1. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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

2. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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,...).

3. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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

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

5. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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

6. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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. —n x a is the four current. The conjugate variables are the four-momenta.

7. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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.

8. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 The currents and momenta The quantities  x a 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 v s = p/m and are not the physical velocities of the average motion of the particles.

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

10. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Equations of motion with an elastic component —Define  ab 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

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

12. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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

13. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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

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

15. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Approximate frequencies Fundamental: Overtones: Crust thickness: Leads to:

16. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Application: Flares in Soft Gamma-Ray repeaters —SGRs: persistent X-ray sources envisaged as magnetars –B ~ 10 15 G –P ~ 1-10 s —Key property: E mag >> E kin —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

17. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Observations SGRf QPO (Hz)f mode ln 1806-2018?-- 26?-- 29 20 939260 150151100 626308?1 ? 1840??? 1900+1428 20 545940 848860 155159110 Newtonian limit, homogeneous stars, no dripped neutrons: Fundamental mode (n = 0): Overtones (n > 0): = crust thickness ~ 0.1 R

18. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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 —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? Where is the Alfvén velocity and G

19. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Toy model I [Glampedakis, LS, Andersson, MNRAS 371, L74 (2006)] Assume: uniform density, shear modulus and magnetic field, ideal MHD Correct MHD conditions at interface R c couple crust and core provided Key effect: crust-core resonance at the crustal frequencies:

20. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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: —Our model naturally predicts the presence of excitable modes below the fundamental crustal frequency —Low frequency QPOs: Example: SGR 1806-20. Identify Hz Then: Consistent with QPO data Hz

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

22. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 60 100 ?1 20 Seismology – exemplified by SGR 1806-20 frequencyln 18 26 29 93 150 626 (720) 1837 (2387) T. Strohmayer & A. Watts, astro-ph/0608463 M=0.96M o R=11.4 km

23. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 60 100 ?1 20 Seismology – exemplified by SGR 1806-20 frequencyln 18 26 29 93 150 626 (720) 1837 (2387) T. Strohmayer & A. Watts, astro-ph/0608463 M=1.05M o R=12.5 km

24. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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...

25. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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

26. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 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.

27. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008

28. NORDITA The Complex Physics of Compact Stars Ladek Zdrój 28 February 2008 Corrections —Magnetic field: Sotani et al [astro-ph/0608626, 0611666] –Main effect: —  : effect of EOS in inner core - Douchin & Haensel [A&A 380, 151 (2001)], Baym, Bethe & Pethick, [Nucl Phys A 175, 225 (1971)], Negele & Vautherin, [Nucl Phys A 207, 298 (1973)] –Main dependence: p/  at interface –limits: —Anisotropy and reduction of the shear moduli in the “pasta” phases. Pethick & Potekhin [Phys Lett B 427 (1998)]. (Eagerly avaiting results of N. Chamel & W. G. Newton & J. R. Stone)