A Fractal Origin for the Mass Spectrum of Interstellar Clouds

Interstellar molecular clouds have power-law size L and mass M distributions of the form n(L) dL = L^−α_L dL and n(M) dM = M^−α_M dM, where M ∝ L^k is also a power law. These relations are shown to result from the fractal and scale-free nature of interstellar gas with power indices that are independent of distance. The results are α_L = 1 + D and α_M = 1 + D/κ for interstellar fractal dimension D = 2.3 ± 0.3 and a value of κ in the range 2.4-3.7, as determined from cloud surveys in the literature. The same fractal dimension also results from the expected relation D = κ when the M(L) correlation includes many different surveys, spanning a range of 10¹⁰ in mass. These results imply that interstellar CO clouds are the unresolved parts of a pervasive fractal structure in the interstellar gas. The similarity between n(M) for interstellar clouds and n(M) for globular clusters suggests that the clusters formed inside fractal progenitor clouds at a nearly constant efficiency.