IINT-PUB-11-054
Strangelet dwarfs
Mark G. Alford, Sophia Han ( 韩 君 ) Physics Department, Washington University, St. Louis, MO 63130, USA
Sanjay Reddy
Institute for Nuclear Theory, University of Washington, Seattle, Washington 98195-1550, USA (Dated: 7 March 2012)If the surface tension of quark matter is low enough, quark matter is not self bound. At sufficientlylow pressure and temperature, it will take the form of a crystal of positively charged strangelets ina neutralizing background of electrons. In this case there will exist, in addition to the usual familyof strange stars, a family of low-mass large-radius objects analogous to white dwarfs, which we call“strangelet dwarfs”. Using a generic parametrization of the equation of state of quark matter, wecalculate the mass-radius relationship of these objects.
PACS numbers: 25.75.Nq, 97.20.Rp, 26.60.-c, 97.60.Jd,
I. INTRODUCTION
The matter that is directly observed in nature consistsof atoms, whose nuclei are droplets of nuclear mattercomposed of up and down quarks. Nuclear matter isvery stable: the most stable nuclei have lifetimes longerthan the age of the universe. However, it has been hy-pothesized [1–3] that nuclear matter may actually bemetastable, and the true ground state of matter con-sists of a combination of roughly equal numbers of up,down, and strange quarks known as “strange matter”.Strange matter is hypothesized to exist as (kilometer-sized) pieces, known as “strange stars” (reviewed in [4]),or as small nuggets, known as “strangelets” [3]. It has fur-ther been hypothesized that dark matter could be someform of quark matter, trapped in strangelets or strangestars before the era of nucleosynthesis [5–7]. However,even if strange matter is not invoked as a dark mattercandidate, there could still be a population of strangematter objects, from strange stars to strangelets, manyof which would be relatively non-luminous. In this articlewe show that the masses and radii of such objects can ex-tend in to the range expected for planets. Recent surveyssuch as the Microlensing Observations in Astrophysics(MOA) and the Optical Gravitational Lensing Experi-ment (OGLE) to detect such low mass non-luminiouslow mass objects by gravitational lensing have yieldedinteresting results [8]. The hypothetical compact objectswe predict could be detected by such methods and suchsurveys could place stringent bounds or perhaps hint attheir possible existence.It is generally assumed that strange stars are compactobjects, with sizes in the 10 kilometer range [4], end-ing at a sharp surface of thickness ∼ σ ofthe interface between quark matter and the vacuum isless than a critical value σ crit (of order a few MeV/fm intypical models of quark matter) then large strangelets areunstable against fission into smaller ones [12–14], and theenergetically preferred state is a crystal of strangelets: a mixed phase consisting of nuggets of positively-chargedstrange matter in a neutralizing background of electrons.In this “low surface tension” scenario, strange starsare not self-bound: they require gravitational attractionto bind the strangelets. Stars made of strange matterare then qualitatively similar to those made of nuclearmatter: in each case the mass-radius relation has twobranches, one compact and the other diffuse. For nu-clear matter, the compact branch contains neutron stars,which consist of gravitationally bound nuclear matter,with an outer crust that is a crystal of nuclei in a back-ground of electrons; the diffuse branch contains whitedwarfs, which are a gravitationally bound cold plasma ofnuclei (ions) and electrons, forming, at sufficiently lowtemperature, a crystalline structure (see, e.g., [15]). Forstrange matter with a low surface tension, there are simi-larly two branches. The compact branch contains strangestars with a crust that consists of strangelets in a back-ground of electrons; this “strangelet crystal crust” wasstudied in Ref. [14]. In this paper we study the diffusebranch, which has no core of uniform quark matter: thesestars consist entirely of strangelets in a background of de-generate electrons, so by analogy with white dwarfs wecall them strangelet dwarfs.The strangelet-crystal phase is a charge-separatedphase. Charge separation is favored by the internal en-ergy of the phases involved, because a neutral phase is al-ways at a maximum of the free energy with respect to theelectrostatic potential (see [16, 17]; for a pedagogical dis-cussion see [18]). The domain structure is determined bycompetition between surface tension (which favors largedomains) and electric field energy (which favors smalldomains). Debye screening plays a role in determiningthe domain structure, because it redistributes the elec-tric charge, concentrating it in the outer part of the quarkmatter domains and the inner part of the surroundingelectron gas, and thereby modifying the internal energyand electrostatic energy contributions. Our parameteri-zation (1) of the electrostatic properties of quark matteris generic, but is not appropriate for strangelets in thecolor-flavor locked (CFL) phase [19], which is a degener- a r X i v : . [ nu c l - t h ] A p r ate case requiring separate treatment (see Sec. IV A).To obtain the M ( R ) relation of strangelet dwarfs, wesolve the Tolman Oppenheimer Volkoff equation [20, 21],using the equation of state of the mixed phase. We obtainthe equation of state by assuming that the strangeletlattice can be divided into unit cells (“Wigner-Seitz cells”)and calculating the pressure of a cell as a function of itsenergy density. Our approach is similar to that used inprevious studies of the strangelet crystal [13, 14] (exceptthat in this paper we include electron mass effects) andin studies of mixed phases of quark matter and nuclearmatter in the interior of neutron stars [22].The main assumptions that we make are:1) We assume that the strangelets in the plasma forma regular lattice of Wigner-Seitz cells, which we treat asrotationally invariant (spherical). In reality the cells willbe unit cells of some regular lattice. We do not considerlower-dimensional structures (rods or slabs) because inRef. [14] we found that such structures were never ener-getically favored.2) Within each Wigner-Seitz cell we use a Thomas-Fermiapproach, solving the Poisson equation to obtain thecharge distribution, energy density, and pressure. Thisis incorrect for very small strangelets, where the energylevel structure of the quarks becomes important [23, 24].3) We treat the interface between quark matter and thevacuum as a sharp interface which is characterized by asurface tension. We assume there is no charge localizedon the surface. (Thus we neglect any surface charge thatmight arise from the reduction of the density of states ofstrange quarks at the surface [25–28].)4) We neglect the curvature energy of a quark matter sur-face [29, 30], so we do not allow for “Swiss-cheese” mixedphases, in which the outer part of the Wigner-Seitz cellis filled with quark matter, with a cavity in the center,for which the curvature energy is crucial.5) We work at zero temperature.In our calculations we use units (cid:126) = c = (cid:15) = 1 , so α = e / (4 π ) ≈ / . II. PHENOMENOLOGICAL DESCRIPTION OFQUARK MATTER
We use the fact that in most phases of quark matterthe chemical potential for negative electric charge µ e ismuch less than the chemical potential for quark number µ . This allows us to write down a model-independentparameterization of the quark matter equation of state,expanded in powers of µ e /µ [13], p QM ( µ, µ e ) ≈ p ( µ ) − n Q ( µ ) µ e + χ Q ( µ ) µ e + . . . (1)Note that the contribution of electrons to the pressure ofquark matter is O ( µ e ) , and is neglected. This is a verygood approximation for small strange quark mass, whichcorresponds to small n Q . (For the largest value of n Q thatwe study, µ e in neutral quark matter is close to 100 MeV,and the assumption is still reasonable.) As noted in Sec. I, we assume that the interface be-tween quark matter and vacuum has a surface tension σ ,and we neglect any curvature energy.The quark density n and the electric charge density q QM (in units of the positron charge) are n = ∂p QM ∂µ , q QM = − ∂p QM ∂µ e = n Q − χ Q µ e . (2)So in uniform neutral quark matter the electron chem-ical potential is µ neutral e = n Q /χ Q . Eq. (1) is a genericparametrization if µ neutral e (cid:28) µ , which is typically thecase in three-flavor quark matter.The bag constant enters in p ( µ ) , and we will fix it byrequiring that the first-order transition between neutralquark matter and the vacuum occur at quark chemicalpotential µ crit , i.e. p ( µ crit , µ neutral e ) = 0 . Because we areassuming that the strange matter hypothesis is valid, werequire µ crit (cid:46) MeV, since at µ ≈ MeV there isa transition from vacuum to neutral nuclear matter. Inthis article we will typically use µ crit = 300 MeV . Thevalue of µ inside our quark matter lumps will always bevery close to µ crit , so we can also expand in powers of µ − µ crit , and write p QM ( µ, µ e ) ≈ n ( µ − µ crit ) + χ ( µ − µ crit ) + n Q χ Q − n Q µ e + χ Q µ e . (3)A quark matter equation of state can then be expressedin terms of 6 numbers: µ crit , the charge density n Q andcharge susceptibility χ Q evaluated at µ = µ crit , the quarknumber density n and susceptibility χ evaluated at µ = µ crit , and the surface tension σ .We will restrict ourselves to values of the surface ten-sion that are below the critical value [13] σ crit = 0 . n Q λ D χ Q = 0 . n Q √ παχ / Q , (4)where λ D is the Debye screening length in quark matter λ D = 1 (cid:113) παχ Q . (5)If the surface tension is larger than σ crit then the ener-getically favored structure at low pressure will not be astrangelet crystal, and there will be no strangelet dwarfs.Rough estimates of surface tension from the bag modelare in the range to
10 MeV / fm [31, 32], and for typicalmodels of quark matter, σ crit is of order to
10 MeV / fm [13], so it is reasonable to explore the possibility thatstrange quark matter could have a surface tension below σ crit . A. Specific equations of state
When we show numerical results we will need to vary n Q and χ Q over a range of physically reasonable values.To give a rough idea of what values are appropriate,we consider the example of non-interacting three-flavorquark matter, for which n Q and χ Q become functions of µ and the strange quark mass m s , while p is in additiona function of the bag constant B . Expanding to lowestnon-trivial order in m s , p ( µ ) = 9 µ π − B ,n Q ( µ, m s ) = m s µ π ,χ Q ( µ, m s ) = 2 µ π . (6)We emphasize that these expressions are simply meantto give a rough idea of reasonable physical values for n Q and χ Q . Our treatment does not depend on an expansionin powers of m s . To tune the transition between neutralquark matter and the vacuum so it occurs at µ = µ crit (see previous subsection), we set B so that p ( µ crit ) = n Q ( µ crit ) /χ Q ( µ crit ) .In the regions between lumps of strange matter, wewill assume that there is a degenerate electron gas, whosepressure, and charge density in units of e , are p e − ( µ e ) = 124 π (cid:18) (2 k F e − m ) k F e µ e + 3 m ln (cid:16) k F e + µ e m (cid:17)(cid:19) ,q e − ( µ e ) = − π k F e . (7)where µ e = k F e + m e . Note that at low pressures thisis more accurate than the electron gas equation of stateused in Ref. [14], where the electron mass was set to zero. III. EQUATION OF STATE OF STRANGELETCRYSTALA. Wigner-Seitz cell
Following the approach of [14], we analyze a sphericalWigner-Seitz cell of radius R cell , with a sphere of quarkmatter at the center of radius R . We use the Thomas-Fermi approximation to calculate µ e ( r ) , ∇ µ e ( r ) = − παq ( r ) , (8)where q ( r ) is the electric charge density in units of thepositron charge e , and µ e is the electrostatic potentialdivided by e .The boundary conditions are that there is no electricfield in the center of the cell (no δ -function charge there),and no electric field at the edge of the cell (the cell iselectrically neutral), dµ e dr (0) = 0 , dµ e dr ( R cell ) = 0 . (9) We also need a matching condition at the edge of thequark matter. Since we assume that no charge is localizedon the surface, we require continuity of µ e and its firstderivative (the electric field) at r = R .The value of µ inside the strange matter will be slightlydifferent from µ crit because the surface tension com-presses the droplet. To determine the value of µ , werequire the pressure discontinuity across the surface ofthe strangelet to be balanced by the surface tension: p QM ( µ, µ e ( R )) − p e − ( µ e ( R )) = 2 σR . (10)Once these equations are solved, we can obtain theequation of state of matter made of such cells. The totalenergy of a cell is E = 4 π (cid:90) R r dr (cid:16) µn ( µ e ) − µ e q QM ( µ e ) − p QM ( µ, µ e ) (cid:17) + 4 π (cid:90) R cell R r dr (cid:16) − µ e q e − ( µ e ) − p e − ( µ e ) (cid:17) + 4 πR σ , (11)The − µ e q terms in (11) come from combining − µ e q (from the relationship between energy density and pres-sure) with the electric field energy density + µ e q . Thepressure of the cell is simply the pressure of the electronsat the edge of the cell, p cell = p e − (cid:0) µ e ( R cell ) (cid:1) . (12)The total number of quarks is N = 4 π (cid:90) R r dr n ( µ, µ e ) . (13)The volume of the cell is V = (4 / πR cell3 .By varying R and R cell we generate a two-parameterfamily of strangelets. However, there is really only asingle-parameter family of physical configurations, pa-rameterized by the external pressure p cell . On each line ofconstant p cell in the ( R, R cell ) parameter space, we mustminimize the enthalpy per quark, h = E + p cell VN , (14)to find the favored value of R and R cell . We assume zerotemperature so h is also the Gibbs free energy per quark.We now have a well-defined way to obtain the equationof state of the mixed phase of quark matter, namely theenergy density ε = E/V as a function of the pressure p cell . B. Numerical solution
Inside the quark matter, the solution to the Poissonequation (8) that obeys the boundary condition at theorigin is µ e ( r ) = n Q χ Q + Arλ D sinh( rλ D ) , (15)where A will be determined by matching conditions.In the degenerate electron gas region outside thestrange matter, from (7) and (8) the Poisson equationbecomes ∇ µ e ( r ) = 4 α π ( µ e − m e ) / , (16)which must be solved numerically. For a given value of A we find from (15) the value and slope of µ e ( r ) at r = R ,and use these as initial values to propagate µ e ( r ) out to r = R cell using (16). We vary A until we obtain a solutionthat obeys the boundary condition of no electric field atthe edge of the cell. C. Low-pressure approximations
If the pressure is not too high, the strangelet crys-tal consists of large Wigner-Seitz cells ( R cell (cid:29) R ). Inthis regime one can obtain approximate analytic expres-sions for the equation of state of the crystal by assum-ing that the electrons have a roughly constant densityoutside the strangelet. We give these expressions below,and in later sections we use them to calculate mass ra-dius relations for large strangelet dwarf stars. However,we expect these approximations break down at ultra-lowpressures, when the cell size becomes so large that screen-ing cannot be ignored, and the electrons are clumpedaround the strangelets, forming atoms, rather then beingroughly uniformly distributed between the strangelets.This will happen when R cell approaches the Bohr ra-dius a = 1 / ( αm e ) , i.e. when p cell (cid:46) α Z / m e ≈ (10 − MeV ) Z / . At these ultra-low pressures oneshould use an atomic matter equation of state: we donot do this, since we expect it will only affect a verysmall surface layer of the star, without any appreciableeffect on the mass-radius relationship.The equation of state ε ( p cell ) is found by writing theenergy density ε of the cell and its pressure as a functionof the size of the cell. For now we will treat the size R and charge Z of the central strangelet as unknowns; laterwe will estimate their values.Since the pressure inside a large cell is very low the en-ergy density of the quark matter is approximately nµ crit ,so ε ≈ nµ crit R R cell3 . (17)To obtain the pressure at the edge of the cell we need toestimate the density distribution of the electrons outsidethe strangelet.
1. Constant potential approximation
The simplest approximation is to ignore screening, tak-ing the electron Fermi momentum k F e to be independentof r outside the strangelet (Sec. I of Ref. [33]). Imposingneutrality of the cell fixes the Fermi momentum of theelectrons, k F e = 9 πZ R cell3 . (18)Using (17), we obtain the equation of state ε ( p cell ) of thestrangelet crystal ε ≈ µ crit n k F e R ) πZ (19)where we use (7) to relate the electron Fermi momentumto p cell .Because the constant potential approximation gives afairly simple expression we can use it to understand howthe strangelet crystal EoS depends on the parameters ofthe quark matter EoS, and hence how the M ( R ) curvefor strangelet dwarf stars depends on those parameters.Note that in (19) the dependence of the energy density onthe pressure is via a universal and monotonically increas-ing function k F e ( p ) ; dependence on the quark matter pa-rameters enters via the factor that multiplies this func-tion. To make the dependence on quark matter parame-ters explicit we use results for R and Z from Sec. III C 3below, and rewrite (19) for the EoS of the strangelet crys-tal as ε ( p cell ) ∼ S (cid:0) k F e ( p cell ) (cid:1) ,S = µ crit n π n Q ξ ( x (¯ σ )) , (20)where all dependence on the quark matter parameterscomes through the prefactor S , which has units of en-ergy. S can be explicitly obtained using (25), (27), and(28) for the ξ function. One could informally think of S as a “softness” parameter of the strangelet crystal EoS:as S increases, the pressure becomes a more slowly-risingfunction of energy density. We expect that softer equa-tions of state will yield smaller stars with lower maximummasses. In Table I we give the value of S for a range ofvalues of the parameters of the underlying quark matterEoS.At low enough pressures, the electrons become nonrel-ativistic. Then p cell ≈ k F e / (15 π m e ) , and (19) simplifiesto an analytic expression for the equation of state, ε NR ≈ R Z (cid:16) π m e (cid:17) / nµ crit p / (21)This is a reasonable approximation when k F e (cid:46) m e , i.e.when p cell (cid:46) m e / (24 π ) ≈ . . However, as wewill see below, at the very lowest pressures the constantpotential approximation becomes inaccurate.
2. Coulomb potential approximation
We can improve on the constant potential approxima-tion by including the Coulomb energy of the electrons inthe calculation of the pressure. The equation of state isstill given by (19), but now the relationship between p cell and k F e is modified by the addition of a Coulomb energyterm (Ref. [33], (5)), yielding p cell = p e − − α (cid:16) Z π (cid:17) / k F e . (22)Unlike the constant potential approximation, theCoulomb potential approximation gives an energy den-sity that goes to a non-zero value at zero pressure, ε Coul (0) = µ crit n Z ( αm e R ) π . (23)Comparing with (17) we see that this corresponds to theenergy of cells with size of order / ( αm e ) ∼ − m .This is the energy density of a lattice of zero-pressureatomic matter with strangelets in place of nuclei, whichis a reasonable guess for the low-pressure configuration ofstrangelets. We will therefore use the Coulomb approxi-mation as the low-pressure extension of our equation ofstate. As we will see, this leads to a “planet” branchin the mass-radius relation for configurations of strangematter.
3. Radius and charge of strangelet at low pressure
The low-pressure approximation expressions givenabove depend on the size R and charge Z of the strangeletat the center of a large cell. This is approximately an iso-lated strangelet, whose radius can be calculated by min-imizing the isolated strangelet free energy given in eqn(25) of Ref. [13], ∆ g ( x ) = − x − tanh xx + 3¯ σx , (24)where x is the radius of the strangelet in units of λ D , and ¯ σ = σ παn Q λ D . (25)So the strangelet radius R as a function of the parametersof the quark matter equation of state is R = x λ D , where d ∆ gdx ( x ) = 0 . (26)We are interested in values of ¯ σ up to 0.13, since forhigher surface tension the strangelet crystal is no longerstable [13]. An approximate expression for the solutionto (26), accurate to about 0.2% for ¯ σ (cid:46) . , is x approx0 = (cid:18) σ (cid:19) / + 2 .
174 ¯ σ − .
982 ¯ σ , (27) where the first term is the leading-order analytic expres-sion for x in the limit of small ¯ σ .The charge Z of the central strangelet is given byeqn. (17) of Ref. [13], which can be written Z ≈ πR n Q ξ ( R/λ D ) ,ξ ( x ) ≡ x ( x − tanh x ) , (28)where ξ is a correction for the effects of screening insidethe quark matter; it is an even function with ξ (0) = 1 . IV. NUMERICAL RESULTSA. Range of parameters studied
Our assumption that the strange matter hypothesisis valid requires that µ crit must be less than the quarkchemical potential of nuclear matter, about 310 MeV,so we fix µ crit = 300 MeV . The value of µ inside ourstrange matter lumps will always be within a few MeVof µ crit , because if the surface tension is small enough tofavor the strangelet crystal it will not cause significantcompression.We will perform calculations for λ D = 4 .
82 fm and λ D = 6 .
82 fm , corresponding to χ Q ≈ . µ (appro-priate for unpaired quark matter (6)) and χ Q ≈ . µ (appropriate for 2SC quark matter [13]).Typical values of n Q will be around . µ crit m s (6),and a reasonable range would correspond to varying m s over its physically plausible range, from about 100 to 300MeV. (To have strange matter in the star, m s must beless than µ crit .) In this paper we use n Q = 0 . , . ,and .
124 fm − , which would correspond to m s = 150 ,200, and 250 MeV in (6).There is another widely-discussed phase of quark mat-ter, the color-flavor locked (CFL) phase, but it is a de-generate case where n Q = χ Q = 0 . CFL strangelets havea surface charge, but it does not arise from the mecha-nism studied here, Debye screening, and has a differentdependence on the size of the strangelet [26]. We hopeto study CFL strangelet matter in a separate work. B. Testing approximations to the equation of state
In Fig. 1 we show the equation of state for thestrangelet crystal, for critical quark chemical poten-tial µ crit = 300 MeV , quark matter screening distance λ D = 6 .
82 fm , quark charge density parameter n Q =0 . − , and quark matter surface tension σ =1 . − . The dots were obtained numerically fol-lowing the procedure of Sec. III B. The solid line is theCoulomb-potential approximation (Sec. III C 2). On thisplot the constant potential approximation (Sec. III C 1) λ D n Q σ crit Softness prefactor S (MeV) at(fm) (fm − ) (MeV fm − ) σ = 0 . σ = 1 . σ = 3 . σ = 10 . λ D and n Q , specify the quark matter equation of state (3) (via (5)). The third column gives the maximum surface tension for whicha strangelet crystal will occur (4). The last four columns give the softness prefactor S for different values of the surface tension σ (given in MeV fm − ) of the interface between quark matter and vacuum. pressure (MeV ) e n e r gy d e n s it y ( M e V ) C o u l o m b n o n -r e l a ti v i s ti c FIG. 1: Equation of state of the mixed phase (strangelet crys-tal) for strange matter with µ crit = 300 MeV , λ D = 6 .
82 fm , n Q = 0 . − , σ = 1 . − . The dots were ob-tained numerically following the procedure of Sec. III B. Thesolid line is the Coulomb-potential approximation (Sec. III C 2).The dashed line is the non-relativistic electron (ultra-low pres-sure) limit (21). Above p ≈ , uniform quark matterbecomes favored over the mixed phase. line would be indistinguishable from the Coulomb-potential line, so we do not show it. The dot-dashed lineis the non-relativistic electron (ultra-low pressure) limit(21) of the constant potential approximation. Above p ≈ , uniform quark matter becomes favoredover the mixed phase. On this very expanded logarith-mic scale, the Coulomb approximation appears reason-ably accurate up to pressures of order 1 MeV.To achieve more discrimination between the differentapproximations, we show in Fig. 2 a magnified versionof the low-pressure end of the plot in Fig. 1, where wehave divided out the non-relativistic scaling of the en-ergy density, ε ∼ p / . We can see that, down to thelowest pressures for which we can perform the numeri-cal Wigner-Seitz calculation of the equation of state, the pressure (MeV ) e n e r gy d e n s it y / p . Coulombapproxconstantapproxnon-relativistic
FIG. 2: Equation of state of the mixed phase for the same pa-rameters as in Fig. 1, zoomed in on the low pressure region, andwith the energy density divided by p . . The dots were obtainednumerically following the procedure of Sec. III B. The Coulomb-potential approximation (Sec. III C 2) is the most accurate, fol-lowed by the constant-potential approximation (Sec. III C 1),and then the non-relativistic electron approximation (21). Coulomb approximation gives the most accurate semi-analytic approximation, although the constant potentialapproximation is accurate to within about 10%.We then have to decide which approximation to usefor lower pressures, where numerical calculations are notavailable. In the low-pressure limit, the Coulomb approx-imation to ε ( p ) tends to a fixed value, while the constantand nonrelativistic approximations to ε ( p ) tend to zeroas p / . So in Fig. 2 the Coulomb approximation willdiverge at p (cid:28) − MeV , while the constant and non-relativistic approximations will tend to the same constantvalue. As discussed in Sec. III C 2, it seems reasonable toexpect that at the lowest pressures there will be a crystalof “strange atoms”, each consisting of electrons bound toa strangelet, and the Coulomb approximation gives a rea-sonable estimate of the energy density of such matter, so R (km) M ( s o l a r) Compactbranch (strange stars)
Diffusebranch (strangeletdwarfs)(strangeletplanets) stablestability uncertain
FIG. 3: The full mass-radius curve for stars made of quarkmatter with the equation of state plotted in Fig. 1, using theCoulomb approximation (22) to extrapolate to lower pressures.The compact branch contains strange stars with a strangeletcrystal crust. The diffuse branch contains stars consisting en-tirely of strangelet crystal matter. Solid lines represent con-figurations that are stable; stability of the other branches isdiscussed in the text. at low pressure we will use the Coulomb approximation.
C. Mass-radius relation of strange stars
In Fig. 3 we show the full mass-radius curve for starsmade of quark matter with the equation of state plottedin Fig. 1. The compact branch contains strange starswith a strangelet crystal crust. The diffuse branch con-tains stars consisting entirely of strangelet crystal mat-ter. It includes two segments: the lighter one is planetsof dilute strange matter whose the mass increases withradius. This joins to the strangelet dwarf branch wherethe mass decreases with radius as the strangelet crystalis compressed by the pressure due to gravity. We usethe numerically calculated equation of state (Sec. III)except that at very low pressure (the planetary branch)the Wigner-Seitz cells become so large that our numer-ical methods break down, so as discussed in Sec. III C 2we use the Coulomb approximation (22) to extrapolatedown to zero pressure.Fig. 3 shows the whole M ( R ) curve, not all of whichcorresponds to stable configurations. The usual stabilitycriterion for stars [34] is that one radial mode becomeseither stable or unstable at each extremum in the M ( R ) function. A stable mode becomes unstable at each ex-tremum where the curve bends counterclockwise as thecentral density increases; a stable mode becomes un-stable at each extremum where the curve bends clock-wise as the central density increases. However, Glen-denning et.al. [35] report that at some extrema there isno change in stability: the squared frequency of one of R (km) M ( s o l a r) σ =1.0 σ =3.0 µ crit =300 MeV, λ D =6.82 fm, n Q =0.079 fm Numeric+CoulombCoulomb approxConstant approx
FIG. 4: Mass-radius relation for strangelet dwarfs made ofstrangelet crystal matter, comparing different approximationsto the equation of state. Upper (blue) curves are for thesame parameters as in Figs. 1 and 2. Lower (red) curvesare for a larger surface tension, σ = 3 MeVfm − . Thedots were obtained using the full numerical equation of state(Sec. III B). The solid lines use the Coulomb-potential approx-imation (Sec. III C 2), and the dashed lines use the constant-potential approximation (19). R (km) M ( s o l a r) µ crit =300 MeV, λ D =4.82 fm, σ =0.3 MeV fm -2 n Q =0.124 fm -3 n Q =0.045 fm -3 Numeric+CoulombCoulomb approx FIG. 5: Mass-radius relation for strangelet dwarfs made ofstrangelet crystal matter, comparing different approximationsto the equation of state. the fundamental radial modes may touch zero, but notchange sign. We defer a detailed study of the stability ofradial modes of strange stars to future work, and in Fig. 3we show as “stable” (solid curves) the parts of the M ( R ) curve that both Ref. [34] and Ref. [35] agree are stable.We note that Ref. [35] is a study of stars that have a coreof uniform strange matter surrounded by a crust of nu-clear matter: these are similar to the configurations alongthe dashed part of the mass-radius curve in Fig. 3, wherewe have a core of uniform strange matter surrounded bya crust of strangelets, with a density discontinuity at theboundary. If Ref. [35]’s stability argument is correct andapplicable to our stars, then some of these configurationsmay also be stable. In the remainder of this paper wewill focus on the strangelet dwarf branch, which consistsof a simple crystal of stranglets with no uniform core, sothere is no controversy about the appropriate stabilitycriterion. D. Mass-radius relation of strangelet dwarfs
To investigate the sensitivity of the masses and radii ofstrangelet dwarfs to the parameters of the quark matterequation of state, we show in Fig. 4 and 5 the strangeletdwarf part of the mass-radius curve, excluding the com-pact and planetary branches, for various values of thequark matter parameters.In Fig. 4 we explore the effects of varying the sur-face tension, and we compare the different approxima-tions to the equation of state. The upper curves arefor the same equation of state as was shown in Fig. 1and 2; the lower curves use a larger surface tension, σ = 3 MeVfm − . In both cases the solid curves are ob-tained from the Coulomb-potential approximation to theequation of state, and the dashed lines are obtained fromthe constant-potential approximation. The dots use theequation of state that is obtained numerically followingthe procedure of Sec. III B, except that at very low pres-sures, where the numerical calculation becomes too diffi-cult, the Coulomb approximation is used.We see that, as one might have expected from Fig. 1,using the Coulomb approximation over the entire pres-sure range of the mixed phase yields reasonably accurateresults. However, as noted in Sec. III C 1, the constantpotential approximation is still useful for gaining an un-derstanding of how the M ( R ) curve for strangelet dwarfsdepends on the parameters of the EoS, because in therange of pressures that is important for strangelet dwarfsit gives a good indication of the M ( R ) curve. (At ultra-low pressures, relevant for the strange planet branch, thisis no longer the case: one has to use the Coulomb approx-imation instead.) As discussed in Sec. III C 1, the con-stant potential approximation to the EoS can be writtenin terms of a “softness prefactor” S (20). To understandhow the M ( R ) curve in Fig. 4 changes with σ , note that x (¯ σ ) is a monotonically increasing function and ξ ( x ) is a monotonically decreasing function, so as the surfacetension σ increases at fixed values of the other parame-ters, the softness prefactor S of the strangelet crystal EoSincreases (one can see this in Table I). Since the EoS isbecoming softer, the M ( R ) curve moves down and to theleft, giving smaller stars with a lower maximum mass.In Fig. 5 we explore the effects of varying the chargedensity parameter n Q in (3) while keeping the other pa-rameters constant. As in Fig. 4, solid lines are for theCoulomb approximation to the equation of state, dots are for the numerically calculated equation of state usingthe Coulomb approximation to extrapolate to the lowestpressures. We see that increasing n Q yields heavier, largerstrangelet dwarf stars. Again, this can be understood interms of the constant potential approximation and itssoftness prefactor S (20). As n Q increases, it causes S todecrease through two effects. Firstly via the explicit fac-tor of n Q in the denominator of (20), and secondly via therelationship (25) between σ and ¯ σ . The sensitivity of S to changes in n Q can be seen in Table I: for the two valuesof n Q used in Fig. 5 the values of S are near the extremesof its range in the parameter set we studied: S ≈ and S ≈ for n Q = 0 . and n Q = 0 . respectively.Consequently, the M ( R ) curve for n Q = 0 . is char-acteristic of a soft equation of state, with low radius at agiven mass and a low maximum mass, whereas the M ( R ) curve for n Q = 0 . is characteristic of a hard equationof state, with large radius at a given mass and a highmaximum mass. V. DISCUSSION
We have shown that, if the strange matter hypothesis iscorrect and the surface tension of the interface betweenstrange matter and the vacuum is less than a criticalvalue (4), there is at least one additional stable branch inthe mass-radius relation for strange stars, correspondingto large diffuse objects that we call “strangelet dwarfs”,consisting of a crystal of strangelets in a sea of electrons.This is easily understood, since if σ < σ crit then uniformstrange matter is unstable at zero pressure, and under-goes charge separation to a crystal of positively-chargedstrangelets surrounded by electrons, just as normal mat-ter at zero pressure is a mixed phase consisting of dropletsof nuclear matter surrounded by electrons. Strangeletdwarfs are then the strange matter equivalent of whitedwarfs.We emphasize that in this low-surface-tension scenario,strange matter is not self bound. Like nuclear matter, itis only bound by gravitational forces. Every strange starwill have a strangelet crystal crust, and strangelet dwarfsare those strange stars that are “all crust”.The natural production mechanism by whichstrangelet dwarfs might be produced is a collisionbetween a strange star and another compact object. Insuch collisions, up to . M (cid:12) may be ejected [36], whichis in the mass range we are predicting for strangeletdwarfs. There are two ways a collision could producestrangelet dwarfs. Firstly, part of the crust of thestrange star might be ejected to become a isolatedobject, which would be a strangelet dwarf. Secondly,if a sufficiently light piece of the uniform quark mattercore were ejected in the collision, it would be unableto exist on the compact branch, and would evaporateinto a configuration on the diffuse branch. For example,for the equation of state studied in Fig. 3, the lightestcompact configuration of strange matter is . M (cid:12) .A lighter piece of strange matter could only exist on thediffuse branch, and would spontaneously evaporate tobecome a strangelet dwarf. Strangelet dwarfs producedby these mechanisms could then bind gravitationally, toform heavier strangelet dwarfs.It should be noted that our proposed mechanism forthe production of strangelet dwarfs is also a mecha-nism for creating a diffuse cosmic flux of strangelets(“strangelet pollution”), which might be expected to con-vert all neutron stars to strange stars [37]. Although ob-servations of glitches and magnetar oscillations [38] seemconsistent with some compact stars having nuclear mat-ter crusts, there remains some uncertainty. Crystallinephases of quark matter could allow strange stars to glitch[39], and in our low-surface-tension scenario strange starshave crusts that could be hundreds of meters thick [14].A cosmic flux of strangelets may seem unlikely but untilit is ruled out experimentally (as may happen soon fromthe AMS experiment [40]) it remains useful to analyzethe full observational consequences of the strange matterhypothesis.Our analysis assumes that at any given pressure thestrangelet crystal consists of the most energetically fa-vorable strangelet configuration (in terms of strangeletsize and charge and cell size). However, other configu-rations will in general be metastable with long lifetimes.If one compresses a piece of strangelet crystal then thecharge of the strangelets can readily change via absorp-tion or emission of electrons, but it is very difficult forthe quark matter to rearrange itself in to strangelets ofthe now-energetically-favored size: it is more likely thatthe strangelets will stay the same size and the radial den-sity profile of the electrons will change. The sizes of thestrangelets will be determined more by the history of theobject than by the pressure. Taking this point further, itis quite possible to have a crystal consisting of a mixtureof strangelets and ordinary nuclei, held apart by theirelectrostatic repulsion but also bound together in to acrystal by the degenerate electron gas that neutralizesthem, forming a hybrid strangelet/white dwarf star.Detection of strangelet dwarfs requires an observationmethod that can find non-luminous objects with typical masses of − to − M (cid:12) and radii in the range to km. An example is gravitational microlensing sur-veys, such as those conducted by the Microlensing Obser-vations in Astrophysics (MOA) and the Optical Gravita-tional Lensing Experiments (OGLE) groups, which lookfor lensing events in the galactic bulge, and are capable ofdetecting Jupiter-mass objects. It is intriguing that suchsurveys now report the existence of an abundant pop-ulation of unbound distant planetary masses, suggest-ing that such objects may be twice as common as mainsequence stars [8]. Although models of planet forma-tion indicate that mechanisms exist for unbinding planetsthrough disk instabilities and planet interactions [41], wesuggest that a possible alternative is formation of strangedwarfs from matter ejected in strange star mergers. Onewould expect that sometimes a strangelet dwarf producedin a merger might be unable to escape the gravitationalfield of the remaining compact object, and this wouldexplain the presence of dense planet-mass objects in thevicinity of compact stars. An example is the millisecondpulsar PSR J1719-1438, which has a Jupiter-mass com-panion whose inferred central density ( ρ >
23 g cm − ) isfar in excess of what is expected in a planet [42]. Weexpect that in the near future further light will be caston this question, as microlensing surveys help us betterunderstand the distribution of planetary mass compactobjects and as strategies are devised to provide informa-tion about both mass and radius. Acknowledgments
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