Date post: | 26-Mar-2015 |
Category: |
Documents |
Upload: | landon-klein |
View: | 215 times |
Download: | 0 times |
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