Time-dependent Force-free Pulsar Magnetospheres: Axisymmetric and Oblique Rotators

By A. Spitkovsky. 2006

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Magnetospheres of many astrophysical objects can be accurately described by the low-inertia (or "force-free") limit of MHD. We present a new numerical method for the solution of equations of force-free relativistic MHD based on the finite-difference time-domain approach, with a prescription for handling the spontaneous formation of current sheets. We use this method to study the time-dependent evolution of pulsar magnetospheres in both aligned and oblique magnetic geometries. For the aligned rotator, we confirm the general properties of the time-independent solution of Contopoulos et al. For the oblique rotator, we present the three-dimensiional structure of the magnetosphere and compute, for the first time, the spin-down power of pulsars as a function of the inclination of the magnetic axis. We find that the pulsar spin-down luminosity is L ≈ (μ2Ω/c3)(1 + α), where μ is the stellar dipole moment, Ω* is the rotation frequency, and α is the magnetic inclination angle. We also discuss the effects of current sheet resistivity and reconnection on the structure and evolution of the magnetosphere.

Subfields: Astronomy and Astrophysics

View Paper: https://www.semanticscholar.org/paper/24b886ea509a19889a947561622954dd7eb19892

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EFFICIENCY OF MAGNETIC TO KINETIC ENERGY CONVERSION IN A MONOPOLE MAGNETOSPHERE (2009, background): “…of a dipolar pulsar magnetosphere, and various studies have indicated that the low efficiency of the split-monopole magnetosphere carries over to the dipolar problem (Contopoulos et al. 1999; Uzdensky 2003; Gruzinov 2005; McKinney 2006c; Spitkovsky 2006; Komissarov 2006; Bucciantini et al. 2006).”
Millisecond Magnetar Birth Connects FRB 121102 to Superluminous Supernovae and Long-duration Gamma-Ray Bursts (2017, background): “…& Bildsten (2010), which however differs in normalization from the force-free spin-down rate, which is likely more applicable in the plasma-dense environment of a young neutron star (e.g. Spitkovsky 2006). ejecta be comparable, or moderately less than, the timescale of the SN optical peak of weeks.”
Spacetime approach to force-free magnetospheres (2014, background): “A current sheet of nearly identical shape and dynamics is observed outside the light cylinder in simulations of inclined dipolar magnetospheres (Spitkovsky 2006; Kalapotharakos, Contopou-los & Kazanas 2012).”
Three-dimensional numerical simulations of the pulsar magnetosphere: preliminary results (2008, methodology): “Another interesting issue has to do with the pulsar spindown rate L as a function of the inclination angle θ as obtained by Spitkovsky (2006), namely L(θ) = B2∗r6∗Ω4∗ 4c3 (1 + sin2 θ), (9) where B∗, r∗ are the polar dipole magnetic field and radius of the central neutron star respectively.”
Time-dependent 3D magnetohydrodynamic pulsar magnetospheres: oblique rotators (2012, background): “…of axisymmetric (Contopoulos, Kazanas & Fendt 1999; Gruzinov 2005; McKinney 2006b; Timokhin 2006; Parfrey, Beloborodov & Hui 2012) and oblique (Spitkovsky 2006, S06 hereafter; Kalapotharakos & Contopoulos 2009; Li, Spitkovsky & Tchekhovskoy 2012b; Pétri 2012a) pulsar magnetospheres were…”
A burst in a wind bubble and the impact on baryonic ejecta: high-energy gamma-ray flashes and afterglows from fast radio bursts and pulsar-driven supernova remnants (2016, methodology): “Based on recent magnetohydrodynamics simulations (Gruzinov 2005; Spitkovsky 2006; Tchekhovskoy, Spitkovsky & Li 2013), the initial spindown power is Lsd,i ≈ B 2 ∗R 6 ∗(2π/Pi) 4 4c3 (1 + sin2 χ) ≃”
Gamma-ray binaries and related systems (2013, background): “Particles moving along open field lines escape, forming a pulsar wind carrying away energy at a rate (Spitkovsky 2006; Pétri 2012, and references therein) Ė ≈ B 2 nsR 6 nsΩ 4 c3 ( 1 + sin2χ ) (5) where χ is the obliquity (inclination of the magnetic field axis with respect to the spin axis, χ= 0◦…”
Time-dependent pair cascades in magnetospheres of neutron stars - I. Dynamics of the polar cap cascade with no particle supply from the neutron star surface (2010, methodology): “Recently the force-free pulsar magnetosphere model has been studied in great detail (e.g. Contopoulos, Kazanas & Fendt 1999; Gruzinov 2005; Timokhin 2006, 2007a; Spitkovsky 2006; Kalapotharakos & Contopoulos 2009; Bai & Spitkovsky 2010).”
A Decade of Gamma-Ray Bursts Observed by Fermi-LAT: The Second GRB Catalog (2019, background): “…electromagnetic spin-down power of a neutron star with surface (equatorial) dipole magnetic field Bdip and spin period P is given by (Spitkovsky 2006): ĖFF ≈ 4π4B2dipR 6 P 4c3 ∼ 1049 ( Bdip 1015 G )2( P 1 ms )−4( R 10 km )6 erg/s, (11) 52 The Fermi-LAT collaboration where R is the…”
A NICER View of PSR J0030+0451: Evidence for a Global-scale Multipolar Magnetic Field (2019, background): “Solutions of the global force-free (FF) pulsar magnetosphere (Contopoulos et al. 1999; Spitkovsky 2006) require specific current patterns that depend on magnetic inclination angle (Bai & Spitkovsky 2010; Timokhin & Arons 2013).”
Constraining the Maximum Mass of Neutron Stars from Multi-messenger Observations of GW170817 (2017, background): “A NS of radius Rns, rotation frequency Ω = 2π/P , and spin period P loses energy to a magnetic wind at a rate (Spitkovsky 2006) Ėmag = µ2Ω4 c3 (1 + sin2 χ), (2) where Bd, µ = BdR 3 ns, and χ are the surface magnetic dipole field strength, dipole moment, and angle between the rotation and dipole…”
Formation of Double Neutron Star Systems (2017, positive): “A combined model was derived by Spitkovsky (2006) (see also Li et al. 2012) and applying his relation between B and α, Tauris et al. (2012) obtained5: B= √ c3IP Ṗ 4π2R6 1 1 + sin2 α (1) ≃ 1.3× 1019G √ PṖ ( M 1.4M⊙ )3/2√ 1 1 + sin2 α .”
Unifying the observational diversity of isolated neutron stars via magneto-thermal evolution models. (2013, methodology): “The spin-down formula for force-free magnetospheres, and valid for oblique rotators (Spitkovsky 2006), gives the following relation IΩΩ̇ ≈ − B2pR 6Ω4 4c3 (1 + sin2 χ) (6) where R is the NS radius, χ is the angle between the rotational and the magnetic axis, c is the speed of light, Ω = 2π/P is the…”
MODELING OF GAMMA-RAY PULSAR LIGHT CURVES USING THE FORCE-FREE MAGNETIC FIELD (2009, methodology): “In this paper, we use the force-free field from 3D timedependent simulations by Spitkovsky (2006) (hereafter, S06), and present the calculation of pulsar gamma-ray light curves using various theoretical models of emission.”
The Proto-Magnetar Model for Gamma-Ray Bursts (2010, background): “Indeed, in the limit that σ0 ≫ 1 we assume that Ė approaches the force-free spin-down rate (Spitkovsky 2006), which depends only on φ and Ω.”
Pulsar Wind Nebulae as Cosmic Pevatrons: A Current Sheet’s Tale (2012, background): “…analytically minded theorists to guess the detailed answer uniformly failed over 30 years of trying), first for the aligned rotator (Contopoulos et al. 1999; Gruzinov 2005; Komissarov 2006; Timokhin 2006), then for the full 3D oblique rotator (Spitkovsky 2006; Kalapotharakos and Contopoulos 2009).”
ECHO: a Eulerian conservative high-order scheme for general relativistic magnetohydrodynamics and magnetodynamics (2007, methodology): “Moreover, different methods other than Godunov-type have also been proposed for GRMHD (Koide et al. 1999; Wilson & Mathews 2003; Koide 2003; De Villiers & Hawley 2003; Anninos et al. 2005) and (GR)MD (Spitkovsky 2006).”
High-Altitude Emission from Pulsar Slot Gaps: The Crab Pulsar (2008, background): “While the vacuum dipole solution does not accurately apply to a real pulsar magnetosphere that is filled with charges, and more realistic ideal MHD solutions now exist (Spitkovsky 2006; Timokhin 2006), the MHD solutions are numerical and timeconsuming to simulate.”
AN ATLAS FOR INTERPRETING γ-RAY PULSAR LIGHT CURVES (2008, background): “Numerical models of force-free magnetospheres do exist (Spitkovsky 2006), but such models do not have the resolution required for the sort of beaming computations pursued here and in any case do not, by definition, include acceleration gaps.”
TOWARD A REALISTIC PULSAR MAGNETOSPHERE (2011, methodology): “Thus, Spitkovsky (2006) implemented a scheme that evolves the fields considering only the first term of Eq.”