1. Non-axisymmetric modes of differentially rotating neutron stars Andrea Passamonti Southampton, 13 December 2007 University of Southampton In collaboration with: Kostas Kokkotas and Adam Stavridis

2. Outline Motivation. Description of the rotating background and fluid perturbations. Dependence of the non-axisymmetric spectrum on the rotation, compactness and degree of differential rotation. Conclusions and future works.

3. Motivation Differetially rotating configurations: a) proto-neutron stars, b) in a component of a binary system due to accretion, c) Remants of Hypermassive star merger. Non-axisymmetric instabilities:  =T / |W| - secular instabilities: driven by gravitational radiation (CFS) or viscosity  s  = 0.14 (polar modes). - dynamical instabilities: i) classical bar mode instabilities  d = 0.25-0.27 ii) low T / W instabilities (m=1,2) for  ~ 0.01-0.09. Possible interpretation: shear instabilities of inviscid fluid due to the presence of corotation modes (Watts et al. 2005 ).  Asteroseismology: to infer the rotational rates and the differential law from the splitting of non-axisymmetric modes.

4. Motivation QNMs in GR for rotating stars: In perturbation theory: problems for the BC at infinitity. In Numerical Relativity, is a 3D problem, then high performance computers and large simulation time are required. Aims of our project: 1) Determine the spectral properties of non-axisymmetric adiabatic oscillations both for barotropic and non-barotropic relativistic stars. 2) Study the low T / W instability and its connection with corotation modes. Assumptions: 1) Slow rotation approximation. It can describe quite accurately also fast uniformly rotating NS (Hartle 1967) 2) Cowling approximation: we focus on the fluid modes and neglect all the metric perturbations.

5. Diff. Rot. Background model   Slowly rotating approximation at first order in   r   TOV Equations + ODE for frame dragging function  r   J-constant rotation law: satisfies the Rayleigh’s stability A : Differential rotation parameter. Uniform rotation for A >> 1 Ω c : Angular velocity at rotation axis.

6. Diff. Rot. Background models  Spherical model: TOV Equations +barotropic relativistic EoS:  B sequence: the norotating member B0 has  Differentially rotating model: we vary the diff. parameter A. B-model : each member of the sequence has Ωc=const BJ-model : each member of the sequence has J=const.  Models with different compactness with the same EoS:

7. DR Background model - Solve up to l=3 Background corrections : Ω1, Ω3, ω1, ω3 - Uniform rotation: Ω3 = ω3 = 0. A. Stavridis, AP, K. Kokkotas Phys.Rev. D75 064019

8. Comparison with RNS code B1 model: Polytropic model M / R = 0.146 T = 1.719 ms A = 12 km  e  =   e  k  = 0.161

9. Fluid Perturbation Equations   fluid perturbations: Polar and Axial perturbations  Perturbed Conservation Equations: (+ 1 Eq. for  in case of non-barotropic stars) Integration over the angles leads to a system of 4 coupled PDEs. where

10. Fluid Perturbation Equations

11. Frequency Domain Analysis  Non-axisymmetric oscillations  Splitting of the non-axisymmetric modes due to rotation  We study the dependence of this rotational splitting on the M / R,  e, A. AP, A. Stavridis, K. Kokkotas [gr-qc] arXiv:0706.0991  We set up a Boundary-Value Problem with regularity conditions at the origin of the star and  p=0 at the surface.  For uniformly rotating stars:

12. Dependence on the number of couplings - Perturbative Eqs. contain some intrinsecally 2nd order corrections in , due to the axial variable u 3 lm - We neglect these 2nd order corrections in order to be consistent with a first order slowly rotation approximation.

13. Mode splitting B - models: Polytropic model M / R = 0.146 R = 14.15 km A = 12 km

14. Mode splitting / compactness Stellar models: - A sequence of Polytropic models with different M / R - A is such that  e  c = 0.357

15. Mode splitting / A (Diff. Rot.) B and BJ stellar models. - Dependence of non- axisymmetric modes on the parameter A.

16. Corotation band  Low T / W instability T / W ~ 0.01 (Shibata, Karino, Eriguchi 2002, 2003) appears in rotating stars with a high degree of differential rotation.  Association with corotation modes.  Pattern mode speed:

17. Corotation band  Rotational parameters in order to have corotation modes.

18. Time Domain Analysis  The numerical code has been recently stabilized for A > 10 km, and tested with the frequency of the eigenvalue problem.  Numerical method: Method of Line with a RK3 and second order finite difference scheme in the spatial coordinate. Implementation of a fourth order Kreiss-Oliger numerical dissipation.  Inertial and g-modes by using both the time and frequency domain codes.  It seems that some inertial modes can be identified even in the continuous spectrum band.  Low T/W instability: further improvement of the code are needed for star rotating with a high degree of differential rotation, i.e. A < 10 km.

19. Summary We have studied the spectral properties of differentially and slowly rotating stars. In the Cowling approximation, the rotational splitting of the non- axisymmetric modes has been studied with respect to the compactness and the degree of differential rotation. For a sequence of relativistic polytropic models, we have determined the necessary rotational configurations in order to have corotation modes. We are currently studying the inertial modes and g-modes of barotropic and non-barotropic stars. We are currently working on the full 1-order slow-rotation approximation problem, i.e. without Cowling approximation.