0 1.39 11.57 5.8X10 14 0 0.0001 1.40 11.59 5.8X10 14 0.25

1. Charged Compact Stars Beatriz B. Siffert, João R. T. de Mello Neto, Maurício O. Calvão Instituto de Física - UFRJ I. Introduction The possibility that stars could actually contain a non-vanishing net charge was first pointed out by Rosseland [1] in 1924. He modeled the star as a gas of positive ions and electrons and concluded that, due to their greater thermal energy, the electrons tend to escape the star more often than the ions. The star will then acquire a net positive charge. The process will be carried on until the electric field induced in the star stops more electrons from escaping. Recently other mechanisms to induce electric charge in stars, in particular in black holes, have been proposed. We can obtain an upper limit for the charge a black hole can acquire by demanding that the singularity is not naked. In the Reissner-Nordstrøm spacetime, this requirement sets Q2 M2 and the maximum charge a black hole can have is Qmax ~ 1020(M/M) C. If a black hole could really acquire such huge charge and stay stable, it would be a very strong candidate for an ultra high energy cosmic ray accelerator. In this work we study what effects a non-vanishing net charge could cause in the structure of compact stars in general. We also analyze the stability of such charged objects, trying to determine if they could really exist in a stable configuration in nature. III. Results We show below the results obtained for five different choices of charge fraction f (in units of (MeV/fm3)1/2/km), including the case f=0 (no charge). In each of the five curves, the central pressure varies from 0.5 MeV/fm3 (9.2X1011 g/cm3) to 5X103 MeV/fm3 (9.2X1015 g/cm3). We can see in fig. 1 that both the total mass and radius of the stars are increased by the presence of charge. Fig. 2 shows how much charge these stars can have. Table 1 shows these results for the maximum mass configuration [4]. f M(M) R(km) Pc(g/cm3) Q(X1020C) 0 1.39 11.57 5.8X1014 0 0.0001 1.40 11.59 5.8X1014 0.25 0.0005 1.66 13.31 3.8X1014 1.49 0.0008 2.39 18.06 1.4X1014 3.36 0.001 4.27 30.66 0.2X1014 7.38 Total Mass (M) f = 0.001 Total Charge Q (X1020 C) f = 0.001 f = 0.0008 f = 0.0008 f = 0.0005 f = 0.0005 f = 0 f = 0.0001 f = 0.0001 FIG. 2: Total charge X total radius for different values of f. FIG. 1: Total mass X total radius for different values of f. The stars with larger Pc are the ones on the left side. TABLE 1: Results obtained for the maximum mass stars. The pressure for which the maximum occurs is also shown. FIG. 3, 4 and 5 show how the electric field, the mass and pressure vary from the center to the surface of the maximum mass stars for all five different cases of f. II. The Structure of Charged Compact Stars We follow the steps developed by Ray et al. [2], reproducing their results and making an analysis of their dependence with the model used. We use the line element for a spherically symmetric and static star: and model the matter inside the star as a perfect fluid plus an electromagnetic field: where P is the pressure and  is the energy density of the fluid. A spherical surface of radius r, within the star, presents an electric field and encloses an electric charge: where ch is the star’s charge density. The mass inside a radius r is therefore: Using these expressions, the four differential equations for the equilibrium of a charged stars turn out to be: Since we have 6 variables and 4 equations, we need 2 other equations in order to solve the system. One of them arises from the hypothesis that charge density is linear with the energy density: The other will be a polytropic equation of state for the fluid, relating its pressure and energy density: where  = /c2 is the mass density. Following [2], we choose the exponent  as 5/3, which describes a non-relativistic Fermi gas. If the gas is constituted of particles of mass m, the constant k is given by [3]: where mN is the nucleon mass and Z/A is the ratio between the atomic number and mass of the element that is more abundant in the star. We choose k = 0,05 fm8/3. We solve numerically these equations with the boundary conditions M(r=0)=0, U(r=0)=0, P(r=0)Pc and (r=0)=0. Charge Q (X1020 C) Mass (M) Electric Field U (MeV/fm3)1/2 Radius inside the star (km) FIG. 4: Mass X radius for maximum mass stars. FIG. 3: Electric field X radius for maximum mass stars. FIG. 5: Charge X radius for maximum mass stars. From Fig. 3 and using: 1 (MeV/fm3)2 = 1.22 X 1021 V/m we see that the electric field induced in the stars are huge. These fields are larger than the critical electric field for pair creation: Ec = 1.3 X 1018 V/m This fact suggests that these charged stars would actually be unstable, discharging as soon as the pair creation begins. Fig. 6-9 show the dependence our results have on the choices of the constant k and of charge fraction f. We see that in both cases the variations are appreciable. In Fig. 8 and 9 both charge and mass presented huge instabilities for values of f above the ones shown in the graphs. This issue is under investigation. Total Mass (M) Total Charge Q (X1020 C) k (fm8/3) k (fm8/3) FIG. 6: Total mass X k for stars with the same central pressure. FIG. 7: Total charge X k for stars with the same central pressure. Total Mass (M) Total Charge Q (X1020 C) f (MeV/fm3)1/2 / km f (MeV/fm3)1/2 / km FIG. 8: Total mass X f for stars with the same central pressure. FIG. 9: Total charge X f for stars with the same central pressure. IV. Conclusions and Prospects The model adopted allows stable stars to acquire huge charges. However, if we take into account the effects of the electromagnetic fields induced and the surrounding plasma, we come to the conclusion that they are not stable after all. The charge should thus be neutralized through at least these two processes. In our opinion, the aforementioned neutralization as well as the initial charging mechanism are the key issues which must be addressed in this subject. Our analysis also shows that the results obtained are very dependent on the choices we make for f and k. Both these parameters come from the hypotheses we make for the kind of matter constituting the star. We intend to extend this preliminary work by using other equations of state and by adopting a multi-layered model for the stars. References [1] S. Rosseland, Mont. Not. Royal Astronomical Society 84, 720 (1924). [2] S. Ray, A. L. Espíndola, M. Malheiro, J. P. S. Lemos, V. T. Zanchin, Phys. Rev. D 68, 084004 (2003); astro-ph/0307262. [3] R.R. Silbar, S. Reddy, Am. J. Phys. 72, 892 (2004). [4] N.K. Glendenning, Compact Stars: Nuclear Physics, Particle Physics and General Relativity, 2nd ed. Springer Verlag, New York, USA (2000).