We study the structure of Milky Way (MW)- and cluster-sized halos in a Lambda Cold Dark Matter (CDM) cosmology with self-interacting (SI) dark particles. The cross section per unit of particle mass has the form sigma = sig_0(1/v_100)^alpha, where sig_0 is a constant in units of cm^2/gr and v_100 is the relative velocity in units of 100 km/s. Different values for sigma with alpha= 0 or 1 were used. For small values of sigma = const. (sig_0<0.5), the core density of the halos at z=0 is typically higher at a given mass for lower values of sig_0 or, at a given sig_0, for lower masses. For values of sig_0 as high as 3.0, the halos may undergo the gravothermal catastrophe before z=0. When alpha = 1, the core density of cluster- and MW-sized halos is similar. Using sigma = 0.5-1.0x(1/v_100), our predictions agree with the central densities and the core scaling laws of halos both inferred from the observations of dwarf and LSB galaxies and clusters of galaxies. The cumulative Vmax-functions of subhalos in MW-sized halos with (sig_0,alpha) = (0.1,0.0), (0.5,0.0) and (0.5,1.0) agree roughly with observations (luminous satellites) for Vmax > 30 km/s, while at Vmax = 20 km/s the functions are a factor 5-8 higher, similar to the CDM predictions. The halos with SI have slightly more specific angular momentum at a given mass shell and are rounder than their CDM counterparts. We conclude that the introduction of SI particles with sigma propto 1/v_100 may remedy the cuspy core problem of the CDM cosmogony, while the subhalo population number remains similar to that of the CDM halos.
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