A necessary condition for best fitting analytical to simulated density profiles in dark matter haloes

Analytical and geometrical properties of generalized power-law (GPL) density profiles are reviewed, and special effort is devoted to the special cases where GPL density profiles reduce to (i) a double power-law (DPL), and (ii) a single power-law (SPL). Then GPL density profiles are compared with simulated dark haloes (SDH) density profiles, and nonlinear least-squares fits are prescribed, involving five parameters (a scaling radius, r0, a scaling density, ρ0, and three exponents, α, β, γ), which specify the fitting profile (RFSM5 method). More specifically, the validity of a necessary condition for the occurrence of an extremal point, is related to the existence of an intersection between three surfaces in a 3-dimension space. Using the algorithm makes also establish that the extremal point is a fiducial minimum, while the explicit calculation of the Hessian determinant is avoided to gain in simplicity. In absence of a rigorous proof, the fiducial minimum can be considered as nothing but a fiducial absolute minimum. An ∗Astronomy Department, Padua Univ., Vicolo Osservatorio 2, I-35122 Padova, Italy email: [email protected]