1. Effects of r-mode induced differential rotation on the long-term evolution and gravitational wave radiation of neutron stars (preliminary results) Yun-Wei Yu （俞云伟） IOA, Huazhong Normal University

2. Outline Differential rotation induced by r-modes A phenomenological model for r-mode evolution Long-term evolution of isolated and accreting NSs Gravitational radiation from NSs Outlook

3. 1 、 Differential rotation induced by r-modes R-modes in a perfect fluid star with arbitrary rotation arise due to the action of the Coriolis force with positive feedback (Andersson 1998; Friedman & Morsink 1998), succumbing to gravitational radiation driven CFS instability (Chandrasekhar 1970; Friedman & Schutz 1978). R ossby waves on the earth

4. The r-modes of rotating barotropic Newtonian stars are solutions of the perturbed fluid equations having (Eulerian) velocity perturbations (Lindblom et al. 1998, PRD, 80, 4843) where is the unperturbed angular velocity and is the dimensionless amplitude of the perturbation. In spherical coordinates, the three components are Which is obtained by solving the linear fluid equations at the first order of the r-mode amplitude

5. Second-order r-modes At the second order, the perturbed Euler, continuity, and Poisson equations in an inertial frame are given, respectively, by (Sa 2004)

6. A and N are two constants determined by the initial condition. Sa & Tome (2005) suggested N = 2l -1 and redefined A by introducing a new free parameter K as This second-order solution gives a differential rotation, producing large scale drifts of fluid elements along stellar latitudes. Rezzolla et al. (2000, 2001); Sa (2004); Sa & Tome (2005)

7. 2 、 A phenomenological model for r-mode evolution The physical angular momentum and energy of the l=2 r- mode can be calculated up to the second order in alpha as (Sa & Tome 2005) For K = -2, J (2) vanishes and the expressions of J r and E r return to their canonical forms (Owen et al. 1998). Namely, the case of K = -2 corresponds to the non-differential (uniform) rotation case

8. Both the physical angular momentum and energy of r- modes are increased by gravitational radiation back reaction and decreased by viscous damping, which yields

9. The total angular momentum of the star contains two components where I is the moment of inertial of the star. For isolated magnetized (10 10 -10 12 G) NSs For accreting NSs in LMXBs with a weak magnetic field

10. For isolated magnetized NSs For accreting NSs in LMXBs with a weak magnetic field The key problem is to fix the five timescales

11. As a preliminary work, we adopt the simplest EOS of, Owen et al. (1998) strongly dependent on the EOS and the stellar temperature physicists

12. Thermal evolution also strongly dependent on the composition of the stellar matter

13. 3 、 Long-term evolution of isolated NSs 1 prolongs duration 2 stops spin-down 3 enhances the heating effect

15. 3 、 Long-term evolution of accreting NSs

17. 4 、 Gravitational radiation from NSs Using the obtained r-mode amplitude and angular velocity, we can estimate the amplitude of the emitted gravitational waves as follows The frequency-domain gravitational wave amplitude

19. Outlook It is necessary to adopt some more realistic EOSs (Hyperon? Pion? Quarks? Superfluid? CFL?...) of the stellar materials to give the composition and structure of NSs. The interactions between the r-modes and the magnetic field in the presence of the differential rotation. On one hand, the differential rotation can distort the magnetic field and increases its energy. On the other hand, the distortion of the magnetic field would prevent the r-mode oscillation.

20. Thank you for your attention !