THERMAL EVOLUTION OF NEUTRON STARS:
Theory and observations
D.G. Yakovlev
Ioffe Physical Technical Institute, St.-Petersburg, Russia
1. Formulation of the Cooling Problem
2. Superlfuidity and Heat Capacity
3. Neutrino Emission
4. Cooling Theory versus Observations
Catania, October 2012,
FERMI SYSTEMS
Electrons, muons, neutrons, protons, hyperons, quarks
= fermions = Fermi-Dirac statistics
n
2
(2 )3
 dp f ( ),
E
1
f 
exp((   ) / T )  1
Low temperatures: T
 Fermi-Dirac distribution
 , f ( )  1 at    ;
T 0
f
2V
dp  f ( ),
3 
(2 )
f
T
f ( )  0 at   

Thermal width of
Fermi level
T
1
Fermi sea
Fermi sea




FERMI SEA AND FERMI SURFACE
(Many fully occupied states;
Many “dead” energetic particles)

FERMI-SEA

Bulk properties of matter:
particle number density, energy density, pressure
n
pF p
pF3
3 2
For neutrons in a neutron star core at  =0
pZ
nn  n0  0.16 fm=1.6 1038 cm -3
n  60 MeV; pFn  340 MeV/c  1.68 fm 1
pF
P  6.5 1033 dyn/cm 2
pY
pX
3
 pF  (3 2 n)1/ 3  Fermi momentum
For protons and electrons:
n p  ne ~ 0.04 nn ; pFp  pFe ~ 100 MeV/c
e  n ;  p  a few MeV
n ~  2 / 3 ;  p ~  4 / 3
FERMI SEA AND FERMI SURFACE
FERMI-SURFACE
(Smaller amount of exchangable states
and “reacting” particles )
IMPORTANT FOR:
• Second-order thermodynamic properties:
entropy, heat capacity
• Kinetic properties:
thermal and electric conductivity
• Neutrino processes
• Generally, any reactions
• Superfluid processes
Superfluidity – neutron stars
Mechanism of superfluidity: Cooper pairing of
degenerate neutrons and/or protons due to nuclear attraction
Any superfluidity is defined by
critical temperature TC, that depends on density
Pairing type: singlet-state (1S0 )
or triplet state (3P2 )
Inner crust of neutron star:
Singlet-state pairing of free neutrons
Singlet-state pairing of nucleons in atomic nuclei
Neutron star core (typically):
Singlet-state pairing of protons
Triplet-state pairing of neutrons
SCHEME
Superfluidity – neutron stars
Low densities
Weak pairing
Medium densities
Strong pairing
High densities
Repulsion, no pairing
r ~  ~ / pF
pF  (3 2n)1/3
Superfluidity – microscopic manifestations
Creates gap (  , T )
in energy spectrum
near Fermi level

T=0
2
p
2m
Microscopic calculations
of the gap are very model
dependent
(nuclear interaction;
many-body effects)
Free Fermi gas
Superfluid
Fermi gas
Temperature dependence
0
Free Fermi gas T>Tc
  kT
Superfluid
Fermi gas T~Tc
  kT
Superfluid
Fermi gas T<<Tc
SUPERFLUIDITY IN NEUTRON STARS
0 ~ 1 MeV
Density dependence of the gap
Tc ~ 1010 K
After Lombardo & Schulze (2001)
A=Ainsworth, Wambach, Pines (1989)
S=Schulze et al. (1996)
W=Wambach, Ainsworth, Pines (1993)
C86=Chen et al. (1986)
C93=Chen et al. (1993)
Our task is to study
Tcn ( ), Tcp ( )
in neutron star core
Effects of superfluidity
Cooper pairing at T<Tc:
• Modifies heat capacity
• Suppresses ordinary neutrino
processes
• Creates a new process: neutrino
emission due to Cooper pairing
HEAT CAPACITY OF NEUTRON STAR CORES
Mixture of strongly degenerate fermions (the simplest version – n, p, e)
C  Cn  Cp  Ce
Cj 
m*j pF k B2T
3
3
~ n j kB
(per cm3)
kBT
j
In superfluid matter:
n0  0.16 fm3 = number density
Takes into account
superfluidity
of nucleons in saturated nuclear matter
Heat Capacity in Nucleon Neutron Star Cores
Non-superfluid core
Superfluid core
Summary on superfluidity and heat capacity
Neutrons, protons and other baryons in neutron star interiors
can be in superfluid state
Superfluidity is very model dependent (too many different
microscopic models)
Superfluidity is a Fermi surface phenomena which affects
thermodynamics and kinetics of neutron star matter
Superfluidity can strongly affect heat capacity of neutron stars
What are the effects of superfluidity on neutrino emission
and neutron star cooling?
 Next lecture
REFERENCES
U. Lombardo, H.-J. Schulze. Superfluidity in neutron star matter.
In: Physics of Neutron Star Interiors, edited by D. Blaschke,
N. Glendenning, A. Sedrakian, Berlin: Springer, 2001, p. 30.
D.G. Yakovlev, K.P. Levenfish, Yu.A. Shibanov. Cooling of neutron
stars and superfluidity in their cores. Physics – Uspekhi 42, 737, 1999.
D.G. Yakovlev, A.D. Kaminker, O.Y. Gnedin, P. Haensel.
Neutrino emission from neutron stars. Phys. Rep. 354, Nums. 1,2, 2001.