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On the Density Distribution in Star-forming Interstellar Clouds
Alexei G. Kritsuk, Michael L. Norman, Rick Wagner
TL;DR
The paper examines how gravity changes molecular-cloud density distributions that are otherwise expected to be lognormal. Using deep AMR simulations of isothermal self-gravitating supersonic turbulence, it finds high-density power-law tails and interprets them with self-similar collapse solutions. The predicted mass-density slopes range from −7/4 to −3/2 and broadly agree with simulations and observations.
Problem
The paper investigates whether high-density power-law tails in self-gravitating molecular-cloud density PDFs have a universal power index and what determines their slope.
Method
The authors analyze a deep AMR simulation of isothermal, self-gravitating supersonic turbulence and compare its density PDFs with self-similar collapse solutions.
Results
Star-forming clouds develop high-density power-law tails rather than purely lognormal density distributions, with predicted mass-density indices m ∈ [−7/4,−3/2].
Takeaways & Limitations
Self-similar r−2 and r−12/7 collapse laws provide a basis for interpreting density-PDF tails across simulations and observations.
Takeaways & Limitations
The model uses a low virial parameter, which can exaggerate self-gravity; higher virial parameters should produce weaker, shallower tails over a more extended density range.
Abstract
from arXiv · showhide
We use deep adaptive mesh refinement simulations of isothermal self-gravitating supersonic turbulence to study the imprints of gravity on the mass density distribution in molecular clouds. The simulations show that the density distribution in self-gravitating clouds develops an extended power-law tail at high densities on top of the usual lognormal. We associate the origin of the tail with self-similar collapse solutions and predict the power index values in the range from -7/4 to -3/2 that agree with both simulations and observations of star-forming molecular clouds.
1. INTRODUCTION
Density PDFs are lognormal in non-self-gravitating turbulence, but high-density power-law tails appear in self-gravitating simulations and star-forming molecular clouds. The paper investigates whether these tails have a universal power index and what determines its slope.
- Non-self-gravitating isothermal supersonic turbulence is generally expected to produce a lognormal mass-density PDF.
- High-density power-law tails have been reported in simulations of self-gravitating turbulent interstellar media and observations of star-forming molecular clouds.
- The paper asks whether self-gravitating isothermal turbulence has a universal power index and what determines the slope.
- A deep AMR simulation follows star formation from turbulent initial conditions on scales of a few parsecs down to a few AU.
2. NUMERICAL EXPERIMENT
The numerical experiment simulates driven, isothermal, self-gravitating turbulence in a molecular-cloud-scale domain, resolving collapse with adaptive mesh refinement down to 2 AU. Turbulence is first driven to a statistically developed state, then forcing is stopped while gravity and refinement proceed.
- The simulation solves hydrodynamics with self-gravity and large-scale random forcing using an isothermal equation of state and periodic boundaries.
- Adaptive mesh refinement resolves the Jeans length in collapsing regions, using a 512^3 root grid, five refinement levels, and 2 AU minimum scales in a 5 pc box.
- The model represents a 3.4 × 10^3 M⊙ molecular cloud with mean density n0(H2) = 500 cm^-3 and conditions prone to gravitational collapse.
- Turbulence is driven for 4.8tdyn with 40% dilatational and 60% solenoidal power, then forcing is turned off for about 0.29tdyn ≈ 0.43tff of self-gravitating AMR evolution.
3. EFFECTS OF SELF-GRAVITY
Self-gravity transforms an initially lognormal density distribution into an extended high-density power-law tail, while collapse solutions connect the observed slopes to self-similar density profiles. Rotation and numerical resolution affect the highest-density regime.
- Density-PDF evolution: After gravity begins, the initially lognormal PDF develops an extended high-density tail with slope about −1.7, reaching across nearly 10 dex in probability.The tail begins near ρ/ρ0 ∼ 10 and extends over more than 6 dex in density.
- Rotational support and resolution: Above ρ/ρ0 > 10^7, a shallower slope near −1 indicates density pile-up associated with additional rotational support.The paper notes that the enforced maximum of five refinement levels may also contribute to this high-density break.
- Hierarchical collapse: Dense-core PDFs show power-law segments with slopes from −1.25 to −1.75, supporting a hierarchical collapse origin for the global tail.
- Self-similar collapse: Self-similar collapse solutions develop an early ρ ∼ r−2 profile and later an expansion-wave profile ρ ∼ r−3/2 near the center.For a spherical profile ρ ∝ r−n, the mass-density PDF is itself a power law, linking collapse profiles to PDF slopes.
- Self-similar collapse: The predicted mass-density PDF slopes include mLP = −3/2 for r−2 collapse and mEW = −2 for the later r−3/2 profile.
- Rotational support and resolution: Disk structure remains uncertain because the simulated disks have steeper ρ ∼ r−3 profiles than the equilibrium singular-disk scenario and require higher resolution for proper modeling.
- Density-PDF evolution: At the simulation end, the density-PDF slope is −1.67 and slowly evolves toward shallower values.It may remain steeper than −1.5, the value associated with an r−2 density profile.
- Projected density: The projected-density PDF has slope −2.50±0.03, between the predicted PF and LP values and inconsistent with the EW prediction p = −4.Individual projections show bumps and slopes ranging broadly from about −2 to steeper than −3.
4. DISCUSSION
The simulation produces high-density power-law tails, while their slopes depend on collapse conditions, magnetization, virial state, and numerical or driving assumptions.
- Driving and evolution: The lack of turbulent driving is expected to have only weak influence by t = 0.43tff because the free-fall time is shorter than the energy-decay timescale.The authors state that driven and decaying models should not differ significantly during this fraction of the first free-fall time.
- Virial state: A small virial parameter can exaggerate self-gravity; α ≈ 1 should produce a weaker effect, a shallower tail, and a later departure from lognormality.Cloud-to-cloud variations in α, cloud age, and projection effects may account for observed tail diversity.
- Magnetic fields: Magnetization variations make little or no difference to the high-density end of the density PDF in the cited isothermal and multiphase MHD calculations.Without self-gravity, the high-density distribution remains lognormal.
- Magnetic fields: −7/4 to −3/2 is the predicted mass-density power-index range when B ∼ ρ^γ with γ ∈ [1/2,2/3].The corresponding magnetic-field PDF prediction is based on the same density–field correlation.
5. CONCLUSIONS
Star-forming clouds show high-density power-law tails unlike non-star-forming clouds, and self-similar collapse laws provide predicted indices consistent with simulations and observations.
- Observed and simulated PDFs: Star-forming clouds show strong high-density deviations from lognormality, whereas clouds without active star formation display purely lognormal density distributions.This agrees with the density distributions found in the deep AMR simulation and observations.
- Physical interpretation: r−2 isothermal and r−12/7 pressure-free collapse laws are attributed as the origin of the density-PDF tails.These self-similar laws control the density profiles of collapsing structures.
- Predictions: m ∈ [−7/4,−3/2] and p ∈ [−2.8,−2] are the predicted indices for mass-density and projected-density power-law tails.The predictions depend on parent-cloud conditions such as Mach number and virial parameter and broadly agree with simulations and observations.
- Implications: The results may help reconcile phenomenological theories of star formation and interpret numerical simulations through the proposed phenomenologies.
DENSITY PDF IN STAR-FORMING CLOUDS 5
This section contains references to prior work on turbulence, magnetic fields, collapse, numerical methods, and observed molecular-cloud density distributions.
- Turbulence and density structure: The cited literature includes studies of interstellar turbulence and density distributions by Vázquez-Semadeni, von Weizsäcker, and related authors.
- Star formation and collapse: The references include theoretical and numerical studies of star formation, gravitational collapse, and molecular-cloud turbulence.
- Numerical methods: The bibliography cites work on adaptive mesh refinement, numerical stability, and computational astrophysics.
- Magnetized turbulence: The cited studies also cover magnetic fields, MHD turbulence, and phenomenological models of star formation.