Draft version February 5, 2008
A Preprint typeset using L TEX style emulateapj v. 6/22/04
MAGNETIC BURIAL AND THE HARMONIC CONTENT OF MILLISECOND OSCILLATIONS IN THERMONUCLEAR X-RAY BURSTS
D. J. B. Payne and A. Melatos
School of Physics, University of Melbourne, Parkville, VIC 3010, Australia. Draft version February 5, 2008
arXiv:astro-ph/0607214v1 11 Jul 2006
ABSTRACT Matter accreting onto the magnetic poles of a neutron star spreads under gravity towards the magnetic equator, burying the polar magnetic ?eld and compressing it into a narrow equatorial belt. Steady-state, Grad-Shafranov calculations with a self-consistent mass-?ux distribution (and a semiquantitative treatment of Ohmic di?usion) show that, for Ma 10?5 M⊙ , the maximum ?eld strength and latitudinal half-width of the equatorial magnetic belt are Bmax = 5.6 × 1015 (Ma /10?4M⊙ )0.32 G ˙ and ?θ = max[3? (Ma /10?4 M⊙ )?1.5 , 3? (Ma /10?4M⊙ )0.5 (Ma /10?8 M⊙ yr?1 )?0.5 ] respectively, where ˙ Ma is the total accreted mass and Ma is the accretion rate. It is shown that the belt prevents north-south heat transport by conduction, convection, radiation, and ageostrophic shear. This may explain why millisecond oscillations observed in the tails of thermonuclear (type I) X-ray bursts in low-mass X-ray binaries are highly sinusoidal: the thermonuclear ?ame is sequestered in the magnetic hemisphere which ignites ?rst. The model is also consistent with the occasional occurrence of closely spaced pairs of bursts. Time-dependent, ideal-magnetohydrodynamic simulations con?rm that the equatorial belt is not disrupted by Parker and interchange instabilities. Subject headings: accretion, accretion disks — stars: magnetic ?elds — stars: neutron — stars: rotation — X-rays: bursts
Thermonuclear (type I) X-ray bursts are observed from 70 of the 160 low-mass X-ray binaries (LMXBs) discovered to date (Strohmayer & Bildsten 2003). They recur every few hours to days, when the accreted surface layer of the neutron star ignites and is incinerated by hydrogen and helium burning. Brightness oscillations with millisecond periods are observed during thermonuclear X-ray bursts in 13 LMXBs (Muno et al. 2002; Piro & Bildsten 2004). They arise during burst onset because the stellar photosphere is temporarily patchy while the thermonuclear ?ame spreads from its ignition point to cover the star. A surprising property of the burst oscillations is that they often persist into the tails of bursts, after the ?ame is expected to have engulfed the star (Strohmayer et al. 1997). Equally surprising is how sinusoidal the oscillations are. Muno et al. (2002) analyzed the harmonic content of 59 oscillations from six sources, observed with the Rossi X-Ray Timing Explorer (RXTE ), and found that the Fourier amplitudes of integer and half-integer harmonics are less than 5 and 10 per cent of the maximum signal respectively. These data imply that if there is one hot spot on the surface, it must lie near the rotational pole or cover an entire hemisphere, whereas if there are two, antipodal hot spots, they must lie near the rotational equator (Muno et al. 2002). Most theories that explain why the oscillations persist into the tails of the bursts, e.g. uneven heating/cooling during photospheric uplift (Strohmayer et al. 1997), or cyclones driven by zonal shear in a geostrophic ?ow (Spitkovsky et al. 2002), are hard pressed to account for the pattern of hot spots implied by the Fourier data. Global r-modes in the neutron star ocean can divide the photosphere into symmetElectronic address: firstname.lastname@example.org
ric halves (Heyl 2004), but the physical mechanism converting r-mode density perturbations to brightness oscillations is unclear (Muno et al. 2002). Cumming (2005) showed that di?erential rotation between the pole and equator of 2 per cent can also excite unstable modes. In this paper, we show that an equatorial belt of intense magnetic ?eld, compressed by accreted material spreading away from the magnetic poles, can impede thermal transport between the hemispheres of the star. In §2, we review the physics of magnetic burial and compute the maximum ?eld strength and width of the equatorial magnetic barrier. In §3, we estimate the e?ciency of thermal transport across the barrier by conduction, convection, radiation, and ageostrophic shear, and investigate whether cyclonic ?ows can disrupt the barrier. The implications for the harmonic content of burst oscillations are explored in §4.
2. MAGNETIC BURIAL 2.1. Grad-Shafranov equilibria
In the process of magnetic burial, material accreting onto a neutron star accumulates in a column at the magnetic polar cap, until the hydrostatic pressure at the base of the column overcomes the magnetic tension and matter spreads equatorward, dragging along frozen-in polar magnetic ?eld lines to form an equatorial magnetic belt or ‘tutu’. Figure 1(a), reproduced from Payne & Melatos (2004), illustrates the equilibrium con?guration obtained for Ma = 10?5 M⊙ , where Ma is the total accreted mass. The polar mountain of accreted material (dashed contours) and the pinched, ?aring, equatorial magnetic belt are evident (Melatos & Phinney 2001; Payne & Melatos 2004). The equatorial magnetic ?eld strength increases in inverse proportion to the surface area of the equatorial belt, by ?ux conservation. The Lorentz force per unit volume exerted by the compressed equatorial ?eld [Fig-
2 ure 1(b)] balances the thermal pressure gradient [Figure 1(c)], and gravity, preventing the accreted matter from spreading to the equator. In the steady state, the equations of ideal magnetohydrodynamics (MHD) reduce to the force balance equation (CGS units) ?p + ρ?φ ? (4π) c2 ρ, s
(? × B) × B = 0,
2 GM? r/R?
where B, ρ, p = and φ(r) = denote the magnetic ?eld, ?uid density, pressure, and gravitational potential respectively, cs is the isothermal sound speed, M? is the mass of the star, and R? is the stellar radius. In spherical polar coordinates (r, θ, φ), for an axisymmetric ? ?eld B = ?ψ(r, θ)/(r sin θ) × eφ , equation (1) reduces to the Grad-Shafranov equation ?2 ψ = F ′ (ψ) exp[?(φ ? φ0 )/c2 ], s where ?2 = 1 2 sin2 θ ?0 r sin θ ? ?2 + ?r2 r2 ?θ 1 ? sin θ ?θ (3) (2)
is the Grad-Shafranov operator, F (ψ) is an arbitrary function of the magnetic ?ux ψ, and we set φ0 = φ(R? ). In this paper, as in Payne & Melatos (2004), we ?x F (ψ) uniquely by connecting the initial dipolar state (with Ma = 0) and ?nal distorted state [e.g. Figure 1(a)] via the integral form of the ?ux freezing condition of ideal MHD, viz. dM ds ρ = 2π . (4) dψ C |B| Here, C is any magnetic ?eld line, and the mass-?ux distribution dM/dψ is chosen to capture the magnetospheric geometry, e.g. the accretion stream is funneled magnetically onto the pole, with dM/dψ ∝ exp(?ψ/ψa ), where ψa is the polar ?ux. We also assume north-south symmetry and adopt the boundary conditions ψ = dipole at r = R? (line tying), ψ = 0 at θ = 0, and ?ψ/?r = 0 at large r. The line-tying approximation arti?cially prevents accreted material from sinking, so the computed ρ is an upper limit. Equations (2) and (3) are solved numerically using an iterative relaxation scheme and analytically by Green functions, producing equilibria like Figure 1.
2.2. Equatorial magnetic belt
(Payne & Melatos 2004). Also plotted in Figure 2 is the half-width half-maximum thickness of the belt, ?tted by ?θ = 3? (Ma /Mc )?1.5±0.03 . We ?nd that ?θ decreases as Bmax increases, as expected from magnetic ?ux con?1 servation, but not exactly as ?θ ∝ Bmax , because Br is underestimated numerically at the equator by ? 10 per cent (?ux loss due to ?nite grid resolution). Note that equation (5) and the above scalings of ?θ versus Ma do not include Ohmic di?usion, which is discussed further in §2.3 and §4. Grad-Shafranov equilibria are di?cult to compute directly from (2) and (4) for Ma 1.6Mc , because the magnetic topology changes abruptly: magnetic bubbles form which are disconnected from the surface, hinting at time-dependent processes which the steady-state theory cannot describe. In addition, the iterative relaxation scheme struggles to handle steep ?eld gradients. Therefore the results in Figure 2 for Ma ≥ 1.6Mc are computed by direct numerical simulation using ZEUS3D, a multipurpose, time-dependent, ideal-MHD code for astrophysical ?uid dynamics which uses staggeredmesh ?nite di?erencing and operator splitting in three dimensions (Stone & Norman 1992). We load the GradShafranov equilibrium for Ma = 1.6Mc into ZEUS-3D; double Ma quasistatically over 250 Alfv?n times with e in?ow (and hence radial B) boundary conditions; stop the in?ow and allow B to relax to a dipole at large r; then iterate to reach Ma ? 10?3 M⊙ (Payne & Melatos 2006). An isothermal equation of state is chosen and self-gravity is switched o?. The experiment is performed for h/R? = 2 × 10?2 (for computational e?ciency) and scaled to neutron star conditions (h/R? = 5 × 10?5 ) according to Bmax ∝ R? /h; this scaling is veri?ed numerically for the range of Ma in Figure 2.
2.3. Stability and Ohmic relaxation
The compressed magnetic ?eld in the equatorial belt emerges approximately perpendicular to the stellar surface, with opposite sign in the two hemispheres. Near the equator, B falls o? roughly as exp[?0.7(π/2 ? θ)/?θ] exp[?(r ? R? )/h], where h = c2 R? /GM? is the s hydrostatic scale height, ?θ is the belt thickness, and the factor 0.7 comes from empirically ?tting to the numerical results. The maximum magnetic ?eld strength in the belt, Bmax , computed numerically as a function of Ma , is plotted in Figure 2. It is ?tted by Bmax = 2.0 × 1016 (Ma /Mc )0.91±0.06 G Ma 6.3 × 1015 (Ma /Mc )0.32±0.01 G Ma 0.4Mc 0.4Mc
(5) 2 2 with Mc = GM? B0 R? /8c4 ? 10?4 M⊙ , where B0 is the s polar magnetic ?eld strength prior to accretion. The scaling (5) agrees with analytic theory for small Ma
Distorted ideal-MHD equilibria are often disrupted by Parker and interchange instabilities. Remarkably, however, this is not true for the equilibrium in Figure 1. Figure 3 shows the results of an experiment in which the equilibrium is loaded into ZEUS-3D, perturbed, and evolved for 500 Alfv?n times (Payne & Melatos 2006). e [The Alfv?n time is de?ned as h/vA , where vA is the e Alfv?n speed averaged over the grid.] It is marginally stae ble; Bmax oscillates via magnetosonic (phase speed ≈ cs ) and Alfv?n (phase speed ≈ vA ) modes which are damped e by numerical dissipation. The con?guration already represents the end point (reached quasistatically) of the nonlinear Parker instability of an initially dipolar ?eld overlaid with accreted material. It is not interchange unstable because line tying prevents ?ux tubes from squeezing past each other. That said, we caution that we have not yet investigated the full gamut of three-dimensional MHD instabilities; cross-?eld mass transport cannot be categorically ruled out. Another pathway to cross-?eld mass transport is if the magnetic belt relaxes by Ohmic di?usion, either during quiescence or during a burst itself. During quiescence, di?usion occurs most rapidly at the base of the accreted layer. In the relaxation time approximation, with Coulomb logarithms set to 10, the electron-phonon conductivity for a hydrogen-helium mixture is σ = 6.3 × 1024 (ρ/1011 g cm?3 ) s?1 , the density at the base of the accreted layer is ρ = 6.2 × 1010 (Ma /10?5 M⊙ )3/4 g cm?3
3 (Brown & Bildsten 1998), and hence the Ohmic di?usion time-scale across the equatorial belt (at the base 2 of the accreted layer) is td = 4πσR? ?θ2 /c2 = 2.6 × 7 ?5 ?9/4 8 ?1 10 (Ma /10 M⊙ ) (T /10 K) yr, much longer than the burst time-scale, where we rewrite ?θ in terms of Ma . During a burst, di?usion occurs most rapidly in the burning layer. The temperature of the burning layer rises isobarically until the radiation pressure dominates the hydrostatic pressure, reaching T ? 108 K for a typical ignition column (Brown 2004). The elevated temperature and reduced density lower σ and accelerate Ohmic di?usion. In the burning layer (ρ ≈ 106 g cm?3 ), we ?nd td = 1.5 × 103 yr, still much longer than the burst time-scale. Ohmic di?usion will be modelled self-consistently in a future paper.
3. THERMAL TRANSPORT ACROSS THE EQUATOR
(Brown & Bildsten 1998), ?nding tcond = 27(Ma /Mc )?3 (T /108 K)?1 (B/1015 G)2 yr , (6) which is safely longer than the duration of the burst. Note that this estimate does not include corrections due to electron-electron scattering and impurity scattering. Furthermore, our ?t smoothes over the wiggles in the solid curves of Figures 5 and 6 in Potekhin (1999), which arise from quantization into Landau orbitals.
Does the equatorial magnetic belt impede thermal transport enough to stop the thermonuclear ?ame in a type I X-ray burst from spreading from one hemisphere to another? A burst is initiated locally by a thin shell thermal instability (Schwarzschild & H¨rm a 1965). As the nuclear burning time-scale is much shorter than the time to accrete the minimum column for ignition, the accreted layer ignites at a single point, most likely at the equator where gravity is rotationally reduced, and the thermonuclear ?ame spreads away either as a de?agration front (Fryxell & Woosley 1982; Bildsten 1995), by detonation (Fryxell & Woosley 1982; Zingale et al. 2001), or as a cyclone driven by zonal shear (Spitkovsky et al. 2002). Detonation, which occurs when the nuclear burning time-scale is less than the vertical sound crossing time-scale, requires a thick (? 100 m) column of fuel and hence a low accretion rate ( 10?11.5 M⊙ yr?1 ). De?agration occurs most commonly, with the front propagating at a speed set by the heat ?ux and convection (Fryxell & Woosley 1982). Note that temperature ?uctuations leading to single-point ignition are di?cult to create because the sound crossing time around the star is very short, but Coriolis forces can assist by balancing sideways pressure gradients when the star is rapidly rotating (Spitkovsky et al. 2002). We estimate below how the equatorial magnetic belt modi?es heat transport in these scenarios.
In a strong magnetic ?eld, photons polarized perpendicular to B dominate radiative transport. In the burning (helium) layer, where Thomson scattering dominates, the Rosseland mean opacity ⊥ B is given by κR = 1.3 × 10?6(T /108 K)2 (B/1015 G)?2 g?1 cm2 , decreasing in inverse proportion to the square of the cyclotron frequency (ωc ) (Joss & Li 1980; Fryxell & Woosley 1982). Hence the optical depth of the barrier is τR = κR ρR? ?θ, i.e. τR = 6.8 × 104 (Ma /Mc )?3/2 (T /108 K)2 (B/1015 G)?2 . (7) The optical depth (7) is a lower limit obtained by assuming that radiation is transported from one hemisphere to the other near the surface, at the depth of the burning layer (ρ ? 106 g cm?3 ) rather than at the base of the accreted column (ρ ? 1011 g cm?3 ). Hence the peak ?ux 2 penetrating the barrier is Fburst = Lburst e?τR /4πR? = 8 × 1024 e?τR erg s?1 cm?2 (for Lburst = 1038 erg s?1 ), which is insu?cient to heat the other (quiescent) hemisphere above its ignition temperature (? 108 K) (assuming thermal equilibrium and applying the StefanBoltzmann law). Interestingly, once Bmax exceeds 1017 G, we ?nd τR 1 and the magnetic belt becomes optically thin. Vertical heat propagation is not considered here. Equation (7) can be generalized in several ways. Mode coupling (Miller 1995) can cause a fraction ? 0.2ω/ωc of the perpendicular mode photons to convert into parallel mode photons, so that the opacity scales ∝ B ?1 , increasing the ?eld strength required for the belt to become opti¨ cally thin. Vacuum polarization (Ozel 2003), along with proton-cyclotron resonance, gives sharp spikes in the frequency response of the atmospheric opacity. However, the resonant densities for vacuum polarization (ρ ? 10?3 g cm?3 ) are achieved at shallow depths, well above the burning layer.
The thermal conductivity κ of degenerate electrons at the atmosphere-ocean boundary of a magnetized neutron star, where H/He burning occurs, was calculated by Potekhin (1999). We extract approximate values for the conductivity perpendicular to the magnetic ?eld from Figures 5 and 6 of that paper, obtaining κ⊥ ≈ 107.5 (ρ/106 g cm?3 ) (T /108K)2 (B/1015 G)?2 in units of erg s?1 cm?1 K?1 in the regime 104 g cm?3 ρ 109 g cm?3 , 1012 G B 1015 G, and 106 K T 108 K. Note that κ⊥ at the equator is reduced 106 times relative to its value before accretion commences (∝ B ?2 ) and, for reference, one has κ⊥ /κ ? 10?5 at T = 108 K, B = 1012 G (Potekhin 1999). We estimate the conduction time-scale across the magnetic barrier from tcond = 2 R? ?θ2 Cρ/κ⊥ , where C = 104 (T /108K) erg g?1 K?1 is the speci?c heat capacity of a degenerate Fermi gas
Bildsten (1995) estimated the convective speed in the burning layer in terms of the mixing length, lm , and thermal time-scale, tth , ?nding vc ≈ 2 cs (lm /h)1/3 (cs R? /GM? tth )1/3 ≈ 106 cm s?1 . This is consistent with the upper bound 107 cm s?1 obtained if 3 Lburst is transported entirely by the mechanical ?ux ρvc . It is also consistent with the observed burst rise time R? /vc 1 s (Spitkovsky et al. 2002). Convection is stabilized magnetically if the magnetic tension exceeds the 2 2 ram pressure ρvc , which occurs for Bmax (8πρvc )1/2 ≈ 5×109 G. This condition is met comfortably in the equatorial belt. The magnetic ?eld can also quench convection (Gough & Tayler 1966).
3.4. Ageostrophic shear ?ow Spitkovsky et al. (2002) suggested that inhomogeneous cooling drives zonal currents which are unstable to the formation of cyclones, as in planetary atmospheres. This may explain why the coherent oscillations observed in the tails of some type I X-ray bursts persist for many rise times. The drift in oscillation frequency (? Hz) during the burst is attributed to the Coriolis drift of the cyclone in the frame of the star, although theory predicts larger frequency drifts (? 10 Hz) than those observed. The oscillation amplitude in the burst tails (? 10%) is governed by processes other than magnetic burial, e.g. small-scale magnetic ?elds generated by an MHD dynamo in the burning front, which are con?ned to the ashes after the burst, while the freshly accreted matter remains unmagnetized (Spitkovsky et al. 2002). Ageostrophic shear ?ow moves hot material ahead of the burning front and draws fresh fuel into it at the ?ame speed v?ame ≈ 2 × 107 cm s?1 (hhot /103 cm)(f /0.32kHz)?1 (tnuc /0.1 s)?1 , where hhot is the scale height of the incinerated ocean, f = 2? cos θ is the local Coriolis parameter, ? is the angular frequency of the star, and tnuc is the nuclear burning time-scale. Magnetic tension stabilizes 2 ageostrophic shear for B (8πρv?ame )1/2 ≈ 4.5 × 1010 G ? Bmax . In other words, when the cyclonic ?ame runs into the equatorial magnetic belt, it is re?ected; conversely, ageostrophic shear cannot disrupt the belt. Latitudininal shear instabilities, a possible source of burst oscillations (Cumming 2005), may disrupt the magnetic belt and provide one motivation for extending our burial model to three dimensions in the future. 4. DISCUSSION
Polar magnetic burial creates an intense, equatorial belt of magnetic ?eld which can thermally isolate the magnetic hemispheres of an accreting neutron star. The maximum magnetic ?eld strength in the belt, Bmax = 5.6 × 1015 (Ma /10?4 M⊙ )0.32 G, is su?cient to prevent heat transport by conduction, radiation, convection, and ageostrophic shear. The conduction time-scale tcond ? 27 yr exceeds the cooling time of the incinerated material; the magnetic belt is opaque (optical depth ? 7 × 104 ); 2 2 convection is stabilized magnetically (ρvc ? Bmax /8π); 2 2 and so is ageostrophic shear (ρv?ame ? Bmax /8π). However, the conclusion that the hemispheres are thermally isolated by the magnetic belt is less secure at large accreted masses (Ma ? 0.1M⊙ ), where the Grad-Shafranov and ZEUS-3D calculations in section 2 are hampered by numerical di?culties, and Ohmic di?usion (which we do not incorporate self-consistently) becomes important. Thermal isolation of the magnetic hemispheres is consistent with the highly sinusoidal light curves of millisecond oscillations in thermonuclear bursts in LMXBs (Muno et al. 2002). During the rise of the burst, oscillations are caused by the spreading of a hot spot, probably in the form of a cyclone (Spitkovsky et al. 2002). In the tail of the burst, the hot spot is sequestered in one hemisphere (Muno et al. 2002), and misalignment of the magnetic and spin axes guarantees that we observe persistent oscillations at the spin frequency. Furthermore, it is observed that bursts occasionally occur in quick succession, separated by a 5 ? 10 minute interval, compared
to an interval of several hours between typical bursts (Lewin et al. 1993). Such burst pairs are consistent with the model of magnetic burial: if a burst ignites one hemisphere, the other remains dormant and can, in principle, ignite shortly thereafter when more material accretes. If this explanation is valid, one would expect that there are no triple bursts, that the ?uences emitted in each half of a burst pair are roughly equal, and that the number of burst pairs relative to normal bursts increases as the ignition probability per unit time increases (relative to ˙ Ma ).1 A more detailed calculation is required to resolve whether the 10-min intervals are caused by delayed heat propagation through the magnetic belt. Muno et al. (2002) showed that the bright burning region must cover 80? ? 110? in latitude in order to match the upper limit on the observed ratio of harmonic to fundamental amplitudes, which implies ?θ 10? . If the scaling of ?θ versus Ma in §2.2 is taken at face value, this requires Ma > 0.5Mc . Furthermore, for the belt to be optically thick, we need Ma 2Mc (§3.2). Apparently then, only a narrow range of accreted masses (0.5 Ma /Mc 2) can account for the observations. However, the above ?θ scaling, which depends steeply on Ma , does not tell the whole story, because Ohmic di?usion is not incorporated self-consistently. Equatorward hydromagnetic spreading is arrested when the accretion time-scale exceeds the Ohmic di?usion time-scale (Brown & Bildsten 1998), softening the dependence of Bmax and ?θ on Ma in the manner described by Melatos & Payne (2005). Ohmic di?usion arrests magnetic compression for Ma > Md , where ˙ Md = 3.4×10?7(T /108 K)?2.2 (Ma /10?8 M⊙ yr?1 )0.44 M⊙ (Melatos & Payne 2005) is the accreted mass at which the accretion time-scale exceeds the Ohmic di?usion time-scale. Including this e?ect, ?θ is modi?ed to the maximum of 3? (Ma /10?4 M⊙ )?1.5 and ˙ 3? (Ma /10?4 M⊙ )0.5 (Ma /10?8 M⊙ yr?1 )?0.5 ], and the belt remains optically thick even for Ma 2Mc (§3.2). Only 70 out of 160 LMXBs undergo bursts, and only 13 exhibit millisecond oscillations. Does the equatorial belt model respect these statistics? If accretion occurs ˙ at the Eddington rate, Ma ≈ 10?8 M⊙ yr?1 , it takes less than 104 yr to achieve Ma > Mc and screen the polar magnetic ?eld, allowing bursts to ignite and creating a thermally insulating equatorial belt. However, for Ma 2 × 103 Mc , the belt becomes optically thin to Xrays and tcond drops below the duration of the burst, potentially allowing the ?ame to engulf the entire star in LMXBs older than a few times 107 yr. If accretion ˙ occurs at Ma ? 10?11 M⊙ yr?1 , as in accreting millisecond pulsars (Chakrabarty et al. 2003), it takes 107 yr to achieve Ma > Mc , screen the polar magnetic ?eld, and allow bursts to ignite. Therefore, we do not expect bursts from all accreting millisecond pulsars, but when bursts do occur, we expect to detect oscillations at some level because we have τR ? 1 and tcond ? (burst time-scale) for Ma 10?3 M⊙ . Such is the case, within current observational sensitivity, for SAX J1808.4-3658
1 If the ignition probability is higher (given a speci?c M ), it ˙a is more likely that the second hemisphere will ignite shortly after the ?rst (before the ?rst has a chance to re?ll and ignite again). A faster burst rate may be associated with more pairs, but gaps in the satellite data make it di?cult to be sure.
5 and XTE J1814-338. Sources like XTE J1814-338, whose oscillations contain harmonics exceeding 25 per cent of the peak amplitude (Strohmayer et al. 2003), may have semi-transparent equatorial belts; perhaps this millisecond pulsar has experienced interruptions in its accretion history resulting in Ma < Mc . Note that, for ˙ Ma 10?10 M⊙ yr?1 , Ohmic di?usion may magnetize the freshly accreted material (Cumming et al. 2001), increasing the polar magnetic ?eld strength and suppressing bursts. Also, ignition is more likely near the equator, where gravity is rotationally reduced, but this is where the magnetic ?eld is strongest. The issues of conservative mass-transfer (Tauris et al. 2000) and di?erential rotation leading to shear instabilities (Cumming 2005) are not considered here. On the face of it, high-mass X-ray binaries (HMXBs) accrete many times Mc , yet a strong magnetic ?eld survives. However, one needs to be cautious. While the accretion rate in HMXBs can reach 10?10 –10?8 M⊙ yr?1 during either atmospheric Roche lobe over?ow (close binaries with orbital period < 5 days) or stellar wind accretion (wider binaries), the typical lifetime of HMXBs as strong X-ray sources is 104 –105 yr (Urpin et al. 1998). Sub-Eddington accretion rates (10?13 –10?10 M⊙ yr?1 ) characterize the rest of the companion’s main-sequence evolution for 106 –107 yr. These scenarios yield roughly Ma ? Mc . Mass transfer can also be nonconservative, further reducing Ma , e.g., in intermediate-mass X-ray binaries (Tauris et al. 2000). Of course, despite these cautionary remarks, it may well be that some HMXBs do accrete many times Mc . If so, then either HMXBs are a counterexample to the simple magnetic burial model we have calculated, or else cross-?eld transport by Ohmic di?usion (which we do not incorporate self-consistently) becomes important. A recent model for doubly peaked bursts which are too weak to cause photospheric expansion (Bhattacharyya & Strohmayer 2006) proposes that a burning front forms quickly after ignition at or near a pole and propagates quickly towards the equator where it stalls. After a delay, the burning front speeds up again into the opposite hemisphere. While the time between burst peaks (? 10 seconds) is less than that estimated for burst propagation across the magnetic belt, a weak magnetic belt may explain why the burning front stalls at the equator. Numerical di?culties associated with steep ?eld gradients prevent us from verifying whether our model scales up to Ma ? 0.1M⊙ . However, if we continue to respect mass-?ux conservation as in §2.1, it is probable that accreting extra mass does not eliminate the magnetic barrier. Even if, for example, there are instabilities which disrupt the equatorial magnetic belt above ? 10?4 M⊙ , the belt does not stay disrupted; magnetic burial ensures that it reforms as soon as a further ? 10?5 M⊙ is accreted (Payne & Melatos 2004). Given that 0.1M⊙ is typically accreted in LMXBs, the chances of catching the belt in its disrupted state are slim.
We thank Duncan Galloway for pointing out to us that sinusoidal light curves are a signature of hemispheric emission, and alerting us to the existence of burst pairs. We thank an anonymous referee for pointing out to us that the burial model may imply the existence of pairs of bursts in quick succession, and for comments which improved the treatment of the crust and burning physics in the manuscript. This research was supported in part by an Australian Postgraduate Award.
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Fig. 1.— (top) Equilibrium magnetic ?eld lines (solid curves) and density contours (dashed curves) for Ma = 10?5 M⊙ and ψa = 0.1ψ? . Coordinates measure altitude. Density contours are drawn for ηρmax (ρmax = 2.52 × 1014 g cm?3 ), with η = 0.8, 0.6, 0.4, 0.2, 10?2 , 10?3 , 10?4 , 10?5 , 10?6 , 10?12 . Convergence residuals are less than 10?3 . [From Payne & Melatos (2004).] (middle) Contours of Lorentz force per unit volume for the same η values. (bottom) Contours of pressure gradients for the same η values. Note that colatitude = 0? at the pole.
Fig. 2.— Maximum magnetic ?eld in the equatorial belt, Bmax , computed numerically for h/R? = 2×10?2 (crosses) and h/R? = 5×10?5 (triangles) and scaled using Bmax ∝ h?1 , plotted together with the half-width half-maximum thickness, ?θ (squares), as a function of accreted mass, Ma .
Fig. 3.— Maximum magnetic ?eld strength, Bmax , as a function of time (in units of the Alfv?n time) for Ma /Mc = e 0.16, 0.32, 0.64, 1.12, 2.4, 4.0 (bottom to top) when the equilibrium in Figure 1 is loaded into ZEUS-3D and perturbed slightly. The con?guration is stable.