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Neutron  stars  as  laboratory  for  dense  ma1er  

Laura  Tolós  

Outline

Neutron star (I) •  first observations by the Chinese in 1054 A.D. and prediction by Landau after discovery of neutron by Chadwick in 1932

•  produced in core collapse supernova explosions

•  encompass not only “normal” stars but also “strange quark” stars

•  usually refer to compact objects with M≈1-2 M and R≈12 Km

•  extreme densities up to 5-10 times nuclear density ρ0 (ρ0=0.16 fm-3= 31014 g/cm3)

pulsar  (rota+ng  neutron  star)  in  middle  Crab  Nebula  

•   usually observed as pulsars

•   magnetic field : B ~ 10 8..16 G

•  electric field: E ~ 1018 V/cm

•  temperature: T ~ 10 6…11 K

•  shortest rotational period observed: P B1937+21 ~ 1.58 ms

•  accretion rates: 10 -10 to 10 -8 M/year

Neutron star (II)

Observational data: Masses

•  > 1700 pulsars known

•  best determined masses: Hulse-Taylor pulsar M=1.4414 ± 0.0002 M #Weisberg and Taylor, 2004

•  mass of PSR J0751+1807 corrected from M=2.1 ± 0.2 M#

to M=(1.14-1.40) M #Nice et al, 2008

Observational data: Cooling  

Lattimer and Prakash, Science ’ 04

Fe,  superfluidity    Fe,  no  superfluidity  

-­‐-­‐-­‐  H,  superfluidity  -­‐-­‐-­‐  H,  no  superfluidity  

models  with  dURCA  including  superfluidity  

Internal Structure and Composition  •  Atmosphere few tens of cm, ρ ≤104 g/cm3 made of atoms •  Outer crust or envelope few hundred m’s, ρ=104-41011 g/cm3

made of free e- and lattice of nuclei •  Inner crust 1-2 km, ρ=41011-1014 g/cm3 made of free e-, neutrons and neutron-rich atomic nuclei ρ0/2: uniform fluid of n,p,e-

•  Outer core ρ0/2-2ρ0 is a soup of n,e-,µ and possible neutron 3P2 superfluid or proton 1S0 superconductor •  Inner core 2-10 ρ0 with unknown interior made of hadronic, exotic or deconfined matter

Fridolin Weber

The Core of a Neutron Star

Prerequisites and definitions

•  Important quantities:

•  The composition of matter is determined by chemical and electrical equilibrium. The equilibrium conditions are governed by weak processes (β-equilibrium)

•  Matter is at temperatures much lower than typical Fermi energies

•  Typical values for nuclear matter (matter made of neutrons and protons in equal proportion)

E = B/A

� = B/V = Eρ

P = ρ2∂E/∂ρ = ρ∂�/∂ρ− �

µi = ∂�/∂ρi

xi = ρi/ρ

S(ρ) = E(ρ, xp = 0)− E(ρ, xp = 1/2)

K = 9∂P/∂ρ

(vs/c)2 = dP/d�

ρ0 = 0.16± 0.02 fm−3

E(ρ0) = −15.6± 0.2 MeV

K(ρ0) ≈ 220− 250 MeV

S(ρ0) = 32.5 MeV

q1 → q2 + L+ ν̄L

q2 + L → q1 + νL

Hadronic phase: bosons vs baryons  

Bosonic matter: Kaon condensation

Kaon condensation will occur if

but interaction of kaons with matter is more complicated due to many-body dynamics!!

µe−

Heiselberg & Hjorth-Jensen ’00  

mixed  phase  of  quark  maFer  

Hartree:  r0<<R  

Lenz:  r0>>R  

WS:  Finite  volume  

�K− = µe−Kaplan & Nelson ‘86

r0:  interpar+cle  spacing  R:  range  interac+on  

mK- ~ 500 MeV

but more exotic degrees of freedom are expected, in particular hyperons, due to the high value of density at the center and the rapid increase of the nucleon chemical potential with density

Baryonic matter: Nucleons vs Hyperons

Traditionally neutron stars were modeled by a uniform fluid of neutron rich nuclear matter in equilibrium with respect to weak interactions

Hyperons are expected at ρ~(2-3)ρ0

•  NN: well constrained up to k≈ 350 MeV/c •  NΛ: Λ-hypernuclei for A=3-209, UΛ(ρ0)= -30 MeV •  NΣ: 4ΣHe bound, Σ- atoms: UΣ(ρ0)= -30 or 30 MeV ? •  NΞ: 7 Ξ hypernuclear events and quasi-free production UΞ(ρ0)= -18/-28 MeV ? •  ΛΛ: 5 ΛΛ hypernuclear events, more attractive than ΛN ? •  YY: Y=Λ, Σ, Ξ unknown!

Experimental Data for Baryon-Baryon interaction

Theoretical models for hyperons in neutron stars

•  Relativistic mean field models Glendenning ’85; Knorren, Prakash & Ellis ’95; Schaffner & Mishustin ’96 •  Non-relativistic potential model Balberg & Gal ‘97 •  Quark-meson coupling model Pal et al ‘99 •  Chiral effective lagrangians Hanauske et al. ‘00 •  Density dependent hadron field model Hofmann, Keil & Lenske ‘01 •  DBHF/BHF approaches Brockmann & Machleidt ’90; Baldo, Burgio,Schulze ’00; Vidana et al. ’00; Jong and Lenske ’98 •  Low-momentum interactions Schwenk, Pethick, Hebeler, Friman, LT, Djapo..

Since Ambartsumyan & Saakyan ‘60

from RG approach with low momentum interactions

First study of EoS for neutron matter at finite temperature from low-momentum interactions, providing theoretical error bands for future analysis of error estimates in (M,R) predictions for neutron stars and properties of protoneutron stars

Bogner et al., ’05; LT, Friman and Schwenk ‘08

neutron  maFer    T  finite  

Only nucleons..  

Including hyperons..  within the Brueckner-Hartree-Fock approach

including  hyperons  

Hulse-­‐Taylor  pulsar  

The inclusion of hyperons induces a softening of the EoS and, therefore, a too low Mmax(< 1.44 M). Need of transition to “quark” matter?

Schulze, Polls, Ramos and Vidana ‘06

Superfluidity  

Cooper pairing of baryons close to the Fermi surface

BCS: Theory of Superconductivity Bardeen, Cooper and Schrieffer ‘57

neutron  star  maFer  

BCS gap equation

outer  core  

crust  

Lombardo and Schulze ’00: BHF with V14 and n,p,e,µ

Relevant for formation of glitches Sauls ‘89 and Pines, Tamagaki,Tsuruta ‘92 and cooling Tsuruta ‘98, Heiselberg and Hjorth-Jensen ‘00

glitches  

cooling    

Shear viscosity due to phonons in neutron stars Manuel, LT ‘11

Mean free path of phonons: establish when phonons become hydrodynamic

0.1 MeV < T < Tc = 1 MeV

< p >: thermal average

n : phonon density

lH=R  

Shear viscosity due to phonons (η) scales as η α 1/T5 and has to be compared to other processes such as viscosity due to electron-electron processes

Structure equations for neutron stars Since neutron stars have masses M ~1-2 M and radii R ~ 10-20 Km, the value of the gravitational potential on the neutron star surface is of order 1

with escape velocities of order c/2. Therefore, general relativistic effects become very important and we have to solve Einstein’s field equations, Gµν, with the energy-density tensor of the stellar matter, Tµν(ε,P(ε)):

For spherically symmetric non-rotating star, the Einstein’s equations reduce to the Tolman-Oppenheimer-Volkoff (TOV) equations:

Gµν = 8πTµν(�, P (�))

P (r = 0) = P (�c)

P (r = R) = 0

m(r = 0) = 0

m(r = R) = M

dP

dr= −Gm�

c2r2

�1 +

P

�

��1 +

4πr3P

c2m

��1− 2Gm

c2r

�−1

dm

dr=

4πr2�

c2

“Recipe” for neutron star structure calculation

•  Energy/particle

•  Chemical potentials

•  β equilibrium and charge neutrality

•  Composition and EoS

•  TOV equations

•  Structure of the neutron star

�(ρ, xe, xp, xΛ, ..);xi =ρi

ρ

µi =∂�

∂ρiµi = biµn − qiµe�

i

xiqi = 0

xi(ρ) ; P (ρ) = ρ2 d(�/ρ)dρ

(ρ, xi(ρ))

ρ(r), M(R), ..Shulze@Compstar07

dP

dr= −Gm

r2

(� + P )(1 + 4πr3P/m)1− 2Gm/r

dm

dr= 4πr2�

Mass-Radius Relation

Lattimer and Prakash, Science ’04

TOV  equa+ons  (non  rota+ng)  

•  solve TOV equations for a given EoS: P(ρ) or ε(ρ)=E/N(ρ)

•  some constraints: - Schwarzschild limit (GR):

R > 2 GM - causality limit for EoS:

R > 3 GM - spin rate from PSRB 1937+21 implies R< 15.5 Km for M=1.4 M

•  large variation of (M,R) due to uncertainties in symmetry energy Sv(ρ) in EoS with same constituents near and above ρ0:

E(ρ,x)=E(ρ,x=1/2)+Sv(ρ)(1-2x)2

x: proton fraction

•  need a simultaneous (M,R) measurement!!

dP

dr= −Gm

r2

(� + P )(1 + 4πr3P/m)1− 2Gm/r

dm

dr= 4πr2�

What about cooling?  

Lattimer and Prakash, Science ’ 04

Pethick ‘92, Yakovlev ‘04

Cooling curves (T∞ vs age)

•   Fast neutrino reactions: direct URCA process only in inner core and have density thresholds

•  Slow neutrino reactions: modified URCA process & NN bremsstrahlung everywhere in core, particularly in outer core (low-mass stars)

Neutrino emission processes:

suppressed due to superfluidity!!

N + p + e− → N + n + νe

N + n→ N + p + e− + ν̄e

Fe,  superfluidity    Fe,  no  superfluidity  

-­‐-­‐-­‐  H,  superfluidity  -­‐-­‐-­‐  H,  no  superfluidity  

models  with  dURCA  including  superfluidity  

n→ p + e− + ν̄e ; p + e− → n + νe

Y → (Y,N) + e− + ν̄e

N + N → νν̄

Back-to-Earth Lab: Heavy-ion collisions

S. Bass

Fuchs, Faessler, Zabrodin and Zheng ‘01

Kaons in nuclear collisions as a probe for EoS

•   K+ are a good probe to study dense and hot matter because they are created in a high density phase and not reabsorbed in the medium

•  Calculation: QMD transport model using soft/hard EoS for nuclear matter

•  KaoS data favours soft EoS

EoS from KaoS data and implications in neutron stars Sagert, LT, Chatterjee, Schaffner-Bielich, Sturm ‘11

Calculate the highest possible mass using the restriction that the hadronic EoS should fulfill the analysis of KaoS data for ncrit ≈ (2-3)n0 : compare nucleon potential (UN) for the different EoS to the Skyrme parametrization with K0< 200 MeV used for KaoS data

Conclusion: maximum mass limit of 3 M. To explain pulsar of 2.7 M*, need a prompt transition from soft EoS (KaoS data) to the stiffest EoS at 2.5 n0

*  Freire et al ’08

Danielewicz, Lacey and Lynch, Science ‘02

Flow of matter in nuclear collisions can help to constrain EoS

•   Flow: compression and expansion in nucleus-nucleus collisions

•  Flow and EoS are linked:

•  Data and flow calculations constrain EoS for symmetric matter : hard EoS with phase transition (quark matter) at ρ > 3ρ0

•  For neutron matter, need of symmetry energy Sv(ρ), which is uncertain!

te ≈ R/cs

cs = c�

∂P/∂�

rare isotope accelerators?

Summary

  Neutron stars are an excellent laboratory for testing matter under extreme conditions and a lot of effort has been invested in understanding their interior and the EoS of the different phases

  We have reviewed results for kaonic, nucleonic and hyperonic matter as well as superfluidity in neutron stars

  We have analyzed the consequences of those different phases for the mass-radius relation and the cooling of neutron stars

  We have commented on possible constraints from back-to-Earth experiments, such as heavy-ion collisions.