50 100 150 R 99 = 20 0 0.02 0.04 0.06 0.08 0.01 0.02 0.03 0.04 0.05 2M/R 99 3 its ground state. We can observe that there is no time dependence in (8). Furthermore, there are new numeri- cal factors that appear due to the averaging performed in ˆ T$_{µν}$ [19]. For instance 〈 Φ $^{4}$〉 = 2 R 4 and 〈 Φ $^{6}$〉 = 5 R $^{6}$, in such a way that we cannot recover the original cos(Φ /f$_{a}$ ) from which we departed. Following a similar procedure but now applied to the scalar wave equation (3), with a potential (7) and the spherically symmetric metric (4), the Einstein-Klein-Gordon system is obtained: Following standard definitions [8], we rewrite the system (9) in dimensionless variables: x = rm , R = $_{σ/}$√$_{4}$$_{πG}$ and ˜ B = m $^{2}$B/E$^{2}$, and we have found it convenient to define the dimensionless self-interaction term Then, the true expansion parameter of (1) is ΛΦ, an it is always a small parameter . Another interesting issue is that the dependence of the radius R$_{99}$ (defined as the ra- dius where 99% of the total gravitational mass is reached) on the value of Λ is weak, as it is shown in upper panel of Fig. 2. Combining the invariance of the radius as Λ increases, with the decrease in the mass, means that the self-gravitating system made of a scalar field that has an axion-like potential has a lower compactness (2 M/R$_{99}$ ) as the self-interaction term increases. This “newtoniza- tion” of the system is shown on the lower panel of Fig. 2. A ′ A $^{2}$r + 1 r 2 ( 1 − 1 A ) = − 8 πG 〈 T 0 〉 , B ′ ABr − 1 r 2 ( 1 − 1 A ) = 8 πG 〈 T 1 〉 , R ′′ + ( 2 r + B ′ 2 B − A ′ 2 A ) R ′ + A [ ( E 2 B − m 2 2 ) R + + m $^{2}$R3 6 f 2 a − m $^{2}$R5 48 f 4 a ] = 0 . (9) Λ = 1 24 π ( m$_{p}$ f$_{a}$ ) . (10) FIG. 2. R 99 and compactness for different configurations. The dependence of R 99 on Λ is negligible. For a given value of σ (0), the compactness decreases as Λ increases. By imposing regularity at the origin and flatness at infin- ity, system (9) is solved using a shooting method. Even though the set of equations (9) is very similar to the case for typical BS, [8], the behavior we found for the fam- ily of solutions with zero-nodes is completely different. A full set of equilibrium configurations is shown in Fig. 1, where the gravitational mass is plotted for different values of σ (0) and Λ. The equilibrium configurations have a maximum mass M$_{max}$ at some σ (0) = σ$_{c}$ for each value of Λ. But the switch in the potential sign of the Φ 4 term produces a significant change in the behavior of M$_{max}$ in comparison with standard BS [8]. The relation M$_{max}$ ∼ Λ 1 / 2 is no longer satisfied. Instead of increasing M$_{max}$ as we increase the value of Λ, we found a decreas- ing M$_{max}$ . This effect is expected since instead of adding a repulsive interaction between the particles of the sys- tem, the change in the sign due to the cosine-like poten- tial (1) implies an attractive potential. Thus the total number of particles needed to form an equilibrium con- figuration that balance the gravitational collapse against the quantum pressure is lower than the case of a repul- sive potential. One can think that this effect is apparent and as soon as the complete potential (1) is implemented in the EKG system, a different behavior would be seen. But the decrease in the mass of the equilibrium configu- rations is a robust behavior. The masses of equilibrium configurations including up to the fourth power of Φ in the Taylor series are plotted also in Fig. 1, illustrating this robustness. The previous results where obtained assuming arbi- trary values of the mass m of the scalar field associated with the axion as well as free values for the decay con- stant f$_{a}$ . But the mass of the axion is constrained by as- trophysical and cosmological considerations to lie in the range 10 − 5 eV ≤ m ≤ 10 − 3 eV and the decay constant is related to the axion mass by m = 6 µ eV ( 10 12 GeV f a ) [4, 20] With these two restriction we have 10 13 < Λ < 10 17 and then, the previous selection of dimensionless variables { x,σ,A. ˜ B } ) is now inadequate in order to numerically solve the system (9). After some frustrated attempts, we found that a more suitable set of variables to solve the system (9) is the following: II. AXION-STAR R = f$_{a}$ √$_{m}$ σ , r = m$_{p}$ f$_{a}$ √ m 4 π x, 1 ˜ B = E 2 m $^{2}$B . (11) The bigger the value of Λ, the lower the differences on the masses. This is because when Λ is increased, σ$_{c}$ de- creases (and equivalently Φ( r ) where we are interested). Since Λ >> 1, it is natural to think that the result- ing axion star will have a small compactness and low