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\begin{document}
\title{Incompressibility of strange matter\footnote{Work supported  in  part  by DST  grant no. SP/S2/K-03/01, Govt. of India.}.}

\author{ Monika Sinha $^{1,2,3}$}
\email{sinha_monika@hotmail.com}
\author{Manjari Bagchi$^{1,2}$} 
\author{Jishnu Dey $^{2,4,**}$}
\email{deyjm@giascl01.vsnl.net.in}
\author{Mira Dey $^{1,4,**}$}
\email{deyjm@giascl01.vsnl.net.in}
\affiliation{$^1$ Dept. of Physics, Presidency College, 86/1 College Street, Kolkata 700073,~India; \\$^2$ Dept. of Physics, Maulana Azad College, 8  Rafi Ahmed Kidwai Road, Kolkata 700013,~India;\\ $^3$ CSIR NET Fellow; \\ $^4$ Associate, IUCAA, Pune, India; \\$^{**}$ Permanent address: 1/10 Prince Golam Md. Road, Kolkata 700 026, India}

\author{Subharthi Ray}
\email{sray@if.uff.br}
\affiliation{Instituto de Fisica, Universidade Federal Fluminense, Niteroi, RJ, Brasil, FAPERJ Fellow(E-26/152.117/2001),}
  
\author{Siddhartha Bhowmick}
\affiliation{ Department of Physics, Barasat Govt. College, Kolkata 700 124, India.}

\date{today}

\begin{abstract}
  {Strange stars (ReSS) calculated from a realistic equation of
state (EOS) \cite{d98} show  compact objects in the mass radius
curve, when they are solved for gravitational fields via TOV
equation. Many of the observed stars seem to fit in with this kind
of compactness irrespective of whether they are X-ray pulsars,
bursters or soft $\gamma$ repeaters or radio pulsars. Calculated
incompressibility of the ReSS matter shows continuity with that of
nuclear matter. This is important in the cosmic separation of
phase scenario.
We compare our calculations of incompressibility with that of a
nuclear matter EOS. This EOS has a continuous transition to
ud-matter at about five times normal density. From a look at the
consequent velocity of sound it is found that the transition to
ud-matter seems necessary. }
\end{abstract}

\keywords{ compact stars~--~realistic strange stars~--~dense
matter~--~elementary particles ~--~ equation of state}

\pacs{97.10.Q -- 97.10.Cv -- 97.60.-s -- 12.39.-x -- 12.40.y -- 51.30.+i -- 51.35.+a.}

\maketitle

\section{Introduction}
    A most exciting aspect of modern astrophysics is the
possible existence of a family of compact stars made entirely of
deconfined u,d,s quark matter or ``strange matter" (SM) and thereby
denominated strange stars (SS). They differ from neutron stars,
where quarks are confined within neutrons, protons and eventually
within other hadrons (hadronic matter stars). The possible
existence of SS is a direct consequence of the so called strange
matter hypothesis\cite{bodmer}, according to which the energy per
baryon of SM would be less than the lowest energy per baryon found
in nuclei, which is about 930 MeV for $Fe^{56}$. Also, the
ordinary state of matter, in which quarks are confined within
hadrons, is a metastable state. Of course, the hypothesis does not
conflict with the existence of atomic nuclei as conglomerates of
nucleons, or with the stability of ordinary matter \cite{farhi,madsen,Bombaci}.

    The best observational evidence for the existence of quark stars
come from some compact objects, the X-Ray burst sources
SAX~J1808.4$-$3658 (the SAX in short) and 4U~1728$-$34, the X-ray
pulsar Her X-1 and the superburster 4U~1820$-$30. The first is
the most stable pulsating X-ray source known to man as of now.
This star is claimed to have a mass $M* \sim 1.4 ~M_\odot~ $ and
a radius of about 7 kms \cite{Li99a}. Coupled to this claim are
the various other evidences for the existence of ReSS, such as the
possible explanation of the two kHz quasi-periodic oscillations
in 4U 1728~-~34 \cite{Li99b} and the quark-nova explanation for
$\gamma$ ray bursts \cite{boma}.

The expected behaviour of SS is directly opposite to that of a
neutron star as Fig.(\ref{MR}) shows. The mass of 4U~1728$-$34 is
claimed to be less than  1.1 $M_\odot$ in Li et al. \cite{Li99b},
which places it much lower in the M-R plot and thus it could be
still gaining mass and is not expected to be as stable as the SAX.
So for example, there is a clear answer \cite{singhini} to the
question posed by Franco \cite{fr01}: why are the pulsations of
SAX not attenuated, as they are in 4U~1728$-$34 ?

    From a basic point of view the equation of state for SM should
be calculated solving QCD at finite density. As we know, such a
fundamental approach is presently not feasible even if one takes
recourse to the large colour philosophy of 't Hooft \cite{'t
Hooft}. A way out was found by  Witten \cite{wit} when he
suggested that one can borrow a phenomenological potential from
the meson sector and use it for baryonic matter. Therefore, one
has to rely on phenomenological models. In this work, we use
different equations of state (EOS) of SM proposed by Dey et al
\cite{d98} using the phenomenological Richardson potential. Other
variants are now being proposed, for example the chromo dielectric
model calculations of Malheiro et al.\cite{mal}.

\begin{figure}[htbp]
\centerline{\psfig{figure=ssvbbb.eps,height=5cm}}\vskip .5cm
\caption{\label{MR} Mass and radius of stable stars with the strange
star EOS (left curve) and neutron star EOS (right curve), which
are solutions of the Tolman-Oppenheimer-Volkoff (TOV) equations
of general relativity. Note that while the self sustained strange
star systems can have small masses and radii, the neutron stars
have larger radii for smaller masses since they are bound by
gravitation alone.}
\end{figure}

\begin{figure}[htbp]
\centerline{\psfig{figure=ebya.eps,height=5cm}}
\vskip .5cm
\caption{Strange matter EOS employed by D98 show respective
stable points. The solid line for is EOS1, the dotted line for
EOS2 and the dashed line for EOS3. All have the minimum at energy
per baryon less than that of $Fe^{56}$} \label{efit}
\end{figure}

Fig.(\ref{efit}) shows the energy per baryon for the EOS of
\cite{d98}. One of them (eos1, SS1 of \cite{Li99a}) has a minimum
at E/A = 888.8 $MeV$ compared to 930.4 of $Fe^{56}$, i.e., as
much as 40 $MeV$ below. The other two have this minimum at 911
$MeV$ and 926 $MeV$, respectively, both less than the normal
density of nuclear matter. The pressure at this point is zero and
this marks the surface of the star in the implementation of the
well known TOV equation. These curves clearly show that the
system can fluctuate about this minimum, so that the zero
pressure point can vary.

\section{Incompressibility : its implication for Witten's Cosmic
Separation of Phase scenario.}

    In nuclear physics incompressibility is defined as
\be K~=~ 9 \f{\p} {\p n}\left(n^2\f{\p\varepsilon}{\p n}\right)~,
\label{com}\ee where $\va~=~E/A$ is the energy per particle of
the nuclear matter and $n$ is the number density. The relation of K with
bulk modulus B is \be K=\f{9B}n~~.\label{bulk}\ee $K$ has been
calculated in many models. In particular, Bhaduri et al
\cite{bdp} used the non - relativistic constituent quark model,
as well as the bag model, to calculate $K_A$ as a function of $\
n$ for the nucleon and the delta. They found that the nucleon has
an incompressibility $ K_N$ of about 1200 MeV, about six times
that of nuclear matter. They also suggested that at high density
$K_A$ matches onto quark gas incompressibility.

The velocity of sound in units of light velocity c is given by
\be v = \sqrt{K/9 \va}\label{vel}.\ee  There is a common
misconception that $dp/d\ep$ is the square of sound velocity -
this is equal to $B/(\ep +
p)$ except at the stable point for the system, where the pressure p is
equal to zero when it is equal to $B/\ep~=~K/9\va~=~v^2$. Here $\ep$ is
the energy density.

The simple models of quark matter considered in \cite{d98} use a
Hamiltonian with an interquark potential with two parts, a scalar
component (the density dependent mass term) and a vector
potential originating from gluon exchanges. In the absence of an
exact evaluation from QCD, this vector part is borrowed from meson
phenomenology \cite{richardson}. In common with the
phenomenological bag model, it has built in asymptotic freedom
and quark confinement (linear). In order to restore the
approximate chiral symmetry of QCD at high densities, an {\it
ansatz} is used for the constituent masses, viz.,
\begin{equation}
M(n) = M_Q~sech\left(\nu\f{n}{n_0}\right), \label{sech}
\end{equation}
where $n_0$ is normal nuclear matter density and $\nu$ is a
parameter. In figure (\ref{incmas}), $K$ with three values of
$M_Q$ implying different running masses, $M(n)$, is plotted as a
function of the density expressed by its ratio to $n_0$. Given
for comparison, is the incompressibility $K_q$ of a perturbative
massless three flavour quark gas consisting of zero mass current
quarks \cite{bdp} using the energy expression given in \cite{baym}
to order $\alpha_s ^2$.

\begin{table}[htbp]
\caption{\label{eos}Parameters for the three EOS}
\vskip .5cm
\begin{ruledtabular}
\begin{tabular}{lccccccr}
EOS&$\nu$&$\al_s$&$M_G/M_{\odot}$&R& $n_s/n_o$\\
&&&&(km)&\\
\hline EOS1&0.333&0.20&1.437&7.055&4.586\\
\hline EOS2&0.333&0.25&1.410&6.95&4.595\\
\hline EOS3&0.286&0.20&1.325&6.5187&5.048\\
\end{tabular}
\end{ruledtabular}
\end{table}


\begin{figure}[htbp]
\vskip .5cm
\centerline{\psfig{figure=inmass.eps,height=5cm}}
\vskip .5cm
\caption{\label{incmas} Incompressibility as a function of density ratio for
EOS's with different constituent mass as parameter. 
Dashed lines correspond to
perturbative massless three flavour quark gas with different values of
$\alpha_s$ (see \cite{bdp,baym}).}
\end{figure}

The general behaviour of the curves is relatively insensitive to
the parameter $\nu$ in $M(n)$ as well as the gluon mass, as
reflected in the EOS. It can be seen that as $M_Q$ decreases,
the nature of the relation approaches the perturbative case of
\cite{bdp}. At high density our incompressibility and that due to
Baym \cite{baym} matches, showing the onset of chiral symmetry restoration. In
EOS1 for uds matter, the minimum of $\varepsilon$ occurs at about
$4.586 ~ n_0$. nucleation may occur at a density less than this
value of $n$. This corresponds to a radius of about $0.67~fm$ for
a baryon. For EOS1 we find $K$ to be $1.293~GeV$ per quark at the
surface.

It is encouraging to see that this roughly matches with the
compressibility $K_N$ so that no `phase expands explosively'. In
the Cosmic Separation of Phase scenario, Witten \cite{wit} had
indicated at the outset that he had assumed the process of phase
transition to occur smoothly without important departure from
equilibrium. If the two phases were compressed with significantly
different rates, there would be inhomogenieties set up.

But near the star surface at $n~\sim~4$ to $5~n_0$ the matter is
more incompressible showing a stiffer surface. This is in keeping
with the stability of strange stars observed analytically with
the Vaidya-Tikekar metric by Sharma, Mukherjee, Dey and Dey
\cite{smdd}.

    The velocity of sound, $v_s$ peaks somewhere around the middle of the star
and then falls off. We show it for three different EOS in
fig.(\ref{vels}) with parameters given in Table (\ref{eos}).

\begin{figure}[htbp]
\vskip .5cm
\centerline{\psfig{figure=vels.eps,height=5cm}}
\vskip .5cm
\caption{Velocity of sound, $v_s$ as a function of density
ratio. The solid line is for EOS1, the dotted for EOS2 and the
dashed for EOS3.} \label{vels}
\end{figure}

In a recent paper \cite{krein} Krein and Vizcarra (KV in short)
have put forward an EOS for nuclear matter which exhibits a
transition from hadronic to quark matter. KV start from a
microscopic quark-meson coupling Hamiltonian with a density
dependent quark-quark interaction and construct an effective
quark-hadron Hamiltonian which contains interactions that lead to
quark deconfinement at sufficiently high densities. At low
densities, their model is equivalent to a nuclear matter with
confined quarks, i.e., a system of non-overlapping baryons
interacting through effective scalar and vector meson degrees of
freedom, while at very high densities it is ud quark matter. The
$K_{NM}$ at the saturation density is fitted to be $248~MeV$.
This EOS also gives a smooth phase transition of quark into
nuclear matter and thus, conforms to Witten's assumption.
Interestingly enough, the transition takes place at about $\sim 5
n_0$.

The KV model does not incorporate strange quarks so that
comparison with our EOS is not directly meaningful. However it is
quite possible that the signal results of the KV calculations
mean that quark degrees of freedom lower the energy already at
the ud level and once the possibility of strange quarks is
considered the binding exceeds that of $Fe^{56}$. An extension of
KV with strange quarks is in progress\footnote{G. Krein, e-mail}.

\begin{figure}[htbp]
\centerline{\psfig{figure=krin.eps,height=5cm}}
\vskip .5cm
\caption{Incompressibility as a function of density ratio for
pure nuclear matter and quark-nucleon system.} \label{krin}
\end{figure}

Results obtained from the KV calculation are presented
simultaneously. The incompressibility shows softening and the
velocity of sound decreases when quark degrees of freedom open
up, as expected. At $\sim~5n_0$ $K_{qN}$ is about $2~GeV$ in
qualitative agreement with our value.

Note that in our EOS we also have strange quarks reducing the
value of $K$. Fig (\ref{krvs1}) shows the $v_s$ as a function of
$n_B/n_0$ for both kinds of EOS. The EOS with quarks shows a
lowering of $v_s$.

\begin{figure}[htbp]
\centerline{\psfig{figure=krvs.eps,height=5cm}}
\vskip .5cm
\caption{The velocity of sound in pure nuclear matter and in
quark-nuclear system as a function of density ratio. For pure 
nuclear matter at $9 n_0$ the sound velocity is too close
to that of light c, whereas for quark-nuclear system it is much less,
about 0.5 c.}
\label{krvs1}
\end{figure}


\begin{thebibliography}{}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem{d98} M. Dey, I. Bombaci, J. Dey, S. Ray  \& B. C.
Samanta, Phys.  Lett. B438 (1998) 123; Addendum  B447 (1999) 352;
Erratum B467 (1999) 303;  Indian J. Phys. 73B (1999) 377.

\bibitem{bodmer} N. Itoh, Prog. Theor. Phys. 44, 291 (1970),
A. R. Bodmer, Phys. Rev. D 4, 1601 (1971); H. Terazawa, INS Rep.
336 (Univ. Tokyo, INS) (1979); E. Witten, Phys.Rev. D 30,
272(1984).

\bibitem{farhi} E. Farhi and R. L. Jaffe, Phys. Rev. D 30,2379
(1984)

\bibitem{madsen} J. Madsen, Lecture notes in Physics, Springer verlag, 516 (1999).

\bibitem{Bombaci} I. Bombaci, Strange Quark Stars: structural
properties and possible signatures for their existence, in
"Physics of neutron star interiors", Eds. D. Blaschke, et al.,
Lecture notes in Physics, Springer verlag, 578, (2001).

\bibitem{Li99a} X-D. Li, I. Bombaci, M. Dey, J. Dey  \& E.P.J. van den
Heuvel, Phys. Rev. Lett. 83 (1999) 3776.

\bibitem{Li99b} X-D. Li, S. Ray, J. Dey, M. Dey  \& I.  Bombaci,
Astrophys. J. Lett 527 (1999) L51.

\bibitem{boma} I. Bombaci and B. Datta, Astrophys. J. Lett 530 (2000) L69; R.
Ouyed, J. Dey and M. Dey, v3, Astron. \&
Astrophys. Lett. (in press).


\bibitem{singhini} M. Sinha, J. Dey, M. Dey, S. Ray and
S. Bhowmick, ``Stability of strange stars (SS) derived
from a realistic equation of state", Mod. Phys. Lett. A (in
press).

\bibitem{fr01} L. M. Franco, The Effect of Mass Accretion Rate on
the Burst Oscillations in 4U~1728-34.  Astrophys. J. Lett
(2001).

\bibitem{'t Hooft} G. 't Hooft, Nucl. Phys. B72 (1974) 461;
B75 (1974) 461.

\bibitem{wit} E. Witten,  Nucl. Phys  B160 (1979) 57.

\bibitem{mal} M. Malheiro,  E. O. Azevedo,  L. G. Nuss,  M.
Fiolhais  and  A. R. Taurines,  v1.

\bibitem{bdp} R. K. Bhaduri, J. Dey and M. A. Preston, Phys. Lett.
B 136 (1984) 289.

\bibitem{richardson} J. L. Richardson, Phys. Lett. B82 (1979) 272.

\bibitem{baym} G. Baym, Statistical mechanics of quarks and
hadrons. ed. H. Satz, (North holland, Amsterdam, 1981) p.17.

\bibitem{smdd} R. Sharma, S. Mukherjee, M. Dey and  J. Dey, Mod.
Phys. Lett. A, 17 (2002) pp. 827-838.

\bibitem{krein} G. Krein \& V. E. Vizcarra, arXiv:
v1. Paper submitted to Proceedings of JJHF, University of
Adelaide, Australia.









\end{thebibliography}
\end{document}

