the Einstein tensor and d’Alembertian of $g_{*\mu\nu}$, respectively. $T_*$ and $\alpha(\varphi)$ are defined as $$\begin{aligned}
      T_* &\equiv& T_{*\mu}^{~~\mu} 
      \equiv T^{\mu\nu}_* g_{*\mu\nu},\\
      \alpha(\varphi) &\equiv& \frac{d\ln A(\varphi)}
      {d\varphi}.\end{aligned}$$

The parameters in the parametrized post-Newtonian framework are given by [@will1993; @de1992] $$\begin{aligned}
      & & 1-\gamma_{Edd} = \frac{2\alpha_0^2}{1+\alpha_0^2}, \\
      & & \beta_{Edd}-1 = \frac{\beta_0\alpha_0^2}{2(1+\alpha_0^2)^2},\\
      & & \xi = \alpha_1 = \alpha_2 = \alpha_3 = 0,   \end{aligned}$$ where $\beta_{Edd}$ and $\gamma_{Edd}$ are the so-called Eddington parameters. We have defined $$\begin{aligned}
  \label{eq:defofalpha0}
      \alpha_0 &\equiv& \alpha(\varphi_0), \\
      \label{eq:defofbeta0}
      \beta_0 &\equiv& \frac{d\alpha}
      {d\varphi}(\varphi_0),\end{aligned}$$ and $\varphi_0$ is the value of the scalar field $\varphi$ in the spatial asymptotic region. We assume that the cosmological evolution of the scalar field is sufficiently slow in comparison with the characteristic time scale of the local gravitational process of the isolated object considered here. From this assumption $\varphi_0$ is regarded as the cosmological value of the scalar field. On the other hand, we can identify the asymptotic value $\varphi_0$ to the value of the scalar field in the matching region in the matching approach to the $N$-compact-body problem (see Appendix A of [@de1992]). Then, the solar-system experimental constraints are [@lebach1995] $$\label{eq:constraint1}
      \gamma_{Edd}= 0.9996\pm0.0017$$ and [@williams1995] $$\label{eq:constraint2}
  4\beta_{Edd}-\gamma_{Edd}-3=-0.0007\pm0.0010.$$

We summarize equations for the structure 