From strangelets to strange stars: A unified description
The conventionally separated treatments for strangelets and strange stars are now unified with a more comprehensive theoretical description for objects ranging from strangelets to strange stars. After constraining the model parameter according to the Witten-Bodmer hypothesis and observational mass-radius probability distribution of pulsars, we investigate the properties of this kind of objects. It is found that the energy per baryon decreases monotonically with increasing baryon number and reaches its minimum at the maximum baryon number, corresponding to the most massive strange star. Due to the quark depletion, an electric potential well is formed on the surface of the quark part. For a rotational bare strange star, a magnetic field with the typical strength in pulsars is generated.
Keywords:strangelets strange stars strange quark matter unified description Witten-Bodmer hypothesis
It was pointed out long ago that strange quark matter (SQM) might be the ground state of strongly interacting matter, which is nowadays called the Witten-Bodmer hypothesis Bodmer1971_PRD4-1601 (); Witten1984_PRD30-272 (). If true, there should exist stable objects of SQM with the baryon number ranging from a few to . Customarily, small SQM nuggets with are often referred to as strangelets Berger1987_PRC35-213 (); Madsen1993_PRL70-391 (); Farhi1984_PRD30-2379 (); Greiner1987_PRL58-1825 (); Gilson1993_PRL71-332 (); He1996_PRC53-1903 (); Wen2007_JPG34-1697 (); Xia2014_SCPMA57-1304 (), or slets Peng2006_PLB633-314 (), while stars consisting of SQM are called strange (quark) stars Alcock1986_ApJ310-261 (); Weber2005_PPNP54-193 (); Itoh1970_PTP44-291 (); Haensel1986_AA160-121 (); Perez-Garcia2010_PRL105-141101 (); Herzog2011_PRD84-083002 (); Dexheimer2013_EPJC73-2569 (); Chang2013_SCPMA56-1730 (); Xia2014_PRD89-105027 (); Xu2015_PRD92-025025 (), being possible candidates for pulsars.
Lumps of SQM are expected to be produced in the collision of binary compact stars containing SQM Madsen2005_PRD71-014026 (); Bauswein2009_PRL103-011101 (). Further collisions among those lumps may create slets, nuclearites Rujula1984_Nature312-734 (); Lowder1991_NPB24-177 (), meteorlike compact ultradense objects Rafelski2013_PRL110-111102 () etc., and some of them may eventually make their way to our Earth Monreal2007_JHEP02-077 (). Due to the special characteristics of these objects such as the lower charge-to-mass ratio Sandweiss2004_JPG30-S51 (); Han2009_PRL103-092302 (), the larger mass Herrin2006_PRD73-043511 (), the highly ionizing tracks in the interstellar hydrogen cloud (e.g., pulsar scintillations) Pcuteerez-Garccuteia2013_PLB727-357 (), and the characteristic gamma rays through heavy ion activation Isaac1998_PRL81-2416 (), there are possibilities to observe them. However, despite decades of efforts, no compelling evidence for the existence of stable SQM is found (for reviews, see, e.g., Refs. Klingenberg1999_JPG25-R273 (); Finch2006_JPG32-S251 ()).
This case is due to the extreme complexity of an SQM system which involves all the fundamental interactions, i.e., the strong, weak, electromagnetic, and gravitational interactions. In the conventional theoretical treatments, significantly different simplifications were adopted for slets and strange stars. For a slet, electrons were ignored since the Compton wavelength is much larger than the size of the quark part Madsen1999_LNP516-162 (), and quarks were assumed to be uniformly distributed. For strange stars, gravity has to be considered. The normal way is to first get an equation of state of SQM by assuming the local charge neutrality, and then obtain the mass-radius (-) relation by solving the Tolman-Oppenheimer-Volkov equations.
However, according to recent studies, effects such as the charge screening, electron-positron pair creation, and nonzero charge densities in strange stars have important implications on the properties of SQM. For example, taking into account the electrostatic effects, Alford et al. Alford2006_PRD73-114016 () found that, for a small enough surface tension, large slets are unstable to fragmentation and strange star surfaces fragment into a crystalline crust made of slets and electrons. For quark-hadron phase transition, the finite-size effect turns out to be very important Voskresensky2002_PLB541-93 (); Tatsumi2003_NPA718-359 (); Voskresensky2003_NPA723-291 (); Endo2005_NPA749-333 (). It was shown that the geometrical structures may be destabilized by the charge screening effect Maruyama2007_PRD76-123015 (). Due to the electron-positron pair creation, an upper bound on the net charge of slets or strange stars was found Madsen2008_PRL100-151102 (). The local charge neutrality in compact stars is also in question Rotondo2011_PLB701-667 (). In the case of a neutron star, an overcritical electric field was found in the transitional region from the core to the crust Belvedere2012_NPA883-1 (). For a bare strange star, an electric dipole layer may be formed on the surface and results in an electric field of Alcock1986_ApJ310-261 (). Due to the presence of a critical electric field, the electron-positron production may be induced and results in some astrophysical observables Ruffini2010_PR487-1 (). The mass and radius of a strange star are increased by and , respectively, if the star possesses a net charge on the surface Negreiros2009_PRD80-083006 ().
Meanwhile, the possibility of pulsars being strange stars may give us an insight into the properties of SQM. Up till now, around 2,500 pulsars have been observed and among them about 70 pulsars’ masses were measured Manchester2005_AJ129-1993 (); Manchester2015 (). At the same time, more than 10 pulsars provide us the - probability distributions with photospheric radius expansion bursts as well as quiescent low-mass X-ray binaries Lattimer2012_ARNPS62-485 (); Steiner2010_ApJ722-33 (); Guillot2013_ApJ772-7 (); Lattimer2014_EPJA50-40 (); Li2015_ApJ798-56 (). If SQM is absolutely stable, those pulsars may be strange stars Page2006_ARNPS56-327 (), then the properties of SQM can be constrained with the - relations.
In the present paper, we study the SQM system ranging from slets to strange stars in a unified description. After constraining the only model parameter, the bag constant , according to the Witten-Bodmer hypothesis and the observational - probability distribution of pulsars, it is found that the ratio of charge to baryon number of a slet is different from previous findings, while the size is significantly smaller than that of a nucleus with the same mass number. In addition, rotation of a bare strange star generates a strong magnetic field with the typical strength in pulsars. This paper is organized as follows. In Section 2, the unified description for SQM objects is presented, where the effects of gravity and electrostatic interactions are treated on the macroscopic scale while the strong and weak interactions are considered locally. Based on this description, the properties of SQM objects ranging from slets to strange stars are investigated in Section 3. A summary is presented in Section 4.
2 Theoretical framework
The internal structure of a spherically symmetric, charged, and static object should fulfill the thermodynamic equilibrium condition, which can be obtained by minimizing the energy of the system for given total particle number and entropy. We consider the gravity and electrostatic interactions on the macroscopic scale. The metric for the SQM sphere reads
where , , and are the standard spherical coordinates with the metric elements satisfying
Here we use the natural system of units, with and being the gravitational and fine-structure constants. The total mass, particle number, and entropy are obtained with
Then the total charge is given by with , , and . Based on the Thomas-Fermi approximation, the pressure , energy density , particle number density , and entropy density are given locally by incorporating both the strong and weak interactions.
By minimizing the mass with respect to the particle distribution and entropy distribution at the fixed total particle number and entropy , we immediately have
with and being the chemical potential and temperature.
For the local properties of SQM, we adopt the bag model and consider only zero temperature, where the thermodynamic potential density is given by
in the ideal Fermi-gas approximation. To reach the lowest energy, SQM undergoes weak reactions and reach the chemical equilibrium
Then the internal structure of an SQM sphere can be determined by solving Eq. (7). Since electrons are not confined by the vacuum pressure, an atom-like structure of the SQM system is formed, i.e., a positively charged SQM core with a cloud of electrons surrounding it.
The quark-vacuum interface on the surface of the SQM core needs to be treated with special care. We consider the number of depleted quarks on the interface by adopting the multiple reflection expansion (MRE) method Berger1987_PRC35-213 (); Madsen1993_PRL70-391 ()
Here is the momentum of quark flavor (). Its upper bound corresponds to the Fermi momentum on the surface with being the radius of the SQM core. Note that Eq. (11) only gives the average number of depleted quarks, while for smaller systems shell corrections may be important Madsen1994_PRD50-3328 (). Then the energy contribution and pressure are given by
Under the influence of gravity, the energy contribution to the mass is .
The quark-vacuum interface is obtained when the pressure of quarks is in balance with the vacuum pressure, i.e.,
Then we have the total quark number , mass , and charge of the core .
3 Results and discussions
For a given core radius , the structure of an SQM sphere is determined by solving the differential equation (7) under the boundary conditions, i.e., , , and Eq. (14). To illustrate our results, we present a colored contour plot in Fig. 1 for an SQM sphere with the core radius . A rich charge profile on the surface is found. At the region, there exists an electron cloud which neutralizes the positively charged core and expands by . Respectively, the SQM sphere contains -quarks, -quarks, -quarks, and electrons, which gives the total core charge and mass MeV. Due to the quark depletion on the core surface (, , and ), there exists a surface charge , while the corresponding mass modification is MeV.
Note that the current masses of quarks and the electron mass are taken as MeV, MeV, MeV, and MeV Olive2014_CPC38-090001 (), leaving only the bag constant undetermined. For SQM to stably exist at zero external pressure, the bag constant should meet the requirement of the Witten-Bodmer hypothesis, which gives MeV. We take the three typical values , 152, and 159 MeV. It is worth mentioning that if exceeds the upper bound, SQM is unstable and may only exist at the core of a compact star, i.e., hybrid star Weber2005_PPNP54-193 (); Wang2013_CSB58-3731 ().
A full calculation from slets to strange stars is carried out. In Fig. 2, the energy per baryon is given, which decreases monotonously with increasing radius or baryon number. When the energy per baryon reaches 930 MeV, the minimum baryon numbers for absolutely stable slets are determined, i.e., , 80, and 394849, which increases dramatically as approaches to its upper limit. In the region with , the variation of the energy per baryon is infinitesimal, and the radius is related to the baryon number by with , 0.901, and 0.862 fm. When , gravity starts to reduce the energy per baryon and a minimum value is obtained, corresponding to the maximum mass and baryon number of strange stars. The obtained - relations, as shown in the inset of Fig. 2, are in good agreement with the - probability distribution obtained by averaging the observed values of eight pulsars (Lattimer2012_ARNPS62-485, , Fig. 10).
The charge-to-mass ratio in Fig. 3(a) is defined as ignoring the electrons surrounding the SQM core. The obtained values are larger than previous results Berger1987_PRC35-213 (); Heiselberg1993_PRD48-1418 (). For smaller , the SQM core carries slightly more charge. Note that when , SQM within slets is positively charged. However, for , as shown in Fig. 1, the SQM carries negative charge to compensate the positive surface charge. Then an electric potential well for negatively charged particles is formed due to the quark depletion on the quark-vacuum interface. These particles may be trapped in the potential well and give a distinct photon spectrum when excited, which have significant implications for the experimental searches of SQM. As indicated in Fig. 3(b), when fm, charges are mostly located on the core surface and can be described by a constant surface charge density with . It is found that , 0.0135, and 0.0128 for , 152, and 159 MeV, which are much larger than the upper bound considering the electron-positron pair creation Madsen2008_PRL100-151102 ().
When the SQM core rotates, a magnetic field may be generated. For strong enough field strength, the electrons are expected to be trapped along the magnetic field lines. Then we simply assume the electron cloud stays still while the core rotates. It is straightforward to write out the magnetic field at the pole area on the surface Negreiros2010_PRD82-103010 ():
Here represents the vacuum permeability while is the rotational frequency. Then in Fig. 4 the magnetic field of a rotating strange star is obtained, with the field strength approaching to the typical value of pulsars, e.g., Gauss at Hz. It is found that the variation of the bag constant barely affects the field strength.
In conclusion, we propose a unified description from strangelets to strange stars considering the gravity and electrostatic interactions on the macroscopic scale while incorporating the strong and weak interactions locally. The quark-vacuum interface is treated with the multiple reflection expansion method. The model parameter follows the Witten-Bodmer hypothesis and is confronted with the observational - probability distributions of pulsars. Then the properties of SQM systems with all possible baryon numbers are investigated. It is found that the energy per baryon decreases monotonously for increasing baryon number, while the obtained charge-to-mass ratio of the SQM core is larger than previous predictions. On the core surface, due to the quark depletion, an electric potential well is formed for negatively charged particles and may give some unique observables for SQM detection. When fm, charges are mostly located on the core surface and a constant surface charge density is obtained. Then for a rotational bare strange star, a magnetic field comparable to the typical strength of pulsars is generated.
Acknowledgements.We are grateful to Professors Lie-Wen Chen, Thomas Papenbrock, Michael Smith, and She-Sheng Xue for fruitful discussions. This work was supported by National Natural Science Foundation of China (Grant Nos. 11135011, 11120101005, 11275248, 11475110, 11475115, 11575190 and 11525524), National Key Basic Research Program of China (Grant No. 2013CB834400), and the Knowledge Innovation Project of the Chinese Academy of Sciences (Grant No. KJCX2-EW-N01). The computation of this work was supported by the HPC Cluster of SKLTP/ITP-CAS and the Supercomputing Center, CNIC of CAS.
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