[@bunta2]. Notice that other possibilities for this model with $\sigma-\rho$ and $\omega-\rho$ couplings have already been discussed in the literature as in [@piek06], for instance.

The mesonic equations of motion in the Thomas-Fermi approximation become

$$\nabla^2 \phi = m_s^2\phi- g_s \rho_s + \frac{\kappa}{2} \phi^2 +
\frac{\lambda}{6} \phi^3
\label{elphi2}$$

$$\nabla^2 V_0 = m_v^2 V_0 - g_v \rho + 2 \Lambda_v g_v^2\, V_0\, g_{\rho}^2
b_0^2,
\label{elV02}$$

$$\nabla^2 b_0 =m_\rho^2 b_0 -\frac{g_\rho}{2} \rho_3 + 2 \Lambda_v
g_{\rho}^2 b_0 g_v^2 V_0^2,
\label{elb02}$$

$$\nabla^2 A_0 =-e \rho_p,\label{elA02}$$

and the expression for the energy reads

$$E= \sum_i \int \d^3r \left(
\gamma \int_0^{k_{Fi}(r)}  \frac{\d^3p}{(2\pi)^3}\,
\sqrt{{\mathbf p}^2 + {M^*}^2} \right.$$ $$+\frac{1}{2} \left[ (\nabla \phi)^2 + m_s^2 \phi^2 
-(\nabla V_0)^2 - m_v^2 V_0^2 \right.$$ $$- (\nabla b_0)^2 - m_\rho^2 b_0^2  - (\nabla A_0)^2$$ $$\left. + g_v V_0 \rho + \frac{g_\rho}{2} \rho_3 b_0 + e A_0 \rho_p \right]$$ $$\left. +\frac{\kappa}{6} \phi^3 + \frac{\lambda}{24} \phi^4 
-\Lambda_v g_v^2 V_0^2 g_\rho^2 b_0^2 \right).$$

All other expressions are very similar to the ones obtained from the TW model and can be read off from them bearing in mind that the density dependent couplings have to be replaced by the constant couplings. In particular the chemical potentials do not contain the rearrangement term $\Sigma_0^R$.

Numerical 