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### Properties of Neutron Star and its oscillations

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

陈列文

Institute of Theoretical Physics, Shanghai Jiao Tong University

广州大学 天文学学术论坛 2011.11

Outline

- Research history and observations
- EOS constrained by terrestrial data and non-Newtonian gravity in neutron star
- Gravitational radiations from oscillations of neutron star

Theory prediction

- 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.

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.

First observation of NS- 验证引力理论，包括引力辐射；
- 高密度核物质的研究。

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

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

Science, 304(2004)536

Observations: (1) Period and its derivation

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)

(2) Observation of pulsar masses.

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

Phys.Rev.Lett. 94 (2005) 111101

(5) Constraints on the Equation-of-State of neutron starsfrom nearby neutron star observations

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.

arXiv:1111.0458v1

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

Possible EOSs of NS

APJ, 550(2001)426

Mass-Radius of neutron star

Physics Reports, 442(2007) 109

δ

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

Energy per nucleon in symmetric matter

Energy per nucleon in asymmetric matter

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

Constrain by the flowdata 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

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

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。

Many models predict that the symmetry energy first increases and then decreases above certain supra-saturation densities.

The symmetry energymay even become negative at high densities.

According to Xiao et al. (Phys. Rev. Lett. 102, 062502 (2009)), constrained by the recent terrestrial nuclear laboratory data, the nuclear matter could be described by a super softer EOS — MDIx1.

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).

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

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

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

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.

Hep-ph\0810.4653v3

EOS of MDIx1+WILB

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

- It is shown that the super-soft nuclear symmetry energy preferred by the FOPI/GSI experimental data can support neutron stars stably if the non-Newtonian gravity is considered;
- Observations of pulsars constrain the g2/2 in a rough range of 50~150 GeV-2.

(I). Gravitational Radiation and detection

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 Universe

Gravitational Waves = “Ripples in space-time”

Possible Sources of Gravitational Waves From Neutron star

Supernovae

“Mountain” on neutron star

Compact binary

Orbital decay of the Hulse-Taylor binary neutron star system (Nobel prize in 1993)

is the best evidence

so far.

Oscillating neutron star

GEO600, Hanover Germany [UK, Germany]

VIRGO: Pisa, Italy [Italy/France]

AIGO, Jin-Jin West Australia

TAMA300, Tokyo [Japan]

LIGO’s International PartnersThe importance for astrophysics

(II). Oscillation 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

1. Axial w-modes of static neutron stars

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

Axial w-mode: not accompanied by any matter motions and

only the perturbation of the space-time.

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

the minimum compactness for the existence of the wII-mode

to be M/R ≈ 0.1078.

Eigen-frequency of the wI-mode scaled by the gravitational energyWen D.H. et al., Physical Review C 80, 025801 (2009)

Euler equations in the rotating frame

(1). Background

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.

(2). Motivations(3). Basic equations for calculating 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:

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 viscous timescale

The gravitational radiation timescale

Wen, et al, 1110.5985v1

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

Wen, et al, 1110.5985v1

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

Wen, et al, 1110.5985v1

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 under the Kepler frequencyvaries with transition density for 1.4Msun (except for ploy2.0) neutron star

Wen, et al, 1110.5985v1

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

Conclusion

- 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;
- A massive neutron star has a wider instability window;
- Giving the constraint on the critical temperature.

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