Stability and Size of Galaxies from Planck's Constant

From a classical point of view, there are no arguments capable of completely explaining the stability and the size of galaxies: the only assumption is that they are considered to be relaxed (and virialized) systems where gravity is the overall interaction [1]. Such a force is considered as a “Newtonian” interaction and the confining potentials, due to the mutual attractions of stars (and the other components as dust and gas clouds) can have several forms. For example, logarithmic potentials well describe the regular motion of stars before the onset of chaos [2]. In any case, the “stability is an assumption” and sizes are deduced from observations. Actually, the problem is extremely involved since galaxies undergo environmental effects, being never isolated systems; they always belong to large gravitationally bound systems as loose and tight groups, associations or clusters of galaxies and the observational times are so short that the overall dynamics can be only inferred [1],[3]. Besides, galaxies have to be related to some cosmological model and, due to cosmological evolution of large scale structures, they should be connected to some theory of primordial perturbations [4],[5]. For these reasons, it is not senseless to ask for some quantum signature in the today observed galaxies [6]. The main point, however, is to connect the estremely large size of galaxies (∼ 10kpc) with the extremely small numbers of quantum mechanics (h ∼ 10 erg sec). In this letter, we want to show that, for a given galaxy, a minimal characteristic action is of the order of h and, furthermore, the onset of chaos [7] is prevented if and only if the characteristic sizes of a galaxy are related to the Planck constant. In other words, it is the quantum signature which stabilizes the galaxies; furthermore it gives rise to their characteristic sizes, where, by “sizes”, we intend geometrical and kinematical quantities which assignes a galaxy. This result is not particular since the collective features (in particular the stability and the confinement properties) of several mesoscopic and macroscopic systems can be explained only by invoking quantum