Contraction of cold neutron star due to in the presence a quark core
Abstract
Motivated by importance of the existence of quark matter on structure of neutron star. For this purpose, we use a suitable equation of state (EoS) which include three different parts: (i) a layer of hadronic matter, (ii) a mixed phase of quarks and hadrons, and, (iii) a strange quark matter in the core. For this system, in order to do more investigation of the EoS, we evaluate energy, Le Chatelier’s principle and stability conditions. Our results show that the EoS satisfies these conditions. Considering this EoS, we study the effect of quark matter on the structure of neutron stars such as maximum mass and the corresponding radius, average density, compactness, Kretschmann scalar, Schwarzschild radius, gravitational redshift and dynamical stability. Also, considering the mentioned EoS in this paper, we find that the maximum mass of hybrid stars is a little smaller than that of the corresponding pure neutron star. Indeed the maximum mass of hybrid stars can be quite close to the pure neutron stars. Our calculations about the dynamical stability show that these stars are stable against the radial adiabatic infinitesimal perturbations. In addition, our analyze indicates that neutron stars are under a contraction due to the existence of quark core.
1 Introduction
Neutron stars which are born in the aftermath of core-collapsing supernova (SN) explosions, are a cosmic laboratory and the best environment for the studying dense matter problems. It is notable that, in the center of neutron star because of high densities, the matter is envisaged to have a transition from hadronic matter to strange quark matter, see Refs. [1, 2, 3, 4], for more details. Also, Glendenning in Ref. [5], showed that proper construction of phase transition between the hadron and quark, inside the neutron stars implies the coexistence of nucleonic matter and quark matter over a finite range of the pressure. Accordingly, a mixed hadron-quark phase exists in the neutron star, so that its energy is lower than that of the quark matter and nucleonic matter. Phase transition between quark matter core and hadronic external layers (hadron-quark phase) in neutron star is an interesting subject which has been investigated by many authors [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18]. For example, the effects of quark-hadron matter in the center of neutron stars have been studied in Refs. [19, 20], and the obtained results have been shown that for \(PSR~J1614-2230\), and \( PSR~J0348+0432\) with masses about \(2M_{\,\odot }\), may contain a region of quark-hybrid matter in their center. Plumari et al. have investigated the effects of a quark core inside neutron star by considering the quark-gluon EoS in the framework of field correlator model [21]. They have found an upper limit for the mass of neutron stars by adjusting some parameters. This limit was in the range \(M_{\max }\simeq 2M_{\,\odot }\). In addition, Chen et al. have studied cold dense quark matter and hybrid stars with a Dyson-Schwinger quark model and various choices of the quark-gluon vertex [22]. They have showed that hadron states have the maximum mass lower than the pure nucleonic neutron stars, but higher than two solar masses. Their results depended on parameters of EoS. Also, Lastowiecki et al. have found that compact stars masses of about \(2M_{\,\odot }\) such as \( PSR~J1614-2230\) and \(PSR~J3048+0432\) were compatible with the possible existence of deconfined quark matter in their core [23]. Neutrino emissivity in the quark-hadron mixed phase of neutron stars have been investigated in Ref. [24]. According to importance of existence of quark matter in the neutron stars [15, 25, 26, 27, 28, 29, 30, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49], we consider a neutron star to be composed of a hadronic matter layer, a mixed phase of quarks and hadrons, and in the core of star, a quark matter. One of our aims in this work is determining the structure of neutron star with a quark core and comparing it with observation data.
In order to study the structure of stars and their phenomenological properties, we must use the hydrostatic equilibrium equation (HEE). Indeed, this equation is based on the fact that a typical star will be in equilibrium when there is a balance between the gravitational force and the internal pressure. The first HEE equation was introduced by Tolman, Oppenheimer and Volkoff (TOV) [50, 51, 52] in the Einstein gravity which is known as TOV equation. Considering TOV equation, the structure of compact stars have been evaluated by many authors in Refs. [53, 54, 55, 56, 57, 58, 59, 60, 61].
According to this fact that, there are some massive neutron stars with mass more than two times of solar mass, \(M\ge 2M_{\,\odot }\), for example, \( 4U1700-377\) with \(M=2.4M_{\,\odot }\) [62], and \(J1748-2021B\) with \( M=2.7M_{\,\odot }\) [63], one of our goals in this work is related to answer this question: is there a quark core inside neutron stars with mass more than \(2M_{\,\odot }\) (\(M\ge 2M_{\,\odot }\)), by considering a suitable EoS which obtained by combining three different parts, a layer of hadronic matter, a mixed phase of quarks and hadrons, and a strange quark matter in core? In order to evaluate a suitable EoS, we will study energy, Le Chatelier’s principle and stability conditions of this EoS. Another our goal is related to the investigation of the effects of quark core in the structure of hybrid star. For this goal, we study the difference between the structure of hybrid stars and pure neutron stars.
The outline of the paper is as follows; First, in order to investigate the structure of neutron stars, we evaluate a suitable EoS which includes three different layers. Then, we compare the structure of neutron stars by considering the EoS with and without a quark core. Indeed, we study the effects of EoS by applying a quark core on the structure of neutron stars. Next, we compare our obtained results with those of observational data. The last section is devoted to closing remarks.
2 Equation of state
A neutron star with a quark core composed of a hadronic matter layer, a mixed part of quarks and hadrons and a quark matter in core. Thus we calculate the EoS of different parts of this star in the following subsections.
2.1 Hadron phase
2.2 Quark phase
2.3 Mixed phase
2.4 Energy conditions
Energy conditions for neutron star with a quark core
Type of star | \({\mathcal {E}}_c\left( {10^{14}\,\mathrm{g/cm}^3}\right) \) | \(P_c\left( { 10^{14}\,\mathrm{g/cm}^3}\right) \) | NEC | WEC | SEC | DEC |
---|---|---|---|---|---|---|
\(NS+Quark Core\) | 25.8 | 7.8 | \(\surd \) | \(\surd \) | \(\surd \) | \(\surd \) |
2.5 Stability
Structure properties of neutron star without quark matter (NS) and with quark core (NS + Q)
Type of star | \(M_{\max }\left( M_{\,\odot }\right) \) | R (km) | \( {\mathcal {E}}_c\left( {10^{14}\,\mathrm{g/cm}^3}\right) \) | \(R_{Sch}\) (km) | \({\overline{\rho }}\left( {10^{14}\,\mathrm{g/cm}^3}\right) \) | \(\sigma \) | \(K(10^{-8}\) m\(^{-2})\) | z |
---|---|---|---|---|---|---|---|---|
NS | 1.98 | 9.8 | 27.1 | 5.84 | 9.99 | 0.59 | 2.15 | 0.57 |
NS + Quark Core | 1.8 | 10 | 25.8 | 5.31 | 8.55 | 0.53 | 1.84 | 0.46 |
2.6 Le Chatelier’s principle
Le Chatelier’s principle is defined as: the matter of star satisfies \(dP/d {\mathcal {E}}\ge 0\) which is a essential condition of a stable body both as a whole and also with respect to the non-equilibrium elementary regions with spontaneous expansion or contraction [77]. According the Fig. 1, the Le Chatelier’s principle is established.
The above results show that we encounter with a suitable EoS for studying the neutron stars with a quark matter in the core. Therefore, we consider our EoS and investigate the structure of neutron star with three different layers.
3 Structure of the neutron star with and without quark matter
As one can see, considering the quark matter in the core of neutron stars change their structure properties. Indeed, there are some interesting results when we consider the quark matter in the calculation of structure of neutron stars. For example, by considering the quark matter, the maximum mass decreases (see Table 2 and Fig. 2, for more details). In order to do more investigation of the effect of quark matter, we calculate another properties of the neutron star such as the average density, compactness, Kretschmann scalar, gravitational redshift and dynamical stability.
3.1 Average density
3.2 Compactness
The compactness of a spherical object is usually defined as the ratio of Schwarzschild radius to the radius of object (\(\sigma =\frac{R_{Sch}}{R}\)), which may be indicated as the strength of gravity of compact objects. Our results for the compactness are presented in Table 2. Similar to the average density, by applying the quark matter in the calculation of structure of neutron star, the compactness from the perspective of a distant observer (or a observer outside the neutron star) decreases.
3.3 Kretschmann scalar
3.4 Gravitational redshift
3.5 Dynamical stability
Properties of neutron star with the gravitational mass equal to \( 1.4M_{\,\odot }\) for NS and NS + Q
Type of star | R (km) | \(\sigma \) | \(K(10^{-8}\) m\(^{-2})\) | z |
---|---|---|---|---|
NS | 10.96 | 0.37 | 1.09 | 0.26 |
NS + Quark Core | 10.76 | 0.38 | 1.15 | 0.27 |
Properties of neutron star with the gravitational mass equal to \( 1.8M_{\,\odot }\) for NS and NS+Q
Type of star | R (km) | \(\sigma \) | \(K(10^{-8}\) m\(^{-2})\) | z |
---|---|---|---|---|
NS | 10.56 | 0.50 | 1.56 | 0.42 |
NS + Quark Core | 10.00 | 0.53 | 1.84 | 0.46 |
Another interesting results is related to contraction of neutron stars due to the quark core (see Tables 3 and 4, for more details). In the other words, the obtained results in Tables 3 and 4 and Figs. 2 and 3 show that the radius of neutron stars without the quark matter (NS) and with the gravitational mass equal to \(1.4M_{\,\odot }\) (or \( 1.8M_{\,\odot }\)) are greater than the radius of neutron stars with a quark core (NS + Q). Indeed, for the same gravitational masses of NS and NS + Q, the compactness, the Kretschmann scalar and the gravitational redshift increase due to the reduced radius. In addition, this difference appears for the neutron stars with the gravitational mass higher than the solar mass (\(M\ge M_{\,\odot }\)) (see Figs. 2 and 3).
Briefly, we can see that the existence of quark matter inside the neutron stars leads to decreasing for the maximum mass and so it contracts them. These results indicate that by applying the mentioned EoS in Eq. (9), the cores of neutron stars with mass more than \(2M_{\,\odot }\), do not have any quark matter. Indeed, the presented EoS in paper (Eq. 9), can not predict the existence of massive hybrid stars with more than \( 2M_{\,\odot }\), because by adding the quark matter to the structure of these stars, the maximum masses decreases.
4 Summary and conclusion
- (i)
the EoS derived in this work satisfied the energy, Le Chatelier and stability conditions.
- (ii)
considering the EoS introduced in this paper, we found that the maximum mass of cold hybrid stars could not be more than \(2M_{\,\odot }\)(\(M_{\max }\le 2M_{\,\odot }\)). In other word, there are no cold massive hybrid stars in the mass range \(M_{\max }>2M_{\,\odot }\), when we applied our EoS. Therefore, our results showed that inside the neutron stars such as \( 4U~1700-377\) [62] with the mass about \(2.4M_{\odot }\), and \( J1748-2021B\) [63] with the mass about \(2.7M_{\odot }\), there is no any quark matter.
- (iii)
the maximum mass, the average density, compactness, the Kretschmann scalar, and gravitational redshift of neutron stars decrease owing to the existence of quark matter in the structure them.
- (iv)
the neutron stars are contracted due to the presence of quark matter in their center.
- (v)
the obtained results indicated that the studied hybrid stars in this paper are stable against the radial adiabatic infinitesimal perturbations.
- (vi)
for neutron stars with the gravitational mass more than one solar mass (\( M\ge M_{\odot }\)), there was a difference between NS and NS + Q. In other words, there was no any difference between the properties of NS and NS + Q in the range \(M<M_{\odot }\).
Notes
Acknowledgements
We thank an anonymous referee for useful comments. We wish to thank Shiraz University Research Council. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/5750-56. TY wishes to thank the Research Council of Islamic Azad University, Bafgh Branch.
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