Download Neutron stars: the densest state of condensed matter

Neutron stars: the densest state of
condensed matter
Nicolas Chamel
Institut d’Astronomie et d’Astrophysique
Université Libre de Bruxelles, Belgique
Toulouse, 15 April 2011
What is a neutron star?
Astrophysicist’s answer: a neutron star is the compact
remnant of a type II supernova explosion.
Neutron stars are tiny stars
Neutron stars are the smallest and
densest stars in the Universe:
M ∼ 1 − 2M⊙
R ∼ 10 km
⇒ ρ̄ ∼ 1014 − 1015 g.cm−3
RCW 103 (from ESA)
What is a neutron star?
Nuclear physicist’s answer: a neutron star is a giant nucleus
containing A ∼ 1057 nucleons (90% being neutrons).
Neutron stars as nuclear
liquid drops
Treating a neutron star as a
drop of incompressible nuclear
matter at saturation density
ρ0 ≃ 2.8 × 1014 g.cm−3
M ∼ Am ∼ 1 − 2M⊙
R ∼ r0 A1/3 ∼ 10 km
What is a neutron star?
Condensed-matter physicist’s answer: a neutron star is the
largest quantum system in the Universe! The interior of neutron
stars is cold and strongly degenerate.
Condensed-matter issues
structure and composition of
dense matter
phase transitions (crystallization,
frustration)
superfluidity, superconductivity
strong magnetic fields
Cassiopeia A (from NASA)
Internal constitution of neutron stars
From the surface of a neutron star to its center the density
spans more than 14 orders of magnitude!
Chamel&Haensel, Living Reviews in Relativity 11 (2008), 10
http://relativity.livingreviews.org/Articles/lrr-2008-10/
Outline
Historical introduction to neutron stars
Neutron star magnetic fields
Superfluidity and superconductivity in neutron stars
Brief history of neutron stars
Compact stars in the 1930’s
The history of neutron stars began in
the 1930s. At this time a few white
dwarfs were known. In 1915, Walter
Adams spectral observations of Sirius
B had revealed that white dwarfs are
very compact stars.
Sirius B
In the 1920s, Arthur Eddington
developed the theory of stellar
structures. Later astrophysicists
started to speculate about the
ultimate fate of stars using the newly
developed quantum theory.
Sir Arthur Eddington
Degenerate matter in white dwarfs
The central density in white dwarfs was
found to be much higher than that in
ordinary matter. At such densities atoms
are fully ionized and all electrons are free.
However electrons are fermions and due to
the Pauli exclusion principle (1925), they
cannot occupy the same quantum state.
Only four months after Dirac published his
paper about the statistics of fermions, R.H.
Fowler realized that this is the electron
degeneracy pressure which resists the
gravitational collapse.
MNRAS 87, 114 (1926).
First ideas about neutron stars
In February-March 1931, Landau, Bohr and
Rosenfeld discussed the possible existence
of compact stars as dense as atomic
nuclei. Landau published a paper in
January 1932.
In February 1932, the neutron (which was
predicted by Rutherford in 1920) was
discovered by James Chadwick. He was
awarded the Nobel prize in 1935.
Talk of D. G. Yakovlev from Ioffe Institute in St Perterburg
http://www.ift.uni.wroc.pl/~karp44/talks/yakovlev.pdf
Baade and Zwicky prediction
In December 1933, during a meeting of the American Physical
Society at Stanford, Baade and Zwicky predicted the existence
of neutron stars as supernova remnants
William Baade and Fritz Zwicky
Phys. Rev. 45 (1934), 138
Neutron core
In 1937, Gamow and Landau proposed independently that a
possible stellar energy source could be the accretion of
matter onto a dense neutron core.
picture from K.S. Thorne
George Gamow and Lev Landau
But very soon it was shown that stars are powered by
thermonuclear reactions (as suggested in the 20s by
Eddington and others). The interest in neutron stars declined.
Connection between white dwarfs and neutron stars
Baade and Zwicky were apparently unaware of the work about
the maximum mass of white dwarfs. This is Gamow who first
made the connection in 1939 (Phys. Rev.55, 718). At a
conference in Paris in 1939, Chandrasekhar also pointed out
“If the degenerate core attain sufficiently high
densities, the protons and electrons will combine to
form neutrons. This would cause a sudden diminution
of pressure resulting in the collapse of the star to a
neutron core.”
A neutron star should thus have a mass close to the
Chandrasekhar limit, i.e. M ∼ 1.4M⊙ .
Compactness criterion
The compactness of an object of mass M
and radius R can be estimated from the
dimensionless parameter
Ξ≡
Earth
Sun
White dwarf
Neutron star
stellar black hole
2GM
Rc 2
10−10
10−6
−4
10 − 10−3
∼ 0.2 − 0.4
1
PSR J1846-0258 (Chandra)
⇒ Neutron stars have to be described using Einstein’s theory
of General Relativity
Global structure of neutron stars
In 1939, Richard Tolman, Robert
Oppenheimer and his student George
Volkoff solved the equations describing
static spherical stars in General Relativity
Richard Tolman
They found Mmax ≃ 0.7M⊙ by
considering a non-interacting gas of
degenerate neutrons. Since this is
smaller than the maximum mass of
supernova cores, they concluded that
neutron stars could not exist.
Robert Oppenheimer and
George Volkoff
Theoretical developments after the Second World War
The first realistic equation of state of dense matter was
constructed in the 50s by John Wheeler and his collaborators
(in 1939 he elaborated a liquid drop model of fission with Bohr).
fission of a liquid drop
Wheeler, Ann. Rev.Astr.Astrophys. 4
(1966), 393.
It was later found that Mmax ∼ 1 − 2M⊙ : neutron stars could thus
be formed as proposed by Baade&Zwicky. Born in supernova
explosions, neutron stars were expected to be "hot".
X-ray observations
In the 60s, theoretical efforts focused on modeling the cooling
of neutron stars motivated by the hope of detecting their
thermal emission.
Cooling calculations predicted T ∼ 106 K for
neutron stars ∼ 103 year old (x-ray
emission).
e.g. Chiu and Salpeter, PRL12(1964),413.
X-ray observations in space started in the
60’s with pioneer experiments by Riccardo
Giacconi (Nobel 2002).
Several sources were discovered but their
nature remain elusive.
Giacconi with Uhuru
satellite, 1970
Compact X-ray sources
In 1967, Iosif Shklovsky correctly proposed that Scorpius X-1
(found in 1962) is a neutron star accreting matter from a
normal star. But its work attracted little attention among
astrophysicists.
Iosif Shklovsky
By 1968, about 20 compact x-ray sources were known.
Early observations of the Crab nebula and first
speculations
Already in 1942, Baade and Minkowski found that the central
region of the Crab nebula (supernova remnant) contains an
unusual star. Later a strong linearly polarized radio
emission was detected.
In 1953, Shklovsky interpreted this as
being due to synchrotron radiation by
relativistic electrons spiraling along a
strong magnetic field.
From Carroll and Ostlie
Subsequent theoretical efforts were focused on understanding
the origin of the energy powering the Crab nebula.
Search for a neutron star in the Crab nebula
The Crab nebula was observed during a lunar occultation on 7
July 1964. The size of the x-ray source was estimated as 1
light-year∼ 1013 km (size of the nebula 11 ly). This was much
larger than the typical size of a neutron star (10-20 km).
In 1965 Anthony Hewish and his student
found a scintillating radio source and
speculated that it "might be the remains of
the original star which had exploded".
John Wheeler in 1966 and Franco Pacini in
1967 proposed that a rapidly rotating
neutron star with a strong dipole
magnetic field could power the Crab nebula
and could explain Hewish observations.
Fortuituous discovery of pulsars
In 1965, Jocelyn Bell started a PhD
under the supervision of Anthony
Hewish at the Cavendish Laboratory
in Cambridge. Her research was
about scintillation of radio sources.
They constructed a 3.7m
radiotelescope with a very good
temporal resolution. The telescope
(which consisted of an array of 2048
dipole antenna) was completed in
July 1967.
Jocelyn Bell in 1966
Fortuituous discovery of pulsars
In August 1967, Jocelyn Bell discovered a pulsating radio
source with a period of about 1 second.
The source was later found to be extremely
regular. In December its period was
accurately measured : 1.3373012 seconds.
By Februray 1968 when the results were
published, three other sources had been
found.
These new pulsating stars were dubbed "pulsars" by a
journalist of the Daily Telegraph. Anthony Hewish was awarded
the Nobel Prize in 1974.
Unmasking pulsars
Pulsars are magnetized rotating neutron stars emitting a
highly focused beam of electromagnetic radiation oriented long
the magnetic axis. The misalignment between the magnetic
axis and the spin axis leads to a lighthouse effect.
Crab pulsar
A pulsar (PSR B0531+21) was
eventually found in the Crab nebula in
1968 by astronomers of Green Bank
observatory. Its period is only 33
milliseconds!
Vela pulsar
In the same year,
astronomers from the
Sydney University
discovered another
pulsar in a supernova
remnant with a period of
89 ms : the Vela pulsar
(PSR B0833-45).
The discovery of the Crab and Vela pulsars definitevely
established the nature of pulsars and confirmed the predictions
of Baade and Zwicky 35 years earlier that neutron stars are
the compact remnants of supernova explosions.
Pulsar properties
Since 1967, ∼ 2000 pulsars have been discovered.
http://www.atnf.csiro.au/research/pulsar/psrcat/
Their period P ranges from 1.396 ms for PSR
J1748−2446ad up to 8.5 s for PSR J2144−3933.
Their period increases gradually with time at a rate given
by
10−20 . Ṗ ≡
dP
. 10−12
dt
Note that for the best atomic clocks Ṗ & 10−16 (this
corresponds to a delay of 1 second every 300 millions
years).
Each pulsar has a specific pulse profile (with different
polarization angles).
Pulsar glitches
Sometimes P may suddenly decrease. The variations are tiny
but observable.
∆Ω
∼ 10−9 − 10−5
Ω
Example of a glitch in the Crab pulsar
Pulsar glitches
Sometimes P may suddenly decrease. The variations are tiny
but observable.
∆Ω
∼ 10−9 − 10−5
Ω
Example of a glitch in the Crab pulsar
⇒ Small pulsar glitches are interpreted as starquakes thus
providing a direct proof for the existence of a solid crust
Pulsar glitches
Sometimes P may suddenly decrease. The variations are tiny
but observable.
∆Ω
∼ 10−9 − 10−5
Ω
Example of a glitch in the Crab pulsar
⇒ Small pulsar glitches are interpreted as starquakes thus
providing a direct proof for the existence of a solid crust
⇒ Large glitches and the observed long relaxation times
bring strong evidence of superfluidity inside neutron stars
Schematic pulsar emission mechanism
From Maxwell-Faraday’s law of induction, the rotating magnetic
field induces a huge electric field at the surface of the star.
Charged particles are ejected and
accelerated to relativistic speeds
forming a magnetosphere. Electrons
emit very energetic γ rays which are
converted into electron-positron pairs,
producing more radiations and
leading to a cascade of
pair-production.
Goldreich & Julian, ApJ157(1969),869.
From Carroll and Ostlie
When particles reach the light cylinder Rc = cP/2π, they leave
the star producing a pulsar wind. This can be seen in x-rays.
X-ray observations of the Crab pulsar
X-ray observations can reveal new phenomena that cannot be
seen in radio or in optical range
Neutron star magnetic fields
Basic pulsar model
The standard model consists of a rotating neutron star with a
strong dipole magnetic field in vacuum
The loss of energy due to
electromagnetic dipole radiation (in
cgs units) is given by
Ė ≡
From Carroll and Ostlie
dE
8π 4 B 2 R 6 sin2 θ
=−
≤0
dt
3c 3 P 4
B is the field strength at the magnetic
pole, R is the radius of the star and P
its spin period.
Pulsar magnetic field and characteristic age
As the neutron star spins down, it loses kinetic energy at a rate
given by Ėkin = IΩΩ̇. Assuming that this is entirely due to
magnetic dipole radiation, we can infer the magnetic field
strength
p
6c 3 IP Ṗ
8π 2 B 2 R 6 sin2 θ
⇒
B
=
P Ṗ =
3c 3 I
2πR 3 sin θ
If the initial period at birth is infinitively short and that B and I
remain constant, we can further obtain the characteristic age
τ of the pulsar
Z
P
PdP = P Ṗ
0
Z
τ
dt ⇒ τ =
0
P
2Ṗ
Pulsar ages
For the Crab pulsar, τ ≃ 1.2 × 103 years, in good agreement
with the known age of the supernova (1054 AD).
The validity of the rotating magnetic dipole model can be better
tested by measuring higher order time derivatives of the pulsar
angular frequency Ω.
The dipole model predicts n = 3.
Braking index
Ω̈Ω
n≡−
Ω̇2
PSR B1509−58
Crab
Vela
2.84
2.5
1.4
Other spin-down mechanisms
magnetospheric processes, pulsar wind, magnetic field decay,
gravitational radiation, fallback disk, neutrino emission,
electromagnetic radiation from superfluid neutron vortices,
interstellar medium, etc.
Pulsar magnetic fields
Most pulsars have a magnetic field B ∼ 1012 G = 108 T.
Seiradakis and Wielebinski, Astron.Astrophys.Rev.12(2004), 239.
Discovery of millisecond pulsars
The first millisecond pulsar was found in 1982 at Arecibo by
Backer’s team. Today ∼ 200 millisecond pulsars are known.
The fastest one PSR J1748−2446ad discovered in 2005, has a
period of only 1.396 ms!
Millisecond pulsars are
characterised by
1.4 ≤ P ≤ 30 ms
Ṗ ≤ 10−19
Most of them belong to a binary
system and are found in
globular clusters.
Globular cluster Terzan 5 (ESO)
Millisecond pulsars in the P − Ṗ diagram
In the standard scenario of neutron star formation from
supernova explosions, the fastest pulsars are the youngest.
Millisecond pulsars (open
circles) are thought to be old
recycled pulsars that have
been spun up by accretion from
a companion star. This is
supported by the fact that the
fastest ones are not associated
with any supernova remnant.
Cyclotron emission line in Hercules X-1
The discovery of cyclotron line emission in the spectrum of the
X-ray pulsar (accreting neutron star) in the binary Hercules X-1
provided the first direct measurement of neutron star
magnetic fields.
The spectral feature can be identified to
relativistic electrons making a transition from
the first excited Landau level E1 to the
ground state E0
p
EN = me c 2 ( 1 + 2(B/Bc )N − 1)
Bc ≡ me2 c 3 /e~ ≃ 4.4 × 109 T
⇒ B ≃ 5.3 × 108 G
Trümper, ApJ 219(1978), L105 .
Cyclotron lines in other neutron stars
Cyclotron lines have been detected in ∼ 15 x-ray pulsars
⇒ B ∼ 1 − 5 × 108 T.
Makishima et al. ApJ 525(1999), 978.
Coburn et al. ApJ 580(2002), 394.
Cyclotron harmonics have been reported in a few of them:
three in 4U 0115+63
two harmonics in X0331+63
one harmonic in Hercules X-1, Vela X-1, 4U1907+09,
A0535+26.
Enoto et al., PASP 60 (2008), S57.
Evidence for a cyclotron line in the isolated neutron star
CCO 1E 1207.4−5209 ⇒ B ∼ 8 × 106 T.
Bignami et al., Nature 423 (2003), 725.
Suleimanov et al., ApJ 714 (2010), 630.
The intriguing case of RX J1856.5-3754
RX J185635−3754 is one of the nearest isolated neutron stars
(400 ly). Its age is estimated to be . 106 years.
Walter and Lattimer, Astr. J.576 (2002), L145.
The intriguing case of RX J1856.5-3754
X-ray observations of RX J1856.5-3754 by Chandra
The intriguing case of RX J1856.5-3754
X-ray observations of RX J1856.5-3754 by Chandra
Trümper (2005),
astro-ph/0502457
The intriguing case of RX J1856.5-3754
X-ray observations of RX J1856.5-3754 by Chandra
Trümper (2005),
astro-ph/0502457
The best fit is obtained for a pure black body spectrum (no
spectral lines!)
The intriguing case of RX J1856.5-3754
X-ray observations of RX J1856.5-3754 by Chandra
Trümper (2005),
astro-ph/0502457
⇒ condensed magnetic surface?
Origin of neutron-star magnetic fields
In 1964 (before the discovery of pulsars),
Lodewijk Woltjer argued that neutron stars
could have very strong magnetic fields. This
was also independently shown by Ginzurg.
If we assume that the magnetic flux Φ = B4πR 2 is conserved
during the gravitational collapse of massive stars eventually
giving birth to a neutron star, we have
Φi = Φf ⇒ Bf = Bi
Ri
Rf
2
This implies neutron-star magnetic fields up to Bf ∼ 1012 T.
Woltjer, Astrophys. J. 140,1309 (1964).
Ginzburg, Sov. Phys.Doklady 9, 329 (1964).
Fossile fields
The fossile-field scenario is supported by the fact that peculiar
main sequence Ap/Bp stars, magnetic white dwarfs and
neutron stars have similar magnetic fluxes as pointed by
Ruderman in 1972.
Ap/Bp
WD
NS
R (in R⊙ )
∼1
∼ 10−2
∼ 10−5
B (in T)
∼3
∼ 105
∼ 1011
The initial magnetic field might be have been produced in the
convective core of main sequence stars by a dynamo
mechanism.
Alternatively the field might have been produced in the neutron
star itself.
Theory of magnetars
The interior of a neutron star is an
almost perfect electrical conductor.
Duncan and Thompson showed that
strong magnetic fields ∼ 1012 T can
be generated via dynamo effects in
hot newly-born neutron stars with
initial periods of a few milliseconds.
Huge amount of magnetic energy can
be occasionally released in
crustquakes producing γ-ray bursts.
Thompson & Duncan, ApJ 408, 194 (1993).
The March 5, 1979 event
The theory of magnetars was proposed in 1992 by Robert
Duncan, Christopher Thompson and Bohdan Paczynski to
explain Soft-Gamma Repeaters (SGR). SGRs are repeated
sources of x- and γ-ray bursts. The first such object called SGR
0525−66 was discovered in 1979.
A very intense gamma-ray
burst was detected on March
5, 1979 by two Soviet satellites
Venera 11 and Venera 12.
The burst lasted about 3 minutes and showed a periodic
modulation of 8 seconds.
Mazets et al., Nature 282 (1979), 587.
The March 5, 1979 event
The source was later found to
lie inside a supernova remnant
in the Large Magellanic Cloud
(N49) thus suggesting that it
might be a young isolated
neutron star. But it was difficult
at that time to explain the origin
of the bursts.
ROSAT
Other burst sources have been found. 9 SGRs (7 confirmed, 2
candidates) are currently known (April 2011).
http://www.physics.mcgill.ca/~pulsar/magnetar/main.html
Anomalous X-ray pulsars
Anomalous X-ray pulsars (AXP) are isolated sources of
pulsed x-rays. Their periods range from 2 to 12 s and their
spin-down rate Ṗ ∼ 10−11 so that B ∼ 1010 T. Some of them
are bursters.
SGR and AXP have much in
common. Their observed x-ray
luminosity is much larger than their
kinetic energy loss rate suggesting
these objects are powered by
magnetic field decay. SGR and AXP
are thought to belong to the same
class of neutron stars: magnetars.
CXO J164710.2-455216 (Chandra)
12 AXPs (9 confirmed, 3 candidates) are currently known.
http://www.physics.mcgill.ca/~pulsar/magnetar/main.html
Magnetar seismology
Quasi Periodic Oscillations (QPO) have
been discovered in the x-ray flux of giant
flares from SGR 1806−20, SGR 1900+14
and SGR 0526−66.
These QPOs coincide reasonably well
with seismic crustal modes thought
to arise from the release of
magnetic stresses.
Thompson and Duncan, MNRAS 275,
255 (1995)
The huge luminosity variation suggests B & 1011 T at the
star surface thus lending support to the magnetar scenario.
Vietri et al., ApJ 661, 1089 (2007).
Cyclotron lines in SGR and AXP
Evidence for proton cyclotron lines have been found in the
spectra of a few SGR and AXP during bursts:
SGR 1900+14
SGR 1806−20
1E 1048−59
XTE J1810−197
4U 0142+61
Bspec (in T)
2.6 × 1011
∼ 1011
2.1 × 1011
2 × 1011
4.75 × 1010
Bspin (in T)
7 × 1010
2 × 1010
4.2 × 1010
2.1 × 1010
1.3 × 1010
Mereghetti, Astron. Astrophys. Rev. 15, 225 (2008).
The magnetic fields inferred from both spin-down and
spectroscopic studies (not only cyclotron lines but also
continuum) are consistent with the magnetar scenario:
B>
me2 c 3
e~
The puzzling case of SGR 0418+5729
SGR 0418+5729, discovered on 5 June 2009, has a period
∼ 9.1 s but a spin-down rate Ṗ < 6 × 10−15 implying
B < 7.5 × 108 T like ordinary pulsars!
Rea et al., Science 330, 944 (2010).
Much stronger internal fields
could power magnetar activity
whereas the existence of a
fallback disk might explain the
low external field.
Trümper et al., Astron.
Astrophys.518, 46 (2010)
But recent spectral analysis of the x-ray emission from SGR
0418+5729 suggests that its surface magnetic field
B ≃ 1.1 × 1010 T.
Güver et al., arXiv:1103.3024.
Radio emission in strong magnetic fields
Radio emission in pulsars is believed to be produced by
the creation of a cascade of electron-positron pairs in
the pulsar magnetosphere.
In strong magnetic fields B > me2 c 3 /e~, a photon can
split into two lower energy photons thus preventing the
creation of pairs. Radio emission is therefore expected to
be suppressed.
However radio emissions have been detected from several
high-magnetic field rotation-powered pulsars and even in three
“magnetars” (although the radio emission is quite distinct from
that of ordinary pulsars)
Ng and Kaspi, arXiv:1010.4592
Superconductivity and superfluidity in
neutron stars
Are electrons in neutron stars superconducting?
The surface layers of non-accreting neutron stars are mostly
composed of iron which is not superconducting.
It was found in 2001 (Shimizu et al., Nature
412, 316) that iron can become
superconducting at densities ρ ≃ 8.2
g.cm−3 with Tce ≃ 2 K.
But Tce is much smaller than the temperature in neutron stars.
Yakovlev et al, proceedings (2007), arXiv:0710.2047
Are electrons in neutron stars superconducting?
In the deeper layers of neutron stars at densities ρ & 104
gcm−3 , atoms are fully ionised.
rs ≡ d/a0
a0 ≡ ~2 /me e2
d ≡ (3/4πne )1/3
Ceperley and Alder,
PRL45(1980) 566.
In ordinary metals
rs ∼ 2 − 6
In neutron star crust
rs ∼ 10−5 − 10−2
Electrons in neutron stars form an almost ideal Fermi gas
Are electrons in neutron stars superconducting?
The critical temperature of a uniform non-relativistic electron
gas is given by (Tpi is the plasma temperature)
Tce = Tpi exp −8~vFe /πe2 ⇒ Tce ∝ exp(−ζ(ρ/ρord )1/3 )
with ρord = mu /(4πa03 /3). At densities above ∼ 106 g.cm−3 ,
electrons become relativistic vFe ∼ c so that
(α = e2 /~c ≃ 1/137)
Tce = Tpi exp (−8/πα) ∼ 0
Ginzburg, J. Stat. Phys. 1(1969),3.
Electrons in neutron stars are not superconducting.
Nuclear superfluidity and superconductivity in neutron
stars
The BCS theory was applied to nuclei by Bohr, Mottelson,
Pines and Belyaev
Phys. Rev. 110, 936 (1958).
Mat.-Fys. Medd. K. Dan. Vid. Selsk. 31 , 1 (1959).
N.N. Bogoliubov, who developed a
microscopic theory of superfluidity and
superconductivity, was the first to explore
its application to nuclear matter.
Dokl. Ak. nauk SSSR 119, 52 (1958).
Superfluidity in neutron stars was suggested long ago (before
the discovery of pulsars) by Migdal in 1959. It was first studied
by Ginzburg and Kirzhnits in 1964.
Ginzburg and Kirzhnits, Zh. Eksp. Teor. Fiz. 47, 2006, (1964).
Superfluidity in neutron matter
Most microscopic calculations have been performed in uniform
neutron matter. They show that medium effects act against
pairing but pairing gaps remain poorly known.
Lombardo & Schulze, Lect. Notes Phys. 578 (2001) Springer
Superfluidity and superconductivity in neutron stars
In spite of their names, neutron stars are not only made of
neutrons! As a consequence, they could contain various kinds
of superfluids and superconductors.
traditional neutron star
quark−hybrid
star
N+e
N+e+n
s
to
ns
H
d
Σ,Λ
,Ξ,∆
neutron star with
pion condensate
ui
π−
p
ro
2SC
CFL
color−superconducting
strange quark matter
(u,d,s quarks)
2SC
2SC+s
,µ
rfl
n,p,e
u,d,s
quarks
pe
ond u c t i
r c
n g
p e
su
u
n
n,p,e, µ
hyperon
star
−
K
CFL
CFL−K +
0
CFL−K
0
CFL− π
crust
Fe
6
3
10 g/cm
11
10
g/cm 3
3
10 14 g/cm
Hydrogen/He
atmosphere
strange star
nucleon star
R ~ 10 km
picture from
F. Weber
Superfluidity in neutron star crusts
Density functional theory in a nut shell
The nuclear energy density functional theory is the only method
which allows a tractable and consistent treatment of both
crust and core.
The energy of a lump of matter is expressed as (q = n, p)
Z h
i
E = E ρq (rr ), ∇ ρq (rr ), τq (rr ), J q (rr ), ρ̃q (rr ) d3r
(q)
(q)
where ρq (rr ), τq (rr )... are functionals of ϕ1k (r ) and ϕ2k (r )
!
(q)
hq (r ) − λq
∆q (r )
ϕ1k (r )
(q)
= Ek
(q)
∆q (r )
−hq (r ) + λq
ϕ2k (r )
∇·
hq ≡ −∇
δE
δE
δE
∇+
−i
·∇ ×σ ,
δτq
δρq
δJJ q
(q)
ϕ1k (r )
(q)
ϕ2k (r )
∆q ≡
!
δE
δ ρ̃q
Effective nuclear energy density functional
In principle, one can construct the nuclear functional from
realistic nucleon-nucleon forces (i.e. fitted to experimental
nucleon-nucleon phase shifts) using many-body methods
E=
~2
(τn + τp ) + A(ρn , ρp ) + B(ρn , ρp )τn + B(ρp , ρn )τp
2M
+C(ρn , ρp )(∇ρn )2 +C(ρp , ρn )(∇ρp )2 +D(ρn , ρp )(∇ρn )·(∇ρp )
+Coulomb, spin-orbit and pairing
Drut et al.,Prog.Part.Nucl.Phys.64(2010)120.
But difficult task so in practice, the functional is
parametrized with simple analytical functions
Bender et al.,Rev.Mod.Phys.75, 121 (2003).
Construction of the functional: experimental data
The parameters of the functional are fitted to the 2149
measured nuclear masses with Z , N ≥ 8. The rms deviation for
the best fit so far is σ = 0.581 MeV.
Goriely, Chamel, Pearson, PRL102,152503 (2009).
This fitting procedure ensures that the neutron-rich nuclei in
neutron-star crusts will be properly described.
Construction of the functional: N-body calculations
The functional is free from self-interactions and is constrained
to reproduce various properties of nuclear matter as obtained
from many-body calculations:
isoscalar effective mass Ms∗ /M = 0.8
equation of state of pure neutron matter
1S
0
pairing gaps in symmetric and neutron matter
Landau parameters (stability against spurious instabilities)
Chamel, Goriely, Pearson, Phys.Rev.C80,065804 (2009)
Chamel, Phys. Rev. C 82, 014313 (2010)
Chamel and Goriely, Phys. Rev. C 82, 045804 (2010)
Chamel, Phys. Rev. C 82, 061307(R) (2010)
With these constraints, the functional can be reliably applied to
describe superfluid neutrons in the inner crust and the
neutron-proton mixture in the liquid core
Ground-state composition of the inner crust
Results for BSk14
A
200
460
1140
1215
1485
1590
1610
800
780
0.1
0.05
0
0
-3
Z
50
50
50
40
40
40
40
20
20
nn(r), np(r) [fm ]
nb (fm−3 )
0.0003
0.001
0.005
0.01
0.02
0.03
0.04
0.05
0.06
0.1
0.05
0
0
0.1
0.05
0
0
0.1
0.05
0
0
0.1
0.05
0
0
nb = 0.06
2
4
8
6
10
12
14
nb = 0.04
3
9
6
12
18
15
21
nb = 0.02
4
8
12
20
16
24
nb = 0.005
10
5
20
15
30
25
35
nb = 0.0003
10
20
30
r [fm]
Onsi, Dutta, Chatri, Goriely, Chamel and Pearson,
Phys.Rev.C77,065805 (2008).
40
50
Anisotropic multi-band neutron superfluidity
In the decoupling approximation, the Hartree-Fock-Bogoliubov
equations reduce to the BCS equations
∆αkk = −
Eβkk ′
∆βkk ′
1 X X pair
tanh
v̄αkk α−kk βkk ′ β−kk ′
2
Eβkk ′
2T
′
β
pair
v̄αkk α−kk βkk ′ β−kk ′
=
Z
k
d3 r v π [ρn (rr ), ρp (rr )] |ϕαkk (rr )|2 |ϕβkk ′ (rr )|2
Eαkk =
q
ε2αkk + ∆2αkk
εαkk and ϕαkk (rr ) are obtained from band structure calculations
Chamel et al., Phys.Rev.C81,045804 (2010).
Analogy with terrestrial multi-band superconductors
Multi-band superconductors were first studied by Suhl et al. in
1959 but clear evidence were found only in 2001 with the
discovery of MgB2
Ordinary superconductors generally involve a few bands
whereas the number of bands in neutron star crust can be very
large ∼ up to a thousand
Neutron pairing gaps vs single-particle energies
Example at ρ = 0.06 fm−3 with BSk16
2,2
∆F [MeV]
2
1,8
1,6
1,4
1,2
1
0,8
-40
-30
-20
-10
ε [MeV]
0
10
20
⇒ The presence of clusters reduces ∆αkk but much less than
predicted by previous calculations
Average neutron pairing gap vs temperature
∆(T)/∆(0)
Example at ρ = 0.06 fm−3 with BSk16
1
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,1
0
0
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9
1
T/Tc
⇒ ∆αkk (T )/∆αkk (0) is a universal function of T
⇒ The critical temperature is approximately given by the usual
BCS relation Tc ≃ 0.567∆F
Pairing field and local density approximation
The effects of inhomogeneities on neutron superfluidity can be
directly seen in the pairing field
Λ
X
∆ k
1
|ϕαkk (rr )|2 αk
∆n (rr ) = − v πn [ρn (rr ), ρp (rr )]ρ̃n (rr ) , ρ̃n (rr ) =
2
Eαkk
α,kk
Neutron pairing field for ρ = 0.06 fm−3 at T = 0
90
(c)
80
ρ = 0.06 fm
60
∆(r) [MeV]
ξ (r) [fm]
70
-3
50
40
30
20
10
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
r [fm]
2.4
2.2 (c)
-3
ρ = 0.06 fm
2
1.8
1.6
LDA
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
r [fm]
Observational evidence of superfluidity
in neutron stars
Cooling of isolated neutron stars
During the first tens of seconds, the newly formed
proto-neutron star with a radius of ∼ 50 km stays very hot with
T ∼ 1011 − 1012 K. Within ∼ 10 − 20 s the proto-neutron star
becomes transparent to neutrinos and thus rapidly cools
down by powerful neutrino emission shrinking into an
ordinary neutron star.
After about 104 − 105
years, the cooling is
governed by the
emission of thermal
photons due to the
diffusion of heat from the
interior to the surface.
Puppis A (RX J0822-4300) from Chandra
Cooling of isolated neutron stars
The surface temperature of a neutron star depends on its mass
and composition but also on superfluidity. Superfluidity reduces
the heat capacity and neutrino emissivities but also opens new
channels of neutrino emission.
TC = 1
0 9K
TC = 5
.5x10 8
K
TC = 0
Recent observations of the cooling neutron star in the
Cassiopeia A supernova remnant put stringent contraints on the
critical temperature of the neutron superfluid Tc & 5 × 108 K.
Page et al., PRL 106, 081101 (2011)
X-ray binaries
Neutron stars in x-ray binaries may be heated as a result of
the accretion of matter from the companion star.
The accretion of matter onto the
surface of the neutron star triggers
thermonuclear fusion reactions which
can become explosive, giving rise to
x-ray bursts.
In soft x-ray transients, accretion outbursts are followed by long
period of quiescence during which the accretion rate is much
lower. In some cases, the period of accretion can last long
enough for the crust to be heated out of equilibrium with the
core.
Thermal relaxation of soft x-ray transients
The thermal relaxation during the quiescent state has been
recently monitored for KS 1731−260 and for MXB 1659−29
after an accretion episode of 12.5 and 2.5 years respectively.
Curves 1,4 : crystalline crust with
neutron superfluidity
Curve 2 : crystalline crust without
neutron superfluidity
Curve 5 : amorphous crust with
neutron superfluidity
Shternin et al., Mon. Not. R. Astron. Soc.382(2007), L43.
Brown and Cumming, ApJ698 (2009), 1020.
Pulsar glitches and superfluidity
Sometimes pulsars may suddenly spin up. These glitches are
followed by a relaxation over days to years thus revealing the
superfluidity in neutron stars.
Superfluidity is also expected to play a key role in the
mechanism of large glitches (e.g. catastrophic unpinning of
superfluid vortices). Similar phenomena were observed in
superfluid helium.
Anderson and Itoh, Nature 256, 25 (1975)
Neutron star precession
Long-term cyclical variations of order months to years have
been reported in a few neutron stars: Her X-1 (accreting
neutron star), the Crab pulsar, PSR 1828−11, PSR B1642−03,
PSR B0959−54 and RX J0720.4−3125.
Stairs et al., Nature
406(2000),484.
30
0
∆P (ns)
-30
2
1
0
-1
1
<S>
Example: Time of arrival
residuals, period residuals, and
shape parameter for PSR
1828−11
∆t (ms)
60
0.5
0
49,500
50,000
50,500
Modified Julian Date
These variations have been interpreted as the signature of
neutron star precession.
51,000
Precession and superfluidity
Observations of long-period precession put contraints on the
superfluid properties of neutron star interiors.
For a non-superfluid star with
deformation ǫ = ∆I/I,
Pprec =
For a superfluid star with
pinned vortices
P
≫P
ǫ
Link, Astrophys. Space Sci.308,435 (2007)
Pprec =
Ipin
P≪P
I
Plate tectonics
The strong interaction between neutron superfluid vortices and
proton flux tubes in neutron star cores leads to movement of
crustal plates.
The evolution of the pulsar spin and magnetic field are
intimately related to superfluidity and superconductivity
Ruderman, Astrophys. Space Sci.357, 353 (2009)
Superfluidity and mutual entrainment
In superfluid helium at T > 0, the momentum p and the velocity
v of each atom are not aligned due to the interactions between
atoms and thermal excitations (phonons, rotons...)
momentum vs velocity
p = m⋆v + (m − m⋆ )vvN
vN is the velocity of excitations.
The effective mass m⋆ (T ) can
be experimentally measured
from Andronikashvili’s
experiment
Likewise in superfluid mixtures, momentum and velocity are
not aligned even at T = 0: this is the Andreev&Bashkin effect
Andreev and Bashkin, Sov.Phys.JETP42,164(1975)
Entrainment and dissipation in neutron star cores
Historically the long post-glitch relaxation was the first
evidence of superfluidity in neutron stars. However...
κ
Due to (non-dissipative) mutual entrainment
effects, neutron vortices carry a fractional
magnetic quantum flux
B
vp
vn
V
e−
Sedrakyan and Shakhabasyan, Astrofizika 8
(1972), 557; Astrofizika 16 (1980), 727.
picture from Glampedakis
The core superfluid is strongly coupled to the crust due to
electrons scattering off the magnetic field of the neutron vortex
lines. This leads to a mutual friction force acting on the
superfluid which could be an important dissipative
mechanism for neutron star pulsations.
Alpar, Langer, Sauls, ApJ282 (1984) 533-541
Entrainment effects in neutron star crusts
Even though entrainment effects have been studied for a long
time in neutron-star cores (e.g. Sauls, NATO Proceedings
1989), it has been only recently realized that similar effects
should also occur in the crust.
Like electrons in metals,
unbound neutrons in neutron
star crusts move as if they had
an effective mass mn⋆ > mn .
In the crust rest frame pn = mn⋆vn therefore in another frame
pn = mn⋆vn + (mn − mn⋆ )vvc
Carter, Chamel, Haensel, Int.J.Mod.Phys.D15(2006)777.
Conclusions
Pines theorem
Neutron stars are superstars!
The theorem was formulated at the conference
on “Neutron Stars: Theory and Observation”,
NATO Advanced Study Institute, Greece, 1990.
Proof:
neutron stars are superdense stars
neutron stars are superfast rotators
neutron stars are superprecise clocks
neutron stars have superstrong magnetic fields
neutron stars are superfluid
neutron stars are superconducting