cooling of neutron stars d.g. yakovlev ioffe physical technical institute, st.-petersburg, russia...
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COOLING OF NCOOLING OF NEUTRON STEUTRON STAARRSS
D.G. YakovlevIoffe Physical Technical Institute, St.-Petersburg, Russia
Ladek Zdroj, February 2008,
1. Formulation of the Cooling Problem 2. Superlfuidity and Heat Capacity 3. Neutrino Emission 4. Cooling Theory versus Observations
• Introduction• Physical formulation• Mathematical formulation• Conclusions
Cooling theory: Primitive and Complicated at once
BASIC PROPERTIES OF NEUTRON STARS
Chandraimage of the Velapulsarwind nebulaNASA/PSUPavlov et al
km 10~ ,4.1~ SUN RMM
2 53 2
2 14 2
3 14 30
14 30
57
~ / ~ 5 10 erg ~ 0.2
~ / ~ 2 10 cm/s
3 /(4 ) 7 10 g/cm ~ (2 3)
2.8 10 g/cm standard density of nuclear matter
~ / ~ 10 = the number of baryonsb N
U GM R Mc
g GM R
M R
N M m
Composed mostlyof closely packedneutrons
OVERALL STRUCTURE OF A NEUTRON STAR
Four main layers:1. Outer crust2. Inner crust3. Outer core4. Inner core
The main mystery:1. Composition of the core+2. The pressure of densematter=The problem ofequation of state (EOS)
Heat diffusion with neutrino and photon losses
PHYSICAL FORMULATION OF THE COOLING PROBLEM
Equation of State in Neutron Stars: Main Principles
).~ that (so lnd/ln d
index adiabatic by the described is EOS theof stiffness The 7.
.)/d/d( then );(
calculate and density number baryon theintroduce toconvenient isIt 6.
elements.matter of neutrality electric of condition theimposesusually One 5.
ns).interactio weak and Coulomb, strong, (involving
channelsreaction all orespect t with mequilibriu dynamic thermo
full toequivalent is which te,sta energy)-minimum (ground,
lowest itsin ismatter star neutron e that thassumedcommonly isIt 4.
. particles) of energies mass-rest (includingdensity energy
total theis [erg/cc] where,/ as defined isdensity mass The .3
).( ritesusually w one matter; theofn compositio the
and density mass by the determined is and re temperatu theof
t independenalmost is that dense so ismatter star neutron The 2.
. matter, theof pressure thedetermines (EOS) state ofEquation 1.
2
2
PP
nnEnPnEE
n
EcE
PP
T
P
P
bbbb
b
Mathematical Formulation of the Cooling Problem
Equations for building a model of a static spherically symmetric star:
2
2
(1) Hydrostatic equilibrium: ( )
(2) Mass growth: 4
(3) Equation of state: ( )
(4) Thermal balanc
dP Gmm m r
dr rdm
rdrP P
e and transport: dS
Qdt
{Neutron stars: Hydrostatic equilibrium is decoupled from thermal evolution.
Relativity Generalneglect cannot one 3.0~ :starneutron aFor
km 95.22
:Relativity General of EffectsSun
2
R
r
M
M
c
GMr
g
g
HYDROSTATIC STRUCTURE
THERMAL EVOLUTION
0)()(
?)( ),(
sin
ee 2222
22222222
rr
r
rr
ddd
drdrdtcds
Space-Time Metric
Metric for a spherically -symmetric static star
Metric functions
Radial coordinate
In plane space
2 2 2 2
const, const, / 2, 0 2
2
t r
ds dl r d l r
1
Radial coordinate r determines equatorial length – «circumferential radius»
2
0
0
0
e ,e
0 const, const, const,0
rdrldrdl
rrtr
Proper distance to the star’s center
Variables: , , , t r
2 sin = proper surface elementdS r d d
3 Periodic signal: dN cycles during dt
)( e
0
e ,e
r
dt
dNr
dt
dN
d
dNdtd
r
r
Pulsation frequency
in point r
Frequency detected by a distant observer
Determines gravitational redshift of signal frequency
Instead of it is convenient to introduce a new function m(r):
rcGm
2
2
21
1e
m(r) = gravitational mass inside a sphere with radial coordinate r
)(r
2
2
4
1 2 /
r drdV
Gm c r
= proper volume element
HYDROSTATIC STRUCTURE
tensor)metric velocity,-4 ,(
tensor momentum-energy )(
curvaturescalar tensor;curvature Ricci
8
2
1
2
4
iki
ikkiik
iiik
ikikik
gucE
gPuuEPT
RRR
Tc
GRgR
Einstein Equations for a Star
)( )4(
1 1
(3)
4 )2(
21
41 1 (1)
1
22
2
1
22
3
22
PP
c
P
dr
dP
cdr
d
rdr
dm
rc
Gm
mc
Pr
c
P
r
mG
dr
dP
{Tolman-Oppenheimer-Volkoff (1939)
Einstein Equations
Outside the Star
2 2
2 2 2
The stellar surface: circumferential star radius at ( ) 0.
Gravitational stellar mass: ( ) .
At : e e 1 / and one comes to the
Schwarzschild metric:
g
r R P R
m R M
r R r r
ds c dt
2 2 2 2 2(1 / ) / (1 / ) ( sin ).
Gravitational redshifts of signals from the surface:
( ) 1 / ( ).
g g
g
r r dr r r r d d
r R R
).( radiusapparent the /1/ radius The
energy. binding the 2.0~ :difference The
:Generally
mass.baryon sticcharacteri baryons; ofnumber total
, :star theof massbaryon theintroducesoften One
Sun
RRRrRR
MMMM
MM
mN
mNM
g
b
b
bb
bbb
Non-relativistic Limit
2
2
2
22 2
2
;
4 ;
1
1 4
( )
dP G m
dr rdm
rdrd dP
dr c dr
d d Gr
r dr dr c
r c
) ; ;( 2232 rcGmmcPrcP
Gravitational potential
1. Thermal balance equation:
2. Thermal transport equation
Equations of Thermal Evolution
+Qh
Both equations have to be solved together to determine T(r) and L(r)
Thorne (1977)
At the surface (r=R) T=Ts
Boundary conditions and observables
=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 interior: , b br R
The blanketing envelope: , b bR r R
9 11The boundary: , ~ 10 10 g/ccb br R
Exact solution of transport and balance equations
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
(~100 m under the surface)
Degenerate layerElectron thermal conductivity
Non-degenerate layerRadiative thermal conductivity
Atmosphere. Radiation transfer
THE OVERALL STRUCTURE OF THE BLANKETING ENVELOPE
Nearly isothermal interior
Radiativesurface
T=TF = onset of electron degeneracy
9 11 3~ 10 10 g cm
b
H
ea
t b
lan
ke
t
z
Z=0
Hea
t fl
ux
F
T=TS
T=Tb
TS=TS(Tb) ?
SEMINAR 1
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
2
2
4
1 2 /
r drdV
Gm c r
= 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
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.
REFERENCES