We have calculated the equilibrium configurations of slowly rotating protoneutron stars according to the above EOS in the framework of the Einsteinian theory of gravitation. This task has been done by solving the equations derived by Hartle numerically (Hartle 1967; Hartle & Thorne 1968) in order to describe the structure of slowly rotating relativistic stars (see Appendix for a summary). Hartle's formalism, basically, consists in perturbing the geometry of spacetime around a static and spherical background. Although Hartle's approach was developed for cold EOS, its generalization to isentropic configurations is straightforward.
Tables 1 and 2 summarize the main macroscopic properties of the protoneutron stars for respectively. The table headings show the different magnitudes (see below, and also the Appendix, for definitions) allowing for configuration characterisation. For each value of the entropy, we have selected three models: the one corresponding to the minimum of gravitational mass, the standard , and the maximum of gravitational mass, .
Let us note that the value of has been normalized
to the Newtonian critical one ( in Eq. (1. 1)) in the figures
shown in this paper and to the Haensel and Zdunik fitting
( in Eq. (1. 1)) in the results shown in Tables 1
and 2. Hence, there is an overestimation of
rotation effects (at the limit where the formalism breaks down) but
due to the well-known scaling laws with satisfied by
the different magnitudes, it is easy to extend our results to any
lower value (see Appendix). Glendenning (1997, and references
therein) has reported that the
self-consistency condition for the keplerian angular velocity in
Hartle's perturbative approach satisfies the following useful approximation:
(see Appendix for definition of ). Typical values of are , hence Glendenning's considerations lead to a critical value of the rotational frequency which can be fitted by a formula like Eq. (1. 1) with a value of a few percent lower than the Newtonian one ().
Figure 1 shows the surfaces spanned by the baryonic mass of the equilibrium configurations for as a function of the entropy and the central rest-mass density, , for the non-rotating (left) and rotating protoneutron stars (right). They summarize the main features of : i) increasing function with s and ii) existence of a minimum and a maximum for with which bind the set of stable configurations. Qualitatively similar results are obtained for .
|Figure 1: Baryonic mass (in solar units) as a function of the central rest-mass density (in cgs units and logarithmic scale) and entropy (in units of the Boltzmann constant) for the non-rotating equilibrium configurations (left) and for rotating configurations (right)|
|Figure 2: Baryonic (left) and Gravitational mass (right), in solar units, as a function of the radius (in km) for two extreme values of entropy (labels stand for values s=0.8 and 5, in units of the Boltzmann constant) for the non-rotating (thin) and rotating (thick) equilibrium configurations. Solid (dashed) lines correspond to ()|
|Figure 3: Binding energy (left), in units of , and Gravitational mass (right), in solar units, as a function of the central rest-mass density (in cgs units and logarithmic scale) for two extreme values of entropy (labels stand for values s=0.8 and 5, in units of the Boltzmann constant) for the non-rotating (thin) and rotating (thick) equilibrium configurations. Solid (dashed) lines correspond to ()|
In Fig. 2 we have plotted the baryonic mass (left) and the gravitational mass (right) as a function of radius, for a selection of two extreme values of s, s=0.8, 5. Both the nonrotating and rotating sequences of equilibrium configurations have been displayed for comparison. Curves corresponding to the same entropy value (indicated by labels) have been drawn with the same line type and different thickness (thicker for the rotating configurations and thinner for the non-rotating ones). Concerning this figure, we would like to point out the following: thermal effects have a stronger influence on the radius of the equilibrium configurations than those caused by rotation. The radius is the most sensitive to thermal effects since the atmosphere of the protoneutron star is placed at low densities and its radius increases for weak gravitational fields. At these low densities thermal pressure dominates over other non-thermal sources of pressure.
If we consider a rotating model with a canonical mass of 1.4 , its radius (see Eq. (A7) in the Appendix) satisfies: i) for is when (see Table 1), and ii) for is when (see Table 2). On the other hand, maximum mass depends less on temperature than radius: i) for the maximum gravitational mass is for (see Table 1), and ii) for it is for (see Table 2).
As can be seen in Tables 1 and 2 and Fig. 2, the gravitational mass of the rotating models increases up to relative to the nonrotating ones. The correction to the gravitational mass, , is greater in configurations of intermediate mass and for low entropy values. From Tables 1 and 2, we can verify the following relationship for : and for and , respectively. Corrections to the radius induced by rotation, , are more significant for models with intermediate masses than for those having the maximum mass at any entropy value. For example, the correction for satisfies the relation, and for and , respectively, as can be seen in Tables 1 and 2.
Binding energy (Fig. 3, left) and gravitational mass (Fig. 3, right) as a function of central rest-mass density have been plotted for two values of (0.4 and 0.3) and for two extreme entropy values (0.8 and 5) by comparing nonrotating and rotating configurations. The corrections on binding energy, , induced by rotation are consistent with the fact that the most bounded configurations have the most important corrections. When thermal effects are included, less bounded configurations are obtained (see Fig. 3, left) and the importance of the corrections induced by rotation decreases with entropy. Hence, e.g., quantity of the decreases from 1.5 10-2 to 1.7 10-4 when entropy increases in the interval for (see Tables 1 and 2).
The quantity is the angular velocity of the fluid relative to the local inertial frame (LIF). It measures the dragging of inertial frames. In this expression , the angular velocity of the LIF, is proportional to the star's angular velocity and, physically, stands for the angular velocity of a freely falling particle. In other words, LIFs are frames with rotational angular velocities as measured by an observer at infinity at rest with respect to the star. On the surface, quantity is the effective angular velocity of a mass element at the star's surface and, in some sense, it determines the strength of centrifugal forces from a Newtonian point of view (Glendenning 1997). The function w1 , evaluated at the surface, is shown in Fig. 4 (for and 0.3) in terms of the central rest-mass density. From Fig. 4 and Tables 1 and 2, several features can be addressed: frame dragging varies inversely to the star radius, it increases with central rest-mass density up to , and decreases with entropy.
|Figure 4: Dragging of inertial frames evaluated at the surface. Quantity w1 stands for at r=R. It is plotted as a function of the central rest-mass density (in cgs units and logarithmic scale) for several values of entropy (labels stand for values s=0.8, 1.8, 3.4 and 5, in units of the Boltzmann constant). Solid (dashed) lines correspond to ()|
The moment of inertia (see Appendix & Hartle 1967, for definition) is shown, as a function of the central rest-mass density in Fig. 5 (for and 0.3). In this figure we have plotted the equilibrium values for four isentropic configurations (s=0.8,1.8,3.4,5). The maximum, for a given entropy, appears at central densities which are lower than the ones corresponding to . The moment of inertia is an increasing function of entropy; in particular, the maximum of the moment of inertia varies in a factor two when entropy varies from 0.8 to 3.4. The behaviour of the moment of inertia of hot protoneutron stars can be explained by taking into account that radius depends strongly on entropy. For the model of , the moment of inertia (in units of 1044 g cm2) is when the entropy is for (see Tables 1 and 2). The curves of I for a given value of entropy and two different have an intersection point at some value of the central rest-mass density lower than the corresponding maximum, with their values for lower than for at lower central densities. This is related to the behavior of the gravitational mass (Fig. 3, right).
|Figure 5: Moment of inertia (in units of 1044 cgs) plotted as a function of the central rest-mass density (in cgs units and logarithmic scale) for several values of entropy (labels stand for values s=0.8, 1.8, 3.4 and 5, in units of the Boltzmann constant). Solid (dashed) lines correspond to ()|
The eccentricity evaluated on the surface, e, (see Appendix for definition) has been displayed as a function of the central rest-mass density in Fig. 6 (for and 0.3) and for several values of the entropy (s=0.8,1.8,3.4,5). It has a local maximum at low central densities and intermediate masses. This local maxima of the function was already found by Chandrasekhar & Miller (1974) in their study on slowly rotating homogeneous (energy density constant) masses and by Miller (1977) in his study of polytropic configurations. In Tables 1 and 2 we have shown the values of the quantity ( and are, respectively, the values of the radial coordinate at pole and equator); hence, deviations of this quantity from one -the spherical value- give an idea about the nonsphericity of the configuration. Tables 1 and 2 show that these models reach values as high as of nonsphericity. For increases in the interval when the entropy increases in the interval for (see Tables 1 and 2).
|Figure 6: Eccentricity, [(radius at equator)2/(radius at pole)2 - 1]1/2, evaluated at the surface, plotted as a function of the central rest-mass density (in cgs units and logarithmic scale) for several values of entropy (labels stand for values s=0.8, 1.8, 3.4 and 5, in units of the Boltzmann constant). Solid (dashed) lines correspond to ()|
Deformations induced by rotation are, according to previous discussion, more relevant for intermediate masses than for the most massive objects allowed by the present EOS. Thermal effects decrease this tendency. Figure 7 shows, in a spatial two-dimensional diagram, the isopycnic curves (geometrical loci of points inside the star with the same value of the rest-mass density) of two models with the same baryonic mass () and lepton abundance (). The characteristics of these models are indicated in the figure headings, which correspond to the values of entropy s=0.75 (left) and 4 (right). For the sake of comparison, we have preferred to keep the same ratio for both axis (in km.) in both figures, and, therefore, the corresponding scales differ in a factor of three. We have also indicated (thick continuous line) the surface of the corresponding spherical non-rotating model. Models shown in Fig. 7 have the critical angular velocity according to Eq. (1. 1), with . As can be seen in Fig. 7, the density gradient is sharper at the pole than at the equator. This may enhance the anisotropy of the flux of neutrinos emitted from the protoneutron star and may strengthen the deposit of energy for supernova explosion at polar direction (Janka & Mönchmeyer 1989; Shimizu et al. 1994; Shimizu 1995). In a crude model according to which the neutrinos are radiated - assumed black body - from a Maclaurin spheroid of semiaxis and , the ratio between the local neutrino flux in the polar and equatorial directions, measured by some terrestrial detector, is (Shimizu 1995). In our case, for the models plotted in Fig. 7, that ratio can be as high as .
The star's mass quadrupole moment, Q (see Appendix and Hartle 1967, for definition) measures the deformation of the star's exterior gravitational field. It is maximal for configurations of nearly uniform density and minimum for configurations with diffuse envelopes. When the central density increases the thermal effects become less important. As in the case of eccentricity, and for the model with , quantity Q/MR2 decreases in the interval when entropy increases in the interval for (see Tables 1 and 2).
|Figure 7: Two-dimensional diagram displaying the isopycnic curves for the rotating model with (baryonic mass), entropy 0.75 (in units of the Boltzmann constant) and .The model on the left has an angular velocity: . A total of 29 isocontours have been plotted spanning -at equispaced intervals in logarithmic scale- the total range of density, from the center ( cgs) to the surface ( cgs). The model on the right has an angular velocity: . A total of 29 isocontours have been plotted spanning -at equispaced intervals in logarithmic scale- the total range of density, from the center ( cgs) to the surface ( cgs). For the sake of comparison we have also added the curve showing the position of the surface of the corresponding non-rotating model|
|Figure 8: Entropy (in units of the Boltzmann constant) versus central rest-mass density (in cgs units and logarithmic scale) diagrams. The diagrams on the upper panel show the curves which are the geometrical loci of those non-rotating equilibrium configurations having the same baryonic mass (labels stand for the mass, in solar units) and lepton abundance (left) and (right). The diagrams on the lower panel show the curves which are the geometrical loci of those non-rotating (continuous lines) and rotating (dashed lines), with the maximum angular momentum, equilibrium configurations having the same baryonic mass (labels stand for the mass, in solar units) and lepton abundance (left) and (right)|
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