Analytical expressions for the saturation density of asymmetric nuclear matter as well as its binding energy and incompressibility at saturation density are given up to 4 th-order in the isospin asymmetry \delta = ( \rho _ { n } - \rho _ { p } ) / \rho using 11 characteristic parameters defined by the density derivatives of the binding energy per nucleon of symmetric nuclear matter , the symmetry energy E _ { \text { { sym } } } ( \rho ) and the 4 th-order symmetry energy E _ { \text { { sym, 4 } } } ( \rho ) at the normal nuclear density \rho _ { 0 } .
Using an isospin- and momentum-dependent modified Gogny ( MDI ) interaction and the Skyrme-Hartree-Fock ( SHF ) approach with 63 popular Skyrme interactions , we have systematically studied the isospin dependence of the saturation properties of asymmetric nuclear matter , particularly the incompressibility K _ { \mathrm { sat } } ( \delta ) = K _ { 0 } + K _ { \mathrm { sat, 2 } } \delta ^ { 2 } + K _ { \mathrm { sat, 4 } } % \delta ^ { 4 } + O ( \delta ^ { 6 } ) at saturation density .
Our results show that the magnitude of the higher-order K _ { \mathrm { sat, 4 } } parameter is generally small compared to that of the K _ { \mathrm { sat, 2 } } parameter .
The latter essentially characterizes the isospin dependence of the incompressibility at saturation density and can be expressed as K _ { \mathrm { sat, 2 } } = K _ { \mathrm { sym } } -6 L - \frac { J _ { 0 } } { K _ { 0 } } L , where L and K _ { \mathrm { sym } } represent , respectively , the slope and curvature parameters of the symmetry energy at \rho _ { 0 } while J _ { 0 } is the third-order derivative parameter of symmetric nuclear matter at \rho _ { 0 } .
Furthermore , we have constructed a phenomenological modified Skyrme-like ( MSL ) model which can reasonably describe the general properties of symmetric nuclear matter and the symmetry energy predicted by both the MDI model and the SHF approach .
The results indicate that the higher-order J _ { 0 } contribution to K _ { \mathrm { sat, 2 } } generally can not be neglected .
In addition , it is found that there exists a nicely linear correlation between K _ { \mathrm { sym } } and L as well as between J _ { 0 } / K _ { 0 } and K _ { 0 } .
These correlations together with the empirical constraints on K _ { 0 } , L , E _ { \text { { sym } } } ( \rho _ { 0 } ) and the nucleon effective mass lead to an estimate of K _ { \mathrm { sat, 2 } } = -370 \pm 120 MeV .