Cusp-Core Problem in the Clausius Virial Dynamical Theory of Fundamental Plane

The dynamical theory of the Fundamental Plane is based on the existence of a maximum in the Clausius Virial potential energy (CV) of a stellar component, V_{*} when it is completely embedded inside a dark matter (DM) halo. At its first order approximation (linear approximation) the theory was developed by modeling the two components with two power-law density profiles and two homogeneous cores. To extend the theory to an higher order (non-linear) we explore the effect on an homogeneous stellar component due to a DM halo with selected Zhao' density profile characterized by three exponents (α,β,γ), where γ is the slope at r→, β at r→∞ and α describes the transition region between the above mentioned limits. The aim is to compare the predictions of the theory in two special cases: (2,3,0) and (1,3,1), i.e. the NFW density profile. We follow the general method proposed by tet{dbin_Caimmi93} for two homoidally striated ellipsoids in virial equilibrium described by tensor virial equations extended to two components. The role of the dark to bright mass ratio m and of the halo concentration C_{D} in order to produce the maximum of CV in both cases are taken into account. The relevance of the slope at which the maximum appears inside the halo DM profile is also considered