New Astronomy 7 (2002) 107–112www.elsevier.com/ locate /newast

Timing evolution of accreting strange starsa,b , c d ,2 d ,1*D. Blaschke , I. Bombaci , H. Grigorian , G. Poghosyan

a ¨Fachbereich Physik, Universitat Rostock, D-18051 Rostock, GermanybBogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Russia

c ´Dipartimento di Fisica ‘‘E. Fermi’’, Universita di Pisa and INFN Sezione di Pisa, 56127 Pisa, ItalydDepartment of Physics, Yerevan State University, 375025 Yerevan, Armenia

Received 24 October 2001; received in revised form 10 January 2002; accepted 13 January 2002Communicated by E.P.J. van den Heuvel

Abstract

It has been suggested that the QPO phenomenon in LMXB’s could be explained when the central compact object is astrange star. In this work we investigate within a standard model for disk accretion whether the observed clustering of spinfrequencies in a narrow band is in accordance with this hypothesis. We show that frequency clustering occurs for accretingstrange stars when typical values of the parameters of magnetic field initial strength and decay time, accretion rate arechosen. In contrast to hybrid star accretion no mass clustering effect is found. 2002 Elsevier Science B.V. All rightsreserved.

PACS: 04.40.Dg; 12.38.Mh; 26.60.1c; 97.60.GbKeywords: Accretion, accretion disks; Stars: interiors; Stars: magnetic fields; Pulsars: general; X-rays: binaries

a phase transition to deconfined quark matter have1. Introductionbeen suggested in the form of peculiar changes ofobservables, such as the pulse timing (GlendenningThe recent discovery of new phenomena in theet al., 1997; Chubarian et al., 2000; Glendenning andphysics of X-ray binaries is giving a new andWeber, 2001; Poghosyan et al., 2001), and thepowerful tool to probe dense hadronic matter (Millerthermal evolution of isolated pulsars (Blaschke et al.,et al., 1998) and possibly to infer the existence of a2001a; Page et al., 2000b; Schaab et al., 1997). Evendeconfined phase of quark matter in the cores of themore intriguing than the existence of a quark core incentral accretors in these systems. Various signals ofa neutron star, is the possible existence of a newfamily of compact stars consisting completely of a

*Corresponding author. deconfined mixture of up (u), down (d), and strangeE-mail addresses: david.blaschke@physik.uni-rostock.de (D. (s) quarks, together with an appropriate number ofBlaschke), bombaci@pisa.infn.it (I. Bombaci), electrons to guarantee electrical neutrality. In thehovik@server.physdep.r.am (H. Grigorian), gevorg@su.am (G.

literature, such compact stars have been referred toPoghosyan).1 as strange quark stars or shortly strange stars (SS),Supported by DFG grant No 436 ARM 17/7 /00.2Supported by DAAD. DFG grant No. 436 ARM 17/5 /01. and their constituent matter as strange quark matter

1384-1076/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PI I : S1384-1076( 02 )00087-8

108 D. Blaschke et al. / New Astronomy 7 (2002) 107 –112

(SQM). In a series of recent papers (Dey et al., R , m , mc1998; Li et al., 1999) it has been argued that the r ¯H0 0.52 r , m $ mA ccompact objects in some X-ray sources are likelystrange star candidates. Particularly interesting candi- where m is that value of the magnetic moment of thec

dates are the compact stars in the newly discovered star for which the disc would touch the star surface.millisecond X-ray pulsar SAX J1808.4-3658, and in ´The characteristic Alfven radius for spherical accre-

24 2 21 / 7~ ~ ~the atoll source 4U 1728-34. tion with the rate M 5 mN is r 5s2m GMM dA

In the present paper, we model the spin evolution (Bhattacharya and van den Heuvel, 1991). Since weof an accreting strange star in a binary stellar system, are interested in the case of fast rotation for whichfrom the so-called ‘‘death line’’ up to the millisecond the spin-up torque due to the accreting plasma in Eq.pulsar phase. We explore the dependence of the SS (2) is partly compensated by N , eventually leadingout

spin evolution upon the mass accretion rate and upon to a saturation of the spin-up, we neglect the spin-upthose physical quantities which regulate the temporal torque in N which can be important only for slowout

evolution of the torque acting on the spinning star. rotators (Ghosh and Lamb, 1979).We try to constrain these quantities to have a From Eqs. (1), (2) one can obtain a first orderpopulation clustering of strange stars in agreement differential equation for the evolution of angularwith the spin frequency distribution for observed Z velocitysources in LMXBs with kHz QPOs. Possible signa-

K (N,V ) 2 K (N,V )dV ext inttures to distinguish strange stars from ordinary ] ]]]]]]]]5 , (3)dt I(N,V ) 1 V(≠I(N,V ) /≠V )neutron stars are also briefly discussed. N

where the internal torque term defined as

≠I(N,V )~2. Accretion model and magnetic field evolution S D]]]K (N,V ) 5 V N . (4)int ≠N V

We consider the spin evolution of a strange star Solutions of (3) are trajectories in the V 2 Nunder mass accretion from a low-mass companion plane describing the spin evolution of accretingstar as a sequence of stationary states of configura- compact stars. Since for the hybrid stars I(N,V )tions (points) in the phase diagram spanned by the exhibits characteristic functional dependencesangular velocity V and the baryon number N. The (Blaschke et al., 2001b) at the deconfinement phaseprocess is governed by the change in angular transition line N (V ) we expect observable conse-critmomentum J(N,V ) 5 I(N,V ) V ~quences in the P 2 P plane when this line is crossed.

In the case of SS we have no expectation of the massd] clustering due to absence of phase transition effects(J(N,V )) 5 K , (1)extdt

in strange matter.In our model calculations we assume that both thewhere I(N,V ) is the moment of inertia of the star and

mass accretion and the angular momentum transfer]]]2 processes are slow enough to justify the assumption~K 5 GMM r 2 N (2)œext 0 out

of quasistationary rigid rotation without convection.For a more detailed description of the method andis the external torque due to both the specific angularanalytic results we refer to Chubarian et al. (2000)momentum transferred by the accreting plasma andand the works of Hartle and Thorne (1968), as wellthe magnetic plus viscous stress given by N 5out

2 23 as Sedrakian and Chubarian (1968).km r , k 5 1/3 (Lipunov, 1992). For a star withcThe time dependence of the baryon number for theradius R and magnetic field strength B, the magnetic

3 ~constant accretion rate N is given bymoment is given by m 5 R B. The co-rotating2 1 / 3radius r 5sGM /V d is very large (r 4 r ) forc c 0 ~N(t) 5 N(t ) 1 (t 2 t )N . (5)0 0slow rotators. The inner radius of the accretion disc

For the magnetic field of the accretors we consideris

D. Blaschke et al. / New Astronomy 7 (2002) 107 –112 109

Table 1the exponential decay (Bhattacharya and van denGround state properties of SQM for the equations of state used inHeuvel, 1991) athe present work

B(t) 5 [B(0) 2 B ]exp(2t /t ) 1 B . (6) EoS (E /A) r n` B ` gs gs gs

SS1 888 12.3 0.779We solve the equation for the spin-up evolution (3)7 SS2 926 14.1 0.858of the accreting star for decay times 10 # t [yr] #B B60 836 4.6 0.2959 15010 and initial magnetic fields in the range 0.2 #

a 14(E /A) (in MeV) is the energy per baryon, r ( 3 10gs gsB(0)[TG] # 4.0. The remnant magnetic field is3 23

24 3 g /cm ) the mass density, and n (fm ) the baryon numbergschosen to be B 5 10 TG (Page et al., 2000a).` density. The two equations of state SS1 and SS2 differ for theAt high rotation frequency, both the angular choice of the parameter n entering in the expression of the

momentum transfer from accreting matter and the in-medium quark masses.influence of magnetic fields can be small enough tolet the evolution of angular velocity be determined

23fm is the normal nuclear matter density, and n is aby the dependence of the moment of inertia on theparameter. The effective quark mass M (n ) goesbaryon number, i.e. on the total mass. This case is q B

from its constituent mass value at zero density, to itssimilar to the one with negligible magnetic fieldcurrent mass m as n goes to infinity. Here weconsidered in (Burderi et al., 1999; Chubarian et al., q B

consider two different parameterizations of the EoS,2000; Shapiro and Teukolsky, 1983), where m # mcwhich correspond to a different choice for thein Eq. (3), so that only the so-called internal torqueparameter n. The equation of state SS1 (SS2)term (4) remains.corresponds to n 5 0.333 (n 5 0.286). These twomodels for the EoS give absolutely stable SQMaccording to the strange matter hypothesis (Bodmer,3. Equation of state for strange quark matter1971; Witten, 1984), see Table 1.

Medium dependent mechanisms for confinementTo describe the properties of strange quark matter,and their consequences for the EoS of quark matter,we used a recent model for the equation of statehave been explored by many authors using different(EoS) derived by Dey et al. (1998). This model isQCD motivated phenomenological models (Blaschkebased on a dynamical density-dependent approach toet al., 1999; Blaschke and Tandy, 2000; Blaschke etconfinement. This EoS has asymptotic freedom builtal., 1990; Drago et al., 1996). In addition to thein, shows confinement at zero baryon density, andprevious model for the EoS, we make use of the MITdeconfinement at high density. In this model thebag model EoS (Farhi and Jaffe, 1984) for strangequark interactions is described by a color-Debye-quark matter for non-interacting quarks, with strangescreened interquark potential originating from gluonquark mass m 5 150 MeV, massless u and d quarks,exchange, and by a density-dependent scalar po- s

3and with bag constant B 5 60 MeV/fm (hereaftertential which restores chiral symmetry at high den-the B60 EoS).sity (in the limit of massless quarks). This density- 150

dependent scalar potential arises from the densitydependence of the in-medium effective quark masses

4. Results and discussionM , which in the model of Dey et al. (1998) areq

taken to depend on the baryon number density nBIn Fig. 1 we show evolutionary paths of accretingaccording to

strange stars in the mass-radius (MR) plane (topnB] panels) corresponding phase diagrams (lower panels)M (n ) 5 m 1 310 (MeV) sech n , (7)S Dq B q n0 for two different EsoS (SS1 right panels and B60150

left panels). In each panel we show two trajectorieswhere q( 5 u,d,s) is the quark flavor index, n 5 0.160

of a strange star initially rotating with frequencyV(0) 5 0.001 Hz; for which the initial magnetic field

83 121 TG510 G. is B(0) 5 7 TG and its decay time is t 5 10 yr. TheB

110 D. Blaschke et al. / New Astronomy 7 (2002) 107 –112

Fig. 1. Evolutionary paths for strange stars in the mass-radius plane (top panels) and in the frequency-baryon number plane (lower panels)for the equations of state SS1 (left panels) and B60 (right panels), see text.150

29 28~ ~mass accretion rate onto the star is M 5 10 M / which are divided into Z sources with M | 10(210~yr. The solid lines show the evolution of strange star M /yr and A(toll) sources with M | 10 M /yr( (

configurations with initial gravitational mass M(0) 5 (Bhattacharya and van den Heuvel, 1991; Glenden-1.4 M , the dashed lines show that of configurations ning and Weber, 2001; van der Klis, 2000).(

with initial baryon mass N(0) 5 1.4 N . In the insets In Fig. 2 we explore the sensitivity of the model to(

of Fig. 1 we show the stable branches for configura- changes of the parameters with respect to the set8 28~tions rotating with maximal frequency (dash–dotted used in Fig. 1 (B(0) 5 7 TG, t 5 10 yr, M 5 10B

lines) in compare to static ones (dotted lines). M /yr) for accretors with initial baryon mass(

From the evolution paths V(N) one can see that in N(0) 5 1.4 N . In the upper panel we vary the inital(

all cases of initial masses the magnetic braking force magnetic field B(0) 5 5,7,8 TG, in the middle one7 8 9is strong enough to stop fast spin-up of the star and the magnetic field decay time t 5 10 , 10 , 10 yrB

211~saturate the frequency of rotation. This can lead to an and in the lower panel the accretion rate M 5 10 ,29 28effect of frequency clustering. However such effect 10 , 10 M /yr.(

for strange stars is correlated with the accretion rate. We see from this figure that changes of theIn Fig. 2, we show the result of our calculation for accretion parameters not only shift (choice of the

the spin evolution of the accreting strange star with initial magnetic field) and deform (choice of the~EoS SS1 in the P 2 P diagram, where P 5 2p /V is magnetic field decay time) the interval where the

~the period of rotation. The parameters of the accre- spin-up is saturated (dip in P ), but for some casestion model are chosen such as to correspond to this effect can be washed out by a variation of thevalues extracted from observations made on LMXBs, accretion rate. This phenomenon means that the

D. Blaschke et al. / New Astronomy 7 (2002) 107 –112 111

~Fig. 2. P2P diagrams for accreting strange stars with equation of state SS1. We show the dependence on the initial magnetic field B(0)~(upper panel), mass accretion rate M (lower panel), and magnetic field decay time t (middle panel).B

existence of a frequency clustering for strange stars (frequency clustering) in the interval 220 Hz , n ,

380 Hz which corresponds to recent observationsrequires a strong limitation of possible values of the(Glendenning and Weber, 2001), we have to chooseaccretion parameters.

8~ the following parameters: B(0) 5 2.5TG, t 5 10 yr,In Fig. 3 we plot the ‘‘Waiting time’’ t 5 uP/P u B29~M 5 10 M /yr for SS1 model and B(0) 5 3 TG,(Poghosyan et al., 2001) of the strange star as a (8 28~function of the spin frequency for the EsoS SS1 and t 5 10 yr, M 5 10 M /yr for B60 model. ForB ( 150

B60 . In order to obtain an enhanced waiting time both cases the initial gravitational mass is the same150

1.4 M , which corresponds to 1.83 N initial baryon( (

mass for SS1 and 1.71 N for B60 model. The( 150

main difference between both of these scenarios forthe frequency clustering is the initial baryon massand the mass accretion rate. An independent de-termination of these quantities to a sufficient accura-cy could thus rule out one of the compact starmodels. At present, this could be done only for aminority of objects (Lamb and Miller, 2001) andbears some model dependence (Poghosyan et al.,2001).

5. Conclusions

We have investigated the question whether theFig. 3. Waiting times as a function of spin frequency for spin-upevolution of strange stars with equations of state B60 and SS1. clustering of spin frequencies which has been ob-150

112 D. Blaschke et al. / New Astronomy 7 (2002) 107 –112

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I.B., H.G. and G.P. acknowledge the hospitality of Page, D., Geppert, U., Zannias, T., 2000a. A&A 360, 1052.Rostock University. This work was supported in part Page, D., Prakash, M., Lattimer, J.M., Steiner, A., 2000b. PhRvL

85, 2048.by the Deutsche Forschungsgemeinschaft (DFG)Poghosyan, G., Grigorian, H., Blaschke, D., 2001. ApJ 551, L73.under Grant Nos. 436 ARM 17/7 /00, 436 ARMSchaab, Ch., Hermann, B., Weber, F., Weigel, M.K., 1997. ApJ

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