Effects of the superfluid neutrons on the dynamics of the crust

1. Effects of the superfluid neutrons on the dynamics of the crust Lars Samuelsson, Nordita (Stockholm) Nils Andersson Kostas Glampedakis [Karlovini & LS, CQG 20 3613 (2003), Carter & LS CQG 23 5367 (2006) LS & Andersson, MNRAS 374 256(2007)] Umberto Boccioni: Elasticity, 1912 The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

2. Punchlines • We may potentially constrain the high density Eos if the properties of the crust are accurately known. • We need properties beyond the Eos in order to describe neutron star dynamics (shear moduli, entrainment parameters, transport properties,...). The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

3. Outline • Motivation • Equations of motion for continuous matter in GR • Example: axial modes in non-magnetic stars • Application: QPOs in the tails of giant flares and seismology • Conclusions The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

4. Neutron stars Not perfect fluid The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

5. A minimal model • Solid outer crust • Solid inner crust with superfluid neutrons • Superfluids and superconductors coexisting in the core • Huge magnetic fields – possibly bunched (Type I) or in flux tubes (Type II) • Rotation – hence vortices Here I will only consider the crust without magnetic fields The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

6. Continuous matter in GR • Variational approach [Brandon Carter et al.] • Amounts to specifying a Lagrangian masterfunction. • The ... represent “structural” fields describing eg. the relaxed geometry of the solid or the frozen in magnetic field. • nxa is the four current. The conjugate variables are the four-momenta. The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

7. Entrainment For multi-fluids it is convenient to consider the Lagrangian to be a function of the scalars that can be formed from the currents: as well as (x≠y) This leads a momentum given by This illustrates the key fact that the current and the momentum for a given fluid need not be parallel. It is known as the entrainment effect, and is important for superfluid neutron stars. The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

8. The currents and momenta The quantities mxa are both the canonically conjugate and the physical (four) momenta. Note that The four-currents describe the flow of particles and are related to the physical velocity. Due to entrainment the momenta are not parallel to the velocity. Warning: Landau’s superfluid velocities are vs = p/m and are not the physical velocities of the average motion of the particles. The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

9. Equations of motion for Multifluids Assuming that each particle species is conserved, we get (no summationover x) Note: Tab is not the whole story The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

10. Equations of motion with an elastic component • Define hab given by the energy minimum under volume preserving deformations • Define the strain tensor as: The strain tensor measures volume preserving deformations Simplest case: isotropic solid The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

11. Total Stress-energy tensor The magnetic contribution is just: The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

12. The Lagrangian density The EOS contribution is the contribution from the rest mass density and the part of the internal energy that does not depend on relative motion or the state of strain in the solid. : Assuming small relative velocities the entrainment can be represented by The solid contribution can similarly be expanded assuming small strain The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

13. Example: axial modes in the Cowling approximation • Due to the static spherical background the neutron equation of motion become very simple. For non-static perturbations it amounts to • The remaining equation is nearly identical to the purely elastic case. The only difference is that the frequency is multiplied by a factor The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

14. Dynamical equation The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

15. Application: Flares in Soft Gamma-Ray repeaters • SGRs: persistent X-ray sources envisaged as magnetars • B ~ 1015 G • P ~ 1-10 s • Key property: Emag >> Ekin • Three giant flares to date • March 5, 1979: SGR 0526-66 • August 27, 1998: SGR 1900+14 • December 27 2004: SGR 1806-20 • Flares are associated with large scale magnetic activity and crust fracturing • Quasi-periodic oscillations discovered in the data T. Strohmayer & A. Watts, ApJ. 653 (2006) p.593 The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

16. Observations Newtonian limit, homogeneous stars, no dripped neutrons: Fundamental mode (n = 0): Overtones (n > 0): = crust thickness ~ 0.1 R The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

17. Magnetic crust-core coupling • The strong magnetic field threads both the crust and the fluid core (assuming non-type-I superconductor...) • The coupling timescale is the Alfvén crossing timescale Where is the Alfvén velocity and G • Generic conclusion: • If the crust is set to oscillate the magnetar’s core gets involved in less than one oscillation period • Pure crustal modes replaced by global MHD modes • Puzzle: Why do we observe the seismic frequencies? The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

18. Mode excitation • Modes in the vicinity of a crustal mode frequency are preferable for excitation by a “crustquake” as they communicate minimum energy to the core: Consistent with QPO data • Our model naturally predicts the presence of excitable modes below the fundamental crustal frequency • Low frequency QPOs: Example: SGR 1806-20. Identify Hz Then: Hz The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

19. Modelling the QPOs: Input data Eos by Haensel & Pichon, Douchin & Haensel Shear modulus (bcc) by Ogata & Ichimaru: The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

20. 6 0 10 0 M=0.96Mo R=11.4 km Seismology – exemplified by SGR 1806-20 2 0 ? 1 T. Strohmayer & A. Watts, astro-ph/0608463 The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

21. 6 0 10 0 M=1.05Mo R=12.5 km Seismology – exemplified by SGR 1806-20 2 0 ? 1 T. Strohmayer & A. Watts, astro-ph/0608463 The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

22. Conclusions • From a theoretical point of view we have come a long way towards a description of neutron star dynamics • Need better understanding of • Dissipation in GR • Superconductor fluid dynamics • Magnetic field dynamics • We need microscopic calculations providing better understanding on matter properties beyond the equation of state: eg Superfluid parameters, shear modulus, pinning, vortex/fluxtube interactions, dissipation, ... • The potential return is a “point” in the mass radius diagram implying constraints for the high density equation of state but... The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

23. Conclusions continued • We need to understand the dynamics and structure of the magnetic field. • We need accurate Eos of the crust including shear moduli/us and effective neutron mass • In particular the seismology is sensitive to The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

24. Commercial NORDITA (recently moved to Stockholm) provide the opportunity to organizing programmes of 1-2 month duration. Applications for funding are open to the whole theoretical physics community. See http://www.nordita.org/ for details. There will be a 2 week mini-programme next year on the physics of the crust and beyond, tentatively in the spring. The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

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