much less than in the innercrust. Some of the energy that is propagating outward will be reflected by the relatively small gradients in ˜ ρ and ˜ µ in thecore. Treating the core as having constant density slightly overestimates the transmission coefficient. Also, the choice of aconstant field in the z direction is the most favourable geometry for the propagation of energy out of the core, through thecrust, and into the magnetosphere. Any other field configuration will lead to more effective reflection of energy from the crustand the magnetosphere back into the core. For realistic field configurations, this effect could be large. As argued in §
7, thesecalculations of the transmission coefficient probably represent significant overestimates, so the energy transmission efficiencycalculated in this paper is a robust upper limit.
Calculations of the transmission coefficient are shown in Fig. 3 for different values of the magnetic field strength, assumingthat the core neutrons are superfluid. For each solid curve, the core was assumed to be a type II superconductor. For eachdashed curve, the core protons are assumed to be normal - the two curves coincide for B = H c = 10 G. For ν < ∼
100 Hz, k c ∆ R <<
1, and T is nearly independent of frequency. For B < H c , type II superconductivity increases the magnetic stressin the core by a factor ( H c /B ) / relative to the normal case (see eqs. 12 and 13), giving a corresponding increase in T by decreasing the impedance mismatch between the core and the magnetosphere; see eq. [11]. For B > H c , the situationis reversed, and superconductivity decreases T . A field of B = 10 G is close to the upper critical field H c above whichsuperconductivity is destroyed; only the dashed curve is likely be relevant for B = 10 G.Above ν ∼
100 Hz, gradients in the crust of the density and the shear modulus act as an effective potential which partiallyreflects the wave back into the core, reducing T . At ν ∼ c (cid:13)000
100 Hz, gradients in the crust of the density and the shear modulus act as an effective potential which partiallyreflects the wave back into the core, reducing T . At ν ∼ c (cid:13)000 , 1–13 B. Link frequency (cid:105) (Hz) -8 -7 -6 -5 -4 -3 -2 -1 ene r g y t r an s m i ss i on T G10 G 10 G10 G10 G10 G Figure 3.
The transmission coefficient as a function of frequency. For frequencies above 10 −
100 Hz (depending on B ), gradients in ˜ µ and ˜ ρ increase the reflection of the wave back into the core, reducing T . Above ∼ T for a core of superconducting protons, and the dashed curves are for normal protons; thecurves are the same for B = H c = 10 G. The dotted curve for B = 10 shows the effect that perfect nuclear entrainment in the innercrust would have, corresponding to ρ d = ρ . The core neutrons are assumed to be superfluid. The dotted line shows the effect of perfect entrainment by nuclear clusters in the inner crust ( f = 1). Entrainmentincreases the effective mass of nuclear clusters, reducing the shear speed for zero field, and reducing T . Because a fraction ofthe neutron superfluid does not move with the nuclei, the shear speed is increased, and T increases by a factor of ∼ µ c k c in eq. [11] at the base of the evanescent wave zone. Eq.[11] applies only at low frequency, and only for the case that a wave zone exists for z <
0. Blaes et al. (1989) consideredtransmission for frequencies in the range 10 < ν < Hz. Given the different boundary conditions and frequency regimes,a direct comparison to their work is not possible.
Shear waves trapped in the core carry energy across the core at a speed equal to the wave group velocity, c s = c cv forsuperconducting core protons, and c s = c A for normal protons. The wave crossing time is (cid:39) R/c s . The ‘attempt frequency’is c s / R , with an energy transmission probability T ( ν ) per attempt, so the energy transmission rate is ∼ ( c s / R ) T ( ν ) for amode of frequency ν . The associated trapping time for energy in the core is thus t trap (cid:39) Rc s T ( ν ) . (21)The trapping time is shown in Fig. 4. The spin-down rates for both SGR 1900 and SGR 1806 imply a dipole field of strength B (cid:39) G. The trapping time is ∼ ∼
100 s at higher frequencies. These time-scales greatlyexceed the observed rise time of < ∼
10 ms that is seen in both giant flares and in small bursts. Even if B = 10 G, which ismuch larger than the field implied by the observed spin-down rates, energy cannot enter the magnetosphere from the corenearly quickly enough to explain observed rise times. Note that these trapping times represent lower limits, since the mostfavourable magnetic geometry for coupling of the stellar interior to the magnetosphere was assumed. These results stronglysuggest that the flares are powered by the release of magnetic energy directly into the magnetosphere, not in the core. c (cid:13) , 1–13 rigin of Magnetar Flares frequency (cid:105) (Hz) -1 c o r e t r app i ng t i m e ( s ) G10 G10 G Figure 4.
The trapping time for waves in the core. For B = 10 G, the core protons are assumed to be normal. These time-scales arelower limits (see the text).
The analysis so far treats the magnetosphere as having infinite extent. Energy is transferred most efficiently to the magneto-sphere when it excites field lines that are long enough to resonate with the wave that propagates from the core; coupling toshorter field lines will be greatly reduced, as will the transmission coefficient integrated over the stellar surface. For a field lineof length L with two fixed foot-points on the star, the fundamental frequency is c/ L . Field lines with fundamental frequenciesbelow 1 kHz are longer than 150 km. Most of the field lines that emerge from star are much shorter length than this, so themechanical coupling between the stellar interior and the magnetosphere is poor. The energy flux into the magnetosphere willbe reduced by a factor of A/ πR with respect to the spherically symmetric situation, where A is the area of the star thatis connected to field lines that are long enough to resonate with seismic waves. Though the transmission coefficient T ( ν ) hasbeen calculated in a planar approximation for simplicity, a good final estimate for the transmission rate integrated over thestellar surface is T ( ν ) A/ πR .To estimate A , recall that the field line configuration from a magnetic dipole is given by rR = (cid:16) sin θ sin θ L (cid:17) , (22)where ( r, θ ) are spherical coordinates measure with respect to the magnetic dipole moment, and θ L is the angle the field linetakes at r = R (the stellar surface in this simple approximation). Integration of eq. [22] along any given line gives the lengthof the field line L in the limit θ L << L (cid:39) . R sin − θ L . (23)If the power spectrum of excited magnetospheric waves ends at a cut-off ν c , corresponding to a value θ L, max from eq. [23],the area of the magnetic polar region through which Alfv´en waves enter the star is approximately A (cid:39) π ( Rθ L, max ) (cid:39) − πR ν c (Hz). Taking ν c = 1 kHz, indicates that the energy transfer into the magnetosphere could be reduced by a factorof about 10, raising the curves in Fig. 4 by the same factor. If ν is closer to 100 Hz, as suggested by the estimates of § . The trappingtime for B = 10 G becomes ∼
400 s to ∼ ν < That the trapping time for seismic energy in the star is much longer than observed rise times suggests that flares are driven bythe release of energy stored in the magnetosphere, where magnetic energy might be converted to Alfv´en waves and radiationmuch more quickly. If QPOs represent stellar oscillations, they might be excited by the absorption of the star of relativisticAlfv´en waves from the magnetosphere. I now estimate this time-scale. c (cid:13)000
400 s to ∼ ν < That the trapping time for seismic energy in the star is much longer than observed rise times suggests that flares are driven bythe release of energy stored in the magnetosphere, where magnetic energy might be converted to Alfv´en waves and radiationmuch more quickly. If QPOs represent stellar oscillations, they might be excited by the absorption of the star of relativisticAlfv´en waves from the magnetosphere. I now estimate this time-scale. c (cid:13)000 , 1–13 B. Link frequency (cid:105) (Hz) -2 -1 (cid:111) e x ( s ) G10 G10 G Figure 5.
The time-scale over which Alfv´en waves in the magnetosphere can excite stellar modes. The core protons are assumed to benormal for B = 10 G. By energy conservation, the transmission coefficient for the excitation process is the same T ( ν ) calculated above. Since T ( ν ) is much less than unity for 1 Hz < ν < ν supported by closed field lines in the magnetosphere will bounce off the stellar surfaceat a rate ν , with an absorption probability T ( ν ) at each bounce. Ignoring dissipation in the magnetosphere, the absorptionrate by the star of a magnetospheric Alfv´en wave of frequency ν is νT ( ν ) for a planar geometry. As the energy is absorbed,it excites primarily shear waves in the crust and core (see § ν by an Alfv´en wave in the magnetosphere of frequency ν is τ ex ∼ [ νT ( ν )] − (24)This time-scale is shown in Fig. 5. We see that for B ≥ G, most of the energy deposited in the magnetosphere could beabsorbed by the star before the end of a giant flare, and so there could be sufficient time for relativistic Alfv´en waves in themagnetosphere to excite stellar modes and associated QPOs. The strongest QPO seen in the tail of the giant flare in SGR1806, at 92.5 Hz, was estimated to appear about two minutes into the flare (Israel et al. 2005; Strohmayer & Watts 2006).Eq. [24] is a crude estimate. The field geometry near the surface of the star and inside the star is likely to be quitecomplicated, and certainly not everywhere perpendicular to the surface as assumed in this simple planar treatment. If themagnetospheric structure constrains the delivery of Alfv´en energy to only small patches on the stellar surface, the energytransfer rate into the star could be greatly reduced, making τ ex much longer. If this is the case, the interpretation of QPOsas magneto-stellar oscillations could be problematic. Calculations with more realistic field geometries are needed to resolvethis issue; τ ex calculated here is most likely a lower limit.Suppose, however, that the magnetosphere changes its structure instantaneously. Now there is no time-scale in theproblem, and all frequencies will be excited in the magnetosphere; f modes and torsional modes could be excited to largeamplitudes (Levin & van Hoven 2011). More realistically, the time-scale for adjustment of the inner magnetosphere throughan instability is ∼ R/c (cid:39) µ s, implying a cut-off frequency of ∼
30 kHz. The transmission coefficient generally increaseswith frequency, and high frequency waves could enter the star relatively easily.A big uncertainty is the damping rate of Alfv´en waves in the magnetosphere. In a magnetized plasma with a gradientin the Alfv´en velocity, the case in the neutron star magnetosphere, dephasing of Alfv´en waves drives currents and the wavecan be quickly damped by electrical resistivity (Heyvaerts & Priest 1983). If the damping rate is too fast, the magnetospherecannot excite global magneto-elastic modes. This problem merits further study. c (cid:13) , 1–13 rigin of Magnetar Flares The main conclusion of this paper is that the large impedance mismatch between the neutron star interior and the magne-tosphere causes energy exchange between the two to be relatively slow. If the energy that drives a flare, either a small flareor a giant flare, is driven by sudden , global relaxation of the internal magnetic field, the trapped seismic energy takes atleast seconds to minutes to reach the magnetosphere for B = 10 G (see Fig. 4), and possibility as long as minutes to hoursfor a realistic magnetic field geometry, in any case much longer than the observed rise times of <
10 ms. This conclusionrules out models of flares powered by sudden , internal magnetic relaxation ( e.g. , Thompson & Duncan 1995, 2001). Crustfailure cannot remove the fundamental impedance mismatch that limits the transfer of shear-wave energy from the core intothe magnetosphere.
The energy that drives a flare must be stored in the magnetosphere . One way this could happen is if theinternal field gradually untwists, slowly twisting the magnetosphere until it becomes unstable (Lyutikov 2003). A lower limitto the rise time will be determined by the time-scale over which the instability develops. This time-scale could be < ∼
10 msfor the tearing mode (Lyutikov 2006; Komissarov et al. 2007). The rise time of the observed flux could be longer, dependingon the emission processes that accompany the instability.The lower limits on the excitation time-scales of stellar modes by relativistic Alfv´en waves in the magnetosphere givenin Fig. 5 indicate that magneto-elastic oscillations of the star could be excited before the flare ends. Sudden readjustment ofthe magnetospheric configuration could excite frequencies up to ∼
30 kHz, which could deliver energy to the stellar interiorrelatively efficiently, though these frequencies are far higher than those of the observed QPOs. An important question is ifAlfv´en waves have time to excite global magneto-elastic modes before the Alfv´en waves damp. I stress that it is not yet knownif stellar oscillations can produce observable QPOs in the emission, though the mechanism of Timokhin et al. (2008) appearspromising.Realistic microphysical inputs have been used in these analyses, but the conclusions have been drawn using models withsimplified geometries. More realistic energy deposition physics and magnetic field geometries should be considered.
ACKNOWLEDGMENTS
I thank C. D’Angelo, C. Gundlach, Y. Levin, M. Lyutikov, C. Pethick, and A. Watts for enlightening discussions. I am gratefulto A. Watts and Y. Levin for comments on the manuscript. This work was supported by NASA Award NNX12AF88G, NWOVisitor Grant 040.11.403 (PI A. Watts), a Monash Research Enhancement Grant (PI Y. Levin), and a Kevin WestfoldScholarship. I thank the Astronomical Institute Anton Pannekoek, University of Amsterdam, and Monash University, fortheir hospitality.
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