NUCLEAR, PHYSICS B

OCEEGI GSSUPPLEMENTS

ROTATIONAL PROPERTIES OF STRANGE ST!!RS

John MILLER

Osservatorio Astronomico di Trieste, ItalySISSA, Trieste, ItalyAstrophysics Group, Department of Physics, University of Oxford, England

Monica COLPI

Dipartimento di Fisica, University di Milano, Half

Results are presented from studies of the rotational properties of strange stars concerning, in particular,(i) the changes of mass and mean radius resuli,iag iroin rotation ; (ii) the minimum rotation period consistentwith stability to non-axisymmetric perturbations.

1 . INTRODUCTIONIfstrange stars are to be considered as serious mod-

els for pulsars, then it is important to know what theirrotational properties would be. We present here somemain points from a recent study 1 which has concen-trated on two questions : (i) What are the overall ro-tational properties of strange star models and to whatextent do they differ from those of ordinary neutronstars? (ii) If a strange star were to rotate fast enoughfor non-axisymmetric instability modes to become ex-cited, does the onset of these differ from the situationfor neutron stars?

2 . FORMULAE USED FOR STRANGE MATTERThe equation of state used for these calculations

was it simple lip a, model Y:th zero mass for the itrangequark . This gives

p = 13 (p - 48)

for p > 48,

where p and p are the mass-energ.,,- density and pressurerespectively and Bs is the hag constant, for which wetake the value 1014g cm' (closely equivalent to (145McV)4 ) .

For the study of the onset of non-axisymmetric in-stabilities, A is necessary to consider the ways in whichoscillations can be damped . Two of the relevant mechanisms are shear and bulk viscosity. An approximate

Nuclear Physics B (Proc . Suppl .) 24B (?991) 166-169North-Holland

analytic expression for the coefficient of shear viscosityin strange matter is 2

0920-5632/91/$03.50 Uc 1991 - ElseAer Science Publishers II-V.

All rights reserved .

5/3

71 = 7.0 x 1015 0.1 nBOg cm

1 s-1

(2)

where nc is the strong interaction coupling constant(Ior which we take ,,he value 0.15), T1 � is the tem-perature in units of 101° K, ne is the baryon numberdensity and n3o is the normal, nuclear matter density{= 1 .7 x 1038cm.-3) . In our calculations we have usedexpression (2) -__ultiplied by a factor 2.5 in order to im-prove agreement with more accnrate numerical results(see Ref. 2) .

The coefficient of bulk viscosity in strange matterhas been calculated in Ref. 3 and a convenient analyticfit to the numerical results presented there is given by

=1 .2 x 1027(ngnsu

-- 0.18 ) Ti-1

x[(

ŒO

10~ad s-1 l2 +4.9 T~U

Bo

)2]

g cm-1 s-1

(3)

where aro is the oscillation frequency as measured inthe local mean rest frame of an element of the stellaifluid. (This expression follows from the one given in

Ref. 4 and assumes a mass for the strange quark of100 MeV.) The S

. Ior standard neutron star matter hasa completely different temperature dependence (oc TO)and is very much smaller .

3 . OVERALL. PROPERTIES OF PLOTATINCSTRANGE STARSIn considering our first question, Rm note that for all

pulsars so far observed, the rotation may be regardedas slow in the sense that rotational perturbations awayfrom the structure of a non-rotating comparison modelare always quite small . This allows rotational proper-ties to be calculated by means ofa perturbation method(see Ref. 5) : the relativistic structure equations are de-rived by expanding the fluid and field equat?e r� aboutnon-rotating solutions, in powers of the angular veloc-ity fl, and retaining only first and second order terms .The quantities describing the rotational perturbationsare all proportional to either St or 11' and, for any fixedvalue of the central density p,, equilibrium configura-tions for different values of fl can be obtained by scalingthe results from a single calculation.

In the present work, a sequence of slowly-rotatingstrange star models has been generated from a sequenceof non-rotating ones with central densities p, rangingfrom 4B (for which the mass M,, = 0) up to p. "'

1 .92 x 101 s g cm-3 (for which Mo = M,,'z = 2.03 My ) .

Non-rotating models with p, > pmaz would be unstableto radial perturbations .

For presenting reSWaZs, we use as a scale the ICeple-rian angular velocity Sllcl for a test particle on a cir-cular orbit just outside the surface of the non-rotatingcomparison model, (Rh) (GM�/R3)112 where Ro isthe radius of the comparison model) . Figure 1 showsthe mass against radius relation for zero rotation (solidcurve) and for fl set equal to ft (Ko ) (dashed curve - inthis case R refers to the mean radius) . The straightlines between the two curves connect models havingequal central densities . (For smaller values of fl, theperturbations are smaller by a factor (WfI i ; 2 .) It isinteresting to compare these results with those shownire Figure 1 of Ref. 6 which are for standard neutron starmodels . Despite the difference in shape of the carves,in both cases the orientation of the linking straight

J. Milier, M. :7^.'yy l'riutational properties ofstrange stars

2.5

2

.5

5 .5 5 .0log R

167

FIGURE 1Mass M .versus radius R (in units of cm) for rotatingand non-rotating strange star models.

lines changes smoothly from being horizontal to nearlyvertical as one moves upwards in mass through therange of stable models . The fractional mass increase forthe strange star models is first an increasing functionof Al, but then becomes almost constant beyond MQ -1.4 ME) ; the size of the increase is smaller in generalthan for standard neutron stars (cf Ref. 7) . Full detailsof our results for other quantities are given in Ref. 1 .

4 . THE T/JWJ RATIOThe ratio t = T/)W J (where T here denotes the ro-

tational kinetic energy and W is the gravitational po-tential energy) is a quantity which measures "sirCngthof rotation" . The uniformly rotating neutron star inod-els with standard equations of state studied in Ref. 8(fully relativistic calculations with consistent treatmentof rapid rotation) all have values of t at the sheddinglimit (,,Z = QK ) less than 0.14, the value near whichthe m = 2 bar mode is thought likely to go unstable.For models with M -1.4 Mr , their values of t(StK) allcluster around 0.1 with the maximum being 0.120.

Strange stars with masses of - 14Me would haveradii very similar to those of ordinvry neutron starswith the same mass but there =s n striking difference be-tween the density profiles . The average adiabatic index

16 8

i' can be much larger for strange star models (= 5.68for 1.4 Mo in our calculations) than for the standardneutron star models (- 2 - 3) and their density profilesare rather flat right out to the surface, at which thereis then a discontinuity. As discussed in Refs . 1 and 4,this difference in structure leads to much higher valuesof for strange stars and opens the possibility foroperation of the m = 2 bar mode instability in this cast .

5 . STABILITY OF STRANGE STARSIn considering our second question, we have ex-

amined the stability of rotating strange stars to non-*modes (with azimuiha: dependence e ~ m)

and ûZ -ve derived upper limits on the angular velocitybased on this . We discuss here instabilities driven bygravitational radiation reaction (GRR) and moderatedby viscosity. The angular velocities fl .. at which eachmode becomes unstable have been calculated -acing thesame equation of state as in the previous sections andfollowing the approach described in kef. 10 with modi-fications contained in Refs . 11 and 12 .

The interplay between GRR and viscosity deter-mines the value orm for which fl. is °a minimum (cor-responding to the period P being a maximum) andhence which mode would be the first to go unstablewith increasing fl . Since al and t; depend on tempera-ture so also does ft,n .

The critical angular velocities arc: b:vcza :zy

fl ?n n.-(0)~r

m

GRR-

j

1/( m+1)mfrn)+ym(ilm) n

Trn

11 jr2rn-r1)x 1+~

(

Tm ~ 1 - f

ig 4 ]

j

where a*n(0), r~;GRR, T;n and 7-,ç� are the frequency andthe dissipation timescales (due to GRR, shear and bulkviscosity respectively) for the m th mode of a corre-sponding non-rotating model and the corrections forrotation are contained in the functions y,,#1)and E,n (fl) .

For calculating c,.(0), rgRR, -r,.1 and rm we haveconstructed v fully general relativistic computer codeusing broadly the same strategy as that of Ref. 13and taking the formulae for viscous dissipation times

J. Millor, M . Cclpi/Rotational properties ofstrange stars

given in Ref. 4. Following previous practice, the ro-tational corrections have been ir±ciuded by using New-tonian Maclaurin spheroid expressions for a,n and ?',n ;however, dissipatio:. duz In bulk- """ !:^5sity vanishes inthe limit of incompressibility and so there is no Maclau-rin spheroid expression for e�a . For this, we have usedresults from Ref. 12, where values of E,n have been givenfor Newtonian polytropic models . We have used the re-su'ts for the stiffest equation of st :R,,,te considered there(which should lead to a slight overestimate of the effectsof bulk viscosity it our calculations) .

Figure 2 shows the critical periods, as a function oftemperature T, for strange star models of 1.4 MC (up-per curves) and 2.03 NO (lower curves) . For each mass,any point above the corresponding curve lies in the al-lowed range ofrotation speeds compatible with stabilitywhile the region beneath the curves is forbidden . Thesolid lines represent results of the full calculation whilethe dashed ones. are for calculations in the absence ofbulk viscosity. The discontinuities in gradient corre-spond to changes in the critical mode.

At T = 1ä7K, shear viscosity is completely dominantcover bulk viscosity and the critical periods are those forthe m = 2 mode. Between 10sK and 10% bulk viscos-ity first starts to become important and then rapidly

a

FIGURE 2Critical periods P (in units of milliseconds) for strangestar models as; a function of temperature T (in units ofK).

dominates; this change produces a maximum in 'the Pversus T relation which does not appear irl presencec:.ly of 5iiear vl5cosltty. The smallest calculated criticalperiods for the two masses occur at T = 107K andare P = 0.95ms (for 1.4 MO) and Y = 0.79ms (for2.03 M(D) both of which correspond to rotation speedswell below the respective shedding limite .

6 . CONCLUSIONlu this article, we have made a brief presentation of

results from our calculations of the rotational propertiesof strange stars and we conclude by summarizing ov -answers to the two questions raised in the Introduction(but with the proviso that these conclusiorà5 La-e depen-dent, to some extent, on our choice of parameters) .

As far as rotational deformations are concerned,there are a number of detained differences between thestrange star models and standard neutron stars but onthe whole the similarities are more striking than thedifferences .

As far as stability to non-axisymmetric modes is con-cerned, we can distinguish two different regimes of in-terest . (1) The case of young (hot) strange stars bornrapidly r,)ta .ting for which the relevant temperatures areinitially - 10 1°K : here, Lulk viscosity moderates (butdoes not completely suppress) the GRR instability andthe m = 3 mode is preferred . The effect of bulk vis-cosity for strange stars does not appear to be as dra-matic as had previously been predicted' and this can betraced to the very low compressibility of strange mat-ter, particularly near to the surface where the dissipa-tion integral peaks . (2) The case of old (cooler) strangestars spun up by accretion, for which the relevant tem-peratures are probably - 107K or slightly more (seeRefs . 14 and 15) : here, shear viscosity moderates theGRR instability and the preferred mode will be eitherm=2 or m=3.

The situation regarding this last point is worth em-phasizing . For old pulsars spun up by accretion, it ispossible that the limiting rotation speed may be set bythe rn = 2 bar mode if the pulsar is a strange star simi-

J. Miller, M . Colpi/Rotational properties of strange stars

tar to the models considered herewhereas it seems prob-able that no standard neutron star equation of stateworld allow instability to set in by this mode . If futureobservations of gravitational waves were to reveal thesignature of a bar mode in such a pulsar, this wouldbe gaite strong evidence in favour of it being a strangeséarr .

ACKNOWLEDGEMENTSThis research is being carried out with financial sup-

port from the Italian Ministero dell'Università e deltaRico,rca Scientifica e Tecnologica.

REFERENCES

169

1 . M. Colpi and J.C. Miller, submitted to Ap. J.(1991) .

2 . P. Haensel and A.J . Jerzak, Acta Physica Polonica,B20 (1989) 141 .

3 . R.F . Sawyer, Phys . Lett. B, 233 (1989) 412.

4 . C . Cntler, L. Lindblom and R.J . Splinter, Ap . J .,363 (1990) 603 .

5 . J.B . Hartle, Ap . J ., 150 (1967) 1005 .

6 . J.B . Hartle and K.S . Thorne, Ap . J.,153 (1968) 807 .

7 . A . Roy and B . Dotta, Ap . J ., 282 (1984) 542.

8 . J.L . Fried-man, J.R. Ipser and L . Parker, Ap . J ., 304(l986) 115.

9 . J.M . Lattimer, M. Prakash, D. Masak and A. Yahil,An. 3 ., 355 (1990) 241 .

10 . L . Lindblom; Ap. J ., 303 (1986) 146 .

11 . C . Cutler an -l L . Lindblom, Ap. J ., 314 (1987) 234 .

12 . J.R. Ipser and L. Lindblom, Ap . J ., 373 (1991) 213 .

13 . L . Lindblom and S.L . Detweiler, Ap. J . Suppl ., 53(l983) 73.

14 . J . Miralda-Escudé, P. Hoensel and B. Paczynski,Ap . J ., 362 (1990) 572 .

15 . P.M . Pizzoaaero, Phys . Rev . Lett., 66 (1991 2425 .