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