SQM stars around pulsar PSR B1257+12
Marek Kutschera, Joanna Jałocha, Sebastian Kubis, Łukasz Bratek
aa r X i v : . [ a s t r o - ph . E P ] O c t SQM stars around pulsar PSR B1257+12
Marek Kutschera ∗ Institute of Physics, Jagiellonian University, Reymonta 4, PL-30059 Krak´ow, Poland.
Joanna Ja locha, † Sebastian Kubis, and Lukasz Bratek
Institute of Nuclear Physics, Polish Academy of Sciences, Radzikowskego 152, PL-31342 Krak´ow, Poland.
Abstract:
Following Wolszczan’s landmark discovery of planets in orbit around pulsar PSR B1257+12 in 1991,over 300 planets in more than 200 planetary systems have been found. Therefore, the meaning of Wolszczan’sdiscovery cannot be overestimated. In this paper we aim to convince the reader that the objects accompanyingpulsar PSR B1257+12 are more exotic than thought so far. They might not be ordinary planets but dwarf strangequark stars, whereas the pulsar might be a quark star with standard mass, not a neutron star. If this was thecase, it would indicate that strange quark matter is the ground state of matter.
I. INTRODUCTION
The millisecond pulsar PSR B1257+12 with the esti-mated age of 800 millions years, is located at a distanceof 300 parsecs from the Earth. The tiny pulsar is beingorbited by three bodies with masses 0 . .
9, and 4 . .
19, 0 .
36 and 0 . AU [1], [2]. For thatreason, the bodies are considered to be planets. It wasclear from the start, that the planetary system aroundPSR B1257+12 was unusual. Before Wolszczan’s discov-ery, the existence of planets around neutron stars hadnot been considered seriously. If the planets were to beremnants of a planetary system existing around the pro-genitor star of PSR B1257+12, then it would be highlyimprobable for these planets to have survived the super-nova explosion accompanying the neutron star’s birth,since then compact bodies acquire velocities high enoughto escape from the binding potential of the gravitationalwell of the surrounding matter. These planets might alsohad been formed already after the supernova explosionand the neutron star’s formation. In particular, models ofplanetary system formation around pulsar accreting mat-ter from its companion have been suggested [3]. However,this scenario seems also improbable because of strong ra-diation in the pulsar’s vicinity [4]. Therefore, anotherhypothesis seems worth considering. The objects in or-bit around PSR B1257+12 may be just miniature quarkstars. II. STRANGE QUARK MATTER AS ABUILDING MATERIAL OF MINI-STARS
A quark star is a hypothetical object consisting ofstrange quark matter (SQM). SQM is a mixture of quasi-free quarks u , d , s and electrons, all of which formrelativistic fermi gas. If SQM is the ground state of ∗ Also at Institute of Nuclear Physics, Polish Academy of Sciences,Radzikowskego 152, PL-31342 Krak´ow, Poland. † [email protected] strong interactions, then one could expect strange starsto form in space. For SQM to be the ground state,one needs the minimum of energy per baryon to be notgreater than the energy per baryon for the strongestbound nucleus in nature – the nucleus of iron-56. Hence,( ρ E /n B ) SQM ≤ E (Fe ) /
56 = 930 . ρ E is the energy density and n B the baryondensity. The possibility of existence of such objects wasput forward first in [5]. The easiest way to obtain a phe-nomenological equation of state for SQM, is to considerthe following simplified model [6]. Suppose that quarksform ultra-relativistic and electrically neutral quantumgas of non-interacting fermions in the β -equilibrium attemperature T = 0[ K ]. In this case, the pressure p andthe total energy density ρ E are p = 34 ( ρ E − B ) , ρ E = 9 π / ~ cn / B + B, where n B is the baryon density and B is the bag constant, B = 50 ÷
200 [MeV / fm ]. These equations are valid whenthe Fermi energy is much higher than the rest energyof the heaviest quark (in our numerical calculations weincluded also the strange quark mass, but the full formof the resulting equations is not important here).The maximal mass and the corresponding size of aquark star are close to those for a neutron star. Withthe assumed example values B = 70 [MeV / fm ] and thestrange quark mass m s = 150 [MeV /c ], SQM wouldbe the ground state for strong interactions. Then, themaximal mass of a quark star made of SQM wouldbe ≈ . M ⊙ with the corresponding areal radius ofabout 9 . ? ], whereas a neutron star with theUV14+TNI equation of state [7] attains M max = 1 . M ⊙ at R = 9 . . M ⊙ [8]. Unlike for neutron stars, therewould be no such a limit for SQM stars bound mainlyby quark forces. Consequently, there are possible SQMstars of arbitrary low mass and radius. The mass-radiusdiagram for SQM stars is also different, see figure 1(b). SS NS M / M o n B [fm -3 ]
10 100 10000,00,20,40,60,81,01,21,41,61,82,0
SS NS M / M o R[km] (a) (b)
FIG. 1:
Masses of quark stars as a function of central baryon density compared with a similar relation for a neutron starwith a crust shell (a), and the mass-areal radius relation for SQM stars and similar relation for a neutron star with crustshell (b). Both cases assume bag constant B = 70[MeV /fm ], the strange quark’s mass 150[MeV /c ], and the UV14+TNIequation of state for the neutron star. Low mass quark stars have small radii, while the radii ofneutron stars grow with decreasing mass.Confirmation of the presence of quark stars in spacewould indicate that SQM is the ground state for stronginteractions. It is thus natural that astrophysicists havebeen intensively searching for such objects [9],[10]. Un-fortunately, this endeavor is difficult, since quark andneutron stars are hardly distinguishable from each otherin the range of masses close to 1 . M ⊙ , typical of pulsars,in which range the mass-radius relations are similar forboth kinds of stars. On the other hand, it may turn outthat quark stars have been already found. III. SQM AND WOLSZCZAN’S SYSTEMAROUND PSR B1257+12
The pulsar PSR B1257+12 is being orbited by threebodies of masses 0 . . . B = 70 [MeV / fm ] and m s = 150 [MeV /c ]. Since for such small stars the en-ergy density must tend to the energy density on theirsurface ρ ext , and since 2 GM ( r ) / ( rc ) ≪
1, the relationbetween the total mass M and the physical radius R of a low mass SQM star is nothing but M = π ρ ext R ]strange quark mass: 150 [MeV/c ] t o t a l m a ss [ S un m a ss e s ] areal radius [km] FIG. 2:
Comparison of Newtonian M − R relation ( dash-dot line ) and General-Relativistic M − R relation ( solid line ) for a strange quarkstar. The assumed strange quark mass is150 [MeV] and the bag constant is70 (cid:2) MeV · fm − (cid:3) . which, for the assumed parameters, gives the follow-ing mass-radius relation in Newtonian limit M R − ≈ . × − [M ⊙ / km ][ ? ] (comparison between thisrelation and that obtained in the framework of GeneralRelativity is illustrated in figure 2). Therefore, the stars’radii are of about 40, 217, and 225 meters, respectively.The central pulsar of mass 1 . M ⊙ has the areal radiusof about 9 .
75 [km]. Figure 3 shows energy density andpressure as functions of the areal radius for two exampleSQM stars.What could have caused the mini-quark stars to findthemselves in the pulsar vicinity? The first possibilityis detachment of quark matter from the central pulsarduring its formation phase, when the ”boiling” strangematter core could eject some fragments, so that they start P [ e r g / c m ] R[km] p [ e r g / c m ] R[m] a) b)
FIG. 3:
Cross section through a quark star of mass 1 . M ⊙ (a) and 3 . M ⊕ (b). The assumed values for the bag constant is B = 70MeV / fm , and the strange quark mass m s = 150MeV /c ). to orbit around the strange star. However, this mecha-nism does not explain why the orbits of the three planetsare nearly circular and almost coplanar (at least, for twoof them) [14]. These features make the planetary diskorigin of the pulsar planets very likely. The planetarydisk stability and the planet formation present a lot ofunknowns and require ad hoc assumptions [15].Here we propose another scenario. The present rota-tion period of pulsar B1257+12 is of about P = 6 . P = 1 . × − [12].The pulsar has the feature of a pulsar spun up in a binarysystem – it is a millisecond pulsar and, at the same time,it has comparably small deceleration rate and weak mag-netic field. It seems therefore probable the pulsar mighthave had a companion from which it had been accretingmatter. In this case it is difficult to estimate the age andthe initial rotation velocity of a compact object. How-ever, during the accretion phase, the compact object, byassumption being a quark star, could have spun up to the velocities necessary to detach matter from its surfacein the equatorial region. If only the accretion were suf-ficient to accelerate the star to the Keplerian frequency,which is about 1 KHz for strange star [13], then severalbubbles of quark matter could start to orbit the centralstar. In a natural way their trajectories would be cir-cular and coplanar. Being pieces of strange matter theywould be completely resistant to the unfriendly pulsarenvironment heated by pulsar wind or accretion whichmay evaporate the normal planet. IV. SUMMARY