On the probability distribution function of the mass surface density of molecular clouds. II.
Bibliographic record
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) = ρc/ (1 + (r/r0)2)n/ 2 with central density ρc 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 pext and on the overpressure q-1 = pc/pext, where pc 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 Psph(Σ) ∝ Σ-2 to Psph(Σ) ∝ Σ-3. The corresponding asymptote of the PDF of cylinders for the large q-1 is approximately given by Pcyl(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 ∝ Σ−2k2 − 1 (spheres) or ∝ Σ−2k2 (cylinders).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.021 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.003 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.003 | 0.003 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.002 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".