Investigation on the oscillation modes of neutron stars 文德华 Department of Physics, South China Univ. of Tech. ( 华南理工大学物理系 ) collaborators Bao-An Li, William Newton, Plamen Krastev Department of Physics and astronomy, Texas A&M University-Commerce 2012 超重核合成与性质研讨会兰州 2012.8
Outline I. W-modes in neutron stars II. R-modes in neutron stars
I. W-modes in neutron star Introduction of axial w-mode The non-radial neutron star oscillations could be triggered by various mechanisms such as gravitational collapse, a pulsar glitch or a phase transition of matter in the inner core. Axial mode: under the angular transformation θ π θ, ϕ π+ ϕ, a spherical harmonic function with index l transforms as ( 1) l+1 for the expanding metric functions. Polar mode: transforms as ( 1) l Oscillating neutron star
Axial w-mode: not accompanied by any matter motions and only the perturbation of the spacetime, exists for all relativistic stars, including neutron star and black holes. One major characteristic of the axial w-mode is its high frequency accompanied by very rapid damping.
Motivation (1) The w-modes are very important for astrophysical applications. The gravitational wave frequency of the axial w-mode depends on the neutron star s structure and properties, which are determined by the EOS of neutron-rich stellar matter. (2) It is helpful to the detection of gravitational waves to investigate the imprint of the nuclear symmetry energy constrained by very recent terrestrial nuclear laboratory data on the gravitational waves from the axial w-mode.
Key equation of axial w-mode The equation for oscillation of the axial w-mode is give by 1 where 2 d z dr 2 * 2 + [ ω V ( r)] z = 0 ω = ω 0 + iω i d dr * = e ν λ d dr or r * r = 0 e ν λ dr Inner the star (l=2) V 2ν e 3 = [6r + r ( ρ 3 r p) 6m] Outer the star V = 6e r 2ν 3 [ r M ] 1 S.Chandrasekhar and V. Ferrari, Proc. R. Soc. London A, 432, 247(1991) Nobel prize in 1983
EOS constrained by terrestrial laboratory data It was shown that only values of x in the range between 1 (MDIx-1) and 0 (MDIx0) are consistent with the isospindiffusion and isoscaling data at sub-saturation densities. Here we assume that the EOS can be extrapolated to suprasaturation densities according to the MDI predictions. 1. L.W.Chen, C. M. Ko, and B. A. Li, Phys. Rev. Lett. 94, 032701 (2005). 2.B. A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113 (2008).
M-R relation Wen D. H., Li B.A. and Krastev P.G., Phys. Rev. C 80, 025801 (2009)
Numerical Result and Discussion Frequency damping time Wen D. H., Li B.A. and Krastev P.G., Phys. Rev. C 80, 025801 (2009)
Scaling characteristic 1 Wen D. H., Li B.A. and Krastev P.G., Phys. Rev. C 80, 025801 (2009)
Exists linear fit Wen D. H., Li B.A. and Krastev P.G., Phys. Rev. C 80, 025801 (2009) Based on this linear dependence of the scaled frequency, the w II -mode is found to exist about compactness M/R>0.1078.
Conclusion 1. The density dependence of the nuclear symmetry energy affects significantly both the frequencies and the damping times of axial w-mode. 2. Obtain a better scaling characteristic through scaling the eigen-frequency by the gravitational energy. 3. Give a general limit, M/R~0.1078, based on the linear scaling characteristic of w II, below this limit, w II - mode will disappear.
II. R-modes in neutron star (I) Background and Motivation Euler equations in the rotating frame In Newtonian theory, the fundamental dynamical equation (Euler equations) that governs the fluid motion in the co-rotating frame is Acceleration u where is the fluid velocity and Coriolis force external force centrifugal force Φ represents the gravitational potential.
Definition of r-mode For the rotating stars, the Coriolis force provides a restoring force for the toroidal modes, which leads to the so-called r-modes. Its eigen-frequency is 2mΩ ωr = [1 ω2 l( l + 1) 3 R Ω M It is shown that the structure parameters (M and R) make sense for the through the second order of Ω. ω r 2 ] or ω r 2mΩ l ( l + 1) Class. Quantum Grav. 20 (2003) R105P111/p1
CFS instability and canonical energy APJ,222(1978)281 canonical energy (conserved in absence of radiation and viscosity): The function E c govern the stability to nonaxisymmetric perturbations as: (1) if E ( ξ ) 0, stable; (2) if E c ( ξ ) 0, unstable. c 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. σ r = 2Ω l( l+ 1) σ i = σ r + Ω = Ω 2( l 1)( l+ 2) l( l+ 1) gr-qc/0010102v1
Images of the motion of r-modes http://www.phys.psu.edu/people/display/index.html?person id=1484;mode=research;research description id=333 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. 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 Note: The CFS instability is not only existed in GR, but also existed in Newtonian theory.
Viscous damping instability The r-modes ought to grow fast enough that they are not completely damped out by viscosity. 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. Bulk viscosity arises because the pressure and density variations associated with the mode oscillation drive the fluid away from beta equilibrium.
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 0748-676) may have high core temperature (arxiv:1107.5064v1.); which hints that there may exist r-mode instability in the core. (b) The discovery of massive neutron star (PRS J1614-2230, Nature 467, 1081(2010) and EXO 0748-676, Nature 441, 1115(2006)) reminds us restudy the r-mode instability of massive NS, as 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.
(II). Basic equations for r-mode instability window of neutron star with rigid crust The viscous timescale for dissipation in the boundary layer: The subscript c denotes the quantities at the outer edge of the core. Here only considers l=2, I 2 =0.80411. And the viscosity η c is density and temperature dependent: T<10 9 K: T>10 9 K: PhysRevD.62.084030
The gravitational radiation timescale: According to, the critical rotation frequency is obtained: Based on the Kepler frequency, the critical temperature defined as: PhysRevD.62.084030
(III). Numerical Results
Equation of states W. G. Newton, M. Gearheart, and B.-A. Li, arxiv:1110.4043v1. 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. D.H. Wen, W. G. Newton, and B.A. Li,Phys. Rev. C 85, 025801 (2012)
The mass-radius relation and the core radius D.H. Wen, W. G. Newton, and B.A. Li,Phys. Rev. C 85, 025801 (2012)
Comparing the time scale The gravitational radiation timescale D.H. Wen, W. G. Newton, and B.A. Li,Phys. Rev. C 85, 025801 (2012) The viscous timescale
The lower boundary of 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 consistent with experiment. D.H. Wen, W. G. Newton, and B.A. Li,Phys. Rev. C 85, 025801 (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. D.H. Wen, W. G. Newton, and B.A. Li,Phys. Rev. C 85, 025801 (2012) 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, 101101(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, 025801 (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.
THANKS!
The standard axial w-mode is categorized as w I. The high order axial w-modes are marked as the second w- mode (w I2 -mode), the third mode (w I3 -mode) and so on. An interesting additionally family of axial w-modes is categorized as w II.
Constrain by the flow data relativistic heavy-ion reactions P. Danielewicz, R. Lacey and W.G. Lynch, Science 298 (2002) 1592 1.M.B. Tsang, et al, Phys. Rev. Lett. 92, 062701 (2004) 2. B. A. Li, L.W. Chen, and C.M. Ko, Phys. Rep. 464, 113 (2008).
The gravitational energy is calculated from 1 S.Weinberg, Gravitation and cosmology, (New York: Wiley,1972)