1. 1 Shu-Hua Yang ( 杨书华 ) Hua Zhong Normal University The role of r-mode damping in the thermal evoltion of neutron stars

2. 2 Outline Introduction R-mode heating of neutron stars Our results

3. 3 1 、 Introduction Yakovlev et al. (2008), AIPC,983,379

4. 4 The thermal evolution equation with the approximation of isothermal interior

5. 5 Slow and fast neutrino emission processes

6. 6 focus on the r-mode instability heating

7. 7 2 、 R-mode heating of neutron stars  In contrast, the growth of the modes can be suppressed by the viscosity of the stellar matter. Andersson et al (2001)  R-mode in a perfect fluid star with arbitrary rotation due to the action of the Coriolis force with positive feed back, succumbing to CFS instability.

8. 8 The r-modes of rotating barotropic Newtonian stars are solutions of the perturbed fluid equations having velocity perturbations where is angular velocity of the unperturbed the star, is the dimensionless amplitude of the perturbation, is the magnetic-type vector spherical harmonic: R-mode evolution in neutron stars

9. 9 (Owen et al. 1998)  First-order r-modes In spherical coordinates, solving the linear fluid equations at the first order of the r-mode amplitude( ), we can get the r-mode solutions:

10. 10  Second-order r-modes At the second order of the r-mode amplitude (Sa 2004) 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.

11. 11 The physical angular momentum of the l=2 r-mode calculated up to the second order is (Sa & Tome 2005) For K = -2 ， J (2) =0 ， J r return to the first order case. is the magnetic braking timescale. is the total angular momentum of the star,

12. 12 R-mode evolution Spin evolution of NSs Thermal evolution of NSs

13. 13 Long-term evolution of isolated NSs prolongs duration enhances the heating effect

14. 14

15. 15 No superfluidity Proton superfluidity (model 1p ) Yakovlev et al. (2008), AIPC,983,379

16. 16

17. 17 Problem ： Tsuruta et al. (2002) ApJL

18. 18

19. 19 3 、 Our results:

20. 20 Thank you!