\int_{0}^{\infty} dr e^{-\nu/2}\left(1-\frac{2\mu}{r}\right)^{-1/2}
  \zeta \left[-\frac{d^2}{dr_*^2}+V(r)\right]\zeta,$$ where $$\begin{aligned}
  \zeta&\equiv& r\delta\varphi, \\
  dr_* &\equiv& e^{\nu/2}\left(1-\frac{2\mu}{r}\right)^{1/2}dr, \\
  V(r) &\equiv& \frac{1}{r}\left(1-\frac{2\mu}{r}\right)\left[
    \frac{\nu^{\prime}}{2}-\frac{\mu^{\prime}r-\mu}{r(r-2\mu)}\right]
  e^{\nu}-4\pi \beta_0 (-\tilde{\rho}+3\tilde{p})e^{\nu},\end{aligned}$$ and see Appendix B of [@htww1964] for the general relativistic terms. The general relativistic part is positive definite if $\Gamma>\Gamma_c$, and $\Gamma_c \to 4/3$ in the Newtonian limit [@chandrasekhar1964]. The second term is positive definite because the eigenvalues of the operator, $-d^2/dr_*^2+V$, are all positive for an arbitrary coupling function $A(\varphi)$ if $m/r_s$ is sufficiently small, which has been shown in [@harada1997]. Therefore, the general relativistic equilibrium solution in which the central density is sufficiently small is stable for the case $\alpha(\varphi_0)=0$ if $\Gamma > \Gamma_c \simeq 4/3$.

Results
=======

Hereafter we restrict our attention to the coupling function of the quadratic form $$\label{eq:quadratic}
  A(\varphi)=\exp\left(\frac{1}{2}\beta\varphi^2\right).$$ For this model, the solar-system experiments constrain the present cosmological value of the scalar field through Eqs. (\[eq:constraint1\]) and (\[eq:constraint2\]) as $$|\varphi_0|\alt 0.032|\beta|^{-1}$$ and $$|\varphi_0|\alt\cases{
    0.012(1+\beta)^{-1/2}|\beta|^{-1} & for $\beta>-1 $, \cr
    0.029|1+\beta|^{-1/2}|\beta|^{-1} & for $\beta<-1 $, \cr
    }$$ respectively. In particular, if $\varphi_0=0$, the post-Newtonian limit of this theory agrees completely with that of general relativity because $\alpha(\varphi_0)=0$.

$\beta\protect\agt-4.35$ case
-----------------------------

We present here the results of the case $\beta=-4$, but the features are 