Transcript Слайд 1 - University of Wrocław

COOLING OF NEUTRON STARS
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
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Introduction
Physical formulation
Mathematical formulation
Conclusions
Ladek Zdroj, February 2008,
Cooling theory: Primitive and Complicated at once
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)
PHYSICAL FORMULATION OF THE COOLING PROBLEM
Heat diffusion with neutrino
and photon losses
Equation of State in Neutron Stars: Main Principles
1. Equation of stat e (EOS) determines the pressure of thematter, P.
2. T he neutron star matter is so dense that P is almostindependent
of thetemperatu
re T and is determined by themass density  and
thecomposit ion of thematter;one usually writes P  P(  ).
3. T he mass density is defined as   E / c 2 , where E [erg/cc] is thetot al
energy density (including rest - mass energies of particles).
4. It is commonlyassumed that th
e neutron star matter is in its lowest
(ground, minimum- energy) sta te, which is equivalent to full
thermodynamic equilibrium with respect ot all reactionchannels
(involvingstrong,Coulomb, and weak interactions).
5. One usually imposes the condition of electric neutralit y of matterelements.
6. It is convenient tointroduce the baryon number density nb and calculate
E  E (nb ); then P  nb2 d( E / nb )/dnb .
7. T hestiffness of the EOS is described by the adiabat ic index
  d ln P / d ln  (so that P ~   ).
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
t  const,
1
Metric functions
Radial coordinate
In plane space
r  const,
   / 2,
 ds 2  dl 2  r 2 d 2
0    2
 l  2 r
Radial coordinate r determines equatorial length
– «circumferential radius»
dS  r 2 sin  d d = proper surface element
2
t  const,   const,

dl  e dr,
  const,
0  r  r0
r0
l   dr e   r0
0
Proper distance to the star’s center
3
Periodic signal: dN cycles during dt
dN
dN
Pulsation frequency
 e 
in point r
d
dt
dN
Frequency detected by a
r     0  
distant observer
dt
Determines gravitational redshift of
    r e   (r )
d  dt e  ,
r 
signal frequency
Instead of
 (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
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 2 dt 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).
One often introduces the baryon mass of the star : M b  N b mb ,
N b  total number of baryons; mb  characteristic baryon mass.
Generally: M b  M
T he difference: M  M b  M ~ 0.2 M Sun  thebinding energy.
T heradius R  R / 1  rg / R  the apparent radius ( 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
Requirements:
• Should be thin
• No large sources of energy generation and sink
• Should serve as a good thermal insulator
• Should have short thermal relaxation time
THE OVERALL STRUCTURE OF THE BLANKETING ENVELOPE
 SEMINAR 1
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
Nearly isothermal interior
TS=TS(Tb) ?
z
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
dV 
4 r 2 dr
1  2Gm / c r
2
= proper volume element
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 mechanisms
(c) Heat capacity
(d) Thermal conductivity
(e) Superfluidity
• The main problems:
(a) Which physics of dense matter can be tested?
(b) In which layers of neutron stars?
(c) Which neutron star parameters can be determined?
 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.