The influence of the symmetry energy on the structure of hyperon - - PowerPoint PPT Presentation

The influence of the symmetry energy on the structure of hyperon stars.

• I. Bednarek (Katowice)

Matter To The Deepest Recent Developments In Physics Of Fundamental Interactions XXXIX International Conference of Theoretical Physics Ustron 2015

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Study of nuclear matter at densities above saturation density

Motivation: better understanding of the physics of neutron stars examining the possibility of the existence of strange baryons in the very inner part of a neutron star

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Modelling neutron star structure and composition

• P. Haensel et al. 2007

Outer core - n, p, e, µ matter under β equilibrium ε = εN(nn, np) + εl(e, µ) Equilibrium conditions:

1 µn = µp + µe 2 µµ = µe

Schematic structure of a neutron star atmosphere

• uter crust - lattice of

neutron-rich heavy nuclei, degenerate, relativistic electrons - correction to radius ∼ 10 percent inner crust - as above plus degenerate non-relativistic neutrons

• uter core - homogeneous

nucleonic matter inner core - may contain exotic forms of matter

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Modelling neutron star structure and composition

Threshold chemical potentials

• f hyperons
• P. Haensel et al. 2007

Apppearance of hyperons - at 2 − 3n0 Equilibrium conditions - contribution of hyperons to β equilibrium.

1 µΞ− = µΣ− = µn + µe 2 µΞ0 = µΣ0 = µΛ = µn 3 µΣ+ = µp = µn − µe

hyperon onset points - hyperon threshold densities nY lim

nY→0 = ∂ε

∂nb |eq= µ0

Y

For nb > nth

Y hyperon Y become

stable in dense matter.

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Modelling neutron star structure and composition - Tolman-Oppenheimer-Volkoff equation

dP dr = − G c4 (E + P)

• mc2 + 4πr3P
• r (r − 2Gm/c2)

dm dr = 4πr2 E c2 M − R relations details about the internal structure of a neutron star provides data on the impact of a given model on the internal structure of a neutron star Solution of the TOV equations needs supplementation by the equation of state (EoS) of the matter of a neutron star P(E(nB)) Lattimer et al. 2013

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Measured neutron star masses.

Lattimer et al. 2013 There are no precise simultaneous measurements of neutron star mass and radius. Constraints on the mass-radius relation radius - not strong enough mass

1 PSR J1614-2230, NS-WD

binary system, MNS = 1.97 ± 0.4M⊙, MWD = 0.5M⊙ P.Demorest et al. 2010

2 PSR J0348+0432, NS-WD

binary system, MNS = 2.01 ± 0.4M⊙, MWD = 0.172M⊙ Antoniadis et al. 2013

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Hyperon puzzle.

Mmax ≥ Mmeasured ⇒ Mmax ≥ 2M⊙ Massive neutron stars - strong constraint on the equation of state - requires stiff equation of state Hyperons soften the equation of state significantly.

20 40 60 80 100 120 140 160 180 200 100 200 300 400 500 600 700 P (MeV fm-3) ε (MeV fm-3) TM1-npe TM1-weak TM1-ext

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Equation of state of isospin asymmetric nuclear matter

• two component system of N nucleons

The energy differences of the states with different composition

• f protons and neutrons are encoded in the symmetry energy.

Esym(Np, Nn) ≡ E(Np, Nn) − E(Np = N/2, Nn = N/2) δa = Nn − Np NB = 1 − 2Yp Esym(N, δa) ≡ E(N, δa) − E(N, δa = 0)

1 symmetric nuclear matter (SNM) δa = 0 ⇒ Nn = Np 2 pure neutron matter (PNM) δa = 1 ⇒ Np = 0

Esym(nB) = E(nB, δa = 1) − E(nB, δa = 0) E(n0, δa = 1) = Esym(n0) + E(nB, δa = 0)

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Using the expansion E(nB, yp) = E(nB, yp = 1/2) + (1 − 2yp)2S2(nB) + . . . S2(nB) = Sv + L 3 nB − n0 n0 + . . . Sv ≃ 31 MeV, L ≃ 50 MeV

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Symmetry energy- connections to neutron star parameters

Proton fraction µp − µn = ∂ETot ∂Yp = 4Esym(nB)(1 − 2Yp) ETot = E + Ee at saturation nB = n0 Yp ≈ 1 3π2n0 4Sv ℏc 3 ≈ 0.04 Pressure at saturation density pβ(n0) =≃ L 3 n0

• 1 −

4Sv ℏc 3 4 − 3Sv/L 3π2n0 + . . .

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Symmetry energy- connections to neutron star parameters

Pressure- radius correlations R = C(nB, M)(pβ/MeVfm−3)1/4 Coefficients C(nB, 1.4M⊙) M∗/M⊙ n0 1.5n0 2n0 1.3 9.30±0.58 6.99±0.30 5.72±0.25 2.0 9.52±0.49 7.06±0.24 5.68±0.14 Coefficients appropriate for nB = n0 - C(n0, 1.4M⊙) Crust-core transition density and pressure Crust thickness

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Theoretical predictions for symmetry energy

Theoretical considerations predict wide range of symmetry energies for densities below and above saturation density n0 = 0.16fm−3. Density dependence of the symmetry energy predicted by various theoretical calculations. (Shetty, 2010)

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Nuclear matter with strangeness degrees of freedom - system of nucleons and hyperons

Modification of the symmetry energy by the presence of hyperons. EH

sym(nB, δa, yi) = E(nB, δa, yi) − E(nB, δa = 0, yi)

In this case: nB = nN + y and y =

i yi - total hperon number

density Pure neutron matter − → y = 0

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Experimental constraints for symmetry energy parameters.

Constraint for the centroid energy of the giant dipole resonance for 208Pb - S2(0.1) ≃ (24.1 ± 0.9) MeV Consensus agreement of the six experimental constraints 44 MeV < L < 66 MeV Results of neutron matter studies - direct estimates of Sv and L consistent with the results determined from nuclear experiments

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Astrophysical considerations

Measurements of neutron star radii Estimation of neutron star radii - distant measurement and atmospheric modelling required. Photospheric Radius Expansion Bursts Accreation from the companion (MS star) - overflowing the Roche lobe Unstable burning of the accreated material Spread of the nuclear burning accros stellar surface - sudden increase in X-ray luminosity and temperature X-ray bursts The average neutron star mass and radius implied by these results: ¯ M = 1.65 ± 0.12M⊙, ¯ R = 10.77 ± 0.65.

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Quiescent LMXBs in globular clusters - candidate II

QLMXBs Neutron stars in binary system with intermittently accreated matter from evolving companion star. Episodes of accretion separated by long periods of quiescence. Low magnetic field Compression of matter in the crust induces nuclear reactions Sufficient amount of heat is released to warm the star Neutron stars cool via neutrino radiation from their interiors and X-ray from their surfaces The emitted X-ray spectra (for a given composition) depend on: R, Teff, g = GM(1 + z)/R2 (observed spectra - D and NH) J.Lattimer, 2014

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The model

L =

• B

LB + LM + LNL + LL LB = ¯ ψB(γµiDµ − M⋆

B)ψB

M⋆

B = M − gBσσ − gBσ⋆σ⋆

Dµ = ∂µ + igBωωµ + igBφφµ + igBρIBρµ LNL = −1 3g3σ3 − 1 4g4σ4 +

• i,j,k

Cijkωi

µρj µφk µ

Constituents of the model baryons: B ∈ {n, p, Λ, Σ+, Σ0, Σ−, Ξ0, Ξ−} leptons: L ∈ {e−, µ−} mezons: M ∈ {σ, ωµ, ρa

µ}

∪{σ∗, ϕµ} Coupling constants vector meson-hyperon - SU(6) symmetry scalar meson-hyperon - hypernuclear potential in nuclear matter

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The Walecka-type models

Very ”stiff” form of the symmetry energy. To provide additional freedom in varying the density dependence of the symmetry energy the model is supplemented by the term: ΛV(gωω)2(gρρ)2 The density dependence of the symmetry energy Esym(nB) = k2

F

6

• (k2

F + M2 eff)

+ k3

F

12(m2

ρ/g2 ρ + 2ΛV(gωω)2)

for ΛV = 0 the symmetry energy varies linearly with the density. TM1 nonlinear (isovector sector) ΛV 0.014 0.015 0.016 0.0165 gρ 9.264 9.872 9.937 10.003 10.037 L (MeV) 108.58 77.52 75.81 74.16 73.36

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Density dependence of symmetry energy

50 100 150 200 250 0.2 0.4 0.6 0.8 1 1.2 1.4 Esym ( MeV ) nb ( fm-3 ) Λv = 0 Λv = 0.0165 Λv = 0.03 AV14+VII UV14+VII UV14+TNI

Calculations performed for different values of parameter ΛV and compared with the results obtained for the AV14+VII, UV14+VII and UV14+TNI models.(R.B.Wiringa,1988) The inclusion of ω − ρ coupling softens the symmetry energy.

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Modification of the symmetry energy for nuclear matter with hyperons.

16 18 20 22 24 26 28 30 32 34 36 0.4 0.5 0.6 0.7 0.8 0.9 Esym(δa=0.5) [MeV] nB [fm-3] EXT fS=0.4 WEAK fS=0.4 EXT fS=0.2 WEAK fS=0.2 fS=0 20 / 24

Modification of neutron star parameters

Equations of state Results obtained for non-strange and and strangeness-rich matter for different parameterizations.

100 200 300 400 500 600 700 800 900 1000 1100 1200 ε (MeV/fm

3)

50 100 150 200 250 300 P (MeV/fm

3)

TM1 Λv=0.014 TM1 Λv=0.015 TM1 Λv=0.016 TM1 Λv=0.0165 TM1 Λv=0.017 TM1 npeµ TM1 weak NL3 npeµ FSUGold

Mass-radius relations Neutron star matter with hyperons - the maximum mass range: 1.86 − 2.03M⊙

10 11 12 13 14 15 R ( km ) 0.5 1 1.5 2 M ( MΘ ) TM1 npeµ TM1 Λv =0.014 TM1 Λv =0.015 TM1 Λv =0.016 TM1 Λv =0.0165 TM1 Λv =0.017 FSUGold

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Modification of neutron star parameters

Neutron star matter with hyperons - equations of state Pressure and energy density dependence on strangeness fraction.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 400 800 1200 1600 2000 200 400 600 800 1000 1200 1400 P [MeV/fm3] EXT WEAK fS [-] ε [MeV/fm3] P [MeV/fm3]

Density dependence of strangeness fraction fs =

• i ni|si|

nB Hyperon supression at high density

0.1 0.2 0.3 0.4 0.5 0.6 0.7 1 2 3 4 5 6 7 fS u = nb / n0 fS, ΛV=0.014 fS, ΛV=0.0165 fS, TM1-weak

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Modification of the internal structure of a neutron star

Neutron star matter with hyperons Composition of the maximum mass configurations

0.001 0.01 0.1 1 1 2 3 4 5 6 7 8 9 10 11 12 Particle fraction Yi, TM1-weak r ( km ) n p+ e- µ- Λ Ξ-

0.01 0.1 1 2 4 6 8 10 Particle fraction Yi, Λv = 0.0165 R ( km ) e- Λ µ- n p+ Σ- Ξ-

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Conclusions

Different neutron star observables are sensitive to the density dependence of the symmetry energy - some of them depend on symmetry energy at relatively low density Hyperons affects the nuclear symmetry energy Modifies properties and structure of a neutron star

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