Equations of State of Dense Matter in Neutron stars

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Equations of State of Dense Matter in Neutron
stars
S. Cerný Institute of Physics, Silesian
University at Opava J.R.Stone Department of
Physics, Oxford University S.Hledík, Z.Stuchlík
Institute of Physics, Silesian University at Opava

• Berlin
• July 23 29, 2006

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Introduction

• Interior of neutron star consists of various
  types of matter.
• Surface iron crust describe by Feynmann
  Metropolis Teller EOS
• Neutron rich nuclei - describe by Baym Pethick
  Sunderland EOS
• Neutron drip to dissolve nuclei - describe by
  Baym Bethe - Pethick EOS
• Center of neutron star different types of EOS
  with Kaons, Pions and Hyperons

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Neutron Star Model

• The neutron star models describe non-rotating,
  cold (T 0) neutron stars without magnetic
  field. The line element in standart Schwarzschild
  coordinates (t,r,?,f) is

(1)
where ? and ? are fuctions only of r.
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To get the equation of hydrostatic equilibrium,
we need Einstein field equation
(2)
where Gµ? is Einstein tensor, Rµ? is Ricci
tensor, R is Ricci scalar and Tµ? is
stress-energy tensor.
For a perfect fluid,
(3)
The conservation of energy and momentum is
expressed by
(4)
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• The Equation of hydrostatic equilibrium
• can be derived from Eqs (2)-(4) we get

(5)
where
is the mass inside radius r. Eq. (5) is TOV
equation of hydrostatic equilibrium. For
obtaining our neutron-star models, we need to
integrate Eq.(5) from the center out to the
stellar surface (where P?0). For given central
conditions (central energy, density ? and the
corresponding pressure P), we obtain a
neutron-star model and its gross properties (mass
M, radius R, total baryon number A etc.)
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Relativistic Mean Field Model with Kaons (Pions)

• Neither pions or kaons exist as a component
  of nuclear matter under normal circumstances. The
  pion has negative parity and consequently its
  expectation value in the normal ground state,
  which is diagonal matrix element, vanishes.
• The kaon carries strangeness and its
  expectation likewise vanishes.

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• However, conditions may arise at
• higher density that favor a change in
• the structure of the ground state,
• yielding finite expectation values and
• permitting the appearance of a Bose
• condensate.
• The electron chemical potential incre-
• ases with the density of charge neutral
• matter to compensate the growing
• density of protons. The growth of the
• electron chemical potential can be
• arrested if hyperons of negative charge
• become an important component of
• baryon populations.

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• A crucial question with regard to the occurr-
• ence of kaon (pion) condensation is how
• well neutrality can be archived among
• baryon carrying species.
• The Kaon is coupled to the mean field
• using minimal coupling

(6)
where the vector fields are coupled by defining
(7)
The Lagrangians for the kaons describes the
kaon-kaon interaction as well as the
kaon- nucleon interaction.
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• The form of equation of state depends
• sensitively on the chosen optical potential
• of the kaon UK (?0), which reflects its inter-
• action with nuclear medium. The value vary
• between -80 MeV and -140 MeV.
• The radius expecially the limiting mass are
• sensitive functions of UK (?0).
• For the preferred value of UK(?0) lt-120MeV,
• radii are similar to neutron stars without the
• condensate. However the behavior is actu-
• ally continuous, but depends sensitively on
• UK (?0) a pure kaon condensed core deve-
• lops with decreasing values of the optical
• potential below -120 MeV and this
• causes the change of the radius from
• R12.5 km to R8 km for UK(?0) -140MeV.

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Conclusions

• The presence of kaons in neutron stars occurs in
  the inner region up to 5 km.
• Equation of state with hyperons is softened
• and the maximum mass of NS lowered.
• (Some models without hyperons show maxi-
• mum mass 2 solar masses no obser-
• vations)
• We know that, according to present obser-
• vations, it appears that the lover bound on
• the maximum NS mass is 1.5 solar masses.

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Thank you for your attention.