Nucleation of antikaon condensed matter in proto neutron stars
ABSTRACT A first order phase transition from nuclear matter to antikaon condensed
matter may proceed through thermal nucleation of a critical droplet of antikaon
condensed matter during the early evolution of proto neutron stars (PNS).
Droplets of new phase having radii larger than a critical radius would survive
and grow, if the latent heat is transported from the droplet surface to the
metastable phase. We investigate the effect of shear viscosity on the thermal
nucleation time of the droplets of antikaon condensed matter. In this
connection we particularly study the contribution of neutrinos in the shear
viscosity and nucleation in PNS.
arXiv:1108.1892v1 [astro-ph.HE] 9 Aug 2011
Nucleation of antikaon condensed matter in proto neutron
Sarmistha Banik and Rana Nandi
Saha Institute of Nuclear Physics, 1/AF Bidhan Nagar, Kolkata -7000064, India.
Abstract. A first order phase transition from nuclear matter to antikaon condensed matter may proceed through thermal
nucleation of a critical droplet of antikaon condensed matter during the early evolution of proto neutron stars (PNS). Droplets
of new phase having radii larger than a critical radius would survive and grow, if the latent heat is transported from the droplet
surface to the metastable phase. We investigate the effect of shear viscosity on the thermal nucleation time of the droplets of
antikaon condensed matter. In this connection we particularly study the contribution of neutrinos in the shear viscosity and
nucleation in PNS.
Keywords: Neutron stars, antikaon condensation, nucleation
PACS: 26.60.-c, 97.60.Jd, 05.70.Ce, 82.60.Nh
A first order phase transition from hadronic to exotic matter phases may proceed through the nucleation of droplets of
the new phase. The formation of droplets of quark matter [1, 2, 3] and antikaon condensed matter [4, 5] in neutrino-
free neutron stars (NS) was studied using the homogeneous nucleation theory of Langer . Droplets of new phase
may appear in the metastable nuclear matter due to thermal fluctuations. The droplets of the stable phase with radii
larger than a critical radius survives and grows if the latent heat is transported from the surface of the droplet to the
metastable state. This heat transportation occurs through thermal dissipation  and viscous damping .
We have seen that the onset of antikaon condensate influences the shear viscosity of NS matter composed of
neutron(n),proton(p),electron(e)andmuon(µ)that caninteract bystrongor electromagneticinteractions. Effectof
shear viscosity on the nucleation of the antikaon condensed matter was recently been studied  in deleptonised NS,
after the neutrinos are emitted. Here we investigate the contribution of neutrinos to the shear viscosity and nucleation
of antikaon condensates. In the PNS where the temperature is of the order of a few 10s of MeV, neutrinos are trapped
because their mean free paths under these conditions are small compared to the radius of the star. On the other hand,
they are very effective at transporting both heat and momentum because their mean free paths are orders of magnitude
larger than that for other particles.
We adopt the homogeneous nucleation theory of Langer  to calculate the thermal fluctuation rate for a first order
phase transition from the charge-neutral and beta-equilibrated nuclear matter to K−condensed matter in a neutrino-
trapped PNS. The thermal nucleation rate is given by
I = Γ0exp(−∆F(Rc)
where ∆F = is the change in free energy required to activate the formation of the critical droplet. Γ0=
the prefactor of which Ω0=
dynamical prefactor. Here σ is the surface tension for the surface separating the two phases, ξ is the correlationlength
for kaons, ∆w is the difference of the enthalpy of the two phases, λ the thermal conductivity, η and ζ are the shear
and bulk viscosity respectively. The free energy is maximum at this critical radius given by
is the statistical prefactor and κ =
3η +ζ??is the
Figure1: Totalshearviscosity innuclearmatterphaseisplot-
tedwith normalised baryonnumberdensityfori)neutrino-free
NS ii) neutrino-trapped PNS.
Γ 0 (c fm-4)
ξ= 5.0 fm
Figure 2: Prefactor as a function of temperature at a fixed
density and surface tension for PNS matter .
Finally the thermal nucleation time is given by τth= (VnucI)−1, whereVnuc=4π
the thermodynamic variables is assumed to remain constant.
To calculate shear viscosity in the PNS, which is mostly contributed by the neutrinos , we consider the
scattering of neutrinos νe+N → νe+N where N=n, p, e. Shear viscosity of neutrinos due to scattering is calculated
using the coupled Boltzmann transport equation . For the deleptonised NS, total shear viscosity is given by
η = ηn+ηp+ηe+ηµas in Ref. .
nucis the volume of the core, where
The Model EoS.
we construct at finite temperature using the relativistic mean field model [11, 12]. The interaction between baryons
is mediated by the exchange of scalar (σ) and vector (ω,ρ) mesons. This picture is consistently extended to include
the kaons. We use the parameter sets of GM1 model  for nucleon-meson coupling constants. The kaon-meson
coupling constants are determined using quark model, isospin counting rule and the real part of K−potential depth,
that we take as -160 MeV in our calculation .
In order to calculate the shear viscosity and critical radius (Rc), we need to know the EoS, that
The total shearviscosity is shownas a functionof normalisedbaryondensity fordifferenttemperaturesin Fig. 1.We
consider two cases i)the deleptonised NS matter, where the total shear viscosity has contribution from all the species
such as n, p, e and µ; ii)for the neutrino-trapped PNS matter (lepton fraction YL= 0.4), where the major contribution
comes from neutrinos. In both the cases shear viscosity decreases with rising temperature.
The prefactor Γ0is plotted as a function of temperature for PNS in Fig. 2 and is compared when it is approximated
by T4from dimensional analysis. Here we find the shear viscosity term changes the prefactor by a large order of
magnitude compared to T4approximation. This difference is much more pronounced in PNS compared to NS .
In Fig. 3 we display the nucleation time as a function of temperature for a set of values of surface tension at a
fixed baryondensity for lepton-trappedPNS matter and find that both the droplet radii and the thermal nucleationtime
strongly depend on the surface tension. Nucleation may not occur in PNS for surface tension <30 MeV fm−2as the
radius of droplet is less than the correlation length (ξ =∼ 5 fm). We approximate that the radius of the droplet should
be greater than ξ in this calculation[5, 7]. For NS case, nucleation is observed to be possible for σ < 20 MeV fm−2
. Larger viscosity there leads to larger value of T which might melt the condensate.
Finally in Fig. 4 we compare the results of thermal nucleation time taking into account the effect of shear viscosity
in the prefactor with that of the prefactor approximated by T4. For PNS we displayed the results for nb= 3.30n0and
σ = 35MeV fm−2. We find the result of the T4approximation a few orders of magnitude higher than those of our
calculation. These results demonstrate the importance of including the shear viscosity in the prefactor of Eq. (1) in
the calculation of thermal nucleation time. We already obtained similar results for NS matter . Also, it may be
log 10(τ) (sec)
Figure 3: Thermal nucleation time is displayed as a function
of temperature for PNS matter .
log 10τ sec
Figure 4: Our result for thermal nucleation time is compared
with the calculation of T4approximation for PNS.
mentioned that nucleation of antikaon condensates is possible if τth< τcooling(∼ 100s). That is possible for PNS with
σ ≥ 30 MeV fm−2.
We have investigated the role of shear viscosity on the thermal nucleation rate for the formation of a critical droplet
of antikaon condensed matter. For this we considered a first order phase transition from the nuclear to antikaon
the nucleation times for NS and PNS and found that nucleation is possible for a lower value of surface tension σ < 20
MeV fm−2in NS while it may be possible only for higher value in PNS ( σ ≥ 30 MeV fm−2).
S.B. would like to acknowledge the Department of Science & Technology, India for the travel grant to present this
paper in the conference PANIC11.
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