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Investigation on the oscillation modes of neutron stars

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### 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

第十四届全国核结构大会暨第十次全国核结构专题讨论会

湖州 2012. 4. 13

Outline

- 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 space-time, 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 by1

where

or

Inner the star (l=2)

Outer the star

1 S.Chandrasekhar and V. Ferrari, Proc. R. Soc. London A, 432, 247(1991)

Nobel prize in 1983

It was shown that only values of xin the range between −1 (MDIx-1) and 0 (MDIx0) are consistent with the isospin-diffusion and isoscaling data at sub-saturation densities.

Here we assume that the EOS can be extrapolated to supra-saturation densities according to the MDI predictions.

It was shown that only values of xin the range between −1 (MDIx-1) and 0 (MDIx0) are consistent with the isospin-diffusion and isoscaling data at sub-saturation densities.

Here we assume that the EOS can be extrapolated to supra-saturation 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

damping time

Frequency

Wen D. H., Li B.A. and Krastev P.G., Phys. Rev. C 80, 025801 (2009)

Re-wII-linear

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 wII-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 wII, below this limit, wII-mode will disappear.

Euler equations in the rotating frame

(I) Background and Motivation

In Newtonian theory, the fundamental dynamical equation (Euler equations) that governs the fluid motion in the co-rotating frame is

external force

Coriolis force

Acceleration

centrifugal force

where is the fluid velocity and 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

or

It is shown that the structure parameters (M and R) make sense for the through the second order of .

Class. Quantum Grav. 20 (2003) R105P111/p113

CFS instability and canonical energy

canonical energy (conserved in absence of radiation and viscosity):

The function Ec govern the stability to nonaxisymmetric perturbations as: (1) if , stable; (2) if , unstable.

For the r-mode, The condition Ec < 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/0010102v1

Images of the motion of r-modes

- 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.

http://www.phys.psu.edu/people/display/index.html?person_id=1484;mode=research;research_description_id=333

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 109 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 109 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 tgw to be smaller than both tsv and tbv.

For l= m= 2 r-mode of a canonical neutron star (R= 10 km and M= 1.4M⊙and Kepler period PK≈ 0.8 ms (n=1 polytrope)).

Int.J.Mod.Phys. D10 (2001) 381

(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 1.4Msun 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.

Motivations(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, I2=0.80411.

And the viscosity c is density and temperature dependent:

T<109 K:

T>109 K:

PhysRevD.62.084030

The gravitational radiation timescale: neutron star with rigid crust

According to

, the critical rotation frequency is obtained:

Based on the Kepler frequency, the critical temperature defined as:

PhysRevD.62.084030

(III). Numerical Results neutron star with rigid crust

Equation of states neutron star with rigid crust

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 neutron star with rigid crust

D.H. Wen, W. G. Newton, and B.A. Li，Phys. Rev. C 85, 025801 (2012)

Comparing the time scale neutron star with rigid crust

The gravitational radiation timescale

The viscous timescale

D.H. Wen, W. G. Newton, and B.A. Li，Phys. Rev. C 85, 025801 (2012)

The lower boundary of the neutron star with rigid crustr-mode instability window for a 1.4Msun (a) and a 2.0Msun(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 1.4Msun(a) and a 2.0Msun(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 in frequency-temperature space, for a 1Tc 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 1.4Msunand 2.0Msunstars.

D.H. Wen, W. G. Newton, and B.A. Li，Phys. Rev. C 85, 025801 (2012)

Conclusion in frequency-temperature space, for a 1

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

Thanks! in frequency-temperature space, for a 1

The standard axial in frequency-temperature space, for a 1w-mode is categorized as wI. The high order axial w-modes are marked as the second w-mode (wI2 -mode), the third mode (wI3 -mode) and so on. An interesting additionally family of axial w-modes is categorized as wII.

Constrain by the f in frequency-temperature space, for a 1lowdata of relativistic heavy-ion reactions

P. Danielewicz, R. Lacey and W.G. Lynch, Science 298 (2002) 1592

It is worth noting that it’s simply an extrapolationfrom low to high densities. This extrapolation may NOT be right as the pion production data(Z.G. Xiao et al, Phys. Rev. Lett. 102, 062502 (2009))showed that the symmetry energy at high density is actually super-soft.

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 in frequency-temperature space, for a 1

1S.Weinberg, Gravitation and cosmology, (New York: Wiley,1972)

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