a plateau around B 1 / 4 = 161 − 163 MeV between 3 ρ$_{0}$ − 5 ρ$_{0}$ in the standard NJL model. The plateau is due to the partial chiral symmetry restoration of quarks u and d . For the new parametrizations of the cutoff the value B 1 / 4 eff ∼ 162 MeV will occur at lower densities and the plateau disappears in all cases. This effect occurs because the chemical potential dependence of the cutoff starts for a chemical potential below the partial chiral symmetry restoration for the quarks u and d . We have marked the onset of the strange quark on the effective bag curve with vertical lines. The effective bag estabilizes much faster for Λ = Λ$_{2}$ and tends to behave in a similar way to the MIT bag model with the decrease of the µ$_{0}$ . In Fig. 4 we show the quark fractions, Y$_{i}$ = ρ$_{i}$/ (3 ρ$_{B}$ ) [Fig. 4a)], and the constituent masses of the u , d and s quarks in β -equilibrium [Fig. 4b)]. The effect of different choices of the cutoff on the quark s is clear: for the faster increase of Λ( µ ) and smaller values of µ$_{0}$ the appearance of the quark s occurs at lower densities and the its con- stituent mass M$_{s}$ approaches the current quark mass m$_{0}$$_{s}$ at lower densities too. As can be seen from Figs. 3 and 4 the different slopes in the µ dependent cutoff and the different values of the chemical potential µ$_{0}$ change the behavior of the model, namely the constituent quark masses and baryonic den- sity. The EOS becomes softer with these modifications, as seen in Fig. 5, where the pressure is displayed as a function of the chemical potential for the standard NJL model and the different choices of the cutoff. The cutoff dependence proposed in Eq. (11) gives the softest EOS. We also note that decreasing µ$_{0}$ favors a deconfinement phase transition at lower densities for both parametriza- tions of the cutoff, Λ$_{1}$ andΛ$_{2}$. In the same figure we plot the EOS of the hadronic phase (dotted curve). The crossing point between the hadronic and the quark EOS indicates the phase transition from the hadronic phase to the quark phase using a Maxwell construction [24]. The In Fig. 3 we plot the effective bag pressure, Eq.(8), for each choice of the cutoff and for quark matter in β - equilibrium as a function of the baryonic density ρ$_{B}$ = ( ρ$_{u}$ + ρ$_{d}$ + ρ$_{s}$ ) / 3. As shown in references [11, 12] there is FIG. 4: Quark stellar matter in β -equilibrium with different choices of the cutoff: a) fractions and b) constituent masses of the quarks u , d , s as a function of density. FIG. 3: Effective bag pressure defined in Eq. (7) for different parametrizations of the cutoff and stellar quark matter in β - equilibrium. µ$_{s}$ = µ$_{d}$ = µ$_{u}$ + µ$_{e}$, µ$_{e}$ = µ$_{µ}$, ρ$_{e}$ + ρ$_{µ}$ = 1 $_{3}$(2 ρ$_{u}$ − ρ$_{d}$ − ρ$_{s}$ ) . (18) ρ$_{l}$ = 1 3 π $_{2}$k 3 $_{fl}$. (17) FIG. 2: Plot of the thermodynamic condition, Eq. (14), for the two parametrizations proposed (10,11). In β -equilibrium the conditions of chemical equilibrium and charge neutrality are given by The electron and muon densities are 380 400 420 440 460 480 500 0.4 0.6 0.8 1 1.2 1.4 1.6 (− / , ) (fm −3 )  (MeV) − /  −−  1  −−  1 − /  −−  2  −−  2 $_{0}$ = 400 MeV 2 4 6 8 10 0 200 400 / 0 M (MeV) 2 4 6 8 10 0 0.2 0.4 0.6 / 0 Y i b) Down Strange a) Up Down Strange Up 2 4 6 8 10 100 120 140 160 180 200 / $_{0}$ (Mev) B eff 1/4 $_{1}$( $_{0}$ = 400 MeV) $_{1}$( $_{0}$ = 350 MeV) $_{2}$( $_{0}$ = 400 MeV) $_{2}$( $_{0}$ = 350 MeV) NJL 4