Nuclear Physics B (Proc. SuPPL) 24B (1991) 103-109North-Holland
CONVERTING NEUTRON STARS INTO STRANGE STARS
When a neutron and a stable strangelet meet theneutron is readily absorbed, while a proton can coexistwith a strangelet due to the Coulomb barrier betweenthem.' , ' Therefore, if a stable strangelet (a strangeletwith baryon number greater than or equal to some mini-mum baryon number for strange matter stability, ."!.,;n )makes its way to the neutron rich regions of a neutronstar, it will gro -,,v by absorbing neutrori , and, eventually,convert most of the neutron star into a strange star .',3-e
The conversion of a neutron star into a strange staris quite a spectacular event : one stable strangelet canseed the conversion process that would liberate about10 6'ergs in binding energy (for a strange matter boundby 1OMeV) .
The only region of a neutron star that could sur-vive this conversion is the outer crust, since densitiesthere are below the density where neutrons drip out ofnuclei. The Coulomb barrier between strange matterand the ions in the outer crust forms a gap of a fewhundred fermis, which is sufficient to support the entireouter crust of the neutron star above the strange matterinterior.
In what follows, I first review the possible seedingmechanisms of strange matter in neutron stars and dis-cuss the implications of an abundance of strangelets inthe interstellar medium . Then, I will discuss the prop-agation o the conversion front and the possible observ-ables of such a remarkable event .
1 . SEEDING MECHANISMSA variety of different seeding mechanisms have been
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Department of Astronomy and Astrophysics and Enrico Fermi Institute, University of Chicago, Chicago, IL 60637,U.S.A .
If strange rnattcr is formed in the interior of a neutron star, it will convert the entire neutron star into a strangestar . We review the proposed mechanisms for strange matter seeding and the possible strange matter contaminationof neutron star progenitors . We discuss the conversion process that follows seeding and the recent calculations of theconversion timescale.
suggested so far3-8 They can be divided into two maincategories: primary mechanisms, in which the seed isformed inside the neutron star ;T and secondary mech-anisms, in which the seed comes from the interstellarmedium7-s Obviously, there can only be secondary pro-cesses ., if there were some primary ones before.
1 .1. Primary ProcessesThe formation of a strangelet is much more ilkeiy
in a neutron star than in ordinary nuclei . The centralprevrcs i it ac tV~ Iut ;:lYi i 'i11e SIYiILC:CYIL fur iL Phasetransition from neutron matter to two-flavor quark mat-ter to occur. Two-:favor quark matter can then easilydecay into the lower energy strange matter via weakinteractions .' .' . (Unlike the case for nuclei, here theformation of many strange quarks does not need to besimultaneous .) The possible existence of quark matterin the inner core of neutron stars has been widely stud-ied, but no firm conclusion can be reached at present?
Another possible way of forming a seed of strangematter is via an. agglomeration of A particles' At veryhigh densities (p > 1 .6 x 10 1sgm/cm3 ), the neutronchemical potential becomes greater than the A mass andA's start to appears Strangelets can then form directlyby the overlap of A,nin A's . The probability of AminA's overlapping is roughly P(Arni,,) ti (nA1'A_;^ )AAU .,
where nA is the number density of A's and VA_i. is thevolume of a strangelet with A = Ami,, . The enact valueof Ami,, (and, therefore, of VA_ ;,.) is very uncertain, butit is expected to be of the order of 100 or larger. Then,P(A,i,,) jumps from negligible (- 10-100 ) to - 1, whennAVA-;a varies from 0.1 to 1 . The outcome of this pro-
cess is hal :d to determine due to the the strong depen-dence on two uncertain parameters : the density of A'sreached inside neutron stars and the value of A,;,,.
Hadronic condensates with strangeness can also bepresent in neutron stars at high enough densities . Kaoncondensates' and dibaryon condensates" may occur inneutron stars for certain ranges of parameters . Bothtypes of condensate would facilitate the formation ofstrange matter seeds .
Strange matter seeds may also be formed throughthe burning of neutrons into strangelets a Neutron starsare born hot, so neutrons can be heated into higherenergy states such as low baryon number strangelets,which can then form higher baryon number (lower en-ergy) strangelets . The process is analogous to chemicalburning through an activation barrier . In ref. 4, weestimated the rate at which strangelets are made perunit volume. Again. we find that the rate varies from"extremely large" to "vanishingly small," with a smallvariatim: of the temperature and of the mass of the lowbaryon wimber strangelets (for example, a factor of twoin the parameters changes the rate from ti 1045/cm39to - 10-196/cuss) .
Finally, a more contrived way to form strangelets isvia neutrino sparking Here, an ultra-high energy neu-trino (E ti 1014MeV), the secondary from an ultra-high energy cosmic ray, penetrates a neutron star andscatters inelastically off a quark, depositing most of itsenergy in a small volume . This forms a hot quark-gluonplasma, with a pairs . If strangeness separation is effi-cient and cooling is fast enough, strangelets could beformed . This process shares some of the difficultiesof forming stable strangelets (low entropy states) inheavy ion collisions (high entropy environments) . Onthe other hand, it could possibly be tested by futurequark-gluon plasma experiments.
1 .2 Secondary ProcessesIf any of the above primary seeding processes (or
some other we have not envisioned yet) can convertsome neutron stars into strange stars, it is likely thatmil young neutron -stars are strange stars, Tius state-ment is based on recent estimates, by Friedman andCaldwell,' of the abundance of strangelets in the inter-
A . Olinto/ Converting neutron stars into strange stars
stellar medium due to the coalescence of binary systemsof two strange stars or of a strange star and a black hole.
If strange stars exist, they will naturally be foundin binary systems . Some fraction, f, of strange starswill be in binaries with other strange stars or withblack holes . These systems will eventually coalesc,- vi-olently and in the process, some fraction of the lowerdensity strange star will be ejected into the interstel-lar medium. (Neutron star-strange star binary systemswill most likely end in the disruption of the neutronstar not the strange star, since strange stars are ex-pected to be more dense than neutron stars.)a Beforethe final coalescence, the more dense companion willtidally strip some mass from the lower density strangestar . Typical sizes for strange matter lumps that arestripped can be estimated by balancing the tidal pull ofthe companion star against the surface tension of thestrange star. For a 1Mo companion at 30km separa-tion, the typical stripped strang,--: matter lump is around104s in baryon number. Lumps this large will not con-taminate the interstellar medium effectively. If theyescape the binary system, they will most likely escapethe galaxy all together. If they stay bound to the bi-nary system, these lumps will orbit the system and willcollide with each other from time to time . The colli-siou of these strange matter satellites will break themdown into smaller lumps . Of these smaller lumps, thosewith A < 10e remain in their parent galaxy confined bythe galactic magnetic field, and will become part of theinterstellar medium . 8
Neutron star coalescence studies give estimates ofaround 0.1 - 0.01Mo of ejected material per coales-cence . The coalescence rate for nentron stars todayin our galaxy is estimated to be around 10-4/yr . Wecan assume that roughly the same rate Lhould applyto strange stars .' If we further assume that this ratehas not changed significantly for a billion years, than alower limit on the number density of trangelets in ourgalaxy after a billion years is :
n, (A < 108 ) > f10
y fe l0enp 1013LYs
- ffe10`/LY3 ,
where fs is the fraction of the 0.01Mo ejected in stran-gelets with A < 108 per coalescence and the factor frelates to the fraction of strange star-strange star andstrange star-black hole binaries .
The strangelets in the interstellar medium can, inprinciple, seed previously formed neutron star,' butthe probability for this event will depend on a finetuning of f and fs, given that the volume of a neu-tron star is around 10-3sLY3 . These strangelets will bemuch more effective in seeding young neutron stars bycontaminating their progenitor stars which are formedfrom a region with a volume ^= LY3 in the interstel-lar medium. (Friedman and Caldwells expect f and fsnot far from unity.) If there have been strange starsin binary systems for the last billion years, the progen-itors of recent supernovae (formed around ten millionyears ago) will have been contaminated from birth bystrangelets in the interstellar medium . Young neutronstars (like the Crab and Vela) would have been formedwith strangelet seeds already in them and would havebecome strange stars .
Friedman and Caldwell used this argument togetherwith the notion that strange stars cannot exhibit glitchphenomena (present in young pulsars like Crab andVela) to rule out the existence of strange matter . Thisstrong conclusion could bring this conference to an earlyend, were it not for some loopholes .
First, strange matter car. be the ground state ofhadronic matter, but may never be formed in neutronstars (if none of the primary processes occur) . In thiscase, strange matter has no astrophysical implicationsbut could possibly be studied the laboratory . Second,the notion that strange stars cannot glitch is based ona very simplified and approximate model for stars com-posed of strange matter and should not be taken asfinal . Third, the event responsible for the creation ofsmaller lumps, i .e ., the collision of large baryon numbersatellites, might lead to fs < 1. The smaller strangematter nuggets might all evaporate into neutrons, forexample .
Two aspects of the two-component starquake modelfor pulsar glitches challenge the strange star scenario . 12(There have been models proposed where the glitch
A . Olinto/ Converting neutron stars into strange stars 105
event originatcJ in the magnetosphere, in which casestrange stars are on equal footing with neutron stars.)The first is the need for a crust 100 times more massive(i .e ., the outer plus the inner crusts of neutron stars)than the thin outer c_ust of strange stars, in order toaccount for the observed spin-down rate. The secondin-gredient of the phenomenological two-component glitchmodel that challenges a strange star pulsar model is theneed for a neutral superfluid to explain the post-glitchrelaxation behaviour.
In principle, the answers to the questions of the sta-bility of strange matter and the structure of high den-sity matter (as well as the understanding of nucleons)lie probably in QCD . Unfortunately, we are not ableto extract this information at present, and need to relyon simplified models that describe different aspects ofthe strong interactions . Our picture of strange stars isbased on the bag model, a simplified approach to thecomplexity of a strange quark matter system . The lackof internal structure in strange stars may be physicallycorrect or may be due to our oversimplification.
Strange stars may have a more complex intern:dstructure than the simple outer crust-core separation .Within quark matter, different correlations may ariseat different densities . For example, in a model basedon constituent quarks, Donoghue and Sateesh13 showedthat diquark states may be favored at densities justabove the quark-hadron phase transition ; in a bag mo-del with massless quarks, diquark states may condenseat higher densities, in analogy to dibaryon states inneutron stars. 11 (Benvenuto et al."' suggested a strangepulsar model making use of another hypothetical state,that of quark-alphas.)
The need for a neutral superfluid also challenges thepossibility of a strange matter interior for pulsars . Aneutral superfluid in quark matter seems less likely thansome additisnal structure . One possibility is that of aneutral pion-like condensate of quark-antiquark pairs,but this haai not been seriously explored yet .
Pulsar glitch constraints on strange stars are a veryserious challenge to the strange matter hypothesis . Gi-ven the present state of affairs, howevf!r, a firm conclu-sion on this issue at awaits observational confirmations
of pulsar glitch models and tests of neutron star equa-tions of state, and theoretical or experimental insightsinto the existence and possible composition of strangequark matter . The lack of a strange star pulsar glitchmodel may be due to physics or due to lack of imagina-tion.
Finally, the collision of two strange matter satellitesorbiting a coalescing binary system might have a differ-ent outcome than a production peak of strangelets withA =A;..8Amin strangelets may befragile in high tem-perature and high entropy environments.ls ,le,l'
2 . THE CONVERSION PROCESSIf we assume that seeding of strange matter has oc-
curred inside a neutron star we can study the conse-quent conversion of the rest of the star . The first at-tempt to study the conversion timescale (ref. 3) wasconceptually reasonable, but the quantitative resultswere off by several orders of magnitude . A more com-plete treatment of the conversion process was given inref. 6, with subsequent studies done in refs . 18, 19, and20. 1'11 review this approach and the more recent studiesand conclude by discussing the observable consequencesof this conversion process.
In general, for simplicity, the strong coupling con-stant and the strange quark mass are neglected, andwith them the small fraction of electrons present . Inthis case, charge neutrality gives the following relationbetween the baryon cumber density, n, and the num-ber density of each quark nq (q = u, d, e) : n = n andnd +n, = 2n . In equilibrium, Ad = it. and the threenumber densities are equal, n = nd = n, . Out of equi-librium, there is only one flee parameter. (In this pic-ture, p is constant, so we can set uu _ is = 300MeV.)Another simplifying assumption used is that neutronstar matter is composed only of neutrons .
The conversion pre^ess of neutron matter to strangematter is simplerwhen viewed from the rest frame oftheconversionfront . The volume over which strange matterequilibrates was shown's to be much smaller than thatof the total strange-matter region, so that the problemcan be treated one-dimensionally. In thi i frame thefront can be set at z = 0, neutron matter is at x < 0,
A . Olinto/Converting neutron stars into strange stars
strange matter at x > 0, and, asymptotically (z -+oo), strange matter is in equilibrium. The region ofsmall positive i; has an excess of down quarks relativeto strange quarks due to the flux of neutrons (udd) atx = 0 . The excess down quarks will convert into strangequarks via the weak process d -f- u --# a -I- u, as longas lid > As (nd > n,). For large positive values ofx the system is close to equilibrium, so Ad = A, andnd ;n !~:! n.
In terms of a variable a defined by u(z) = n = -n'nd+n . fwe can write : a(z < 0) = 1, and, as z --+ oo, a(z) -+ 0 .As z --+ 0, from the strange matter side (z > 0), a -+ ao,where ao is the maximum value of a for which strangematter is stable . The value of ao corresponds to theminimum number density of strange quarks such thatstrange matter is stable . In our scenario, the conversionofneutron matter to strange matter is direct and cannotproceed unless the strange matter next to the conver-sioa front is stable. [It is possible that an activation-type process happens instead, in which two-flavor quarkmatter is maintained. between. the neutron matter andthe strange matter regions by the energy released inthe conversion process . This possibility has not beenfully explored at the present .] The limit ao --* 0 cor-responds to the case where only strange matter withnd = n, can be stable . Then the front cannot inove,given that the swallowed neutrons do nit have nd = an,, .The case where ao --+ 1 corresponds to stability of two-flavor quark matter ; the neutrons can all convert in-stantaneously. Two-flavor quark matter is known to beunstable; nuclei witx: baryon number A are made of Anucleons and not of 3A quarks in a shtgle bag . There-fore we expect 0 < au < 1 .
The conversion process can be thought of as a fluidof excess down quarks coming from z < 0 at a velocityequal to that ofthe oat and asymptotically (at z > 0)becoming equilibrium strange matter (a = 0) . Thetransformation occurs via two -main processes : the con-version of down quarks into strange quarks via d -E- u --+e + u (in the region where Ad > A.) and the diffusionof the strange quarks from a > 0 to z --+ 0. These two
processes combined give a differential equation for a(x) :
where a' = A, D is the diffusion coefficient for thestrange quarks, and R(a, T) is the rate at which theweak process u + d -* u -i- a converts down quarksinto strange quarks . The boundary conditions for a(a)are: a(0) = ao, a(a -+ oo) --r 0, and conservationof baryon number through the conversion front gives :a(0) = -v(1 - ao)/D s
The velocity v at which the front moves, is deter-mined by solving the differential equation (2) and im-posing I.1he above overdetermined boundary conditions ;only a special value for v lets all the boundary condi-tions be satisfied .
The rate, R, can be calculate3 by integrating thesquare of the matrix element for the pocess d + u -+s + u over the phase space of the four particles withthe appropriate fermi factors . In the zero-temperature,small a limit, we get :s- 19
This can be written as : R(al
- * , where r ^_" 3.4 x10-9 sec .
Substituting the above R(a) into (2) and imposingthe boundary conditions we can find v(ao, r, D) numer-ically. The solution can be well approximated by thefollowing analytic expression
A . Olinto/ Converting neutron stars into strange stars
Da" - va' - R(a) = 0 ,(2)
15a3 GFCOaa9caina6cp6(3 )3 .(3)
D ar 2(1 - ao)
The precise value of ao cannot be well determinedclue to the uncettaint lies present in strong-interactioncalculations . Reasonable values for ao lie in the range0 < ao
108 A . Olinto/ Converting neutron
exdra factor of 24 in the expression for R(a, T) . So staytuned.
Before discussing the possible observable consequen-ces of such a conversion, let me discuss another issuethat has attracted some attention on this subject . Hor-vath et al.'s have studied the stability of the conversionfront and concluded that the front will be hydrodynam-ically unstable . They speculate that this instability willlead to the formation of a detonation front. The mech-anism by which a detonation front could be sustainedis not very clear, given that there are limitations tohow fast strangeness can be formed and supplied to thedetonation front due to the slowness of the weak in-teractions . These instabilities might instead slow downthe conversion process." This issue is not yet settled .Benvenuto et al . 24 suggest that if a detonation front isformed, the energy released by the conversion of neu-tron stars into strange stars could come to the aid ofthe failed supernova explosion simulations. However,it is not clear how the energy released in the conver-sion could get transferred to the supernova shock frontway into the envelope of the progenitor (far from theneutron. tar), as pcinted out by P. Haensel during thismeeting .
In any case, a more careful study of the conver-sion process which addresses the reheating due to thestrange matter binding energy, the possibility of activat-ing the conversion via two-flavor quark matter region,and the consequences of the hydrodynamical instabili-ties should be done.
The conversion of a neutron star into a strange staris more likely to happenjust after a neutron star is born .The conditions for primary seeding of strange matterare better at this stage and the progenitor might havesome strangelet contamination already. In this case, anobservable consequencc e,oul! be an extr. neutrino fluxproduced thermally at the conversion front .6 With thepossibility of detecting the neutrinos from a supernovaexplosion in our galaxy, we will be able to confirm orrule out such a conversion . In generA, the conversionwould be seen as a longer tail on the neutrino pulse ob-served on earth. The exact form of the neutrino emis-sion will depend on more careful studies of the conver-
stars into strange stars
sion already underway.If an older neutron star gets converted, the conver-
sion might be observed by a faster (lower T) neutrinoburst accompanied by a gamma ray burst .6
ACKNOWLEDGEMENTSI would like to thank the organizers and the partic-
ipants of this workshop on "Strange Quark Matter inPhysics and Astrophysics ." This work was supportedby DOE (DE-FG02-90ER40606) and NSF (AST-22595)at the University of Chicago .
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