  We analyze the uncertainties introduced in the determination of the neutron
star matter proton fraction, in a range of densities close to the saturation
density, if the cold $\beta$-equilibrium neutron star matter equation of state
(EoS) is known. In particular, we discuss the effect of neglecting the muon
contribution and of considering that the energy density of nuclear matter is
well described by taking only terms until second order in the proton-neutron
asymmetry. It is shown that two types of uncertainties may be associated with
the extraction of the symmetry energy from the $\beta$-equilibrium equation of
state: an overestimation if terms above the parabolic approximation on the
asymmetry parameter are neglected, or an underestimation if the muon
contribution is not considered. The effect of the uncertainty on the symmetric
nuclear matter EoS on the determination of the proton fraction is discussed. It
could be shown that the neutron star mass-radius curve is sensitive to the
parabolic approximation on the asymmetry parameter.
