Bulk viscosity for interacting strange quark matter and r-mode instability windows for strange stars

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Jian-Feng Xu, Dong-Biao Kang, Guang-Xiong Peng and Cheng-Jun Xia. Bulk viscosity for interacting strange quark matter and r-mode instability windows for strange stars[J]. Chinese Physics C.
Jian-Feng Xu, Dong-Biao Kang, Guang-Xiong Peng and Cheng-Jun Xia. Bulk viscosity for interacting strange quark matter and r-mode instability windows for strange stars[J]. Chinese Physics C. shu
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Bulk viscosity for interacting strange quark matter and r-mode instability windows for strange stars

  • 1. School of Physics & Electrical Engineering, AnYang Normal University, AnYang, 455000, China
  • 2. School of Physics, University of Chinese Academy of Sciences, Beijing 100049, China
  • 3. Theoretical Physics Center for Science Facilities, Institute of High Energy Physics, Beijing 100049, China
  • 4. School of Information Science and Engineering, Ningbo Tech University, Ningbo 315100, China

Abstract: We investigate the bulk viscosity of strange quark matter in the framework of equivparticle model, where analytical formulae are obtained for certain temperature ranges and can be readily applied to those with various quark mass scalings. In the case of adopting a quark mass scaling with both linear confinement and perturbative interactions, the obtained bulk viscosity increases by $1 \sim 2$ orders of magnitude comparing with bag model scenarios. Such an enhancement is mainly due to the large quark equivalent masses adopted in the equivparticle model, which essentially attribute to the strong interquark interactions and are related to the dynamical chiral symmetry breaking. Due to the large bulk viscosity, the predicted damping time of oscillations for canonical 1.4 ${\rm{M}}_\odot$ strange star is less than one millisecond, which is faster than previous findings. Consequently, the obtained $r$-mode instability window for the canonical strange stars well accommodates the observational frequencies and temperatures for pulsars in the low-mass X-ray binaries (LMXBs).

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    1.   Introduction
    • The recent observation of gravitational waves (GWs) emitted by binary neutron star merger, combined with the electromagnetic counterparts, has inaugurated the multi-messenger era of astronomy [1]. If it does not collapse promptly into a black hole, significant post-merger oscillations are expected for the central merger remnant, among which the $ r $-mode oscillations have been identified as viable and promising sources for continuous emission of GWs [2-9] which could be detected by the the advanced Laser Interferometer Gravitational-wave Observatory (aLIGO) and Virgo Interferometer, or next generation gravitational observatories [10-13]. The emission of GWs can in turn drive the $ r $-mode oscillations of compact stars with certain spin frequency and temperature via the Chandrasekhar-Friedman-Schutz (CFS) mechanism [14,15], which may eventually lead the compact stars to an $ r $-mode instability state [16-18] and reduces their spin frequencies.

      In fact, the r-mode amplitude driven by emission of GWs is hindered by different kinds of viscous damping mechanisms [19,20], such as the shear and bulk viscosities. Shear viscosity due to layer surface rubbing and particles scattering determines the relaxation of momentum components perpendicular to the direction of fluid flow, which usually operates effectively at low temperatures [21]. Bulk viscosity, which is originated from the mismatching between the reaction rate of chemical equilibration in matter and the frequency of the volume perturbation, is of decisive importance at high temperatures that is especially important for newly-born pulsars and accreting pulsars in binary systems. It is interesting to note that although the shear viscosity of strange quark matter (SQM) is comparable to that of baryonic matter, the bulk viscosity of SQM could be many orders of magnitude larger than that of baryonic matter [22-24]. Therefore, the studies about bulk viscosity of the composition of compact stars may be a viable way to distinguish strange stars from neutron stars [25,26].

      As noted above, bulk viscosity has close relation to $ r $-mode instability and GWs emission. Due to the competition between GWs driven effects and viscous damping effects, pulsars can finally reach a steady rotation frequency with certain internal temperature, which gives a curve dividing the frequency-temperature ($ \nu-T $) plane into stability region and instability region (or usually regarded as $ r $-mode stability window and instability window). Although old cold pulsars in low-mass X-ray binaries (LMXBs) with long-term stable spin frequencies are expected to locate within the stability window, many of them are actually in the instability window according to various theoretical predictions [27].

      To solve this paradox, many possible solutions have been proposed. In Refs. [28,29], the authors argued that including hyperons may be a viable solution to the $ r $-mode problem. In Ref. [30], in stead of hyperons, the authors adopted realistic equation of state of SQM in modified bag model but obtained similar results as in Ref. [31], where the obtained instability windows are not consistent with the observational data of neutron stars in LMXBs. Nevertheless, they suggested that employing large bulk viscosity may be a reasonable solution to reconcile theory with observations. Indeed, recent investigations have shown LMXBs composed of interacting SQM could be located well in the stability window [32]. Additionally, long-range interactions between quarks was also proposed to overcome this problem [33]. Besides these scenarios, strong magnetic fields have meaningful and profound influence on bulk viscosity of SQM as well [34].

      The bulk viscosity of SQM arises mainly from the nonleptonic weak process $ d+u \leftrightarrow s+u $, which was firstly stressed by Wang and Lu [24], and was extensively studied from then on [17,26,35-44]. The common conclusion is that bulk viscosity of SQM is generally larger than that of baryonic matter. Furthermore, in our previous study of bulk viscosity in enhanced perturbative QCD model [45], we found the interactions between quarks can significantly enlarge bulk viscosity to 1-2 orders of magnitude compared with that in MIT bag model.

      Motivated by the puzzle and conclusion mentioned above, in this paper we study the $ r $-mode instability windows for strange stars with the implementation of bulk viscosity obtained in the equivparticle model adopting a recently proposed quark mass scaling with quark confinement effects and first order perturbative interactions. In Sec. II, we first present a concise introduction of the equivparticle model, then deduce the corresponding expression of bulk viscosity of SQM. In Sec. III, numerical results about bulk viscosity of SQM are displayed and discussed. On application of the obtained bulk viscosity, the damping times of oscillations and $ r $-mode instability windows for 1.4 $ M_\odot $ strange stars are computed in Sec. IV. Finally, the summary is presented in Sec. V.

    2.   Bulk viscosity in equivparticle model

      2.1.   Brief introduction to the equivparticle model

    • We first give a brief introduction to the equivparticle model, and one can refer to Ref. [46] for more details, in which the key point is the real chemical potentials of particles are replaced by the effective ones. Therefore, the thermodynamic potential density of quark agglomerate is seemingly the same in form with the free particles, i.e.,

      $ \begin{aligned}[b] \Omega_0 = & -\sum\limits_i\frac{g_i}{24\pi^2}\bigg[\mu_i^*\nu_i\left(\nu_i^2-\frac{3}{2}m_i^2\right) +\frac{3}{2}m_i^4\\ & \times\ln\frac{\mu_i^*+\nu_i}{m_i}\bigg], \end{aligned} $

      (1)

      where $ \mu_i^* $ and $ \nu_i = \sqrt{\mu_i^{*2}-m_i^2} $ are respectively the effective chemical potential and the Fermi momentum of particle species $ i $, with $ i $ running over $ u, d, s $ and $ e $. The degenerate factor $ g_i $ is 6 for quarks and 2 for electrons. Here $ m_i $ is the corresponding effective mass of particle species $ i $, which can be written in two parts,

      $ m_i = m_{i0}+m_I, $

      (2)

      where $ m_{i0} $ is the quark current mass and $ m_I $ is the interacting part. For $ u, d, s $ and $ e $, $ m_{i0} $ reads 5,10,100 and 0.511 MeV respectively. Originated from strong interactions, $ m_I $ is naturally the same to all quarks, while it vanishes for electrons.

      In previous literatures, $ m_i $ has been parameterized in many different forms. According to bag model assumption, it is originally parameterized as [47]

      $ m_i = m_{i0}+\frac{B}{3\rho_b}, $

      (3)

      where $ \rho_b $ is the baryon number density. Then, a cubic-root scaling is proposed in Ref. [48]

      $ m_i = m_{i0}+\frac{D}{\rho_b^{1/3}}, $

      (4)

      which is derived from the linear confinement and leading-order in-medium chiral condensate. Later, it is extended to include temperature effect [49],

      $ m_i = m_{i0}+\frac{D}{\rho_b^{1/3}}\left[1-\frac{8T}{\lambda T_c} \exp\left(-\lambda\frac{T_c}{T}\right)\right]. $

      (5)

      Recently, a new mass scaling is suggested with linear confinement and leading-order perturbative interactions [46],

      $ m_i = m_{i0}+\frac{D}{\rho_b^{1/3}}+C\rho_b^{1/3}. $

      (6)

      Here, we should emphasize that the parameters $ C $ and $ D $ respectively indicate the strength of perturbative interactions and confinement effects, which are adopted in the following section to investigate their effects on bulk viscosity of SQM.

      The relation between real and effective chemical potential in equivparticle model is

      $ \mu_i = \mu_i^*+\frac{1}{3}\frac{{\rm{d}} m_I}{{\rm{d}}\rho_b}\frac{\partial \Omega_0}{\partial m_I} \equiv\mu_i^*-\mu_I, $

      (7)

      where $ \mu_I $ is the same to all quarks. Consequently, the $ \beta $-equilibrium still holds for effective chemical potentials, i.e.,

      $ \mu_u^*+\mu_e = \mu_d^* = \mu_s^*. $

      (8)

      According to the traditional thermodynamics, particle number density $ \rho_i $ can be derived by the common relation $ \rho_i = -d\Omega_0/d\mu_i^* $, which gives

      $ \rho_i = \frac{g_i}{6\pi^2}(\mu_i^{*2}-m_i^2)^{3/2} = \frac{g_i\nu_i^3}{6\pi^2}. $

      (9)

      The charge neutrality condition and baryon number conservation can be hence respectively written as

      $ \frac{2}{3}\rho_u-\frac{1}{3}\rho_d-\frac{1}{3}\rho_s-\rho_e = 0, $

      (10)

      and

      $ \rho_b = \frac{1}{3}\left(\rho_u+\rho_d+\rho_s\right). $

      (11)

      Then the energy density and pressure are given by

      $ \begin{aligned}[b] E = & \Omega_0-\sum\limits_i\mu_i^{*}\frac{\partial\Omega_0}{\partial\mu_i^*}\\ =& \sum\limits_i\frac{g_i}{16\pi^2}\bigg[\nu_i\mu_i^*(\mu_i^{*2}+\nu_i^2)-m_i^4\\ &\times\ln\left(\frac{\mu_i^*+\nu_i}{m_i}\right)\bigg], \end{aligned} $

      (12)

      and

      $ P = -\Omega_0+\rho_b\frac{\partial m_I}{\partial \rho_b} \frac{\partial\Omega_0}{\partial m_I} = -\Omega_0-3\rho_b\mu_I. $

      (13)

      With a given baryon number density $ \rho_b $, one can simultaneously solve Eqs. (8), (10), and (11) to obtain the values of effective chemical potentials $ \mu_i^*\ (i = u,d,s,e) $. Then according to Eqs. (12) and (13), the equation of state (EoS) of strange quark matter can be obtained by $ P = P(E) $.

      In Fig. 1, we give the EoS of SQM in the equivparticle model with the new quark mass scaling in Eq. (6). From this figure one can readily find the inclusion of interactions between quarks can significantly soften the EoS. Additionally, the effect of quark confinement (dashed line) prevails over that of perturbative interactions (dotted line) in EoS with the specified model parameters.

      Figure 1.  EoS of SQM with new mass scaling in Eq. (6). In comparison of the solid line and dash-dotted line, it's obvious to find the interactions between quarks can significantly soften the EoS

    • 2.2.   The derivation of bulk viscosity in equivparticle model

    • In this section, employing a traditional method [26,50], we will give the bulk viscosity of SQM in equivparticle model. After reproduce the existing results [26], we investigate the effects of quark interactions on bulk viscosity by adopting different mass scalings.

      First, we assume the volume per unit mass of SQM is $ v $, which, due to the vibration of strange quark star, oscillates harmonically according to the following equation,

      $ v(t) = v_0+\Delta v\sin\left(\frac{2\pi t}{\tau}\right)\equiv v_0+\delta v(t), $

      (14)

      where $ \tau $ is the volume vibration period, $ v_0 $ is the equilibrium volume, and $ \Delta v $ is the vibration amplitude which is tiny compared with $ v_0 $, i.e., $ \Delta v/v_0\ll1 $. The vibration of $ v $ could result in the change of particle phase space, and further causes variation of particle number per unit mass

      $ n_i(t) = n_{i0}+\delta n_i(t), $

      (15)

      where unless otherwise stated, $ i $ represents $ u, d, $ and $ s $ quarks here and below, and $ \delta n_i $ can be obtained by integration

      $ \delta n_d = -\delta n_s = \int_0^t\frac{dn_d}{dt}dt. $

      (16)

      As to the reaction rate $ dn_d/dt $, we adopt the result valid in low temperatures ($ T\ll\mu_i $) [26], i.e.,

      $ \frac{{\rm{d}} n_{{\rm{d}}}}{{\rm{d}} t} \approx G_{{\rm{C}}} \mu_{{\rm{d}}}^{5} \delta \mu\left(\delta \mu^{2}+4 \pi^{2} T^{2}\right) v_{0}, $

      (17)

      where $ \delta\mu\equiv\mu_s-\mu_d $ and the constant $ G_c $ is connected to the weak-coupling $ G_F $ and the Cabibbo angle $ \theta_C $ by

      $ G_{{\rm{C}}} \equiv \frac{16}{5 \pi^{5}} G_{F}^{2} \sin ^{2} \theta_{C} \cos ^{2} \theta_{C} = 6.76 \times 10^{-26} {\rm{MeV}}^{-4}. $

      (18)

      Since the bulk viscosity of SQM mainly stems from the nonleptonic weak reaction $ u+d\leftrightarrow u+s $, the pressure $ P $ can be expanded near the $ \beta $- equilibrium pressure $ P_0 $, i.e.,

      $ P(t) = P_0+\left(\frac{\partial P}{\partial v}\right)_0\delta v +\left(\frac{\partial P}{\partial n_d}\right)_0\delta n_d +\left(\frac{\partial P}{\partial n_s}\right)_0\delta n_s. $

      (19)

      Then the mean dissipation rate of the vibration energy per unit mass is given by

      $ \left(\frac{{\rm{d}} w}{{\rm{d}} t}\right)_{{\rm{av}}} = -\frac{1}{\tau} \int_{0}^{\tau} P(t) \frac{{\rm{d}} v}{{\rm{d}} t} {\rm{d}} t. $

      (20)

      In practice, the first two terms on the right-hand side of Eq. (19) do not contribute to energy dissipation in Eq. (20), therefore they can be safely ignored.

      Similarly, the quark chemical potential difference $ \delta\mu $ can also be expanded in terms of $ \delta v $, $ \delta n_d $, and $ \delta n_s $, namely

      $ \delta \mu(t) = \left(\frac{\partial \delta \mu}{\partial v}\right)_{0} \delta v +\left(\frac{\partial \delta \mu}{\partial n_{{\rm{d}}}}\right)_{0} \delta n_{{\rm{d}}}+\left(\frac{\partial \delta \mu}{\partial n_{{\rm{s}}}}\right)_{0} \delta n_{{\rm{s}}}, $

      (21)

      in which $ \displaystyle\frac{\partial\delta\mu}{\partial x} = \displaystyle\frac{\partial\mu_s}{\partial x} -\frac{\partial\mu_d}{\partial x},\ (x = v,n_d,{\rm{or}}\ n_s) $. Specially, $\displaystyle \frac{\partial\mu_i}{\partial v} $ is connected to the third and forth terms in Eq. (19) by the following relation,

      $ \frac{\partial \mu_{i}}{\partial v} = -\frac{\partial P}{\partial n_{i}}. $

      (22)

      In order to get the bulk viscosity, one should give the expressions of derivatives of chemical potentials in terms of $ v $, $ n_d $, and $ n_s $. To this end, we start from the quark number per unit mass $ n_i $,

      $ n_i = \rho_iv = \frac{1}{\pi^2}(\mu_i^{*2}-m_i^2)^{3/2}v, $

      (23)

      from which it's not difficult to obtain the differential form of $ n_i $, i.e.,

      $ \begin{aligned}[b] d n_{i} = & v d \rho_{i}+\rho_{i} d v\\ =& \frac{3v\nu_i}{\pi^{2}}\left(\mu_{i}^{*} d \mu_{i}^{*}-m_{i} d m_{i}\right) +\rho_{i} d v. \end{aligned}$

      (24)

      In equivparticle model, to mimic the complicated strong interactions between quarks, the particle masses are taken as functions of baryon number density, i.e., $ m_i = m_i(\rho_b) $, from which the differential form of $ m_i $ can be written as

      $ d m_i = \frac{d m_i}{d \rho_b}d \rho_b = \frac{d m_I}{d \rho_b}d \rho_b. $

      (25)

      According to the definition of baryon number density $ \rho_b = \displaystyle\frac{1}{3}\sum\nolimits_i \rho_i $, one can obtain $ v\rho_b = \displaystyle\frac{1}{3}\sum\nolimits_i v\rho_i = \displaystyle\frac{1}{3}\sum\nolimits_i n_i $. Then the corresponding differential form is $ d(v\rho_b) = vd\rho_b+\rho_bdv = \displaystyle\frac{1}{3} \sum\nolimits_i dn_i $, from which

      $ d \rho_{b} = \frac{1}{3v} \sum\limits_{i} dn_{i}-\frac{\rho_{b}}{v} d v. $

      (26)

      Substituting Eqs. (25) and (26) into Eq. (24), one can get the differential form of particle number per unit mass $ dn_i $, i.e.,

      $ \begin{aligned}[b] dn_i = & \frac{3v\nu_i\mu_i^*}{\pi^2}d\mu_i^*-\frac{\nu_i m_i}{\pi^2} \frac{dm_I}{d\rho_b}\sum\limits_jdn_j\\ & +\left(\rho_i+\frac{3\nu_im_i\rho_b}{\pi^2} \frac{dm_I}{d\rho_b}\right)dv,\\ & \equiv C_{1i}d\mu_i^*+C_{2i}\sum\limits_jdn_j+C_{3i}dv, \end{aligned} $

      (27)

      where we have changed the dummy index $ i $ in Eq. (26) to $ j $ in Eq. (27) and introduced three notations to give a more tight form of this expression, i.e.,

      $ C_{1i} \equiv \frac{3v\nu_i\mu_i^*}{\pi^2}, $

      (28)

      $ C_{2i} \equiv -\frac{\nu_i m_i}{\pi^2}\frac{dm_I}{d\rho_b}, $

      (29)

      $ C_{3i} \equiv \rho_i+\frac{3\nu_im_i\rho_b}{\pi^2}\frac{dm_I}{d\rho_b}, $

      (30)

      Note that the chemical potentials in Eq. (27) are effective ones which should be linked to the real ones. In order to do this, we use the differential form of Eq. (7), i.e.,

      $ d\mu_i^* = d\mu_i+d\mu_I. $

      (31)

      To facilitate the derivation of bulk viscosity of SQM in the equivparticle model, $ \mu_I $ can be redefined as a new function $ f $ as follows,

      $ \mu_I = -\frac{1}{3}\frac{dm_I}{d\rho_b}\frac{\partial\Omega_0}{\partial m_I} \equiv f(\rho_b,m_i(\rho_b),\mu_i^*). $

      (32)

      Accordingly, the differential form of $ \mu_I $ is

      $ d\mu_I = \left(\frac{\partial f}{\partial\rho_b}+\frac{dm_I}{d\rho_b} \sum\limits_j\frac{\partial f}{\partial m_j}\right)d\rho_b +\sum\limits_j\frac{\partial f}{\partial\mu_j^*}d\mu_j^*. $

      (33)

      Substituting Eqs. (33) and (26) into Eq. (31) and re-arranging the terms, one can write the explicit differential form of effective chemical potential $ d\mu_i^* $ as

      $ \begin{aligned}[b] d\mu_i^* = & \frac{1}{1-\displaystyle\frac{\partial f}{\partial\mu_i^*}}\bigg[d\mu_i+\left(\frac{\partial f}{\partial\rho_b} +\frac{dm_I}{d\rho_b}\sum\limits_j\frac{\partial f}{\partial m_j}\right) \left(\frac{1}{3v} \sum\limits_{j} dn_{j}-\frac{\rho_{b}}{v} d v\right)+\sum\limits_{j \neq i}\frac{\partial f}{\partial\mu_j^*}d\mu_j^*\bigg],\\ =& \frac{1}{1-\displaystyle\frac{\partial f}{\partial\mu_i^*}}d\mu_i+\frac{1}{3v} \frac{1}{1-\displaystyle\frac{\partial f}{\partial\mu_i^*}}\left(\frac{\partial f}{\partial\rho_b} +\frac{dm_I}{d\rho_b}\sum\limits_j\frac{\partial f}{\partial m_j}\right)\sum\limits_{j} dn_{j} -\frac{\rho_b}{v}\frac{1}{1-\displaystyle\frac{\partial f}{\partial\mu_i^*}}\left(\frac{\partial f}{\partial\rho_b} +\frac{dm_I}{d\rho_b}\sum\limits_j\frac{\partial f}{\partial m_j}\right)dv\\ & +\sum\limits_{j \neq i}\frac{\partial f}{\partial\mu_j^*}d\mu_j^*. \end{aligned} $

      (34)

      By taking

      $ \begin{aligned}[b] & A_i = \frac{1}{1-\displaystyle\frac{\partial f}{\partial\mu_i^*}}, \\ & B = \frac{\partial f}{\partial\rho_b} +\frac{dm_I}{d\rho_b}\sum\limits_j\frac{\partial f}{\partial m_j}, \end{aligned} $

      with

      $ \frac{\partial f}{\partial\mu_i^*} = -\frac{m_i\nu_i}{\pi^2}\frac{dm_I}{d\rho_b}, $

      (35)

      $ \frac{\partial f}{\partial m_j} = \frac{g_j}{4\pi^2} \left(\mu_j^*\nu_j-3m_j^2\ln\frac{\mu_j+\nu_j}{m_j}\right), $

      (36)

      $ \frac{\partial f}{\partial\rho_b} = -\frac{d^2m_i}{3d\rho_b^2} \sum\limits_j\frac{g_jm_j}{4\pi^2}\left[\mu_j^*\nu_j-m_j^2\ln\frac{\mu_j^*+\nu_j}{m_j}\right], $

      (37)

      one could rewrite Eq. (34) as

      $ d\mu_i^* = A_id\mu_i+\frac{A_iB}{3v}\sum\limits_jdn_j-\frac{\rho_bA_iB}{v}dv+ \sum\limits_{j \neq i}\frac{\partial f}{\partial\mu_j^*}d\mu_j^*. $

      (38)

      Substituting Eq. (38) into Eq. (27) and applying $ dn_i = \sum\nolimits_j\delta_{ij}dn_j $, where $ \delta_{ij} $ is the Kronecker delta, one can obtain the differential form of the real chemical potential

      $ \begin{aligned}[b] d\mu_i = & \sum\limits_j\left(\frac{\delta_{ij}-C_{2i}}{A_iC_{1i}}-\frac{B}{3v}\right)dn_j +\left(\frac{\rho_bB}{v}-\frac{C_{3_i}}{A_iC_{1i}}\right)dv\\ &-\frac{1}{A_i}\sum\limits_{j \neq i}\frac{\partial f}{\partial\mu_j^*}d\mu_j. \end{aligned} $

      (39)

      From Eq. (39), it immediately gives

      $ \left(\frac{\partial\mu_i}{\partial n_j}\right)_0 = \frac{\delta_{ij}-C_{2i}}{A_iC_{1i}}-\frac{B}{3v}, $

      (40)

      $ \left(\frac{\partial\mu_i}{\partial v}\right)_0 = \frac{\rho_bB}{v}-\frac{C_{3i}}{A_iC_{1i}}. $

      (41)

      Substituting Eqs. (40) and (41) into Eq. (19) and ignoring the first two terms in the right-hand side of Eq. (19), one gets the pressure contributing to the dissipation energy

      $ \begin{aligned}[b] P(t) = & \left(\frac{\partial P}{\partial n_d}\right)_0\delta n_d +\left(\frac{\partial P}{\partial n_s}\right)_0\delta n_s\\ =& \left(\frac{C_{3d}}{A_dC_{1d}}-\frac{C_{3s}}{A_sC_{1s}}\right) \int_0^t\frac{dn_d}{dt}dt. \end{aligned} $

      (42)

      The chemical potential difference $ \delta\mu $, accordingly, can be derived in a similar manner, namely

      $\begin{aligned}[b] \delta\mu(t) = & \left(\frac{\partial\delta\mu}{\partial v}\right)_0\delta v +\left(\frac{\partial\delta\mu}{\partial n_d}\right)_0\delta n_d+ \left(\frac{\partial\delta\mu}{\partial n_s}\right)_0\delta n_s,\\ = &\left(\frac{C_{3d}}{A_dC_{1d}}-\frac{C_{3s}}{A_sC_{1s}}\right)\Delta v\sin(\omega t)\\ &-\left(\frac{1}{A_dC_{1d}}+\frac{1}{A_sC_{1s}}\right)\int_0^t\frac{dn_d}{dt}dt,\end{aligned} $

      (43)

      which naturally gives the differential equation of $ \delta\mu(t) $,

      $ \begin{aligned}[b] \frac{d\delta\mu(t)}{dt} = & \left(\frac{C_{3d}}{A_dC_{1d}}-\frac{C_{3s}}{A_sC_{1s}}\right) \frac{2\pi\Delta v}{\tau}\cos\left(\frac{2\pi t}{\tau}\right)\\ & -\left(\frac{1}{A_dC_{1d}}+\frac{1}{A_sC_{1s}}\right)\frac{dn_d}{dt}. \end{aligned} $

      (44)

      To solve this differential equation numerically, the related expressions of these notations in Eq. (44) are explicitly given as follows,

      $ C_{3i} = \rho_i+\frac{3\nu_im_i\rho_b}{\pi^2}\frac{dm_I}{d\rho_b}, $

      (45)

      $ \frac{1}{A_iC_{1i}} = \frac{1}{3v\mu_i^*} \left(\frac{\pi^2}{\nu_i}+m_i\frac{dm_I}{d\rho_b}\right). $

      (46)

      From these expressions it is obvious to see the mass-density dependent term $ dm_I/d\rho_b $ which explicitly denotes the main feature of bulk viscosity in equivparticle model.

      The mean dissipation rate of the vibration energy per unit mass reads

      $\begin{aligned}[b] \left(\frac{dw}{dt}\right)_{\rm{av}} = & -\frac{1}{\tau}\int_0^\tau P(t)\frac{dv}{dt}dt\\ =& \frac{\Delta v}{\tau}\frac{2\pi}{\tau} \left(\frac{C_{3s}}{A_sC_{1s}}-\frac{C_{3d}}{A_dC_{1d}}\right) \int_0^\tau dt\\ &\times\left(\int_0^t\frac{dn_d}{dt}dt\right)\cos\left(\frac{2\pi t}{\tau}\right). \end{aligned} $

      (47)

      Finally, the bulk viscosity is

      $\begin{aligned}[b] \zeta \equiv & 2\frac{(dw/dt)_{\rm{av}}}{v_0} \left(\frac{v_0}{\Delta v}\right)^2\left(\frac{\tau}{2\pi}\right)^2\\ . =& \frac{1}{\pi}\frac{v_0}{\Delta v} \left(\frac{C_{3s}}{A_sC_{1s}}-\frac{C_{3d}}{A_dC_{1d}}\right) \int_0^\tau dt\\ & \times\left(\int_0^t\frac{dn_d}{dt}dt\right)\cos\left(\frac{2\pi t}{\tau}\right). \end{aligned} $

      (48)

      Due to the cubic term of $ \delta\mu $ ($ \delta\mu \lesssim 0.1 $ MeV) in Eq. (17), it's not possible to solve the bulk viscosity analytically. However, in the relatively high-temperature limit ($ 2\pi T\gg\delta\mu $, or the temperature of interest here satisfies $ T \lesssim 1\ {\rm{MeV}} $), one can ignore the cubic term, and find the analytical form of bulk viscosity as

      $ \zeta_a = \frac{\alpha T^2}{\omega^{2}+\beta T^4}\left[1-\left(1-e^{-\beta^{1/2} T^2 \tau}\right) \frac{2 \beta^{1/2} T^2/ \tau}{\omega^{2}+\beta T^4}\right], $

      (49)

      where

      $ \alpha = G_{c} \mu_{d}^{5} v_{0}^{2} 4 \pi^{2} \left(\frac{C_{3d}}{A_{d} C_{1 d}} -\frac{C_{3 s}}{A_{s} C_{1 s}}\right)^{2}, $

      (50)

      and

      $ \beta = G_{c}^2 \mu_{d}^{10} v_{0}^2 16 \pi^{4} \left(\frac{1}{A_{d} C_{1d}}+\frac{1}{A_{s} C_{1s}}\right)^2. $

      (51)

      In what follows, in order to prove the validity of our results, we will show that in the mass-density-independent case, i.e., parameters $ C = D = 0 $, equations in our model are exactly the same with that in bag model. If $ C = D = 0 $, $ A_i, B, C_{1i}, C_{2i} $ and $ C_{3i} $ can be respectively simplified to

      $ A_i = 1, B = 0, C_{1i} = \frac{3v\nu_i\mu_i}{\pi^2}, C_{2i} = 0, C_{3i} = \rho_i. $

      (52)

      Accordingly, Eq. (40) and Eq. (41) can be reduced to

      $ \left(\frac{\partial\mu_i}{\partial n_j}\right)_0 = \frac{\delta_{ij}\pi^2}{3v\nu_i\mu_i} $

      (53)

      and

      $ \left(\frac{\partial\mu_i}{\partial v}\right)_0 = -\frac{\pi^2\rho_i}{3v\nu_i\mu_i}. $

      (54)

      Therefore, Eq. (39) can be simplified to

      $ d\mu_i = \frac{\pi^2}{3v\nu_i\mu_i}dn_i-\frac{\pi^2\rho_i}{3v\nu_i\mu_i}dv = \frac{\pi^2}{3\nu_i\mu_i}d\rho_i, $

      (55)

      which is exactly the same with the differential form of particle number density $ \rho_i = \frac{\nu_i^3}{\pi^2} $ where $ \nu_i = \sqrt{\mu_i^2-m_i^2} $ with the real chemical potential $ \mu_i $. In this case, the bulk viscosity in equivparticle model given by Eq. (48) is reduced to

      $ \zeta = -\frac{1}{\pi}\frac{v_0}{\Delta v}\frac{m_s^2}{3v_0\mu_d} \int_0^\tau dt\left(\int_0^t\frac{dn_d}{dt}dt\right)\cos\left(\frac{2\pi t}{\tau}\right). $

      (56)

      Meanwhile, although the analytical form of $ \zeta $ in Eq. (49) is not changed, the $ \alpha $ and $ \beta $ are respectively reduced to

      $ \alpha = \frac{4\pi^2}{9}G_c\mu_d^3 m^4_{s0}, $

      (57)

      and

      $ \beta = \frac{64\pi^8}{9}G_c^2\mu_d^6\left(1+\frac{m_{s0}^2}{4\mu_d^2}\right)^2. $

      (58)

      They are respectively consistent with Eqs. (15-18) in Ref. [26]. This limiting case suggests that bulk viscosity in equivparticle model can be treated as a generalization of previous result in bag model.

    3.   Numerical results
    • To calculate the bulk viscosity of SQM in equivparticle model, one should first give the EoS of SQM and then simultaneously solve Eqs. (17), (44), and (48) numerically.

      In Fig. 2, we show the behavior of bulk viscosity $ \zeta $ as function of relative volume perturbation amplitude $ \Delta v/v_0 $ and temperature $ T $. The lines with solid balls calculated by Eq. (48) show the results in equivparticle model, while the lines with stars calculated by Eq. (56), gives the bulk viscosity in mass-density-independent case which are exactly the same with the previous results in bag model [26]. In equivparticle model, the parameters $ C $ and $ D^{1/2} $ in mass scaling Eq. (6) are respectively taken as 0.7 and 129 MeV, which guarantees both that the EoS of SQM can support quark stars with maximum mass larger than 2 $ M_\odot $ and SQM is absolutely stable. In addition, temperatures are respectively given as $ 10^{-5} $, $ 10^{-3} $, and $ 10^{-1} $ MeV, denoted by red, green, and blue lines.

      Figure 2.  Bulk viscosity as function of relative volume perturbation amplitude and temperature. The lines with solid balls and stars respectively indicate the results in equivparticle model and the mass-density-independent case (i.e., the bag model). In numerical calculations, the oscillation period of quark matter is $ \tau = 10^{-3} $ s, and the temperatures $ T $ are $ 10^{-5} $, $ 10^{-3} $, and $ 10^{-1} $ MeV, denoted by red, green, and blue lines respectively

      In numerical calculations, the baryon number density is set to $ \rho = 1.34 $ fm-3. To facilitate comparison of the magnitude of bulk viscosity, we also give the projection of each line on $ Z-Y $ plane. From Fig. 2, it's obvious to see that with different temperatures bulk viscosity in equivparticle model is generally larger than that in bag model by a magnitude of about 2 orders. This is not difficult to understand according to the conclusion we obtained in our previous paper [45] that the inclusion of interactions between quarks can significantly enhance the bulk viscosity of SQM. To understand this aspect better, we introduce a parameter $ \kappa $ in the interaction part of mass scaling, i.e.,

      $ m_i = m_{i0}+\kappa m_I,\ \kappa\in[0,1]. $

      (59)

      If $ \kappa = 0 $ the interactions between quarks vanishes, while $ \kappa $ approaches to unit, the interactions are gradually recovered.

      Fig. 3 reveals the interaction intensity behavior of bulk viscosity, from which one can observe that as interactions between quarks become strong, bulk viscosity becomes large as well. Whether we ignore the quark confinement effects ($ D = 0 $, dashed line) or the perturbative interactions ($ C = 0 $, dotted line), this behavior still holds. From Fig. 3, it also clearly indicates that the quark confinement effects play a more important role on the bulk viscosity than perturbative interactions. This conclusion still holds if we adopt other values of parameter sets, where the approximate relation between $ C $ and $ D $ in absolutely stable region of SQM [46] is given as,

      Figure 3.  Interaction intensity behavior of bulk viscosity. As the interactions between quarks become strong, the bulk viscosity becomes large. In numerical calculations, all the parameter values are the same with that in Fig. 2. Besides that, the values of $ \Delta v/v_0 $ and $ T $ are respectively set to $ 10^{-8} $ and $ 10^{-5} $ MeV

      $ D^{1/2}/{\rm{MeV}} = \left\{\begin{array}{ll} -48C+156, & C\in[-0.5,0],\\ -\displaystyle\frac{27}{0.7}C+156, & C\in(0,0.7]. \end{array}\right. $

      This relation is adopted in Fig. 4 on top and bottom X-axes. To study the perturbative interactions on bulk viscosity, we set $ D = 0 $, and the results are given by the dotted line. Similarly, to study the confinement interactions on bulk viscosity, we set $ C = 0 $, and the results are given by the dashed line. From these two lines, it is observed that with increasing $ C $ or $ D $, bulk viscosity increases. Furthermore, in a large parameter range of $ C $ and $ D $, it shows again that confinement effects on bulk viscosity are much larger than perturbative interactions on bulk viscosity. For the purpose of comparison, we also present the bulk viscosity of SQM exhibited in the solid line with both confinements effects and perturbative interactions.

      Figure 4.  Different interactions to bulk viscosity of absolutely stable SQM. From the dashed and dotted lines, it shows that confinement effects from model parameter $ D $ attribute to the bulk viscosity much more than perturbative interactions from model parameter $ C $

      To show explicitly the effects of quark confinement and perturbative interactions on bulk viscosity, in Fig. 5 we present the bulk viscosity of SQM adopting various model parameters $ C $ and $ D $. If $ C $ and $ D $ both vanish, it corresponds to the bag model predictions (the dash-dotted line). If only the perturbative interactions are accounted, we get the results indicated by the dotted line which is much larger than that of bag model. Moreover, if only the quark confinement effects are included, the results are given in the dashed line, which is consistent with the implications of Fig. 3, namely, quark confinement effects prevails over the perturbative interactions in bulk viscosity of SQM. Finally, as expected, if these two type of interactions are both considered, bulk viscosity showed in solid line becomes the largest. It is interesting to note that, the results showed in dashed line is almost the same with the solid line at large volume vibration amplitudes, which suggests that the perturbative interactions become insignificant in comparison.

      Figure 5.  Effects of interactions between quarks on bulk viscosity of SQM. The solid line gives results in equivparticle model, while the dash-dotted line corresponds to bag model. The dotted line only includes the perturbative interactions, and the dashed line merely includes the quark confinement effects. To check the impact of different mass scaling on bulk viscosity, we also give the results calculated in equivparticle model with mass scaling in Eq.(3)

      To check the influence of different mass scaling on bulk viscosity, we adopt the mass scaling in Eq. (3), which gives the largest bulk viscosity. However, we should emphasize that this situation doesn't contradict to our conclusion, since the mass scaling in Eq. (3) can provide much stronger quark confinement effects.

      Customarily, strange stars have strong magnetic fields, which have significant influence on the transport coefficients, especially the bulk viscosity. In Ref. [34], the authors investigated the bulk viscosity in much detail, and found that in strong magnetic fields the transport coefficients of SQM become anisotropic. However, when the magnetic fields $ B<10^{17} $ G, the effect of magnetic field on bulk viscosity can be ignored, and a comparison with our results at low-B limit is possible. In Fig. 6 we show the results of the bulk viscosity in Ref. [34] and equivparticle model at the same baryon number density $ \rho \sim 0.787 $ fm-3 (or $ \mu_u = \mu_d = 400 $ MeV in Ref. [34]). From Fig. 6, we find that the bulk viscosity in our model is usually larger than that in Ref. [34], except $ 10 \lesssim \omega/{\rm{s}}^{-1} \lesssim 1000 $, where $ \omega = 2\pi/\tau $. In addition, with increasing $ \omega $, the bulk viscosity in equivparticle model decreases much slow, which is because the interactions between quarks are taken into account.

      Figure 6.  Comparison of the bulk viscosity in Ref. [34] and equivparticle model at the same baryon number density $ \rho \sim 0.787 $ fm-3

      To understand how the interactions between quarks can enhance the bulk viscosity of SQM, we give the time behavior of quark chemical difference $ \delta\mu = \mu_s-\mu_d $ with different $ \Delta v/v_0 $ in Fig. 7. In these panels, from top to bottom, $ \Delta v/v_0 $ reads $ 10^{-1} $, $ 10^{-2} $, and $ 10^{-3} $ respectively. The solid lines represent the results including interactions between quarks with $ C = 0.7 $ and $ D^{1/2} = 129 $ MeV, while the dashed lines give the results with vanishing $ C $ and $ D $. It can be seen from Fig. 7 that with increasing $ \Delta v/v_0 $ the amplitudes of $ \delta\mu $ in both cases become large. Furthermore, according to Fig. 7, the amplitude are significantly enhanced once the interactions between quarks are taken into account, which suggests that more energy could be involved in one period of oscillation, and physically means larger bulk viscosity of SQM.

      Figure 7.  $ \delta\mu(t) $ for two cycles with parameters $ \rho_b = 1.34 $ fm-3, $ T = 10^{-5} $ MeV, $ \tau = 10^{-3} $ s, and $ \Delta v/v_0 = 10^{-1}, 10^{-2},\; {\rm{and}}\;\ 10^{-3} $ respectively. The solid lines indicate the results in equivparticle model with parameters $ C = 0.7 $ and $ D^{1/2} = 129 $ MeV, while the dashed lines correspond to the results with $ C = D = 0 $

      Like the amplitude of $ \delta\mu $ in Fig. 7, the same situation can be observed in Table 1, which gives the mean dissipation rate of vibration energy per unit volume $ \frac{1}{v_0}(\frac{dw}{dt})_{av} $. The corresponding value reflects how much energy can be dissipated during one period of oscillation in unit volume. From Table 1, one can find that when the interactions between quarks are considered and $ \Delta v/v_0 $ is relatively large, the dissipation rate would be huge, which implies that large amplitude oscillations in a quark star do not last. Contrary to big amplitude oscillation case, small amplitude oscillation usually persist a long time. As a matter of fact, the damping time of big amplitude oscillation can be shorter than one millisecond, while the small amplitude oscillation can last for years [26,45]. In next section, the oscillation damping times for strange stars are also investigated, which is in accordance with the conclusion obtained here.

      $ (C,D^{1/2}) $ $ \frac{1}{v_0}(\frac{dw}{dt})_{av} $ $ \Delta v/v_0 $ $ 10^{-1} $ $ 10^{-2} $ $ 10^{-3} $
      (0.7,129 MeV) $ 3.626\times10^{13} $ $ 6.599\times10^{11} $ $ 1.955\times10^8 $
      (0,0) $ 4.608\times10^{12} $ $ 4.877\times10^9 $ $ 4.926\times10^5 $

      Table 1.  The mean dissipation rate of vibration energy per unit volume $ \frac{1}{v_0}(\frac{dw}{dt})_{av} $ in unit of g·fm−3·s−1 as functions of model parameters $ (C,D^{1/2}) $ and relative amplitude of vibration $ \Delta v/v_0 $. Other model parameters are the same with that in Fig. 7

      In what follows, we aim to find the reasonable ranges of relative oscillation amplitude $ \frac{\Delta v}{v_0} $ and temperature $ T $ for the analytical expression of bulk viscosity given in Eq. (49). For this purpose, we define a new function $ \Theta $,

      $ \Theta = \frac{\zeta-\zeta_a}{\zeta}, $

      (60)

      which actually reflects the accuracy of $ \zeta_a $ compared to $ \zeta $.

      In Fig. 8, $ \Theta $ is given as a function of $ \frac{\Delta v}{v_0} $ and $ T $, and two isolines of $ \Theta $ are shown with values of $ 10^{-1} $ (solid line) and $ 10^{-2} $ (dashed line) respectively. From this figure, it's not difficult to conclude that when the accuracy $ \Theta = 10^{-1} $, the relation between $ \frac{\Delta v}{v_0} $ and $ T $ can be approximately written as

      Figure 8.  $ \Theta $ as function of relative oscillation amplitude $ \frac{\Delta v}{v_0} $ and temperature $ T $. Two isolines of $ \Theta $ are shown, with values of $ 10^{-1} $ (solid line) and $ 10^{-2} $ (dashed line) respectively

      $ \log_{10}\bigg(\frac{\Delta v}{v_0}\bigg)\approx\log_{10}T-1 \quad (\Theta = 10^{-1}, T\lesssim 10^{-2}\ {\rm{MeV}}), $

      (61)

      and when $ \Theta = 10^{-2} $, the relation should be changed to

      $ \log_{10}\bigg(\frac{\Delta v}{v_0}\bigg)\approx\log_{10}T-1.5 \quad (\Theta = 10^{-2}, T\lesssim10^{-2}\ {\rm{MeV}}). $

      (62)

      In fact, in the enhanced perturbative QCD model studied in Ref. [45], we had derived the similar relation between $ \displaystyle\frac{\Delta v}{v_0} $ and $ T $ given by Eq. (61). Here, we obtain it again in a more visualized way in equivparticle model. Furthermore, we update this relation when the accuracy is improved to $ 10^{-2} $. Additionally, the shaded area in Fig. 8 gives the ranges of parameters $ \displaystyle\frac{\Delta v}{v_0} $ and $ T $ corresponding to $ \Theta $ with accuracy higher than $ 10^{-2} $. Here, we should point out that the relations given in Eq. (61) and (62) are not only valid in the current employed model, but also reasonable in other phenomenological models, for example, the bag model, the quasiparticle model, the perturbative model and so on.

    4.   Astrophysical application of bulk viscosity in equivparticle model

      4.1.   Damping times of strange quark stars

    • Given the EoS of quark matter in equivparticle model, one can obtain the properties of strange quark stars by numerically solving the following Tolman-Oppenheimer-Volkov (TOV) equation,

      $ \frac{{\rm{d}} P}{{\rm{d}} r} = -\frac{G m E}{r^{2}} \frac{(1+P / E)\left(1+4 \pi r^{3} P / m\right)}{1-2 G m / r}, $

      (63)

      with the subsidiary condition

      $ \frac{{\rm{d}} m}{{\rm{d}} r} = 4 \pi r^{2} E, $

      (64)

      where $ G = 6.707 \times 10^{-45} {\rm{MeV}}^{ -2} $ is the gravitational constant.

      In Fig. 9, we give the mass-radius relations of quark stars with model parameter sets $ (C,D^{1/2}/{\rm{MeV}}) = (0.7,129) $ and $ (0,129) $ respectively. As we can see from the figure that the maximum mass (denoted by solid dots) of quark star can be larger than 2 ${\rm{M}}_\odot $, which now is generally taken as a popular constraint on EoS of strange quark matter [51-53]. When model parameter $ D = 0 $, it can be noticed from Fig. 1 that the pressure vanishes when energy density becomes zero, which corresponds to zero particle number density. However, as we all know, strange stars have nonzero surface quark number density. Therefore, for $ D = 0 $ the surface pressure of strange stars is positive, which implies strange stars would fall apart. So, we don't give the mass-radius relation for $ D = 0 $ in Fig. 9. Alternatively, in our model with $ D = 0 $, SQM only exists in the interior of neutron stars or hybrid stars with nuclear matter covering the surface of compact stars. This also can be understood from the mass scaling in Eq. (6), which would not satisfy the basic quark confinement in QCD if $ D = 0 $.

      Figure 9.  Mass-radius relations for strange stars. When model parameter $ D = 0 $, the requirement of quark confinement can not be satisfied, and there is no minus pressure in the EoS. Therefore, the mass-radius relations are not given in this figure for $ D = 0 $

      In Fig. 10, the density profiles for different model parameter sets $ (C,D^{1/2}/{\rm{MeV}}) = (0.7,129) $ and $ (0,129) $ are given, in which the solid lines correspond to the most massive strange star, the dashed lines are for stars with canonical 1.4 ${\rm{M}}_\odot $, and the dotted lines represent low-mass strange quark stars with unit solar mass. Additionally, the horizontal dash-dotted lines in both panels give the surface densities for these two cases. The intersection points of lines and left axis denote the central densities, from which one can observe that the heavier the star is, the larger difference of densities between center and surface can be. In other words, the density distributions become more isotropic for less massive stars. Therefore, it is reasonable to roughly take constant values for low-mass stars' density [54]. For more information, one can refer to Table 2, where values of typical quantities of stars are tabulated. In such cases, the damping time $ \tau_D $ of quark stars can be calculated by

      $ (C,D^{1/2}/{\rm{MeV}}) $ $ M/M_\odot $ $ R $/km $ \rho_c $/fm-3 $ \rho_s $/fm-3 $ \bar{\rho} $/fm-3
      2.13 13.71 0.649 0.089 0.234
      (0.7,129) 1.4 14.38 0.215 0.089 0.134
      1 13.40 0.169 0.089 0.118
      2.38 14.12 0.719 0.129 0.240
      (0,129) 1.4 14.12 0.243 0.129 0.141
      1 12.98 0.204 0.129 0.130

      Table 2.  Characteristic quantities for strange stars with typical model parameter sets $ (C,D^{1/2}/{\rm{MeV}}) = (0.7,129) $ and $ (0,129) $. The second through sixth columns give, respectively, the maximum mass $ M $, the radius corresponding to maximum mass $ R $, the center baryon number density $ \rho_c $, the surface baryon number density $ \rho_s $, and the mean baryon number density $ \bar{\rho} $

      Figure 10.  Density profile for different model parameters $ C $ and $ D $. The solid line corresponds to the most massive quark star, the dashed line denotes the quark star with 1.4 ${\rm{M}}_\odot $, the dotted line gives the star with 1 ${\rm{M}}_\odot $, and the dash-dotted line shows the surface baryon number density of strange quark stars

      $ \tau_D = 30^{-1}\bar{\rho}R^2\zeta^{-1}. $

      (65)

      Employing the values of radius and mean density in Table 2, one can easily obtain the damping times for strange stars with canonical 1.4 ${\rm{M}}_\odot $ and the results are graphically given in Fig. 11.

      Figure 11.  Damping times for quark stars with canonical 1.4 solar mass. From bottom to top, the relative oscillation amplitudes are respectively $ \Delta v/v_0 = 10^{-1} $, $ 10^{-4} $, $ 10^{-7} $, and $ 10^{-10} $ for both the solid and dashed lines

      From this figure, it can be seen that the damping time with nonzero $ C $ is a little shorter than that in the other case, except for large relative oscillation amplitudes, e.g. $ \Delta v/v_0 = 10^{-1} $ with the damping times as short as about $ 10^{-4} $ s. In such cases, if a low-mass strange star experiences a strong starquake, the radial oscillations could be rapidly damped with the star heated by the release of the oscillation energy, which may have profound implications in astrophysics and affect the dynamics of compact star merger [55-58].

    • 4.2.   $ R $-mode instability window for strange quark stars

    • Now we investigate the $ r $-mode instability window for strange quark stars based on the obtained bulk viscosity. Driven by gravitational wave emission, compact star would experience $ r $-mode instability if its spin frequency exceeds one critical value [7,59] which is subject to various dissipation mechanisms, such as the bulk and shear viscosity. Therefore, the $ r $-mode instability window is the consequence of the competition between gravitational radiation and various damping mechanisms.

      For a given stellar configuration, the critical rotation frequency, as a function of temperature, is determined by the following equation

      $ \frac{1}{\tau} = \frac{1}{\tau_{\rm{gw}}}+\frac{1}{\tau_{\rm{sv}}}+\frac{1}{\tau_{\rm{bv}}}+\cdots = 0, $

      (66)

      where $ \tau_{\rm{gw}} $ is the characteristic time scale due to GWs emission, $ \tau_{\rm{sv}} $ and $ \tau_{\rm{bv}} $ are damping time scales due to shear and bulk viscosity, and the ellipsis here denotes other dissipation mechanisms, like the surface rubbing which plays crucial role in determining the $ r $-mode instability window of neutron stars [60,61] and color-flavor-locked strange stars [21,54].

      To very good approximation, the resulting bulk viscosity of SQM in equivparticle model, according to the analytical form of bulk viscosity in Eq. (49), is given by

      $ \zeta = \frac{\alpha T^2}{(\kappa\Omega)^2+\beta T^4}. $

      (67)

      Here $ \kappa = 2/3 $ is for dominant $ r $ mode and $ \Omega $ is angular rotation frequency. Here, $ \alpha $ and $ \beta $, can be rewritten in cgs units as

      $ \alpha = 9.39\times10^{22}\mu_d^5\left(\frac{3C_{3d}v_0}{A_dC_{1d}}-\frac{3C_{3s}v_0}{A_sC_{1s}}\right)^2 \ ({\rm{g}}\cdot {\rm{cm}}^{-1}\cdot {\rm{s}}^{-1}), $

      (68)

      and

      $ \beta = 7.11\times10^{-4}\left[\frac{3\mu_d^5v_0}{2\pi^2} \left(\frac{1}{A_dC_{1d}}+\frac{1}{A_sC_{1s}}\right)\right]^2 \ ({\rm{s}}^{-2}). $

      (69)

      In analogy to previous results of $ \zeta $ in Refs. [26,44], a low-$ T $ limit of $ \zeta $ in cgs units yields

      $ \zeta^{{\rm{low}}}\approx\bar{\zeta}^{{\rm{low}}}\bar{\rho} T^2(\kappa\Omega)^{-2}m_{100}^4, $

      (70)

      with

      $ \bar{\zeta}^{{\rm{low}}} = 3.41\times10^{-20}\alpha \mu_d^{-3} m_s^{-4}, $

      (71)

      where $ m_{100} $ is the mass of strange quark in units of $ 100 $ MeV, and $ \bar{\rho} $ is the mean densities of strange stars. Meanwhile, a high-$ T $ limit of $ \zeta $ in cgs units gives,

      $ \zeta^{{\rm{high}}}\approx\bar{\zeta}^{{\rm{high}}}\bar{\rho}^{-1} T^{-2}m_{100}^4, $

      (72)

      with

      $ \bar{\zeta}^{{\rm{high}}} = 2.88\times10^{35}\alpha\beta^{-1}\mu_d^3m_s^{-4}. $

      (73)

      Note that the only difference between the bulk viscosity here and the one in Ref. [54] is the coefficients in the expressions of $ \zeta^{{\rm{low}}} $ and $ \zeta^{{\rm{high}}} $. Therefore, one can obtain the damping time scales $ \tau_{\rm{bv}}^{{\rm{low}}} $ and $ \tau_{\rm{bv}}^{{\rm{high}}} $ in the same manner, which gives

      $ \tau_{\rm{bv}}^{{\rm{low}}} = \bar{\tau}_{\rm{bv}}^{{\rm{low}}}(\pi G \bar{\rho}/\Omega^2)T_9^{-2}m_{100}^{-4}, $

      (74)

      with the prefactor $ \bar{\tau}_{\rm{bv}}^{{\rm{low}}} = 9.44\times10^{-24}\alpha \mu_d^{-3} m_s^{-4} $, and

      $ \tau_{\rm{bv}}^{{\rm{high}}} = \bar{\tau}_{\rm{bv}}^{{\rm{high}}}(\pi G \bar{\rho}/\Omega^2)^2T_9^{2}m_{100}^{-4}, $

      (75)

      with the prefactor $ \bar{\tau}_{\rm{bv}}^{{\rm{high}}} = 2.03\times10^{-27}\alpha\beta^{-1}\mu_d^3m_s^{-4} $. Here $ T_9 $ is the temperature of strange stars in units of $ 10^9 $ K.

      In literatures [5,62], the driving time scale due to gravity wave emission is

      $ \tau_{\rm{gw}} = -3.26\ {\rm{s}}\ (\pi G \bar{\rho}/\Omega^2)^3, $

      (76)

      where for a assumed polytropic equation of state with $ n = 1 $ the prefactor -3.26 s is adopted. For the damping time scale due to shear viscosity [63], it is

      $ \tau_{\rm{sv}} = 5.37\times 10^8\ {\rm{s}}\ (\alpha_s/0.1)^{5/3}T_9^{5/3}, $

      (77)

      where $ \alpha_s $ is the strong coupling. We take $ \alpha_s = 0.1 $ in the following numerical calculations.

      For the purpose of comparison, in Fig. 12, we show the $ r $-mode instability window of pulsar spin frequency $ (\nu = \Omega/2\pi) $ and temperature $ T $, where a typical compact star with mass $ M = 1.4\ {\rm{M}}_\odot $ and radius $ R = 10 $ km is considered. The observational data (solid dots with error bar) on spin frequency and internal temperature of compact stars in LMXBs are also given [64]. The dashed lines obtained by setting the model parameters $ C = 0 $ and $ D = 0 $ are exactly the same with that shown in Fig. 1 in Ref. [54]. It's found that $ r $-mode instability window is significantly narrowed when the quark confinement effects and perturbative interactions are both taken into account, which is consistent with the observational data of compact stars in LMXBs in the stable regime.

      Figure 12.  $ R $-mode (in)stability window for an assumed typical compact star with mass $ M = 1.4M_\odot $ and radius $ R = 10 $ km. Observational data on spin frequency and internal temperature of compact stars in low-mass X-ray binaries (LMXBs) are also given

      However, we mention that the typical strange star configuration assumed in Fig. 12 is not tolerated in our model, where according to TABLE II the radius of a strange star with $ M = 1.4\ {\rm{M}}_\odot $ is 14.38 km rather than 10 km. In Fig. 13, we thus present the $ r $-mode instability window for realistic strange stars with $ M = 1.4\ {\rm{M}}_\odot $ in our model. The solid line denotes the star with both quark confinement effects and perturbative interactions, whereas the dashed line represents the star merely with quark confinement. According to the Fig. 13, although the absence of perturbative interactions makes one star (SAX J1808.4-3658) locate in the unstable region, it is obvious that with full interactions our numerical results are still consistent well with the observational data.

      Figure 13.  $ R $-mode (in)stability window for real stars in our model. The solid line denotes the star with both quark confinement effects and perturbative interactions, whereas the dashed line corresponds to the star merely with quark confinement

      Nevertheless, we stress that the results obtained here are based on the calculations for bare strange quark stars. If compact stars, like hybrid stars [65], bear an nuclear matter envelop, the surface rubbing effects would play an important role. Also, if strange stars are made of SQM in 2SC phase [66] or CFL phase [21,54] of QCD [67], the shear viscosity due to electron-electron scattering plays a dominant role in dissipating radial oscillations, while the shear viscosity due to quark-quark scattering and bulk viscosity due to the non-leptonic weak interaction is less significant.

    5.   Summary
    • In this paper, we have investigated the bulk viscosity of SQM in equivparticle model, which can be readily applied to those with other quark mass scalings. By taking constant masses for quarks, previous results in bag model were reproduced. With a recently obtained quark mass scaling, we investigated the contributions of the quark confinement effects and perturbative interactions to bulk viscosity of SQM, and found the contributions to bulk viscosity from quark confinement effects significantly prevail over that from quark perturbative interactions. Due to the interactions between quarks, the dissipation rate in equivparticle model is generally larger than that in bag model, which therefore considerably enhances the magnitude of bulk viscosity. This conclusion is in accordance with previous results [36,37,44]. However, we would like to point out that the model we employed here can yield strange quark stars with maximum mass larger than 2 ${\rm{M}}_\odot $, consistent with the recent astrophysical observations [51,52].

      As to the application of the resulting bulk viscosity in astrophysics, we firstly studied the oscillation damping times of strange quark stars, from which we found the damping times can be as short as to a fraction of millisecond for large relative amplitude oscillations. This could lead to rapid heating of strange stars after unstable activities such as starquakes, which may affect star evolutions and have meaningful implications in astronomical observation. At last, we calculated the $ r $-mode instability windows for $ 1.4\ {\rm{M}}_\odot $ strange stars. Unlike those previously obtained in bag model or modified bag model, the $ r $-mode instability windows predicted by the new enlarged bulk viscosity are consistent with the observational data of stars in LMXBs.

      However, it should be pointed out that the conclusion obtained here are based on the assumption of bare strange stars. For compact stars with normal nuclear matter crust, strong magnetic field [34], 2SC or CFL SQM in the core, or superfluid neutron star cores [68], the current conclusion may be altered. Therefore, more efforts should be devoted to these related issues. Additionally, the upper limit on spin frequency of pulsars in LMXBs has been updated from 716 Hz [69] to 1122 Hz [70], although the statistical significance of the later is not very strong. Therefore, it is meaningful to examine the critical spin frequencies predicted by the model employed here with the parameters properly fixed. We leave this subject as one part of our future work.

Reference (70)

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