  The macroscopic properties of compact stars in modified gravity theories can
be significantly different from the general relativistic (GR) predictions.
Within the gravitational context of scalar-tensor theories, with a scalar field
$\phi$ and coupling function $\Phi(\phi)= \exp[2\phi/\sqrt{3}]$, we investigate
the hydrostatic equilibrium structure of neutron stars for the simple potential
$V(\phi)= \omega\phi^2/2$ defined in the Einstein frame (EF). From the scalar
field in the EF, we also interpret such theories as $f(R)$ gravity in the
corresponding Jordan frame (JF). The mass-radius relations, proper mass, and
binding energy are obtained for a polytropic equation of state (EoS) in the JF.
Our results reveal that the maximum-mass values increase substantially as
$\omega$ gets smaller, while the radius and mass decrease in the
low-central-density region as we move further away from the pure GR scenario.
Furthermore, a cusp is formed when the binding energy is plotted as a function
of the proper mass, which indicates the appearance of instability.
Specifically, we find that the central-density value where the binding energy
is a minimum corresponds precisely to $dM/d\rho_c^J = 0$ on the
$M(\rho_c^J)$-curve.
