1. Compact objects. Neutron stars: matter under extreme conditions.
General properties
First stimation of size and mass.
Equation of state of -stable matter
TOV equations and mass, radius, and density profiles.
Maximum mass.
“Compact objects for everyone: a real experiment”, J. Piekarewicz et al. Eur. J. Phys. 26 (2005)
“ Neutron stars for undergraduates “, R. Silbar, S. Reddy, Am. J. Phys. 72, 892 (2004).
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2. What is a neutron star?
The answer depends to whom you ask. Very multidisciplanary subject
For astronomers they are very small stars visible as radio-pulsars or also
as a source of X and  rays.
For particle physicists they are a source of neutrinos (in some early stage
of its evolution) and one of the few places were quark matter can exist.
For nuclear physicist , they are consider as the largest nucleus in the univers
For cosmologists, they are considered as almost black-holes.
For computational physicists, the simulation of their birth and evolution
is a real headache.
In any case they offer fascinating phenomena under extreme
density conditions. Their properties are sensitive to the
equation of state of neutron matter in a wide range of
densities! A very active field of research!

3. A neutron star can be considered as gigantic nucleus,
with a mean density of the order of  1014 – g cm-3.
A neutron star contains  1057 nucleons, 90% being
neutrons.
They are the densest macroscopic objects in the Universe
They are about 10 km diameter, and about 1-2 MS, with and average
density 1014 – g cm-3 to be compared with the average density in
the universe  g cm-3 , the density of the Earth 5.5 g cm-3 and is
of the same order of the density inside nuclei  g cm-3
Because of the small size and the high density a neutron star has a surface gravitational field of about times that of the Earth and a very large gravitational energy
for a star with M= 1.4 MS and R= 10 km.

4. N  neutrons
M  1 MS

5. Now a days we can explore the phase diagram of hadronic matter.
There are several experimental facilities (RHIC and FAIR in the future)
to explore different regions of the phase diagram. The observational
data and study of neutron stars offers a unique possibility to understand
hadronic matter at very high density.

6. Size comparison of a neutron star with a characteristic white dwarf
And the earth.
Characteristic scape velocities
for 1 solar mass, white dwarf
with a radius of 5000 km and
a neutron star with 10 km radius:
Sun Km/s
White dwarf km/s
Neutron star km/s

7. Liquid drop model picture
sphere of constant density
with only neutrons
the energy described by the mass phormula, modified with a
gravitational energy term
For Z=0,
Adding the gravitaional energy :
Whre mn is the mass of the neutron, we get

8. If N is large we can neglect the term N2/3
av= 16 MeV,
asym= 30 MeV,
G= MeV fm (c4/GeV2)
mn= GeV/c2
r0= 1.15 fm
If N is large we can neglect the term N2/3
Gravitation helps to bind the system and we can determine the minimun
number of neutrons to bind the neutron star, is given B(0,Nmin )  0
R is fixed by the number of neutrons!!!!

9. N  1056 – , M  1 MS and R  10 km
And a density ,   1014 – g/cm3
A neutron star looks as gigantic nucleus where the
neutrons are bound by gravity, and the collapse is
avoided by the nuclear forces

10. All signals from the neutron star go through the surface  it is crucial
to have a good knowledge of the crust!!

18. One can use this EoS to study neutron and nuclear matter.

19. - stable neutron star matter
Neutron stars are not made only of neutrons but also include a small
fraction of protons ans electron!
The presence of lectrons guarantees charge neutrality.
There should be chemical equilibrium respect to the weak interaction  as many
neutron decays as electron capture reactions
The equilibrium can be expressed in terms of the chemical potentials.
Now since the mean free path of the neutrinos in a neutron star is larger than
the typical radius of such star, we assume that the neutrino escape freely
from the neutron star, and do not play a role in the equilibrium for these
reactions.

20. n = p + e
- equilibrium conditions
in terms of the chemical
potentials.
ρp = ρe
A possible equivalent way to find the composition of -stable matter consists
in minimizing the total energy density of the system constrained by the
subsidiary conditions that express the conservation of the different charges.
In the case of neutron star we have two conserved charges: total
barionic density and total electric charge!.
For the case that we consider, with p,n, and e- we have:
Minization conditions
We have two Lagrange multipliers
and we should minimize respect to
them

21. In general there will be as many independent chemical potentials as conserved
charges!
Remember

22. It is interesting to realice that the relations between the chemical potentials
is defined by the weak interaction. However, the chemical potentials
of the hadrons are determined by the strong interaction.
We can also include muons! It could be that the chemical potential of
electrons increases rather fast and becomes equal to the bare mass
of the muons. Than the minimization of F says that the system will
prefere to start to populate the Fermi sea of muons.
We still have only two independent chemical potentials
And the minimization produces the follwoing equations

24. The chemical potential
of the electron in
-stable matter is  100 MeV
Once the rest mass of the
muon is exceeded, the
electrons will start to decay
through a weak process,
that would not take place
In the free space.
Composition of -stable neutron star matter. On the left panel
we have only neutrons,protons and electrons. In the right one we
include also muons.

26. As soon as the chemical potential of the neutrons becomes sufficiently large.
Energetic neutrons can decay via strangeness non-conserving weak interactions
into  hyperons leading to a  Fermi sea with  =n .

27. M() = 1115 MeV
M(-) = 1197 MeV
One can also expect that - can appear through e- + n  - + e at lower densities
than than that of the , even though the - is more massive. The reason being,
that the above process removes both an energetic neutron and an energetic
electron. The negative charged hyperons appear in the ground state of -stable
matter when its mass equals n + e .

28. Composition of the -stable neutron star matter.
The composition depends on the model used for the interaction.
The dashed lines correspond to the NO inclusion of YY interactions.
In principle should nof affect the onset density of - .

29. Each curve correspond to a different composition.
Leptonic contributions are included! Treated as relativistic Fermi seas.
The presence of the hyperons produces a softening of the EoS.

30. M(r) is the gravitational mass enclosed in a radius r, and (r) is the
mass-energy density. They contain the bare mass and take care of
the binding energy.
In the limit of c , one recovers Newton’s experssion.
The factors in the numerator represent special relativity
corrections of order v2/c2 , and the factor in the denominator is a general
relativistic correction. All factors tend to 1 in the non-relativistic limit.
All corrections are positive defined, it is as if Newtonian gravity becomes
stronger for any value of r.

31. NN
NN-YN
NN-YN-YY
The radius does not change
so much.
The maximum mass is probably
too small.
The different EoS produce different maximum masses. When the hyperons
are present the maximum mass decreases. If rotation of the star is taken
into account, the maximum mass increases. It is as if the centrifugal effects
increase the pressure or that you need more mass to compensate for the
centrifugal effects.

32. Composition as a function of density (upper panel) and composition inside
T he star. The onset density of the hyperons allows to define an hyperonic radius