Imprint of nuclear symmetry energy on gravitational waves from axial w modes
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Imprint of nuclear symmetry energy on gravitational waves from axial w-modes. Dehua Wen (文德华). Department of Physics, South China Univ. of Tech. Department of Physics, Texas A&M University-Commerce. collaborators. Bao-An Li 1 and Plamen Krastev 2.

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Imprint of nuclear symmetry energy on gravitational waves from axial w modes

Imprint of nuclear symmetry energy on gravitational waves from axial w-modes

Dehua Wen(文德华)

Department of Physics, South China Univ. of Tech.

Department of Physics, Texas A&M University-Commerce

collaborators

Bao-An Li1 and Plamen Krastev2

1Department of Physics and Astronomy, Texas A&M University-Commerce

2Department of Physics, San Diego State University

Outline:

1. Introduction of axial w-mode

2. EOS constrained by recent terrestrial laboratory data

3. Numerical Result and Discussion

Please readPhys. Rev. C 80, 025801 (2009) for details


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 from axial w-modes: 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.


The importance for astrophysics
The importance for astrophysics from axial w-modes

A network of large-scale ground-based laser-interferometer detectors (LIGO, VIRGO, GEO600, TAMA300) is on-line in detecting the gravitational waves (GW).

Theorists are presently try their best to think of various sources of GWs that may be observable once the new ultra-sensitive detectors operate at their optimum level.

GWs from non-radial neutron star oscillations are considered as one of the most important sources.

MNRAS(2001)320,307


Key equation of axial w mode
Key equation of axial w-mode from axial w-modes

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


The standard axial from axial w-modesw-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 low data of relativistic heavy ion reactions
Constrain by the f from axial w-modeslowdata 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).


It was shown that only values of from axial w-modesxin 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 from axial w-modes


Motivation
Motivation from axial w-modes

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.

Heavy-ion reactions provide means to constrain the uncertain density behavior of the nuclear symmetry energy and thus the EOS of neutron-rich nuclear matter.

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.


3 numerical result and discussion
3. Numerical Result and Discussion from axial w-modes

frequency and damping time-M



Scaling characteristic from axial w-modes

wI

1

L. K. Tsui and P. T. Leung, MNRAS 357, 1029 (2005).


The gravitational energy is calculated from 1
The gravitational energy is calculated from from axial w-modes1

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


w from axial w-modesI2

Tsui, et al, MNRAS, 357, 1029(2005)

Eigen-frequency of wI2 scaled by the mass and the gravitational field energy


Re w ii linear
Re-w from axial w-modesII-linear

Exists linear fit

Based on this linear dependence of the scaled frequency, the wII-mode is found to exist about compactness M/R>0.1078.


Conclusion
Conclusion from axial w-modes

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, especially to the wI2-mode 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.

PHYSICAL REVIEW C 80, 025801 (2009)


Thanks
Thanks from axial w-modes


Appendix
Appendix from axial w-modes


The value of the isospin asymmetry from axial w-modesδ at β equilibrium is determined by the chemical equilibrium and charge neutrality conditions, i.e., δ = 1 − 2xp with


Definition of radial and non radial oscillation 1
Definition of radial and non-radial oscillation 1 from axial w-modes

Rezzola, Gravitational waves from perturbed black holes and neutron stars

Radial oscillation:


Definition of radial and non radial oscillation 2
Definition of radial and non-radial oscillation 2 from axial w-modes

Non-radial oscillation(Newton theory):

The main difference is the perturbation function.

Or in spherical harmonic function, the l=0 is radial oscillation, and l>=1 is non-radial oscillation.

PRD, 1999,60,104025


Definition of polar and axial
Definition of polar and axial from axial w-modes

gravitational radiation from neutron star oscillations exhibits certain characteristic frequencies which are independent of the processes giving rise to these oscillations. These “quasi-normal” frequencies are directly connected to the parameters of neutron star (mass, charge and angular momentum) and are expected to be inside the bandwidth of the constructed gravitational wave detectors.

gravitational radiation from neutron star oscillations exhibits certain characteristic frequencies which are independent of the processes giving rise to these oscillations. These “quasi-normal” frequencies are directly connected to the parameters of neutron star (mass, charge and angular momentum) and are expected to be inside the bandwidth of the constructed gravitational wave detectors.

There are two classes of vector spherical harmonics (polar and axial) which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics. The difference between the two families is their parity. Under the parity operator π (under the angular transformation θ→ π − θ, ϕ → π + ϕ), a spherical harmonic with index ℓ transforms as (−1)ℓ, the polar class of perturbations transform under parity in the same way, as (−1)ℓ, and the axial perturbations as (−1)ℓ+1. Finally, since we are dealing with spherically symmetric space-times the solution will be independent of m, thus this subscript can be omitted.

There are two classes of vector spherical harmonics (polar and axial) which are build out of combinations of the Levi-Civita volume form and the gradient operator acting on the scalar spherical harmonics. The difference between the two families is their parity. Under the parity operator π (under the angular transformation θ→ π − θ, ϕ → π + ϕ), a spherical harmonic with index ℓ transforms as (−1)ℓ, the polar class of perturbations transform under parity in the same way, as (−1)ℓ, and the axial perturbations as (−1)ℓ+1. Finally, since we are dealing with spherically symmetric space-times the solution will be independent of m, thus this subscript can be omitted.


Definition of axial oscillation 1
Definition of from axial w-modesAxial oscillation 1

Thorne, APJ, 1967,149, 491


Definition of axial oscillation 2
Definition of from axial w-modesAxial oscillation 2

Note: in fact, now it is already found that there are gravitational waves in axial mode, that is, in the odd-parity pulsations.


Definition of polar oscillation
Definition of from axial w-modespolar oscillation

Polar oscillation:


Definition of quasi normal mode 2
Definition of Quasi-normal mode from axial w-modes2


The meaning of the complex frequency
The meaning of the complex frequency from axial w-modes

The eigen-frequencies of the modes are complex, the real part gives the pulsation rate and the imaginary part gives the damping time of the pulsation as a result of the emission of the gravitational radiation.

MNRAS(1992)255,119


The frequency of different modes
The frequency of different modes from axial w-modes

Typical values of the frequencies and the damping times of various families of modes for a polytropic star (N = 1) with R = 8.86 km and M = 1.27M⊙ are given. p1 is the first p-mode, g1 is the first g-mode, w1 stands for the first curvature mode and wII for the slowest damped interface mode. For this stellar model there are no trapped modes11.


Definition of w mode
Definition of w-mode from axial w-modes

To the perturbation,

Only consider the space-time, that is, V = W = 0

So the perturbation equations for a specific multi-pole ℓ take the following simple form inside the star


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