HEAT BLANKETING ENVELOPES OF HEAT BLANKETING ENVELOPES OF NNEUTRON STEUTRON STAARRSS

D.G. Yakovlev

Ioffe Physical Technical Institute, St.-Petersburg, Russia

Ladek Zdroj, February 2008,

• Outer crust• Density profile• Thermal structure• Main properties of heat blankets

OUTER CRUST

Composition: electrons + ions (nuclei)

Electrons (e): constitute a strongly degenerate,almost ideal gas, give the main contributioninto the pressure

Ions (A,Z): fully ionized by electron pressure, givethe main contribution into the density

Electron background

g/cc 10/ ;)/(009.1)/(

:parameter icrelativistElectron

)3( :momentum FermiElectron

:Pressure

:densty Mass

:neutrality Electric

66

3/16F

3/12F

AZcmpx

np

PP

nm

Znn

e

e

e

ii

ie

)g/cm 104( 311

Equation of state of degenerate electron gas

2 2 4 2 2 1/ 23

0

1/32

24 , ( ) , sinh

(2 )

( , ), 3

Fp

e

VE dp p m c p c p mc

NE V x E E xP x

V V x V mc V

))1(ln(13

2)1(

))1(ln()12()1(

2/1222/120

2/1222/120

xxxxxPP

xxxxxVPE

223

32

54

0 cm

dyn 10801.1

8

cm

P e Frenkel (1928)Stoner (1932)Chandrasekhar (1935)

LIMITING CASES

Non-relativistic electron gas

6 3

2 22 1 1/

1, 10 g/cm ( 10 meters under the surface)

3 1 5E , ~ , , 1.5

10 5 3nF F

e e e ee e

x

p pEm c n n P n n

V m m

Ultra-relativistic electron gas

6 3

1 1/

1, 10 g/cm ( 10 meters under the surface)

3 1 4E , ~ , , 3

4 4 3n

F e F e

x

Ep c n P p c n n

V

2 4 2 2 2 2

2

9 6

Electron chemical potential: 1

Electron degeneracy temperature: ( ) /

( 1.6 10 K at 10 g/cc)

e F e

F e B

m c c p m c x

T m c k

Equation of state of degenerate electron gas

Universal Density Profile in a Neutron Star Envelope

In a thin surface envelope

222222

222

222

22

/1/)( ,/1

/11

11ee , ,

dRdzdcds

RrrRzdtRrd

dRRr

dr

R

rdtcds

R

r

r

rMmRr

gg

g

g

gg

locally flat space

r

r=R z=0

z

13

2 2 2 2

2 3 2 2

2

4 2 1 1 1

, / ~ / 1

1 /

S S

g

dP G m P r P Gm

dr r c mc rc

P c P r mc P c

dP GMg g

dz R r R

surface gravity

14 2Sun1.4 , 10 km 2.43 10 cm/sSM M R g for a «canonical»

neutron star

1/32 2 6

2

22

00 14

2

, , /

/ , 1 , 1.009

При 0 0

1 1 49.3 m

(1 /

e e i i u e

FeS u e

e

e

e

u S S

P P dP n d n m m n A Z

p Zdg m A Z m c x x

dz m c A

z x m c

m c Zz Zx z

z m g A Ag

x z

3/ 2

2 3 60

0 0

) 1, 1.03 2Z z z

z xA z z

In the outer envelope:

depth-scale

the density profile in the envelope

Limiting cases:

360

2/360

~ g/cc 10 ,1 , )2(

~ g/cc 10 ,1 , )1(

zxzz

zxzz

Accumulated mass:

2

2 20

6 10 11 5Sun Sun

( ) ( )

1 / 4 1 /

( ~ 10 g/cc) ~ 10 , ( ~ 4 10 g/cc) ~ 10

z RS S

S S

rg g

g g M rdPg P r g dz dr R

dz R r R R r R

M M M M

14 214 /10 cm/sS Sg g

Density profile in the envelope of a canonical neutron star

THERMAL STRUCTURE OF HEAT BLANKETING ENVELOPES

=F=const

MAIN EQUATIONS

heat transport in a thin envelope without energy sources and sinks

= thermal conductivity (radiative+electron)

= opacity

S

dPg

dz hydrostatic equilibrium;

gS=const – surface gravity

(F)

(H)

Divide (F)/(H):

The basic equation to be solved(TP)

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) ?

NON-DEGENERATE RADIATIVE LAYER

Assume:

Kramers’s radiative thermal conductivity (free-free transitions):

66

3

2, 6.5

~ 1 = Gaunt factor

/(10 K)

in g/cm

effg

T T

Pressure of nondegenerate matter (P=nkBT):

Eq. (TP):

(K)

Integrate with

0 at =0:T

Insert Into (K):

/( ) ~T

Constant thermal conductivity along thermal path

14 214 /10 cm/sS Sg g

Analytic model

Linear growth of T with z zCM=z/(1 cm)

Densityprofile

( )T T

TEMPERATURE AND DENSITY PROFILES IN THE RADIATIVE LAYER

ONSET OF ELECTRON DEGENERACY2 /(2 )F F e BT T p m k

ELECTRON CONDUCTION LAYER Analytic model

Electron thermal conductivity of degenerate electrons (ei-scattering):

~ 1 = Coulomb logarithm

Equation (TP) assuming P=Pe (degenerate electrons):

Integrate within degenerate layer with at ( )d t d r rtT T x x

Temperature profile within degenerate layer

INTERNAL TEMPERATURE VERSUS SURFACE TEMPERATURE

Typically, and

T(z)const=Tb at z>>zd which is the temperature of isothermal interior

The main thermal insulation is provided by degenerate electrons!

TS-Tb RELATION FOR NEUTRON STARS

Our semi-analytic approach:

Exact numerical integration (Gudmundsson, Pethick and Epstein 1983)

For estimates:

8 46 1410 / Kb S ST T g

“DETECTOR OF LIE”

For iron heat blanketing envelopes (A=56, Z=26)

COMPUTER VERSUS ANALYTIC CALCULATIONS

log TS [K] = 5.9 or 6.5 (Potekhin and Ventura 2001) s = radiative surface solid lines – computer d = electron degeneracy dashed lines – analytics t = transition between radiative and electron conduction

THERMAL CONDUCTIVITY OF DEGENERATE ELECTRONS

MAIN PROPERTIES OF HEAT BLANKETING ENVELOPES

• Self-similarity (regulated by gS)

• Dependence on chemical composition (thermal conductivity becomes lower with increasing Z). Envelopes composed of light elements are more heat transparent (have higher TS for a given Tb)

• Dependence on surface magnetic fields (B-fields make thermal conductivity anisotropic). For a given Tb magnetic poles can be much hotter than the magnetic equator – non-uniform surface temperature distribution

• Finite thermal relaxation (heat propagation) times:

• Actual heat blanket is typically thinner than the “computer one” (density <1010 g/cc). When the star cools, the actual heat blanket becomes thinner (as well as degeneracy layer and the atmosphere)

• In very cold stars (TS<<104 K) the blanket disappears (TSTb)

6 10

6 8

For ~ 10 K and ~ 10 g/cc ~1 yr

For ~ 10 K and ~ 10 g/cc ~1 d

s b

s b

T

T

REFERENCES