Nuclear Physics B (Proc . Suppl .) 24B (1991) 119-124North-Holland

MAXIMUM ROTATION FREQUENCY OF STRANGE STARS

Julian L . ZDUNIK

N. Copernicus Astronomical Center, Bartycka 18, PL-00-716 Warszawa, Poland

Two constraints on the rotation rate of strange stars are examined . The first one results from the observedmass of the binary pulsar, the second from the requirement that strange matter is the ground state of matter.Maximum Keplerian frequency is calculated in the bag model of strange matter . Damping of gravitationalradiation reaction instabilities by shear and bulk viscosity is discussed .

1 . INTRODUCTIONFast rotating compact objects, neutron stars or

strange stars, have been a subject of interest of manyastrophysicists for the last two years . The event whichgave rise to the extensive examination of maximum ro-tation frequency of such stars was the announcementreporting the detection of the optical pulsations with aperiod 0.508 ms from the direction of the two years oldremnant of supernova SN 1987A in the Large Magel-lanic Cloud 1 . Although this observation had not beenconfirmed and finally, after one year, turned out to beerroneous z, the possible existence of a half millisec-ond pulsar stimulated detailed investigations of the con-straints limiting the rotation frequency of neutron andquark stars 3`_as .

The problem was to find Euch an equation of statewhich could simultaneously fulfill two requirements :1 . The maximum mass ofthe star built of such a matter

should be greater than the very precisely determinedmass of the binary pulsar PSR 1913 + 16 13

Mmat > M1913+16 = 1.442 :h 0.003 Mo

(1)

2. The maximum rotation frequency of a star shouldbe greater than 0.508 ms (now, after the retractic, .of the claim for the existence of the half millisecondpulsar, the fastest pulsar has the period 1.56 ms,which is not very restrictive) .

In general the first condition means that matter shouldbe sufficiently stiff, soft equations of state giving smallermaximum mass. Or the contrary, it is easier to reach

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high rotation frequency for a star built of the soft mat-ter . For soft equation of state the stars near maximummass have high central and mean density, so that theKeplerian rotation rate, proportional to %rp is high.

These two constraints applied to halfmillisecond pe-riod would rule out nearly all "realistic" equations ofstate of neutron matter. In this situation the naturalcandidate for a star rotating so fast was a strange star,built of strange quark matter being, as suggested byWitten 14, the ground state of the matter. This kindof the matter, composed of roughly equal numbers ofu, d, s quarks, is self-bound at high density, ofthe orderof nuclear matter density. Thus for strange star, hav-ing a bare dense quark surface, it is easier to reach highmean density, and as a consequence high rotation fre-quency. This possibility have been examined by manyauthors 7-12 .

2 . INSTABILITIESThe fundamental problem in the calculation ofmax-

imum rotation frequency of a star is the stability ofsuchan object . One has to answer the question which kindof instability imposes the upper bound on rotation . Ofcourse an absolute upper limit is given by the Keple-r!anfrciuaacy, defïaed ab ü v-_'1--iby of a test particle on acircular orbit at the equator . Stars rotating at velocAwshigher than QK would shed mass from the equator .

The instability which plays here an important roleand actually sets the upper limit on rotation of neu-tron stars is the gravitational instability to nonaxisym-metric perturbations driven by gravitational radiation

120

reaction (CRR) 1s-17. This instability sets in whena nona;dsymmetric mode moving baczcward relative tothe star is dragged by the star rotation forward relativeto an inertial frame . In such situation gravitational ra-diation removes positive angular momentum from themode and as a consequence drives a perturbation . Inreality this instability is moderated by viscosity and forhigher modes (larger m in angular dependence e'm#)dissipation damps out GRR perturbations . For neutronstars viscosity seems to sta'oil :~c 113odes with m > 5 andthe m = 4 or 5 mode probably sets the limit on starrotation 17-2°. The limiting frequency S2GRR dependson viscosity, which is a sensitive function of a tempera-ture, and if one takes into account only shear viscosity,SZGRR can be even 20 % smaller than RK for tempera-tures of the order 1 MeV 3. However, inclusion of theeffects of damping due to bulk viscosity can also stabi-lize hotter neutron stars and difference between S?CRR

and S2K could be smaller 21 .

Frieman and Olinto 7 have estimated the dampingof the GRR-driven instabilities due to shear viscosity ofstrange matter 22

rl, .. . 1 .3 x 10

r a` l

3/217\0.13/

(P15)5/3 TMév g cm-1 s-1(2)

which is, at the same temperature, about one order ofmagnitude larger than that of nuclear matter 23

11n ,. 1 .3 x 1016 (Pl5)9/4 TMév g cm-1 s-1

(3)

They concluded that at the temperature of the order108 K m = 3 mode sets the limit on the stellar rotationand the resulting angular frequency SZCRR is very closeto the Kepler velocity RK.

Furthermore the bulk viscosity of strange matter isexpected to be much larger than that of neutron mat-ter. This fact was first discussed by Wang and Lu 24 .

They pointed out the strangeness changing reactionsu -I- s *-. d -F- u as a very effective source of dampingof the strange star pulsations. The bulk viscosity ofstrange matter due to this process as a function of tem-perature, density, frequency and strange quark mass

.i .i; . Zdunik/Maximum rotation frequency ofstrange stars

war, calculated by Sawyer 25 . It can be written in theform :

(e

1027_a~1VSey

g cm-1 s-1

4w4 + b TMev

( )

where w4 = w/104 s-1 . The dimensionless coefficientsa, b depend on me and p and for m, = 200 MeV areof the order 10 . For temperatures 7y' < 1 MeV bulkviscosity of strange matter is many orders of magnitudelarger than that of ordinary nuclear matter 21 :

n

1.46 x 1022 (pl.)2Wî2

TMev

g cm-1 s-1

(5)

The application cf t;e to the estimation of the max-imum angular velocity of a star leads to the conclusionthat for high temperatures the GRR-driven instabilityis very effectively damped by bulk viscosity and the re-sulting maximum rotation frequency of the strange staris close to the Keplerian velocity lK 25 . Recently, Cut-ler, Lindblom and Splinter 26 have calculate3 the damp-ing times due to bulk viscosity for nonrotating stellarmodels, including strange stars . However, in order todetermine which mode sets the limit on star rotationand what is the precise value of the limiting frequencyfor various temperatures one has to make detailed cal-culations similar to that of Lindblom 18 .

3 . MAXIMUM KEPLERIAN FREQUENCYFOR GIVEN EQUATION OF STATEKeplerian frequency is the most con.:iervative limit

to the maximum rotation frequency of a star . Thusfor any star, well-defined e.g. by total baryon num-ber, one can determine its maximum rotation rate asa Keplerian velocity. Calculating for a given equationof state a sequence of stars labelled by an increasingcentral density pc rotating at StK(pe ) one gets finally aconfiguration with maximum gravitational and baryonmasses . This is the last configuration (with the largestpc) rotating uniformly at the maximum angular veloc-ity and stable against, gravitational collapse . Accord-ing to the turning-point stability theorem 27 this starmarks the onset of instability. Although configurationswith higher central densities can sustain higher rotation

rates they are gravitationally unstable and would col-lapse . Thus for given model of matter we can determineas absolute upper limit to the rotation of star built ofsuch matter as a maximum value of the Keplerian fre-quency 11K,ynaz which is reached for the maximum massstar.

There are two ways to determine maximum Kep-lerian frequency for a given equation of state. Thefirst one is to solve the set of general relativity equa-tions govering the rotation of the star for a sequence ofmodels rotating at 3ZK and find the configuration withmaximum mass. Such calculations have been made byFriedman, Ipser and Parker 4,2s for rather broad rangeof equations of state of neutron matter, mainly fromthe Arnett-Bowers collection `s . Recently, similar cal-culations were performed by Lattimer at al. 11 for sev-eral equations of state including the simple bag modelof strange quark matter . The second way to estimàie11K,maz is to use an approximate "empirical" formulawhich enables us to determine Keplerian frequency asa function of the properties of the nonrotating configu-ration with maximum mass for given EOS s :

7.7 x 103

M

1/2(R(Mm.,.)

_s/2

0

10 km(6)

The above formula reproduces the results of numeri-cal calculations of the uniform rotation of neutron starswith an error less than 4 % and for the strange stargives almost exactly the same result as fully relativisticcalculations 11 . Thus the estimation of StK,,nas fromthis formula is very easy and seems to be reasonable,especially in the case o£ the strange star .

4. MAXIMUM KEPLERIAN FREQUENCYIN THE BAG MODELUp to now, the models of strange matter used in

determining the structure of stars have been calculatedmainly within the framework of the phenomenologicalMIT bag model 14,30-32 in which one mimics the con-finement introducing a constant positive energy of vac-uum, bag constant B. The properties of strange quarkmatter can be studied usingperturbative QCD, but one

J.L. Zdunik/Maximum rotation frequency ofstrange stars

should remember that the use of the perturbative the-ory in the case of strong coupling is rather risky. Thusthis model should be treated as a simple approximationreproducing main properties of strange quark matter.

Wi: shall present results for the model of Faxhi andJaffe 30 characterized by main parameters: B the bagconstant, ma the mass of the strange quark, ac the den-sity independent QCD coupling constant. Using stan-dard methods one can calculate thermodynamical po-tentials and energy per baryon number as a functionof B, m, and ac . The specific form of the resultingequation of state enables us, after introduction of the di-mensionless parameter in, = m,g/(h3c3B)i/4 , to writevery simple scaling relations for global parameters ofthe strange stars b"._iIt of such matter :

where M,nnz and Rmaz are the maximum mass and thecorresponding stellar radius, respectively. According tothe approximate formula (6) the above relations imply

1

~K,maz(B~ ms~ ac) _ flK,maz(S071n

B,. ac) (B

(9)Using these formulae one can obtain the highest

value of the rotation rate of the strange star consistentwith the constraint M,naz > 1.442 Mp:

~K,masl(ma~ ac) _~K,mas ~Mmaz

iL .442 Mo

9.4 x 104C

G Mm-z) ss-~10)

Rm.. c2

All the above. relations refer, however, to a star withbare strange matter surface in accordance with sugges-tion of Witten 14 that strange matter may be the trueground state of matter, energetically prefered over the

Mmaz~B~B

ma, ac) _ " maz(B0,ms, ac)(B

i

(7)

R,nas(B, ma, ac) = Rmas(Boi ma, ac)(LB

) ($)

199

normal, nucleonic one, even at zero pressure . Thatmeans that energy per baryon at zero pressui~e, Eo ,should be less than the nucleon mass :

Eo(B, in�aj < 939 MeV

(11)

Taking into account the dexlen"iracc : ;: 'E'u ora B in ourmodel :

the microscopic stability condition (11) determines forgiven rim, and ac the largest possible value of the bagconstant, B,nax at which strange quark matter is theground state of matter. This value BR.,, yields themaximum rotation frequency SZK,max2 as a function ofm, and ac .

flK,"naz2(ms, ac) = SIK.max r( .939 MeV

2

(13)

Strange star built of the matter with B > Bmax

would be metastable. In such a situation stable config-uration is a strange star with an outer envelope builtof nucleoli matter . In general these stars have largerradii than strange stars with bare surface, so that SZK

;s smaller 11 .

The maximum rotation frequencies S1K,max as a func-tion of th, and a,, are presented in Fig . 1 .

HK,maxl depends very weakly on m. and acing the maximum value of 1.32 x 104 s-1 in a simplestbag -;nodel with free, massless quarks. The changes inQK,maxi due to inclusion of strange quark mass andQCD coupling do not exceed 6 %. The largest valueof flK,max2e 1.23 x 104 s-1 , is also reached for m, =0 and ar = 0 . The microscopic constraint, result-ing from the stability of strange quark matter at zeropressure, is more stringent than macroscopic conditionMmaz > M1913+16 = 1.442 Me and is crucial for esti-mating maximum rotation frequency of strange stars .Glendenning argued in favor of disregarding the for-mer constraint, Eq. (11) A

. In view of the fact thatto answer the. q,>estinn which state of matter, nucleonic

J.L. Zdunik/Maximum rotation frequency ofstrange stars

h .. v-

wIns

FIGURE 1Maximum Keplerian frequency of strange stars versusdimensionless strange quark-mass m,, for several valuesof the QCD coupling constant a, : The solid lines showthe maximum Keplerian rotation rate consistent withthe constraint An.- > M1913+16 . The dashed linescorrespond to the assumption that strange .-,natter "isthe ground state of matter.

or strange, is energetically prefered one has to calcu-late energy per baryon with high accuracy, Ref.g claimsthat the bag model is too crude for this purpose andfe condition (11) should be abandoned . Relaxing thiscondition one can easily increase S1K,max increasing thevalue of bag constant B according to the scaling prop-erty (13) up to the value (10) . However such stars are,stricly streaking, inconsistent with our model of strangequark matter and if they existed, would be metastablein outer regions . The energy per baryon for a strangestar, which can sustain the highest rotation rate allowedby the constraint Mmax > M1913+16 = 1.442 MO, is al-most 40 MeV larger than the nucleon mass of 939 MeV,contrary to the assumption that strange matter is en-ergetically prefered over the nucleonic one.

Although all macroscopic properties of stars, suchas the mass, the radius and the rotation rate, dependon the function P(p), since only pressure and energydensity enter (through stress-energy tensor) the generalrelativity equations of the structure of the star, a phys-ical description of matter and its constituents requirespecification of the another part of the EOS, namelythe dependence of energy density on the baryon den-sity p(n) . It is the function which defines the chemical

potential and the stability condition of matter . Thusthe cost of relaxing the microscopic stability constraintseems to be too high.

5 . CONCLUSIONSWe have presented main results of calculati:ans of

the maximum rotation frequency of strange stars.The effects of the gravitational radiation reaction

instabilities are much less important than in the case ofrapidly rotating neutron stars . This kind of instabilityis very effectively damped by viscosity. At temperaturesbelow 107 K the damping by shear viscosity dominates,at larger temperatures GRR instability is damped bybulk viscosity, many orders of magnitude larger thanthat of nuclear matter. Thus for strange stars maximumrotation rate is close to the Keplerian frequency and thedifference between these two values seems to be muchsmaller than for neutron stars .

Maximum Keplerian frequency have been calculatedin the simple, phenomenological MIT bag model o£strange matter . Strange stars built of such matter andhaving maximum nonrotating mass larger than the ob-served mass of the binary pulsar PSR 1913+16 can sus-tain higher rotation rates than neutron star models con-si,tent with the same constraint . The assumption thatstrange matter is the true ground state of matter evenat zero pressure lowers the maximum Keplerian velocityto the value which can be supported by some modelsof neutron stars . However, the actual maximum rota-tion rate of these neutron stars should be smaller thanthat of strange stars due to larger effects of the GRRinstabilities .

REFERENCES

1 . J . Kristian et al ., Nature 338 (1989) 234 .

2 . G.C . Anderson, Nature 343 (1990) 679 .

J . Kristian et al., Nature 349 (1991) 234 .

3 . J.R . Ipser and L. Lindblom, Phys . Rev. Lett . 62(1989) 2777 .

4 . J.L . Friedman, J.R . Ipser and L. Parker, Phys. Rev.Lett. 62 (1989) 3015 .

J.L. Zdunik/Maximum rotation frequency ofstrange stars 123

5 . S.L . Shapiro, S.A . Teukolsky, and I . Wasserman, Na-ture 340 (1989) 451 .

6 . P. Haensel and J.L. Zdunik, Nature 340 (1989) 617 .

7 . J.A . Frieman and A. Olinto, Nature 341 (1989) 633

S. N.K . Glendenning, Phys . Rev. Lett . 63 (1989)2629 .

9 . N.K . Glendenning, Journal of Phys . G 15 (1989)L255.

10 . J.L . Zdunik and P. Haensel, Phys . Rev. D 42 (1990)710 .

11 . J.M . Lattimer, M. Prakash, D . Masak, and A. Yahil,Astrophys . J . 355 (1990) 241 .

12 . M . Prakash, E. Baron, and M. Prakash, Phys . Lett .B 243 (1990) 175.

13 . J.H . Taylor and J.M . Weisberg, Astrophys. J . 345(1989) 434 .

14. E. Witten, Phys . Rev. D 30 (1984) 272 .

15 . S . Chandrasekhar, Astrophys . J . 161 (1970) 561 .

16 . J.L . Friedman a.-id B.F. Schutz, Astrophys. J . 222(1978) 281 .

17 . J .L . Friedman, Phys. Rev. Lett . 51 (1983) 11 .

18 . L . Lindblom, Astrophys . J . 303 (1986) 146.

19 . C . Cutler and L . Lindblom, Astrophys . J . 314 (1987)234 .

20 F . Weber, N.K . Glendenning, and M.K. Weigel, As-trophys . J . (1991), in print .

21 . R.S. Sawyer, Phys . Rev . D 39 (1989) 3804 .

22. P. Haensel and A.J . Jerzak, Acta . Phys. Pol . B 20(1989) 141 .

23 . E . Flowers and N . Itoh, Astrophys. J ., 230 (1979)847 .

24 . Q.D . Wang and T . Lu, Phys. Lett . 148 B (1984)211 .

25 . R.S . Sawyer, Phys. Lett . 233 (1989) 412 .

26 . C . Cutler, L . Lindblom and R.J . Splinter, Astro-phys . J . 363 (1990) 603 .

27 . J .L . Friedman, J.R. Ipser, and R.D . Sorkin, Astro-phys. J . 325 (1988) 722.

28. J.L. Friedman, J.R. Ipser, and L . Parker, Astrophys .J . 304 (1986) 115 .

124

J.L . Zdunik/Maximum rotat?on frequency ofstrange stars

29. W.D . Arnett and R.L . Bowers, Astrophys. J . Suppi .33 (1977) 415 .

30. E . Farhi and R.L . Jaffe, Phys . Rev. D 30 (1984)2379 .

31 . P. Haensel, J.L . Zdunik, and R. Schaeffer, Astron.Astrophys. 160 (1986) 121 .

32 . C . Alcock, E. Farhi, and A . 4linto, Astrophys . J .310 (1986) 261 .