Bulk viscosity of strange quark matter in a density-dependent quark mass model and dissipationof the r mode in strange stars

Zheng Xiaoping, Liu Xuewen, Kang Miao, and Yang ShuhuaDepartment of Physics, Huazhong Normal University, Wuhan 430079, People’s Republic of China

(Received 15 December 2003; published 19 July 2004)

We study the bulk viscosity of the strange quark matter in the density-dependent quark mass model(DDQM)under the background of self-consistent thermodynamics. The correct formula of the viscosity is derived. Wealso find that the viscosity in the DDQM is larger by two to three orders of magnitude than that in MIT bagmodel. We calculate the damping time scale due to the coupling of the viscosity andr mode. The numericalresults show that the time scale cannot be shorter than 10−1 s.

DOI: 10.1103/PhysRevC.70.015803 PACS number(s): 97.60.Jd, 12.38.Mh, 97.60.Gb

I. INTRODUCTION

Witten conjectured[1] that the strange quark matter(SQM) composed of comparable numbers ofu, d, and squarks might be an absolutely stable or metastable phase ofnuclear matter. Since then many theoretical and observa-tional efforts have been made on its properties and potentialastrophysical significance[2–7]. Because of the difficulty ofquantum chromodynamics(the lattice approach) in the non-perturbation domain, one used to adopt phenomenologicalmodels. One of the famous models is the MIT bag modelwith which many authors study the equation of state(EOS)of SQM and the mass-radius relation of the assumed strangestars [3,4,7]. Another alternative model is the density-dependent quark mass model(DDQM). Using this modelChakrabatyet al. re-discussed the same issues[8–11]. How-ever, Benvenuto and Lugones pointed out that the pioneerinvestigations utilized wrong thermodynamic treatment. Fol-lowing Benvenuto and Lugones’s philosophy, Penget al.[13,14] indeed tried a thermodynamic self-consistent de-scription in the DDQM and studied the structural propertiesof strange stars.

Accordingly, the important effect of the dynamics ofSQM on strange stars’ maximum rotation had also been rec-ognized when the EOS of SQM was gradually uncovered.Wang and Lu[1] first found that the nonleptonic reactionu+s↔u+d dominates the bulk viscosity of SQM[15]. Theiridea was developed soon and fruitful results had been ob-tained based on the MIT bag model[16–18]. Recently weshowed that the medium effect increases the bulk viscosityof SQM considerably[20]. In fact, previous work[21] hadalso studied these issues in the DDQM without consideringthe self-consistent thermodynamics, which immediately im-pelled us to re-investigate the influence of medium effect onthe viscosity of SQM in the DDQM. Here we will concen-trate the problem in the background of the correct thermody-namics and consider the effect of the variable quark mass onthe bulk viscosity of SQM, as seen later in this paper. Thiseffect has twofold contributions. First, the modification ofEOS influences the viscosity. Second, the density-dependentquark mass contributes directly to the dynamic quantities.

We organize this paper as follows. In Sec. II, the self-consistent thermodynamics in the DDQM are recalled. In

Sec. III, the bulk viscosity of SQM arising from the nonlep-tonic weak interaction is derived in the DDQM. In Sec. IV,the numerical results are given and an application to the re-cent astrophysics is discussed. And finally we will present abrief summary in Sec. V.

II. THERMODYNAMICS WITH DENSITY-DEPENDENTPARTICLE MASSES

The self-consistent thermodynamics in DDQM have beendiscussed in many papers[12–14]. Penget al. [14] gave thecorrect treatments. They re-derived the pressure and energyexpressions from the general ensemble theory and found anextra term in the expression of pressure but not in the energyformula. This modified pressure is expressed as[14]

P = − V + nb] V

] nb, s1d

where the termnbs]V /]nbd accounts for the density depen-dence ofV. In the framework of self-consistent thermody-namics, the EOS of SQM and the structural properties ofstrange stars have been studied. At zero temperature, thethermodynamic potential density is

V = − oi

1

8p2Fmismi2 − mi

2d1/2s2mi2 − 5mi

2d

+ 3mi4 ln

mi + Îmi2 − mi

2

miG . s2d

Accordingly, the energy and pressure can be obtained fromEqs.(1) and (2), and the number density is

ni =1

psmi

2 − mi2d3/2. s3d

It has been found that the density-dependent quark massshould take the form[13]

mi = mi0 +D

nb1/3, i = u,d,s, s4d

whereD is a parameter to be determined by stability argu-ments. For given valuenb,

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nb =nu + nd + ns

3, s5d

we can easily determine the quark chemical potentialsmi andevaluate the thermodynamic quantities of the system by thebeta equilibrium and the charge neutrality conditions,

md = ms ; m, ms = mu + me, s6d

2

3nu −

1

3nd −

1

3ns − ne = 0. s7d

In the following sections, we will consider the dynamicsof SQM (bulk viscosity) near the equilibrium state, which isgoverned by the EOS calculated with the self-consistent ther-modynamics.

III. DERIVATION OF THE BULK VISCOSITY

In this section, we derive the bulk viscosity of SQM withthe self-consistent thermodynamics taking the variable quarkmass into consideration. Since the quark mass is the functionof the baryon number density represented in Ref.[14], weassume that the baryon number density, instead of the vol-ume as had been adopted in previous works[15–17,21], os-cillates periodically,

nb = nb0 + DnbsinS2pt

tD = nb0 + dnb, s8d

wherenb0 is the equilibrium baryon number density,Dnb isthe amplitude of the perturbation, andt is the period. Thepresence of viscosity results in the energy dissipation of pul-sations in bulk matter. On one hand, using the standard defi-nition of the bulk viscosity of matter, the energy dissipationrate per unit volume average over the oscillation periodt canbe written as

Ekin = −z

tE

0

t

dtsdivvd2, s9d

wherev is velocity of the fluid. On the other hand, the hy-drodynamic matter deviates from the equilibrium accompa-nied by the time variations of the local pressurePstd. Thedissipation of the energy of hydrodynamic motion due toirreversiblity of the periodic compression-decompressionprocess can be expressed as

Ediss= −1

tVE

0

t

dtPstdV. s10d

According to the continuity equationnb+nb0divv and the

specific volumeV=1/nb, bearing in mind thatEkin=−Ediss,from Eqs.(9) and (10) we have the bulk viscosity,

z =1

pS nb0

DnbDE

0

t

Pstdcos2pt

t. s11d

It is proven that the bulk viscosity of SQM is mainly pro-duced by the nonleptonic reaction as follows:

u + d ↔ s+ u. s12d

Thereby at quasiequilibrium the pressure can be regarded asa function of the number density. From the baryon numberdensity Eq.(5), we know that there are only three indepen-dent variables ofnb,nu,nd,ns. Therefore it is suitable to as-sume thatP=Psnb,nd,nsd. The change in pressure arises dueto both changes in baryon number density and concentrationof various species of particles. Thus we can write

P = P0 + S ] P

] nbDdnb + S ] P

] ndDdnd + S ] P

] nsDdns, s13d

where all derivatives are taken at equilibrium and thechanges ind- ands-quark number density are composed oftwo parts,

dni = dni − nidV

V, s14d

wheredni denotes the number density at a given volume fors andd quark. From the reaction(12), we have the followingrelation:

dnd = − dns =E0

t dnd

dtdt. s15d

The net rate of the reactionu+s↔u+d is given by[21]

dnd

dt=

3

p3GF2sin2u cos2uJT2S1 +

dm2

4p2T2Ddm, s16d

whereJ is given in Ref.[22] and dm=ms−md. Due to thebaryon number conservation, we also have

dnb

nb= −

dV

V. s17d

Finally, we can cast Eq.(13) into

P = P0 + Fnb] P

] nb+ nd

] P

] nd+ ns

] P

] nsGdnb

nb

+ F ] P

] nd−

] P

] nsGE

0

t dnd

dtdt. s18d

When calculating the bulk viscosity from Eq.(11) only thelast term in Eq.(18) contributes. In the DDQM, we regardthe thermodynamic potential and the chemical potential asthe functional form below:

V = Vsm j„ni,nbsnid…,nbsnidd, mi = mi„ni,nbsnjd…, s19d

wherei , j represents different flavors of quarks, respectively.Using the thermodynamic relation

ni = U−] V

] miU

nb

s20d

and taking the derivatives ofP andmi in Eqs.(1) and(3), wehave

] P

] nj= o

i

sni − 3nbd] mi

] nj, s21d

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] mi

] ni=

mi2 − mi

2

3mini,

] mi

] nb=

mi

mi

] mi

] nb, s22d

one gets the formula for the contributing terms in Eq.(18),

dP = Fms2 − md

2

3m+ nbSm2 − ms

2

3mns−

m2 − md2

3mndDGE

0

t dnd

dt,

s23d

where the second term comes directly from the modifiedterm of thermodynamics in Eq.(1) i.e., nbs]V /]nbd.

Analogous to the pressure Eq.(13), the chemical potentialdifference can be expanded as

dm = S ] dm

] nbDdnb + S ] dm

] ndDdnd + S ] dm

] nsDdns. s24d

This can be evaluated using the relations in Eq.(22) and wefind

dm = Sms2 − md

2

3m+ nb

ms − md

m

] md

] nbDdnb

nb− Sm2 − md

2

3mnd

+m2 − ms

2

3mnsDE

0

t dnd

dt, s25d

where the termsms2−md

2d /3m was also obtained in previousworks, from both the MIT bag model[1,17,20] and theDDQM [21]. The extra term includes]md/]nb roots in thatquark mass depending on the baryon number density. Theextra terms appearing in Eqs.(23) and (25) reflect the con-tribution of quark interaction to the transport process, and thechange of the bulk viscosity also arises relative to the inter-action between quarks. Now substituting Eq.(23) into Eq.(1), we finally get the viscosity,

z =1

p

nb0

DnbFms

2 − md2

3m+ nbSm2 − ms

2

3mns

−m2 − md

2

3mndDGE

0

t

dtE0

t dnd

dt. s26d

For the sake of convenience, Eq.(25) can be rewritten as

] dm

] t= Sms

2 − md2

3m+ nb

ms − md

m

] md

] nbDS−

1

nb

dnb

nbD − Sm2 − md

2

3mnd

+m2 − ms

2

3mnsDE

0

t dnd

dt. s27d

IV. RESULTS AND DISCUSSION

In order to give the ratednd/dt, one must solve Eqs.(16)and (27) numerically. In Fig. 1, we have plotted viscosityversus relative perturbationDnb/nb with different param-eters, respectively.(i) D1/2=156 MeV, ms0=80 MeV at nb=0.4 fm−3, and t=10−3 (solid curves), and (ii ) D1/2

=156 MeV, ms0=140 MeV at nb=1.36 fm−3, and t=10−3

(dashed curves).In Fig. 2, we compared the results from our formula(solid

curves) with those in Ref.[21] (dashed curves) for the same

parametersnb=0.4 fm−3, D1/2=156.0 Mev, ms0=80 Mev.Evidently our viscosity is larger.

In Fig. 3, we show the results in the MIT bag model fromZhenget al. [19] for ms0=80 MeV,t=10−3. The solid curvesinclude screening mass effects and the dashed curves corre-spond to the case of ignoring the coupling among quarks inthe MIT bag model. Comparing Fig. 2 with Fig. 3, we noticethat our results are higher by two to three orders of magni-tude than those in the bag model. We also find that the solidcurves in Fig. 2 are also higher than the dashed curves. In

FIG. 1. (Color online) Bulk viscosity as a function of relativebaryon number density perturbation amplitudeDnb/nb. Dashedcurves are fornb=0.4 fm−3, D1/2=156 MeV, ms0=80 MeV, andsolid curves fornb=1.36 fm−3, D1/2=156 MeV,ms0=140 MeV. Inboth cases, we take t=10−3 and temperatures10−5,10−4,10−3,100,10−2,10−1 MeV from bottom to top,respectively.

FIG. 2. (Color online) A comparison of our bulk viscosity(solidcurves) with that in Ref. [21] (dashed curves) for parametersnb

=0.4 fm−3,D1/2=156.0 Mev,ms0=80 Mev, andt=10−3. The corre-sponding temperatures of the curves are as in Fig. 1

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other words, a medium effect increases the bulk viscosity ofSQM.

Recently it has been realized that rotating relativistic starsare generically unstable against ther (otational)-mode insta-bility [23]. The viscous dissipation of dense matter has rel-evancy to the onset ofr-mode instability, gravitational radia-tion, and the evolution of pulsars[24–27]. A main motivationto study viscosity is to calculate the damping time for com-pact star vibrations. For a star of constant density, ther-modeenergy is estimated as in Ref.[23],

E =a2p

2ms2m+ 3dsm+ 1d3s2m+ 1d ! rR5V2. s28d

The mode dissipation energy can be computed in terms of

dE

dt=E zdsds*d3x. s29d

Therefore, the damping time scale dues to bulk viscous dis-sipation is thus

tD = −2E

dE

dt

= 1.313 1015r2V−4z−1. s30d

Figures 4–6 show the damping time as a function of tem-perature for some typical spinning strange stars with periodsT=1.5 ms, 15 ms, and the Keplian limit. In comparison with

FIG. 4. The time scales as functions of temperature of a strangestar forM =1.4M( ,R=10 km,T=1.5 ms and relative baryon num-ber density perturbation amplitude 10−2, 10−1, 100, 10−3, 10−4, 10−5,10−6, 10−7 from bottom to top, respectively.

FIG. 5. As in Fig. 4, butT=15 ms.

FIG. 6. As in Fig. 4, but assuming a Keplian rotation star.

FIG. 3. Bulk viscosity in the MIT bag forms=80 MeV, md

=470 MeV,t=10−3 s. Solid curves are for interacting quark matter,while dashed curves for ideal quark gas and the corresponding tem-peratures of the curves are the same as in Fig. 1.

ZHENG XIAOPING et al. PHYSICAL REVIEW C 70, 015803(2004)

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harmonic oscillations, the damping time scale of ther modewith periodt= 2

3T=10−3 s (Fig. 4) is only short to about 1 s,but that of harmonic oscillation can be short to 10−3 s (seeFig. 4 in Ref.[17]). This implies that the simple radial pul-sation can be damped more efficiently by the bulk viscositythan ther mode. For a Keplian rotation star, the minimumtime scale is about 10−1 s.

V. CONCLUSION

Considering the self-consistent thermodynamics in thedensity-dependent quark mass model, we improve the calcu-lation in Ref.[21] and re-derive the bulk viscosity of SQM.

We show that the viscosity in the DDQM is larger than thatin the MIT bag model. This is because of an inclusion ofmedium effect due to interaction between quarks. The con-sequence is that ther-mode of a strange star will be dampedmore efficiently in the DDQM, which implies that ther-mode instability window of strange stars should be modi-fied.

ACKNOWLEDGMENTS

We would like to thank D. F. Hou for the English writingand the support by the National Natural Science Foundationof China under Grant Nos. 90303007 and 10373007.

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