Rotating hybrid stars with the Dyson-Schwinger quark model

TL;DR

This paper investigates the properties of rapidly rotating hybrid stars using advanced quark and nuclear matter models, analyzing their maximum mass, size, and spin behavior, and comparing results with observational data.

Contribution

It introduces a combined approach using the Dyson-Schwinger model and Brueckner-Hartree-Fock theory to study rotating hybrid stars, including their rotational evolution and stability.

Findings

Maximum gravitational mass and radius determined for stable configurations.

Identification of a spinup phenomenon in supramassive stars.

Comparison of theoretical models with observational constraints.

Abstract

We study rapidly rotating hybrid stars with the Dyson-Schwinger model for quark matter and the Brueckner-Hartree-Fock many-body theory with realistic two-body and three-body forces for nuclear matter. We determine the maximum gravitational mass, equatorial radius, and rotation frequency of stable stellar configurations by considering the constraints of the Keplerian limit and the secular axisymmetric instability, and compare with observational data. We also discuss the rotational evolution for constant baryonic mass, and find a spinup phenomenon for supramassive stars before they collapse to black holes.

Tables

Table 1. Table 1: Several properties of rotating NS for the selected EOSs: Maximum gravitational mass, corresponding central baryon density, and Maximum Keplerian frequency.

EOS		BOB	DS1	DS2	DS3	DS4	DS10
Static	$M_{\text{max}}/M_{\odot }$	2.51	2.30	2.02	1.79	1.60	1.48
	${\rho }_c/{\rho }_0$	5.22	4.96	5.38	5.88	6.14	9.69
Keplerian	$M_{\text{max}}/M_{\odot }$	2.99	2.82	2.47	2.19	1.95	1.76
	${\rho }_c/{\rho }_0$	4.52	4.43	4.64	5.08	5.45	8.64
	$f_K\left[\text{Hz}\right]$	1653	1461	1399	1346	1316	1763

Equations

$${B\over A}={3\over 5}{k^{2}_{F}\over 2m}+{1\over 2\rho}\sum_{k,k^{\prime}<k_{F}}\!\!\!\big{\langle}kk^{\prime}\big{|}G[e(k)+e(k^{\prime});\rho]\big{|}kk^{\prime}\big{\rangle}_{A}\>,$$

$$G[E;\rho]=V+\!\!\!\sum_{k_{a},k_{b}>k_{F}}\!\!\!V{\big{|}k_{a},k_{b}\big{\rangle}Q\big{\langle}k_{a},k_{b}\big{|}\over E-e(k_{a})-e(k_{b})}G[E;\rho]\>,$$

$$e(k)=e(k;\rho )=\frac{k^2}{2m}+U(k;\rho )$$

$$U(k;\rho)=\sum_{k^{\prime}\leq k_{F}}\big{\langle}kk^{\prime}\big{|}G[e(k)+e(k^{\prime});\rho]\big{|}kk^{\prime}\big{\rangle}_{A}\>.$$

$$\frac{B}{A}(\rho ,x)=\frac{B}{A}(\rho ,x=0.5)+(1-2x{)}^2{E}_{\text{sym}}(\rho ),$$

$$E_{\text{sym}}(\rho )=\frac{B}{A}(\rho ,x=0)-\frac{B}{A}(\rho ,x=0.5).$$

$$\epsilon ({\rho }_n,{\rho }_p,{\rho }_e,{\rho }_{\mu })$$

$$P={\rho }^2\frac{d}{d\rho }\frac{\epsilon (\{{\rho }_i(\rho )\})}{\rho }=\rho \frac{d\epsilon }{d\rho }-\epsilon .$$

$$S(p;\mu {)}^{-1}$$

$$\Sigma (p;\mu )$$

$${\Gamma }_{\sigma }(q,p;\mu )={\gamma }_{\sigma },$$

$$g^{2}(\mu)D_{\rho\sigma}(k,\mu)=4\pi^{2}d\frac{k^{2}}{\omega^{6}}e^{-\frac{k^{2}+\alpha\mu^{2}}{\omega^{2}}}\Big{(}\delta_{\rho\sigma}-\frac{k_{\rho}k_{\sigma}}{k^{2}}\Big{)}\>.$$

$$f_q(|\mathbf{p}|;\mu )$$

$${\rho }_q(\mu )$$

$$P_q({\mu }_q)$$

$$B_{\text{DS}}\equiv -\sum _{q=u,d,s}{P}_q({\mu }_{q,0}).$$

$${\mu }_n-{\mu }_p={\mu }_e={\mu }_{\mu },$$

$${\rho }_p={\rho }_e+{\rho }_{\mu },$$

$${\mu }_d={\mu }_u+{\mu }_e={\mu }_u+{\mu }_{\mu }={\mu }_s,$$

$$\frac{2{\rho }_u-{\rho }_d-{\rho }_s}{3}-{\rho }_e-{\rho }_{\mu }=0.$$

$${\mu }_i=b_i{\mu }_B-q_i{\mu }_e,p_H=p_Q=p_M.$$

$$\chi {\rho }_c^Q+(1-\chi ){\rho }_c^H=0,$$

$${\epsilon }_M$$

$${\rho }_M$$

$$d{s}^2=-e^{\gamma +\rho }d{t}^2+e^{2\beta }\left(d{r}^2+r^{2}d{\theta }^2\right)+e^{\gamma -\rho }{r}^2{\sin }^2\theta {\left(d\varphi -\omega dt\right)}^2,$$

$$T^{\mu \nu }=(\epsilon +p)u^{\mu }{u}^{\nu }-p{g}^{\mu \nu },$$

$$f_K=f_0{\left(\frac{M}{M_{\odot }}\right)}^{\frac{1}{2}}{\left(\frac{R_s}{10\text{km}}\right)}^{-\frac{3}{2}},$$

$${\frac{\partial M}{\partial {\rho }_c}|}_J=0.$$

Full text

I Introduction

Neutron stars (NS) are among the densest objects known in the Universe. They contain an extreme environment shaped by the effects of the four fundamental interactions. NSs have the typical mass $M\sim 1.4M_{\odot}$ and radius $R\sim 10$km. Therefore, the mean particle density can reach (2–3)$\rho_{0}$, and the core density (10–20)$\rho_{0}$ Haensel2007 , where $\rho_{0}=0.17\,\text{fm}^{-3}$ is the so-called nuclear saturation density. At this density, the nucleons might undergo a phase transition to quark matter (QM), and a hybrid NS (HNS) with a QM core is formed. This makes NS ideal astrophysical laboratories to study hadronic interactions over a wide range of densities Weber2014 .

Unfortunately, as a key ingredient of the investigation of NS, the equation of state (EOS) remains uncertain. The microscopic theory of the nucleonic EOS has reached a high degree of sophistication gle ; bbb ; akma ; zhou ; Li2008 ; mmy , but the QM EOS is poorly known at zero temperature and at the high baryonic density appropriate for NS, because it is difficult to perform first-principle calculations of QM.

Therefore one can presently only resort to more or less phenomenological models for describing QM, such as the MIT Bag model Chod , the Nambu-Jona-Lasino model Buballa05 ; Schertler99 ; Klahn ; Klahn15 , or the quasi-particle model Tian2012 ; Zhao2015 . In Ref. Chen2011 we developed a Dyson-Schwinger quark model (DSM) for deconfined QM, which provides a continuum approach to QCD that can simultaneously address both confinement and dynamical chiral symmetry breaking Roberts1994 ; Alkofer2000wg . In that work, we considered static and spherical symmetric HNSs, whereas in this paper we include the effects of rotation.

Rotation is a common property of NS. Of the thousands of currently observed pulsars, the fastest one has been discovered in the globular cluster Terzan 5 with a frequency of 716 Hz 716hz . At this rapid rotation, a NS would be flattened by the centrifugal force, and the Tolman-Oppenheimer-Volkoff equation, suitable for a static and spherically symmetric situation, cannot describe correctly the rotating stellar structure. In the present paper we approximate the NS as a axisymmetric and rigid rotating body, and resort to Einstein’s theory of general relativity for a rapidly rotating star. Numerical methods for (axisymmetric) rotating stellar structure have been advanced by several groups Komatsu1989a ; Weber1991a ; Weber1991b ; Cook1992 ; Cook1994 ; Salg1994 ; Stergioulas2003 ; Stergioulas1995 . In this work we utilize the KEH method Komatsu1989a to obtain the properties of rapidly rotating HNSs.

This paper is organized as follows. In Sec. II we briefly discuss the construction of the EOS of a HNS. In Sec. III we present the rotation effects on the HNS; the allowed ranges of gravitational mass, equatorial radius, and Kepler frequency are discussed in this section and compared with observational data. The rotational evolution for a constant baryonic mass is also analyzed. Sec. IV contains our conclusions.

II The equation of state

II.1 Nuclear matter

For nuclear matter we resort to the Brueckner-Hartree-Fock (BHF) many-body theory with realistic two-body and three-body nucleonic forces, which has been extensively discussed in Ref. Baldo1999 . We recall that this theory has also been extended with the inclusion of hyperons, which might appear in the core of a NS. The hyperonic EOS in this theory turns out to be very soft, and this results in too low NS maximum masses Schulze2011 , well below the current observational limit of about two solar masses heavy ; heavy2 ; heavy3 . The presence of strange baryonic matter often inhibits the appearance of QM. In this work we do not discuss this aspect, but limit ourselves to consider only nucleons and leptons in the hadronic phase.

In the BHF theory the energy per nucleon of nuclear matter is given by

[TABLE]

where $G[E;\rho]$ is the solution of the Bethe-Goldstone equation

[TABLE]

$V$ is the bare nucleon-nucleon (NN) interaction, $\rho$ is the nucleon number density, and $E$ the starting energy. The single-particle energy

[TABLE]

and the Pauli operator $Q$ determine the propagation of intermediate baryon pairs. The BHF approximation for the single-particle potential using the continuous choice is

[TABLE]

Due to the occurrence of $U(k)$ in Eq. (3), the above equations constitute a coupled system that has to be solved in a self-consistent manner for several momenta of the particles involved, at the considered densities. The only input quantities of the calculation are the NN two-body potentials. In this work we present results obtained with the Bonn-B (BOB) potential bob as input, supplemented with compatible three-body forces Li2008 ; zuotbf ; uix . The associated EOS yields fairly large maximum masses of about $2.5M_{\odot}$ for purely nucleonic NS (NNS).

For the calculation of the energy per nucleon of asymmetric nuclear matter, we use the so-called parabolic approximation bbb

[TABLE]

where $x=\rho_{p}/\rho$ is the proton fraction and $E_{\text{sym}}(\rho)$ is the symmetry energy, which can be expressed in terms of the difference of the energy per nucleon of pure neutron matter ($x=0$) and symmetric matter ($x=0.5$):

[TABLE]

The parametrized results of pure neutron and symmetric matter with different interactions can be found in Ref. Li2008 . The energy density of baryon/lepton matter as a function of the different partial densities is then

[TABLE]

where $\varepsilon_{e}(\rho_{e})$ and $\varepsilon_{\mu}(\rho_{\mu})$ are the energy densities of electrons and muons. Once the energy density is known, the chemical composition of the beta-equilibrated matter can be calculated and finally the EOS,

[TABLE]

II.2 Quark matter

The quark propagator based on the Dyson-Schwinger equation at finite chemical potential $\mu\equiv\mu_{q}=\mu_{B}/3$ assumes a general form with rotational covariance,

[TABLE]

where $m_{q}$ is the current quark mass, $u=(\bm{0},i\mu)$, and possibilities of other structures, e.g., color superconductivity Alford2003 ; Yuan2006 ; Nickel2006 , are disregarded. The quark self-energy can be obtained from the gap equation,

[TABLE]

where $\lambda^{a}$ are the Gell-Mann matrices, $g(\mu)$ is the coupling strength, $D_{\rho\sigma}(k;\mu)$ the dressed gluon propagator, and $\Gamma_{\sigma}(q,p;\mu)$ the dressed quark-gluon vertex at finite chemical potential.

For the quark-gluon vertex and the gluon propagator we employ the widely-used ”rainbow approximation” Chen2008 ; Chen2011

[TABLE]

and assume the Landau gauge form for the gluon propagator, with an infrared-dominant interaction modified by the chemical potential Chen2011 ; Jiang2013

[TABLE]

The various parameters can be obtained by fitting meson properties and chiral condensate in vacuum Alkofer2002 ; Chang2009 , and we use $\omega=0.5\;\text{GeV}$, $d=1\;\text{GeV}^{2}$. The phenomenological parameter $\alpha$ represents a reduction of the effective interaction with increasing chemical potential. This parameter cannot yet be fixed independently and its value has been amply discussed in previous works Chen2011 ; Chen2015 .

Knowing the quark propagator, the EOS of cold QM can be obtained via the momentum distribution Chen2011 ; Chen2008 ; Klahn2009 ,

[TABLE]

The total density and pressure for pure QM are given by summing the contributions of all flavors. In addition, we define the phenomenological bag constant

[TABLE]

In this work we set the value as $B_{\text{DS}}=90\;{\rm MeV\,fm^{-3}}$, see the discussion in Chen2011 .

II.3 Construction of the hybrid star EOS

In order to study the properties of a rapidly rotating HNS, we should first construct the EOS of the star. We assume that the hadron-quark phase transition is of first order, and perform the Gibbs construction, thus imposing that nuclear matter and QM are betastable and globally charge neutral. This is at variance with the Maxwell construction, where the two phases must be separately charge neutral.

In the purely nucleonic phase, which consists of baryons ($n,p$) and leptons ($e,\mu$), the conditions of beta stability and charge neutrality can be expressed as

[TABLE]

where $\mu_{i}$ are the chemical potentials and $\rho_{i}$ the particle number densities. Similarly the pure QM phase, which contains three-flavor quarks ($u,d,s$) and leptons ($e,\mu$), should satisfy the constraints of beta stability and charge neutrality

[TABLE]

According to the Gibbs construction, there is a mixed phase where the hadron and quark phases coexist, and both phases are in equilibrium with each other gle . This can be expressed as

[TABLE]

where $b_{i}$ and $q_{i}$ denote baryon number and charge of the particle species $i=n,p,u,d,s,e,\mu$ in the mixed phase. To solve those equations, we also need the global charge neutrality condition

[TABLE]

where $\rho_{c}^{Q}$ and $\rho_{c}^{H}$ are the charge densities of quark and nuclear matter, and $\chi$ is the volume fraction occupied by QM in the mixed phase. From these equations, we can derive the energy density $\varepsilon_{M}$ and the baryon density $\rho_{M}$ of the mixed phase as

[TABLE]

In the upper panel of Fig. 1 we show the pressure versus baryon chemical potential $\mu_{B}=\mu_{n}=\mu_{u}+2\mu_{d}$. The solid black curve represents the calculation for beta-stable and asymmetric nuclear matter with BOB EOS; the curves labeled DS$\alpha$ are for pure QM with several choices of the phenomenological parameter $\alpha$. In the lower panel the complete EOSs of HNSs are shown, i.e., pressure vs. baryon density. We can see that the EOS contains three sections: a pure hadronic phase at low density, followed by a mixed phase, and a pure quark phase at high density. We note that the onset of the phase transition is determined by the value of the parameter $\alpha$; larger $\alpha$ produces an increasingly softer QM EOS with a lower phase transition onset density. For high values of $\alpha$ we find that QM appears quite early, e.g., for $\alpha=10$ at a baryon density $\rho\approx\rho_{0}$.

For completeness, we mention that for the calculation of the stellar structure we use the EOSs by Feynman-Metropolis-Teller fey and Baym-Pethick-Sutherland bps for the outer and inner crusts, respectively.

III Results and discussion

The structure of a rapidly rotating NS is different from the static one, since the rotation can strongly deform the star. We assume NS are steadily rotating and have axisymmetric structure. Therefore the space-time metric used to model a rotating star can be expressed as

[TABLE]

where the potentials $\gamma,\rho,\beta$,$\omega$ are functions of $r$ and $\theta$ only. The matter inside the star is approximated by a perfect fluid and the energy-momentum tensor is given by

[TABLE]

where $\varepsilon$, $p$, and $u^{\mu}$ are the energy density, pressure, and four-velocity, respectively. In order to solve Einstein’s field equation for the potentials $\gamma,\rho,\beta$,$\omega$, we adopt the KEH method and use the public RNS code code for calculating the properties of a rotating star.

III.1 Keplerian limit

The rotational frequency is a directly measurable quantity of pulsars, and the Keplerian (mass-shedding) frequency $f_{K}$ is one of the most-studied physical quantities for rotating stars Cook1992 ; Cook1994 ; Haensel1995 ; Benhar2005 ; Haensel2009 ; zhang2013 . In Fig. 2 we show the gravitational NS mass as a function of the central baryon density (left panel) and of the equatorial radius (right panel), using the EOSs displayed in Fig. 1. Results are plotted for both the static configurations (thin curves) and for the ones rapidly rotating at Keplerian frequency (bold curves).

In all cases the maximum masses of HNSs are lower than those of NNSs, because the appearance of QM in the core of the star results in a softening of the very hard nucleonic EOS. Comparing Keplerian and static sequences, rotations increase the maximum mass and equatorial radius substantially. The maximum masses of the static and Keplerian sequences with various EOSs, as well as the corresponding central densities, are listed in Table 1. The maximum masses increase by about $20\%$ from the static to the Keplerian sequence. According to the current observations of massive pulsars heavy ; heavy2 ; heavy3 , the DSM EOSs with $\alpha\gtrsim 2$ are ruled out.

In Fig. 3 we present the Keplerian frequency as a function of gravitational mass for some selected EOSs. We observe that it increases monotonically both for NNSs and HNSs. The Keplerian frequency of HNSs increases more rapidly after QM onset, and is larger than the one of a NNS with the same gravitational mass, because the stellar radius is smaller in the former case due to the presence of a very dense QM core. However, due to the lower maximum mass of HNSs, the maximum Keplerian frequency of HNSs is lower than the one of NNSs, as also listed in Table 1 for the various EOSs discussed above. Our results satisfy the constraint from the observed fast-rotating pulsar PSR J1748-2446ad with $\rm 716\,Hz$ 716hz , or the even more severe constraint from XTE J1739-285 with $\rm 1122\,Hz$ 1122hz , which has not been confirmed, however.

We compare our results with the empirical formula

[TABLE]

proposed in Lattimer2004 , where $M$ is the gravitational mass of the Keplerian configuration, $R_{s}$ is the radius of the nonrotating configuration of mass $M$, and $f_{0}$ is a constant, which does not depend on the EOS. In Ref. Haensel2009 an optimal prefactor $f_{0}=1080\;\text{Hz}$ in the range $0.5\,M_{\odot}<M<0.9\,M_{\text{max}}^{\text{static}}$ was obtained. Rotating HNSs with masses in that range are characterized by a purely nucleonic phase, and therefore the empirical formula cannot be applied. This is at variance with NNS configurations. As displayed in Fig. 3, our results for NNSs below $2.1M_{\odot}$ can be fitted well with the same parameter $f_{0}=1080\,\text{Hz}$, as shown by the thin curve.

III.2 Stability analysis

In order to complete the description of Figs. 2 and 3, one should pay attention to the stability criteria of stars. It is well known that the onset of the instability of the static sequence is determined by the condition $dM/d\rho_{c}=0$, i.e., the curve should stop at its mathematical maximum, which thus gives the maximum mass of the static stable sequence. In the rotating case, the above criterium has to be generalized, i.e., a stellar configuration is stable if its mass $M$ increases with growing central density for a fixed angular momentum $J$ Haensel2007 . Therefore the onset of the instability, which is called secular axisymmetric instability (SAI), is expressed by

[TABLE]

The configurations in the Keplerian sequences shown in Fig. 2 have different angular momenta, and thus the curves do not stop at the mathematical maximum. In the upper panel of Fig. 4 we show, for some selected EOSs, the gravitational mass for the Keplerian sequence vs. the angular momentum (solid black curves), along with the SAI condition, Eq. (29), represented by the dashed red curves. Thus the Keplerian sequence should stop at the intersection with the SAI curves, which is indicated by an open circle. This constraint determines the corresponding endpoints of the curves in Figs. 2 and 3.

Some enlarged details are shown in the insets of Fig. 4. For a given mass $M$, there are two possible values of angular momentum $J$, which correspond to two possible values of radius $R$ in Fig. 2. In the case of NNSs with the BOB EOS, the branch with the lower $R$ has a larger values of $J\sim MR^{2}f_{K}$, because the Kepler frequency $f_{K}$ increases faster than $R^{2}$ diminishes on the Keplerian sequence. In the case of HNSs, the situation is opposite: the branch with the lower $R$ has also a lower value of $J$. Therefore for NNSs the Kepler curve meets the SAI at large $R$, before it reaches the mathematical maximum of the mass. This is different from the case of HNSs, whose curves extend a little further on the unstable branch after they reach their mathematical maximum, before meeting the SAI, and thus the maximum mass of the stable configurations coincides with the mathematical maximum value. The maximum mass and maximum angular momentum, as well as the end point given by the SAI constraint, are obtained with different stellar configurations, and are labelled by the open squares, triangles, and circles, respectively. The discussed effects are however very small, of the order of 0.01$M_{\odot}$ at most.

In order to visualize better the intricate relations between $M$, $R$, and $f_{K}$, we present in Fig. 5 the mass-radius relations of NS with EOS BOB (upper panel) and DS2 (lower panel) at various fixed rotation frequencies (dash-dotted olive curves). The stable configurations are constrained by the Kepler and SAI conditions at large and small radius, respectively. At a low frequency ($f=796\,$Hz for HNSs), the lower boundary of $M$ is fixed by the Kepler condition and the upper boundary by the SAI condition. As the frequency increases ($f=1082\,$Hz), the SAI mark point moves to the left side of the mathematical maximum (MaM), and the upper boundary of $M$ is now fixed by the MaM, but not anymore by the SAI condition. This is indicated by the dotted blue curve that passes through the MaMs for fixed frequency. As the frequency increases further ($f=1194/1273\,$Hz), the lower (upper) boundary values of $M$ are fixed by the SAI (MaM/Kepler) conditions. Finally, at the maximum frequency the Kepler and SAI conditions meet at the same point.

In Fig. 6 we present the allowed domain of NNSs and HNSs in the $R_{\text{eq}}$–$f$ plane (upper panel) and the $M$–$f$ plane (lower panel), together with some observational data. We use the same conventions as in Fig. 5, i.e., dotted blue curves, dashed green curves, and short-dashed red curves represent MaM, mass-shedding, and SAI limits. The allowed region of HNSs with the DS2 EOS is the grey area delimited by the dash-dotted lila curve (PT), which represents the onset of the phase transition. One interesting feature we should mention here is that at high rotation frequency the mass range is small, while the range of radii is still large, corresponding to a flat top of the $M(R)$ curves in Fig. 2. This means the radii are very sensitive to the mass at high rotating frequency.

As discussed above, the current observations on pulsar masses constrain our parameter to $\alpha<2$, hence we present the results of HNSs with the EOS DS2. For smaller $\alpha$ the corresponding (shaded) area of HNSs will shrink and move towards the lower (upper) boundary of NNSs in the upper (lower) panel. The minimum (maximum) mass of HNSs with EOS DS2 is 1.68 (2.02)$\;M_{\odot}$ in the static sequence, and increases as the rotation frequency increases, while the range concentrates to a single value 2.47$\;M_{\odot}$ at the maximum frequency $f=1.4\;\text{kHz}$. Therefore, in the lower panel of Fig. 6, the three stars with lower masses should be conventional NS, and the others could be HNSs in our DS2 model.

The observational data of the radius still suffer large uncertainties. In the upper panel we include the sources 4U1820-30 and SAXJ1808.4-3658, whose mass, radius, and spin are available. One can see that according to their small radii both sources should preferably be high-mass compact HNSs in our model, whereas their masses in the lower panel identify them as preferably “low-mass” NNSs. This can also be seen in Fig. 2, where the same data points are reproduced. However, within the large error bars, both data are still consistent with our model. We expect more accurate observations to constrain our parameters or rule out the model.

III.3 Phase transition caused by rotational evolution

The possibility of a phase transition to QM caused by rotational evolution has been widely discussed in literature Weber1997 ; Spyrou2002 ; Ayvazyan2013 ; Haensel2016 . For a constant baryonic mass, a rotating star loses its rotation energy by magnetic dipole radiation, which makes the star spin down and the central density increase. When the central density of a NNS reaches a critical value, the phase transition from hadronic matter to QM will take place, and the star converts to a HNS. As the star continues spinning down and the central density continues increasing, more and more QM appears in the core of the HNS.

This is clearly shown in Fig. 7, where we display the change of the number density of all particle species with rotational frequency in the interior of a star with baryonic mass $M_{B}=2.0M_{\odot}$ for the DS2 EOS (corresponding to $M=1.74M_{\odot}$ in the static sequence and $M=1.80M_{\odot}$ at the Kepler frequency $f_{K}=1018$Hz, see the lower panel of Fig. 5). One notes that this star at Keplerian frequency has no QM core, but as it spins down, it is compressed to a smaller volume, which enhances the central density, and the star is converted into a HNS. As the frequency decreases further, the QM mixed phase extends outward from the core and the region occupied by the pure hadron phase gets narrower. At the same time, the radius of the star is decreasing.

In Fig. 8 we present the stellar models with DS2 EOS in the $f$–$M_{B}$ plane, where the same labels as in Fig. 6 are used, i.e., the dash-dotted lila curve represents the onset of conversion from a NNS to a HNS. It can be seen that the conversion is possible only in the baryon mass range $1.84<M_{B}/M_{\odot}<2.37$. Examples could be the pulsars J1903+0327 and 4U1820-30, located at the edge of the phase transition boundary in Fig. 6. Above that range, even the fastest rotating stars are already HNSs. In addition, when the star’s baryonic mass is larger than 2.35$M_{\odot}$, the static configuration is unstable, and the star will collapse to a black hole as it loses angular momentum and meets the SAI borderline (dashed red curve). These are supramassive stars max that will be discussed in more detail in the following. The maximum baryonic mass for the DS2 EOS is 2.87$M_{\odot}$. The various limits are indicated by vertical lines in Fig. 8.

For further illustration, we show in the upper panel of Fig. 9 the fraction of QM in HNSs as function of the rotation frequency for several choices of fixed baryonic mass with the DS2 EOS. The trajectories in the $M$–$R_{\text{eq}}$ plane for the same values of $M_{B}$ are reported in Fig. 5. Usually the QM fraction increases with decreasing frequency due to the increasing density and extension of the QM domain in the star, see Fig. 7. The maximum value of $8.39\%$ is reached for the heaviest possible static NS with $M_{B}=2.35M_{\odot}$, see Fig. 8. This value can be increased by choosing larger values of $\alpha$ in the DSM, but then the maximum HNS mass falls below two solar masses. Supramassive HNSs ($M_{B}>2.35M_{\odot}$) have no static limit and collapse when reaching the (dashed red) SAI line. Their QM fraction remains below the maximum static value.

In the lower panel of Fig. 9 we show the angular momentum as a function of rotation frequency for NNSs and HNSs. The conversion points between NNSs and HNSs on the PT line are indicated by markers in some cases. Normal HNSs ($M_{B}<2.35M_{\odot}$) are spinning down when losing angular momentum in the evolution, whereas supramassive stars spin up close to the collapse Cook1994 . A similar backbending phenomenon is often related to the onset of the phase transition from hadronic matter to QM Weber1997 ; Glendenning1997 ; Haensel2016 , but here it occurs for both HNSs and NNSs in supramassive configurations, in the case of NNSs for $3.10<M_{B}/M_{\odot}<3.59$, see Fig. 8.

In more detail, for example for the $M_{B}=2.6\,M_{\odot}$ trajectory in Fig. 9, Fig. 5, and in the inset of Fig. 8, the HNS spins down until it reaches the minimum of the fixed rotation frequency curve ($f=1082$ Hz). Then it spins up until the final SAI point. In fact, in the evolution the maximum angular momentum is given at the Kepler sequence and the minimum angular momentum at the static sequence or the SAI line. Therefore, if the lower boundary of the frequency in Fig. 8 is not at the static sequence or the SAI line, there must be a spinup with loss of angular momentum.

Quantitatively, the difference of angular momentum between NNSs and HNSs with equal baryonic mass is slight at lower baryon mass ($M_{B}<2.35\,M_{\odot}$), but becomes important for larger masses, where the QM content increases and only HNSs exhibit the spinup phenomenon.

IV Conclusion

We have investigated the properties of rotating HNSs, employing an EOS constructed with the BHF approach for nucleonic matter and the DSM for QM, and assuming the phase transition under the Gibbs construction. We computed the properties of HNSs in the Keplerian sequence, respecting the SAI constraint. HNSs are more compact and have lower maximum masses and maximum Kepler frequencies than NNSs. Our results for the maximum mass, maximum rotation frequency, and the equatorial radius range fulfill the current constraints by observational data of the fastest rotating pulsars.

We also investigated the phase transition induced by the spindown of pulsars with a constant baryonic mass. We showed the variation of the QM content under rotational evolution, and found that the QM ratios are small, with the maximum value about $8\%$, in order to respect the current two-solar-mass lower limit of the maximum mass. We also found that in our model the spinup (backbending) phenomenon is not related to the phase transition, but happens in supramassive stars before they collapse to black holes, which is possible in a narrow range of large mass for both HNSs and NNSs.

V Acknowledgments

We acknowledge financial support from the NSFC (11305144,11475149,11303023). Partial support comes from “NewCompStar,” COST Action MP1304.