application/xmlMultifragmentation model for astrophysical strangeletsSayan BiswasJ.N. DePartha S. JoarderSibaji RahaDebapriyo SyamAstrophysical strangeletsStatistical multifragmentation modelPhysics Letters B 715 (2012) 30-34. doi:10.1016/j.physletb.2012.07.055journalPhysics Letters BCopyright © 2012 Elsevier B.V. All rights reserved.Elsevier B.V.0370-26937151-329 August 20122012-08-2930-34303410.1016/j.physletb.2012.07.055http://dx.doi.org/10.1016/j.physletb.2012.07.055doi:10.1016/j.physletb.2012.07.055http://vtw.elsevier.com/data/voc/oa/OpenAccessStatus#Full2014-01-01T00:14:32ZSCOAP3 - Sponsoring Consortium for Open Access Publishing in Particle Physicshttp://vtw.elsevier.com/data/voc/oa/SponsorType#FundingBodyhttp://creativecommons.org/licenses/by/3.0/

Journals

S300.3

PLB28773S0370-2693(12)00806-410.1016/j.physletb.2012.07.055Elsevier B.V.Astrophysics and Cosmology

Fig. 1

(Color online.) Variation of ln ω(A) with the variation in A for the strangelets at different temperatures for a fixed value of the available volume V=5Vo. Here, Ao=1.2×1053.

Fig. 2

(Color online.) ln ω vs. A for different values of the available volumes V. Temperature is taken as 1.0 MeV. Here, Ao=1.2×1053.

Fig. 3

(Color online.) Energy per baryon (E/A) vs. baryon number (A) of the strangelets at different temperatures but for a fixed value of the available volume V=5Vo.

Multifragmentation model for astrophysical strangelets

Sayan

Biswas

sayan@bosemain.boseinst.ac.in

J.N.

jn.de@saha.ac.in

Partha S.

Joarder

partha@bosemain.boseinst.ac.in

Sibaji

Raha

⁎

sibaji@bosemain.boseinst.ac.in

Debapriyo

Syam

syam.debapriyo@gmail.com

aDepartment of Physics, Bose Institute, 93/1 A.P.C. Road, Kolkata 700009, IndiabCentre for Astroparticle Physics and Space Science, Bose Institute, Block EN, Sector V, Salt Lake, Kolkata 700091, IndiacSaha Institute of Nuclear Physics, 1/AF, Bidhannagar, Kolkata 700064, IndiadBarasat Government College, Barasat, N-24 PGS, Kolkata 700124, India⁎Corresponding author at: Department of Physics, Bose Institute, 93/1 A.P.C. Road, Kolkata 700009, India.

Editor: W. Haxton

Abstract

A model for the possible size distribution of astrophysical strangelets, that fragment out of the warm strange quark matter ejected during the merger of binary strange stars in the Galaxy, is presented here by invoking the statistical multifragmentation model. A simplified assumption of zero quark mass has been considered to obtain such mass-spectrum for the strangelets. An approximate estimate for the intensity of such strangelets in the galactic cosmic rays is also attempted by using a diffusion approximation.

Keywords

Astrophysical strangeletsStatistical multifragmentation model

Introduction

In 1984, Witten [1] proposed that strange quark matter (SQM) containing roughly equal numbers of up, down and strange quarks that are confined in a ‘bag’, representing the confinement of quarks through strong interactions, may be the true ground state of quantum chromodynamics (QCD). In the same year, Farhi and Jaffe [2] showed that small ‘nuggets’ of SQM in the form of ‘strangelets’ with baryon number A<107 may also be stable. Useful theoretical predictions, in the context of strangelets, have subsequently been provided by several authors; e.g., references [3–14]. Conjectures on the possible sources of strangelets include highly energetic nuclear collisions [15,16], collisions between strange stars [17,18] and the material ejected during supernova explosions [19]. Several exotic events were earlier reported in cosmic ray experiments at balloon and mountain altitudes [20–22]. Recently, a doubly charged (i.e., Z=2) event with its charge to mass ratio of about 0.1 is claimed to be detected in the Alpha Magnetic Spectrometer (AMS)-01 experiment [23]. All the above events are usually considered to be candidates for strangelets.The existence of SQM nuggets have numerous consequences in cosmology and astrophysics. It is, therefore, important to be able to find the contribution of such strangelets to the cosmic ray flux. Recently, the AMS-02 experiment has begun to probe into the evidence of strangelets at the top of the atmosphere [24]. As an aid to such experiments, Madsen [25] has provided a theoretical estimate for the possible strangelet flux in galactic cosmic rays above Earthʼs atmosphere. In his treatment, Madsen considered that the SQM, ejected due to tidal disruptions of strange stars in a binary system as they spiral towards each other owing to the loss of their orbital energy, would fragment into strangelets of roughly equal baryon numbers. It is, however, well-known in thermodynamics [26,27] that dilute warm matter finds it more convenient to condense into fragments of different sizes. On this consideration, the dilute SQM ejecta is also expected to clump into strangelets of different sizes having different baryon numbers and attain a state with lower free energy. Such size distribution of astrophysical strangelets has not been satisfactorily treated so far. The primary goal of the present Letter is to find the appropriate mass distribution for the strangelets. For this, we take recourse to Statistical Multifragmentation Models (SMM), the variants of which are often used in the analysis of fragmentation of hot nuclear matter in both terrestrial and astrophysical contexts [28–30].The production rate of strangelets in our Galaxy is somewhat speculative. If SQM is absolutely stable, as implied by the SQM hypothesis, then all the compact stars are likely to be the strange stars [17,18]. A major source for the production of strangelets in our Galaxy may thus be the merger of strange stars in binary systems [25,31–33]. Recent numerical simulations [32,33] show that such mergers are likely to produce tidal arms and the SQM ejected from the tips of those arms may become gravitationally unbound. Such simulations also show the formation of small lumps in the ejected material that may be a signature of the initial formation of a strangelet cluster. It is perhaps reasonable to assume that the further fragmentation and separation of those lumps, as the ejected material approaches its thermodynamic and chemical equilibrium, would ultimately yield a strangelet mass distribution that contributes to the cosmic ray flux in the Galaxy. By combining the population-averaged ejected mass of 10−4M⊙ per binary interaction obtained in their simulations and a strange star merger rate of about 10−5–10−4 yr−1[34], which is consistent with modern observations, Bauswein et al. [33] arrived at a production rate of strangelets of total mass of about 10−9M⊙–10−8M⊙ yr−1 in our Galaxy. We may also note that, in a recent communication, Paulucci et al. [35] gave a somewhat different scenario for the galactic production of strangelets from the fragmentation of SQM material ejected by shock waves during SN-II explosions. Fragmentation of the SQM ejecta at temperatures of a few MeV is a common feature in both the above scenarios. In this Letter, we would simply consider the SQM as given and calculate the strangelet-mass spectrum resulting from the fragmentation of that SQM. To evaluate such spectrum, we adopt the SMM. In Section 2, we therefore, summarize some of the essential features of the SMM. In Section 3, we invoke the formalism of nuclear fragmentation model to find the size distribution of the strangelets. Discussion of the results and their observational implication are presented in Section 4.2

Nuclear multifragmentation model

In nuclear SMM [28–30], it is assumed that the initially compressed warm nuclear matter containing No neutrons and Zo protons evolves in thermodynamic equilibrium and undergoes disassembly after reaching the ‘freeze-out volume’ at temperature T, when the mutual interactions between the fragments cease to be important. The expansion of the system is assumed to be quasi-static [36]; i.e., the expansion time scale is large compared to the equilibration time of the expanding complex. It is also assumed that, at the freeze-out volume, the system reaches chemical equilibrium. Moreover, if the system does not have any radial collective flow, then the total thermodynamic potential [37] of the system at freeze-out is given by(1)Ω=E−TS−∑i=1Nsμiωi. Here, E and S denote the internal energy and entropy of the system respectively, i indicates the fragment species, Ns is the total number of species, while ωi specifies the multiplicity and μi=Ziμp+Niμn is the chemical potential of the i-th species with μp being the chemical potential of each proton and μn being the chemical potential of each neutron. The freeze-out volume is a free parameter in this model. It is taken as about 3–10 times the normal volume of the fragmenting system [38]. The results obtained from the models correlate very well with the relevant experimental observables [28].In quantum statistical multifragmentation, the multiplicity of a fragment of specie i is given as(2)ωi=2Vπλi3∑j=0∞gijJ1/2+(ηij), or,(3)ωi=gi01(e−ηi0−1)+2Vπλi3∑j=0∞gijJ1/2−(ηij), depending on whether the fragment is fermion (Eq. (2)) or boson (Eq. (3)). In Eq. (3), the first term on the right hand side is the Bose-condensation contribution. In the above expressions, V refers to the available volume (freeze-out volume minus the volume of the produced fragments) and λi=h/2πmiT is the thermal de Broglie wavelength. Here, mi is the effective mass of the i-th fragment. It is taken as mi=mnAi where Ai is the baryon number of the fragment and mn=938 MeV is the average nucleon mass. The summation over j refers to the summation over all the energy states of the fragment including the ground state and gij refers to the degeneracy of the states. In Eqs. (2) and (3), J1/2±(η) designate the Fermi or the Bose Integral; i.e.,(4)J1/2±(ηij)=∫0∞x1/2ex−ηij±1dx, while the ‘fugacity’ ηij is given as(5)ηij=(μi−Eij)/T with Eij being the energy of the i-th fragment in the j-th state.If η<0 and |η|≫1, then J1/2±≃(π/2)eη; Eqs. (2) and (3) then go over to the Maxwell–Boltzmann distribution for the fragments; i.e.,(6)ωi=Vλi−3eμi/T∑j=0∞gije−Eij/T. The sum in Eq. (6) is the total canonical partition function Zi of the fragment specie i that is defined as Zi=e−Fi/T with Fi being the free energy of the i-th specie. The final expression for the multiplicity of the i-th specie may, therefore, be written as(7)ωi=Vλi−3e(μi−Fi)/T=Vλi−3e(−Ωi/T),Ωi being the thermodynamic potential of the fragment.3

Mass-spectrum for the strangelets

We would use Eq. (7) to find the strangelet-mass spectrum that arises from the fragmentation of SQM material in thermodynamic equilibrium at the freeze-out volume at a temperature T. In this Letter, we make the simplifying assumption that the quarks are massless and also consider each of the strangelets to contain equal numbers of up, down and strange quarks, each having the same chemical potential μq at freeze-out. By constitution, the strangelets are then chargeless so that the Coulomb interactions among the strangelets are absent here. Following SMM, the strangelets are also assumed to have no strong interaction at freeze-out [28]. We further assume that each strangelet is in mechanical equilibrium, with its internal quark pressure exactly balancing the bag pressure, such that each of the i-th specie satisfies a condition [39](8)BVi=[(19π2/36)T4+(3/2)μq2T2+(3/4π2)μq4]Vi−[(41/216)T2+(1/8π2)μq2]Ci, where, B is the MIT bag constant, the value of which is taken to be B1/4=145 MeV. In Eq. (8), Vi and Ci denote the volume and curvature for a spherical strangelet such that Ci=8(3π2Vi/4)1/3, Vi=4πro3Ai/3 with ro being the radius parameter for the strangelets, the value of which is calculated to be 0.96 fm for the entire range of parameter values considered in this Letter.For three massless quark flavors of equal chemical potentials, the expressions for the thermodynamic potential Ωi and the baryon number Ai of the i-th strangelet in the equilibrated cluster may be written as [39](9)Ωi=[(41/108)T2+(1/4π2)μq2]Ci, and(10)Ai=[μqT2+(1/π2)μq3]Vi−(1/4π2)μqCi, respectively.By using Eqs. (7)–(10) and also by imposing the condition for the conservation of baryon number Ao of the initial SQM, namely(11)Ao=∑iAiωi(Ai), we may now calculate the mass-spectrum of the strangelets for values of temperature (T) in the range 1–10 MeV. Such temperature range includes typical temperatures of the ejecta found in the simulations of merger between two neutron stars [41], although the temperature of the material ejected in strange star-mergers is not explicitly mentioned in the published simulation results [32,33]. As noted above, the temperature considered here is also consistent with those of the fragmenting SQM during SN-II explosions [35]. In our calculations, Ao=1.2×1053 corresponding to the average mass of about 10−4M⊙ ejected during each merger-event between two strange stars [32,33]. The range of values for the available volume V is taken to be 2Vo–9Vo in consonance with the standard practice in the calculations of nuclear fragmentation. Here, Vo=4πrb3Ao/3 is the volume of the bulk SQM, where rb3=3/(4πnB); nB=0.7B3/4 being the typical baryon number density of bulk SQM [8].

Fig. 1

shows the mass-spectrum for the strangelets at V=5Vo and for three different values of the temperature at freeze-out, namely T=1.0, 5.0 and 10.0 MeV. In this figure, the multiplicity of the strangelets is seen to decrease with increasing size (or, baryon number) of the strangelets. We also find that light fragments are produced more copiously with an increase in temperature while the production of heavy fragments is suppressed. Similar enhancement in multiplicity of smaller fragments and the suppression of larger fragments is also observed in Fig. 2

as we increase the available volume V at a fixed temperature (T=1.0 MeV). Such pattern of fragmentation is familiar from the nuclear multifragmentation models; e.g., [29].

To address the question of stability of the produced strangelets, we plot the energy per baryon (Ei/Ai with Ei=4BVi) of the strangelets against their baryon number Ai at different temperatures and at V=5Vo. In Fig. 3

, we find that the strangelets with Ai≲4 tend to have energies (per baryon) that are larger than 930 MeV that corresponds to the energy (per baryon) of the nuclei of Fe56. Such strangelets are possibly unstable and are likely to decay into normal nuclei. Similar problem with the stability of light (A⩽6) strangelets was mentioned in Ref. [2]. Such stability properties of the strangelets are, however, strongly parameter dependent [25].

Discussion

In this Letter, our primary goal was to obtain the baryon number distribution of strangelets that may fragment out of the warm SQM ejecta produced in various possible astrophysical processes, such as the merger between binary strange stars or the SN-II explosions. The following observations are, however, pertinent to the fragmentation model presented here.A comment on the value of the quark chemical potential μq must be made in this context. Analytical calculation at T=0 with the condition for the number of quarks in a strangelet to be positive requires that (4π2B/3)1/4<μq<(4π2B)1/4. By substituting B1/4=145 MeV, we thus obtain 275.5 MeV<μq<362.5 MeV. Farhi and Jaffe had earlier pointed out that value of μq should be around 300 MeV [2,42]. The magnitude of chemical potential in the range 275.0–276.2 MeV, that we find in our calculations at finite temperature, is consistent with the above estimates.In writing the expression for fragment multiplicity in Eqs. (6) and (7), we have used classical Maxwell–Boltzmann (MB) distribution rather than the quantum statistical distributions. We checked that the values of the multiplicities do not significantly differ due to the neglect of quantum statistics, particularly in the case of massive strangelets. For example, the value of ωi for Ai=7 obtained by using MB statistics is less by about 13.0% than the corresponding value obtained from quantum statistics at a temperature of 1 MeV. Such deviation decreases rapidly with increasing baryon number to become only about 6.0% for Ai=10 at 1 MeV. For a fixed value of Ai, the above discrepancy decreases with increasing temperature thus being only about 1.0% for Ai=7 at 10 MeV.The assumption of thermodynamic equilibrium for the fragmenting SQM ejecta plays an important role in finding the fragment size distribution. A preliminary examination of recent simulations on strange star merger [32,33] suggests that the relative velocity between the ends of the filamentary ejecta (or, the tidal arm) of total length ∼100 km, that is produced after the merger, may be v∼2π×20 km (ms)−1∼105 kms−1. We have assumed that the distance between the centers of the strange stars in a binary system of orbital period ∼1 ms, is ∼20 km at the instant of the merger. We now consider the relative motion between neighboring elements of the tidal filament. By ‘neighboring elements’, we mean the elements separated by distances of about 10–100 times the approximate collisional mean free-path of the fragmenting strangelets which may be just about a few fm. The relative rate of separation between such neighboring regions of the ejecta thus seems to be ≲(105/100)×10−16∼10−13 kms−1 that is many orders of magnitude smaller than the thermal velocity of ∼103 kms−1 for the strangelets of average size A=16 at a temperature T∼ 1 MeV. The assumption of quasi-static evolution of the ejected material towards its thermodynamic equilibrium considered in Section 2 of this Letter seems to be justified in view of the above arguments.In case of finite temperature (T>0), the quarks are statistically distributed over energy levels and so the color singlet condition comes into play. The color singlet constraint increases the energy (per baryon) for fixed entropy and thus the number of possible configurations are reduced [8,39,40]. We, however, do not take the color singlet condition into account as it may not produce significant effect at temperatures ≲10 MeV[8,39].Finally, the assumption of massless quarks, as a consequence of which we get chargeless strangelets, does not allow us to directly use the derived mass-spectrum in a realistic galactic propagation model. We can, nevertheless, estimate an approximate magnitude for the galactic flux of the strangelets of a certain baryon number if we consider that the strangelets may possess a small charge due to minute differences in their numbers of u-, d- and s-quarks. For the purpose of the present Letter, we further assume that such minute difference between the numbers of quark-flavors does not appreciably change the mass-spectrum of the strangelets. The charged strangelets would then propagate in the inhomogeneous magnetic field of the Galaxy so that we can use a diffusion approximation [43,44] to describe their propagation. In this approximation, we consider that apart from certain aspects, such as their unusually high mass to charge (A/Z) ratio compared to the normal nuclei, the strangelets would mostly behave like ordinary cosmic ray nuclei [25] that are formed predominantly near the galactic plane and the galactic center and then diffuse towards the boundaries of the galactic halo [43]. An approximate order of magnitude estimate for the total flux F(Ai) of the strangelets with baryon number Ai at the solar distance R≈8 kpc from the galactic center is then given by [43](12)F(Ai)=D.dn(Ai,r)dr.4πR2∼L2Rmωi(Ai)2VGR.4πR2particless−1. Here, n(Ai,r) is the number density of strangelets of size Ai in the Galaxy, an average value of which may be given by navg(Ai)≈Rmτωi(Ai)/VG with Rm being the rate of merger between the strange stars in the Galaxy, τ the confinement time and VG the effective confining volume of strangelets in the Galaxy. In Eq. (12), D is the diffusion coefficient of the strangelets and L=(2Dτ)1/2 is the root mean square distance traveled by the strangelets in time τ[43]. After the substitution of L∼10 kpc[8], VG∼1000 kpc3[25] and Rm≈10−5 yr−1[32–34], we finally arrive at an estimate for the intensity of strangelets of size Ai in the neighborhood of the Sun. Such estimation reads(13)I(Ai)∼5×10−48ωi(Ai)particlesm−2sr−1yr−1 with ωi(Ai) given in Eq. (7) above. Considering all the strangelets with Ai⩾5, the integrated strangelet intensity above Earthʼs atmosphere turns out to be ∼105 m−2sr−1yr−1 that is comparable with the intensity obtained by Madsen [25]. We, however, note that the detailed processes, such as the ionization energy loss, decay, spallation and the re-acceleration mechanisms [25] for the strangelets in the Galaxy are absent in the simplified treatment of propagation presented above. Also, the effect of solar modulation as well as the effect of the geomagnetic rigidity cut-off on the strangelets [25] have not been taken here into account. In the near future, we plan to incorporate the effect of finite mass of the strange quarks to examine the effect of finite charge on the mass-spectrum of the strangelets. Existence of finite charge would then allow us to consider the detailed aspects of propagation of such strangelets in the Galaxy. It is to be hoped that the simple model of SQM-fragmentation and the resulting strangelet-mass spectrum, that we present here, would provide a useful guidance to such more involved calculations.

Acknowledgements

JND acknowledges the support from the Department of Science and Technology (DST), Govt. of India. SR and PSJ thank the DST, Govt. of India for support under the IRHPA scheme.

References

[1]

WittenPhys. Rev. D

1984

272

[2]

Farhi

R.L.

JaffePhys. Rev. D

1984

2379

[3]

M.S.

Berger

R.L.

JaffePhys. Rev. C

1987

213

[4]

Greiner

StöckerPhys. Rev. D

1991

3517

[5]

E.P.

Gilson

R.L.

JaffePhys. Rev. Lett.

1993

332

[6]

Y.B.

C.S.

Gao

X.Q.

W.Q.

ChaoPhys. Rev. C

1996

1903

[7]

Wilk

WlodarczykJ. Phys. G

1996

L105

[8]

Madsen

CleymensLecture Notes in Physics: Physics and Astrophysics of Strange Quark Matter, vol. 5161999Springer-VerlagHeidelberg

162

arXiv:astro-ph/9809032v1

1998

[9]

Banerjee

S.K.

Ghosh

Raha

SyamJ. Phys. G

1999

L15

[10]

Banerjee

S.K.

Ghosh

Raha

SyamPhys. Rev. Lett.

2000

1384

[11]

MadsenPhys. Rev. Lett.

2001

172003

[12]

X.J.

Wen

X.H.

Zhong

G.X.

Peng

P.N.

Shen

P.Z.

NingPhys. Rev. C

2005

015204

[13]

R.X.

B.Q.

MaJ. Phys. G

2007

597

[14]

Paulucci

J.E.

HorvarthPhys. Rev. C

2008

064907

[15]

Greiner

Koch

StöckerPhys. Rev. Lett.

1987

1825

[16]

Schaffner-BielichJ. Phys. G

1997

2107

[17]

MadsenPhys. Rev. Lett.

1988

2909

[18]

J.L.

Friedman

R.R.

CaldwellPhys. Lett. B

264

1991

143

[19]

Vucetich

J.E.

HorvathPhys. Rev. D

1998

5959

[20]

Kasuya

Saito

YasuèPhys. Rev. D

1993

2153

[21]

Ichimura

Nuovo. Cim. Soc. Ital. Fis. A

106

1993

843

[22]

P.B.

Price

E.K.

Shirk

W.Z.

Osborne

L.S.

PinskyPhys. Rev. D

1978

1382

[23]

ChoutkoAMS01 Collaboration

Kojita

Proc. 28th Int. Cosmic Ray Conf.2003Universal Academy PressTokyo, Japan

1765

[24]

SandweissJ. Phys. G

2004

S51

[25]

MadsenPhys. Rev. D

2005

014026

[26]

HuangIntroduction to Statistical Physics2002Taylor and FrancisNew York, USA

[27]

J.N.

S.K.

SamaddarPhys. Rev. C

2007

044607

[28]

J.P.

Bondorf

A.S.

Botvina

A.S.

Iljinov

I.N.

Mishustin

SneppenPhys. Rep.

257

1995

133

[29]

Pal

S.K.

Samaddar

J.N.

DeNucl. Phys. A

608

1996

[30]

B.S.

MeyerAnnul. Rev. Astron. Astrophys.

1994

153

[31]

MadsenJ. Phys. G

2002

1737

[32]

Bauswein

Phys. Rev. Lett.

103

2009

011101

[33]

Bauswein

Oechslin

H.-T.

JankaPhys. Rev. D

2010

024012

[34]

Belczynski

Astrophys. J. Lett.

680

2008

L129

[35]

Paulucci

J.E.

Horvath

GrassiPoS

NIC-IX

2006

159

[36]

Stöcker

GreinerPhys. Rep.

137

1986

277

[37]

S.K.

MaStatistical Mechanics1993World ScientificSingapore

[38]

A.S.

Botvina

I.N.

MishustinEur. Phys. J. A

2006

121

[39]

D.M.

Jensen

MadsenPhys. Rev. D

1996

R4719

[40]

M.G.

Mustafa

AnsariPhys. Rev. C

1997

2005

[41]

Oechslin

H.-T.

Janka

MarekAstron. Astrophys.

467

2007

395

[42]

MadsenPhys. Rev. Lett.

1993

391

[43]

V.L.

Ginzburg

S.I.

SyrovatskiiThe Origin of Cosmic Rays1964Pargamon PressOxford, England

[44]

T.K.

GaisserCosmic Rays and Particle Physics1990Cambridge University PressCambridge, England