Superfluid instability in precessing neutron stars

Transcription

1 Superfluid instability in precessing neutron stars Kostas Glampedakis SISSA, Trieste, Italy in collaboration with Nils Andersson & Ian Jones Soton October 2007 p.1/15

2 This talk Precessing neutron stars The 2-fluid model for superfluid neutron stars Unstable inertial waves and superfluid turbulence Implications for precessing neutron stars Conclusions Soton October 2007 p.2/15

3 Pulsars & Precessors Recent years: evidence of freely precessing neutron stars. Best candidate: PSR-B1828 with P = 0.4 s, P pr 500 d and wobble angle θ w 3 (Stairs et al. 2000). Just a handful of precessors: precession is a rare phenomenon as to why, we have no clue really... Basic precession model: a biaxial rigid body with misaligned rotational and deformation axis. Precession period: P pr = P/ɛ Timing data: ɛ log 10 [(df/dt)/hz/s] τ = 10 4 yr τ = 10 7 yr df/dt -- f plane log 10 (f/hz) Pulsars Glitchers Precessors However, realistic neutron stars are much more complex... Soton October 2007 p.3/15

4 Why superfluidity? Cooper-pairing below T c 10 9 K neutron stars aged 1 month should be superfluid (neutrons) and superconductive (protons). Critical temperature for neutron and proton superfluidity crust core 1 S0 proton Key physics: S0 neutron 3 P2 neutron Bulk rotation locked into neutron vortex array. T c (K) For Type II superconductivity: magnetic field locked in fluxtubes. Mutual friction ρ (10 14 g/cm 3 ) ρ (10 14 g/cm 3 ) Precession extremely sensitive to superfluid dynamics. Clue: vortex pinning forbids long-period precession! (Shaham 1977) Soton October 2007 p.4/15

5 The 2-fluid formalism A minimal model for a neutron star interior is a mixture of superfluid neutrons and protons ( = conglomerate of protons + electrons). Labelled as x = {n, p}. Euler and continuity equations come in pairs: ( t + v j n j) `v n i + ɛ n pn i ( t +v j p j) `v p i ɛ p pn i ɛp pn j + ɛn pn j ivn j + iψ n = 2ɛ ijk vn j Ωk n + f i mf ivp j + iψ p = 2ɛ ijk vp j Ωk n (ρ n/ρ p )fi mf +ν ee 2 v p i i v i n = iv i p = 0, i np = vi p vi n Shear viscosity enters only in the proton equation, as the underlying mechanism is electron-electron scattering. Coupling due to relative flow: (i) Entrainment ignored in this work: ɛ n = ɛ p = 0 (ii) Mutual friction f mf i where ω i n = ɛ ijk j v n k. = Bɛ ijk ɛ kml ˆω j nω n m np l + B ɛ ijk ω j n k np Soton October 2007 p.5/15

6 Mutual friction (I) Mutual friction force: f mf = B ˆω n ω n np + B ω n np Coupling coefficients B, B make contact with microphysics: vortex interaction with protons/electrons. proton superconductivity plays a key role. Drag force balanced by Magnus force: ˆκ v n V = R v p V B = R/(1 + R 2 ), B = R 2 /(1 + R 2 ) 1 crust core Proton Superconductivity κ - ratio If type II superconductivity: T cp Local neutron circulation induces proton flow (via entrainment) vortex acquires magnetic field T = 5 x 10 7 K Type II Type I normal If type I superconductivity: Vortex can be located in a normal proton domain, no magnetic field acquired ρ (10 14 gr/cm 3 ) Soton October 2007 p.6/15

7 Mutual friction (II) We assume type II proton superconductivity. κ Weak mutual friction Electrons scattering by the vortex s magnetic field (Alpar et al. 1984). B v p V v n Weak drag: R 10 4 B 1, B B 2 e Strong mutual friction (ii) Interaction with fluxtubes: pinning or cutting (Ruderman et al. 1998). κ Strong drag: R 1 B 1, B 1 B 2 B v p v n V If type I superconductivity, only weak mutual friction. fluxtube Soton October 2007 p.7/15

8 Two-fluid inertial waves Inertial modes unveil the two-fluid dynamics. We study short-wavelength (λ R) plane-waves over a background where both fluids rotate rigidly: v i x0 = ɛijk Ω x j x k and δv i x = A i xe iσt+ik j x j Key point: Ω n and Ω p can be misaligned. Wave propagation along vortices: k i = k ˆn i. Two types of inertial waves: (i) neutron modes δv i n 0, δv i p = 0 R 0 limit. (ii) proton modes δv i n = 0, δv i p 0 R 0 limit. Mode frequencies: σ 1 2Ω n + (ib B )(2Ω n k ) + 2iΩ n ν ee k x p B2 (B ) 2 σ 2 (Ω n + Ω p) + k + 2Ω n (ib B x ) + iν ee k 2 p ( proton modes) with x p = ρ p /ρ n, = vp0 i vi n0 ˆn i. ( neutron modes) Soton October 2007 p.8/15

9 A novel superfluid instability The neutron inertial modes can become unstable provided there is sufficient misalignment β between the angular velocities Ω n and Ω p. Mutual friction coupling is the key element. Instability criterion: > 2Ω n /k with = (v i p0 vi n0 )ˆn i ΩRβ. Ω n β Ωp A two-stream type instability (Sidery et al. 2007). v n0 Typical growth time: τ grow λ/2πb v p0 Dissipation: Unstable modes can only feel viscosity as mediated by mutual friction viscous damping weakened. Soton October 2007 p.9/15

10 Tangled vortices New instability not entirely new: resembles the so-called Glaberson-Donnelly instability in superfluid Helium, discovered in the 70s (Glaberson et al. 1974). Superfluid turbulence: Helium lab experiments: the G-D instability drives an initially rectilinear vortex array to a state where vortices get tangled up. Neutron stars: inertial waves are collective vibrations of the vortex array. Unstable inertial modes likely to generate turbulence. Physics of turbulent superlfuids largely phenomenological, essentially unexplored for neutron stars. Soton October 2007 p.10/15

11 Modelling free precession The standard model of neutron star precession is based on the same two-fluid formalism (Sedrakian et al. 1998): superfluid neutrons (spherical component I n ) coupled with protons+crust (biaxial component I p, I p (1 + ɛ) ) via mutual friction. Both components in rigid rotation. J Slow precession Weak drag R 1 P pr P/ɛ, T d (I p /I n )(P pr /2πB) θ w β/ɛ θ w Ω β n Ω p Fast Shaham precession Strong drag R 1 P pr (I p /I n )P, T d P pr /2πB θ w β(i p /I n ) All candidate precessors seem to be slow. Soton October 2007 p.11/15

12 Unstable precessing neutron stars? Precessing neutron stars is the natural playground for the instability. Requires: τ grow < P pr. Slow precession: Unstable waves below a wavelength: λ max = θw 1 or above a wobble angle: θ w > 1.9 P 1s «ɛ 10 8 cm «1/2 ɛ θ w (degrees) ε = 10-7 ε = 10-6 Instability on ε = 10-5 RX J PSR B PSR B P (sec) Conclusion: PSR-B1828 and other candidates are safely stable. A precessing millisecond pulsar with ɛ likely unstable. Soton October 2007 p.12/15

13 Unstable precessing neutron stars? Fast precession τ grow (λ) for R = 10 3 Fastest growing waves for 10 2 «R 1 «θw λ < Wobble angle constraint: cm τ grow (s) P pr I small λ 10 5 K 10 7 K «1/2 R « K θ w > P 1 s Growth timescale: e+05 wavelength λ (cm) τ grow 140 xp 0.1 θ w 1 «1 P 1 s «R 10 3 ««λ R s Conclusion: Fast precession generically unstable. Superfluid wave dynamics/turbulence need to be accounted for. Soton October 2007 p.13/15

14 A precession conundrum? Precession dynamics could probe neutron star interiors (Link 2003/06). Is type II proton superconductivity consistent with long-period precession? Data: slow precession, which is only possible under weak mutual friction coupling. If Type II: Fluxtube-vortex interaction strong mutual friction Only fast precession possible. If Type I: No fluxtubes, macroscopic magnetic bundles instead weak mutual friction slow precession possible. Our results: fast precession is unstable, hence final state may not be that of fast precession Type II could still be consistent. Soton October 2007 p.14/15

15 Conclusions The two-fluid model of a neutron star core allows for unstable/rapidly growing, short-wavelength inertial waves. As in the case of Helium, this instability is likely to lead to vortex tangles and superfluid turbulence. Implications for precessing neutron stars: Slow precessors (weak mutual friction, P pr P ) are typically stable. PSR-B1828 would require a deformation as large as ɛ 10 5 to become unstable. Millisecond pulsars require only ɛ Fast precessors (strong mutual friction, P pr P ) are generically unstable for all relevant wobble angles and core temperatures. Our results add an element of doubt in the proposal that type II superconductivity is inconsistent with long-period precession. Open issues: Numerical simulations of supefluid flows (Peralta et al. 2005/06). Models with three fluids + magnetic field. Instability in the crust. Precession physics with exotica in the core? Soton October 2007 p.15/15