Evolution of the local power-law index (Γ) of a mass distribution. The influence of the mass transfer rate (ξ) and the fragmentation rate (ϕ) dependences with the mass M of the parental fragment is shown in (a), (b) and (c), (d) respectively. The solid lines, triangles and squares represent respectively the theoretical solutions from Eq. D.3, the Monte Carlo solutions obtained from a sample of 10⁶ objects distributed in a power law of index Γ₀ fragmenting according to Eq. D.8 and Eq. D.12. (a)-(c): Local power-law index (Γ) as a function of the initial slope Γ₀ at scale R₀/2. Dashed green and blue lines indicate the asymptotic limits of Γ(R). (b)-(d): Local power-law index (Γ) as a function of the spatial scale R for different initial slopes. The Monte Carlo solutions do not go below R/R₀ ≈ 0.3 because the number of fragment constituting the fCMF becomes too large to compute.