Download Neutron Stars, first class

rd
3
lecture of “Compact
Object and Accretion”,
Master Programme at Leiden
Observatory
st
I
Neutron Stars
class
study material: Chapter 9, ShapiroTeukolsky
these slides at
http://home.strw.leidenuniv.nl/~emr/COA/
Friday, October 2, 2015
Neutron stars (NSs)
a few facts we already know
•
Final evolutionary stage of stars with mass between
9 Msun and a not well determined mass, 30-40 Msun
(depending, for example the star metallicity)
•
After all the nuclear reaction cycles take place, the
iron core collapses and form a neutron star
•
During the core collapse, the core neutralisation
+
takes place via inverse beta reaction*p → n + e + νe
−
as the neutron beta decay n → p + e + ν¯e is
suppressed because an electrons with an energy
larger than the Fermi energy needs to be produced
in an electron degenerate medium
−
*Note, also electron capture p + e → n + νe
Friday, October 2, 2015
Neutron stars (NSs)
a few facts we already know
•
Matter remains transparent to neutrinos until a
density of ~1015 kg m-3 (R~300 km) and neutrinos
escape and be detected
•
Formation of a neutron stars is accompanied by a
supernova explosion (Type II ou Type I/b)
Friday, October 2, 2015
Neutron stars (NSs)
a few facts we already know
•
Typical values for their physical characteristics
Radius
•
•
Friday, October 2, 2015
Mass
Central mass density
Compactness of
0.2
Gravity is counterbalanced by pressure originated by
strong force between nucleons. We will see that the
equation of state corresponding to this situation is rather
uncertain
NSs: history
Friday, October 2, 2015
Discovery of Neutron
•
Neutrons were discovered by James Chadwick in 1932 in
Cambridge (Nobel prize in 1935). Meanwhile
Chandrasekhar and Landau were working on WD and
showed the existence of a maximum mass, wondering
what was then the fate of more massive stars. In
1931/1932 Landau proposed that this fate would be an
object “where the atomic nuclei were so close to each
other to form a gigantic nucleus”
James Chadwick
Friday, October 2, 2015
NS concept
1934 Baade & Zwicky proposed the existence of
NSs. They had discovered that some “Novea”
were extragalactic and therefore much more
intrinsically luminous than Galactic Novea. They
called them “Supernovae” and propose:
Friday, October 2, 2015
Equation of state
•
1939 first calculations of NS structure by
Oppenheimer & Volkoff. They use general relativity
and an equation of state for a perfect gas of
degenerate neutrons. we will see that this predict a
maximum mass of ~0.7 Msun in contraddiction with
observations. This shows that neutron degenerate
pressure is not counterbalancing gravity in NSs.
WWII stopped momentarily
further advancements...
Friday, October 2, 2015
Equation of state cont.
•
Friday, October 2, 2015
At the end of 1950s theoretical works on NSs restarted, in particular around the question of what is
the equation of state for ultra-dense matter. Notable
works by Harrison, Wikano & Wheeler (1958),
Cameron (1959), Ambartsumyan & Saakyan (1960) &
Hamada & Salpeter (1961). These more realistic
equation of state showed that NS must have very
small radii ~ 10 km. Despite their large surface
temperature, the small size makes direct detection
of surface emission undetectable. NSs during these
years remained a study subject for (a small group of)
theoreticians.
The beginning of X-ray
astronomy
In 1962, Giacconi and co-authors discovered the first
X-ray source beyond the solar system (Sco X-1, Lx~ 6
104 Lsun), following the launch of the USA Arobee
rocket (Nobel prize in 2002). It is interpreted as a young
and hot neutron star (Te ~2 1010 K). The question is then
how fast NSs cool down. At the same time, there is a
debate on whether QSOs (discovered in 1963) are
neutron stars. However, it was soon realised that the
observed redshift in their spectra cannot be of
gravitational origin, given the NS expected
compactness.
We now know that Sco X-I is an accreting neutron star and QSOs
are accreting supermassive black holes
Friday, October 2, 2015
Pulsars discovery
•
In 1964, Hoyle, Narlikar & Wheeler proposed that a
highly magnetised NS (~1010 Gauss) is at the centre of
th Crab nebula.
•
In 1967, Pacini proposed that the source of energy for
the Crab would be a highly magnetised and rotating
NS.
X-ray
Optical
Friday, October 2, 2015
Pulsars discovery
•
In 1967, Hewish and his student Bell
discover the first radio pulsar: a period
radio source, with an extremely short
(modern value P = 1.337301 s) and stable
period. (Nobel prize to Hewish in 1974). The
extreme stability suggested to some an
extra-terrestrial life origin...
•
In 1968, they published their discovery
after much double checking. Name: CP
1919 (Cambridge Pulsar, right ascension 19,
declination 19. The Modern name is “PSR B
1919+21”)
Friday, October 2, 2015
More pulsars...
•
In 1968, discovery of the radio Pulsar Vela (PSR B
0833-45) and Crabe (PSR B 0531+21). They are
both within supernova remnants. These discovery
validate Baade, Zwicky & Pacini’s predictions.
•
In 1968, Gold proposes a pulsar model as
magnetised, rapidly rotating NS, with which he
predicts with precision the Crabe period and its
derivative, that were only measured in 1969 (we
will study it...)
Friday, October 2, 2015
Pulsars, as NS
There are very little alternative to a NS to explain a Pulsar.
Given mass M, radius R and angular velocity Ω
the centrifugal force at the surface, at the equator is
&
The maximum rotation for balance is then
where
So
Friday, October 2, 2015
Pulsars, as NSs
The first radio pulsar has
Ω � 4.7 s
−1
ρ̄ > 7.9 × 1010 Kg m−3
a minimum density of
The Crabe pulsar has
Ω � 0.20 ms
−1
a minimum density of
ρ̄ > 1.2 × 10
14
Kg m
−3
No system is known that is so dense, not even WDs
recall, for WD ρ̄ ≈ 2 × 10 Kg m
9
Friday, October 2, 2015
−3
Rotation as cause for
observed period
Alternatives were discussed but dismissed:
•
Surface radial oscillation: WDs fundamental mode is
too slow, NSs is too fast
•
Orbital period of a binary: for two NSs with 1.4
Msun, one need a separation of 4000 km, for P~1 s. In
this situation gravitational wave emission would be so
intense to make the NSs coalesce in a few hours
It is therefore immediately clear that the observed period is
linked to the NS spin and the agreement with Gold’s model
indicates a strongly magnetised NS
Friday, October 2, 2015
Other notable discovery
•
Friday, October 2, 2015
1969, first X-ray pulsars (no radio detected). They
are binary systems, that allowed to measure the NS
mass. Results in agreement with theory.
Other notable discovery
•
1975, Hulse& Taylor discover
the first binary pulsar PSR B
1913+16 (pulsar period ~56 s;
orbital period of 7.8 h).
Determined that they are
two NSs of ~1.4 Msun. Only
one is a pulsar. Monitoring
the pulsar, they measure a
decrease in the orbital period
consistent with Gravitational
Wave emission. First
(indirect)detection of GW!
Nobel prize in 1993
from Weisberg & Taylor 2004
Currently, agreement with GR within 0.1 %
Friday, October 2, 2015
Other notable discoveries
•
1982, first pulsar discovered with P ~1.56 ms
(millisecond pulsar) . Such a short period is difficult to
justify within the standard formation scenario. We think
now that they were “re-accelerated” via accretion from
a companion. Recycled pulsars
•
In 1992, the first discovery of (3) extrasolar planets,
around a millisecond pulsar by Wolszcan & Frail
•
.....
Friday, October 2, 2015
WDs vs NS
similarities and differences
Friday, October 2, 2015
Similarities
•
•
•
Friday, October 2, 2015
Equation of state under the assumption of T=0
Therefore, fluid barotropic with P=P(ρ)
To calculate the structure, one therefore needs only
eq. mass conservation + eq. hydrostatic equilibrium
Differences
1. NS much higher compactness, therefore
General relativistic frame
GR (solid) and Newtonian (dashed)
WDs
neutralisation,
calculation not valid
NSs
calculations are indistinguishable
Friday, October 2, 2015
Idialised NS
with neutron
degenerate
pressure. In GR,
this is in fact the
Oppenheimer &
Volkoff (1939)
calculation
differ at high ρ, maximum mass just 0.7 Msun
Differences
2. NS much high densities (~8 orders of magnitude
denser)
Equation of state uncertain
Mass and Radius measurements of NSs are used to
try and constrain the equation of state
Friday, October 2, 2015
Differences
3. NS radius is ~100-1000 times smaller. NS cool very rapidly after
their formation and they typically have an effective temperature of a
few 106 K . These imply L~ 10 Lsun but in X-ray (106 k => 0.1 keV)
We cannot detect thermal radiation from the surface, unless
the object is very close e.g. RX J185635-3754 at 120 pc
WD can be observed
in the optical band
Friday, October 2, 2015
Differences
4. NSs
have higher rotation and magnetic field
B ~ 106 G (WDs); B~1012 G (NSs)
P~ a few hr (WDs); P~ seconds (NSs)
white dwarfs
neutron stars
(upper limit as some B and Ω can be lost during collapse)
Lpulsar ∝ Ω R B (power)
4
6
2
WD ‘s pulsar emission is ~10-14 smaller !
Friday, October 2, 2015
cfr. Ch 9 Shapiro Teukolsky
NSs:
Equation of state
Friday, October 2, 2015
Oppenheimer & Volkoff
(1939)
• perfect gas
• degenerate neutrons only
• T=0
The equation of state is the same as that
discussed for WDs, exchanging the electron
mass with the neutron mass in the coefficients
Friday, October 2, 2015
n-p-e
•
1)
2)
gas
A step forward is to consider a perfect gas of
electrons, protons and neutrons at T=0
total density
charge neutrality
If β and inverse-β reactions are at equilibrium
3)
n↔p+e
“chemical equilibrium”
where the chemical potential of neutrinos ~zero (like photons)
T=0 + chemical equilibrium = cold catalysed matter
Friday, October 2, 2015
n-p-e
gas, cont.
At T=0 we have μ=EF
With
We have 3 equations in 3 unknown np, nn,ne
We are going to find np,nn,ne as a function of n, or:
Friday, October 2, 2015
n-p-e
gas, cont.
Pressure an internal energy are just given by the sum of the 3 species
(Dalton’s law for ideal gas)
Reminder:
Friday, October 2, 2015
Fermi-Dirac
neutron
drip
5/3
n
e
n
electron degenerate
pressure dominates
Friday, October 2, 2015
from the chemical
e-
NR
regime
WD
1/8
Neutron rich
above 109 g/cm3 (1011 Kg/m3)
4/3
UR
regime
nuclear density
equilibrium with
x>>1
neutron drip line =
neutron rich nuclei becomes
unstable for neutron ejection:
nucleons drip out of unstable nuclei
neutron degeneracy
pressure dominates
Realistic equation of state
Additional facts to consider:
•
presence of iron nuclei that can survive at law
densities, but will progressively disintegrate beyond
the neutron drip density (4 1011 g/cm3)
•
strong force interaction beyond the nuclear density
(~2.8 1014 g/cm3), when the mean particle distance
is ~a Fermi (10-6 nm). Perfect gas approximation
breaks down (perfect gas = non interacting gas)
•
Strong Magnetic field
Friday, October 2, 2015
Equation of state within a NS
I) Surface, depth of ~100 m. EOS
strongly influenced by magnetic
field
II.)External crust of ~500 m.
density less than neutron drip. EOS
well understood: crystal structure
of neutron rich nuclei (that interact
via Coulomb force) + a fluid of free
electrons and neutrons
V) Core. Most uncertain region.
some possibilities: formation of a
superfluid core of n +p; pion
condensation; formation of a solid
core of neutron; phase transition
towards a strange quark matter
(see Ch. 8 S-T optional)
Friday, October 2, 2015
III) Internal crust of ~1 Km.
density above neutron drip and
below nuclear density. Nuclei
progressively dissolve and
neutron pressure becomes
more important
IV) Interior: neutrons become
superfluid, nuclei completely
dissolved, small fraction of e- + p
Above neutron drip
•
Non relativistic domain. Between nuclear density
and ~1015 g/cm3 there are two complication: 1)
describing strong force potential; 2) finding
appropriate numerical technique for solving a many
body problem. The potential is somewhat
constrained by experimental data (e.g. nucleonnucleon scattering data)
•
Beyond 1015 g/cm3, composition should include
hyperions (baryons that contain at least one strange
quark) and nuclear interaction must be treated
relativistically. Relativistic many-body technique for
many-body interaction not fully developed.
see Box 8.1 in S-T
Friday, October 2, 2015
NS structure:
maximum mass
Note: despite the uncertainty on the equation of state,
it can still be written as a barotropic relation P = P (ρ)
We are thus going to solve the NS structure as we did for
WDs, but in a general relativistic framework
Friday, October 2, 2015
Einstein equation
Source term is the energy-momentum tensor:
µ
dx
µ
u
=
with
dτ
and Einstein tensor:
four-velocity
where the space-time metric
Gravity is a property of space-time, that depends
on the local content of mass & energy
Friday, October 2, 2015
Schwarzschild solution
At the outside of a neutron star of M, we can
assume vacuum
Static, spherical symmetric solution
M is the gravitational mass, as measure
by an observer at infinity
Minkowski space
Friday, October 2, 2015
Tolman-Oppenheimer-Volkhoff
(T.O.V)
Inside a NS. For a static spherical system, only the
diagonal elements of
are non-zero. We thus have
3 independent equations:
with
Friday, October 2, 2015
Tolman-Oppenheimer-Volkhoff
(T.O.V) cont.
In the limit:
2
Gm(r)/r � c and P � ρc
2
They become the non-relativistic equations of 1) mass
conservation, 2) hydrostatic equilibrium and 3) the equation for
the gravitational potential
Boundary conditions
Non relativistic regime
Relativistic regime
Equivalent to dP/dr =0
ensures continuity with solution inside and outside
Friday, October 2, 2015
Solutions
with
&
we can easily solve equations
M/Msun
M/Msun
Unstable configurations for
radial perturbation
there is a maximum mass
for any ϒ
ρ/ρnuc
Friday, October 2, 2015
ρ/ρnuc
M/Msun
For a perfect gas of degenerate neutrons or of a mix of
degenerate n+p+e the maximum mass is 0.7 Msun, in
contradiction with observations
ρ [ g cm-3]
Friday, October 2, 2015
For realistic equation of state the maximum mass is in the range
Salgado et al. 1994
mass measurements can be used to rule out e.o.s
Friday, October 2, 2015