Jeans Mass & Jeans Length Calculator

Find the Jeans mass and Jeans length for any interstellar gas cloud using temperature, density, and mean molecular weight inputs.

💫 Jeans Mass & Jeans Length Calculator
Gas preset
Temperature T
K
Number density n
cm³
Mean molecular weight μ
Adiabatic index γ
Sound speed cs
m/s
Mass density ρ
kg/m³
Sound speed cs
—
Mass density ρ
—
Jeans Length λJ
—
Jeans Mass MJ
—
Jeans Radius λJ/2
—
Free-fall Time tff
—

💫 What is the Jeans Mass?

The Jeans mass is the minimum mass a region of a gas cloud must possess for its self-gravity to overcome the internal gas pressure and trigger gravitational collapse, potentially forming stars. It was first derived by the British astrophysicist James Jeans in 1902 through a stability analysis of a self-gravitating gas. The Jeans criterion provides the physical basis for star formation: regions of a molecular cloud more massive than MJ are gravitationally unstable and will collapse; regions below MJ are pressure-supported and oscillate.

The Jeans length λJ is the critical wavelength at which this instability sets in. Perturbations in the gas with wavelength λ > λJ grow exponentially under gravity; perturbations with λ < λJ propagate as sound waves without collapse. The physical picture is a competition between two timescales: the free-fall time tff = √(3π/(32Gρ)) (how fast gravity collapses the cloud) and the sound-crossing time λ/cs (how fast pressure can respond). When tff < λ/cs, gravity wins and the region collapses. Setting these equal gives λJ = cs√(π/(Gρ)).

In practice, real molecular clouds have temperatures of 10–30 K and number densities of 10–10,000 particles per cm³. The sound speed at 10 K in a molecular cloud (cs ≈ 0.19 km/s for isothermal H2) is much lower than the turbulent velocity dispersion (1–10 km/s), so turbulent pressure provides additional support. The effective Jeans mass incorporating turbulent pressure is the turbulent Jeans mass, which can be 10–100 times the thermal MJ. Magnetic fields also provide support: only super-Alfvénic turbulence allows efficient collapse. This calculator computes the thermal Jeans mass assuming no turbulent or magnetic support.

The Jeans mass depends strongly on temperature and density: MJ ∝ T3/2 ρ−1/2. Cold, dense regions have low Jeans masses (stellar mass range, 1–100 M☉), while warm, diffuse gas has enormous Jeans masses (millions of M☉). As a molecular cloud collapses, it fragments: the increasing density lowers MJ, allowing sub-regions to become independently unstable, eventually producing the stellar initial mass function. This hierarchical fragmentation, first conceptualized by Fred Hoyle in 1953, is a key mechanism linking molecular cloud structure to the observed mass distribution of new-born stars.

📐 Formula

λJ = √(π cs² / (G ρ))     MJ = (π/6) ρ λJ³
cs = √(γ kB T / (μ mH)) — isothermal sound speed (m/s); γ = 1.0 isothermal, 5/3 monatomic, 7/5 diatomic H2
ρ = n × μ × mH × 106 kg/m³ (n in cm³ converted to m³)
G = 6.674 × 10−11 N m² kg−2; kB = 1.381 × 10−23 J/K; mH = 1.674 × 10−27 kg
MJ = (π/6) ρ λJ³ — mass of a sphere of diameter λJ; in solar masses: MJ / 1.989 × 1030 kg
tff = √(3π / (32 G ρ)) — free-fall collapse timescale (seconds); divide by 3.156 × 1013 for Myr
Example (GMC): T = 10 K, n = 100 cm³, μ = 2.33, γ = 1 → cs = 188 m/s, ρ = 3.90 × 10−19 kg/m³ → λJ = 2.12 pc, MJ = 28.7 M☉

📖 How to Use This Calculator

Steps

1
Select a mode — Cloud Parameters mode accepts temperature (K), number density (cm³), mean molecular weight μ, and adiabatic index γ; the calculator derives the sound speed and mass density internally. Direct Sound Speed mode accepts cs (m/s) and ρ (kg/m³) directly.
2
Use a preset or enter custom values — select Giant Molecular Cloud (T = 10 K, n = 100 cm³, μ = 2.33, γ = 1.0), Dense Protostellar Core (n = 10&sup4; cm³), Cold Neutral Medium (T = 80 K, μ = 1.27, γ = 5/3), or Warm Neutral Medium (T = 8000 K). Adjust any input for custom scenarios.
3
Read the Jeans length, mass, and free-fall time — compare the Jeans mass to the cloud's estimated mass. If the cloud mass > MJ, the cloud is unstable and will collapse. The free-fall time sets the timescale for collapse.

💡 Example Calculations

Example 1 — Giant Molecular Cloud (T = 10 K, n = 100 cm³)

Standard GMC conditions: T = 10 K, n = 100 cm³, μ = 2.33, γ = 1.0 (isothermal). The Jeans mass sets the characteristic mass for stellar cores.

1
Sound speed: cs = √(1.0 × 1.381×10−23 × 10 / (2.33 × 1.674×10−27)) = √(35,404) = 188.2 m/s = 0.188 km/s.
2
Mass density: ρ = 100 × 106 × 2.33 × 1.674×10−27 = 3.899×10−19 kg/m³. Jeans length: λJ = √(π × 188.2² / (6.674×10−11 × 3.899×10−19)) = √(4.274×1033) = 6.54×1016 m = 2.12 pc.
3
Jeans mass: MJ = (π/6) × 3.899×10−19 × (6.54×1016)³ = 5.71×1031 kg = 28.7 M☉. Free-fall time: tff = √(3π / (32 × 6.674×10−11 × 3.899×10−19)) = 3.37 Myr.
λJ = 2.12 pc (6.91 ly), MJ = 28.7 M☉, tff = 3.37 Myr. A 2-pc core in this GMC must accumulate >28.7 M☉ to collapse.
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Example 2 — Dense Protostellar Core (T = 10 K, n = 10,000 cm³)

Inner dense core of a collapsing cloud: n = 10,000 cm³ (100× the GMC), same T = 10 K. Lower Jeans mass means individual stars can form.

1
Sound speed unchanged at cs = 188.2 m/s (T and μ are the same). Mass density 100× higher: ρ = 3.899×10−17 kg/m³.
2
λJ = λJ(GMC) / √100 = 2.12 pc / 10 = 0.212 pc (0.69 ly). Because λJ ∝ ρ−1/2, doubling the density halves λJ/√2.
3
MJ = MJ(GMC) × (ρGMC/ρ)1/2 = 28.7 × (100)−1/2 = 28.7 / 10 = 2.87 M☉. Free-fall time: tff = 3.37 Myr / √100 = 337 kyr. Dense cores fragment to stellar mass.
λJ = 0.212 pc, MJ = 2.87 M☉, tff = 337 kyr. Individual low-to-intermediate mass stars can form from these dense cores.
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Example 3 — Warm Neutral Medium (T = 8,000 K, n = 0.5 cm³)

Diffuse warm neutral ISM: T = 8,000 K, n = 0.5 cm³, μ = 1.27 (atomic H + He), γ = 5/3. The enormous Jeans mass shows why the WNM cannot fragment to form stars.

1
Sound speed: cs = √(5/3 × 1.381×10−23 × 8000 / (1.27 × 1.674×10−27)) = √(8.71×107) = 9330 m/s = 9.33 km/s. This is the thermal sound speed; the WNM also has turbulent motions at similar speeds.
2
ρ = 0.5 × 106 × 1.27 × 1.674×10−27 = 1.063×10−21 kg/m³. λJ = √(π × 9330² / (6.674×10−11 × 1.063×10−21)) = 6.21×1019 m = 2.01 kpc.
3
MJ = (π/6) × 1.063×10−21 × (6.21×1019)³ = 1.33×1038 kg = 6.7 × 107 M☉ (67 million solar masses). This is so large that only galaxy-scale structures are Jeans-unstable in the WNM — not individual stars.
λJ = 2.01 kpc, MJ = 67 million M☉, tff = 64.5 Myr. The WNM must cool to ~100 K to reach stellar Jeans masses — cooling is the gateway to star formation.
Try this example →

❓ Frequently Asked Questions

What is the Jeans mass?+
The Jeans mass M_J is the minimum mass a gas cloud region must have for gravity to overcome pressure and trigger collapse. Derived by James Jeans (1902), it equals the mass of a sphere of diameter equal to the Jeans length: M_J = (π/6)ρλ_J³. Cloud regions more massive than M_J are gravitationally unstable; less massive regions oscillate as sound waves. For a typical GMC at T=10 K, n=100 cm⁻³, M_J ≈ 28.7 M☉ — within the range of stellar masses.
What is the Jeans length?+
The Jeans length λ_J = √(πc_s²/(Gρ)) is the critical scale above which gravitational instability grows. Perturbations with wavelength λ > λ_J collapse; those with λ < λ_J are pressure-supported and propagate as acoustic waves. It is set by the condition that the free-fall time equals the sound-crossing time. For GMC conditions (T=10 K, n=100 cm⁻³), λ_J ≈ 2.1 pc — comparable to the size of individual molecular cloud cores observed in CO emission surveys.
What is the free-fall time?+
The free-fall time t_ff = √(3π/(32Gρ)) is the time for a uniform, pressure-free sphere to collapse to a point. It depends only on the initial density: t_ff ≈ 3.4 Myr for n=100 cm⁻³ (GMC), 0.34 Myr for n=10⁴ cm⁻³ (dense core), and 64 Myr for n=0.5 cm⁻³ (WNM). Real collapse is slower due to turbulent and magnetic support. The ratio of the turbulent crossing time to t_ff (the virial parameter) determines whether a cloud is bound and collapsing or unbound and dispersing.
What mean molecular weight should I use?+
Use μ = 2.33 for a standard molecular cloud (H₂ + ~10% He by number fraction). This accounts for the mass of H₂ molecules (mass 2m_H each) and helium (mass 4m_H) in the correct proportion. For purely atomic neutral gas (H + He, 25% He by mass): μ ≈ 1.27. For fully ionized hydrogen plasma: μ ≈ 0.6 (free electrons reduce the mean mass). For the primordial gas before molecule formation (just H and He atoms): μ ≈ 1.22. The sound speed scales as c_s ∝ 1/√μ, so heavier gas is easier to collapse.
What adiabatic index should I use?+
Use γ = 1.0 (isothermal) for cold molecular clouds where efficient line cooling (CO, H₂O, CII) keeps the gas at constant temperature during collapse. Use γ = 7/5 = 1.4 for molecular H₂ in adiabatic conditions. Use γ = 5/3 ≈ 1.67 for monatomic ideal gas (atomic H, He, or hot plasma). The isothermal approximation (γ=1) is appropriate for the outer, optically thin envelope of a collapsing core; the inner region becomes adiabatic at densities above ~10¹⁰ cm⁻³ when the cloud becomes opaque to its own cooling radiation.
How does the Jeans mass change during collapse?+
During isothermal collapse (γ=1), the temperature stays constant while ρ increases, so M_J ∝ ρ^(-1/2) decreases. This allows hierarchical fragmentation: as the overall cloud collapses, sub-regions individually exceed the (shrinking) local Jeans mass and fragment further. When the collapsing region becomes optically thick (n ≈ 10¹⁰ cm⁻³), cooling switches off, the collapse becomes adiabatic (γ=5/3), and the Jeans mass begins increasing again: M_J ∝ ρ^(-1/2) × T^(3/2) ∝ ρ^(-1/2) × ρ^(γ-1 × 3/2). For γ > 4/3, M_J increases, halting fragmentation and defining the opacity limit for fragmentation at about 0.001 M☉.
Why is star formation inefficient in molecular clouds?+
In any given free-fall time, only about 0.5–2% of the GMC gas converts to stars (the star formation efficiency per free-fall time, ε_ff). The reasons include: (1) turbulent pressure — the effective turbulent Jeans mass is much larger than the thermal M_J, so only rare over-dense regions collapse; (2) magnetic field support — flux-frozen fields prevent collapse in sub-Alfvénic regions; (3) stellar feedback — ionizing radiation, stellar winds, and supernovae from newly-formed massive stars inject energy and disperse the parent cloud before more stars can form. The observed ε_ff ≈ 0.5–2% is consistent with turbulent fragmentation models.
Can the Jeans mass explain the stellar initial mass function?+
Partially. The Jeans mass sets a characteristic collapse scale, but the stellar initial mass function (IMF) spans from 0.1 to >100 M☉. The thermal Jeans mass in a typical GMC core (2–30 M☉) is too large to directly explain the peak of the IMF at ~0.3 M☉ from pure Jeans fragmentation. Turbulent fragmentation (Padoan-Nordlund, Hennebelle-Chabrier models) produces a distribution of Jeans masses through the turbulent density PDF, reproducing an IMF-like distribution. Competitive accretion adds further mass segregation. The opacity limit for fragmentation (~1 Jupiter mass) sets the minimum stellar mass.
What is the primordial Jeans mass?+
In the early universe before the first stars formed (cosmic dawn, z ≈ 20–150), the gas temperature was set by Compton cooling off CMB photons. At z = 100, T ≈ 270 K, n_H ≈ 10³ cm⁻³. With μ ≈ 1.22 (primordial H + He without molecules), the Jeans mass was about 10⁵ M☉ — the scale of mini-dark-matter halos. Population III stars (the first stars) formed in these halos and are predicted to have been very massive (10–1000 M☉) due to the lack of efficient metal-line cooling. The transition to modern-day low-mass star formation required the first supernova explosions to enrich the gas with metals.
What is the definition of M_J used in this calculator?+
This calculator uses M_J = (π/6) ρ λ_J³, which is the mass of a sphere of diameter λ_J. This is the most common definition in the literature (Jeans 1902, Bonnell & Bate 2006). Some authors define M_J using a sphere of radius λ_J (giving M_J = (4π/3) ρ λ_J³, a factor 8 larger) or using different prefactors from more refined analyses. The definition used here gives a Jeans mass that is typically a few tens of solar masses for standard GMC conditions.
How are turbulence and magnetic fields included?+
This calculator computes the purely thermal Jeans mass. To include turbulent support, replace the sound speed c_s with an effective speed c_eff = √(c_s² + σ_v²), where σ_v is the one-dimensional turbulent velocity dispersion (typically 0.5–3 km/s in GMCs). This gives the turbulent Jeans mass, which is 10–100 times larger than the thermal M_J for typical GMCs. Magnetic support requires comparing the cloud mass to the magnetic critical mass M_Φ = c_Φ Φ / G^(1/2), where Φ is the magnetic flux. Magnetically subcritical clouds cannot collapse even if they exceed M_J.
Tags: Jeans Mass Calculator Jeans Length Calculator Star Formation Threshold Gravitational Collapse Molecular Cloud Interstellar Medium Jeans Instability Astrophysics Calculator Free

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What is the Jeans mass?

The Jeans mass M_J is the minimum mass a region of a gas cloud must have for gravitational self-attraction to overcome the gas pressure and cause gravitational collapse. It was derived by the British astronomer James Jeans in 1902. For a spherical region of diameter equal to the Jeans length λ_J, M_J = (π/6) ρ λ_J³. Cloud fragments more massive than M_J will collapse to form stars; less massive fragments will oscillate as pressure waves. The Jeans mass sets the characteristic scale for star formation and is typically 1–100 solar masses in molecular clouds.

What is the Jeans length?

The Jeans length λ_J is the critical wavelength above which density perturbations in a gas cloud grow exponentially (the Jeans instability). It is the scale at which the gravitational potential energy equals the thermal kinetic energy of the gas: λ_J = √(π c_s² / G ρ), where c_s is the sound speed and ρ is the mass density. Perturbations with wavelength λ > λ_J are gravitationally unstable and collapse; perturbations with λ < λ_J are pressure-supported and oscillate as sound waves.

What is the Jeans instability?

The Jeans instability is the gravitational instability of a self-gravitating gas cloud. A uniform gas cloud is Jeans-unstable if its size exceeds λ_J: gravity can overcome pressure support and the cloud collapses. The instability arises because in a sufficiently large cloud, the time for a sound wave to cross the cloud (setting up a pressure response) exceeds the free-fall time. Jeans (1902) showed this using a perturbation analysis of the fluid equations with self-gravity, finding that modes with k < k_J = 2π/λ_J are unstable.

What are the assumptions of the Jeans mass formula?

The classic Jeans analysis assumes: (1) a uniform, infinite, initially static gas — the famous Jeans swindle (a uniform density cannot be truly static in its own gravity, but the perturbation analysis still gives a valid criterion); (2) an ideal gas with a fixed equation of state (isothermal γ = 1 or adiabatic γ = 5/3); (3) no magnetic fields, turbulence, rotation, or external radiation pressure. Real molecular clouds are turbulent, magnetized, and heated by radiation, so the actual fragmentation mass can be several times M_J.

What is the free-fall time?

The free-fall time t_ff = √(3π/(32Gρ)) is the time for a pressure-free, uniform sphere to collapse to infinite density under its own gravity. For typical Giant Molecular Cloud conditions (n = 100 cm⁻³, μ = 2.33), t_ff ≈ 3.4 Myr. For a dense protostellar core (n = 10⁴ cm⁻³), t_ff ≈ 0.34 Myr. The free-fall time gives a minimum collapse timescale; real clouds collapse more slowly due to magnetic and turbulent support.

What is the sound speed in a molecular cloud?

The sound speed in an ideal gas is c_s = √(γ k_B T / (μ m_H)). For a cold molecular cloud (T = 10 K, μ = 2.33, γ = 1 isothermal): c_s = √(1.381×10⁻²³ × 10 / (2.33 × 1.674×10⁻²⁷)) = √(35,400) ≈ 188 m/s = 0.19 km/s. This is the characteristic turbulent velocity of quiescent molecular cloud cores. For the warm neutral medium (T = 8000 K, γ = 5/3): c_s ≈ 9.3 km/s, which is why WNM fragments have Jeans masses of millions of solar masses.

What is the mean molecular weight μ?

The mean molecular weight μ is the average mass of a gas particle in units of the hydrogen atom mass m_H. For a gas of pure molecular hydrogen H₂: μ = 2. For a realistic molecular cloud with 90% H₂ and 10% He by number: μ ≈ (2 × 0.9 + 4 × 0.1) = 2.2 to 2.33 (the exact value depends on the helium mass fraction, typically 25%). For atomic neutral gas (H + He): μ ≈ 1.27. The sound speed c_s ∝ 1/√μ, so a heavier gas has a lower sound speed and is easier to collapse.

Why does M_J decrease as density increases?

M_J ∝ ρ^(-1/2) c_s³. As density increases at fixed temperature (and hence fixed c_s), the Jeans mass decreases. A denser cloud region needs less total mass to become gravitationally unstable because the gravitational potential energy scales faster with density than the thermal energy. This explains the hierarchical fragmentation of molecular clouds: as a collapsing cloud core becomes denser, the Jeans mass drops, allowing sub-regions to become independently unstable and fragment further, eventually forming individual stars.

How does temperature affect the Jeans mass?

M_J ∝ T^(3/2) (since c_s ∝ T^(1/2) and M_J ∝ c_s³). Hotter gas has a higher sound speed and therefore a larger Jeans mass. This is why the warm neutral medium (T = 8000 K) has a Jeans mass of millions of solar masses (forming galaxy-scale structures cannot collapse from WNM directly) while cold molecular clouds (T = 10 K) have Jeans masses of tens of solar masses (stellar mass range). Cooling is therefore crucial to star formation: gas must cool below ~100 K to allow stellar-mass fragments to become gravitationally unstable.

What is the difference between isothermal and adiabatic Jeans mass?

The isothermal Jeans mass uses γ = 1, meaning the gas is efficiently cooled and stays at constant temperature during collapse (c_s = √(k_BT/μm_H)). The adiabatic case uses γ = 5/3 for monatomic gas or 7/5 for H₂, meaning compression heats the gas. At the same T, the adiabatic sound speed is √γ times larger, giving M_J(adiabatic) = γ^(3/2) × M_J(isothermal). For γ = 5/3: M_J(adia) = (5/3)^(3/2) × M_J(iso) ≈ 2.15 × M_J(iso). The first cores of protostellar collapse (optical depth τ > 1) are adiabatic; the outer envelope stays isothermal.

What is the Bonnell-Bate fragmentation criterion?

The Bonnell-Bate (or competitive accretion) model focuses on competitive accretion in cluster-forming clouds rather than Jeans fragmentation. In this scenario, many fragments (each near the Jeans mass) form simultaneously and then compete for gas in a common reservoir. The most favorably placed fragments (near the center) accrete more gas and become massive stars. This contrasts with the turbulent core model (McKee-Tan 2003) where massive stars form from individually massive, turbulence-supported cores. Both mechanisms may operate in real molecular clouds.

Can the Jeans criterion apply to cosmological structure formation?

Yes. The Jeans instability is central to how the first structures formed in the universe. In the early universe at z ≈ 150 (before the first stars formed), the gas temperature was ~150 K and the baryon density was ~3 × 10⁻²⁰ kg/m³. The Jeans mass in this primordial gas was about 10⁵ M☉ — the typical mass of the first gravitationally collapsing structures (mini-halos). The first stars (Population III) likely formed in such halos. Today's molecular cloud Jeans masses (10–100 M☉) are much smaller because the gas is much denser.

📌 Quick Tips

💡Use γ = 1 (isothermal) for cold molecular clouds where efficient cooling keeps the gas at constant temperature during collapse. Use γ = 5/3 (monatomic) for atomic gas or γ = 7/5 for molecular H₂ in adiabatic conditions.
💡The mean molecular weight μ = 2.33 is standard for a molecular hydrogen cloud (H₂ + 10% He by number). For atomic neutral gas, use μ = 1.27.
💡A cloud fragment larger than λ_J and more massive than M_J will collapse under self-gravity. Sub-Jeans fragments bounce as pressure waves.
💡The free-fall time t_ff = √(3π/(32Gρ)) gives the timescale for collapse once a region exceeds the Jeans mass. For typical GMCs, this is 3–10 Myr.
💡Magnetic fields and turbulence in real clouds provide additional support against collapse, requiring a mass several times M_J for actual star formation.