Magnetic field evolution in superconducting neutron stars arXiv:1505.00124 Vanessa Graber 1 In collaboration with N. Andersson 1 , K. Glampedakis 2 and S. Lander 1 1 Mathematical Sciences and STAG Research Centre, University of Southampton, UK 2 Departamento de Fisica, Universidad de Murcia, Spain Annual NewCompStar Conference 18th June 2015 – Budapest
Magnetic Fields in Neutron Stars Inferred magnetic dipole field strengths reach up to 10 15 G for magnetars . Such fields strongly influence the dynamics. Long-term field evolution could explain observed field changes in pulsars (Narayan & Ostriker, 1990) high activity of magnetars (Thompson & Duncan, 1995) neutron star ‘metamorphosis’ Figure 1: Artistic impression of a neutron star and its magnetic dipole field. (Vigan` o et al., 2013) Mechanisms causing magnetic field evolution are poorly understood. (Goldreich & Reisenegger, 1992; Glampedakis, Jones & Samuelsson, 2011, e.g.) What happens if we take core superconductivity into account? Budapest 2015 Magnetic field evolution in superconducting neutron stars 1 / 9
Standard Resistive MHD In a proton-electron plasma, the frictional coupling force is given by e = n e m e ≡ − m e � � F i v i e − v i J i , (1) p τ e e τ e with the coupling timescale τ e and macroscopic current J i . The electron Euler equation provides a generalised Ohm’s law for the macroscopic electric field. Together with Faraday’s and Amp` ere’s laws this leads to the resistive MHD induction equation , c 2 � m p c � ∂ t B i = ǫ ijk ∇ j ǫ klm p B m − ∇ l B m − v l ǫ lst ( ∇ s B t ) B m , (2) 4 πσ e 4 π e ρ p ⇒ Flux freezing, Ohmic decay and conservative Hall evolution with τ Ohm = 4 πσ e L 2 τ Hall = 4 π e ρ p L 2 ≈ 2 . 4 × 10 13 yr , ≈ 1 . 9 × 10 10 yr . (3) c 2 m p cB Budapest 2015 Magnetic field evolution in superconducting neutron stars 2 / 9
Quantum Condensates – Type-II Superconductivity Equilibrium stars with 10 6 − 10 8 K are cold enough to contain superfluid neutrons and superconducting protons. The macroscopic quantum states influence the stars’ dynamics. Type of superconductivity depends on the characteristic lengthscales. Estimates predict a type-II state (Baym, Pethick & Pines, 1969; Mendell, 1991, e.g.) � − 5 / 6 � T cp � x p κ = λ � 1 ξ ≈ 2 ρ − 5 / 6 > √ . (4) 14 10 9 K 0 . 05 2 Flux is allowed to enter fluid in form of quantised fluxtubes . They are arranged in a hexagonal array. Each fluxline carries a unit of flux, φ 0 = hc 2 e ≈ 2 × 10 − 7 G cm 2 . (5) Figure 2: Vortex array in a rotating, dilute BEC of Rubidium atoms (Engels et al., 2002). Budapest 2015 Magnetic field evolution in superconducting neutron stars 3 / 9
Superconducting MHD The macroscopic Euler equations for the superfluid neutrons and the combined proton-electron fluid are (Glampedakis, Andersson & Samuelsson, 2011) � � � + ∇ i ˜ ∂ t + v j Φ n + ε n w j j = f i mf + f i v i n + ε n w i � pn ∇ i v n n ∇ j mag , n , (6) np j = − n n � � � � + ∇ i ˜ np ∇ i v p ∂ t + v j v i p + ε p w i Φ p + ε p w j f i mf + f i p ∇ j mag , p , (7) pn n p with w i xy ≡ v i x − v i y . The equations are modified by additional force terms , f i mf and f i mag , x , due to vortices/fluxtubes and entrainment , ε x . They are supplemented by continuity equations and the Poisson equation � � n x v i ∇ 2 Φ = 4 π G ρ. ∂ t n x + ∇ i = 0 , (8) x Evolution equation for the magnetic field of a type-II superconductor? Budapest 2015 Magnetic field evolution in superconducting neutron stars 4 / 9
Conventional Mutual Friction Standard ‘resistivity’ due to electrons scattering off magnetic fields of individual fluxtubes (Alpar, Langer & Sauls, 1984) results in macroscopic force � � F i e = N p f i v i e − u i d = N p ρ p κ R , (9) p with fluxtube density N p , electron drag f i d and fluxtube velocity u i p . The dimensionless drag coefficient is (Mendell, 1991) � x p � 1 / 6 R ∼ 2 . 3 × 10 − 4 ρ 1 / 6 ≪ 1 . (10) 14 0 . 05 Rewrite u i p in terms of fluid variables to obtain a macroscopic equation. We use a mesoscopic force balance for an individual fluxtube Figure 3: Fluxtubes can be envisaged as tiny, (Hall & Vinen, 1956) . rotating tornadoes. Different forces determine their motion (NOAA Photo Library). Budapest 2015 Magnetic field evolution in superconducting neutron stars 5 / 9
Results I - Superconducting Induction Equation Eliminating u i p leads to the force R e ≈ − H c1 B � B i + ǫ ijk ˆ � F i R ˆ B j ∇ j ˆ B j ˆ B l ∇ l ˆ B k , (11) 1 + R 2 4 π with the lower critical field of superconductivity H c1 and B i = B ˆ B i . As before, combine (11) with Euler equation and Faraday’s law to obtain a superconducting induction equation for standard mutual friction, R � − κ B m p B m � � ∂ t B i ≈ ǫ ijk ∇ j � p B m � � B l ˆ R ˆ B l ∇ l ˆ B k + ǫ klm ˆ B s ∇ s ˆ v l ǫ klm . m ∗ 1 + R 2 2 π p For R ≪ 1 , the inertial term dominates and the field is frozen to the protons; on large scales electrons, protons and fluxtubes are comoving. Budapest 2015 Magnetic field evolution in superconducting neutron stars 6 / 9
Results II - Conservative/Dissipative Contributions Nature of terms in both induction equations is determined by looking at the evolution of the magnetic energy . For standard MHD, we obtain ∂ B i p B k − J 2 ∂ E mag = B i = 1 c J i ǫ ijk v j − ∇ i Σ i . (12) ∂ t 4 π ∂ t σ e The Hall term vanishes, while Ohmic diffusion causes energy loss ∝ J 2 . In the superconducting case , we have E mag , sc = H c1 B / 2 π and � � ∂ E mag , sc = B ∂ H c1 + H c1 p B k − κ B m p R J i ⊥ ǫ ijk v j 1 + R 2 J 2 ⊥ − ∇ i Σ i , (13) m ∗ ∂ t 2 π ∂ t 2 π 2 π p with J i ≡ ǫ ijk ∇ j ˆ B k decomposed into J i ≡ J � ˆ B i + J i ⊥ . The first term shows that changing the superconducting properties alters the magnetic energy. Budapest 2015 Magnetic field evolution in superconducting neutron stars 7 / 9
Results III - Timescales � − κ B m p R B m � � ∂ t B i ≈ ǫ ijk ∇ j � p B m � � B l ˆ v l R ˆ B l ∇ l ˆ B k + ǫ klm ˆ B s ∇ s ˆ ǫ klm . m ∗ 1 + R 2 2 π p Similar to the Hall evolution of resistive MHD, the second term is conservative . The last term is dissipative (like Ohmic decay) and decreases the magnetic energy of the superconducting mixture ∝ J 2 ⊥ . Extract the dominant timescales from the induction equation m ∗ τ diss = 2 π L 2 1 + R 2 τ cons = τ 1 ≈ 3 . 1 × 10 11 yr , R ≈ 1 . 3 × 10 15 yr . p (14) κ R m p τ diss τ cons Comparison gives: ≈ 1 . 3 × 10 − 2 , ≈ 6 . 8 × 10 4 . (15) τ Ohm τ Hall Budapest 2015 Magnetic field evolution in superconducting neutron stars 8 / 9
Conclusions and Open Questions Analogous to standard MHD, we chose one mesoscopic effect to derive a macroscopic induction equation for the superconducting mixture ⇒ flux freezing, dissipative/conservative contributions are present. τ diss and τ cons are notably longer than the typical spin-down ages ⇒ conventional mutual friction cannot explain observed field changes due to minimum dissipation timescale τ min ≈ 1 . 4 × 10 8 yr for R = 1 . For shorter timescales, different dissipative mechanisms are necessary ⇒ typical candidate for strong coupling is vortex-fluxtube ‘pinning’ Key issue : We discuss bulk fluid evolution but neglect surface terms ⇒ effects due to the crust-core interface are not included. Physics are poorly understood but could be very important for neutron stars. Budapest 2015 Magnetic field evolution in superconducting neutron stars 9 / 9
Magnetic Energy In standard MHD, the Lorentz force contains tension and pressure term L = 1 � B j ∇ j B i − 1 B k B k �� 2 ∇ i � F i . (16) 4 π The work is then given by � B 2 � � r i F i W L = L d V = 8 π d V ≡ E mag d V . (17) For a superconductor , the total magnetic force has to be changed to (Easson & Pethick, 1977; Glampedakis, Andersson & Samuelsson, 2011) mag = 1 � � ρ p B ∂ H c1 �� F i B j ∇ j H i c1 − ∇ i , (18) 4 π ∂ρ p where H i c1 = H c1 ˆ B i . Integration gives for the energy of the bulk fluid � H c1 B � � r i F i W mag = mag d V = d V ≡ E mag , sc d V . (19) 2 π Budapest 2015 Magnetic field evolution in superconducting neutron stars 1 / 2
References Alpar M. A., Langer S. A., Sauls J. A., 1984, ApJ, 282, 533 Baym G., Pethick C. J., Pines D., 1969, Nature, 224, 673 Easson I., Pethick C. J., 1977, Phys. Rev. D, 16, 275 Engels P., Coddington I., Haljan P. C., Cornell E. A., 2002, Phys. Rev. Lett., 89, 100403 Glampedakis K., Andersson N., Samuelsson L., 2011, MNRAS, 410, 805 Glampedakis K., Jones D. I., Samuelsson L., 2011, MNRAS, 413, 2021 Goldreich P., Reisenegger A., 1992, ApJ, 395, 250 Hall H. E., Vinen W. F., 1956, Proc. of the Royal Soc. A, 238, 215 Mendell G., 1991, ApJ, 380, 515 Narayan R., Ostriker J. P., 1990, ApJ, 352, 222 Thompson C., Duncan R. C., 1995, MNRAS, 275, 255 Vigan` o D., Rea N., Pons J. A., Perna R., Aguilera D. N., Miralles J. A., 2013, MNRAS, 434, 123 Budapest 2015 Magnetic field evolution in superconducting neutron stars 2 / 2
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