A&A 387, 710–713 (2002) Astronomy DOI: 10.1051/0004-6361:20020395 & c ESO 2002 Astrophysics

The surface electric field of bare strange

J. Hu1 and R. X. Xu2,1

1 Center for Astrophysics and Physics Department, Tsinghua University, Beijing 100084, PR China 2 School of Physics, Peking University, Beijing 100871, PR China Received 7 January 2002 / Accepted 4 March 2002 Abstract. The surface electric characteristics of bare strange stars are investigated with the inclusion of boundary effects. The thickness of the electron layer where pairs can be spontaneously created is calculated as a function of the bag parameters. We find that previous estimates are representative for bag parameters within a rather wide range, and therefore our results favor the thermal radiation mechanism of bare strange stars advanced by V. V. Usov.

Key words. radiation mechanisms: thermal – elementary particles – stars: neutron

1. Introduction see Zdunik et al. 2001 for the recent developments with the inclusion of rotating and general-relativistic effects). It is of great importance to identify strange stars; a new Also it should be noted that this electric field has some im- window to distinguish neutron stars and bare strange stars portant implications on radio emission mechanisms (BSSs) has been proposed recently based on their sharp (Xu & Qiao 1998; Xu et al. 1999; Xu et al. 2001). differences in surface conditions (Xu et al. 2001; Usov In fact the Usov mechanism of pair production de- 2001a). It is therefore essential to study the surface prop- pends on many parameters; it is therefore imperative to erties of BSSs in detail, e.g., the degree of the thermal study the dependence of the process on these parame- of a hot BSS. ters. With some typical parameters chosen by Usov in his The surface electric field should be very strong calculations, the resultant thickness of the electron layer (∼1017 V/cm) near the bare quark surface of a strange with the electric field E ∼> 1.3 × 1016 V/cm (the criti- because of the mass difference of the cal field necessary for pair production), ∆r ,is∼500 fm. and the up (or down) quark (Alcock et al. 1986), which E However the proper determination of the thickness ∆rE could play an important role in producing the thermal should be done with the dynamical theory of the strange emission of BSSs by the Usov mechanism (Usov 1998; quark matter. Because of the intractable nature of quan- Usov 2001b), because strange quark matter is a poor ra- tum chromodynamics, some phenomenological models, diator of thermal photons at frequencies lower than its i.e., the MIT bag model (e.g., Jensen & Madsen 1996), plasma frequency (∼20 MeV) (Alcock et al. 1986). The  the quark mass-density-dependent model (e.g., Lugones basic idea of the Usov mechanism is that e pairs are cre- & Benvenuto 1995), and the quark potential model (e.g., ated rapidly in a few empty quantum states with energy 2 Dey et al. 1998), have been applied to the descriptions <F − 2mc (F is the Fermi energy, m the electron of the strange quark matter. In the bag model, ∆rE is mass) due to the very strong electric field in an electron a function of α (the coupling constant for strong inter- layer (with a height of ∼500 fm above quark surface); the c actions), m (the strange quark mass), and B (the bag pairs subsequently annihilate into photons which are then s 1 constant). Also it should be noticed that the quark num- thermalized in the electron layer around a BSS. This ra- ber densities, n (i = u, d, s for up, down and strange diative mechanism recently has been applied tentatively to i quarks, respectively), are assumed to be uniform below soft γ ray repeaters (Usov 2001c; Usov 2001d). In addition, the quark surface, and therefore the quark charge density the strong electric field plays an essential role in forming a (2n −n −n )/3=V 3/(3π2) is constant near the surface. possible crust around a strange star, which has been inves- u d s q The potential Vq is usually chosen as 20 MeV for typical tigated extensively by many authors (e.g., Martemyanov cases. However, the quark number densities should not be 1992; Kettner et al. 1995; Huang & Lu 1997; Phukon 2000; uniform near the surface if boundary effects are included, Send offprint requests to:R.X.Xu, since charge neutrality is broken there. e-mail: [email protected] In this paper we improve the calculation of the electric 1 This layer may be optically thick for BSSs with tempera- field in the vicinity of a BSS surface, using the popular bag ture T ∼> 109 K. model and the Thomas-Fermi model. We investigate the

Article published by EDP Sciences and available at http://www.aanda.org or http://dx.doi.org/10.1051/0004-6361:20020395 J. Hu and R. X. Xu: The surface electric field of bare strange stars 711 electric characteristics of BSSs for different values of αc. Table 1. The particle number densities far below the quark −∞ Initially we use typical parameters for ms = 200 MeV and surface, and the potentials of V0 = V (z = )andVc = 1/4 B ∼ 145 MeV. Then the thickness, ∆rE, of the electron V (z = 0), for different values of αc. layer where pairs can be created is computed as functions of B, ms,andαc. It is found that Usov’s estimates are αc 0 0.1 0.3 0.5 0.7 0.9 representative for bag-model parameters within a rather B1/4/MeV 146 144 139 133 127 120 44 −3 wide range. Boundary effect are also considered in this nu/10 m 2.86 2.71 2.41 2.10 1.80 1.48 calculation. 44 −3 nd/10 m 3.95 3.68 3.14 2.60 2.07 1.54 44 −3 ns/10 m 1.77 1.74 1.68 1.61 1.52 1.42 40 −3 2. Calculation of the electric field near the quark ne/10 m 13.9 11.9 7.77 4.03 1.21 0.30 surface V0/MeV 31.65 29.99 26.05 20.93 14.01 4.08 V /MeV 30.69 29.11 25.34 20.44 13.77 4.06 The interesting electric properties of strange stars were c first noted by Alcock et al. (1986), who presented numer- ical calculations of the electric potentials of strange stars replace µ with µ − qV in Eq. (4), where q is the particle with or without crusts. Analytical solutions of the electron i i charges. Finally we come to, number density and the electric field were also formulated,  h   which could be helpful in dealing with some physical pro-  V 3 1 1 1 1 3  + n (V )+ µ + V cesses near the quark surfaces of strange stars (Xu & Qiao  3π2 3 s π2 3 3 1999). In this section, the bag model is applied to calcu-    i  d2V 2 2 3 2α late the quark charge density; the Thomas-Fermi Model is = − − − c (6) 2  µ V 1 ,z<0 dz  3 3 π employed to find ∆rE dependence on αc, ms,andB, with   V 3 the inclusion of boundary effects.  ,z≥ 0 3π2

2.1. Equations where the complex term ns(V ) can be derived from chem- ical potentials the Ωi. In our calculation, the quarks and electrons near the sur- The boundary conditions for Eq. (6) are face of the strange star keep chemical equilibrium locally; the relation between charge density and electric potential z →−∞: V → V0, dV/dz → 0; is described by the classical Poisson equation. The thermo- z → +∞: V → 0, dV/dz → 0. dynamic potentials Ωi as functions of chemical potentials

µi (i=u,d,s,e),ms and αc can be found in the literature where V0 is the electric potential in the deep core of the (Alcock et al. 1986). In the chemical equilibrium of weak strange star, which is determined from Eq. (6) by letting interaction, the left hand side equal zero.

µd = µs = µ, (1) 2.2. Results µe + µu = µ, (2) d2V 1 1 2 By integrating Eq. (6) in (−∞,V ]and[V , +∞) respec- = n + n + n − n , (3) c c dz2 e 3 d 3 s 3 u tively with the boundary conditions, we get the expres- ∂Ωi sion of the electric field E = −dV/dz,whereVc = V |z=0. ni = − , (4) ∂µi Obviously Vc is a function of αc, which can be obtained from the continuity of E at z = 0. Substituting Vc into where z is a measured height above the quark surface. the equations of E and then integrating them, we get the The quark number densities drop to zero on the sur- expression of V .Forz ≥ 0, we find face, but they are not uniform below the surface if the V boundary effects are considered. The chemical potential V (z)= c , (7) µ can be determined by the condition that the pressure √Vcz | 1+ on theP quark surface is zero, P z=0 = 0. The pressure is 6π − − 2 P = i Ωi B (Alcock et al. 1986). dV 1 V − √  c  The kinetic energy of electrons is equal to the elec- E(z)= = 2 , (8) dz 6π Vcz tric potential, p = V , in the Thomas-Fermi model. The 1+√ e 6π electron number density reads (Alcock et al. 1986) 3 3 V 1 Vc · ne(z)= =  3 (9) p3 V 3 3π2 3π2 Vcz n = e = · (5) 1+√ e 3π2 3π2 6π

Note that the thermodynamical potential Ωi is defined Several numerical results are obtained, which are shown when the electric potential V is assumed as zero. We thus in Table 1 and Figs. 1–4. First, we study the electric 712 J. Hu and R. X. Xu: The surface electric field of bare strange stars

20 35 10 α =0 c 18 30 10

α =0.3 c 16 10 25 α =0.5 α =0.9 c c α =0.5 14 c 10

20 −1 V 6 12 10 V / 10 α

15 =0.7 E / V cm c 10 10 α =0 10 c 8 10

5 α =0.9 6 c 10

4 0 10 −20 0 20 40 60 80 100 −50 −45 −40 −35 −30 −25 −20 −15 −10 −5 0 z / fm z / fm

Fig. 1. Electric potential as a function of z with αc =0,0.3, Fig. 3. Electric field varies sharply below the quark surface. 0.5, 0.7, 0.9. αc =0,0.5,0.9.

19 10 as functions of the bag parameters ms, B,andαc.The results are shown in Fig. 4. It is found that the Usov mechanism can work for bag parameters within a rather

18 wide range. The thickness ∆rE could be large enough (i.e., 10 ∆rE ∼> 500 fm) as long as (1) B is not too large and ms

α =0 is not too small, or (2) B is not too small and ms is not

−1 c too large. Increasing αc has an adverse influence on the Usov mechanism. For instance in Fig. 2, the field E cannot 17 E / V cm 10 α =0.7 16 c exceed the critical field Ec =1.3 × 10 V/cm (necessary for pair production) if αc > 0.94. Nevertheless previous α =0.9 c dynamical calculations show that αc ∼> 0.9islesspossible

E=1.3×1016 V cm−1 for stable strange quark matter. In conclusion we could 16 10 expect that the pair emission process proposed by Usov α =0.94223 c might appear in nature for acceptable bag parameters.

0 200 400 600 800 1000 1200 z / fm 3. Conclusion and discussion Fig. 2. Electric field changes slowly above the quark surface. The coupling constant αc is chosen to be 0, 0.7, 0.9, 0.94. We have improved the calculation of the electric character- istics of bare strange stars with the inclusion of boundary effects (i.e., the effects of non-local neutrality near and be- characteristics of a BSS for different values of αc,as- low the quark surface). From our calculation, we find that 2 suming µ = 300 MeV, ms = 200 MeV, and the renor- the Usov mechanism can work for bag parameters within malization point ρR = 313 MeV. The quark and elec- a rather wide range. { } →−∞ tron number densities ni when z ,aswellas As shown in Table 1, both Vc and V0,aswellastheir potentials of V0 and Vc, are listed in Table 1 for αc = very small difference, decrease as αc increases. Our results 0, 0.1, 0.3, 0.5, 0.7, 0.9, respectively. The electric potential on the electric potential V (z)forz<0 is quite different V and field E are also plotted in Figs. 1–3 for this case. to the previous calculation given by Alcock et al. (1986) We see from Fig. 1 that the potential curve becomes flatter where the boundary effects were not included (see Fig. 1). when the coupling constant αc increases. Also it should be It is shown that the strong electric field resides only about noted from Figs. 2 and 3 that the electric field E changes ∼10 fm below the quark surface (see Fig. 3), rather than rather slowly for z>0, but varies rapidly for z<0. This ∼102 fm obtained by Alcock et al. (1986). indicates the non-symmetric nature of the electric field We can proof that E is almost an exponential function with respect to the quark surface of z =0. of z below the quark surface. We denote the right hand Secondly, we investigate numerically the thickness, side of Eq. (6) as f(V ). As z →−∞, namely V → V0,we 0 ∆rE, of the electron layer where pairs can be created, have f(V ) → 0. Approximatingp f(Vp)asf (V0)(V − V0), we can obtain E = − f 0(V ) exp[ f 0(V )]z and V = 2 Generally there is a relation between µ and B, µ = µ(B). p 0 0 − 0 In this assumption B1/4 ∼ 145 MeV. V0 exp[ f (V0)z]forz<0. J. Hu and R. X. Xu: The surface electric field of bare strange stars 713

350 350

300 α =0 1050 300 c α =0.3 c 1030 250 250

1030 1000 200 200 / MeV / MeV s s m m 1000

150 150 900

900 100 100 500 500 50 50 50 100 120 140 160 180 200 100 120 140 160 180 200 B1/4 / MeV B1/4 / MeV

350 350

α =0.9 α =0.6 c c 300 1030 300

250 1000 250 1000

200 200 / MeV / MeV

s 900 s m 900 m

150 150 500

500 100 100 50

50 50 50 100 120 140 160 180 200 100 110 120 130 140 150 160 170 180 190 200 B1/4 / MeV B1/4 / MeV

Fig. 4. A set of calculated thickness ∆rE, as functions of bag parameters of B and ms, for different coupling constants αc =0, 0.3, 0.6, 0.9. Indicated numbers denote the correspondence values of ∆rE.

Acknowledgements. This work is supported by National Kettner, Ch., Weber, F., Weigel, M. K., & Glendenning, N. K. Nature Sciences Foundation of China (10173002) and the 1995, Phys. Rev. D 51, 1440 Special Funds for Major State Basic Research Projects Martemyanov, B. V. 1992, Phys. Rev. D 49, 4293 of China (G2000077602). The authors sincerely thank Dr. Usov, V. V. 1998, Phys. Rev. Lett., 80, 230 Shuangnan Zhang for his comments and the improvement of Usov, V. V. 2001a, invited talk at the Conf. on Compact the language. Stars in the QCD Phase Diagram, Copenhagen, Aug. 15–18 [astro-ph/0111442] Usov, V. V. 2001b, ApJ, 550, L179 References Usov, V. V. 2001c, Phys. Rev. Lett., 87, 021101 Usov, V. V. 2001d, ApJ, 559, L135 Alcock, C., Farhi, E., & Olinto, A. 1986, ApJ, 310, 261 Xu, R. X., & Qiao, G. J. 1999, Chin. Phys. Lett., 16, 778 Day, M., Bombaci, I., Dey, J., Ray, S., & Samanta, B. C. 1998, Xu, R. X., & Qiao, G. J. 1998, Chin. Phys. Lett., 15, 934 Phys. Lett. B 438, 123 Xu, R. X., Qiao, G. J., & Zhang, B. 1999, ApJ, 522, L109 Jensen, D. M., & Madsen, J. 1996, Phys. Rev. D 53, R4712 Xu, R. X., Zhang, B., & Qiao, G. J. 2001, Astroparticle Phys., Lugones, G., & Benvenuto, O. G. 1995, Phy. Rev. D 52, 1276 15, 101 Phukon, T. C. 2000, A&A, 355, 1009 Zdunik, J. L., Haensel, P., & Gourgoulhon, E. 2001, A&A, 372, Huang, Y. F., & Lu, T. 1997, A&A, 325, 189 535