# Structure of Spin Polarized Strange Quark Star in the Presence of Magnetic
Field at Finite Temperature

G. H. Bordbar1,2 111Corresponding author. E-mail<EMAIL_ADDRESS>F. Kayanikhoo 1 and H. Bahri 1 Department of Physics, Shiraz University,
Shiraz 71454, Iran
and
Research Institute for Astronomy and Astrophysics of Maragha, P.O. Box
55134-441, Maragha 55177-36698, Iran

###### Abstract

In this paper, we have calculated the thermodynamic properties of spin
polarized strange quark matter at finite temperature in the presence of a
strong magnetic field using MIT bag model. We have also computed the equation
of state of spin polarized strange quark matter in the presence of strong
magnetic field and finally, using this equation of states we have investigated
the structure of spin polarized strange quark star at different temperatures
and magnetic fields.

##  I Introduction

One of the interesting features of the compact objects is the strange quark
star (SQS) composed of the strange quark matter (SQM). The composition of SQS
was first proposed by Itoh rk3 ; simultaneously with formulation of Quantum
chromo dynamics (QCD). Later, Bodmer discussed the fate of an astronomical
object collapsing to such a state of matter rk4 . The perturbative
computations of the equation of state of the SQM was developed after the
formation of QCD, but the region of validity of these calculations was
restricted to very high densities rk50 . The concept of SQS was also discussed
by witten. He proposed that SQM composed of light quarks, and is more stable
than the nuclei rk5 . Therefore the SQM can be the ground state of matter.
With this point of view, other authors proposed the concept of SQS. The SQM is
composed of 3-flavors of quarks (up, down and strange) and a small number of
electrons to ensure the charge neutrality rk6 ; rk6p . A typical electron
fraction is less than $10^{-3}$ $(fm^{-3})$ and it decreases from the surface
to the center of SQS.

The collapsing star of a supernova will turn into a neutron star only if its
mass is about $1.4-3$ $M_{\odot}$. After formation of the neutron star, if
density of core is high enough ($10^{15}\ gr/cm^{3}$) the nucleons dissolve to
their components, quarks, and a hybrid star (neutron star with a core of SQM)
is formed . But if after the explosion of supernova, in the stage of proto-
neutron, the density of the core is high enough ($10^{15}\ gr/cm^{3}$), the
pure SQS may be formed, directly. Making a quark-novae. If after collapse the
leftover is more than $3M_{\odot}$ mass the star will be transform to a black
hole. For a neutron star, the radius decrease with mass, or in other word
$M\propto\ 1/R$; while for an SQS, the mass-radius relation is as $M\propto\
R^{3}$. Some result of observations show that for a typical SQS, the mass is
more than $2.1\pm 0.28\ M_{\odot}$ and the radius is more than $13.8\pm 1.8\
km$ rk7 . Besides, recent observations indicate that objects RX J185635-3754
and 3C58 may be SQS rk51 .

The maximum magnetic field on the surface of magnetars is about $10^{15}\ G$,
however the highest possible magnetic field in the center of SQS that was
estimated by virial theorem is between $10^{18}-10^{20}\ G$ rk75 ; rk62 .
Therefore, investigating the effect of an strong magnetic field on the SQM
properties is important in astrophysics. Some studies that investigate the
effect of magnetic field on quark star and quark matter are available.
Chakrabarty rk63  studied the effects of an strong magnetic field on quark
matter. He showed that the equation of state of SQS changes significantly at
the presence of strong magnetic fields. Anand et.al. rk64  expressed that the
maximum mass, the radius and gravitational red-shift of the SQS are increasing
as a function of the magnetic field.

Recently, we have calculated the structure of polarized SQS at zero
temperature rk8  and unpolarized SQS at finite temperature rk9 . We have also
calculated the structure of the neutron star with the quark core at zero
temperature rk10  and at finite temperature rk11 . In this paper we calculate
some properties of polarized SQS at finite temperature in the presence of an
strong magnetic field. The plan of this work is as follows. In section II, we
calculate the energy and equation of state of SQM by the MIT bag model at
finite temperature in the presence of a strong magnetic field and in section
III, we present the results of our calculations for different temperatures and
magnetic fields.

##  II Calculation of energy and equation of state of strange quark matter

The equation of state plays an important role in obtaining the structure of a
star. There are different statistical approaches for calculation of the
equation of state of quark matter (SQM) and all of them are based on the QCD,
for example the MIT bag model rk20 ; rk21 , NJL model rk22 ; rk23  and
perturbation QCD model rk24 ; rk25 . In this paper we calculate the equation
of state of SQM by MIT bag model with a density dependent bag constant. The
MIT bag model considers free particles confined to a bounded region by a bag
pressure $B$. This pressure depends on the quark-quark interaction rk55 . In
other words, the bag constant ($B$) is the difference between the energy
densities of the noninteracting quarks and the interacting ones. For the bag
constant in the MIT bag model, different values such as $55$ and $90\
MeV/fm^{3}$ are considered.

To obtain the equation of state of SQM, at first, we should calculate the
total energy of the system. We do this work using the MIT bag model with a
density dependent bag constant in the next section.

###  II.1 Energy of spin polarized strange quark matter at finite temperature
in the presence of magnetic field

At densities high enough ($10^{15}\ gr/cm^{3}$) the hadronic phase is allowed
to undergo a phase transition to the strange quark matter phase. Strange quark
matter consists of $u$, $d$ and $s$ quarks, as well as electrons in weak
equilibrium;

$d\rightarrow u+e+\overline{\vartheta}_{e},$ (1) $u+e\rightarrow
d+\vartheta_{e},$ (2) $u+d\rightarrow u+s,$ (3) $s\rightarrow
u+e+\overline{\vartheta}_{e}.$ (4)

To calculate the energy of SQM, we need to know the quark densities in term of
the baryonic number density. To compute these densities, we use the beta
equilibrium and charge neutrality conditions in the above reactions. The beta
equilibrium and charge neutrality conditions lead to the following relations
for the chemical potentials and densities of relevant quarks,

$\mu_{d}=\mu_{u}+\mu_{e},$ (5) $\mu_{s}=\mu_{u}+\mu_{e},$ (6)
$\mu_{s}=\mu_{d},$ (7)
$\frac{2}{3}n_{u}-\frac{1}{3}n_{s}-\frac{1}{3}n_{d}-n_{e}=0,$ (8)

where $\mu_{i}$ and $n_{i}$ are the chemical potential and the number density
of particle $i$ respectively. It should be noted that because the neutrinos
escape from the interior matter of star, the chemical potential of neutrinos
vanishes ($\mu_{\vartheta_{e}}=\mu_{\overline{\vartheta}_{e}}=0$). Therefore,
we get the above relations between the chemical potentials rk6 ; rk6p . As it
was mentioned in the previous section, we consider the system as pure SQM and
ignore the electrons($n_{e}=0$) rk13 ; rk14 ; rk15 , therefore Eq. (8) becomes
as follows,

$n_{u}=\frac{1}{2}(n_{s}+n_{d}).$ (9)

Now, we want to calculate the energy of SQM in the presence of an strong
magnetic field. Therefore, we consider the spin polarized SQM composed of the
$u$, $d$ and $s$ quarks with up and down spins. We show the number density of
spin-up quarks by $n_{i}^{+}$ and spin-down quarks by $n_{i}^{-}$. The
parameter of polarization is defined by the following relation,

$\zeta_{i}=\frac{n_{i}^{+}-n_{i}^{-}}{n_{i}},$ (10)

where $0\leq\zeta_{i}\leq 1$ and $n_{i}=n_{i}^{+}+n_{i}^{-}$. The chemical
potential $\mu_{i}$ for any value of the temperature ($T$) and number density
($n_{i}$), using the following constraint,

$n_{i}=\sum_{P=\pm}\frac{g}{2\pi^{2}}\int_{0}^{\infty}f(n_{i},k^{(p)},T)k^{2}dk,$
(11)

where

$f(n_{i},k^{(p)},T)=\sum_{P=\pm}\frac{1}{exp\left(\beta((m_{i}^{2}c^{4}+\hbar^{2}(k^{(p)})^{2}c^{2})^{1/2}-\mu_{eff})\right)+1},$
(12)

is the Fermi-Dirac distribution function. In the above equation
$\beta=1/k_{B}T$ and $g$ is degeneracy number of the system. In this equation
$\mu_{eff}$ is the effective chemical potential in the presence of magnetic
field. The effective chemical potential is equal to $\mu_{i}\mp\mu_{s}B$ where
$\mu_{s}$ is magnetic moment and $B$ is the magnetic field. In other word, the
single particle energy is
$\varepsilon_{i}=\sqrt{m_{i}^{2}c^{4}+\hbar^{2}k^{p}}\pm\mu_{s}B$.

The energy of spin polarized SQM in the presence of the magnetic field within
the MIT bag model is as follows,

$\varepsilon_{tot}=\varepsilon_{u}+\varepsilon_{d}+\varepsilon_{s}+\varepsilon_{M}+B,$
(13)

where

$\varepsilon_{i}=\sum_{P=\pm}\frac{g}{2\pi^{2}}\int_{0}^{\infty}(m_{i}^{2}c^{4}+\hbar^{2}(k^{(P)})^{2}c^{2})^{1/2}f(n_{i},k^{(p)},T)k^{2}dk,$
(14)

here $k^{\pm}=(\pi^{2}n_{i})^{1/3}(1\pm\zeta_{i})^{1/3}$. In our calculations,
we suppose that $\zeta=\zeta_{u}=\zeta_{d}=\zeta_{s}$. In Eq. (13),
$\varepsilon_{M}=\frac{E_{M}}{V}$ is the magnetic energy density of SQM where
$E_{M}=-M.B$ is the magnetic energy. By considering a uniform magnetic field
along $z$ direction, the contribution of magnetic energy of the spin polarized
SQM is given by

$E_{M}=-\sum_{i=u,d,s}M_{z}^{(i)}B,$ (15)

where $M_{z}^{(i)}$ is the magnetization of the system corresponding to
particle $i$ which is given by

$M_{z}^{(i)}=N_{i}\mu_{i}\zeta_{i}.$ (16)

In the above equation $N_{i}$ and $\mu_{i}$ are the number density and
magnetic moment of particle $i$. Finally, the magnetic energy density of SQM
can be obtained by the following relation,

$\varepsilon_{M}=-\sum_{i}n_{i}\mu_{i}\zeta_{i}B.$ (17)

We obtain the thermodynamic properties of the system using the Helmholtz free
energy,

$F=\varepsilon_{tot}-TS_{tot},$ (18)

where $S_{tot}$ is the total entropy of SQM,

$S_{tot}=S_{u}+S_{d}+S_{s}.$ (19)

In Eq. (19), $S_{i}$ is entropy of particle $i$,

$\displaystyle S_{i}(n_{i},T)$ $\displaystyle=$
$\displaystyle-\frac{3}{\pi^{2}}k_{B}\int_{0}^{\infty}f(n_{i},k,T)ln(f(n_{i},k,T))$
$\displaystyle+$ $\displaystyle(1-f(n_{i},k,T))ln(1-f(n_{i},k,T))]k^{2}dk.$

###  II.2 Equation of state of spin polarized strange quark matter

For deriving the equation of state of strange quark matter (SQM), the
following equation is used,

$P(n,T)=\sum_{i}(n_{i}\frac{\partial F_{i}}{\partial n_{i}}-F_{i}),$ (21)

where $P$ is the pressure of system, $F_{i}$ is the free energy of particle
$i$ and $n_{i}$ is the numerical density of particle $i$.

##  III Results and discussion

###  III.1 Thermodynamic properties of spin polarized strange quark matter

In Fig. 1, we present the total free energy per volume versus the polarization
parameter $\zeta$, for $B=5\times 10^{18}\ G$, at different densities and
temperatures. From Fig. 1, we can see that at each density and temperature,
the energy is minimum at a particular value of the polarization parameter.
This indicates that at each density and temperature, there is a meta stable
state. We have found that for a fixed temperature, the polarization parameter
corresponding to the meta stable state reaches zero by increasing the density.
Similarly, we can see that at a fixed density, in the meta stable state the
polarization parameter decreases by increasing the temperature. These results
agree with those of reference rk94 .

In Fig. 2, we have plotted the polarization parameter versus baryonic density
at different temperatures in the presence of magnetic field $B=5\times
10^{18}\ G$. We can see that the polarization parameter decreases by
increasing the baryonic density. Fig. 2 shows that at a fixed density, the
polarization parameter decreases by increasing the temperature. This is due to
the fact that contribution of magnetic energy is more significant at low
temperature. The polarization parameter has been also drawn versus baryonic
density at a fixed temperature ($T=30\ MeV$) for different magnetic fields in
Fig. 3. It is seen that for each density, the polarization parameter increases
by increasing the magnetic field. As it can be seen from Fig.3, at high
baryonic densities, for the magnetic fields lower than about $5\times 10^{18}\
G$ the polarization parameter nearly becomes zero. In other words, for lower
magnetic fields at higher densities, the strange quark matter (SQM) becomes
nearly unpolarized. In Fig. 4, we have shown the polarization parameter as a
function of the magnetic field for density $n=0.5\ fm^{-3}$ at different
temperatures. We can see that at lower values of the magnetic field, the
polarization parameter is nearly zero, and at the magnetic fields greater than
about $5\times 10^{17}\ G$, the polarization parameter increases as the
magnetic field increases. Also, this figure shows that for each value of the
magnetic field, the polarization parameter increases by decreasing the
temperature. This is because at a higher temperatures the contribution of the
kinetic energy is higher compared to that of the magnetic energy.

We have shown the total free energy per volume of the spin polarized SQM as a
function of the baryonic density in Fig. 5 for the magnetic field $B=5\times
10^{18}\ G$ at different temperatures. From this figure,we have found that the
free energy of the spin polarization SQM has positive values for all
temperatures and densities. For each density, it is shown that the free energy
decreases by increasing the temperature.

The pressure of spin polarized SQM at different temperatures in the presence
of magnetic field ($B=5\times 10^{18}\ G$) has been plotted in Fig. 6. This
figure shows that the pressure of SQM increases by increasing the density. We
can also see that the pressure increases by increasing temperature. These
results indicate that the equation of state of spin polarized SQM becomes
stiffer by increasing the temperature.

###  III.2 Structure of spin polarized strange quark star

The structures of compact objects such as white dwarfs, neutron stars, pulsars
and strange quark stars (SQS), are determined from the Tolman-Oppenheimer-
Volkov equations (TOV) rk16 ,

$\frac{dP}{dr}=-\frac{G\left[\varepsilon(r)+\frac{P(r)}{c^{2}}\right]\left[m(r)+\frac{4\pi
r^{3}P(r)}{c^{2}}\right]}{r^{2}\left[1-\frac{2Gm(r)}{rc^{2}}\right]},$ (22)
$\frac{dm}{dr}=4\pi r^{2}\varepsilon(r).$ (23)

Using the equation of state which was obtained in the previous section, we
integrate the TOV equation to compute the structure of SQS.

In Fig. 7, we have plotted the gravitational mass of spin polarized strange
quark star (SQS) versus energy density at different temperatures for a
magnetic field $B=5\times 10^{18}\ G$. We can see that for all temperatures,
the gravitational mass increases rapidly by increasing the energy density and
finally reaches a limiting value (maximum gravitational mass). Fig. 7 shows
that this limiting value decreases by increasing the temperature. From this
result, we can conclude that the stiffer equation of state leads to the lower
values for the gravitational mass of SQS.

We have shown the gravitational mass of spin polarized SQS as a function of
the radius at different temperatures in Fig. 8. This figure shows that the
gravitational mass increases by increasing radius. We can see that the
increasing rate of gravitational mass versus radius increases by increasing
the temperature. According to Fig. 6, by increasing temperature, the equation
of state becomes stiffer leading to the conclusion that by increasing
temperature, mass and the radius of spin polarized SQS decrease. Fig. 9 shows
the gravitational mass of SQS as a function of the energy density at a fixed
temperature ($T=30\ MeV$) for different magnetic fields. It can be seen that
as the magnetic field increases, the rate of increasing of gravitational mass
versus central energy density decreases, getting to a lower value at the
limiting case. In Table 1, we have shown the maximum mass and the
corresponding radius of an spin polarized SQS at different temperatures for
$B=5\times 10^{18}\ G$. It is shown that by increasing the temperature, the
maximum mass and radius of spin polarized SQS decreases. We have also shown
the maximum mass and the corresponding radius of the spin polarized SQS for
different magnetic fields at a fixed temperature $T=30\ MeV$ in Table 2. We
see that the maximum mass and radius of the spin polarized SQS decreases by
increasing the magnetic field. One of the method to compare the mass and
radius in our calculation with the observational values is calculation of the
surface red-shift value. The maximum surface red-shift calculated in our work
is $z_{s}=0.38$ (for $T=30\ MeV$ $B=5\times 10^{19}\ G$ ) that is $55.34$
lower than the upper bound on the surface red-shift for sub-luminal equation
of states (i.e. $z_{s}^{CL}=0.8509$). Furthermore the minimum amount of red-
shift in our calculations is equal to $z_{s}=0.37$ that is almost near the
red-shift of RX J185635-3754 is $z_{s}=0.35\pm=0.37$.

##  IV Summary and conclusions

In this paper, we investigated the thermodynamic properties of the strange
quark matter (SQM) composed of the spin-up and spin-down of the up, down and
strange quarks at finite temperature in the presence of an strong magnetic
field using the MIT bag model. We computed the total free energy of the spin
polarized SQM at finite temperature in the presence of an strong magnetic
field. We found that the free energy gets a minimum at a particular value of
the polarization parameter showing a meta-stable state. We have also shown
that the polarization parameter corresponding to the meta-stable state reaches
zero by increasing both density and temperature. In other words, by increasing
the density and the temperature, the spin polarized SQM becomes nearly
unpolarized. We calculated the equation of state of the spin polarized SQM at
different temperatures for $B=5\times 10^{18}\ G$. We showed that by
increasing temperature, the equation of state of spin polarized SQM becomes
stiffer. Finally, using the general relativistic TOV equation, we studied the
structure of the spin polarized strange quark star (SQS). We calculated the
maximum mass and the corresponding radius of spin polarized SQS at different
temperatures and magnetic fields. We have shown that the gravitational mass of
spin polarized SQS rapidly increases by increasing the central energy density.
We also observed that the mass and the radius of the spin polarized SQS
decreases by increasing both temperature and magnetic field.

###### Acknowledgements.

We wish to thank the Research Institute for Astronomy and Astrophysics of
Maragha and Shiraz University Research Council for financial support.

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Table 1: Maximum mass and the corresponding radius of SQS for $B=5\times 10^{18}\ G$ at different temperatures. The results of $T=0\ MeV$ (Bordbar & Peyvand rk8 ) have been also given for comparison. $T\ (MeV)$ | $M_{max}\ (M_{\odot})$ | $R\ (km)$
---|---|---
$0$ | 1.33 | 7.44
$30$ | 1.17 | 7.37
$70$ | 1.16 | 7.21
Table 2: Maximum mass and the corresponding radius of SQS for different magnetic fields at $T=30\ MeV$. $B\ (G)$ | $M_{max}\ (M_{\odot})$ | $R\ (km)$
---|---|---
$0$ | 1.40 | 8.08
$5\times 10^{18}$ | 1.17 | 7.37
$5\times 10^{19}$ | 1.16 | 7.16

Figure 1: Total free energy per volume as a function of the polarization
parameter at different temperatures and densities for $B=5\times 10^{18}\ G$.

Figure 2: The polarization parameter versus baryonic density at different
temperatures for $B=5\times 10^{18}\ G$.

Figure 3: The polarization parameter versus baryonic density at $T=30\ MeV$
for different magnetic fields.

Figure 4: The polarization parameter as a function of magnetic field at
$n=0.5\ fm^{-3}$ for different temperatures.

Figure 5: The total free energy per volume of the spin polarized SQM as a
function of the baryonic density for $B=5\times 10^{18}\ G$ at different
temperatures.

Figure 6: The pressure of the spin polarized SQM for $B=5\times 10^{18}\ G$ at
different temperatures.

Figure 7: The gravitational mass of spin polarized SQS versus energy density
at different temperatures for $B=5\times 10^{18}\ G$. The results of $T=0\
MeV$ (Bordbar & Peyvand rk8 ) have been also given for comparison.

Figure 8: The gravitational mass of spin polarized SQS as a function of the
radius at different temperatures for $B=5\times 10^{18}\ G$. The results of
$T=0\ MeV$ (Bordbar & Peyvand rk8 ) have been also given for comparison.

Figure 9: The gravitational mass versus energy density at $T=30\ MeV$ for
different magnetic fields.


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