FRACTAL STRUCTURES DRIVEN by SELF-GRAVITY: Molecular clouds and the Universe

Fractals have been introduced by Mandelbrot (1975) to define geometrical ensembles, or mathematical sets, that have a fractional dimension. He pioneered the study of very irregular mathematical sets, where the methods of classical calculus cannot be applied. Fractals are not smooth nor differentiable; they are characterized by self-similarity. Their geometrical structure has details at all scales, and the details are representative of the whole. Fractals are very rich, since they give the best approximation for many natural phenomena, that cannot be represented by regular and smooth geometries (natural systems are only fractal between two boundaries, with upper and lower cut-offs, and not true mathematical fractals, of course). There can exist a large part of randomness in fractals, and the fractal dimension and exponents of scaling laws are the tools to quantify the hidden order in them. There are many definitions of the dimension, trying to quantify how much space a system fills. We will use mainly the Hausdorff dimension D, based on the Hausdorff measure, that generalizes the notion of length, area and volume. When the length scale is magnified by λ, a regular and smooth set’s area and volume are respectively scaled by λ2, λ3, while the fractal set measure is scaled by λD. For the astrophysical systems that we describe below, the mass contained within a scale r is M∝ rD, with D a fractional number between 1 and 3.