# Superfluid instabilities and neutron star dynamics

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Superfluid instabilities and neutron star dynamics

Superfluid instabilities and neutron star dynamics Kostas Glampedakis University of Tuebingen Superfluid instabilities and neutron star dynamics 1/16 Setting the scene We observe aspects of neutron star dynamics, like glitches and precession, that we do not really understand. A key point: the observed dynamics is very sensitive to the presence of superfluids in neutron star interiors → could be used to probe the properties of neutron star matter. Existing work on glitches/precession is based on phenomenological “rigid-body” models, and/or studying the local vortex dynamics (eg. vortex pinning/creep ...). This is an obvious restriction. What can we learn if we address these problems using global superfluid hydrodynamics ? Superfluid instabilities and neutron star dynamics 2/16 Why superfluidity? • Neutron stars cool down below Tc soon after birth. Critical temperatures for neutron and proton superfluidity core crust 10 10 1 10 1 10 • At least two distinct fluids in the interior (likely more in the presence of ’exotica’ in the inner core). Tc (K) 3 10 10 10 • A “realistic” neutron star model should be based on multifluid hydrodynamics. 9 10 8 1 14 3 ρ (10 g/cm ) Superfluid instabilities and neutron star dynamics S0 neutron P2 neutron 9 10 7 S0 proton 10 10 8 7 1 14 3 ρ (10 g/cm ) 3/16 10 The two-fluid model • “Minimal” model for a superfluid neutron star: neutron fluid = superfluid neutrons, ’proton’ fluid = superconducting protons and electrons. • Hydrodynamics: (∂t + (∂t + vnj ∇j ) vpj ∇j ) vpi vni + ǫn w − ǫp w i i i + ǫn wj ∇i vnj + ∇i (µ̃n + Φ) = fmf i − ǫp wj ∇i vpj + ∇i (µ̃p + Φ) = −(ρn /ρp )fmf i ∂t ρn,p + ∇i (ρn,p vn,p )=0 where wi = vpi − vni . • Couplings: i , and Entrainment parameters {ǫn , ǫp }, mutual friction force fmf chemical potentials µ̃n,p (through the EoS) Superfluid instabilities and neutron star dynamics 4/16 Mutual friction • Mutual friction is a vortex-mediated force: f~mf R2 R ω̂n × (~ωn × w) ~ + (~ ωn × w) ~ , = 2 2 1+R 1+R ωn = ∇ × ~vn ~ . • The drag R makes contact with the dynamics of a single vortex: drag force R∼ Magnus force R → 0 zero coupling limit, R → ∞ perfect pinning limit. • Diversity in drag forces: – Weak drag R ≪ 1 (electron-vortex scattering). – Strong drag R ≫ 1 (interaction with fluxtubes (core) or lattice (crust)). Superfluid instabilities and neutron star dynamics 5/16 Exploring some new dynamics • Lets consider linear oscillations ( vni → vni + δvni etc) including two key properties: (1) In the background the fluids are in rigid-body rotation with different spin frequencies: ~ n,p × ~r, ~vn,p = Ω ~ n 6= Ω ~ p. Ω (2) Mutual friction is present. • For simplicity we ignore entrainment and assume a uniform j background (⇒ ∇j δvn,p = 0). • Main result: We find two new instabilities associated with inertial modes. Superfluid instabilities and neutron star dynamics 6/16 A new r-mode instability • Global mode analysis: we obtain axial r-mode solutions (for ℓ = m). • One of the r-mode solutions becomes unstable provided: R≫1 Ωn − Ωp 6 0 = and ∆ = Ωp • The instability appears for: m > mc ≈ 320 0.05 xp 1/2 10 ∆ −4 1/2 • The instability is dynamical. Growth timescale: τgrow P x 1/2 10−4 1/2 p ≈ 2.5 0.05 ∆ Superfluid instabilities and neutron star dynamics 7/16 Pulsar glitches • Crust & proton fluid: spin-down together (Ωp = Ωc ) spin frequency • Neutron fluid: unable to follow spin down if vortices are pinned (Ωn > Ωp ). pinned superfluid Ωs Ωc crus t+p spin dow n lasm a relaxation time • At a critical spin-lag vortex unpinning is forced, and angular momentum is transferred to the crust: Ic ∆Ωc Ωs − Ωc ≈ ≈ 5 × 10−4 ∆g = Ωc Is Ωc • Data for Vela-like glitches: ∆Ωc /Ωc ∼ 10−6 , Superfluid instabilities and neutron star dynamics Is /Ic . 0.02 8/16 A glitch-trigger mechanism • The r-mode instability needs to overcome viscous dissipation, (dominated by electron shear viscosity). Combining τgrow < τsv and m > mc leads to a critical ∆ at which the instability first appears: ∆c ≈ 6 × 10−5 P 0.1 s 2/3 T 108 K −4/3 • Estimate glitch size: ∆g = ∆ ≥ ∆c ⇒ ∆Ωc & 10−6 Ωc Is /Ic 0.02 P 0.1 s 2/3 T 108 K −4/3 • Given an estimated (“observed”) core temperature, and the moment of inertia ratio Is /Ic we can confront observations. • Remark: Expect a glitch to occur after the object has “evolved” into the instability region (i.e. τgrow minimum): m & 1.3mc ⇒ ∆ & 1.7∆c Superfluid instabilities and neutron star dynamics 9/16 Confronting glitch observations 10 -4 2.5mc B2334+61 mc -5 ∆Ωc/Ωc 10 1.3 mc B1706-44 0.01 Vela 10 0.005 -6 Crab -7 10 0 10 5 2/3 8 (P/0.1s) (T/10 K) -4/3 • For systems without T data we use a standard URCA cooling law. • Curves for Is /Ic = 0.02, 0.01, 0.005 Superfluid instabilities and neutron star dynamics 10/16 The “Glaberson-Donnelly” instability • We carry out a local plane-wave analysis: i δvn,p = i iσt+ikj xj An,p e , Ain,p = constant • We find that inertial waves could go unstable when there is sufficient ~ n, Ω ~ p. misalignment between Ω • Instability similar to the Glaberson-Donnelly instability in superfluid Helium (where it leads to superfluid turbulence). • The instability naturally applies to precessing neutron stars where the precession modes are derived by assuming rigid body rotation and ~ n, Ω ~ p. misaligned Ω Superfluid instabilities and neutron star dynamics 11/16 Precession of superfluid neutron stars • Nature of precession modes decided by mutual friction R. (Sedrakian et al, ’99) Ppr ≈ 3 P 1s −8 10 yr ǫ fast mode 2 13 10 10 12 10 timescales / P • Weak drag precession: slow mode τd 10 10 1 Ppr 11 10 10 Ppr τd 0 10 9 10 -1 8 10 10 or 7 5 Ppr ≈ 10 In /Ip 10 P 1s −4 10 s R 10 weakly damped precession weakly damped precession -2 10 6 10 -4 10 -2 10 0 10 2 10 4 10 -4 10 -2 10 0 10 2 10 mutual friction drag R • Strong drag precession: Ppr ≈ 0.1 10 In /Ip P 1s s Superfluid instabilities and neutron star dynamics 12/16 4 10 Is slow precession unstable? • Slow precession is only possible for weak mutual friction drag R ≪ 1. • The instability is active above a critical wobble angle: θw > 1900 1◦ P 1s 1/2 10 ǫ −8 10 km R • Conclusion: – Long period precession is typically stable (e.g for a precessor like PSR B1828 − 11). – A notable exception: the instability could be relevant for millisecond-period neutron stars. Superfluid instabilities and neutron star dynamics 13/16 Is fast precession unstable? • Strong drag leads to fast “Shaham” precession. 3 P = 1 s, R = 10 10 2 10 1 5 • Wobble angle constraint: P 1s 1/2 R 103 τgrow (s) θw > 0.014 ◦ 1 10 10 10 Ppr 7 10 K I -3 9 -4 10 • Growth timescale: -1 -2 10 10 10 K 0 10 K -5 -6 10 0.1 τgrow ≈ 140 θw 1◦ −1 P 1s R 103 λ R 1 10 100 1000 10000 1e+05 wavelength λ (cm) s • Conclusion: Fast rigid-body precession is generically unstable. Superfluid instabilities and neutron star dynamics 14/16 Yet another twist: the magnetic field • Our model does not account for hydromagnetic stresses. – This simplification should be obviously relaxed, since the magnetic restoring force may dominate small-wavelength perturbations. • In general, the magnetic field will play a stabilising (?) role. • Local analysis: can be easily modified to include the magnetic field (van Hoven & Levin ’08). The constraint on fast precession becomes weaker, but the instability is still astrophysically relevant: θw >3 1◦ P 1s B 1012 G xp 0.1 • Global analysis: not so easy to modify ... but there is no reason why our proposed glitch mechanism should not work in magnetised neutron stars. Superfluid instabilities and neutron star dynamics 15/16 Outlook • The two fluid model for superfluid neutron stars predicts new dynamical instabilities provided there is relative motion in the background and mutual friction coupling. • These instabilities may well be relevant for “real life” neutron star dynamics (glitches, precession ...). • Obviously, the model is far from being “realistic”. In order to make more reliable predictions, one should worry about things like: – Additional restoring forces (magnetic and crustal stresses) – The phenomenology assumed in the mutual friction force For example, vortex pinning in the core and the crust is poorly understood → need for a consistent description of vortex dynamics. Superfluid instabilities and neutron star dynamics 16/16