  We adopt two- and three-body nuclear forces derived at the
next-to-next-to-leading-order (N2LO) in the framework of effective chiral
perturbation theory (ChPT) to calculate the equation of state (EOS) of
$\beta$-stable neutron star matter using the Brueckner--Hartree--Fock many-body
approach. We use the recent optimized chiral two-body nuclear interaction at
N2LO derived by \cite{ekstrom1} and two different parametrizations of the
three-body N2LO interaction: the first one is fixed to reproduce the saturation
point of symmetric nuclear matter while the second one is fixed to reproduce
the binding energies of light atomic nuclei. We show that in the second case
the properties of nuclear matter are not well determined whereas in the first
case various empirical nuclear matter properties around the saturation density
are well reproduced. We also calculate the nuclear symmetry energy $E_{sym}$ as
a function of the nucleonic density and compare our results with the empirical
constraints obtained using the excitation energies of isobaric analog states in
nuclei and the experimental data on the neutron skin thickness of heavy nuclei.
We next calculate various neutron star properties and in particular the
mass-radius and mass-central density relations. We find that the adopted
interactions based on a fully microscopic framework, are able to provide an EOS
which is consistent with the present data of measured neutron star masses and
in particular with the mass $M=2.01\pm0.04 M_\odot$ of the neutron star in PSR
J0348+0432. We finally consider the possible presence of hyperons in the
stellar core and we find a softening of the EOS and a substantial reduction of
the stellar maximum mass in agreement with similar calculations present in the
literature.
