THERMAL EVOLUION OF NTHERMAL EVOLUION OF NEUTRON STEUTRON STAARRS:S:Theory and observationsTheory and observations

D.G. Yakovlev

Ioffe Physical Technical Institute, St.-Petersburg, Russia

Catania, October 2012,

1. Formulation of the Cooling Problem 2. Superlfuidity and Heat Capacity 3. Neutrino Emission 4. Cooling Theory versus Observations

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)

Equation of State in Neutron Stars

1. Equation of state (EOS) determines the pressure of the matter, .

2. The neutron star matter is so dense that is almost independent

of the temperature and is determined

P

P

T

2

by the mass density and

the composition of the matter; one usually writes ( ).

3. The mass density is defined as / , where [erg/cc] is the total

energy densit

P P

E c E

y (including rest-mass energies of particles) .

What information on NS parameters and properties of dense matter can be extracted from observations of thermal radiation emergent from NS surface?

Heat diffusion with neutrino and photon losses withpossible heat sources

PHYSICAL FORMULATION OF THE COOLING PROBLEM

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

1Radial coordinate r determines equatorial length – «circumferential radius»

Variables: , , , t r

2 Periodic signal:

e ( )

r

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 r dr r r r d d

r R

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

(~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) ?

For estimates: 8 46 1410 / Kb S ST T g 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

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