Impact of the symmetry energy on the neutron star oscillation Department of Physics, South China Univ. of Tech. （文德华 华南理工大学物理系） collaborators Bao-An Li, William Newton, Wei-kang Lin De-Hua Wen Department of Physics and astronomy, Texas A&M University-Commerce The third International Workshop on Nuclear Dynamics in heavy-ion reactions 2012 － 12 － 16 ～ 19 中国深圳

Outline I.Background and Motivation II. Basic equations for r-mode instability window of neutron star with rigid crust III.Results and discussion IV.The f-mode oscillation

In Newtonian theory, the fundamental dynamical equation (Euler equations) that governs the fluid motion in the co-rotating frame is Acceleration Coriolis force centrifugal force external force where is the fluid velocity and represents the gravitational potential. Euler equations in the rotating frame I. Background and Motivation Int.J.Mod.Phys. D10 (2001) 381

For the rotating stars, the Coriolis force provides a restoring force for the oscillation modes, which leads to the so-called r- modes. Its eigen-frequency is It is shown that the structure parameters (M and R) make sense for the through the second order of. Definition of r-mode Class. Quantum Grav. 20 (2003) R105P111/p113 or

CFS instability and canonical energy The function E c govern the stability to nonaxisymmetric perturbations as: (1) if, stable; (2) if, unstable. For the r-mode, The condition E c < 0 is equivalent to a change of sign in the pattern speed as viewed in the inertial frame, which is always satisfied for r-mode. gr-qc/ v1 canonical energy (conserved in absence of radiation and viscosity): APJ,222(1978)281

Seen by a non-rotating observer (star is rotating faster than the r-mode pattern speed) seen by a co-rotating observer. Looks like it's moving backwards The fluid motion has no radial component, and is the same inside the star although smaller by a factor of the square of the distance from the center. Fluid elements (red buoys) move in ellipses around their unperturbed locations. Note: The CFS instability is not only existed in GR, but also existed in Newtonian theory. Images of the motion of r-modes

Viscous damping instability It is the viscosity stabilizes the amplitude growth of the r-mode driven by the gravitational radiation. Two kinds of viscosity, bulk and shear viscosity, are normally considered. At low temperatures (below a few times 10 9 K) the main viscous dissipation mechanism is the shear viscosity arises from momentum transport due to particle scattering. At high temperature (above a few times 10 9 K) bulk viscosity is the dominant dissipation mechanism. The r-modes ought to grow fast enough that they are not completely damped out by viscosity.

The r-mode instability window Condition : To have an instability we need t gw to be smaller than both t sv and t bv. For l = m = 2 r-mode of a canonical neutron star (R = 10 km and M = 1.4M ⊙ and Kepler period P K ≈ 0.8 ms (n=1 polytrope)). Int.J.Mod.Phys. D10 (2001) 381

Motivations (a) Old neutron stars (having crust) in LMXBs with rapid rotating frequency (such as EXO ) may have high core temperature (arXiv: v1.) ; which hints that there may exist r-mode instability in the core. (b) The discovery of massive neutron star ( PRS J , Nature 467, 1081(2010) and EXO , Nature 441, 1115(2006) ) reminds us restudy the r-mode instability of massive NS (most of the previous work focused on the 1.4M sun neutron star). (c) The constraint on the symmetric energy at sub-saturation density range and the core-crust transition density by the terrestrial nuclear laboratory data could provide constraints on the r-mode instability.

PhysRevD Here only considers l=2, I 2 = And the viscosity  c is density and temperature dependent: The subscript c denotes the quantities at the outer edge of the core. T<10 9 K: T>10 9 K: The viscous timescale for dissipation in the boundary layer: II. Basic equations for r-mode instability window of neutron star with rigid crust

The gravitational radiation timescale: According to, the critical rotation frequency is obtained: Based on the Kepler frequency, the critical temperature defined as: PhysRevD

III. Results and discussion

D.H. Wen, W. G. Newton, and B.A. Li ， Phys. Rev. C 85, (2012) The EOSs are calculated using a model for the energy density of nuclear matter and probe the dependence on the symmetry energy by varying the slope of the symmetry energy at saturation density L from 25 MeV (soft) to 105 MeV (stiff). The crust-core transition density, and thus crustal thickness, is calculated consistently with the core EOS. Equation of states W. G. Newton, M. Gearheart, and B.-A. Li, arXiv: v1.

The mass-radius relation and the core radius D.H. Wen, W. G. Newton, and B.A. Li ， Phys. Rev. C 85, (2012)

The viscous timescale Comparing the time scale The gravitational radiation timescale D.H. Wen, W. G. Newton, and B.A. Li ， Phys. Rev. C 85, (2012)

The r-mode instability window for a 1.4M sun (a) and a 2.0M sun (b) neutron star over the range of the slope of the symmetry energy L. D.H. Wen, W. G. Newton, and B.A. Li ， Phys. Rev. C 85, (2012)

The location of the observed short-recurrence-time LMXBs in frequency-temperature space, for a 1.4M sun (a) and a 2.0M sun (b) neutron star. The temperatures are derived from their observed accretion luminosity and assuming the cooling is dominant by the modified Urca neutrino emission process for normal nucleons or by the modified Urca neutrino emission process for neutrons being super-fluid and protons being super-conduction. Phys. Rev. Lett. 107, (2011)

The critical temperature Tc for the onset of the CFS instability vs the crust-core transition densities over the range of the slope of the symmetry energy L consistent with experiment for 1.4M sun and 2.0M sun stars. D.H. Wen, W. G. Newton, and B.A. Li ， Phys. Rev. C 85, (2012)

Conclusion (1)Smaller values of L help stabilize neutron stars against runaway r-mode oscillations; (2) A massive neutron star has a wider instability window; (3)Treating consistently the crust thickness and core EOS, and concluding that a thicker crust corresponds to a lower critical temperature.

The perturbed metric tensor for a non-rotating neutron star is written as The perturbation of the fluid elements is described by the polar components of Lagrangian displacement, VI. The f-mode oscillation

According to the Einstein’s equation and continuity equation the following 4th order coupling linear differential equations is obtained: The function X is used instead of V, and X is defined as

The Non-Newtonian gravitational is described by In the boson exchange picture,

F-mode frequencies. It is shown that the Yukawa potential significantly bring down the mode frequencies Xu, J., Chen, L. W., Ko, C. M., & Li, B. A. 2010, Phys. Rev. C, 81,

The spectrum of the oscillation of non-rotating neutron stars. The p2 mode frequencies have even more discrepancy.

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