Maximum Mass and Radius of Strange Stars in the Linear Approximation of the EOS

A&A 385, 947–950 (2002) Astronomy DOI: 10.1051/0004-6361:20020260 & c ESO 2002 Astrophysics

Maximum mass and radius of strange stars in the linear approximation of the EOS

T. Harko and K. S. Cheng

Department of Physics, The University of Hong Kong, Pok Fu Lam Road, Hong Kong, PR China Received 9 August 2001 / Accepted 4 February 2002 Abstract. The Chandrasekhar limit for strange stars described by a linear equation of state (describing quark matter with density-dependent quark masses) is evaluated. The maximum mass and radius of the star depend on the fundamental constants and on the energy density of the quark matter at zero pressure. By comparing the expression for the mass of the star with the limiting mass formula for a relativistic degenerate stellar conﬁguration one can obtain an estimate of the mass of the strange quark.

Key words. dense matter – equation of state – stars: fundamental parameters

1. Introduction temperatures, as suggested by Witten (1984). In the theo- ries of strong interaction, quark bag models suppose that One of the most important characteristics of compact rel- breaking of physical vacuum takes place inside hadrons. ativistic astrophysical objects is their maximum allowed As a result, vacuum energy densities inside and outside a mass. The maximum mass is crucial for distinguishing hadron become essentially different and the vacuum pres- between neutron stars and black holes in compact bi- sure on the bag wall equilibrates the pressure of quarks, naries and in determining the outcome of many astro- thus stabilizing the system. If the hypothesis of the quark physical processes, including supernova collapse and the matter is true, then some neutron stars could actually merger of binary neutron stars. The theoretical value of be strange stars, built entirely of strange matter (Alcock the maximum mass for white dwarfs and neutron stars et al. 1986; Haensel et al. 1986). For a review of strange was found by Chandrasekhar and Landau and is given 3/2 star properties, see Cheng et al. (1998). ∼ hc¯ −4/3 Most of the investigations of quark star properties have by Mmax G mB (Shapiro & Teukolsky 1983), been done within the framework of the so-called MIT bag where mB is the mass of the baryons (in the case of white dwarfs, even pressure comes from electrons; most model. Assuming that interactions of quarks and gluons of the mass is in baryons). Thus, with the exception of are sufficiently small, neglecting quark masses and sup- composition-dependent numerical factors, the maximum posing that quarks are confined to the bag volume (in the mass of a degenerate star depends only on fundamental case of a bare strange star, the boundary of the bag co- incides with the stellar surface), the energy density ρc2 physical constants. The radius Rmaxof the degenerate 1/2 and pressure p of a quark-gluon plasma at temperature T ≤ h¯ hc¯ star obeys the condition Rmax 2 , with m and chemical potential µ (the subscript f denotes the var- mc GmB f being the mass of either electron (white dwarfs) or neu- ious quark flavors u, d, s etc.) are related, in the MIT bag tron (neutron stars) (Shapiro & Teukolsky 1983). White model, by the equation of state (EOS) (Cheng et al. 1998) dwarfs are supported against of gravitational collapse by (ρ − 4B)c2 the degeneracy pressure of electrons whereas for neutron p = , (1) stars this pressure comes mainly from the nuclear force 3 between nucleons (Shapiro & Teukolsky 1983). For non- where B is the difference between the energy density of rotating neutron stars with the central pressure at their the perturbative and non-perturbative QCD vacuum (the 2 center tending to the limiting value ρcc , an upper bound bag constant). Equation (1) is essentially the equation of of around 3 M has been found (Rhoades & Ruffini 1974). state of a gas of massless particles with corrections due The quark structure of the nucleons, suggested by to the QCD trace anomaly and perturbative interactions. quantum cromodynamics, opens the possibility of a These corrections are always negative, reducing the en- hadron-quark phase transition at high densities and/or ergy density at given temperature by about a factor of two (Farhi & Jaffe 1984). For quark stars obeying the bag Send offprint requests to:T.Harko, model equation of state (1) the Chandrasekhar limit has e-mail: [email protected] been evaluated, from simple energy balance relations, in

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20020260 948 T. Harko and K. S. Cheng: Maximum mass and radius for linear EOS

(Bannerjee et al. 2000). In addition to fundamental con- where a and ρ0 are non-negative constants. ρ0 is the en- stants, the maximum mass also depends on the bag con- ergy density at zero pressure. stant. Such an equation of state has been proposed mainly More sophisticated investigations of quark-gluon in- to describe the strange matter built of u, d and s quarks teractions have shown that Eq. (1) represents a limiting (Gondek-Rosinska et al. 2000; Zdunik 2000). The physical case of more general equations of state. For example, MIT consistency of the model requires ρ0 > 0. bag models with massive strange quarks and lowest- or- The particle number density and the chemical poten- der QCD interactions lead to some corrections terms in tial corresponding to EOS (2) are given respectively by the equation of state of quark matter. Models incorporat- (Zdunik 2000) ing restoration of chiral quark masses at high densities and a +1 p 1/(a+1) giving absolutely stable strange matter can no longer be n (p)=n0 1+ 2 , (3) accurately described by using Eq. (1). On the other hand, a ρ0c in models in which quark interaction is described by an a/(a+1) a +1 p interquark potential originating from gluon exchange and µ (p)=µ0 1+ 2 , (4) by a density-dependent scalar potential which restores the a ρ0c chiral symmetry at high densities (Dey et al. 1998), the where n0 is the particle number density at zero pressure 2 equation of state P = P (ρ) can be well approximated by a and µ0 = ρ0c /n0. linear function in the energy density ρ (Gondek-Rosinska The parameters a and ρ0 can be calculated, for realis- et al. 2000). It is interesting to note that Frieman & Olinto tic equations of state, by using a least squares fit method (1989) and Haensel & Zdunik (1989) have already men- (Gondek-Rosinska et al. 2000; Zdunik 2000). For the equa- tioned the approximation of the EOS by a linear functions of state incorporating restoration of chiral quark tion (see also Prakash et al. 1990; Lattimer et al. 1990). masses at high densities proposed in Dey et al. (1998) one 15 3 Recently Zdunik (2000) has studied the linear approxima- obtains the values a =0.463, ρ0 =1.15 × 10 g/cm 15 3 tion of the equation of state, obtaining all parameters of and a =0.455, ρ0 =1.33 × 10 g/cm , respectively the EOS as polynomial functions of strange quark mass, (Gondek-Rosinska et al. 2000). The standard bag model 14 3 the QCD coupling constant and bag constant. The scal- corresponds to a =0.333 and ρ0 =4×10 g/cm (Cheng ing relations have been applied to the determination of et al. 1998). the maximum frequency of a particle in a stable circular From Eqs. (3)–(4) it follows that the particle number orbit around strange stars. and chemical potential are related by the equation It is the purpose of this paper to obtain, by using a 2 ρ0c simple phenomenological approach (which is thermody- µ = na. (5) na+1 namical in its essence), the maximum mass and radius 0 (the Chandrasekhar limits) for strange stars obeying a From the numerical studies of strange star models we linear equation of state. Of course the maximum mass of know that the density profile of this type of astrophysi- compact astrophysical objects is a consequence of General cal object is quite uniform (Glendenning 1996). Therefore Relativity and not of the character of motion of matter we can approximate n ≈ N/V ,whichleadsto constituents. However, the formulae for maximum mass − µ 4π a N a and radius, due to their simple analytical form, give a = R−3a, (6) better insight into the underlying physics of quark stars, µ0 3 n0 also allowing us to obtain some results which cannot be where N is the total number of particles in a star of radius obtained by numerical methods. For example, from the ob- R and volume V . tained relations one can find the scaling relations for the With the use of Eqs. (2)–(6) one obtains the energy maximum mass and radius of strange stars in a natural density of the star in the form way. − − ρ 4π a 1 N a+1 a The present paper is organized as follows. The maxi- ρ = 0 R−3(a+1) + ρ . (7) a +1 3 n a +1 0 mum mass and radius of quark stars with a general linear 0 equation of state is obtained in Sect. 2. In Sect. 3 we dis- TheR total mass M of the star is defined according to M = R 2 ≈ 4π 3 cuss our results and conclude the paper. 4π 0 ρr dr 3 ρR and is given by −a a+1 ρ0 4π N −3a 4π a 3 M = R + ρ0R , (8) 2. Maximum mass and radius for strange stars a +1 3 n0 3 a +1 in the linear approximation of the EOS where we assumed that the energy density is approxi- mately constant inside the star. We assume that the strange star obeys an equation of Extremizing the mass with respect to the radius R by state that can be obtained by interpolation with a linear means of ∂M/∂R = 0 gives the relation function of density in the form: −a a+1 ρ0 4π N −3a 4π 1 3 − 2 R = ρ0R . (9) p = a (ρ ρ0) c , (2) a +1 3 n0 3 a +1 T. Harko and K. S. Cheng: Maximum mass and radius for linear EOS 949

Substituting Eq. (9) into Eq. (8) we obtain the maximum The maximum mass of the star can be calculated from mass of the strange star in the linear approximation of the Eq. (10) and is: EOS: 4 R3 c3 1 M = 0 √ · (16) 4π 3 max 3/2 M = ρ R . (10) 3 (a +1) G πGρ0 3 0 This expression is very similar to the expression for the In Eqs. (15) and (16) R0 is a numerical factor of the order ≈ maximum mass of the quark star obtained assuming that R0 0.474. the star is composed of three-flavour masslesss quarks, The maximum radius of the quark star given by confined in a large bag (Bannerjee et al. 2000; Cheng & Eq. (15) is the radius corresponding to the maximum Harko 2000). From a physical point of view, Eq. (10) de- mass. On the other hand for the existing models of strange scribes a uniform density zero pressure stellar type con- stars, the configuration with maximum mass has a radius figuration. which is lower than the maximum radius. For example, The maximum equilibrium radius corresponds to a for strange stars described by the bag model equation of minimum total energy of the star (including the gravi- state, the maximum radius is 11.40 km, while the radius tational one), for any radius. For ordinary compact stars, corresponding to the maximum mass is 10.93 km, which the mass is entirely due to baryons, and the correspond- is 4% lower than the maximum radius. This difference is ing (Newtonian) gravitational potential energy is of the neglected in Eq. (15). 2 The maximum mass and radius of the star are strongly order EG ∼−αGM /R (α = −3/5 for constant density Newtonian stars). For quark stars, assumed to be dependent on the numerical value of the coefficient R0 and formed of massless quarks, the total mass can be calcu- estimations based on other physical models could lead to lated from the total (thermodynamic as well as confin- different numerical estimates of the limiting values of the ment) energy in the star. One possibility for the estima- basic parameters of the static strange stars. tion of the gravitational energy per fermion is to define an effective quark mass incorporating all the energy contri- 3. Discussions and final remarks butions (Bannerjee et al. 2000). In the present paper we have shown that there is a max- The gravitational energy per particle (the strange star imum mass and radius (the Chandrasekhar limits) for is assumed to be formed from fermions) is quark stars whose equation of state can be approximated GMmeff by a linear function of the density. We have also obtained EG = − , (11) R the explicit expressions for Mmax and Rmax. With respect to the scaling of the parameter ρ of the where m is the effective mass of the particles inside 0 eff form ρ → kρ , the maximum mass and radius have the the star, incorporating also effects such as quark confin- 0 0 following scaling behaviors: ment. For a star with N particles one can write M = −1/2 −1/2 Nmeff = ρ0V ,ormeff = ρ0/n. On the other hand one Rmax → k Rmax,Mmax → k Mmax. (17) can assume µ = ρ0/2n (Bannerjee et al. 2000), leading to 2 For the maximum mass of the strange stars this scaling meff =2µ/c . Hence, with the use of Eqs. (6), (9) and (10) we can express the gravitational energy per particle as relation has also been found from the numerical study of the structure equations in the framework of the bag model 4π 2 ρ2 (Witten 1984; Haensel et al. 1986). E = −2 G 0 R5. (12) G 3 N A rescaling of the parameter a of the form a +1 → K (a + 1), with ρ0 unscaled, leads to a transformation of The energy density per particle of the fermions follows the radius and mass of the form from Eq. (7) and is given by: −1/2 −3/2 Rmax → K Rmax,Mmax → K Mmax. (18) 4π 1 ρ0 3 EF = R . (13) → 3 a +1N A simultaneous rescaling of both a and ρ0, with a +1 K (a +1),ρ0 → kρ0 gives The total energy E per particle is −1/2 −1/2 Rmax → k K Rmax, 2 2 4π 1 ρ0 3 4π ρ0 5 −1/2 −3/2 E = R − 2 G R . (14) Mmax → k K Mmax. (19) 3 a +1N 3 N The maximum mass and radius of strange stars with lin- Extremizing the total energy with respect to the ra- ear EOS is strongly dependent on the numerical value of dius (with the total particle number kept constant), ρ0, the mass decreasing with increasing ρ0.Forρ0 =4B, ∂E = 0, it follows that the maximum radius of − ∂R N=const. with the bag constant B =1014 g/cm3 (56 MeVfm 3) the equilibrium configuration in the linear approximation we obtain Mmax =1.83 M , a value that must be com- of the EOS is given by: pared to the value Mmax =2M obtained by numerical c methods (Witten 1984; Haensel et al. 1986). The differ- Rmax = R0 p · (15) π (a +1)Gρ0 ence between the numerical and theoretical predictions is 950 T. Harko and K. S. Cheng: Maximum mass and radius for linear EOS

15 −3 −25 around 10%. For ρ0 =1.33 × 10 gcm the maximum gives mqeff ∼ 4.9 × 10 g ≈ 275 MeV. The mass given mass of the star is about 1 M . by Eq. (21) can be considered as the minimum mass of Generally our formulae (15) and (16) underestimate the stable quark bubble. It is of the same order of mag- the maximum values of the mass and radius because we nitude as the mass ms of the strange quark. Therefore have assumed that the density inside the star is uniform. the Chandrasekhar limit applies also for quark stars if we It is obvious that near the surface the density is much take mqeff for the mass of the elementary constituent of lower than at the center of the compact object. Due to the star. the approximations and simplifications used to derive the In the present paper we have considered the maximum basic expressions, reflected mainly in the uncertainties in mass and radius of strange stars in the linear approxima- the exact value of the coefficient R0, Eqs. (15) and (16) tion of the equation of state and the dependence of these cannot provide high precison numerical values of the max- quantities on the parameter a has been found. We have imum mass and radius for linear EOS stars, which must also pointed out the existence of scaling relations for the be obtained by numerically integrating the gravitational maximum radius of strange stars, an aspect that has not field equations. been mentioned in previous investigations (Witten 1984; For the the linear EOS, Mmax and Rmax depend Haensel et al. 1986; Bannerjee et al. 2000; Zdunik 2000). mainly on the fundamental constants c and G and on Our formulae also lead to the transformation relations for the zero pressure density ρ0 (the bag constant). The the maximum mass and radius of strange stars with re- Chandrasekhar expressions for the same physical param- spect to separate and simultaneous scaling of the parameters involve two more fundamental constants,h ¯ and the eters a and ρ0. On the other hand the possibility of es- mass of the electron or neutron. timation of the mass of the strange quark from general For quark stars, usually one assume they are composed astrophysical considerations can perhaps give a better un- of a three-flavour system of massless quarks, confined in derstanding of the deep connection between micro- and a large bag. Hence the mass of the quark cannot play any macro-physics. role in the mass formula. But the linear EOS with arbi- trary a can describe quark matter with non-zero quark Acknowledgements. This work is partially supported by a RGC ≈ masses (the mass of the strange quark ms 200 MeV), grant of Hong Kong Government and T.H. is supported by a forming a degenerate Fermi gas (Gondek-Rosinska et al. studentship of the University of Hong Kong. The authors are 2000; Zdunik 2000). Therefore this system should also be very grateful to the anonymous referee whose comments helped described by the same formulae as white dwarfs or neu- to improve an earlier version of the manuscript. tron stars, not only by Eqs. (15)–(16). Generally ρ0 is a function of the mass of the strange quark, so this mass im- References plicitly appears in the expression of the maximum mass and radius. But on the other hand we can assume that Alcock, C., Farhi, E., & Olinto, A. 1986, AJ, 310, 261 the Chandrasekhar limit also applies to quark stars with Bannerjee, S., Ghosh, S. K., & Raha, S. 2000, J. Phys. G: Nucl. the baryon mass substituted by an effective quark mass Part. Phys., 26, L1 Cheng, K. S., Dai, Z. G., & Lu, T. 1998, Int. J. Mod. Phys. D, mqeff , representing the minimum mass of the quark bub- bles composing the star. Hence we must have 7, 139 Cheng, K. S., & Harko, T. 2000, Phys. Rev. D, 62, 083001 3/2 3 Dey, M., Bombacci, I., Dey, J., Ray, S., & Samanta, B. C. 1998, ¯hc −4/3 ∼ c √ 1 · mqeff (20) Phys. Lett. B, 438, 123 G G πGρ0 Farhi, E., & Jaffe, R. L. 1984, Phys. Rev. D, 30, 2379 Equation (20) leads to the following expression of the ef- Frieman, J. A., & Olinto, A. 1989, Nature, 341, 633 fective mass of the “elementary” quark bubble: Glendenning, N. K. 1996, Compact stars: nuclear physics, particle physics and general relativity (Springer, New York) ! 1/3 3/4 Gondek-Rosinska, D., Bulik, T., Zdunik, L., et al. 2000, A&A, ¯hρ0 363, 1005 mqeff ∼ · (21) c Haensel, P., Zdunik, J. L., & Schaeffer, R. 1986, A&A, 160, 121 The effective quark mass is determined only by elementary Haensel, P., & Zdunik, J. L. 1989, Nature, 340, 617 particle physics constants and is independent of G. From Lattimer, J. M., Prakash, M., Masaak, D., & Yahil, A. 1990, its construction mqeff should be relevant when the system AJ, 355, 241 is quantum mechanical and involves high velocities and Prakash, M., Baron, E., & Prakash, M. 1990, Phys. Lett. B, energies. With respect to a scaling of the zero pressure 243, 175 Rhoades, C. E., & Ruffini, R. 1974, Phys. Rev. Lett., 32, 324 density of the form ρ0 → kρ0, the effective quark mass 1/4 Shapiro, S. L., & Teukolsky, S. A. 1983, Black Holes, White has the scaling behavior mqeff → mqeffk , similar to the scaling of the strange quark mass (Zdunik 2000). Dwarfs, and Neutron Stars (John Wiley & Sons, New York) 14 −3 Witten, E. 1984, Phys. Rev. D, 30, 272 For ρ0 =4× 10 gcm we obtain mqeff ∼ 3.63 × −25 15 −3 Zdunik, J. L. 2000, A&A, 359, 311 10 g ≈ 204 MeV. For ρ0 =1.33 × 10 gcm , Eq. (21)