Transcript Слайд 1 - unict.it

THERMAL EVOLUION 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,
BASIC PROPERTIES OF NEUTRON STARS
Chandra
image of
the Vela
pulsar
wind nebula
NASA/PSU
Pavlov et al
M ~ 1.4M SUN ,
U ~ GM 2 / R ~ 5 1053 erg ~ 0.2 Mc 2
g ~ GM / R 2 ~ 2 1014 cm/s 2
Composed mostly
of closely packed
neutrons
  3M /(4 R 3 )  7 1014 g/cm3 ~ (2  3) 0
0  2.8 1014 g/cm3  standard density of nuclear matter
N b ~ M / mN ~ 1057 = the number of baryons
R ~ 10 km
OVERALL STRUCTURE OF A NEUTRON STAR
Four main layers:
1. Outer crust
2. Inner crust
3. Outer core
4. Inner core
The main mystery:
1. Composition of the core+
2. The pressure of dense
matter=
The problem of
equation of state (EOS)
Equation of State in Neutron Stars
1. Equation of state (EOS) determines the pressure of the matter, P.
2. The neutron star matter is so dense that P is almost independent
of the temperature T and is determined by the mass density  and
the composition of the matter; one usually writes P  P(  ).
3. The mass density is defined as   E / c 2 , where E [erg/cc] is the total
energy density (including rest-mass energies of particles) .
PHYSICAL FORMULATION OF THE COOLING PROBLEM
Heat diffusion
with neutrino
and photon
losses with
possible heat
sources
What information on NS parameters and
properties of dense matter can be
extracted from observations of thermal
radiation emergent from NS surface?
Mathematical Formulation of the Cooling Problem
Equations for building a model of a static spherically symmetric star:
{
(1)
(2)
(3)
(4)
dP
Gm
Hydrostatic equilibrium:
 2
dr
r
dm
HYDROSTATIC STRUCTURE
Mass growth:
 4 r 2
dr
Equation of state:
P  P(  )
dS
Thermal balance and transport:
Q
THERMAL EVOLUTION
dt
m  m(r )
Neutron stars: Hydrostatic equilibrium is decoupled from thermal evolution.
Effects of General Relativity:
For a neutron star :
rg
R
rg 
2GM
M

2
.
95
km
2
c
M Sun
~ 0.3  one cannot neglect General Relativity
Space-Time Metric
Variables: t , r ,  , 
ds2  c 2 dt 2 e 2  e 2 dr 2  r 2 d 2
Metric for a spherically
-symmetric static star
d 2  d 2  sin 2  d 2
(r ),  (r )  ?
r
(r )   (r )  0
1
Metric functions
Radial coordinate
In plane space
Radial coordinate r determines equatorial length
– «circumferential radius»
2
r

Periodic signal:
= pulsation frequency in point r
= frequency detected by a distant observer
    r e
Instead of

( r )
 (r ) it is convenient to introduce a new function m(r):
1
e 
2Gm
1 2
cr
4 r 2 dr
dV 
1  2Gm / c 2 r
2
= determines gravitational redshift of
signal frequency
m(r) = gravitational mass inside a
sphere with radial coordinate r
= proper volume element
HYDROSTATIC STRUCTURE
1
8G
Einstein equations
Rik  g ik R  4 Tik
2
c
Rik  Ricci curvature tensor; R  Rii  scalar curvature
Tik  ( P  E ) ui uk  P g ik  energy- momentum tensor
( E  c 2 , u i  4 - velocity, g ik  metric tensor)
Einstein equations for a star
{
dP
Gm 
P   4r 3 P   2Gm 
 1 2 
(1)
  2 1  2  1 
2 
dr
r  c  
mc 
rc 
dm
(2)
 4r 2 
dr
d
1 dP 
P 
1  2 
 2
dr
c dr  c 
(4) P  P(  )
(3)
1
1
TolmanOppenheimerVolkoff (1939)
Outside the Star
The stellar surface: r  R  circumferential star radius at P( R)  0.
Gravitational stellar mass: m( R)  M .
At r  R :
e 2  e 2   1  rg / r and one comes to the
Schwarzschild metric:
ds 2  c 2dt 2 (1  rg / r )  dr 2 / (1  rg / r )  r 2 ( d 2  sin 2  d 2 ).
Gravitational redshifts of signals from the surface:
   1  rg / R  R .
Non-relativistic Limit
(P  c2 ; r 3 P  mc2 ; Gm  rc 2 )
Gm
dP
 2 ;
r
dr
dm
 4 r 2  ;
dr
1 dP
d

 2
 c dr
dr
4 G 
1 d 2 d


r

2
2
c
dr
r dr
  (r )c 2  gravitational potential
Equations of Thermal Evolution
Thorne (1977)
1. Thermal balance equation:
+Qh
2. Thermal transport equation
Both equations have to be solved together to determine T(r) and L(r)
Boundary conditions and observables
At the surface (r=R) T=Ts
=local effective surface temperature
=redshifted effective surface temperature
=local photon luminosity
=redshifted photon luminosity
HEAT BLANKETING ENVELOPE AND INTERNAL REGION
To facilitate simulation one usually subdivides the problems artificially
into two parts by analyzing heat transport in the outer heat blanketing
envelope and in the interior.
The boundary: r  Rb ,   b ~109 1011 g/cc
(~100 m under the surface)
The interior: r  Rb ,   b
Exact solution of transport and balance equations
The blanketing envelope: Rb  r  R,   b
Is considered separately in the static plane-parallel approximation
which gives the relation between Ts and Tb
THE OVERALL STRUCTURE OF THE BLANKETING ENVELOPE
Z=0 T=TS
Non-degenerate layer
Radiative thermal conductivity
Radiative
surface
T=TF = onset of
electron degeneracy
Heat flux F
Degenerate layer
Electron thermal conductivity
Heat blanket
Atmosphere. Radiation transfer
T=Tb
  b
~ 109  1011 g cm3
TS=TS(Tb) ?
Nearly isothermal interior
z
For estimates:
Tb  108
TS46 / g S 14 K
affected by chemical composition
and magnetic fields
ISOTHERMAL INTERIOR AFTER INITIAL THERMAL RELAXATION
In t=10-100 years after the neutron star birth its interior becomes isothermal
Redshifted internal temperature becomes
independent of r
Then the equations of thermal evolution greatly simplify and
reduce to the equation of global thermal balance:
=redishifted total neutrino luminosity, heating power and heat capacity
of the star
CONCLUSIONS ON THE FORMULATION OF THE COOLING PROBLEM
• We deal with incorrect problem of mathematical physics
• The cooling depends on too many unknowns
• The main cooling regulators:
(a) Composition and equation of state of dense matter
(b) Neutrino emission
(c) Heat capacity
(d) Thermal conductivity
(e) Superfluidity
• The main problem:
Which physics of dense matter can be tested?
 Next lectures
REFERENCES
N. Glendenning. Compact Stars: Nuclear Physics, Particle Physics,
and General Relativity, New York: Springer, 2007.
P. Haensel, A.Y. Potekhin, and D.G. Yakovlev. Neutron Stars 1:
Equation of State and Structure, New York: Springer, 2007.
K.S. Thorne. The relativistic equations of stellar structure and
evolution, Astrophys. J. 212, 825, 1977.