Transcript Essential of Ultra Strong Magnetic field and Activity For Magnetars Qiu-he Peng ([email protected]) (Dept.

Essential of Ultra Strong
Magnetic field and Activity
For Magnetars
Qiu-he Peng ([email protected])
(Dept. of Astronomy, Nanjing University, China)
< CSQCD II>, KIAA-PKU,Bejing, China, May 20-24, 2009
Observation
Ultra strong magnetic field
For AXPs (Anormalous X-ray Pulsars)
and SGRs (Soft Gamma Repeaters )
 Long spin period and spin-down rate
P ~ 5  12
s
P  (1011  1012 )

ss 1
Bp ,14  0.32 PP12
B ~ 1014  1015
 Absorption line at 10 keV
Gauss
Gauss
Activity of Magnetars
Abnormal high X-ray luminosity For AXP
Lx ~ (10  10 )ergs / sec
34
36
•Decaying pulsating tails of giant flares with energy 1044 ergs
It can not be explained by loss of the rotational energy.
It is guessed by transformed from the energy of the ultra strong
magnetic field of the magnetars.
Surface temperature (by observation):
T > 107 K
for AXP
T~105-106 K
(for usual pulsars)
Why??
Proposed Models for the ultra strong B
• Duncan & Thompson (1992, 1993): α-Ωdynamo with
initial spin period less than 3ms
• Ferrario & Wickrammasinghe(2005)suggest that the
extra-strong magnetic field of the magnetars descends
from their stellar progenitor with high magnetic field
core.
 
• Vink & Kuiper (2006) suggest that the magnetars
originate from rapid rotating proto-neutron stars.
• Iwazaki(2005)proposed the huge magnetic field of the
magnetars is some color ferromagnetism of quark
matter.
The question is still open!
Idea of mine:
1) Origin of Ultra strong B:
 Ferromagnetism of Anisotropic ( 3P2 )
neutron superfluid
2) Activity:
Instability due to high Fermi energy of
electrons under the ultra strong B
Part I
Origin of Ultra strong Magnetic Field
Is from Ferromagnetism
of Anisotropic ( 3P2 ) neutron superfluid
Structure of a neutron star
= (g/cm3)
10
1011
5×1014
Core
(1km)
1014
3P NSV
2
(anisotropic)
7
10
4
Inner
crust
1S NSV (isotropic)
0
Protons (5-8)% ( Type II
superconductor?)
(normal) electron Fermi gas
nuclei
with
Extraneutron
rich
Quarks ??
NSV: Neutron superfluid vortices
outer crust
(crystal of heavy metal)
Two kinds of Neutron Cooper pairs
A neutron has a spin :

1
s

2
Vector addition :
(Pauli matrix)
1 1
  0,1
2 2
Total spin of a Cooper pair of neutrons:
S 0

(1S0-Cooper pair of neutrons)
S  1 


3P -Cooper
2
 (Or )
(Projection along the external magnetic field)
pair of neutrons:
1S &3PF
0
2
Neutron superfluid
(BEC based superfluidity )
1S
0
neutron Cooper pair: S=0, isotropic
Energy gap :
△(1S0) ≥ 0,
1011 < ρ(g/cm3) < 1.4×1014
△(1S0)≥2MeV 7×1012 <ρ(g/cm3)< 5×1013
3P
2
neutron Cooper pair: S =1, anisotropic,
abnormal magnetic moment ~10-23 c.g.s.
Energy gap:
△ n(3P2) ～0.05MeV
(3.31014 <  (g/cm3) < 5.21014)
nuc  2.8 1014
g / cm3
Anisotropic 3P2 neutron superfluid
Critical temperature:
Tc (3 P2 (n))  max (3 P2 (n)) / 2k  2.78 108 K
 Magnetic moment:
2 n
n ~ 0.96610
23
erg / gauss
2 n
A magnetic moment tends to point at the direction of applied
magnetic field with lower energy due to the interaction of the
magnetic field with the magnetic moment of the 3P2 neutron Cooper
pair.
Energy distribution of neutrons in the presence of a magnetic field
Statistics
H  2n  B  2n, z B
• Hamiltonian:
•Ensemble average:
.
The Brillouin function
f ( x)  4 x / 3
f ( x)  1
x  1
x  1
 2n  2n f (
n B
kT
2sin h(2 x)
f ( x) 
1  2cos h(2 x)
( i.e.
(i.e.
n B
kT
n B
kT
 1)
 1)
)
Energy gap --- Combing energy
of a Cooper pair
A key idea: The energy gap, Δ, is a combining energy of couple of
of neutrons (the Cooper pair). It is a real energy, rather than the
variation of the Fermi energy due to the variation of neutron
number density.
   EF
Corresponding momentum of the combing energy of the neutron
Cooper pair is (in non-relativity)
p  2mn 
q Value
How many neutrons have been combined into the 3P2
Cooper pairs?
Since only particles in the vicinity of the Fermi surface
contribute (Lifshitz et al. 1999), there is a finite
probability q for two neutrons being combined into a
Cooper pair.
4 pF 2  p

q
3
 0.087
4 3
EF
pF
3
N ( P2  pair )  q  N A m( P2 ) / 2
3
3
 ( P2 )  N ( P2  pair )  2n 
3
3
Total induced magnetic field by the 3P2 superfluid
B (in )
2 (3 P2 (n)) 2m(3 P2 (n)) N A n


qf ( n B / kT )
3
3
R
R
or
Bmax
B
(in )
 Bmax f (n B / kT )
2n qN Am(3 P2 )
14


2.02

10

3
RNS
3
m( P2 ) 3

RNS ,6
0.1mSun
gauss
From paramagnetism to ferromagnetism
B(in)  Bmax f (n B / kT )
b  f ( x)
f ( x)  4 x / 3
B  B (in)  B ( 0)
x  1
n B  kT
(B < 1013 gauss , T > 107 K )
1.40 
x
(b  b(0 )
T7
Set
b(0)  0

B (in )
b
Bmax
b(0)
Tc  2 10 K
7
B(0)

Bmax
(Curea Temparature)
Paramagnetism is not important when T>Tc
Phase Transition From paramagnetism to ferromagnetism
 When T down to T→T and the induced magnetic is very
c
strong
Increase of magnetic field of NS
a) The induced magnetic field for the anisotropic
neutron superfluid increases with decreasing
temperature due to More and more neutron 3P2
Copper pairs transfer into paramagnetic states.
b) The region and then mass of anisotropic neutron
superfluid is increasing with decreasing
temperature
Energy gap of the 3P2 neutron pair (Elgagøy et al.1996, PRL, 77, 1428-1431)
The up limit of the magnetic field for magnetars
2n qN Am( P2 )
14

 2.02 10 
3
RNS
3
Bmax
3
m( P2 ) 3

RNS ,6
0.1mSun
Bmax  3 10
15
gauss
mmax ( NS )  2.5mSun
mmax ( P2 )  1.5mSun
3
gauss
Conclusion:
All assumptions with B > 31015 gauss are unrealistic.
Part II:
Instability and activity of magnetars
are Caused by
The increase of Fermi energy of electrons
under ultra strong magnetic field
Landau
quantization
n=6
n=5
n=4
pz
n=3 n=2
n=1
n=0
p
pz
p
The overwhelming majority of
neutrons congregate in the lowest
levels n=0 or n=1,
When
B  Bcr
The Landau column is a rather
long cylinder along the magnetic
filed, but it is very narrow.
The radius of its cross section is p .
Under the ultra strong magnetic field
The Landau energy level is quantized when B  Bcr
( Bcr =4.4141013 gauss)
2 e B
pz 2
E 2
(
) ( pz , B, n,  )  1  (
)  (2n  1   )
2
me c
me c
me c 2
pz 2
 1 (
)  (2n  1   )b
me c
b  B / Bcr
B (e) ~ 0.927 1020
2 e Bcr
1
2
me c
erg / gauss
n: quantum number of the Landau energy level
n=0, 1,2,3……
Landau column
pz
p
Total Occupied number of electrons(1)
We may calculate the total occupied number by two different ways:
1) Total electron number density:
Ne  N AYe 
Ye: electron fraction
The total number of state occupied by the electrons in a
complete degenerate electron gas should be equal to the
electron number due to the Pauli’s Exclusion Principle
Total Occupied number of electrons(2)
2) Method in Statistical physics:
State number in a volume element of phase space:
 Ntotal
2
Ne  3
h
nmax ( pz ,b , )
pF

1
 3 dxdydzdpx dp y dpz
h
dpz
0
me c 3
 2 (
)
h

n 0
EF / me c 2

0


1
1/ 2

((
p
/
m
c
)

[(2
n

1


)
b
]
) p dp
  e
pz nmax ( pz ,b , 1)
d(
){

me c
n 0

Landau quantization
nmax ( pz ,b , 1)

n0
p
p
p
  ( me c  2nb )( me c )d ( me c )
p
p
p
  ( me c  2(n  1)b )( me c )d ( me c )}
E 2  m2c4  pz2c2  p2 c2
p 2
(
)  (2n  1   )b
me c
b  B / Bcr
EF 2
pz 2
1
nmax ( pz , b,   1)  Int{ [(
) 1  (
) ]}
2
2b mec
mec
EF 2
pz 2
1
nmax ( pz , b,   1)  Int{ [(
) 1  (
) ]  1}
2
2b me c
mec
Cont.
me c 3
N e  2 (
)
h
EF / me c 2

0
nmax ( pz ,b , 1)
pz nmax ( pz ,b , 1)
d(
){
2nb 
2( n  1)b}


me c
n 0
n 0
nmax ( pz , b,   1)  nmax ( pz , b,   1)  nmax ( pz , b)
EF 2
pz 2
1
 [(
) 1 (
) ]
2
2b me c
me c
Result
me c 3
Ne  3  2  b (
)
h
3/ 2
N e  3  23/ 2  b1/ 2 (
EF / me c 2

1/ 2
me c 3
1
)  ( )3/ 2
h
2b
3 me c 3
Ne 
(
)
b h
0
n
0
EF / me c 2
EF / me c 2

3/ 2
max

[(
0
EF 2
pz 2 3/ 2 pz
)

1

(
) ] d(
)
2
me c
me c
me c
EF 2
2 3/ 2
[(
)  x ] dx
2
me c
(3 ) me c 3 1 EF 4

(
) (
)
2
16
h
b me c
2
pz
( pz , b)d (
)
me c
Fermi energy of electrons
in ultra strong magnetic field
(3 )2 me c 3 EF 4
Ne 
(
)(
)  N AYe 
2
16b h
mec
Ye 1/ 4  1/ 4
EF
1/ 4
 77.15  b (
) (
)
2
0.08
nuc
me c
EF  40  b1/ 4 MeV
b  B / Bcr
(b  1)
EF  40  b MeV
(b  1)
1/ 4
B (1014 Gauss)
b
EF (MeV)
1.0
2.415
49.86
3.0
7.246
65.63
5.0
12.077
74.57
10.0
24.155
88.68
15.0
36.232
98.14
20.0
48.309
105.45
Magnetars are unstable
due to the ultra high
Fermi energy of electrons
Basic idea
When the energy of the electron near the Fermi surface is
rather high (EF>60 MeV)
e  p  n  e

Energy of the resulting neutrons will be rather high and
they will react with the neutrons in the 3P2 Cooper pairs
and will destroy these 3P2 Cooper pairs .
n  (n , n )  n  n  n
It will cause the anisotropic superfluid disappear and then
the magnetic field induced by the 3P2 Cooper pairs will
also disappear.
Total thermal energy
Average energy of outgoing neutrons after
breakdown of the 3P2 Cooper pairs
1
 (n)  [ EF (e)  EF ( p)  (mn  m p )c 2  (3 P2 )]
3
It will be transfer into thermal energy.
Total thermal energy will be released after all 3P2
Cooper being breaked up
3
m( P2 )
E  qN A m( P2 )   (n)  3.35 10
0.1mSun
3
50
ergs
Life time of magnetar activity
3
m( P2 )
E  3.35  10
0.1mSun
50
ergs
X ray luminosity of AXPs
Lx  10  10 ergs / sec
34
It may be maintained ~ 107 -108 yr
36
Phase Oscillation
Afterwards,
n  p  e  e

Revive to the previous state just before formation of the 3P2 neutron
superfluid.
 Phase Oscillation .
Questions?
1. Detail process:
The rate of the process
e  p  n  e

Time scale ??
2. What is the evolution of the previous instable process
of the magnetars?
3. What is the real maximum magnetic field of the magnetars?
4. How long is the period of oscillation above?
5. How to compare with observational data
6. Estimating the appearance frequency of AXP and SGR ?
A further work of mine:
“Heating by 3P2 Neutron Superfluid Votexes and Glitch of
Pulsars Due to the Oscillation between A and B phase of
Anisotropic superfluid”.
(It Will be contributed to the conference on <Pulsars> next year)
Thanks