Department of Physics, South China Univ. of Tech. Properties of Neutron Star and its oscillations 2011.11 collaborators Bao-An Li, William Newton, Plamen.

Department of Physics, South China Univ. of Tech.（华南理工大学物理系）

文德华

Properties of Neutron Star and its oscillations

广州大学 天文学学术论坛 2011.11

collaborators

Bao-An Li, William Newton, Plamen Krastev

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

Institute of Theoretical Physics, Shanghai Jiao Tong University陈列文

Outline

I. Research history and observations

II. EOS constrained by terrestrial data and

non-Newtonian gravity in neutron star

III. Gravitational radiations from oscillations

of neutron star

I. Research History and Observations

• A year following Chadwick’s 1932 discovery of neutron, Baade and Zwicky conceived the notion of neutron star in the course of their investigation of supernovae.

• But no searches for neutron stars were mounted immediately following their work. No one knew what to look for, as the neutron star was believed to be cold and much smaller than white dwarfs.

• In 1939 (about 30 years before the discovery of pulsars), Oppenheimer, Volkoff and Tolman first estimated its radius and maximum based on the general relativity.

Theory prediction

•First observation of NS

In 1967 at Cambridge University, Jocelyn Bell observed a strange radio pulse that had a regular period of 1.3373011 seconds, which is believed to be a neutron star formed from a supernova.

Nature, 17(1968)709

初至和元年五

月晨出东方守

天关书见为太

白芒角四出色

青白凡见二十

三日

《宋會要輯

稿》

公元 1054 年

Science, 304(2004)536

研究中子星重要目的：1. 验证引力理论，包括引力辐射；2. 高密度核物质的研究。

中子星的观测和研究与诺贝尔奖1. 1974 年： A. Hewish 发现脉冲

星；2. 1993 年 ： R. A. Hulse & J. H.

Taylor 根据脉冲双星的周期变化间接验证引力辐射的存在。

太阳半径： ６．９６ × １０５千米太阳平均密度： 1.4 g/cm3

地球平均磁场： 6x10-5 T

太阳赤道自转周期约 25 日

Distribution of known galactic disk pulsars in the period–period-derivative plane. Pulsars detected only at x-ray and higher energies are indicated by open stars; pulsars in binary systems are indicated by a circle around the point. Assuming spin-down due to magnetic dipole radiation, we can derive a characteristic age for the pulsar t=p/(2dp/dt), and the strength of the magnetic field at the neutron star surface, Bs= 3.2*1019 (P*dp/dt)0.5 G. Lines of constant characteristic age and surface magnetic field are shown. All MSPs lie below the spin-up line. The group of x-ray pulsars in the upper right corner are known as magnetars.

R. N. Manchester, et al. Science 304, 542 (2004)

Observations: (1) Period and its derivation

Phys.Rev.Lett. 94 (2005) 111101

(2) Observation of pulsar masses.

Demorest, P., Pennucci, T., Ransom, S., Roberts, M., & Hessels, J. 2010, Nature, 467, 1081

(3) The ten fastest-spinning known radio pulsars.

Science, 311(2006)190

(5) Distribution of the Ms NS

0903.0493v1

(4) Distribution of the millisecond NS

(5) Constraints on the Equation-of-State of neutron stars

from nearby neutron star observations

arXiv:1111.0458v1

Radius Determinations for NSs, namely for RXJ1856 and RXJ0720, provide strong constraints for the EoS, as they exclude quark stars, but are consistent with a very stiff EoS.

(7) Observational Constraints for Neutron Stars

II. EOS constrained by terrestrial data and non-Newtonian gravity

in neutron star

TOV equation

From Lattimer 2008 talk

1.

M-R constraint from observation

2006NuPhA.777.497

2. Equation of state of neutron star matter

Stiffest and softest EOS

Possible EOSs of NS

APJ, 550(2001)426

Physics Reports, 442(2007) 109

Mass-Radius of neutron star

0 )) (, (( ) sn ymp

nn

p pE E E

symmetry energy

Energy per nucleon in symmetric matter

Energy per nucleon in asymmetric matter

δIsospin asymmetry

matternuclear symmetricmatterneutron puresym )()()( EEE

(1) Symmetry energy and equation of state of nuclear matter constrained by the terrestrial nuclear data

B. A. Li et al., Phys. Rep. 464, 113 (2008)

Constrain by the flow data of relativistic heavy-ion reactions

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

Promising Probes of the Esym(ρ) in Nuclear Reactions

At sub-saturation densities Sizes of n-skins of unstable nuclei from total reaction cross sections Proton-nucleus elastic scattering in inverse kinematics Parity violating electron scattering studies of the n-skin in 208Pb at JLab n/p ratio of FAST, pre-equilibrium nucleons Isospin fractionation and isoscaling in nuclear multifragmentation Isospin diffusion/transport Neutron-proton differential flow Neutron-proton correlation functions at low relative momenta t/3He ratio

Towards supra-saturation densities π -/π + ratio, K+/K0 ? Neutron-proton differential transverse flow n/p ratio of squeezed-out nucleons perpendicular to the reaction plane Nucleon elliptical flow at high transverse momenta

目前世界上已建立了多个中高能重离子碰撞实验室来测定对称能的密度依赖。包括中国兰州重离子加速器国家实验室、密歇根州立大学国家超导回旋加速器实验室 (NSCL) 、德国重离子物理研究所 (GSI) 的 FAIR 装置等。

B. A. Li et al., Phys. Rep. 464, 113 (2008)

1. R. B. Wiringa et al., Phys. Rev. C 38, 1010 (1988).

2. M. Kutschera, Phys. Lett. B 340, 1 (1994).

3. B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000).

4. S. Kubis et al, Nucl. Phys. A720, 189 (2003).

5. J. R. Stone et al., Phys. Rev. C 68, 034324 (2003).

6. A. Szmaglinski et al., Acta Phys. Pol. B 37, 277(2006).

7. B. A. Li et al., Phys. Rep. 464, 113 (2008).

8. Z. G. Xiao et al., Phys. Rev. Lett. 102, 062502 (2009).

Many models predict that the symmetry energy first increases and then decreases above certain supra-saturation densities. The symmetry energy may even become negative at

high densities.According to Xiao et al. (Phys. Rev. Lett. 10

2, 062502 (2009)), constrained by the recent

terrestrial nuclear laboratory data, the nucl

ear matter could be described by a super so

fter EOS — MDIx1.

)()1(2

1])()0,([

4

1)(),( sym

2sym

22

EEEE

P ee

Can not support the observations of neutron stars!

1. E. G. Adelberger et al., Annu. Rev. Nucl. Part. Sci. 53, 77(2003).2. M.I. Krivoruchenko, et al., hep-ph/0902.1825v1 and references there in.

The inverse square-law (ISL) of gravity is expected to be violated, especially at less length scales. The deviation from the ISL can be characterized effectively by adding a Yukawa term to the normal gravitational potential

In the scalar/vector boson (U-boson ) exchange picture,

and

Within the mean-field approximation, the extra energy density and the pressure due to the Yukawa term is

(2). Super-soft symmetry energy encountering non-Newtonian gravity in neutron stars

Hep-ph\0810.4653v3

PRL-2005,94,e240401 Hep-ph\0902.1825

Constraints on the coupling strength with nucleons g2/(4) and the mass μ (equivalently and ) of hypothetical weakly interacting light bosons.

EOS of MDIx1+WILB

22 / g

D.H.Wen, B.A.Li and L.W. Chen, Phys. Rev. Lett., 103(2009)211102

M-R relation of neutron star with MDIx1+WILB

D.H.Wen, B.A.Li and L.W. Chen, Phys. Rev. Lett., 103(2009)211102

Conclusion

1. It is shown that the super-soft nuclear symmetry energy preferre

d by the FOPI/GSI experimental data can support neutron stars

stably if the non-Newtonian gravity is considered;

2. Observations of pulsars constrain the g2/2 in a rough range of 5

0~150 GeV-2.

V. Gravitational Radiation from oscillations of neutron star

•Why do We Need to Study Gravitational Waves?

1. Test General Relativity:probe of strong-field gravity

2. Gain different view of Universe:(1) Sources cannot be obscured by dust /

stellar envelopes(2) Detectable sources are some of the most

interesting, least understood in the UniverseGravitational Waves = “Ripples in space-time”

(I). Gravitational Radiation and detection

Compact binary

Orbital decay of the Hulse-Taylor binary neutron star system (Nobel prize in 1993)is the best evidence so far.

Supernovae “Mountain” on neutron star

Oscillating neutron star

Possible Sources of Gravitational Waves From Neutron star

Hanford Washington

Livingston, Louisiana

(Laser Interferometer Gravitational-Wave Observatory )

LIGO

From 0711.3041v2

LIGO’s International Partners

VIRGO: Pisa, Italy [Italy/France] GEO600, Hanover Germany [UK, Germany]

TAMA300, Tokyo [Japan]AIGO, Jin-Jin West Australia

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.

MNRAS(2001)320,307

•The importance for astrophysics

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

(II). Oscillation modes

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.

Axial w-mode: not accompanied by any matter motions and only the perturbation of the space-time.

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.

Polar mode: transforms as (−1)ℓ

1. Axial w-modes of static neutron stars

Key equation of axial w-mode

Inner the star (l=2)

Outer the star

The equation for oscillation of the axial w-mode is give by1

dr

de

dr

d *

drerr

0*

where

or

]6)(6[ 33

2

mprrr

eV

]66[3

2

Mrr

eV

0)]([ 22

*

2

zrVdr

zd

ii 0

1 S.Chandrasekhar and V. Ferrari, Proc. R. Soc. London A, 432, 247(1991) Nobel prize in 1983

Eigen-frequency of the wI -mode scaled by the gravitational energy

Wen D.H. et al., Physical Review C 80, 025801 (2009)

the minimum compactness for the existence of the wII -mode

to be M/R ≈ 0.1078.

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.u

Φ

dt

ud

•Euler equations in the rotating frame

2. R-modes(1). Background

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

]1[)1(

2 23

2 M

R

ll

mr

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

•Definition of r-mode

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

)1(

2

ll

mror

•CFS instability and canonical energy APJ,222(1978)281

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

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

0)( cE 0)( cE

)1(

2

llr

)1(

)2)(1(2

ll

llri

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.

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

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

• 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

(2). Motivations

(a) Old neutron stars (having crust) in LMXBs with rapid rotating fr

equency (such as EXO 0748-676) may have high core temperature (ar

Xiv: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, 10

81(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 nucl

ear laboratory data could provide constraints on the r-mode inst

ability.

PhysRevD.62.084030

Here only considers l=2, I2=0.80411. And the viscosity c is density and temperature dependent:

The subscript c denotes the quantities at the outer edge of the core.

T<109 K:

T>109 K:

The viscous timescale for dissipation in the boundary layer:

(3). Basic equations for calculating 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.62.084030

Equation of states

W. G. Newton, M. Gearheart, and Bao-An Li, 1110.4043v1

The mass-radius relation and the core radius

Wen, et al, 1110.5985v1

Comparing the time scale

The gravitational radiation timescale

The viscous timescale

Wen, et al, 1110.5985v1

Constraints of the symmetric energy and the core-crust transition density on the r-mode instability Wi

ndows

Wen, et al, 1110.5985v1

The location of the LMXBs in the r-mode instability windows

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)

Wen, et al, 1110.5985v1

The critical temperature under the Kepler frequency varies with transition density for 1.4Msun (except for ploy2.0) neutr

on star

The critical temperatures should be constrained in the shaded area by the constrained symmetric energy.

Wen, et al, 1110.5985v1

Conclusion

(1)Obtained the constraint on the r-mode instability

windows by the symmetric energy and the core-crust

transition, which are constrained by the terrestrial

nuclear laboratory data;

(2) A massive neutron star has a wider instability window;

(3)Giving the constraint on the critical temperature.

Thanks