1. Astrophysical Constraints on the Nuclear Equation of State
Evgeni Kolomeitsev
Matej Bel University, Banska Bystrica

2. Neutron Star Zoo
>1400 neutron stars in isolated rotation-powered pulsars ~ 30 millisecond pulsars
>100 neutron stars in accretion-powered X-ray binaries ~ 50 x-ray pulsar intense X-ray bursters (thermonuclear flashes)
short gamma-ray bursts neutron star -- neutron star, neutron star -- black-hole mergers
soft gamma-ray repeaters – magnetars (super-strong magnetic fields)

3. Measuring pulsar mass m2 m1
Pulsar mass can be measured only in binary systems
direction of observation m1 m2
Newton gravity Keplerian orbital parameters:
orbital period, semi-major axis length,excentricity, …
Orbital period P; projected semi-major axis x; eccentricity e; longitude and the time of periastron passage w,T0
Do not determine individual masses of stars and the orbital inclination.
Einstein gravity potentially measurable post-Keplerian parameters:
orbit precession, Shapiro delay, gravitational redshift, ….
Measuremnt of any 2 post-Keplerian parameters allows to determine the mass
of each star.

4. White dwarf -- neutron star binaries
Lattimer ARNPS62 (2012)
For 28 pulsars of this type the masses are measured

5. Measuring pulsar mass X-ray binaries 14 masses are measured Low mass
Lattimer ARNPS62 (2012)

6. Measuring pulsar mass Double neutron star binaries
1974 PSR B Hulse-Taylor pulsar
First precise test of Einstein gravitation theory
2003 J first double pulsar system
Pulsar A: P(A)=22.7 ms, M(A)=1.338 Msol
Pulsar B: P(B)=2.77 ms, M(A)=1.249±0.001 Msol
Orbiting period 2.5 hours
[Nature 426, 531 (2003), Science 303, 1153 (2004)]

7. Double neutron star binaries

8. Pulsar J1614-2230 Highest well-known mass of NS
Measured Shapiro delay with high precision
Time signal is getting delayed
when passing near massive object.
Highest well-known mass of NS
there are heavier, but far less precisely measured candidates)
P.Demorest et al., Nature 467, (2010)

9. Cross section of a neutron star
crust
Nuclei and electrons
Neutron-rich nuclei and electrons
Nuclei, electrons and neutrons
outer core
inner core
neutrons, protons,
electrons, muons
Exotics:
hyperons,
meson-condensates,
quark matter
~10 km

10. Crust Nucleus melting density growth Pasta structure
interplay of Coulomb energy
and surface tension
Negele & Vautherin (1973)
saturation density
Mcrust ~0.1Msol
Rcrust~102—103 m

11. relativistic corrections
Equilibrium condition for a shell in a non-rotating neutron star
Newton’s Law
INPUT: equation of state (EoS)
boundary conditions:
OUTPUT:
neutron star density profile, radius R and mass M
relativistic corrections

12. HHJ EoS
limitting mass
nlim=0.1n0
nlim~10-5 n0
stable
unstable
uncertainty in R~103 m

13. Ab inito calculations of the EoS starting from NN potential
auxiliary field diffusion Monte Carlo technique
[Gandolfi et al PRC 79, (2009)]
variational chain-summation methods (Urbana)
matters is superluminal
at higher densities
[Akmal, Pandharipande, Ravenhall PRC58(98)1804]
non-relativistic EoS!

14. Lagrangian
equations of motion

15. field Ansatz
RMF model with density dependent coupling see talk K. Petrik

16. strong dependence on m*N
(pure) Walecka model
U(s)=0
modified Walecka U(s)=as3+bs4
maximal mass of NS
PSR J
weak dependence on K !
strong dependence on m*N
Hardest EoS among RMF models

17. Relativistic Mean Field Models
Nuclear EoS:
Examples
Relativistic Mean Field Models
T. Gaitanos, M. Di Toro, S. Typel, V. Baran,
C. Fuchs, V. Greco, H.H. Wolter
NLr, NLrd
scalar-field dependent couplings
[Nucl. Phys. A 732, 24 (2004)]
KVR, KVOR
E.E. Kolomeitsev, D.N.Voskresensky
reduction of hadron masses in dense medium is included
[Nucl. Phys. A 759, 373 (2005)]
density dependent couplings
DD, D3C, DD-F
S. Typel
[Phys. Rev. C 71, (2005) ]
Dirac- Bruekner-Hartree-Fock
DBHF
E.N.E. van Dalen, C. Fuchs, A. Faessler

18. within empirical range similar isospin diffusion in HIC
atomic
nuclei
ab initio
Urbana
Argonne
small
large
within empirical range
similar
isospin diffusion in HIC
62 MeV < L < 107 MeV
B.A. Li, A.W. Steiner
nucl-th/
similar
surface in atomic nuclei
spin-orbit splitting
RMF Models
single nucleon spectra

21. - Boltzmann kinetic equation
- Mean-field potential
fitted to directed & elliptic flow
[Danielewicz, Lacey, Lynch, Science 298, 1592 (2002)]

23. mass, size, dynamics of SN explosion
integral quantity
equation of state of dense matter
phase transitions
changed in degrees of freedom
We want to learn about properties of microscopic excitations in dense matter
How to study response function of the NS?
How to look inside the NS?

24. Measuring pulsar temperature
Measuring pulsar age
for non-accreting systems
period increases with time
power-law spin-down
1) age of the associated SNR
2) pulsar speed and position w.r. to the
geometric center of the associated SNR
debris of supernova
explosion;
accreted “nuclear trash”
Crab : 1054 AD
Cassiopeia A: 1680 AD
Tycho’s SN: 1572 AD
3) historical events

25. Neutron star cooling data
Given:
EoS
Cooling scenario [neutrino production]
Mass of NS
Cooling curve

26. neutron star is transparent for neutrino
CV - specific heat, L - luminosity
emissivity
each leg on a Fermi surface / T
neutrino phase space ´ neutrino energy

27. standard: modified Urca
exotic: direct Urca
starts at some critical density, i.e. in stars with M>MDUcrit

28. EoS should produce a large DU threshold in NS matter !
DU process schould be „exotics“ (if DU starts it is dificult to stop it)
[Blaschke, Grigorian, Voskresensky A&A 424 (2004) 979]
EoS should produce a large DU threshold in NS matter !
[EEK, Voskresensky NPA759 (2005) 373]

29. no muons:
relativistic limit ( ):

30. The universal “DU curve”
“DU curve” for the symmetry energy
30MeV<J<35MeV
Low density expansion
Analysis of 36 RMF models gives
[Dong, et al PRC85, (2012)]

31. Nuclear matter equations of state can be constraint
at high densities using the neutron star phenomenology and
a flow data analysis of heavy-ion collisions
The EoS should be stiff enough to support massive neutron star
The threshold for direct Urca reactions should high