Pulsar glitch dynamics in general relativity

Transcription

1 Pulsar glitch dynamics in general relativity Jérôme Novak Laboratoire Univers et Théories (LUTH) CNRS / Observatoire de Paris / Université Paris-Diderot Sourie, Novak, Oertel & Chamel Phys. Rev. D 93, (2016) & Month. Not. Roy. Astron. Soc. 464, 4641 (2017) CoCoNuT meeting 2017, Garching, October, 27 th

2 Pulsar glitches 1RXS J Angular momentum loss through emission of electromagnetic waves slowing down of the pulsar with P Kaspi & Gavriil, ApJ, 2003 tiny changes in this slowing down = glitches

3 Pulsar glitches 1RXS J Angular momentum loss through emission of electromagnetic waves slowing down of the pulsar with P Kaspi & Gavriil, ApJ, 2003 tiny changes in this slowing down = glitches

4 Pulsar glitch observations glitch amplitude are low: Ω/Ω rise time is quite short : τ r < 30 s Vela Wong, Backer & Lyne, ApJ, 2001 exponential relaxation during several days, up to months. glitches are driven by internal processes

5 Different glitch types Wang et al., Ap&SS, 2012 giant glitches quasi-periodic narrow amplitude distribution standard glitches randomly spaced in time various amplitudes Different glitch models moment of inertia reduction, with crustquakes transfer of angular momentum between two components, with superfluidity

6 Different glitch types Wang et al., Ap&SS, 2012 giant glitches quasi-periodic narrow amplitude distribution standard glitches randomly spaced in time various amplitudes Different glitch models moment of inertia reduction, with crustquakes transfer of angular momentum between two components, with superfluidity

7 Different glitch types Wang et al., Ap&SS, 2012 giant glitches quasi-periodic narrow amplitude distribution standard glitches randomly spaced in time various amplitudes Different glitch models moment of inertia reduction, with crustquakes transfer of angular momentum between two components, with superfluidity

8 Different glitch types Wang et al., Ap&SS, 2012 giant glitches quasi-periodic narrow amplitude distribution standard glitches randomly spaced in time various amplitudes Different glitch models moment of inertia reduction, with crustquakes transfer of angular momentum between two components, with superfluidity

9 Numerical models Rotating neutron stars in GR hypotheses General relativity to describe gravity Need to describe rotation axisymmetry Glitch time-scale hydro time-scale stationarity Contrary to spherical symmetry no matching to any known vacuum solution is possible (no Birkhoff theorem). Only boundary condition at r : flat metric. Numerical solution obtained using spectral methods (Grandclément & Novak 2009) and the lorene library (

10 Two-fluid approach Anderson & Itoh 1975 Two-fluid model charged particles : Ω p = Ω pulsar superfluid neutrons : Ω n Ω p angular velocity Ω p Superfluid vortices can pin into the crust nuclei When a critical threshold is reached in terms of δω = Ω n Ω p, some vortices unpin and can freely move in radial direction Transfer of angular momentum between both fluids and glitch time

11 Two-fluid approach Anderson & Itoh 1975 Two-fluid model charged particles : Ω p = Ω pulsar superfluid neutrons : Ω n Ω p angular velocity Ω n Ω p Superfluid vortices can pin into the crust nuclei When a critical threshold is reached in terms of δω = Ω n Ω p, some vortices unpin and can freely move in radial direction Transfer of angular momentum between both fluids and glitch time

12 Two-fluid approach Anderson & Itoh 1975 Two-fluid model charged particles : Ω p = Ω pulsar superfluid neutrons : Ω n Ω p angular velocity Ω n Ω p Superfluid vortices can pin into the crust nuclei When a critical threshold is reached in terms of δω = Ω n Ω p, some vortices unpin and can freely move in radial direction Transfer of angular momentum between both fluids and glitch time

13 Two-fluid approach Anderson & Itoh 1975 Two-fluid model charged particles : Ω p = Ω pulsar superfluid neutrons : Ω n Ω p angular velocity Ω n Ω p Superfluid vortices can pin into the crust nuclei When a critical threshold is reached in terms of δω = Ω n Ω p, some vortices unpin and can freely move in radial direction Transfer of angular momentum between both fluids and glitch time

14 Hypotheses Prix, Novak & Comer 2005 Equilibrium configurations: uniform composition : n, p, e crust is neglected rigid rotation : Ω n and Ω p = const. stationary and axisymmetric spacetime + isolated star. T T F, and no magnetic field. dissipation effects are neglected.

15 Hypotheses Prix, Novak & Comer 2005 Equilibrium configurations: uniform composition : n, p, e crust is neglected rigid rotation : Ω n and Ω p = const. stationary and axisymmetric spacetime + isolated star. T T F, and no magnetic field. dissipation effects are neglected.

16 two-fluid relativistic hydrodynamics Carter 1989; Carter & Langlois 1998; Langlois, Sedrakian & Carter 1998 System made of two perfect fluids : superfluid neutrons n α n = n n u α n, protons & electrons n α p = n p u α p. Energy-momentum tensor 1 fluid : T αβ = (E + P ) u α u β + P g αβ

17 two-fluid relativistic hydrodynamics Carter 1989; Carter & Langlois 1998; Langlois, Sedrakian & Carter 1998 System made of two perfect fluids : superfluid neutrons n α n = n n u α n, protons & electrons n α p = n p u α p. Energy-momentum tensor 1 fluid : T αβ = (E + P ) u α u β +P g αβ }{{} n αp β

18 two-fluid relativistic hydrodynamics Carter 1989; Carter & Langlois 1998; Langlois, Sedrakian & Carter 1998 System made of two perfect fluids : superfluid neutrons n α n = n n u α n, protons & electrons n α p = n p u α p. Energy-momentum tensor 2 fluids : T αβ = n nα p n β + n pαp p β + Ψg αβ

19 two-fluid relativistic hydrodynamics Carter 1989; Carter & Langlois 1998; Langlois, Sedrakian & Carter 1998 System made of two perfect fluids : superfluid neutrons n α n = n n u α n, protons & electrons n α p = n p u α p. Energy-momentum tensor 2 fluids : T αβ = n nα p n β + n pαp p β + Ψg αβ conjugate momenta { p n α u n α p p α u p α without entrainment

20 two-fluid relativistic hydrodynamics Carter 1989; Carter & Langlois 1998; Langlois, Sedrakian & Carter 1998 System made of two perfect fluids : superfluid neutrons n α n = n n u α n, protons & electrons n α p = n p u α p. Energy-momentum tensor 2 fluids : T αβ = n nα p n β + n pαp p β + Ψg αβ conjugate momenta { p n α u n α p p α u p α without entrainment { p n α = K nn n n α + K np n p α p p α = K pn n n α + K pp n p α entrainment effect Entrainment matrix:

21 New, realistic equations of state E ( n n, n p, 2) Ψ ( µ n, µ p, 2) Relativistic mean field model : nucleon - nucleon interactions effective meson exchange DDH DDHδ exp. constraints [ units ] Typel & Wolter (1999) Avancini et al. (2009) Oertel et al. (2017) n ± [ fm 3 ] B sat ± 0.3 [ MeV ] K ± 40 [ MeV ] J ± 3.2 [ MeV ] L ± 28.1 [ MeV ] M max,0 G [ M ]

22 Numerical results entrainment vs. Lense-Thirring The (Komar) angular momentum J X is such that dj X = I XX dω X + I XY dω Y Total coupling coefficient ˆε X = I XY / (I XX + I XY ) depends entrainment due to strong interaction between nucleons measured with the global entrainment coefficient ε (integration of ε over the star) Lense-Thirring effect due to GR dragging of inertial frames by each fluid measured with the metric term g tϕ DDH: εˆp ε~ p DDHδ: εˆp ε~ p M G (M )

23 Mutual friction No external torque exchange of angular momentum between neutrons and protons through mutual friction torque Γ mf Angular velocity Ω n Ω p δω 0 From Langlois et al. (1998), with straight vortices parallel to the rotation axis: interplay between Magnus force due to neutron fluid drag force caused by charged particles time Γ mf = R 1 + R 2 Γ nn n ϖ n χ 2 dσ (Ω n Ω p ) = B 2ÎnΩ n ζ δω

24 Evolution equations: { J n = + Γ mf, J p = Γ mf. Rise time Sidery et al δ Ω δω = ÎÎn I nn I pp Inp 2 2 BζΩ n Analytic approximation: ) δω(t) = δω 0 exp ( tτr τ r = Îp Î 1 ˆε p ˆε n 2ζ BΩ n Numerical modeling : Ω n (t) and Ω p (t) are determined by integration from Ω n,0 > Ω p,0 parameters M G, Ω, Ω/Ω, EoS, B

25 Time evolution Ω/Ω = 10 6, Ω f n = Ω f p = 2π Hz, M G = 1.4 M & B = 10 4 [ Ω X (t) - Ω 0 p ] / Ω0 p Ω 0 n Ω n Ω p Ω f n = Ωf p Ω n -Ω p (rad.s -1 ) DDH - numerique ajust. : τ r = 4.23 s DDHδ - numerique ajust. : τ r = 2.92 s DDHδ Ω 0 p DDH t (s) t (s) Rise times can be estimated with high accuracy without time integration, using only equilibrium models.

26 Ω/Ω = 10 6, Ω f n = Ω f p = 2π Hz Vela pulsar 10 2 DDH DDHδ τ r = 0.1 s B = τr B τ r = 1 s τ r = 10 s Constraints on B : τ r < 30 s B > τ r = 30 s B < 1/2 τr > 0.6 ms M G (M ) a glitch event is a quasi-stationary process, from the hydro viewpoint

27 Ω/Ω = 10 6, Ω f n = Ω f p = 2π Hz Vela pulsar 10 2 DDH DDHδ τ r = 0.1 s B = τr B τ r = 1 s τ r = 10 s Constraints on B : τ r < 30 s B > τ r = 30 s B < 1/2 τr > 0.6 ms M G (M ) a glitch event is a quasi-stationary process, from the hydro viewpoint

28 Influence of general relativity Comparison using two analytic EoSs from Prix et al. (2005) 0.5 [ τ GR τnewt ] / τgr r τnewt r ] / τgr r EoS I EoS II Ξ Compactness defined as Ξ = GM G R circ c 2 with Ξ NS depends on Ω, too. impact of general relativity on glitch dynamics can be quite strong!

29 Conclusions perspectives Precise models of rotating neutron stars in GR Realistic EoS for 2 fluids, including entrainment Quasi-stationary approach, with analytic formula for rise time Additional coupling between fluids due to Lense-Thirring effect Strong overall influence of GR on glitch rise time For the future: Looking for accurate data to constrain rise time Local modeling of glitch unpinning and movement Taking into account crust in global models?

30 References Anderson, P.W. and Itoh, N., Pulsar glitches and restlessness as a hard superfluidity phenomenon, Nature, 256, 25 (1975). Bonazzola, S., Gourgoulhon, E., Salgado, M., and Marck, J.-A., Axisymmetric rotating relativistic bodies: a new numerical approach for exact solutions, Astron. & Astrophys., 278, 421 (1993). Carter, B., Covariant theory of conductivity in ideal fluid or solid media, Lect. Notes Math., 1385 (1989). Relativistic models for superconducting-superfluid mixtures, Nucl. Phys. B, 531, 478 (1998). Grandclément, Ph. and Novak, J., Spectral Methods for numerical Relativity, Liv. Rev. Relat., 12, 1 (2009). Kaspi, V.M. and Gavriil, F.P., A second glitch from the anomalous X-ray pulsar 1RXS J , Astrophys. J., 576, L71 (2003). Langlois, D., Sedrakian, D. and Carter, B., Differential rotation of relativistic superfluid in neutron stars, Month. Not. Roy. Astron. Soc., 297, 1189 (1998). Relativistic numerical models for stationary superfluid neutron stars, Phys. Rev. D 71, (2005). Swesty, F.D., Thermodynamically Consistent Interpolation for Equation of State Tables, J. Comp. Phys., 127, 118 (1996). Wang, J., Wang, N., Tong, H. and Yuan, J., Recent glitches detected in the Crab pulsar, Astrophys. Space Sci., 340, 307 (2012). Wong, T., Backer, D.C. and Lyne, A.G., Observations of a series of six recent glitches in the Crab pulsar, Astrophys. J., 548, 447 (2001).

31 with entrainment p α X = K XX n X u α X + K XY n Y u α Y Numerical results entrainment Dynamic effective mass: p i X = m X u i X i {1, 2, 3} in the fluid-y rest-frame In the u i Y = 0 frame: relativity DDH DDHδ ε p 0 ) m X = µ X (1 ε X ε X ε n 0 entrainment n B (fm -3 )

32 with entrainment p α X = K XX n X u α X + K XY n Y u α Y Numerical results entrainment Dynamic effective mass: p i X = m X u i X i {1, 2, 3} in the fluid-y rest-frame In the u i Y = 0 frame: relativity DDH DDHδ ε p 0 ) m X = µ X (1 ε X ε X ε n 0 entrainment n B (fm -3 )

33 with entrainment p α X = K XX n X u α X + K XY n Y u α Y Numerical results entrainment Dynamic effective mass: p i X = m X u i X i {1, 2, 3} in the fluid-y rest-frame In the u i Y = 0 frame: relativity DDH DDHδ ε p 0 ) m X = µ X (1 ε X ε X ε n 0 entrainment n B (fm -3 )