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VOLUME 77, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 AUGUST 1996
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Conversion of Neutron Stars to Strange Stars as a Possible Origin ofg-Ray Bursts
K. S. Cheng1 and Z. G. Dai21Department of Physics, University of Hong Kong, Hong Kong
2Department of Astronomy, Nanjing University, Nanjing 210093, China(Received 21 November 1995)
We propose that some neutron stars in low-mass x-ray binaries can accrete sufficient mass to undea phase transition to become strange stars. The energy released per conversion event satisfierequirements of cosmologicalg-ray bursts, and the Lorentz factor of the resultant expanding fireballmay exceed5 3 103 because the strange star has very low baryon contamination. The model burst rais consistent with observations. [S0031-9007(96)00892-7]
PACS numbers: 98.70.Rz, 12.38.Mh, 26.60.+c, 97.60.Jd
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The recent observational results from the BATSdetector on the Compton Gamma-Ray Observatorystrongly suggest that the sources of weakg-ray burstsare at cosmological distances [2]. Prior to the BATSobservations, several authors [3] proposed thatg-raybursts arise at cosmological distances in the mergebinaries consisting of either two neutron stars or a neutstar and black hole, while other authors [4] suggestedg-ray bursts result from phase transition of normal nuclmatter to matter with pion condensation in neutron staNow the cosmological scenarios also include the collaof a white dwarf to a neutron star with an extremely stromagnetic field and the formation of a transient accretdisk around a black hole resulting from a failed typesupernova [5]. In this Letter we argue that the conversof neutron stars to strange stars is another possible oof g-ray bursts. Such converting stars may be neutstars in the binaries with low-mass companions.
When nuclear matter is squeezed to a sufficiently hdensity, it turns into uniform two-flavor quark (u and d)matter. But the quark matter is unstable, and subsequeconverts to three-flavor (strange) quark matter, duethe fact that strange matter may be more stable tnuclear matter [6]. The properties of strange stars hbeen studied [7]; however, their existence is doubtfulseveral reasons.
First, it has not yet been confirmed that strangquark matter is a stable form of matter. The resufrom ultrarelativistic heavy-ion collisions are expectedprovide solid proof [8], and, in fact, recent experimenresults do suggest some evidence of the existencestrange-quark condensation [9]. However, the conclusis still controversial, and further theoretical studies in tproperties of strange-quark matter, especially the exconditions for the phase transition and experiments inhigher energy range, are necessary [10].
Second, other objections against the existencestrange stars result from astrophysical arguments.has been argued that the disruption of a single strastar can contaminate the entire galaxy and essentially“neutron” stars are strange stars [11]. This conflicts w
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the relaxation behavior of pulsar glitches, which is wdescribed by the neutron-superfluid vortex creep the[12], and current strange-star models scarcely explainobserved pulsar glitches. Furthermore, ifg-ray burstsare the consequence of the merger of two compact seven a single merger event involves a strange star, anpulsars are then strange stars [13]. However, we wancomment that these arguments do not necessarily dispthe existence of strange stars for the following reaso(i) In order to disrupt a strange star, its companion malso be a very compact object, e.g., a neutron staa black hole but not a white dwarf because the tiforce must be strong enough to disrupt the strange sAccording to the standard evolution model of millisecopulsars, which is expected to be the progenitor of strastars in our model, its companion is a low-mass whdwarf. The subsequent evaporation process can furreduce the mass of the white dwarf and may eventutotally remove this companion star [14]. At least in thmodel, disruption will not take place. (ii) Even in thcase of neutron-star–neutron-star merger or neutron-sblack-hole merger, how much material will be shedstill an open question [15]. (iii) If the core of a strangstar consists of quantized fluxoids and vortex lines, ththe relaxation of a strange-star glitch will be very similto that of ordinary neutron stars [16].
Third, the conversion of a neutron star to a strangerequires the formation of a strange-matter seed in thewhich is produced through the deconfinement of neutmatter at a density, sufficiently larger than the centdensity of the1.4MØ star with a rather stiff equation ostate [17]. In view of these uncertainties, we should oregard strange stars as strong possible stellar objects.
It is thought that the density for deconfinementneutron-star matter with a moderately stiff (or stifequation of state to two-flavor quark matter is near8r0
(where r0 is the nuclear density). For a soft equatioof state, the deconfinement density is lower. Hereassume that the equations of state in neutron starsmoderately stiff or stiff. This is because soft equatioof state at high densities are ruled out by the postgli
© 1996 The American Physical Society
VOLUME 77, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 AUGUST 1996
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recovery in four pulsars [18]. More detailed analysof the postglitch curves of the Crab and Vela pulsalso draw similar conclusions [19]. In addition, the soequation of state, as in kaon condensation, seemsto occur in stable neutron stars [20]. The neutron stwith 1.4MØ, based on the modern equations of st[21] named UV14+UVII, AV14+UVII, and UV14+TNI,must accrete matter of,0.6MØ, 0.5MØ, and 0.4MØ inorder that their central densities reach the deconfinemdensity. Once this condition is satisfied, strange-maseeds are formed in the interiors of the stars.
After a strange-matter seed is formed, the stranmatter will begin to swallow the neutron matterthe surroundings. While it has been proposed [that the combustion corresponds to the slow mosubsequent work [23] shows that this mode appearbe hydrodynamically unstable. Thus the conversionneutron matter should proceed in a detonation mode.total kinetic reaction of the detonation mode has tstages: the formation of two-flavor quark matter andweak decays that form strange matter. Since the secprocess enhances the thermal energy at the expensthe chemical energy of two-flavor quark matter [24], ttemperature in the stellar interior will increase to mothan 10 MeV. In addition, the time scale [23] for thconversion of a neutron star to a strange star is smathan 1 s.
The resulting strange star [25] has a thin crust wmass ,2 3 1025MØ and thickness,150 m, but be-cause the internal temperature is so highs,1011 Kd,the nuclei in this crust may decompose into nucleoApproximating strange matter by a free Fermi gas,obtain the total thermal energy of the star,Eth , 5 3
1051 ergss ryr0d2y3R36T 2
11, wherer is the average masdensity,R6 the stellar radius in units of106 cm, andT11
the temperature in units of1011 K. Adopting r 8r0,R6 1, andT11 1.5, we haveEth , 5 3 1052 ergs.
The star will cool by the emission of neutrinos and atineutrinos, and because of the huge neutrino numbersity, the neutrino pair annihilation processnn ! e1e2
operates in the region close to the strange-star surfThe total energy [26] deposited due to this processE1 , 2 3 1048 ergssT0y1011 Kd4 , 1049 ergs (whereT0
is the initial temperature) and the time scale for dposition is of the order of 1 s. On the other hanthe processes forn 1 ne ! p 1 e2 and p 1 ne !n 1 e1 play an important role in the energy depostion, and the integrated neutrino optical depth [27] duethese processes ist , 4.5 3 1022r
4y311 T2
11 (wherer11 isthe crust density in units of1011 g cm23). So the de-position energy is estimated byE2 , Eths1 2 e2td ,2 3 1052 ergs. Here we have used the neutron-drip dsity s r11 , 4.3d, and have assumed that the thermal eergy of the star is wholly lost in neutrinos. The procegg $ e1e2, inevitably leads to the creation of a fireball. However, the fireball must be contaminated by
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baryons in the thin crust of the strange star. If we defih E0yM0c2, whereE0 E1 1 E2 is the initial radia-tion energy produced (e1e2, g) andM0 is the conservedrest mass of baryons with which the fireball is loadthen, since the amount of the baryons contaminatingfireball cannot exceed the mass of the thin crust, we hh $ 5 3 103 and the fireball will expand outward. Thexpanding shell (having a relativistic factorG , h) in-teracts with the surrounding interstellar medium andkinetic energy is finally radiated through nonthermal pcesses in shocks [28].
What mechanism results in the conversion of neutstars? Here we propose that accretion in binaries wlow-mass companions can lead to the conversion.has been shown [29] that the amount of matter accr(DM) by the 18 radio pulsars in these binary systeexceeds0.5MØ. If this is true, some of the milliseconpulsars should have masses over2MØ, and they may bestrange stars. By assuming that the number of galaxiecosmological distances isN , and the number of neutrostars in low-mass x-ray binaries which will convert instrange stars in the characteristic accretion time ss,DMy ÙMd is NB, we have the burst rate
R ,N NB
DMy ÙM
, 10 day21
µN
1010
∂ µNB
10
∂ µDM
0.5MØ
∂21√
ÙMÙMEdd
!, (1)
where ÙM is the accretion rate andÙMEdd is the Eddingtonaccretion rate. In order to determine the conversion rwe need to estimate the value ofNB.
The direct criterion for the conversion of neutron stto strange stars is that the total mass of the accretingtron stars should exceed2MØ. Observationally, only themass function can be determined precisely but notmass of the individual star. The one exception is thein the Hulse-Taylor binary system, because the high prsion of pulsar measurements combined with the relativhigh orbital velocity of the system has allowed measuments of the general relativistic periastron advancesecond order Doppler shift [30]. However, there are inrect ways to estimate the conversion rate. First, if a gfraction of neutron stars in luminous low-mass x-raynaries (LMXBs) is converted into strange stars, thenNB
is close to the current observed number of LMXBss,10d[31]. Second, We expect that possible strange-star cadates are very weak-field millisecond pulsars which ha lifetime near the Hubble time. If the radio beams ofmillisecond pulsars are fan beams [32], there are ab103 to 104 weak-field millisecond pulsars in our galaxas estimated by scaling from the present observed va[33]. Thus, the conversion rate is abouts3 30d 3 10210
per day per galaxy. This conversion rate seems consiswith the g-ray burst rate. Here we want to add two rmarks. (1) The companion of the strange star is a w
1211
VOLUME 77, NUMBER 7 P H Y S I C A L R E V I E W L E T T E R S 12 AUGUST 1996
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dwarf in the accretion induced spun-up scenario, andpulsar wind may evaporate this white dwarf eventua[14]. (2) The fact that the surface magnetic fields of mlisecond pulsars seem independent of the amount of ation material [29] and that there is a gap in the magnfield distribution of pulsars [31,34] are suggested asdence of the phase transition from neutron stars to strastars, because these two kinds of compact objectsdifferent minimum magnetic fields which can be suported by the stellar crusts [35].
It is well known that the merging of two neutron stahas been proposed as a possible origin for cosmologg-ray bursts [3]. Our converting model differs from tmerging model as follows. First, the merging should pduce observable gravitational waves [36], but thereno gravitational radiations in our model if the conversis spherically symmetric. Future observation of gravtional waves may distinguish between the convertingmerging processes. Second, the formation rate of combinaries is quite uncertain, but a current estimation [lies in the range1025 1024 yr21 per galaxy. Thus, themerging rate seems to be much larger than the obseburst rate. In our scenario, the estimated rate is content with the burst rate. Third, because the strangejust formed during the conversion has a very thin crthe resultant fireball is contaminated by a small amoof baryons#1025MØ, but in the merging model the number of baryons loaded with the fireball is unlikely tosmall [27]. Therefore, the evolution of the fireball in tconversion model is somewhat different from that inmerging model.
We would like to thank M. A. Alpar, S. BludmanH. F. Chau, M. C. Chu, J. Fry, S. Ichimaru, W. KluzniaF. Lamb, T. Piran, M. A. Ruderman, N. Shibazaki, W.Suen, J. W. Truran, and E. P. J. van den Heuvel for menlightening discussions, and T. Boyce for a critireading of our paper.
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