Jeans Instability

Jeans Instability

A gas cloud collapses under its own gravity if it is large enough that gravitational potential energy overcomes the thermal kinetic energy of the gas. This threshold is set by the Jeans length λ_J.

The Jeans Wavenumber

The Jeans wavenumber separates stable (oscillating) modes from unstable (collapsing) modes:

$k_J = \sqrt{\frac{4\pi G \rho}{c_s^2}}$

where $G = 6.674 \times 10^{-11}$ m³/(kg·s²), $\rho$ is the gas density in kg/m³, and $c_s$ is the sound speed in m/s.

The Jeans Length

The Jeans length is the critical wavelength:

$\lambda_J = \frac{2\pi}{k_J} = c_s \sqrt{\frac{\pi}{G \rho}}$

Perturbations with wavelength $\lambda > \lambda_J$ are gravitationally unstable and collapse. Those with $\lambda < \lambda_J$ oscillate as sound waves.

The Jeans Mass

The mass enclosed within a sphere of radius $\lambda_J / 2$:

$M_J = \frac{4\pi}{3} \rho \left(\frac{\lambda_J}{2}\right)^3$

Typical Molecular Cloud

For a cold molecular cloud with $c_s \approx 200$ m/s (T ≈ 10 K) and $\rho \approx 10^{-18}$ kg/m³:

• $\lambda_J \approx 4.3 \times 10^{16}$ m ≈ 1.4 parsecs
• $M_J \approx 4.3 \times 10^{31}$ kg ≈ 21 solar masses

These are the seeds of star formation!

Your Task

Implement three functions. All constants must be defined inside each function.

• jeans_wavenumber(c_s_m_s, rho_kg_m3) — returns $k_J$ in m⁻¹
• jeans_wavelength_m(c_s_m_s, rho_kg_m3) — returns $\lambda_J = c_s \sqrt{\pi / (G \rho)}$ in m
• jeans_mass_kg(c_s_m_s, rho_kg_m3) — returns $M_J$ in kg (compute $\lambda_J$ inline)

Use $G = 6.674 \times 10^{-11}$ m³/(kg·s²).