Estimating the nuclear saturation parameter via low-mass neutron star asteroseismology

We examine the fundamental ($f$-) and the 1st pressure ($p_1$-) mode frequencies in gravitational waves from cold neutron stars constructed with various unified realistic equations of state. With the calculated frequencies, we derive the empirical formulae for the $f$- and $p_1$-mode frequencies, $f_f$ and $f_{p_1}$, as a function of the square root of the stellar average density and the parameter ($\eta$), which is a combination of the nuclear saturation parameters. With our empirical formulae, we show that by simultaneously observing the $f$- and $p_1$-mode gravitational waves, when $1.5\lesssim f_{p_1}/f_f\lesssim 2.5$ (which corresponds to neutron star models with the mass of $\lesssim 0.9M_\odot$), one could estimate the value of $\eta$ within $\sim 10\%$ accuracy, which makes a strong constraint on the EOS for neutron star matter. In addition, we find that the maximum $f$-mode frequency is strongly associated with the minimum radius of neutron star. That is, if one would observe a larger frequency of the $f$-mode, one might constrain the upper limit of the minimum neutron star radius.