On the probability distribution function of the mass surface density of molecular clouds. II.^⋆

Canadian Institute for Theoretical Astrophysics, University of Toronto, 60 St. George Street, Toronto, ON M5S3H8, Canada
e-mail: fischera@cita.utoronto.ca

Received: 14 February 2014
Accepted: 11 September 2014

Abstract

The probability distribution function (PDF) of the mass surface density of molecular clouds provides essential information about the structure of molecular cloud gas and condensed structures out of which stars may form. In general, the PDF shows two basic components: a broad distribution around the maximum with resemblance to a log-normal function, and a tail at high mass surface densities attributed to turbulence and self-gravity. In a previous paper, the PDF of condensed structures has been analyzed and an analytical formula presented based on a truncated radial density profile, ρ(r) = ρ_c/ (1 + (r/r₀)²)^{n/ 2} with central density ρ_c and inner radius r₀, widely used in astrophysics as a generalization of physical density profiles. In this paper, the results are applied to analyze the PDF of self-gravitating, isothermal, pressurized, spherical (Bonnor-Ebert spheres) and cylindrical condensed structures with emphasis on the dependence of the PDF on the external pressure p_ext and on the overpressure q^-1 = p_c/p_ext, where p_c is the central pressure. Apart from individual clouds, we also consider ensembles of spheres or cylinders, where effects caused by a variation of pressure ratio, a distribution of condensed cores within a turbulent gas, and (in case of cylinders) a distribution of inclination angles on the mean PDF are analyzed. The probability distribution of pressure ratios q^-1 is assumed to be given by P(q^-1) ∝ q^−k₁/ (1 + (q₀/q)^γ)^{(k₁ + k₂) /γ}, where k₁, γ, k₂, and q₀ are fixed parameters. The PDF of individual spheres with overpressures below ~100 is well represented by the PDF of a sphere with an analytical density profile with n = 3. At higher pressure ratios, the PDF at mass surface densities Σ ≪ Σ(0), where Σ(0) is the central mass surface density, asymptotically approaches the PDF of a sphere with n = 2. Consequently, the power-law asymptote at mass surface densities above the peak steepens from P_sph(Σ) ∝ Σ^-2 to P_sph(Σ) ∝ Σ^-3. The corresponding asymptote of the PDF of cylinders for the large q^-1 is approximately given by P_cyl(Σ) ∝ Σ^−4/3(1 − (Σ/Σ(0))^2/3)^−1/2. The distribution of overpressures q^-1 produces a power-law asymptote at high mass surface densities given by ${}^{\mathrm{⟨}}\mathit{P}_{\mathrm{sph}}\mathrm{(}\mathrm{\Sigma }{\mathrm{)}}^{\mathrm{⟩}}$∝ Σ^{−2k₂ − 1} (spheres) or ${}^{\mathrm{⟨}}\mathit{P}_{\mathrm{cyl}}\mathrm{(}\mathrm{\Sigma }{\mathrm{)}}^{\mathrm{⟩}}$∝ Σ^−2k₂ (cylinders).