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)