Compact static stars in minimal dilatonic gravity

In the version \cite{Fiziev14} of this paper we presented for the first time the basic equations and relations for relativistic static spherically symmetric stars (SSSS) in the model of minimal dilatonic gravity (MDG). This model is {\em locally} equivalent to the f(R) theory of gravity and gives an alternative description of the effects of dark matter and dark energy using the Branse-Dicke dilaton $\Phi$. To outline the basic properties of the MDG model of SSSS and to compare them with general relativistic results, in the present paper we use the relativistic equation of state (EOS) of neutron matter as an ideal Fermi neutron gas at zero temperature. We overcome the well-known difficulties of the physics of SSSS in the f(R) theories of gravity \cite{Felice10,Berti} applying novel highly nontrivial nonlinear boundary conditions, which depend on the global properties of the solution and on the EOS. We also introduce two pairs of new notions: cosmological-energy-pressure densities and dilaton-energy-pressure densities, as well as two new EOS for them: cosmological EOS (CEOS) and dilaton EOS (DEOS). Special attention is paid to the dilatonic sphere (in brief -- disphere) of SSSS, introduced in this paper for the first time. Using several realistic EOS for neutron star (NS): SLy, BSk19, BSk20 and BSk21, and current observational two-solar-masses-limit, we derive an estimate for scalar-field-mass $m_\Phi \sim 10^{-13} eV/c^2 \div 4\times 10^{-11} eV/c^2$. Thus, the present version of the paper reflects some of the recent developments of the topic.