🔭 astrophysics

Infall onto the Protoplanetary Disk during the Gravitational Collapse of a Molecular Cloud

This paper proposes and validates modifications to the widely used Nakamoto and Nakagawa (1994) model for matter infall during molecular cloud collapse, demonstrating that these enhancements lead to a shorter predicted duration for the embedded phase of young stellar object evolution, particularly in regions with high initial density perturbations.

Original authors: E. R. Redkin, E. I. Vorobyov

Published 2026-02-25

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CC BY 4.0

Original authors: E. R. Redkin, E. I. Vorobyov

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: How Stars and Their Planetary Nurseries Are Born

Imagine a giant, cold cloud of gas and dust floating in space. This is a molecular cloud. Sometimes, something happens (like a shockwave from a nearby exploding star) that makes a small patch of this cloud collapse under its own gravity.

As this patch collapses, it spins faster (like an ice skater pulling in their arms). Most of the gas falls straight into the center to become a baby star (protostar). But because of that spinning, not all the gas can fall straight in. Some of it gets flung out into a flat, spinning pancake around the star. This is the protoplanetary disk, the nursery where planets will eventually form.

The Problem: The Old Map Was a Bit Flawed

Scientists have been using a specific "map" (a mathematical model created by Nakamoto and Nakagawa in 1994) to simulate how this gas falls from the cloud onto the disk. It's a popular map because it's simple to use.

However, the authors of this paper, Redkin and Vorobyov, realized this map has a few glitches:

The Cloud Was Infinite: The old model assumed the cloud was infinitely big. In reality, clouds have edges.
The Density Was "Perfect": The old model assumed the gas was packed in a mathematically perfect, singular way right at the center. Real clouds are a bit more "lumpy" and have a flat center before they get dense.
The Rain Never Stopped: The old model assumed gas fell onto the star at a constant rate forever, until the cloud was empty. In reality, as the cloud collapses, the "rain" of gas should slow down and stop eventually.

The Fix: Updating the Simulation

The authors proposed three main upgrades to fix these glitches, making the simulation more realistic without making it too complicated for computers to handle.

1. Giving the Cloud a Boundary (The "Fence" Analogy)

Imagine the cloud as a crowd of people in a field. The old model assumed the field went on forever. The new model puts a fence around the field.

Why it matters: Once the collapse starts, a "wave" of emptiness travels from the center out to the fence. When that wave hits the fence, the supply of gas gets cut off. This allows scientists to calculate how much energy the cloud has (rotation vs. gravity), which is crucial for understanding how big the resulting disk will be.

2. Smoothing the Center (The "Muffin" Analogy)

The old model said the center of the cloud was infinitely dense, like a mathematical point. The new model says the center is more like the top of a muffin—it's flat and dense, but not infinitely so.

Why it matters: This matches what we see in real 3D computer simulations of star formation. It changes how fast the gas falls in during the early stages.

3. The "Slow-Down" Button (The "Leaky Bucket" Analogy)

The old model was like a bucket with a hole that leaked at a constant speed until it was empty. The new model realizes that as the bucket empties, the water pressure drops, and the leak slows down.

Why it matters: They created a new formula where the rate of gas falling onto the star starts high, stays steady for a while, and then exponentially decays (fades out) as the cloud runs out of material. This is much more physically accurate.

The Results: Stars Grow Up Faster Than We Thought

When the authors ran their new, improved simulations, they found something surprising about the timeline of a baby star's life.

Astronomers classify baby stars by how much gas is still falling on them:

Class 0: The "newborn" phase. The star is buried deep in gas; 50% of the cloud is still falling in.
Class I: The "toddler" phase. Most of the gas has fallen in, but there's still a thick envelope.
Class II: The "teenager" phase. The gas is mostly gone, and we can clearly see the disk (where planets form).

The Finding:
Using the new, more realistic model, the transition from Class 0 to Class I happens much faster than previous models suggested.

The Analogy: Imagine a teenager growing up. The old models said, "It takes 500,000 years to go from a baby to a toddler." The new models say, "Actually, it only takes 100,000 years."

Why the difference?
The new model shows that if the initial cloud is a bit "lumpy" (high density perturbation), the gas falls in very quickly. The star grows up fast.

The "So What?"

This matters because astronomers look at real stars in the sky and try to guess their age based on how much gas is around them. If our computer models say "this process takes 500,000 years," but the real stars seem to be doing it in 100,000 years, we might be misjudging how old those stars are.

The authors suggest that maybe the real stars are just growing up faster than we thought, or perhaps the way we classify them is tricky (like trying to tell if a teenager is 13 or 15 just by looking at their height).

Summary

The paper takes an old, popular tool for simulating star birth and patches the holes. They added a boundary, smoothed out the center, and made the gas flow slow down naturally. The result? A more realistic picture of how stars and their planetary disks are born, suggesting that the "embedded" phase (where the star is hidden in gas) is shorter than we previously believed.

1. Problem Statement

The formation of protoplanetary disks relies on accurate models describing how matter falls from a collapsing molecular cloud onto a central protostar and its surrounding disk. The widely used Nakamoto and Nakagawa (NN) model (based on the work of Cassen and Moosman) provides a simplified approach to calculating this infall rate. However, the authors identify three critical limitations in the original NN model that affect the physical realism of numerical simulations:

Idealized Density Distribution: The model assumes a Singular Isothermal Sphere (SIS) density profile ( $\rho \propto r^{-2}$ ), which implies infinite density at the center and lacks a physical outer boundary.
Unlimited Cloud Size: The original model does not define a finite cloud radius, making it impossible to calculate key physical parameters like the ratio of rotational to gravitational energy ( $\beta$ ) or to account for the finite mass reservoir.
Constant Infall Rate: The model assumes a constant mass infall rate ( $\dot{M}_c$ ) until the cloud is exhausted, ignoring the physical reality that the infall rate must decay once a rarefaction wave from the cloud's outer boundary reaches the center.

These limitations lead to discrepancies between simulation results and observational data regarding the duration of the "embedded phase" (Class 0 and Class I stages) of young stellar objects (YSOs).

2. Methodology

The authors propose a series of modifications to the original NN model to address the identified shortcomings while maintaining the model's computational simplicity for use in one-dimensional viscous evolution codes.

A. Introduction of a Finite Cloud Boundary

The authors introduce a calculation for the outer radius of the molecular cloud ( $r_{out}$ ) based on the total cloud mass ( $M_c$ ). This allows for the calculation of the ratio of rotational energy to gravitational energy ( $\beta$ ), a crucial parameter for disk formation.

Result: For a standard solar-mass cloud, this yields $r_{out} \approx 8337$ au and $\beta \approx 0.19\%$ , consistent with observed prestellar cores.

B. Modification of Density Distribution

The singular isothermal sphere (SIS) profile is replaced with a Bonnor-Ebert sphere-like profile (specifically the Larson-Penston profile) characterized by a central plateau:
$\rho(r) = \frac{\rho_0}{1 + (r/r_0)^2}$

Justification: This profile better represents the physical conditions in the central regions of collapsing clouds (avoiding the singularity at $r=0$ ) and matches 3D hydrodynamic simulation data more closely than the SIS profile.
Impact: This change alters the relationship between time and the centrifugal radius ( $R_{cf}$ ), requiring a new derivation for $R_{cf}(t)$ .

C. Time-Dependent Infall Rate

The constant infall rate assumption is replaced with a time-dependent function that accounts for the finite mass of the cloud.

Mechanism: The infall rate remains constant ( $\dot{M}_c = m_0 a^3/G$ ) until a rarefaction wave traveling from the outer boundary reaches the center at time $t_d = r_{out}/2a$ . After $t_d$ , the rate decays exponentially.
Refinement: The authors propose a specific decay function (Modification M2) that ensures the total integrated mass over time equals the initial cloud mass ( $M_c$ ), preventing "locked" mass in the cloud.
Coefficient Adjustment: The paper provides a polynomial approximation for the coefficient $m_0$ as a function of the density perturbation amplitude $A$ , correcting overestimations found in previous literature for $2 < A < 5$ .

D. Disk-Mediated Accretion

The authors refine the infall equation to distinguish between matter falling directly onto the protostar versus the disk. A correction factor $\xi(t)$ is introduced to ensure that mass integration begins at the disk's inner boundary ( $R_{in}$ ) rather than zero, once the centrifugal radius exceeds $R_{in}$ .

3. Key Contributions

Physical Realism: The modified model (NN-1 to NN-7) incorporates finite cloud boundaries and realistic density profiles, allowing for the calculation of $\beta$ and more accurate initial conditions.
Analytical Decay Law: The derivation of an analytical expression for the decaying infall rate that strictly conserves the total cloud mass, resolving the issue of "infinite" or "locked" mass in previous models.
Generalized Infall Equation: The provision of a generalized equation for surface density evolution that accounts for sub-Keplerian infall velocities and the specific inner boundary of the disk.
Parameter Sensitivity Analysis: A systematic study of how the density perturbation amplitude ( $A$ ) and cloud temperature ( $T_c$ ) influence the duration of the embedded phase.

4. Results

The authors ran simulations using the modified models (NN-1 through NN-7) with varying parameters ( $A$ , $T_c$ , $\omega_c$ ) and compared the results with observational estimates for the duration of Class 0 and Class I YSOs.

Embedded Phase Duration: The simulations generally predict a shorter duration for the embedded phase compared to observational estimates (0.15–0.48 Myr).
- Models with high density perturbation amplitudes ( $A > 3$ ) and standard temperatures ( $T_c = 10-15$ K) show very rapid collapse, transitioning from Class 0 to Class II in less than 0.2 Myr.
- Only models with low temperatures ( $T_c \approx 7$ K) or very low perturbation amplitudes ( $A \approx 2.2$ ) produce transition times comparable to observations (e.g., Model NN-7: $t_{C0} \approx 231$ kyr, $t_{CI} \approx 706$ kyr).
Mass Distribution: In all models, the majority of the mass (approx. 80%) falls onto the disk rather than directly onto the star, confirming the "disk-mediated accretion" scenario.
Discrepancy Analysis: The authors suggest the mismatch between simulation and observation may stem from:
- Observational Bias: Edge-on systems may appear as Class 0 due to dust obscuration, artificially inflating the observed duration of early stages.
- Model Limitations: The assumption of an isolated cloud ignores mass accretion from surrounding filaments, which could prolong the infall phase in reality.

5. Significance

This work significantly improves the physical reliability of the widely used Nakamoto-Nakagawa model without sacrificing its computational efficiency. By correcting the density profile, introducing finite boundaries, and modeling the decay of the infall rate, the authors provide a more robust tool for studying the early evolution of protoplanetary disks.

The findings highlight that the initial density perturbation amplitude is a critical factor in determining the timescale of star formation. The paper sets the stage for future research (the second in a series) that will utilize these improved infall models to simulate the viscous evolution of protoplanetary disks and planet formation, offering a more accurate bridge between molecular cloud collapse and planetary system architecture.