14.9 SUPERDENSE MATTER

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14.9 SUPERDENSE MATTER
Throughout this course we have seen that the
existence of stable astronomical objects has
depended on the ability of nature to find a force
to compete against the unrelenting attractive
force of gravity. During each phase an often
temporary solution has been found to maintain the
object in a unique configuration. Usually in the
process of supporting itself the object has had
to expend energy, and when this energy reserve
has been exhausted the object has been obliged to
shrink to a more compact form. Sometimes
violently much of its material has been blown
into space, ready to start all over again. white
dwarfs and neutron stars seem to be long term
cinders. However the question does arise as to
how compact can matter become. For the rest of
this course we explore the ultimate stages of
compactness that are found in nature as revealed
to us through astronomical observations. First we
must understand what we mean by the term
compact.
dr
If we fill a sphere with matter at a uniform
density r, then the total work done assembling
this sphere within its own gravitational field is
r
r
R
This is the binding energy of the object and can
be defined as a mass defect
giving
RS is called the Schwarzschild radius, and
corresponds to the mass defect being the total
mass
If we define
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COSMIC STRUCTURES
The Universe
R lt RS
48
40
Galaxies
Regime of black holes
Nuclei of Active Galaxies
32
Main Sequence Stars
24
White Dwarfs
Earth, Moon, Asteroids
16
Neutron Stars
8
Log M (kg)
Humans
0
-8
Protons
-16
Atoms
-24
-8
0
8
16
32
24
Log R (m)
Black Holes. Earlier, in connection with white
dwarfs, the red shift of photons emitted from the
surface of the star was estimated. If R RS then
the wavelength shift will be infinite and no
radiation can emerge. Hence objects having the
Schwartzschild radius are called black holes.
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14.10 THE MASS, SIZE AND DENSITIY OF NEUTRON
STARS

• A survey of all the objects existing within the
  Universe in terms of their compactness, as
  defined by the R/RS ratio above shows that in the
  context of stellar evolution neutron stars are
  the most compact objects observed to date. Hence
  if we want to investigate the ultimate fate of
  compact matter then it here that we should start
  looking. We have already discussed much of the
  relevant background
• White dwarfs are supported by electron
  degeneracy pressure
• The cores of massive stars were left in
  free-fall after a type II supernova explosion.
  Neutronisation had taken place. Is there anything
  to stop it falling? Will neutron degeneracy do
  the job?

The Approximate Density of Neutron Stars
The precise forces between nucleons is less well
understood than electromagnetic forces and, as a
consequence, the neutron degeneracy conditions
under such extreme gravitational fields are less
well defined. The detailed studies of neutron
stars (i.e. masses and radii - the equation of
state) will therefore not only be of interest to
the astrophysicist, but also may be used as a
test of particle physics. Simplistically we may
assume that the distances between neutrons are
typically
so that the density will be
The Nominal Maximum Mass
No comprehensive equation of state holds for
neutron stars, due to the lack of a real
understanding of the precise forces involved.
Equations similar to the relationships derived
for white dwarfs
For the non-relativistic and relativistic states
respectively are likely to provide a reasonable
approximation
4
Let us make an estimate of the maximum mass in a
simplistic way, similar to the Chandrasekhar
limit derived for white dwarfs
Now the Fermi energy is
If we have N particles
so that
The gravitational potential well per neutron has
a depth
We assume mn mp
Thus if the gravitational force is capable of
overcoming the support of the degeneracy
pressure, the limiting case will be when
giving
This yields a limiting mass for neutron stars as
Similar to the Chandrasekhar limit for white
dwarfs
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The Nominal Size of Neutron Stars
Since
we may estimate the rough size
Maximum Temperature of Neutron Stars
The condition for degeneracy pressure to dominate
is that the Fermi Energy should be greater than
the thermal energy of the particles
i.e.
From our studies of degeneracy pressure, the
number density of degenerate particles is
giving
Thus
More Realistic Theoretical Models
Unstable neutron drip
The above figure shows the gravitational mass vs
central density for the case of a model assuming
a pure, ideal neutron gas. The stable white dwarf
and neutron star configurations are represented
by the heavy solid lines. NOTE Mmax 0.7 M0
6
The above figure shows the gravitational mass vs
central density for various equations of state.
The letters labelling the various curves are
defined in Shapiro and Teukolsky. Note that the
maximum masses predicted range from about 1.5 to
2.7 M0, and that stable neutron star
configurations are predicted for objects with
masses from about 0.2 M0.
The gravitational mass vs radius for the same set
of models. It can be seen that neutron stars will
have sizes typically in the range 10 to 15 km,
and that as for white dwarfs their sizes are
inversely related to their masses.
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14.11 THE INTERNAL STRUCTURE OF NEUTRON STARS
The adjacent figure shows the likely internal
structure as determined from a representative
model of a 1.4 M0 neutron star. The layering of
the internal structure is a direct result of the
onset of different regimes in the equation of
state as one proceeds to higher densities. The
various zones in the models may be described as
follows

• The surface layers (r lt 109 kg m-3). The strong
  surface magnetic fields effect the equation of
  state, making the conductivity high parallel to
  the magnetic field and negligible in the
  orthogonal direction
• The outer crust (109 lt r lt 4 1014 kg m-3) The
  nuclei are extremely close to one another and
  they form a very stiff (1017 x steel) body
  centred Coulomb lattice and exist in
  b-equilibrium with the relativistic degenerate
  electron gas. When the energies of the electrons
  in the degenerate sea around the nuclei are high
  enough inverse b-decay takes place and results in
  the production exotic neutron rich nuclei (e.g.
  126Fe) which would be unstable in the normal
  physical environment.
• Normally such nuclei would b-decay
• i.e.
• In the presence of a degenerate sea of energetic
  (Ee0.5 MeV) electrons the b-decay process is
  blocked and in fact the neutron enrichment is
  caused by the reverse process
• The forces involved and the melting point may be
  gauged if we simplistically relate
• kT (1/4pe0)Z2e2/r where r
  is the distance between the nuclei
• This gives T 1010 K as the likely melting point

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• The inner crust (4 1014 lt r lt 2 1017 kg m-3).
  This consists of a lattice of neutron-rich nuclei
  together with free degenerate neutrons and a
  degenerate relativistic electron gas.. As the
  density increases, more and more of the nuclei
  begin to dissolve, and the neutron fluid provides
  most of the pressure.
• The neutron liquid interior. (r gt 2 1017 kg m-3)
  As the density further increases the material of
  the star contains chiefly a sea of degenerate
  neutrons with a few electrons and protons
  remaining. All nuclear structure has vanished.
  The neutrons form pairs which have extremely weak
  interactions with other pairs, thus making a
  superfluid with near zero viscosity
• The hyperon core. (r gt 3 1018 kg m-3) As the
  density increases so the Fermi energy of the
  neutrons increases. When the neutrons have Fermi
  energies comparable with their rest masses then
  hyperons and other particles can be created. Many
  of these are charged and we may expect that any
  such core will be a solid once more. However,
  because of our basis lack of full understanding
  the particle processes any models of the core
  region (if indeed it can exist) are necessarily
  even more speculative. If a hyperon core can
  exist then it is clear that its mass will be
  directly related to the mass of the neutron star.
  Low mass neutron stars are unlikely to produce a
  hyperon core.

It is possible to think of a neutron star as a
heavy nucleus i.e.

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14.12 THE COOLING OF NEUTRON STARS - PULSAR
GLITCHES
It is generally believed that pulsars will be
formed with extremely high internal temperatures
(T gt 1011 K) in the core of the supernova
explosion. The predominant cooling mechanism
immediately after formation will be due to
neutrino emission derived from the variety of
particle interactions which take place in such a
high temperature - high density environment.
The neutrino cooling is very rapid with an
initial timescale of seconds. After about a day
the temperature drops into the range 109 - 1010
K, i.e. below the melting point of the crust.
Photon emission overtakes neutrino emission when
the temperature drops to about 108 K, after about
1000 years or more. The relative importance of
the various mechanisms is summarised in the
adjacent figure.
Each curve gives T(t) for each process separately
(all except the photons are methods of generating
neutrinos) assuming the others are absent. The
most effective cooling process at any time will
be the one with the lowest T(t).
When neutron stars are born we have seen that
they are likely to be spinning rapidly with W
104. At these speeds the centrifugal force
outwards at the equator will be comparable to the
gravitational pull in wards. Now the object will
be liquid when it forms and thus take up a
surface profile natural to the forces it
experiences - i.e. an oblate spheroid. Since
neutron stars cool rapidly to the solidification
point within 1 day then they will solidify in
the natural shape of an object with W 104. As
time passes they slow down and their natural
shape will converge towards a spheroid.
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SUDDEN CHANGES IN THE SIZE OF NEUTRON STARS -
GLITCHES
Sudden spin-up period changes have been observed
in pulsar periods and are thought to be related
to crust quakes instigated by cracks in the rigid
crystalline material as it attempts to become
more spherical in shape as the neutron star slows
down. The accurate measurement of the timing of
the pulses enables great sensitivity to be
obtained in terms of the measurement of very
small changes in radius.
Period P
Time
The moment of inertia is I a MR2 so that
Since J IW Constant
Since we can measure period changes at about ten
nano-second level we can detect very small
changes in the size of the neutron star (at kpc
distances!)
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Detailed Study of the W Changes
The decrease in W after the glitch exhibits a
more complex time structure than is expected from
a simple radius change. It can be explained in
terms of the viscous coupling between the solid
and liquid components within the neutron star. We
assume that we have IS ISolid IL
ILiquid IT ITotal Let the ratio of the
components be Q IL/IT
W
QDW0
DW0
DW0(1 - Q)
Time
If we have a glitch, all the spin-up is initially
taken up by the solid material
Finally we will have the entire star spinning
together
Since
If we define the ratio of the liquid moment of
inertia to the total moment of inertia by Q
IL/IT
12
Then we will have
so that
Thus at
If we assume an exponential decay we obtain a
good fit to the observational data by
When we look at data from Crab and Vela glitches
we find very different values

• We can see that the Crab and Vela neutron stars
  are clearly very different. The Crab must have a
  lot of liquid and the Vela must be mostly solid.
• One scenario is that the Crab is typically one
  solar mass or slightly more and the Vela a very
  low mass object, and hence nearly all crust
• Alternatively the Vela neutron star could be
  very massive and close to the upper mass limit.
  In this case the large amount of solid would be
  derived from a solid hyperon core
• The greater relative change in DW for the Vela
  supports the latter hypothesis

Note the value of t is a measure of the viscous
coupling between the liquid and solid components.
We can see that the study of glitches provides a
powerful probe for the understanding of the
internal structures of neutron stars.