Fractal dimensions and scaling laws in the interstellar medium: A new field theory approach.

Abstract

We develop a field theoretical approach to the cold interstellar medium (ISM). We show that a nonrelativistic self-gravitating gas in thermal equilibrium with a variable number of atoms or fragments is exactly equivalent to a field theory of a single scalar field $\ensuremath{\varphi}(\stackrel{\ensuremath{\rightarrow}}{x})$ with an exponential self-interaction. We analyze this field theory perturbatively and nonperturbatively through the renormalization group approach. We show a scaling behavior (critical) for a continuous range of the temperature and of the other physical parameters. We derive in this framework the scaling relation $\ensuremath{\Delta}M(R)\ensuremath{\sim}{R}^{{d}_{H}}$ for the mass on a region of size $R$, and $\ensuremath{\Delta}v\ensuremath{\sim}{R}^{q}$ for the velocity dispersion where $q=\frac{1}{2}({d}_{H}\ensuremath{-}1)$. For the density-density correlations we find a power-law behavior for large distances $\ensuremath{\sim}{|{\stackrel{\ensuremath{\rightarrow}}{r}}_{1}\ensuremath{-}{\stackrel{\ensuremath{\rightarrow}}{r}}_{2}|}^{2{d}_{H}\ensuremath{-}6}$. The fractal dimension ${d}_{H}$ turns out to be related with the critical exponent $\ensuremath{\nu}$ of the correlation lenght by ${d}_{H}=\frac{1}{\ensuremath{\nu}}$. The renormalization group approach for a single component scalar field in three dimensions states that the long-distance critical behavior is governed by the (nonperturbative) Ising fixed point. The corresponding values of the scaling exponents are $\ensuremath{\nu}=0.631\dots{}$, ${d}_{H}=1.585\dots{}$, and $q=0.293\dots{}$. Mean field theory yields for the scaling exponents $\ensuremath{\nu}=\frac{1}{2}$, ${d}_{H}=2$, and $q=\frac{1}{2}$. Both the Ising and the mean field values are compatible with the present ISM observational data: $1.4l~{d}_{H}l~2$, $0.3l~ql~0.6$. As typical in critical phenomena, the scaling behavior and critical exponents of the ISM can be obtained without dealing with the dynamical (time-dependent) behavior.