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Ch. C. Moustakidis Department of Theoretical Physics, Aristotle University of Thessaloniki, Greece
Nuclear Symmetry Energy Effects on the r-mode
Instabilities of Neutron Stars
HEP2012: Recent Developments in High Energy HEP2012: Recent Developments in High Energy Physics and Cosmology Physics and Cosmology
Ioannina, Greece, April 508 2012Ioannina, Greece, April 508 2012
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A. Introduction on the r-mode instability of neutron stars
B. Equation of state of neutron stars matter
C. Relative formulae
D. Presentation of the results
Outline
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Properties of Neutron Stars
•Radius: R ~ 10 kmMass: M ~ 1.4 – 2.5 MsunMean density: r ~ 4x1014 g/cm3Period: ~ a few ms – a few sMagnetic field: B ~ 108 – 1015 G
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r-mode instability
• Neutron stars suffer a number of instabilities with a common feature: they can be directly associated with unstable modes of oscillation
• Non-axisymmetric stellar oscillations will inevitably lead to the production of gravitational radiation
• If an unstable oscillation mode grows it may reach a sufficiently large amplitude that the emerging gravitational waves can be detected
• The first prediction for the r-mode made by Chandrasekhar (1970), Friedeman and Schutz (1978) (CFS Instability)
• The r-modes are always retrograde in the rotating frame and prograde in the inertial frame (satisfy the CFS criterion at all rates of rotation ----> the r-modes are generically unstable in rotating perfect fluid (interior neutron stars))
• The CFS instability allows some oscillation modes of a fluid body to be driven rather than damped by radiation reaction (gravitational waves), essentially due to a disagreement between two frames of reference.
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Mechanics of r-mode instability and gravitational radiation
• In rotating stars, gravitational radiation reaction drives the r-mode toward unstable growth
• In hot, rapidly-rotating neutron stars, this instability may not be
suppressed by internal dissipative mechanics (viscosity e.t.c)
• The dimensionless amplitude α of the r-mode will grow to order unity within ten minutes of the formation of a neutron star
• The emitted GWs carry away angular momentum, and will cause the newly-formed neutron star to spin down over time
• The spin-down timescale and the strenght of the GWs themselves are directly dependent on the maximum value αmax to which the amplitude is allowed to grow
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Gravitational radiation instability in the r-modes
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Gravitational radiation instability in the r-modes
The r mode corresponds to the value m=2
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Viscous dissipation mechanisms in the r-modes
• In neutron star colder than about 10^9 K the shear viscosity dominate by electron-electron scattering
• For temperature above 10^9 K, neutron-neutron scattering provides the dominant dissipation mechanics
• The bulk viscosity arises as the mode oscillation drives the fluid away from β-equilibrium (strong at very high temperature)
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Times scales- Critical angular velocity- Critical temperature
The fiducial viscous time scale defined as The fiducial gravitational radiation time scale defined as
The critical angular velocity, above which the r-mode is unstable is defined by the condition
The angular velocity of a neutron star can never exceed some maximum value Ωmax=ΩKepler=2/3Ω0. So there is a critical temperature below which the gravitational radiation instability is completed suppressed by
viscosity
The critical angular velocity marks the boundary between stability and instability
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The nuclear modelThe energy per baryon of neutron star matter
Nuclear symmetry energy
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Nuclear Equation of state
Proton fraction
The pressure Energy
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Core-crust interface
The determination of nt is very complicated problem
The main theoretical approaches are:
•The dynamical method
•The thermodynamical method
•The random phase approximation (RPA)
The location of the inner edge of the
crust is a Phase Transition Problem Inner Crust <========>Outer Core
Solid Phase <========>Liquid Phase
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We consider uniform matter consists mainly by neutrons, few percent of protons and electrons under beta equilibrium
The first low of thermodynamics reads:
The stability of the uniform phase requires that the energy density u(v,q) is a convex functionu(v,q) is a convex function with the following two constraints:
Thermodynamical method
For a given equation of state, the quantity C(n) is plotted as a function of the baryon density n and the equation C(n)=0 defines the transition density nt
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Tolman-Oppenheimer-Volkov (TOV) Equations
m(r): Mass inside a sphere of radius rρ(r): Mass densityε(r): Energy density (ε(r)=ρ(r)c2)P(r): Pressure
Nuclear Matter EOS : P(ε)=P(ε(r))Integration up to r=R where P(R)=0 M(R)
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Conclusions-Outlook •We tried to clarify the nuclear equation of state effect on the possible r-mode instabilities
•The slope parameter L has been connected with the main quantities characterize the r-mode instability (time scales, critical angular velocity and critical temperature)
•The nuclear equation of state affects the r-mode in two ways: a) affect the location of the crust and b) the structure of the density of the neutron star core
•The parameter L has been varied in the interval 65-110 MeV
•Accordingly the fiducial time scales τGR, τee, τnn, increases by a factor 2.5, 1.9 and 2 (for M=1.4 Mo) and 3.1, 1.7 and 2 (for M=1.8 Mo)
• Accordingly Ωc/Ωο increases 10% (for M=1.4 Mo) and 13% (for M=1.8 Mo)
•Tc increases by a factor 1.8 (for M=1.4 Mo) and 2.2 (for M=1.8 Mo)
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Open problems
• The role of magnetic field in the evolution of the r-modes (suppress the
instability ? Limits its growth ? Merely change the values of the frequency and growth times ? )
• The role of the crust: in the present work we consider perfectly rigid crust (the viscosity is maximal between core and crust). Considering elastic crust the oscillation of the core could partially penetrate into the crust
• Do superfluid effects suppress the r-mode instability completely in neutron stars ?