Exploring interior of neutron star through neutron star cooling

I. IntroductionII. Thermal evolution of neutron stars -Basic concepts of cooling curve of neutron starsIII. Neutrino luminosity as a probe of new form of matter inside neutron stars IV Observation of Cas A and nucleon superfluidityV Summary and concluding remarks

T. Tatsumi (Kyoto U.)

I. Introduction 3 14 30 0.16fm 2.8 10 g cm

crust core

Structure of neutron stars

M-R relation (Bulk properties of neutron stars)Cooling curve (Thermal evolution)

curveP P (Magnetic evolution)

EOS (Equation of state) gives

Can microphysics understand or explain these observables ?

・ There have been measured various observables about neutron stars , and great progress in observational technique.・ Unfortunately, most of phenomena occurs near the surface , and can provide us with indirect information about interiors of neutron stars, especially the core region, except their bulk properties.・ Among them neutron star cooling can give direct information of properties of matter at high densities through neutrino emission.

Magnetars　 (1985)　

Dipole radiationFastest pulsar: PSR 1987+21

=1.557 806 448 872 75 0.000 000 000 000 03 msP ±

J. Lattimer, arXiv:1305.3510

(P. B. Demorest, Nature 467, 1081, 2010)

J. Antoniadis et al. Science 340 (2013) 6131

1.97 0.04M

R.A. Hulse and J.H. Taylor, Ap J. !95(1975) L51.

(P. B. Demorest, Nature 467, 1081, 2010)

D.Page, arXiv:1206.5011Comparison with observation

Crab

Cas A

3C 58

Vela

1010 K 1MeV

2 810 (10 K) (0.01MeV)in eT T O O

Cas A

Young pulsars

II. Thermal evolution of neutron stars

(+H)

99 /10 (K)T T

(crust)

Cold catalyzed matter:　　・ chemical equilibrium　　・ charge neutrality

nFp

pFp

eFp

triangle condition:

Ex) n,p,e matter

Direct URCA (b decay) cycle is strongly suppressed in normal neutron star matter.

,e

e

n p e

p e n

e

n

N N

Modified URCA

( ),n p e

p e

n p e

n n

2 2( ) ( )

2 2

n peF FF

N N

p pp

m m

p eF Fp p

3 14 3

0

2/30

2/30

0.16fm 2.8 10 g cm

60( / ) MeV,

340( / ) MeV,

e pF F

nF

p p

p

( )O T

For free particles

' ,

'

e

e

N n N p e

N p e N n

2 810 (10 K) (0.01MeV)in eT T O O

dEthdt

CV

dT

dt L L

dT

dt

q0

cV 0

T 7 t t0 A1

T 6

1

T06

T t 1/6

CV 43 R

3 cV0 T

L 43 R

3 q0 T8

L 4R2 Te4 T 2 [1]

Neutrino Cooling era: Ln >> Lg

Photon Cooling era: L n << Lg

dT

dt T 1 t t0 B

1

T 1

T0

T t 1/

Basic Cooling: neutrino vs photon cooling eras

No superfluidMURCA

(slow cooling)

D.G. Yakovlev and C.J. Pethick, Ann. Rev. Astron. Astrophys. 42 (2004) 169.

3C58

Relaxation Neutrino cooling Photon cooling

8Q T 4Q T “Standard”

scenario

III. Neutrino luminosity as a 　　 probe of new form of matter inside neutron stars

Fast coolingExotic cooling

New form of matter

Standard cooling

Modified URCA+photon(+superfluidity) Slow cooling

for 3C58, Vela

New form of matter or Various phases inside neutron stars

Strange Quark Matter

Boson Condensate

Hyperonic Matter

Quark Matter

-30 0.16 fm

02-3

0

Kπ

ΣΛ

uds

ud

s

Inner cores of massive neutron stars:

Nucleons,hyperons

Pioncondensates

Kaoncondensates

Quarkmatter

e

e

nep

epn

e

e

nep

epn

~~

~~

e

e

qeq

eqq

~~

~~

e

e

deu

eud

scmerg

TQ 36

927103~

scmerg

TQ 36

9262410~

scmerg

TQ 36

9242310~

scmerg

TQ 36

9242310~

s

ergTL 6

94610~

s

ergTL 6

9444210~

s

ergTL 6

9424110~

serg

TL 69

424110~

Everywhere in neutron star cores. Most important in low-mass stars.

ModifiedUrca process

Brems-strahlung

e

e

NnNep

NepNn

NNNN

scmerg

TQ 38

9222010~

scmerg

TQ 38

9201810~

serg

TL 89

383610~

,,e

serg

TL 89

403810~

Fast cooling vs slow cooling

Exotic cooling – Impact of 3C58

3C58 is the remnant of a supernova observed in the year 1181 by Chinese and Japanese astronomers. A long look by Chandra shows that the central pulsar - a rapidly rotating neutron star formed in the supernova event - is surrounded by a bright torus of X-ray emission. An X-ray jet erupts in both directions from the center of the torus, and extends over a distance of a few light years. Further out, an intricate web of X-ray loops can be seen.

(NASA,2004)

3C58

CONCLUSIONSabout the

THEORY • EOS quite well determined

• The mass of the star has little impact

• The dominant neutrino emission process is from the formation and breaking of Cooper pairs from the neutron 3P2 gap (unless this gap is very small)

• Possibility of the presence of light elements in the envelope allows to accommodate a range of Te at a given age

S. Tsuruta et al., Ap.J. 571 (2002) L143.c

e

n

h’

h

6pionQ aT

K.G.Elshamouty et al.,arXiV:1306.3387

NASA

W.C.G. Ho et al, Nature 462 (2009) 71

IV. Observation of Cas A and nucleon superfluidity

/ several %/10years!T T

D.Page, arXiv:1206.5011

Cas A

3C58

Predictions for the NEUTRON 1S0 gap

Medium polarizationeffect O(1/3)

Important feature:

WE DO NOT REALLY KNOW WHAT IT IS

Medium polarization effects were expected to increase the 3P2 gap while they probably strongly suppress it.

32neutron gapP

D. Page et al., astro-ph/0508056D.G. Yakovlev and C.J. Pethick, Ann. Rev. Astron. Astrophys. 42 (2004) 169.

Cooling of compact stars and superfluidity

・ Enhancement of neutrino luminosity・ Suppression by the pairing

New form of matter

/Te

norma

*2normal

2

paired normal

pa

l /

//normae lir d

e.g. :

C (0) , (0)3

( / )

( ex) Durca process):

specific heat

luminosit

y

, pn

TV

FV

V

TT

V c

e e

m pN T N

C C M T T

n p e e

C e

n

L e e

p

L

Neutrino cooling era Photon cooling era

Note: , (modified URCA)

is suppressed by the factor,

exp( / ), for each Fermion

through the suppression of the phase space,

while receives no effec

/

t.

V

VC L T t

C L

T

L

Neutrino emission through the

formation and breaking of Cooper pairs (PBF)

Flowers, Ruderman & Sutherland, Ap. J. 205 (1976), 541

Voskresenskii & Senatorov, Zh. Eksp. Teor. Fiz. 90 (1986), 1505 [JETP 63 (1986), 885]

Voskresenskii & Senatorov, Yad. Fiz. 45 (1987), 657 [Sov. J. Nucl. Phys. 45 (1987), 411]

D.G. Yakovlev et al., A&A 343(1999) 650.

c.f.Quasiparticle recombination time (life-time) in a superconductor

Cooper pair Quasi-particles

0T

See also J.R. Schriefer and D.M. Ginsberg, PRL 8 (1962) 207.

Coope

2 /

r pair by the breaking probability of

the Boltzman factor

.pE Te

NN N N

5 5( ) (1 )2 l l

FW n V A n

GH c c

Neutral-current weak interaction

2 2

22 2

1 1(1 ), (1 ),

2 2 2

( ) ,2

p pp p p p

p p p

p p p

u v u vE E E

pE

m

2 43 3

12

3 331 2

10 20 10 2010

2 2

20

4(2 )( ) ( )

(2 )2

( ) ( )| | | |2 2

FV p p

p pn

Gc d pd p f E f E

d q d qq q E E q q

qM M

q

1 2p - p - q - q

Emissivity (singlet pairing case)

2 2 210 20| | 8( ), | | ( )n p p p pM q q M u v u v 1 2q q

† †( ) exp( ) ( ) exp( ) ( )n x i u i v p pp p p pp p

p x p x

Quasi-particle op.

(Flowers, Ruderman & Sutherland, Ap. J. 205 (1976), 541)ex) Singlet pairing

2

7

2

3

(density of states at

64(0)(

(

0)

)

,15 2

Fermi surfa e)

( )

/

cF

FV

nm k

Gc kT F

/ 1.7

( CS)

5

BCT

E.Flowers et al.,ApJ 205(1976)541.

Emissivity

89

21MUrca

79

22Coop

T 10

T )phase;/( 10

q

TcTFqc.f.

52

22 2

1( )

( 1)xy

xF y y dx

ex y

below Tc

23 2 22For pairing, 2V AVP c c c

Cooper cooling

Cooper Pair Neutrino Luminosities vs MURCA and Photons

Cas A is around here?

Cas A is around here?

D.Page, arXiv:1206.5011

Cas A

3C58

Comments about neutral current processes:

Bremsstrahlung:

(core), Nucleus Nucleus (crust).

Scattering opacity (Supernovae, protoneutron stars):

( ) ( ) , ( ) Nucleus ( ) Nucle

{ }

us

N N N N e e

N

N N NN

N

25

2 25 5

,

oc

( )

octet 3 3 2 3 8

GIM 8 0

tet GI

8 0

5

M

(1 8sin / 3 ) 1 4sin / 3

,2 2

12sin ,

3

1 1 1

3 3 3

(1 ) 1 4sin ,

,

W Wd

NC F

W

l l Wl

s

J

GJ j

J V A V V

J V V A

u u q q

J J

s

j l l

A

L

5(1 ) .s

octet octet

0 0 si

o

ngle

ctet

t( ' 5

' 5

)' ( )

' ( ) ,B V A B

A V B B

B J

B A V B

u u

u

C

u

B C

C

0.80,

0.47.

D

F

well known

unknown

Spin content of proton, especiallydue to ss sea

-(T.T., T. Takatsuka, R. Tamagaki,PTP 110 (2003) 179.)

~ 0.3 ~1.47

Ratio of the reaction rate

2

2 singlet

0lim ( ) AQ

Q C

V Summary and concluding remarks

・ Cooling of neutron stars has provided us with information of high-density matter through the neutrino emission mechanism.

・ Recent observation of Cas A may give information of nucleon superfluidity.

・ Can we catch an evidence about Quark Matter through cooling of “neutron stars”?

・ Simultaneous observation of surface temperature and other observables such as mass, radius … is desired to extract definite conclusions.

・ Surface temperature of some pulsars has already suggested 　 a fast cooling, which may need exotic cooling.

平均場近似と Bogoliubov-Vanatin 変換

† † †

†

†

† † † †

, ,

† † † † †

†

,

,

,

=

k

k

k k kl k k l l k k l l l l k kk k l

k k k k k kk k k k k kk k

k kl l l

kk k k

k kk k k

l

k

H C C V C C C C C C C C C C C C

C u

C C C C C C

v

C

C v u

C

B

C

V

V C

平均場近似：

変換

† † † †0

† † † †0

0BCS BCSk

BCS l lk k l ll k

BCS l lk k l ll k

C u v C C

C u v C C

Leptonic tensor:

'

' 5,

ex) quark decay (or direct

ˆ ˆ ˆtr ( ) ( )

URCA)

( ) ( ') ( ) ( ),

Hadronic tensor can be also written as

ˆ

(1 )

8

tr ( )

l l

e

d

l l

u

d p u p e q k

L q O k O O

q k g q k q k

H p O

i q k

,

2

, , ,

ˆ( ')

8 ' ' ' ' ,

Thus averaged sum gives

1 164( ' )( ) (Iwamoto,1980)

2 2

( decay, e scatt.,...)

d u

d u e

fi

p O

p p g p p p p i p p

M H L p q p k

ランダウ・リフシッツ相対論的量子力学

Vector current only:

2

00 10 208( )VL M q q 1 2q q

22 †00

2† † †

'

† * † † † † † † †' ' '

'

* † † *' ' '

'

* † †'

'

' '

( ) , '

n

k k

k k kk k k k

k k

H M BCS

BCS BC

v u

S

u v

v u v u

p p

k k k k k k k kk,k

k k k k, ,

p p

' ' '

† † † * *' ''

22 * *00 ' '

.

Thus

( )

,

k

p p p

n p p p p

pBCS BCS

H

v u v

M v u v u

u

k k k k

p p

† † †

, ,

† †0

2 2

1/22 2

2 2

with 1.

0

( )

1

2

11 ,

2

,

k k k kl k k l lk k l

BCS k k k

k kl l ll

k k k

lk kl

l l

kk

k k

BCS B

kk

k

CS k k

k

H C C V C C C C

u v C C

u v

V u v

E

VE

u v

N u

E

H v

エネルギーギャ

対ハミルトニア

ップ

準粒子エネルギー

ギャップ方程式

ン

11

2k

kE

BCS 理論ミニマム

クーパー対の凝縮状態

2 22 2 2 2

3 3(

(4

4)

'

3

0

2

0

) 3 3

'

with '.

To determine , , consider

4 / 2, ( ) /

'' '

4,

where

'' '

(

'

)

2

q q

q q

qq q

P p p

A B

I A B

q qI p p q q d qd

P IP I P P A B P I P

d qd qI p p q q

d

q

Ag P BP P

qP

P

P q

2 2 2 200 0

0 0 0

2 20

0 0

2

| | | |,

| | 2 | | 2 | |

1 1 ( | | ) 2

|

2

| | | | |

Finally,

6

P Pqdq P qq

P q q P P

PP P

I P g P P

P P

P P P

PP P P

Lenard integral [A.Lenard (1953), Landau & Lifshitz]

For other application, e.g. muon decay:

ee

3 32 31 2

10 2010 20

2

2 43 3

12

3 32 2 31 2

10 20 10 2010 20

Using the Lenard integral,

| | ( ) ( )2 2

4,

3

4(2 )( ) ( )

(2 )2

| | | | ( ) (2 2

p p

FV p p

n p p

d q d qM E E q q

q q

Gc d pd p f E f E

d q d qq q M M E E q q

q q

1 2

1 2

p - p - q - q

P

p - p - q - q

2 4

3 3 2 2'12

4(2 ) 4( ) ( ) ( ') | |

(2

) 32

)

FV p p p p n

Gc d pd p f E f E E E M

p p

2 4 22 2 2 2

12 2

2 4 22 2 2 2 2 2

0 012 2

2 4 22 2

12

4(2 ) 4 ( )2 ( )

(2 ) 32

4(2 ) 4 1 ( )2 2

(2 ) 3 42

4(2 ) 4 ( )2

(2 ) 3 22

FV F p p z z

p

FV F p p z z z

p

FV F p p

p

Gc p dkd f E E P dP dP d P P

E

Gc p dkd f E E dP P P P P

E

Gc p dkd f E E

E

2 2 2 22

4 ( ') 4 ( ')

, ' '

p p

F F

dk E k k E k k

k p p k p p

2 22 2 22 22

85

( ') ( ')4 4

2

5

F

F

zv

zvF F

F

x x x xd x z z

v v

z v

', ' , ,pF F

Ev k v kx x z

T T T

7T

Neutral current

2

:

1, ( (3))

:

4sin 1 0.08, C ( (3))

V A A

V W A A

n

C C SU g

p

C SU g

D.G. Yakovlev et al., A&A 343(1999) 650

(Non-rela.)

' , ' ,

2 ,

q q q

q

Fermi’s Golden rule:

2 200

2 2

8 ,

0 for singlet pairing

16 for triplet pairing

xx yy zz

I u v

I I I I

u v

p p

p p

2 2 , /z x y y T

II. Thermal evolution of neutron stars(I. Sagert et al., arXiv:0809.4225)

(T. Fischer, CSQCDII)