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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 that the real chemical potentials of particles are replaced by effective ones. Therefore, the thermodynamic potential density of a quark agglomerate appears the same in form as the free particles, i.e.,
Ω0=−∑igi24π2[μ∗iνi(ν2i−32m2i)+32m4ilnμ∗i+νimi],
(1) where
μ∗i andνi=√μ∗2i−m2i are the effective chemical potential and the Fermi momentum of particle speciesi , respectively, withi running overu,d,s ande . The degenerate factorgi is 6 for quarks and 2 for electrons. Here,mi is the corresponding effective mass of particle speciesi , which can be written in two parts,mi=mi0+mI,
(2) where
mi0 is the quark current mass, andmI is the interacting part. Foru,d,s ande ,mi0 values are 5,10,100 and 0.511 MeV, respectively. Originating from strong interactions,mI is naturally the same for all quarks, whereas it vanishes for electrons.In previous literature,
mi has been parameterized in many different forms. According to the bag model assumption, it is originally parameterized as [47]mi=mi0+B3ρb,
(3) where
ρb is the baryon number density. Then, a cubic-root scaling is proposed in Ref. [48],mi=mi0+Dρ1/3b,
(4) which is derived from the linear confinement and leading-order in-medium chiral condensate. Later, it is extended to include the temperature effect [49],
mi=mi0+Dρ1/3b[1−8TλTcexp(−λTcT)].
(5) Recently, a new mass scaling has been suggested with linear confinement and leading-order perturbative interactions [46],
mi=mi0+Dρ1/3b+Cρ1/3b.
(6) Here, we should emphasize that the parameters
C andD respectively indicate the strength of perturbative interactions and confinement effects, which are adopted in the following section to investigate their effects on the bulk viscosity of SQM.The relation between real and effective chemical potential in the equivparticle model is
μi=μ∗i+13dmIdρb∂Ω0∂mI≡μ∗i−μI,
(7) where
μI is the same for all quarks. Consequently, theβ -equilibrium still holds for effective chemical potentials, i.e.,μ∗u+μe=μ∗d=μ∗s.
(8) According to traditional thermodynamics, the particle number density
ρi can be derived by the common relationρi=−dΩ0/dμ∗i , which givesρi=gi6π2(μ∗2i−m2i)3/2=giν3i6π2.
(9) The charge neutrality condition and baryon number conservation can hence be respectively written as
23ρu−13ρd−13ρs−ρe=0,
(10) and
ρb=13(ρu+ρd+ρs).
(11) Then, the energy density and pressure are given by
E=Ω0−∑iμ∗i∂Ω0∂μ∗i=∑igi16π2[νiμ∗i(μ∗2i+ν2i)−m4iln(μ∗i+νimi)],
(12) and
P=−Ω0+ρb∂mI∂ρb∂Ω0∂mI=−Ω0−3ρbμI.
(13) With a given baryon number density
ρb , one can simultaneously solve Eqs. (8), (10), and (11) to obtain the values of effective chemical potentialsμ∗i (i=u,d,s,e) . Then, according to Eqs. (12) and (13), the equation of state (EoS) of SQM can be obtained byP=P(E) .In Fig. 1, we present the EoS of SQM in the equivparticle model with the new quark mass scaling in Eq. (6). From this figure, one can readily find that 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 the EoS with the specified model parameters.
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In this section, employing a traditional method [26,50], we give the bulk viscosity of SQM in the equivparticle model. After reproducing the existing results [26], we investigate the effects of quark interactions on the bulk viscosity by adopting different mass scalings.
First, we assume that the volume per unit mass of SQM is
v , which, due to the vibration of a strange quark star, oscillates harmonically according to the following equation:v(t)=v0+Δvsin(2πtτ)≡v0+δv(t),
(14) where
τ is the volume vibration period,v0 is the equilibrium volume, andΔv is the vibration amplitude, which is small compared withv0 , i.e.,Δv/v0≪1 . The vibration ofv could result in the change in particle phase space and further cause variation in the particle number per unit mass,ni(t)=ni0+δni(t),
(15) and unless otherwise stated,
i representsu,d, ands quarks here and below;δni can be obtained by integration:δnd=−δns=∫t0dnddtdt.
(16) As for the reaction rate
dnd/dt , we adopt the result valid in low temperatures (T≪μi ) [26], i.e.,dnddt≈GCμ5dδμ(δμ2+4π2T2)v0,
(17) where
δμ≡μs−μd , and the constantGc is connected to the weak couplingGF and the Cabibbo angleθC byGC≡165π5G2Fsin2θCcos2θC=6.76×10−26MeV−4.
(18) Because the bulk viscosity of SQM mainly stems from the nonleptonic weak reaction
u+d↔u+s , the pressureP can be expanded near theβ -equilibrium pressureP0 , i.e.,P(t)=P0+(∂P∂v)0δv+(∂P∂nd)0δnd+(∂P∂ns)0δns.
(19) Then, the mean dissipation rate of the vibration energy per unit mass is given by
(dwdt)av=−1τ∫τ0P(t)dvdtdt.
(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
δμ can also be expanded in terms ofδv ,δnd , andδns , namelyδμ(t)=(∂δμ∂v)0δv+(∂δμ∂nd)0δnd+(∂δμ∂ns)0δns,
(21) in which
∂δμ∂x=∂μs∂x−∂μd∂x, (x=v,nd,or ns) . Importantly,∂μi∂v is connected to the third and fourth terms in Eq. (19) by the following relation:∂μi∂v=−∂P∂ni.
(22) To obtain the bulk viscosity, one should give the expressions of derivatives of chemical potentials in terms of
v ,nd , andns . To this end, we start from the quark number per unit massni ,ni=ρiv=1π2(μ∗2i−m2i)3/2v,
(23) from which it is not difficult to obtain the differential form of
ni , i.e.,dni=vdρi+ρidv=3vνiπ2(μ∗idμ∗i−midmi)+ρidv.
(24) In the equivparticle model, to mimic the complicated strong interactions between quarks, the particle masses are taken as functions of baryon number density, i.e.,
mi=mi(ρb) , from which the differential form ofmi can be written asdmi=dmidρbdρb=dmIdρbdρb.
(25) According to the definition of baryon number density
ρb=13∑iρi , one can obtainvρb=13∑ivρi=13∑ini . Then, the corresponding differential form isd(vρb)=vdρb+ρbdv=13∑idni , from whichdρb=13v∑idni−ρbvdv.
(26) Substituting Eqs. (25) and (26) into Eq. (24), one can obtain the differential form of the particle number per unit mass
dni , i.e.,dni=3vνiμ∗iπ2dμ∗i−νimiπ2dmIdρb∑jdnj+(ρi+3νimiρbπ2dmIdρb)dv,≡C1idμ∗i+C2i∑jdnj+C3idv,
(27) where we have changed the dummy index
i in Eq. (26) toj in Eq. (27) and introduced three notations to give a tighter form of this expression, i.e.,C1i≡3vνiμ∗iπ2,
(28) C2i≡−νimiπ2dmIdρb,
(29) C3i≡ρi+3νimiρbπ2dmIdρb.
(30) Note that the chemical potentials in Eq. (27) are effective ones, which should be linked to the real ones. To do this, we use the differential form of Eq. (7), i.e.,
dμ∗i=dμi+dμI.
(31) To facilitate the derivation of bulk viscosity of SQM in the equivparticle model,
μI can be redefined as a new functionf as follows,μI=−13dmIdρb∂Ω0∂mI≡f(ρb,mi(ρb),μ∗i).
(32) Accordingly, the differential form of
μI isdμI=(∂f∂ρb+dmIdρb∑j∂f∂mj)dρb+∑j∂f∂μ∗jdμ∗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
dμ∗i asdμ∗i=11−∂f∂μ∗i[dμi+(∂f∂ρb+dmIdρb∑j∂f∂mj)(13v∑jdnj−ρbvdv)+∑j≠i∂f∂μ∗jdμ∗j],=11−∂f∂μ∗idμi+13v11−∂f∂μ∗i(∂f∂ρb+dmIdρb∑j∂f∂mj)∑jdnj−ρbv11−∂f∂μ∗i(∂f∂ρb+dmIdρb∑j∂f∂mj)dv+∑j≠i∂f∂μ∗jdμ∗j.
(34) By taking
Ai=11−∂f∂μ∗i,B=∂f∂ρb+dmIdρb∑j∂f∂mj,
with
∂f∂μ∗i=−miνiπ2dmIdρb,
(35) ∂f∂mj=gj4π2(μ∗jνj−3m2jlnμj+νjmj),
(36) ∂f∂ρb=−d2mi3dρ2b∑jgjmj4π2[μ∗jνj−m2jlnμ∗j+νjmj],
(37) one can rewrite Eq. (34) as
dμ∗i=Aidμi+AiB3v∑jdnj−ρbAiBvdv+∑j≠i∂f∂μ∗jdμ∗j.
(38) Substituting Eq. (38) into Eq. (27) and applying
dni=∑jδijdnj , whereδij is the Kronecker delta, one can obtain the differential form of the real chemical potentialdμi=∑j(δij−C2iAiC1i−B3v)dnj+(ρbBv−C3iAiC1i)dv−1Ai∑j≠i∂f∂μ∗jdμj.
(39) From Eq. (39), we immediately obtain
(∂μi∂nj)0=δij−C2iAiC1i−B3v,
(40) (∂μi∂v)0=ρbBv−C3iAiC1i.
(41) Substituting Eqs. (40) and (41) into Eq. (19) and ignoring the first two terms on the right-hand side of Eq. (19), one obtains the pressure contributing to the dissipation energy,
P(t)=(∂P∂nd)0δnd+(∂P∂ns)0δns=(C3dAdC1d−C3sAsC1s)∫t0dnddtdt.
(42) The chemical potential difference
δμ , accordingly, can be derived in a similar manner, namelyδμ(t)=(∂δμ∂v)0δv+(∂δμ∂nd)0δnd+(∂δμ∂ns)0δns,=(C3dAdC1d−C3sAsC1s)Δvsin(ωt)−(1AdC1d+1AsC1s)∫t0dnddtdt,
(43) which naturally gives the differential equation of
δμ(t) ,dδμ(t)dt=(C3dAdC1d−C3sAsC1s)2πΔvτcos(2πtτ)−(1AdC1d+1AsC1s)dnddt.
(44) To solve this differential equation numerically, the related expressions of these notations in Eq. (44) are explicitly given as follows:
C3i=ρi+3νimiρbπ2dmIdρb,
(45) 1AiC1i=13vμ∗i(π2νi+midmIdρb).
(46) From these expressions, we can clearly see the mass-density dependent term
dmI/dρb , which explicitly denotes the main feature of the bulk viscosity in the equivparticle model.The mean dissipation rate of the vibration energy per unit mass reads
(dwdt)av=−1τ∫τ0P(t)dvdtdt=Δvτ2πτ(C3sAsC1s−C3dAdC1d)×∫τ0dt(∫t0dnddtdt)cos(2πtτ).
(47) Finally, the bulk viscosity is
ζ≡2(dw/dt)avv0(v0Δv)2(τ2π)2=1πv0Δv(C3sAsC1s−C3dAdC1d)×∫τ0dt(∫t0dnddtdt)cos(2πtτ).
(48) Due to the cubic term of
δμ (δμ≲0.1 MeV) in Eq. (17), it is not possible to solve the bulk viscosity analytically. However, in the relatively high-temperature limit (2πT≫δμ , or the temperature of interest here satisfiesT≲1 MeV ), one can ignore the cubic term and find the analytical form of bulk viscosity asζa=αT2ω2+βT4[1−(1−e−β1/2T2τ)2β1/2T2/τω2+βT4],
(49) where
α=Gcμ5dv204π2(C3dAdC1d−C3sAsC1s)2,
(50) and
β=G2cμ10dv2016π4(1AdC1d+1AsC1s)2.
(51) In what follows, to prove the validity of our results, we show that in the mass-density-independent case, i.e., parameters
C=D=0 , the equations in our model are exactly the same as those in the bag model. IfC=D=0 ,Ai,B,C1i,C2i , andC3i can be respectively simplified toAi=1,B=0,C1i=3vνiμiπ2,C2i=0,C3i=ρi.
(52) Accordingly, Eq. (40) and Eq. (41) can be reduced to
(∂μi∂nj)0=δijπ23vνiμi
(53) and
(∂μi∂v)0=−π2ρi3vνiμi.
(54) Therefore, Eq. (39) can be simplified to
dμi=π23vνiμidni−π2ρi3vνiμidv=π23νiμidρi,
(55) which is exactly the same as the differential form of the particle number density
ρi=ν3iπ2 , whereνi=√μ2i−m2i with the real chemical potentialμi . In this case, the bulk viscosity in the equivparticle model given by Eq. (48) is reduced toζ=−1πv0Δvm2s3v0μd∫τ0dt(∫t0dnddtdt)cos(2πtτ).
(56) Meanwhile, although the analytical form of
ζ in Eq. (49) is not changed,α andβ are respectively reduced toα=4π29Gcμ3dm4s0,
(57) and
β=64π89G2cμ6d(1+m2s04μ2d)2.
(58) They are respectively consistent with Eqs. (15-18) in Ref. [26]. This limiting case suggests that the bulk viscosity in the equivparticle model can be treated as a generalization of previous results in the bag model.
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To calculate the bulk viscosity of SQM in the 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 the bulk viscosity
ζ as function of relative volume perturbation amplitudeΔv/v0 and temperatureT . The lines with solid balls calculated by Eq. (48) show the results in the equivparticle model, while the lines with stars calculated by Eq. (56) indicate the bulk viscosity results in the mass-density-independent case, which are exactly the same as the previous results in the bag model [26]. In the equivparticle model, the parametersC andD1/2 in mass scaling Eq. (6) are respectively taken as 0.7 and 129 MeV, which guarantees that the EoS of SQM can support quark stars with maximum mass larger than 2M⊙ , and that the SQM is absolutely stable. In addition, temperatures are given as10−5 ,10−3 , and10−1 MeV, denoted by red, green, and blue lines, respectively.Figure 2. (color online) Bulk viscosity as a function of relative volume perturbation amplitude and temperature. The lines with solid balls and stars indicate the results in the equivparticle model and the mass-density-independent case (i.e., the bag model), respectively. In numerical calculations, the oscillation period of quark matter is
τ=10−3 s, and the temperaturesT are10−5 ,10−3 , and10−1 MeV, denoted by red, green, and blue lines, respectively.In numerical calculations, the baryon number density is set to
ρ=1.34 fm-3. To facilitate comparison of the magnitude of bulk viscosity, we also give the projection of each line on theZ−Y plane. From Fig. 2, it is clear to see that at different temperatures, the bulk viscosity in the equivparticle model is generally larger than that in the bag model by a magnitude of approximately 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 better understand this aspect, we introduce a parameterκ in the interaction part of mass scaling, i.e.,mi=mi0+κmI, κ∈[0,1].
(59) If
κ=0 , the interactions between quarks vanishes; whileκ approaches unity, the interactions are gradually recovered.Figure 3 reveals the interaction intensity behavior of the bulk viscosity, from which one can observe that as interactions between quarks become strong, the 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 is also clearly indicated that the quark confinement effects have a more important influence on the bulk viscosity than the perturbative interactions. This conclusion still holds if we adopt other values of parameter sets, where the approximate relation betweenC andD in the absolutely stable region of SQM [46] is given asFigure 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 as in Fig. 2. In addition, the values of
Δv/v0 andT are set to10−8 and10−5 MeV, respectively.D1/2/MeV={−48C+156,C∈[−0.5,0],−270.7C+156,C∈(0,0.7].
This relation is adopted in Fig. 4 on the top and bottom X-axes. To study the perturbative interactions on the bulk viscosity, we set
D=0 , and the results are given by the dotted line. Similarly, to study the confinement interactions on the bulk viscosity, we setC=0 , and the results are given by the dashed line. From these two lines, it is observed that with increasingC orD , the bulk viscosity increases. Furthermore, in a large parameter range ofC andD , it is shown again that the confinement effects on the bulk viscosity are much larger than the perturbative interactions. For the purpose of comparison, we also present the bulk viscosity of SQM with both confinements effects and perturbative interactions, shown by the solid line.Figure 4. Different interactions with bulk viscosity of absolutely stable SQM. From the dashed and dotted lines, it is shown that the confinement effects from model parameter
D contribute to the bulk viscosity much more than the perturbative interactions from model parameterC .To explicitly show the effects of quark confinement and perturbative interactions on the bulk viscosity, in Fig. 5 we present the bulk viscosity of SQM adopting various model parameters
C andD . IfC andD both vanish, it corresponds to the bag model predictions (the dash-dotted line). If only the perturbative interactions are accounted for, we obtain the results indicated by the dotted line, which are much higher than those of the bag model. Moreover, if only the quark confinement effects are included, the results are given by the dashed line, which is consistent with the implications of Fig. 3, namely that the quark confinement effects prevail over the perturbative interactions in the bulk viscosity of SQM. Finally, as expected, if these two types of interactions are both considered, the bulk viscosity shown by the solid line becomes the highest. It is interesting to note that the results shown by the dashed line are almost the same as those shown by 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 the bulk viscosity of SQM. The solid line gives the results in the equivparticle model, whereas the dash-dotted line corresponds to those in the bag model. The dotted line only includes the perturbative interactions, and the dashed line only includes the quark confinement effects. To verify the impact of different mass scaling on bulk viscosity, we also give the results calculated for the equivparticle model with mass scaling in Eq. (3).
To study 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 does not contradict our conclusion, as the mass scaling in Eq. (3) can provide much stronger quark confinement effects.
Customarily, strange stars have strong magnetic fields, which have a significant influence on the transport coefficients, especially the bulk viscosity. In Ref. [34], the authors investigated the bulk viscosity in greater detail, and found that in strong magnetic fields, the transport coefficients of SQM become anisotropic. However, for the magnetic fields
B<1017 G, the effect of magnetic field on bulk viscosity can be ignored, and a comparison with our results at the low-B limit is possible. In Fig. 6, we show the results of the bulk viscosity from Ref. [34] and the equivparticle model at the same baryon number densityρ∼0.787 fm-3 (orμu=μd=400 MeV in Ref. [34]). From Fig. 6, we find that the bulk viscosity in our model is usually higher than that in Ref. [34], except for10≲ω/s−1≲1000 , whereω=2π/τ . In addition, with increasingω , the bulk viscosity in the equivparticle model decreases much more slowly, because the interactions between quarks are considered.Figure 6. Comparison of the bulk viscosity in Ref. [34] and equivparticle model at the same baryon number density
ρ∼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
δμ=μs−μd with differentΔv/v0 in Fig. 7. In these panels, from top to bottom,Δv/v0 values are10−1 ,10−2 , and10−3 , respectively. The solid lines represent the results including interactions between quarks withC=0.7 andD1/2=129 MeV, whereas the dashed lines give the results with vanishingC andD . It can be seen from Fig. 7 that with increasingΔv/v0 , the amplitudes ofδμ in both cases become large. Furthermore, according to Fig. 7, the amplitudes are significantly enhanced once the interactions between quarks are considered, which suggests that more energy could be involved in one period of oscillation and physically means a larger bulk viscosity of SQM.Figure 7.
δμ(t) for two cycles with parametersρb=1.34 fm−3,T=10−5 MeV, andτ=10−3 s andΔv/v0=10−1,10−2,and 10−3 , respectively. The solid lines indicate the results in the equivparticle model with parametersC=0.7 andD1/2=129 MeV, whereas the dashed lines correspond to the results withC=D=0 .A situation similar to that of the amplitude of
δμ in Fig. 7 can be observed in Table 1, which gives the mean dissipation rate of vibration energy per unit volume1v0(dwdt)av . The corresponding value reflects how much energy can be dissipated during one period of oscillation per unit volume. From Table 1, one can find that when the interactions between quarks are considered andΔv/v0 is relatively large, the dissipation rate is very high, which implies that large amplitude oscillations in a quark star do not last. Contrary to the large amplitude oscillation case, small amplitude oscillation usually persists for a long time. In fact, the damping time of large amplitude oscillations can be shorter than one millisecond, while small amplitude oscillations can last for years [26,45]. In the next section, the oscillation damping times for strange stars are also investigated, which are in accordance with the conclusion obtained here.Δv/v0 10−1 10−2 10−3 (C,D1/2/MeV) (0.7, 129) (0, 0) (0.7, 129) (0, 0) (0.7, 129) (0, 0) 1v0(dwdt)av 3.626×1013 4.608×1012 6.599×1011 4.877×109 1.955×108 4.926×105 Table 1. Mean dissipation rate of vibration energy per unit volume
1v0(dwdt)av in units of g·fm−3·s−1 as functions of model parameters(C,D1/2) and relative amplitude of vibrationΔv/v0 . Other model parameters are the same as in Fig. 7.In what follows, we aim to find the reasonable ranges of relative oscillation amplitude
Δvv0 and temperatureT for the analytical expression of bulk viscosity given in Eq. (49). For this purpose, we define a new functionΘ ,Θ=ζ−ζaζ,
(60) which reflects the accuracy of
ζa compared toζ .In Fig. 8,
Θ is given as a function ofΔvv0 andT , and two isolines ofΘ are shown with values of10−1 (solid line) and10−2 (dashed line), respectively. From this figure, it is not difficult to conclude that when the accuracyΘ=10−1 , the relation betweenΔvv0 andT can be approximately written asFigure 8.
Θ as a function of relative oscillation amplitudeΔvv0 and temperatureT . Two isolines ofΘ are shown, with values of10−1 (solid line) and10−2 (dashed line), respectively.log10(Δvv0)≈log10T/MeV−1(Θ=10−1,T≲10−2 MeV),
(61) and when
Θ=10−2 , the relation should be changed tolog10(Δvv0)≈log10T/MeV−1.5(Θ=10−2,T≲10−2 MeV).
(62) In fact, in the enhanced perturbative QCD model studied in Ref. [45], we had derived the similar relation between
Δv/v0 andT given by Eq. (61). Here, we obtain it again in a more visualized way in the equivparticle model. Furthermore, we update this relation when the accuracy is improved to10−2 . Additionally, the shaded area in Fig. 8 gives the ranges of parametersΔv/v0 andT corresponding toΘ with accuracy higher than10−2 . Here, we should point out that the relations given in Eqs. (61) and (62) are not only valid in the current employed model but also reasonable in other phenomenological models, for example, the bag model, quasiparticle model, or perturbative model. -
Given the EoS of quark matter in the equivparticle model, one can obtain the properties of strange quark stars by numerically solving the following Tolman-Oppenheimer-Volkov (TOV) equation:
dPdr=−GmEr2(1+P/E)(1+4πr3P/m)1−2Gm/r,
(63) with the subsidiary condition
dmdr=4πr2E,
(64) where
G=6.707×10−45MeV−2 is the gravitational constant.In Fig. 9, we give the mass-radius relations of quark stars with model parameter sets
(C,D1/2/MeV)= (0.7, 129) and (0, 129), respectively. As we can see from the figure, the maximum mass (denoted by solid dots) of a quark star can be larger than 2M⊙ , which now is generally taken as a popular constraint on the EoS of SQM [51-53]. When model parameterD=0 , it can be seen from Fig. 1 that the pressure vanishes when the energy density becomes zero, which corresponds to zero particle number density. However, as is well established, strange stars have nonzero surface quark number density. Therefore, forD=0 the surface pressure of strange stars is positive, which implies strange stars would fall apart. Therefore, we do not give the mass-radius relation forD=0 in Fig. 9. Alternatively, in our model withD=0 , SQM only exists in the interior of neutron stars or hybrid stars with nuclear matter covering the surfaces of compact stars. This can also be understood from the mass scaling in Eq. (6), which does not satisfy the basic quark confinement in QCD ifD=0 .Figure 9. Mass-radius relations for strange stars. When model parameter
D=0 , the requirement of quark confinement cannot be satisfied, and there is no minus pressure in the EoS. Therefore, the mass-radius relations are not given in this figure forD=0 .In Fig. 10, the density profiles for different model parameter sets
(C,D1/2/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 1.4M⊙ , 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 the left axis denote the central densities, from which one can observe that the heavier the star is, the larger the difference in densities between the center and surface can be. In other words, the density distributions become more isotropic for less massive stars. Therefore, it is reasonable to take approximately constant values for low-mass star densities [54]. For more information, one can refer to Table 2, in which values of typical parameters of stars are tabulated. In such cases, the damping timeτD of quark stars can be calculated by(C,D1/2/MeV) M/M⊙ R /kmρc /fm−3ρs /fm−3ˉρ /fm−32.13 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,D1/2/MeV)=(0.7,129) and(0,129) . The second through sixth columns give the maximum massM , radius corresponding to maximum massR , center baryon number densityρc , surface baryon number densityρs , and mean baryon number densityˉρ , respectively.Figure 10. Density profile for different model parameters
C andD . The solid line corresponds to the most massive quark star, the dashed line denotes the quark star with 1.4M⊙ , the dotted line gives the star with 1M⊙ , and the dash-dotted line shows the surface baryon number density of strange quark stars.τD=30−1ˉρR2ζ−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
M⊙ , 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
Δv/v0=10−1 ,10−4 ,10−7 , and10−10 , respectively, 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.,Δv/v0=10−1 , with damping times as short as approximately10−4 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 the compact star merger [55-58]. -
We now investigate the
r -mode instability window for strange quark stars based on the obtained bulk viscosity. Driven by gravitational wave emission, a compact star experiencesr -mode instability if its spin frequency exceeds one critical value [4,7] that is subject to various dissipation mechanisms, such as the bulk and shear viscosity. Therefore, ther -mode instability window is the consequence of 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:
1τ=1τgw+1τsv+1τbv+⋯=0,
(66) where
τgw is the characteristic time scale due to GW emission,τsv andτbv are damping time scales due to shear and bulk viscosities, respectively, and the ellipsis denotes other dissipation mechanisms, such as surface rubbing, which plays a crucial role in determining ther -mode instability window of neutron stars [59,60] and color-flavor-locked strange stars [21,54].To a very good approximation, the resulting bulk viscosity of SQM in the equivparticle model, according to the analytical form of bulk viscosity in Eq. (49), is given by
ζ=αT2(κΩ)2+βT4.
(67) Here
κ=2/3 is for dominantr mode, andΩ is angular rotation frequency. Here,α andβ can be rewritten in cgs units asα=9.39×1022μ5d(3C3dv0AdC1d−3C3sv0AsC1s)2 (g⋅cm−1⋅s−1),
(68) and
β=7.11×10−4[3μ5dv02π2(1AdC1d+1AsC1s)]2 (s−2).
(69) In analogy to previous results of
ζ in Refs. [26,44], a low-T limit ofζ in cgs units yieldsζlow≈ˉζlowˉρT2(κΩ)−2m4100,
(70) with
ˉζlow=3.41×10−20αμ−3dm−4s,
(71) where
m100 is the mass of a strange quark in units of100 MeV, andˉρ is the mean density of strange stars. Meanwhile, a high-T limit ofζ in cgs units givesζhigh≈ˉζhighˉρ−1T−2m4100,
(72) with
ˉζhigh=2.88×1035αβ−1μ3dm−4s.
(73) Note that the only difference between the bulk viscosity here and the one in Ref. [54] is the coefficients in the expressions of
ζlow andζhigh . Therefore, one can obtain the damping time scalesτlowbv andτhighbv in the same manner, which givesτlowbv=ˉτlowbv(πGˉρ/Ω2)T−29m−4100,
(74) with the prefactor
ˉτlowbv=9.44×10−24αμ−3dm−4s , andτhighbv=ˉτhighbv(πGˉρ/Ω2)2T29m−4100,
(75) with the prefactor
ˉτhighbv=2.03×10−27αβ−1μ3dm−4s . Here,T9 is the temperature of strange stars in units of109 K.In the literature [5,61], the driving time scale due to gravity wave emission is
τgw=−3.26 s (πGˉρ/Ω2)3,
(76) where for an assumed polytropic EoS with
n=1 , the prefactor -3.26 s is adopted. For the damping time scale due to shear viscosity [62], it isτsv=5.37×108 s (αs/0.1)5/3T5/39,
(77) where
αs is the strong coupling. We takeαs=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(ν=Ω/2π) and temperatureT , where a typical compact star with massM=1.4 M⊙ and radiusR=10 km is considered. The observational data (solid dots with error bars) on spin frequency and internal temperatures of compact stars in LMXBs are also given [63]. The dashed lines obtained by setting the model parametersC=0 andD=0 are exactly the same as those shown in Fig. 1 in Ref. [54]. It is found that ther -mode instability window is significantly narrowed when the quark confinement effects and perturbative interactions are both considered, 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 massM=1.4 M⊙ and radiusR=10 km. Observational data on spin frequency and internal temperature of compact stars in 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 2, the radius of a strange star with
M=1.4 M⊙ is 14.38 km rather than 10 km. In Fig. 13, we thus present ther -mode instability window for realistic strange stars withM=1.4 M⊙ 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 Fig. 13, although the absence of perturbative interactions locates one star (SAX J1808.4-3658) 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 [64], bear a nuclear matter envelope, surface rubbing effects can play an important role. Moreover, if strange stars are made of SQM in the 2SC phase [65] or CFL phase [21,54] of QCD [66], the shear viscosity due to electron-electron scattering plays a dominant role in dissipating radial oscillations, whereas the shear viscosity due to quark-quark scattering and bulk viscosity due to the nonleptonic weak interaction is less significant.
Bulk viscosity for interacting strange quark matter and r-mode instability windows for strange stars
- Received Date: 2020-05-21
- Accepted Date: 2020-08-18
- Available Online: 2021-01-15
Abstract: We investigate the bulk viscosity of strange quark matter in the framework of the equivparticle model, where analytical formulae are obtained for certain temperature ranges, which 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