Electron-MHD Turbulence in Neutron Stars Toby Wood Newcastle University (previously University of Leeds) Wood, Hollerbach & Lyutikov 2014, Physics of Plasmas 21, 052110
Outline What is a neutron star? What is Hall / Electron MHD? What is the preferred field geometry in a neutron star? How does the field affect the star s evolution? Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 2 / 18
What is a neutron star? Types of dead star: M M white dwarf, supported by electron degeneracy pressure M M 3M neutron star, supported by neutron degeneracy pressure 3M M black hole Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 3 / 18
Neutron star structure M 1.5M R 10km Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 4 / 18
Typical parameters B 10 15 G Ω 10 3 s 1 T 10 8 K Do all neutron stars look like this? Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 5 / 18
Neutron star evolution Age Rotation Period Magnetic Field Magnetars 10 3 years 2 10 s 10 14 15 G Classical Pulsars 10 3 7 years 5 ms 5 s 10 11 13 G Millisecond Pulsars 10 8 10 years 1 10 ms 10 8 9 G Need to explain: magnetic field decay spindown? cooldown glitches magnetic spots? Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 6 / 18
Electron MHD / Hall MHD In a neutron star crust, ions are held fixed within the lattice, but electrons can still flow (EMHD). In Gaussian cgs units, ( v m t +v v B t = c E J = c B = env ) 4π = 1 n P e (E+ v ) c B Electromagnetism negligible on scales d ( mc 2 ) 1/2 d = 4πne 2 10 11 cm. Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 7 / 18
A non-linear induction equation [ B c t = 4πen } B ( B) {{} Hall term η B }{{} diffusion ] c + en P Important parameter is R B = c B 4πenη 102. Hall term has 2 derivatives, but is non-dissipative: d 1 dt 2 B 2 = η B 2. V V Could the Hall term enhance the magnetic diffusion? (Goldreich & Reisenegger 1992) Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 8 / 18
EMHD turbulence [ B t = B ( B) 1 ] B R B Well-defined energy spectrum and forward cascade But no dissipative cut-off... and interactions are non-local in spectral space Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 9 / 18
EMHD turbulence AND temporal coherence in real space Wareing & Hollerbach 2009, 2010 Is this really turbulence at all? Is the system even really unstable?? Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 10 / 18
EMHD stability Are there instabilities in EMHD? Yes, for nonuniform density (Gordeev & Rudakov 1968) Yes, for finite resistivity (Gordeev 1970) Yes, for finite inertia (Bulanov et al. 1992) What about for uniform density, zero resistivity, and zero inertia? B t = [B ( B)] Yes. Drake et al. (1994) Yes. Rheinhardt & Geppert (2002) No. Lyutikov (2013) Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 11 / 18
EMHD stability Regular MHD: B t = [u B] Du Dt = ( B) B P. Electron MHD: B t = [B ( B)] For a mode with growthrate λ, For a mode with growthrate λ, λ 2 dv ξ 2 = dv ξ F(ξ). (λ+λ ) dv ξ F(ξ) = 0. Stable iff F is negative definite (Lundquist 1951). Stable if F is negative definite. So electron MHD is at least as stable as regular MHD (Wood, Hollerbach & Lyutikov 2014). Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 12 / 18
EMHD stability For nonuniform density, EMHD is at least as stable as anelastic MHD. So the density-gradient instability in EMHD must have an anelastic analog. In fact, this is the anelastic zero-gravity magneto-buoyancy instability! Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 13 / 18
More realistic numerical modeling [ ] B t = c 4πen(r) B ( B) η(r) B A Hall attractor in 2D. 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.2 1.0 0.8 0.6 0.4 0.2 0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Is this stable to 3D perturbations? Wood & Hollerbach (in prep.) Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 14 / 18
More realistic numerical modeling RB = 50 RB = 200 Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 15 / 18
More realistic numerical modeling Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 16 / 18
Coupled magneto-thermal evolution [ B 1 t = 0 = B ( B) η B + S T n ( k T +(1/nη)2 (B T)B+(1/nη)B T 1+(1/nη) 2 B 2 ] ) Induction equation has a source term from electron baroclinicity. Temperature diffusion is inhibited across B field lines. Can lead to electron convection. Pons et al. (2009) Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 17 / 18
Implications and future work Non-uniform density can drive EMHD instability in neutron stars. But realistic 3D simulations generally converge to a large-scale equilibrium. Strong jets of warm electrons near the surface. Possibility of electron convection automatic dynamo! Toby Wood (Newcastle) Electron-MHD Turbulence in Neutron Stars 18 / 18