Download Effect of the symmetry energy on gravitational waves from axial

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
1Department
Bao-An Li1 and Plamen Krastev2
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 read Phys. Rev. C 80, 025801 (2009) for details
1.
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.
The importance for astrophysics
A network of large-scale ground-based laserinterferometer 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
The equation for oscillation of the axial w-mode is give by1
d 2z
2

[

 V (r )] z  0
2
dr*
where
  0  ii
Inner the star (l=2)
Outer the star
1
d
  d
e
dr*
dr
or
r
r*   e  dr
0
e 2
V  3 [6r  r 3 (   p)  6m]
r
6e 2
V  3 [r  M ]
r
S.Chandrasekhar and V. Ferrari, Proc. R. Soc. London A, 432, 247(1991)
Nobel prize in 1983
The standard axial w-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 .
2. EOS constrained by recent
terrestrial laboratory data
Constrain by the flow data
of 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).
It was shown that only
values of x in the range
between −1 (MDIx-1) and 0
(MDIx0) are consistent with
the isospin-diffusion and
isoscaling data at subsaturation 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
Motivation
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 neutronrich 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
Frequency and damping time of axial w-mode
Scaling characteristic
1
L. K. Tsui and P. T. Leung, MNRAS 357, 1029 (2005).
The gravitational energy is calculated from1
1S.Weinberg, Gravitation and cosmology, (New York:
Tsui, et al, MNRAS, 357, 1029(2005)
Eigen-frequency of wI2 scaled by the mass and the gravitational field energy
Exists linear fit
Based on this linear dependence of the scaled
frequency, the wII -mode is found to exist about
Conclusion
1. The density dependence of the nuclear
symmetry energy affects significantly both the
frequencies and the damping times of axial wmode.
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, wIImode will disappear.
PHYSICAL REVIEW C 80, 025801 (2009)
Thanks
Appendix
The value of the isospin asymmetry δ 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
Rezzola, Gravitational waves from perturbed black holes and neutron stars
Radial oscillation:
Definition of radial and non-radial oscillation 2
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
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.
Definition of Axial oscillation 1
Thorne, APJ, 1967,149, 491
Definition of Axial 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
Polar oscillation:
Definition of Quasi-normal mode2
The meaning of the complex frequency
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
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
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