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arXiv:1108.1892v1 [astro-ph.HE] 9 Aug 2011

Nucleation of antikaon condensed matter in proto neutron

stars

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

INTRODUCTION

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 [6]. 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 [7] and viscous damping [8].

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[9]. Effectof

shear viscosity on the nucleation of the antikaon condensed matter was recently been studied [10] 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.

FORMALISM

We adopt the homogeneous nucleation theory of Langer [6] 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=

3√3

T

ξ

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

T

),

(1)

κ

2πΩ0, is

2

?σ

?3/2?RC

?4

is the statistical prefactor and κ =

2σ

R3c(∆w)2

?λT +2?4

3η +ζ??is the

Rc(T) =

2σ

(PK−PN).

(2)

Page 2

01234

nb/n0

10

15

20

25

30

log10η[g cm

-1s

-1]

T=1MeV

T=10MeV

T=30 MeV

T=100MeV

YL=0.4

YL=0

Figure1: Totalshearviscosity innuclearmatterphaseisplot-

tedwith normalised baryonnumberdensityfori)neutrino-free

NS ii) neutrino-trapped PNS.

5

10

Temperature (MeV)

15

20

25

30

1e-06

1e+00

1e+06

1e+12

1e+18

Γ 0 (c fm-4)

nb=3.3n0

ξ= 5.0 fm

T4

YL=0.4

σ=35MeV/ fm2

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 [13], 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 [13]. For the deleptonised NS, total shear viscosity is given by

η = ηn+ηp+ηe+ηµas in Ref. [9].

3R3

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 [14] 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 [9].

In order to calculate the shear viscosity and critical radius (Rc), we need to know the EoS, that

Results

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 [10].

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

[10]. 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 [10]. Also, it may be

Page 3

0 10

Temperature (MeV)

20 3040

0

50

100

150

200

log 10(τ) (sec)

σ=30MeV/ fm2

σ=35MeV/ fm2,

R=6.8 fm

R=5.85 fm

YL=0.4

nb=3.3n0

Figure 3: Thermal nucleation time is displayed as a function

of temperature for PNS matter .

1020

Temperature (MeV)

3040

0

50

100

150

200

250

log 10τ sec

T4

Ours

σ=35 MeV/fm2

YL=0.4

{

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.

Summary

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

condensedmatterinPNS.We haveseenthatthedropletradiiincreasewithincreasingsurfacetension.Wealsocompare

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).

ACKNOWLEDGMENTS

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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