Jeans Length & Jeans Mass Formula (Star-Formation Cloud Collapse)

Use one consistent Jeans convention

For a uniform ideal gas in the isothermal linear-perturbation model, cs² = kB T/(μ mH), kJ² = 4πGρ/cs² and λJ = 2π/kJ = cs √(π/(Gρ)). Define MJ as the mass of a sphere of radius λJ/2.

The previous example mixed a radius scale from an energy-balance estimate with the wavelength-based mass definition. Here both outputs consistently use the linear Jeans wavelength. Different conventions can have different numerical coefficients; label the convention before comparing values.

Enter cloud properties and constants

B2 is temperature in kelvin, B3 dimensionless mean particle mass in units of hydrogen-atom mass, and B4 mass density in kg/m³. The fictional cloud uses 10 K, μ=2.33 and 3.9×10⁻¹⁷ kg/m³. E2:E6 show the adopted constants and unit conversions; they are fixed numerical approximations for this exercise.

B6 requires positive numeric cloud inputs and constants. B8 calculates isothermal sound speed, without an adiabatic heat-capacity multiplier.

=IF(B6<>"OK",B6,SQRT(E2*B2/(B3*E4)))

Calculate wavelength in metres and parsecs

B9 retains the unrounded wavelength in metres; B10 converts it to parsecs. Initially sound speed is 188.1670 m/s and wavelength 0.2119 pc. Use display formatting for readable decimals rather than rounding the value before cubing it.

=IF(B6<>"OK",B6,B9/E5)

Calculate mass from half the wavelength

B11 uses MJ = (4π/3)ρ(λJ/2)³ = (π/6)ρλJ³ and divides by the adopted solar-mass conversion. Initially MJ is 2.8688 solar masses. At 30 K, wavelength is 0.3669 pc and mass 14.9065 solar masses.

=IF(B6<>"OK",B6,PI()/6*B4*B9^3/E6)

Distinguish particle density from molecular-hydrogen density

B12 computes n = ρ/(μmH), then converts m⁻³ to cm⁻³. It is approximately 10,000 particles/cm³ for the initial inputs. This is total particle number density under the adopted mean mass, not specifically H2 number density; deriving n(H2) requires a composition convention.

=IF(B6<>"OK",B6,B4/(B3*E4)/1E6)

Explore the scaling and limits

At fixed μ, wavelength scales as √T/√ρ and mass as T^(3/2)/√ρ. Increase density fourfold: both wavelength and mass halve. Increase temperature fourfold: wavelength doubles and mass increases eightfold.

In the ideal linear model, perturbations longer than the critical wavelength are unstable; equality marks the boundary. This does not predict a final stellar mass or guarantee collapse of a real cloud. Magnetic fields, turbulence, rotation, external pressure and non-isothermal evolution are outside this example.

Further reading

• Uppsala University — Isothermal critical Jeans wavelength (equation 17)
• University of Rochester FLASH — Jeans instability test and critical wavenumber
• Jog, 2013 — Jeans instability, spherical half-wavelength mass convention