The interdependence between density PDF, CMF and IMF and their relation with Mach number in simulations

The initial mass function (IMF) of stars and the corresponding cloud mass function (CMF), traditionally considered universal, exhibit variations that are influenced by the local environment. Notably, these variations are apparent in the distribution’s tail, indicating a possible relationship between local dynamics and mass distribution. Our study is designed to examine how the gas PDF , the IMF and the CMF depend on the local turbulence within the interstellar medium (ISM). We run hydrodynamical simulations on small star-forming sections of the ISM under varying turbulence conditions, characterized by Mach numbers of 1, 3.5, and 10, and with two distinct mean densities. This approach allowed us to observe the effects of different turbulence levels on the formation of stellar and cloud masses. The study demonstrates a clear correlation between the dynamics of the cloud and the IMF. In environments with lower levels of turbulence likely dominated by gravitational collapse, our simulations showed the formation of more massive structures with a powerlaw gas PDF, leading to a top-heavy IMF and CMF. On the other hand environment dominated by turbulence result in a lognormal PDF and a Salpeter-like CMF and IMF. This indicates that the turbulence level is a critical factor in determining the mass distribution within star-forming regions.

💡 Research Summary

This paper investigates the intertwined relationships among the gas density probability distribution function (PDF), the core mass function (CMF), and the stellar initial mass function (IMF) in turbulent star‑forming environments. Using the adaptive‑mesh refinement code RAMSES, the authors performed a suite of high‑resolution hydrodynamic simulations of small interstellar medium (ISM) patches with two mean densities (10³ cm⁻³ and 10⁴ cm⁻³) and three turbulent Mach numbers (𝓜 = 1, 3.5, 10). The initial conditions consist of a uniform density field, a solenoidal turbulent velocity field (b≈0.4), and an isothermal temperature of 10 K. Sink particles are created when the local density exceeds 10⁸ cm⁻³ and the Jeans length is resolved by at least four cells, thereby representing newly formed stars.

The analysis focuses on how the shape of the gas density PDF changes with turbulence strength and how these changes propagate into the CMF and IMF. In the low‑Mach (𝓜≈1) runs, gravity dominates; the high‑density tail of the PDF follows a power‑law ∝ ρ⁻³ᐟ², consistent with analytic models of gravity‑driven collapse. In contrast, the moderate and high‑Mach runs (𝓜 ≥ 3.5) produce the classic log‑normal PDF, with a dispersion σₛ that scales as σₛ²≈ln(1+b²𝓜²). The transition density (tₘᵢₙ) between the log‑normal peak and the power‑law tail is identified via maximum‑likelihood estimation (MLE) and shifts to lower densities as Mach number increases.

Core identification is performed using a density threshold of 10⁵ cm⁻³ and a connectivity algorithm; stellar masses are taken directly from sink particles. Both the CMF and IMF are examined via histograms and, more robustly, through complementary cumulative distribution functions (CCDFs). Power‑law slopes (α) are estimated with the MLE method of Clauset et al. (2009), avoiding biases inherent in binned linear fits.

Key results:

• Low‑Mach (𝓜 = 1): Both CMF and IMF exhibit shallow slopes (α≈1.6–1.8) and peaks near 1 M⊙ (CMF) and 0.8 M⊙ (IMF). The high‑mass tail is top‑heavy, reflecting the dominance of large‑scale gravitational collapse.
• Moderate‑Mach (𝓜 = 3.5): Slopes steepen to α≈2.3–2.4, and the peaks shift to ≈0.4 M⊙ (CMF) and 0.3 M⊙ (IMF), closely matching the Salpeter value (α=2.35).
• High‑Mach (𝓜 = 10): The slopes remain Salpeter‑like (α≈2.35–2.38) with peaks around 0.3 M⊙ (CMF) and 0.2 M⊙ (IMF). The log‑normal PDF fully governs the density structure.

When the mean density is increased to 10⁴ cm⁻³, the CMF peak moves to higher masses (≈2–3 × IMF peak), reproducing the observed offset between core and stellar mass functions. However, this offset diminishes in the low‑Mach regime, indicating that strong gravitational collapse can increase the efficiency of mass conversion from cores to stars.

Statistical validation includes χ² minimization for PDF fits, Kolmogorov–Smirnov tests for the CCDFs, and comparison of MLE‑derived α values with those obtained from traditional histogram fitting. The log‑normal description is statistically favored for 𝓜 ≥ 3.5, while the power‑law description is only justified for the 𝓜 = 1 case.

The authors conclude that the turbulent Mach number is a primary control parameter linking the gas density PDF to the CMF and IMF. Low Mach numbers produce power‑law PDFs and top‑heavy mass functions, whereas high Mach numbers generate log‑normal PDFs and Salpeter‑like mass functions. This finding challenges the notion of a universal IMF and provides a physically motivated framework to interpret observed IMF variations across different galactic environments. The study suggests that observational proxies for Mach number (e.g., CO line widths) could be used to predict local IMF shapes. Future work should incorporate magnetic fields, radiative feedback, and larger-scale galactic dynamics to test the robustness of these conclusions.